Handbook of Mathematical Functions

!

HANDBOOK OF MAT HE.MAT1CA L F U NCT I0NS

4

with Formulas, Graphs, and Mathematical Tables

1

4

-..

6

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HANDBOOK OF MATHEMATICAL FUNCTIONS W I T H FORMULAS, GRAPHS, A N D MATHEMATICAL TABLES

Edited by Milton Abramowitz and Irene A. Stegun

The text relating to physical constants and conversion factors (page 6) has been modified to take into account t h e newly adopted Systeme International d’Unites (SI).

r

I

1

ERRATA NOTICE The original printing of this Handbook (June 1964) contained errors t h a t have been corrected in t h e reprinted editions. These corrections are marked with an asterisk (*) for identification. The errors occurred on t h e following pages: 2-3,6-8,10,15,19-20,25,76,85,91,102, 187, 189-197,218,223) 225,233,250,255, 260-263,268,271-273,292,302, 328,332,333-337,362,365,415,423,438-440,443,445,447,449,451,484, 498, 505-506,509-510,543,556) 558,562,571,595, 599,600,722-723,739, 742, 744, 746, 752, 756, 760-765, 774, 777-785,790, 797,801,822-823,832, 835, 844, 886-889,897, 914,915,920,930-931, 936,940-941,944-950,953, 960,963,989-990, 1010, 1026. Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London WC 2.

This Dover edition, first published in 1965, is an unabridged and unaltered republication of the work originally published by the National Bureau of Standards in 1964. This ninth Dover printing conforms to the tenth (Deceniber 1972) printing by the Government Printing Ofice, except that additional corrections have hecn made on pages 18, 79, 80, 82, 408, 450, 786, 825 and 934.

Standard Book Number: 486-61272-4 Library of Congress Catalog Card Number: 65.12253

Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N.Y. 10014

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Preface The present volume is an outgrowth of a Conference on Mathematical Tables held a t Cambridge, Mass., on September 15-16, 1954, under the auspices of the National Science Foundation and the Massachusetts Institute of Technology. The purpose of the meeting was to evaluate the need for mathematical tables in the light of the availability of large scale computing machines. It was the consensus of opinion that in spite of the increasing use of the new machines the basic need for tables would continue to exist. Numerical tables of mathematical functions are in continual demand by scientists and engineers. A greater variety of functions and higher accuracy of tabulation are now required as a result of scientific advances and, especially, of the increasing use of automatic computers. In the latter connection, the tables serve mainly for preliminary surveys of problems before programming for machine operation. For those without easy access to machines, such tables are, of course, indispensable. Consequently, the Conference recognized that there was a pressing need for a modernized version of the classical tables of functions of Jahnke-Emde. To implement the project, the National Science Foundation requested the National Bureau of Standards to prepare such a volume and established an Ad Hoc Advisory Committee, with Professor Philip M. Morse of the Massachusetts Institute of Technology as chairman, to advise the staff of the National Bureau of Standards during the course of its preparation. In addition to the Chairman, the Committee consisted of A. ErdBlyi, M. C. Gray, N. Metropolis, J. B. Rosser, H. C. Thacher, Jr., John Todd, C. B. Tompkins, and J. W. Tukey. The primary aim has been to include a maximum of useful information within the limits of a moderately large volume, with particular attention to the needs of scientists in all fields. An attempt has been made to cover the entire field of special functions. To carry out the goal set forth by tbe Ad Hoc Committee, it has been necessary to supplement the tables by including the mathematical properties that are important in computation work, as well as by providing numerical methods which demonstrate the use and extension of the tables. The Handbook was prepared under the direction of the late Milton Abramowitz, and Irene A. Stegun. Its success has depended greatly upon the cooperation of many mathematicians. Their efforts together with the cooperation of the Ad Hoc Committee are greatly appreciated. The particular contributions of these and other individuals are acknowledged at appropriate places in the text. The sponsorship of the National Science Foundation for the preparation of the material is gratefully recognized. It is hoped that this volume will not only meet the needs of all table users but will in many cases acquaint its users with new functions. ALLENV. ABTIN,Director. Washington, D.C. m

Preface to the Ninth Printing The enthusiastic reception accorded the “Handbook of Mathematical Functions” is little short of unprecedented in the long history of mathematical tables that began when John Napier published his tables of logarithms in 1614. Only four and one-half years after the first copy came from the press in 1964, Myron Tribus, the Assistant Secretary of Commerce for Science and Technology, presented the 100,OOOth copy of the Handbook to Lee A. DuBridge, then Science Advisor to the President. Today, total distribution is approaching the 150,000 mark at a scarcely diminished rate. The success of the Handbook has not ended our interest in the subject. On the contrary, we continue our close watch over the growing and changing world of computation and to discuss with outside experts and among ourselves the various proposals for possible extension or supplementation of the formulas, methods and tables that make up the Handbook. In keeping with previous policy, a number of errors discovered since the last printing have been corrected. Aside from this, the mathematical tables and accompanying text are unaltered. However, some noteworthy changes have been made in Chapter 2: Physical Constants and Conversion Factors, pp. 6-8. The table on page 7 has been revised to give the values of physical constants obtained in a recent reevaluation; and pages 6 and 8 have been modified to reflect changes in definition and nomenclature of physical units and in the values adopted for the acceleration due to gravity in the revised Potsdam system. The record of continuing acceptance of the Handbook, the praise that has come from all quarters, and the fact that it is one of the most-quoted scientific publications in recent years are evidence that the hope expressed by Dr. Astin in his Preface is being amply fulfilled. LEWISM. BRANSCOMB, Director National Bureau of Standards November 1970

Foreword This volume is the result of the cooperative effort of many persons and a number of organizations. The National Bureau of Standards has long been turning out mathematical tables and has had under consideration, for a t least 10 years, the production of a compendium like the present one. During a Conference on Tables, called by the NBS Applied Mathematics Division on May 15, 1952, Dr. Abramowitz of that Division mentioned preliminary plans for such an undertaking, but indicated the need for technical advice and financial support. The Mathematics Division of the National Research Council has also had an active interest in tables; since 1943 it has published the quarterly journal, “Mathematical Tables and Aids to Computation” (MTAC) , editorial supervision being exercised by a Committee of the Division. Subsequent to the NBS Conference on Tables in 1952 the attention of the National Science Foundation was drawn to the desirability of financing activity in table production. With its support a 2-day Conference on Tables was called at the Massachusetts Institute of Technology on September 15-16, 1954, to discuss the needs for tables of various kinds. Twenty-eight persons attended, representing scientists and engineers using tables as well as table producers. This conference reached consensus on several conclusions and recommendations, which were set forth in tbe published Report of the Conference. There was general agreement, for example, “that the advent of high-speed computing equipment changed the task of table making but definitely did not remove the need for tables”. It was also agreed that “an outstanding need is for a Handbook of Tables for the Occasional Computer, with tables of usually encountered functions and a set of formulas and tables for interpolation and other techniques useful to the occasional computer”. The Report suggested that the NBS undertake the production of such a Handbook and that the NSF contribute financial assistance. The Conference elected, from its participants, the following Committee: P. M. Morse (Chairman), M. Abramowitz, J. H. Curtiss, R. W. Hamming, D. H. Lehmer, C. B. Tompkins, J. W. Tukey, to help implement these and other recommendations. The Bureau of Standards undertook to produce the recommended tables and the National Science Foundation made funds available. To provide technical guidance to the Mathematics Division of the Bureau, which carried out the work, and to provide the NSF with independent judgments on grants for the work, the Conference Committee was reconstituted as the Committee on Revision of Mathematical Tables of the Mathematics Division of the National Research Council. This, after some changes of membership, became the Committee which is signing this Foreword. The present volume is evidence that Conferences can sometimes reach conclusions and that their recommendations sometimes get acted on. V

FOREWORD

Active work was started at the Bureau in 1956. The overall plan, the selection of authors for the various chapters, and the enthusiasm required to begin the task were contributions of Dr. Abramowitz. Since his untimely death, the effort has continued under the general direction of Irene A. Stegun. The workers at the Bureau and the members of the Committee have had many discussions about content, style and layout. Though many details have had to be argued out as they came up, the basic specifications of the volume have remained the same as were outlined by the Massachusetts Institute of Technology Conference of 1954. The Committee wishes here to register its commendation of the magnitude and quality of the task carried out by the staff of the NBS Computing Section and their expert collaborators in planning, collecting and editing these Tables, and its appreciation of the willingness with which its various suggestions were incorporated into the plans. We hope this resulting volume will be judged by its users to be a worthy memorial to the vision and industry of its chief architect, Milton Abramowitz. We regret he did not live to see its publication. P. M. MORSE,Chairman.

A. E R D ~ L Y I M. C. GRAY N. C. METROPOLIS J. B. ROSSER H. C. THACHER, Jr. JOHN TODD C. B. TOMPKINS J. W. TUKEY.

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Mathematical Constants . . . . . . . . . . . . . . . . . . . DAVIDS.LIEPMAN 2. Physical Constants and Conversion Factors . . . . . . . . . . A. G. MCNISH 3. Elementary Analytical Methods . . . . . . . . . . . . . . . MILTONABRAMOWITZ 4. Elementary Transcendental Functions . . . . . . . . . . . . ,. Logarithmic, Exponential, Circular and Hyperbolic Functions RUTHZUCKER 5. Exponential Integral and Related Functions . . . . . . . . . . WALTERGAUTSCHI and WILLIAMF. CAHILL 6. Gamma Function and Related Functions. . . . . . . . . . . . PHILIPJ. DAVIS 7. Error Function and Fresnel Integrals . . . . . . . . . . . . . WALTERGAUTSCHI 8. Legendre Functions . . . . . . . . . . . . . . . . . . . . . IRENE A. STEGUN 9. Bessel Functions of Integer Order . . . . . . . . . . . . . . . F. W. J. OLVER 10. Bessel Functions of Fractional Order. . . . . . . . . . . . . . H. A. ANTOSIEWICZ 11. Integrals of Bessel Functions . . . . . . . . . . . . . . . . . YUDELLL. LUKE 12. Struve Functions and Related Functions . . . . . . . . . . . . MILTONABRAMOWITZ 13. Confluent Hypergeometric Functions . . . . . . . . . . . . . SLATER LUCYJOAN 14. Coulomb Wave Functions . . . . . . . . . . . . . . . . . . MJLTONABRAMOWITZ 15. Hypergeometric Functions . . . . . . . . . . . . . . . . . . FRITZ OBERHETTINQER 16. Jacobian Elliptic Functions and Theta Functions . . . . . . . . L. M. MILNE-THOMSON 17. Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . L. M. MILNE-THOMSON 18. Weierstrass Elliptic and Related Functions . . . . . . . . . . . THOMAS H. SOUTHARD 19. Parabolic Cylinder Functions . . . . . . . . . . . . . . . . . J. C. P. MILLER VLI

Page I11

V IX

1 5 9 65

227 253 295 331 355 435 479 495 503 537 555 567 587 627 685

CONTENTS

VIII

20. Mathieu Functions

. . . . . . . . . . . . . . . . .. . . .

GERTRUDE BLANCH 21. Spheroidal Wave Functions, . . . . . . . . . . . . . . . ARNOLDN. LOWAN 22. 0r t hogon a1 Polynomials . . . . . . . . . . . . . . . URS W. HOCHSTRASSER 23. Bernoulli and Euler Polynomials, Riemann Zeta Function . . and KARLGOLDBERG EMILIEV. HAYNSWORTH 24. Combinatorial Analysis . . . . . . . . . . . . . . . . . M. NEWMAN and E. HAYNSWORTH K. GOLDBERG, 25. Numerical Interpolation, Differentiation and Integration . . + PHILIPJ. DAVISand IVAN POLONSKY 26. Probabilit,y Functions . . . . . . . . . . . . . . . . . . MARVINZELENand NORMAN C. SEVERO 27. Miscellaneous Functions . . . . . . . . . . . . . . . . IRENE A. STEGUN 28. Scales of Notation. . . . . . . . . . . . . . . . . . . . S. PEAVY and A. SCHOPF 29. Laplace Transforms . . . . . . . . . . . . . . . . . . . . . Subject Index . . . . . . . . . . . . . . . . . . . . . Index of Notations . . . . . . . . . . . . . . . . . . . . . I

Page

721

. .

751

. .

803

. .

821

.

875

. .

925

. .

997

.

1011

. . . .

1019 1031 1044

.

*

. .

Handbook of Mathematical Functions with

Formulas, Graphs, and Mathematical Tables Edited by Milton Abramowitz and Irene A. Stegun

1. Introduction The present Handbook has been designed to provide scientific investigators with a comprehensive and self-contained summary of the mathematical functions that, arise in physical and engineering problems. The well-known Tables of Functions by E. Jahnke and F. Emde has been invaluable to workers in these fields in its many editions during the past half-century. The present volume ext,ends the work of these authors by giving more extensive and more accurate numerical tables, and by giving larger collections of mathematical properties of the tabulated functions. The number of functions covered has also been increased. The classification of functions and organization of the chapters in this Handbook is similar to that of An Index of Mathematical Tables by A. Fletcher, J. C. P. Miller, and L. Rosenhead.’ In general, the chapters contain numerical tables, graphs, polynomial or rational approximations for automatic computers, and statements of the principal mathematical properties of the tabulated functions, particularly those of computa-

tional importance. Many numerical examples are given to illustrate the use of the tables and also the computation of function values which lie outside their range. At the end of the text in each chapter there is a short bibliography giving books and papers in which proofs of the mathematical properties stated in the chapter may be found. Also listed in the bibliographies are the more important numerical tables. Comprehensive lists of tables are given in the Index mentioned above, and current information on new tables is to*be found in the National Research Council quarterly Mathematics of Computation (formerly Mathematical Tables and Other Aids to Computation). The mathematical notations used in this Handbook are those commonly adopted in standard texts, articularly Higher Transcendental Functions, volumes 1-3, by A. ErdBlyi, W. Magnus, F. Oberhettinger and F. G. Tricomi (McGrawHill, 1953-55). Some alternative notations have also been listed. The introduction of new symbols has been kept to a minimum, and an effort has been made to avoid the use of conflicting notation.

2. Accuracy of the Tables The number of significant figures given in each table has depended to some extent on the number available in existing tabulations. There has been no attempt to make it uniform throughout the Handbook, which would have been a costly and laborious undertaking. In moat t’ables a t least five significant figures have been provided, and the tabular’ intervals have generally been chosen to ensure that linear interpolation will yield. fouror five-figure accuracy, which suffices in most physical applications. Users requiring higher 1 The most recent the sixth with F. Loesch added as co-author, was published in 1960 b; McGraw:Hill, U.S.A., and Teubner, Germany. 8 The second edition with L J Comrie added asco-author was published in two volumes in leb2 by Addison-Wesley, U.S.A., and bcientiflc Computing Service Ltd., Great Britain.

precision in their interpolates may obtain them by use of higher-order interpolation procedures, described below. In certain tables many-figured function values are given at irregular intervals in the argument. An example is provided by Table 9.4. The purpose of these tables is to furnish “key values” for the checking of programs for autoim atic computers; no question of interpolation arises. The maximum end-figure error, or “tolerance” in the tables in this Handbook is 761 of 1 unit everywhere in the case of the elementary functions, and 1 unit in the case of the higher functions except in a few cases where it has been permitted to rise to 2 units. Is

INTRODUCTION

X

Auxiliary Functions and Arguments

3.

One of the objects of this Handbook is to provide tables or computing methods which enable the user to evaluate the tabulated functions over complete ranges of real values of their parameters. In order to achieve this object, frequent use has been made of auxiliary functions to remove the infinite part of the original functions a t their singularities, and auxiliary arguments to CO e with infinite ranges. An example will make t e procedure clear. The exponential integral of positive argument is given by

!i

Ei(x) =S_Zm;du

x

xa

22

=y+lnz+-+-+-+ 1.1! 2.2! 3.3!

...

The logarithmic singularity precludes direct interpolation near s=O. The functions Ei(z)--In z and &iEi(z)-ln z-71, however, are wellbehaved and readily interpolable in this region. Either will do as an auxiliary function; the latter was in fact selected as it yields slightly higher accuracy when Ei(z) is recovered. The function &/Ei(z)-ln 2-71 has been tabulated to nine decimals for the range 02~13.For 3 1 x 2 2 , Ei(z) is sufficiently well-behaved to admit direct tabulation, but for larger values of z, its exponential character predominates. A smoother and more readily interpolable function for large z is ze-ZEi(z); this has been tabulated for 2 1 2 1 1 0 . Finally, the range 10 1 2 1CO is covered by use of the inverse argument z-l. Twenty-one entries of ze-”Ei(z), corresponding to x-l= . l (- .005)0, suffice to produce an interpolable table.

4. Interpolation The tables in this Handbook are not provided with differences or other aids to interpolation, because it was felt that the space they require could be better employed b the tabulation of additional functions. Admitte ly aids could have been given without consuming extra space by increasing the intervals of tabdation, but this would have conflicted with the requirement that linear interpolation is accurate to four or five figures. For applications in which linear interpolation is insufficiently accurate it is intended that Lagrange’s formula or Aitken’s method of iterative linear interpolation3 be used. To help the user, there is a statement at the foot of most tables of the maximum error in a linear inter olate, and the number of function values nee ed in Lagrange’s formula or Aitken’s method to interpolate to full tabular accuracy. As an example, consider the following extract from Table 5.1.

B

B

z

7. 5 7. 6 7.7 7.8 7.9

xezEl(z)

.a9268 .a9384 .89497 .89608 . 89717

7854 6312 9666 8737 4302

5

8.0 8. 1 8. 2 8.3 8. 4

xezEl(z) .a9823 7113 ,89927 7888 .go029 7306 .go129 6013 .go227 4695

“-”I

Let us su pose that we wish to compute the r ) x=7.9527 from this table. value of ~ e ’ % ~ (for We describe in turn the application of the methods of linear interpolation, Lagrange and Aitken, and of alternative methods based on differences and Taylor’s series. (1) Linear interpolation. The formula for this process is given by

.f,=

(1-P)fo+Pf1

where fo,f l are consecutive tabular values of the function, corresponding to arguments zo, zl, respectively; p is the given fraction of the argument interval p= (z-zo)/(z1-z0> and j , the required interpolate. instance, we have fo=.89717 4302

In the present

f1=.89823 7113

p=.527

P

The most convenient way to evaluate the formula on a desk calculating machine is.to set and f l in turn on the keyboard, and carry out t e multiplications by l-p and p cumulatively; a pwtial check is then rovided by the multiplier dial reading unity. e obtain

%

The numbers in the square brackets mean that the maximum error in a linear interpolate is 3 X 10-e, and that to interpolate to the full tabular accuracy five points must be used in Lagrange’s and Aitken’s methods. 8 A . C. Aitken, On inter olation b iteration of roportional parts, with. out the use of differences, #rw. Edingurgh Math. 3, 56-76 (1932).

8oc.

f .627= (1 - ,527)(.89717 4302) f .527(.89823 7113) = A9773 4403. Since it is known that there is a possible error of 3 X 10-e in the linear formula, we round off this result to ,89773. The maximum possible error in this answer is composed of the error committed

XI

INTRODUCTION

by the last rounding, that is, .4403X10-5, plus 3 X 10-0, and so certainly cannot exceed .8 X 10-6. (2) Lagrange’s formula. In this example, the relevant formula is the 5-point one, given by

Tables of the coefficients Ak(p)are given in chapter 25 for bhe range p=O(.O1)1. We evaluate the formula for p=.52, .53 and .54 in turn. Again, in each evaluabion we accumulate the A&) in the multi lier register since their sum is unity. We now ave bhe following subtable.

g

7.952

z@Ei(2) .89772 9757

7.953

.89774 0379

z

10622

n

zn

1 7.9 2 8. 1 3 7. 8 4 8. 2 5 7. 7

yn=zemEl(x)

Yo, n

. 89823 7113 ,89717 4302 ,89773 44034 . 89927 7888 . 89774 48264 . 89608 8737 2 90220 . 90029 . 89497

7306 9666

jp=.3(.89772 9757)+.7(.89774 0379) =A9773 7192.

I

In cases where the correct order of the Lagrange polynomial is not known, one of the preliminar interpolations may have to be performed wit polynomials of two or more different orders as a check on their ade uacy. (3) Aitken’s met od of iterative linear interpolation. The scheme for carrying out this process in the present example is as follows :

4 98773 2 35221

I

I

1

-2

.89775 0999

3 a0

I

g

10620 7.954

The numbers in the third and fourth columns are the first and second differences of the values of se”El(z)(see below); the smallness of the second difference provides a check on the three inter olations. The required value is now obtainei by linear interpolation :

Yo. I , n ,

YO, I , 2.3. n

Y0.1.2.n

89773 71499 2394 .89773 71938 1216 16 2706 43

I

i

Xn-Z

i

. 0473

-. 0527 . 1473 -. 1527 . 2473

89773 71930 30

-.

I

I

I

2527

Here

i I

I

Here

If the quantities xn--2 and xm--2 are used as multipliers when forming the cross-product on a desk machine, their accumulation (x,-x) - (zm-z) in the multiplier register is the divisor to be used at that stage. An extra decimal place is usually carried in the intermediate interpolates to safeguard against accumulation of rounding errors. The order in which the tabular values are used is immaterial to some extent, but to achieve the maximum rate of convergence and at the same time minimize accumulation of rounding errors, we begin, as in this example, with the tabular argument nearest to the given argument, then take the nearest of the remaining tabular arguments, and so on. The number of tabular values required to achieve a given precision emerges naturally in the course of the iterations. Thus in the present example six values were used, even though it was known in advance that five would suffice. The extra row confirms the convergence and provides a valuable check. (4) Difference formulas. We use the central difference notation (chapter 25),

afiis=fi-fo,

..

ajm=fa-fi,

1,

-6ji/a=.fa- 2j1 +fo a’ji =j 3 - 3ja 3.fi -j o S4j2= S8jm- a8f3/z=jr- 4f3+

6% =a.f3/z 8af,iz= aafa-

+

6fa- 4jlSjo

and so on. In the present exam le the relevant part of the difference table is as fop1ows, the differences being written in units of the last decimal lace of the function, as is customary. The sma ness of the high differences provides a check on the function values 5 zesEl(x) aaf ?f

H

7.9 8. 0

. 89717

. 89823

4302 7113

- 2 2754 - 2 2036

-34 -39

Applying, for example, Everett’s interpolation formula

+

+ ..

f p = (1 - p)fo E2 (P)WO+Ei(p)WO

+ p f i + ~ ~ ( p ) ~ l f i + F ~ ( ~ ) ~* ‘ f l* f

and takin the numerical values of the interpolation coe cients E&), E, ), F2(p) and Fd(p) from Table 25.1, we find t at

ff

k)

I

XI1

INTRODUCTION

+

10gf,6a= .473(89717 4302) .061196(2 2754) - .012(34) .527(89823 7113) .063439(2 2036) - .012(39) =89773 7193.

+

can be used. We first compute as many of the derivatives f'"'(x,,) as are significant, and then evaluate the series for the given value of x. An advisable check on the computed values of the derivatives is to reproduce the adjacent tabular values by evaluating the series for x=x-l and xl.

+

We may notice in passing that Everett's formula shows that the error in a linear interpolate is approximately

rft)

2) =zezEl(s)

Since the maximum value of IE,(p)+Fz(p)I in the range O
that is,

1

= (1 +z-')f(z)

f"5)

(5) Taylor's series. In cases where the successive derivatives of the tabulated function can be computed fairly easily, Taylor's expansion

With Xo:0=7.9and x-xo= A527 our computations are as follows; an extra decimal has been retained in the values of the terms in the series to safeguard against accumulation of rounding errors.

3f"'(~0) + 3 !

f ' 'k ($0) /k! 39717 4302 .01074 0669 -.ooii3 7621 .00012 1987

k 0

1 2 3

'(50) (z-zo)* f" (50) f(4=f(zo) + ( 5 - 5 0 ) fr+ 2 ! +(z--zo)

+

Y"(z) = (1 -i-z-')y'(z) - 22-2f' (5) 2 ~ - 3 f ( 4 .

1

If~-f~-fo+f-~l.

-1

= (l+z-')f'(z) -5-*f(z)

...

(5-30)kf'b'(~o)/k!

.89717 4302 .msi . ...- - m - - -8- 3-.ooooo 3159 5 .OOOOO 0017 9 .89773 7194

5. Inverse Interpolation With linear interpolation there is no difference in principle between direct and inverse interpolation. In cases where the linear formula rovides an insufficient1 accurate answer, two met ods are may interpolate directly, for available. example, b Lagrange's formula to prepare a new table at a ne interval in the neighborhood of the approximate value, and then apply accurate inverse linear interpolation to the subtabulated values. Alternatively, we may use Aitken's method or even possibly the Taylor's series method, with the roles of function and argument interchanged. It is important to realize that the accuracy of an inverse interpolate may be very different from that of a direct interpolate. This is particularly true in regions where the function is slowly varying, for example, near a maximum or minimum. The maximum precision attainable in an inverse interpolate can be estimated with the aid of the formula

g

d

K

The desired x is therefore Z=SO+

~(ZI-ZO)

= 8.1

+ .708357(.1)=8.17083 57

To estimate the possible error in this answer, we recall that the maximum error of direct linear interpolation in this table is Af=3X10-e. An approximate value for dfldx is the ratio of the f i s t difference to the argument interval (chapter 25), in this case ,010. Hence the maximum error in xis approximately 3X10-6/(.010), that is, .0003. (ii) Sub tabulation method. To improve the ap roximate value of x just obtained, we interPO ate directly for p=.70, .71 and .72 with the aid of Lagrange's 5-point formula,

P

xezEl(2) 3683

5

8. 170

. 89999

8. 171

. 90000 3834

8. 172

. 90001

8

1 0151 1 0149

-2

3983

Inverse linear interpolation in the new table gives in which A j is the maximum possible error in the function values. Example. Given xeZE1(x)=.9, find x from the table on page X. (i) Inverse linear interpolation. The formula for p is

P = cf,-fo)

'=

.9- .89999 3683= .6223 .00001 0151

Hence x-8.17062 23. is

An estimate of the maximum error in this result

IVI -fo>.

In the present example, we have *'.90029

.9- ,89927 7888 7306- 39927 7888=1-= 72 2112

.708357.

(iii) Aitken's method. This is carried out in the same manner as in direct interpolation.

xm

INTRODUCTION

The estimate of the maximum error in this result is the same as in the subtabulation method. An indication of the error is also provided by the

discrepancy in the highest interpolates, in this case q ,1 ,2 .3 , 4 , and q . 1 .2 .3 ,b.

Bivariate interpolation is generally most simply performed as a sequence of univariate interpolations. We carry out the inter olation in one direction, by one of the methods a ready described, for several tabular values of the second argument in the neighborhood of its given value. The interpolates are differenced as a check, and

interpolation is then carried out in the second direction. An alternative procedure in the case of functions of a complex variable is to use the Taylor's series expansion, provided that successive derivatives of the function can be computed without much dficulty.

P

7. Generation of Functions from Recurrence Relations Many of the special mathematical functions which depend on a parameter, called their index, order or degree, satisfy a linear difference equation (or recurrence relation) with respect to this parameter. Examples are furnished by the Legendre function P,(z), the Bessel function Jn(s) and the exponential integral E,(%),for which we have the respective recurrence relations

(iii) the direction in which the recurrence is being applied. Examples are as follows. Stability-increasing

Qn(z),

2n

Jn+ Jn-I

=0

nEn+l+zEn= e-s.

Particularly for automatic work, recurrence relations provide an important and powerful computing tool. If the values of F'Jz) or J,(z) are known for two consecutive values of n, or E,($) is known for one value of n, then the function may be computed for other values of n by successive applications of the relation. Since generation is carried out perforce with rounded valuos, it is vital to know how errors may be propagated in the recurrence process. If the errors do not grow relative to the size of the wanted function, the process is said to be stable. If, however, the relative errors grow and will eventually overwhelm the wanted function, the process is unstable. It is important to realize that stability may depend on (i) the particular solution of the difference equation being computed; (ii) the values of x or other parameters in the difference equation;

Q."(z) @ < I )

Yn(z) Kn(z) J-n-%(Z),

I-n-%(z)

(n
(n+ l ) P n + i - (2n+ l)~Pn+nP,-l=O Jn+l-

n

P n ( 4 , P.,(z)

Stability-decreasing

n

Pn(z), P."(z) (z
Q:(z)

Jn(z),

In(%)

Jn+%(z), In+%(z)

E n b ) (n>4 . Fn(q,p ) (Coulomb wsve function)

Illustrations of the generation of functions from their recurrence relations are given in the pertinent chapters. It is also shown that even in cases where the recurrence process is unstable, it may still be used when the starting values are known to sufficient accuracy. Mention must also be made here of a refinement, due to J. C. P. Miller, which enables a recurrence process which is stable for decreasing n to be applied without any knowledge of starting values for large n. Miller's algorithm, which is wellsuited to automatic work, is described in 19.28, Example 1.

INTRODUCTION

XIV

8.

Acknowledgments

The production of this volume has been the result of the unrelenting efforts of many persons, all of whose contributions have been instrumental in accomplishing the task. The Editor expresses his thanks to each and every one. The Ad Hoc Advisory Committee individually and together were instrumental in establishing the basic tenets that served as a guide in the formation of the entire work. In particular, special thanks are due to Professor Philip M. Morse for his continuous encouragement and support. Professors J. Todd and A. Erddlyi, panel members of the Conferences on Tables and members of the Advisory Committee have maintained an undiminished interest, offered many suggestions and carefully read all the chapters. Irene A. Stegun has served effectively as associate editor, sharing in each stage of the planning of the volume. Without her untiring efforts, completion would never have been possible. Appreciation is expressed for the generous cooperation of publishers and authors in granting permission for the use of their source material. Acknowledgments for tabular material taken wholly or in part from published works are given on the first page of each table. Myrtle R. Kellington corresponded with authors and ublishers to obtain formal permission for incluiing their material, maintained uniformity throughout the

bibliographic references and assisted in preparing the introductory material. Valuable assistance in the preparation, checkin and editing of the tabular material was receive from Ruth E. Capuano, Elizabeth F. Godefroy, David S. Liepman, Kermit Nelson, Bertha H. Walter and Ruth Zucker. Equally important has been the untiring cooperation, assistance, and patience of the members of the NBS staff in handling the myriad of detail necessarily attending the publication of a volume of this magnitude. Especially appreciated have been the helpful discussions and services from the members of the Office of Technical Information in the areas of editorial format, graphic art layout, printing detail, preprinting reproduction needs, as well as attention to promotional detail and financial support. In addition, the clerical and typing st& of the Applied Mathematics Division merit commendation for their efficient and patient production of manuscript copy involving complicated technical notation. Finally, the continued support of Dr. E. W. Cannon, chief of the Applied Mathematics Division, and the advice of Dr. F. L. Alt, assistant chief, as well as of the many mathematicians in the Division, is gratefully acknowledged.

d

M. ABRAMOWITZ.

1. Mathematical Constants DAVIDS. LIEPMAN

Contents ,

. . . . . . . . . . . . . . . +,nprime
Table 1.1. Mathematical Constants

T)

* National Bureau

Page

2 2 2 2 2 2

2

3 3 3 3 3 3

3 3 3 3

of Standards. 1

2

MATHEMATICAL CONSTANTS

TABLE 1.1. MATHEMATICAL CONSTANTS n(prime) 2 3 5

7

11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

&

1.4142 1. 7320 2. 2360 2.. 6457 3.3166 3. 6055 4. 1231 4.3588 4. 7958 5.3851 5. 5677 6.0827 6.4031 6. 5574 6. 8556 7. 2801 7. 6811 7. 8102 8. 1853 8. 4261 8. 5440 8. 8881 9. 1104 9.4339

13562 50807 67977 51311 24790 51275 05625 98943 31523 64807 64362 62530 24237 38524 54600 09889 45747 49675 52771 49773 03745 944 17 33579 81132

37309 56887 49978 06459 35539 46398 61766 54067 31271 13450 83002 29821 43284 30200 40104 28051 86860 90665 87244 17635 31753 31558 14429 05660

7182 3890 0085 4598 4841 0342 0966 9809 1030 2026

81828 56098 53692 15003 31591 87934 33158 57987 83927 46579

em 45904 93065 31876 31442 02576 92735 42845 04172 57538 48067

n

1 2 3 4 5

6 7 8 9 10

3' I4 2-ln ( 3-10 ( -

5-*n ern er/4 e-rP e-r/4 e10 e-1 I2 e1 e-I 13

i-

(-

I.-

93320 37219 28012 34853 88924 01112 08040 25077 31648 83823 10667 24608 52440 76451 93928

4.8104 2. 1932 1 ) 2.0787 1) 4.5593 i: 6487

96535 73801 50761 65996 70oi2

1) 6.0653 06597 12633 1. 3956 12425 08608 1 ) 7. 1653 13105 73789

16555 54566 90855 23677 81468 42360 95286 25043

e-" 94411 71442 52832 36612 06836 78639 63888 87341 46999 08546

32159 69189 42979 80293 70966

- I

((-

n

52353 02272 67740 39078 60342 12260 85992 82747 40077 16516

60287 30427 92853 11026 11156 83872 63720 43592 09997 95790

1 2 3 4 5

6 7 8 9 10

(((-

(I-

3.6787 1.3533 4.9787 1.8315 3'1 6.7379

1) 1) 2) 2j

77380 80050 95763 81277 21270

( - 3j 21 4787 52176 66635 (-

(-

f-

4 ) 9.1188

19655 !6279 4 j 1.2340 98040 5 ) 4. 5399 92976

54516 02511 86679 24848

55949 29216 06320

49774 88144 99239 54918 46107 07282 27211 08397 85552 38387

26224 14792 1.7810 72417 99019

64190 79852

e-* e-7

((-

2) 6. 5988 03584 53125 1 ) 5.6145 94835 66885

37077 16982

In n 55994 668 10 11989 43410 22805 05531 67983 33621 99404 79837 46153 05621 16644 92914 98647 48514 64422 70430 69356

53094 96913 06188 03746 50008 33051 59282 93827 56840 05440 67360 60802 04600 96908 40271 62459 44443 78038 24234

( 1) 1. 5154

n

0.6931 1.0986 1.3862 1. 6094 1. 7917 1.9459 2.0794 2. 1972 2. 3025 2.3978 2. 5649 2. 8332 2.9444 3. 1354 3.3672 3.4339 3.6109 3. 7135 3.7612

47180 12288 94361 37912 59469 10149 41541 24577 85092 95272 49357 13344 38979 94215 95829 87204 17912 72066 00115

enr 27792 24764 79166 36653 34113 95446

84704

n

4.3213 1.8674 8.0699 3. 4873 (1.5070 i6. 5124 ( - i o j i. 8142 (-11) 1.2161 (-13) 5.2554 (-14) 2.2711

n

172321 952452 344642 007593 124774 053527 516964 904905 179915 619436 534874 495346 090274 067528 832720 291643 680957 667634 728425

2 3 4 5 6 7 8 9 10 11 13 17 19 23 29 31 37 41 43

(-1) 3.0102 (-1) 4.7712 ( - i j 6.0205 (-1) 6. 9897 (-1) 7.7815 (-1) 8.4509 I308 1.

_____

12503 80400 99869 25094 00000 92685 43352 48921 53600 27836 97997 61693

1. 1139 1.2304 1. 2787 1.3617 1.4623 i. 4913 - - .- 1. 5682 01724 1. 6127 83856 1.6334 68455

~

*8ee page

uuuu

i. 0413

log10 n 99956 63981 12547 19662 99913 27962

11.

* *

55238 39995 34242 71802 36048

20800 31361 83882 13891 54949 76367 51535 59152

e- n r 91826 37722 42731 70798 51757 03045 42356 20899 17275 39006 12136 07990 68457 48555 55670 94093 85176 00644 01068 32409

69263 16555 64780 ( 5) 2.8675 13131 ( 6) 6. 6356 23999 ( 8) 1. 5355 29353 ( 9 j 3.5533 21280 (10) 8.2226 31558 (12) 1.9027 73895 (13) 4.4031 50586

*

84230 45167

2) 3) 5) 6) 7) 9j

e*

9

2 1 /4

16837 03188 03892 46111 61277 50958 90349 53497 89487 30740 -. . __ 00272 95249 86547 89625 99957

((((-

( 1) 2.3140 ( 2) 5.3549 ( 4) 1.2391

10 11 13 17 19 23 29 31 37 41 43

31n

3. 1622 77660 2. 1544 34690 1. 7782 79410 1. 5848 93192 4. 6415 88833 2. 5118 86431 5. 6234 13251 3.9810 71705 1. 2599 21049 1. 4422 -. - ~ - - 49.570 _ _ -. 1. 1892 07115 1.3160 74012 1 ) 7.0710 67811 1) 5. 7735 02691 i j 4.4721 35954

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

2 3 4 5 6 7 8

10'0 101n 101/4 10'/6 1001n loo'/' 1000~/4 10001/5 2'

69006 73650 97482 29975 42333 69392 43597 95275 12917 2901 1

n

eV

50488 72935 96964 05905 98491 92931 05498 35522 95416 40313 19221 96890 86865 06523 41249 82711 81758 43941 99700 86306 11679 88501 88819 38113 47217

19521 43729 39042 80478 ...__ 83643 63250 14256 83071 91943 58564 39324 87459 00000 00000 15822 50407 30683 67692 37827 39285 95282 89615 01759 28788 89895 60873 83437 26796 06699 49968 ..... 71973 54945 57958 65264

37389 50279 74778 62611 87668 22163 12167 00558 00000 50200 06505 40170 36333 67777 32847 66704 08451 09412 05088

3

MATHEMATICAL CONSTANTS

TABLE 1.1. MATHEMATICAL CONSTANTS-Continued n 47 53 59 61 67 71 73 79 83 89 97

3. 8501 3. 9702 4. 0775 4. 1108 4. 2046 4. 2626 4. 2904 4. 3694 4.4188 4.4886 4. 5747

1LH

lnd2s

(-1)

47601 91913 37443 73864 92619 79877 59441 . . 47852 40607 36369 10978

1. 1447 29885 84940 9.1893 85332 04672

n 1 2 3 4 5 6 7 8 9

In n 71005 85868 209507 55212 i8341 444691 90571 94506 160504 17331 12487 513891 39096 60596 700720 04131 54213 294545 14839 _ ~ __ ~ 11290 . . ~ 921089 46702 14941 729455 79659 79234 754722 73213 98383 178155 50338 28221 167216

1. 6720 1. 7242 1. 7708 1. 7853 1. 8260 1. 8512 1. 8633 1. 8976 1.9190 1.9493 1. 9867

01741 43427 74178 03296

97857 75869 52011 29835 74802 58348 22860 27091 78092 90006 71734

(-1) 4. 9714 98726 94133 ( - 1 ) 4. 3429 44819 03251

4

S/3 d4 *1/2 ?r

'A

T 1 I4

$A Ira 14 ?,a12

(2H ) 'I2 (H/2)'I2 H(2)-ln

;: Y

nr 58979 17958 76937 43591 79489 15387 51285 87183 23081

n In 10

(

( ( (

3. 1415 6. 2831 9. 4247 1. 2566 1.5707 1. 8849 2.1991 2. 5132 2. 8274

92653 85307 77960 37061 96326 55592 14857 74122 33388

14219 32992 60656 85749 49132 92829 74387 94821 32760 23543 84362

85435 12683 82765 11289 32384 64769 97153 72953 66192 59430 52669 45907 39146

99404 98809 98213 97618 49702 79642 09583 39523 69464

56840 13680 70520 27360 28420 74104

1) 1)

85092 70185 55278 40371 92546 51055 09565 68074 26583

1) 1) 2) 2) 3) 3) 4) 4)

3.1415 9. 8696 3. 1006 9. 7409 3. 0601 9. 6138 3.0202 9. 4885 2. 9809 9.3648

92653 04401 27668 09103 96847 91935 93227 31016 09933 04747

58979 08935 02998 40024 85281 75304 77679 07057 34462 60830

32384 86188 20175 37236 45326 43703 20675 40071 11666 20973

62643 34491 47632 44033 27413 02194 14206 28576 50940 71669

(-5) (-5)

98861 11836 53443 98225 63643 61473 36801 03916 80357 27922

83790 42337 31994 46843 05338 29585 77566 53493 20886 68615

67153 77675 77144 38795 89184 _ . _42205 _ ~ 35189 15278 54726 28250 22960 89838 76432 59528 66633 17287 91287 39854 33662 04078

1. 5707 1.0471 . 1 ) 7. 8539 1. 7724 1. 4645 1. 3313 2. 1450 2. 3597 5. 5683 1) 2. 2459 ' 2. 5066 1.2533 2.2214

96326 97551 81633 53850 91887 35363 29397 30492 27996 15771 28274 14137 41469

79489 19659 97448 90551 56152 80038 11102 41469 83170 83610 63100 31550 07918

66192 77461 30961 60272 32630 97127 56000 68875 78452 45473 05024 02512 31235

31322 54214 56608 98167 ioi43 97535 77444 78474 84818 42715 i5765 07883 07940

4. 7123 88980 4. 1887 90204 4: 4428 82938 ( - 1 ) 5. 6418 95835 (-1) 6. 8278 40632 (-1) 7.5112 55444 ( - i j 4. 6619 40770 ( - 1 ) 4. 2377 72081 ( - 1 ) 1. 7958 71221 ( -2) 4. 4525 26726 ( 1) 3. 9894 22804 ( - 1 ) 7. 9788 45608 ( - 1 ) 4. 5015 81580

38468 78639 15836 47756 55295 64942 3541 1 23757 25166 69229 01432 02865 78553

98576 09846 62470 28694 68146 48285 61438 59679 56168 06151 67793 35587 03477

1) 1)

ij

17991 35983 53974 71966 08996 10795 19788 i2594 65472 14393 11156 16192

(

1) 1) 1)

( (

1) 1)

( (

i

ij

H-

( ( ( ( ( ( ( (

74644 90456 41902 70338 64341 52860 59010 14279 39038 27847 48517

2. 3025 4. 6051 6.9077 9.2103 1. 1512 1. 3815 1.6118 1.8420 2.0723

n 1 2 3 4 5 6 7 8 9 10

93571 60078 64214 01076 70082 71907 12045 29044 37607 64491 rY624

57. 2957 79513 08232 0. 0174 53292 51994 0.5772

15664 90153

1. 7724 2.6789 i. 3541 3. 6256 1. 2254 0. 8929 0. 9027 0. 9064 0. 9190 0. 9854 0. 3031 1. 2880 0. 2032

53850 905516 38534 707748

(-1) (-1) (-2) (-2) (-3) (-3) (-4)

i- 4 j

-

221908 465178 569249 950934 055477 848883 927767 147523 698077 431296

"

0. 0002 90888 20866 0. 0000 04848 13681

08767 98155' 32957 692371-

-0. 5495 39312 98164

28606 06512

i7939 426406

09908 16702 79511 45292 02477 62526 20646 50275 22524 80951

3.1830 1,0132 3.2251 1.0265 3.2677 1.0401 3.3109 1.0539 3. 3546 1. 0678

0. 5641 0. 3732 0. 7384 0. 27.58

89583 82173 88111 lrjS62

1. 1198 1.1077 1. 1032 1. 0880 -0. 1131 -0. 1023 -0. 0982 -0.0844

46521 32167 62651 65252 91641 14832 71836 01121

0: SiSO 489%

11r(714) In r ( q 3 ) In r(513) In r(5/4) In r(7/4)

62643 25287 87930 85057 31322 77586 23850 70115 16379

547756 907395 621648 830209 098263 722186 432472 320837 131017 740343 960640 421813 020486

93965 16858 15881 80795 70208 87030 19885 10077 90820 35273 99461 98921 75996

57215 96154r 10953 59936r 48223 37662

*

2, Physical Constants and Conversion Factors A. G. MCNISH

Contents Table 2.1. Common Units and Conversion Factors . . . . . . . . . Table 2.2. Names and Conversion Factors for Electric and Magnetic Units . . . . . . . . . . . . . . . . . . . . . . . Table 2.3. Adjusted Values of Constants . . . . . . . . . . . . . Table 2.4. Miscellaneous Conversion Factors. . . . . . . . . . . . Table 2.5. Conversion Factors for Customary U.S. Units to Metric Units . . . . . . . . . . . . . . . . . . . . . . . Table 2.6. Geodetic Constants . . . . . . . . . . . . . . . . . . .

Page

6

6 7 8 8 8

I

1

National Bureau of Standards. 6

2. Physical Constants and Conversion Factors The tables in this chapter supply some of t h e more commonly needed physical cons t a n t s and conversion factors. All scientific measurements in t h e fields of mechanics and 'heat a r e based upon four international arbitrarily adopted units, t h e magnitudes of which a r e fixed by four agreed on standards: Length - t h e meter - fixed by t h e vacuum wavelength of radiation corresponding to t h e transition 2PI0- 5Dj of krypton 86

rise to t h e CGS system, often used in physics and chemistry. Table 2.1. Common Units and Conversion

5 Factors

Quantity

Force, F Energy, W Power P

(1 meter - 1650763.73A).

Mass - t h e kilogram - fixed by t h e international kilogram at Sevres, France. Time - t h e second- fixed as 1/31,556,925.9747 of t h e tropical year 1900 at 12" ephemeris time, or t h e duration of 9,192,631,770 cycles of t h e hyperfine transition frequency of cesium 133. Temperature - t h e degree - fixed on a thermodynamic basis by taking t h e temperature for t h e triple point of natural water as 273.16 OK. (The Celsius scale is obtained by adding -273.15 to t h e Kelvin scale.) Other units a r e defined in terms of them by assigning the value unity to t h e proportionality constant in each defining equation. The entire system, including electricity units, is called t h e Systkme International d'Unit6s (SI). Taking t h e 1/100 part of t h e meter as t h e unit of length and t h e 1/1000 part of t h e kilogram as t h e unit of mass, similarly, gives

newton joule w a tt

erg

.................... 107 The SI unit of electric current is t h e ampere defined by t h e equation 2rrnZlZz/4.rr=F giving t h e force in vacuo per unit length between two infinitely long parallel conductors of infinitesimal cross-section. If F is in newtons, and h a s t h e numerical value 477 X lO-', then ZI and Z2 are in amperes. The customa r y equations define t h e other electric and magnetic units of SI such as t h e volt, ohm, farad, henry, etc. The force between electric charges in a vacuum in this system is given by Q1Q 2 / 4 r r e r= 2 F, re having t h e numerical value 107/4,rrc2where c is t h e speed of light in meters per second (re=8.854 x 10-12). The CGS unrationalized system is obtained by deleting 477 in the denominators in these equations and expressing F in dynes, and r in centimeters. Setting r,,,equal to unity defines t h e CGS unrationalized electromagnetic system (emu), re then taking t h e numerical value of 1/c2. Setting re equal to unity defines t h e CGS unrationalized electrostatic system (esu), Trn then taking t h e numerical value of 1/c2.

rrn

Table 2.2. Names and Conversion Factors for Electric and Magnetic Units

!

Quantity

Current Charge Potential Resistance Inductance Capacitance Magnetizing force Magnetomotive force Magnetic flua Magnetic flux density Electric displacement

ampere coulomb volt ohm henry farad amp. turns/ meter amp. turns weber tesla

emu name

1

esu name

1

SI unit1 esu unit

SI unit1 emu unit

10-1 statampere abampere 10-1 statcoulomb abcoulomb 108 statvolt abvolt 109 statohm abohm _ _ _ _ _ _ _ _ _ _ - -109 centimeter 10-9 _ _ _ _ _ - - - - - _- _ centimeter 4 n 10-3* ~ oersted

-3 X log -3 X loo ~ ( 1 1 3X) 10-2 N(1/9) x 10-11 -(1/9) x 10-1' -9 x 10" - 3 x 109*

-3 / 108* 4rXlO-'* -(1/3)X10-2 108 -(l/3)X10-e _ _ . _ _ . 104 ____._ __ - 3.x_105* __ __ _ io-5* _ __ __ _ _ _ _ . _ __ _. _ __ _._

gilbert maxwell gauss

__.____.___...

_ _ _ _ - - - _ _ . - - - -

~~

Example: If the value assigned t o a current is 100 amperes its value in abamperes is lOOX 10-I= 10. *Divide this number by 4s if unrationalized system is involved; othe r numbers a r e unchanged.

6

7

PHYSICAL CONSTANTS AND CONVERSION FACTORS

The values of constants given in Table 2.3 are based on an adjustment by Taylor, Parker, and Langenberg, Rev. Mod. Phys. 41, p.375 (1969). They are being considered for adoption by the Task Group on Fundamental Constants of the Committee on Data for Science and Technology, International Council of Scientific Unions. The uncertainties given are standard errors estimated from the experimental data included in the adjustment. Whore applicable, values are based on the unified scale of atomic masses in which the atomic mass unit (U) is defined a s 1/12 of the mass of the atom of the 1*C nuclide.

2%-Systeme Irternational I

Constant

(SI)

$

I

k10

~ 1 0 8 m/s

~

Nn U 1

me

Proton rest mass ..............................

m,

Neutron rest mass ..........................

m,

Faraday constant ............................

,F

Compton wavelength of electron ....

A0

xc/2u

Compton wavelength of proton ....

xc.p xo,p/2c

...................................... Electron radius ................................ Gyromagnetic ratio of proton ........

T.

7

r/2r

(uncorrected for diamagnetism, H10) .............................................. Bohr magneton .................................. Nuclear magneton ............................ Proton moment ..................................

Centimeter-gram-second (CGS)

_____

Speed of light in vacuum ................ c Elementary charge ............................ e Avogadro constant .......................... Atomic mass unit ............................ Electron rest mass ..........................

Unit

{ r'/2r 7'

BB PN PP

PPlPN

(uncorrected for diamagnetism, Hs0) ................................................ P'PIPN Gas constant ...................................... R Normal volume perfect gas ............ T.', Boltzmann constant .......................... k First radiation constant (8rhc) .... c, Second radiation constant .............. cz Stefan-Boltzmann constant ............ U Gravitational constant .................... G

1.672614 1 1.007 276 611 1.674 920 1.008 665 20 9.648 670 2.892 599 6.626 196 1.054 591 9 7.297 351 1.370 360 2 1.758 802 8 6.272 759 4.135 708 1.3795234 1 2.426 309 6 3.861 592 1.321 440 9 2.103 139 1.097 373 12 5.291 771 5 2.817 939 2.675 196 5 4.257 707 2.675 127 0 4.257 597 9.274 096 5.050 951 1.410 620 3 2.792 782 2.792 709 8.314 34 2.241 36 1.380 622 4.992 579 1.438 833 5.669 61 6.673 2

17 35 39 59 38 61 96 31

$Based on 1 std. dev; applies to last digits in preceding column.

........................ J K-1 mol-1 10-2 m3/m01 10-23 J/K 10-24 J a m 10-2 m-K 10-8 W . m-2K-4 10-11 N m2/kgt 100 100

x10'0 10-20 10-10 1023 10-2' 10-28 10-4 10-24 100 10-84 100 103 1014 10-21 10-2' 10-3 ...................... 102 cm1/2/g1/2 * 107 cm3/2g-Ws-l t 1017 10-1 cmS/2g1/2s-1 * 10-17 cmW2gW t 10-10 cm 10-11 cm cm 10-13 10-14 cm cm-1 105 10-9 cm 10-13 cm rad 8-1G-1 * 104 rlG-1 * 103 rad s-1G-1 * 104 rlG-1 * 103 10-21 erg/G * 10-24 erg/G * 1043 erg/G * 100 100 107 104 10-18 10-15 100 10-5 10-9

*Electromagnetic system.

...................... erg K-1 mol-1 crns/mol erg/K erg. cm cm K e r g cm-*s-lK-4 dyn cmVg9

tElectrostatic system.

PHYSICAL CONSTANTS AND CONVERSION FACTORS

8

Table 2.4. Miscellaneous Conversion Factors

Standard gravity, go Standard atmospheric pressure, PO 1 thermodynamic calorie,l cal, 1 IT calorie2,cal. 1 liter, 1 1 angstrom unit, A 1 bar

= 9.806 65 meters per second per second* = 1.013 25 x 105 newtons per square meter*

= 1.013 25 x 106 dynes per square centimeter* = 4.1840 joules* = 4.1868 joules"

= 10-3cubic meter* = 1O-lo meter* = 105 newtons per square meter*

= 106 dynes per square centimeter* 1 gal

= 10-2 meter per second per second*

1 astronomical unit, AU 1 light year 1 parsec

= 1 centimeter per second per second* = 1.496 x 10" meters = 9.46 x 1015meters = 3.08 x 10IGmeters = 3.26 light years

1 curie, the quantity of radioactive material undergoing 3.7 x 101O disintegrations per second". 1 roentgen, the exposure of x- or gamma radiation which produces together with its secondaries 2.082 X log electron-ion pairs in 0.001 293 gram of air.

The index of refraction of the atmosphere for radio waves of frequency less than 3 X 1010 Hz is given by ( n - 1)106= (77.6/t) ( p 4810e/t), where n is the refractive index; t, temperature in kelvins; p , total pressure in millibars; e, water vapor partial pressure in millibars.

+

Factors for converting the customary United States units to units of the metric system are given in Table 2.5. Table 2.5. Factors for Converting Customary U.S. IJnits to SI Units 1 yard 1 foot 1 inch 1 statute mile 1 nautical mile (international) 1 pound (avdp.) 1 oz. (avdp.) 1 pound force 1 slug 1 poundal 1 foot pound Temperature (Fahrenheit) 1 British thermal unitS

0.914 4 meter* 0.304 8 meter* 0.025 4 meter* 1609.344 meters* 1852 meters* 0.453 592 37 kilogram* 0.028 349 52 kilogram 4.448 22 newtons 14.593 9 kilograms 0.138 255 newtons 1.355 82 joules 32 (9/5) Celsius temperature* 1055 joules

+

Geodetic constants for the international (Hayford) spheroid are given in Table 2.6. The gravity values are on the basis of the revised Fotsdam value. They are about 14 parts per million smaller than previous values. They are calculated for the surface of the geoid by the international formula.

Length of 1' of longitude

Latitude

Meters

Oo

15 ~

1855.398 1792.580 1608 174 131:175 930.047 481.725

I 1 j ~

~

Length of 1' of latitude

B

Meters

mlsr

1842.925 1844.170 1 847.580 1852.256 1856.951 1860.401 1861.666

9.780350 9.783 800 9.793 238 9.806 154 9.819 099 9.828 593 9.832 072

Used principally by chemists. Used principally by engineers. 3 Various definitions are given for the British thermal unit. This represents a rounded mean value differing from none of the more important definitions by more than 3 in 104. * Exact value.

3. Elementary Analytical Methods MILTONABRAMOWITZ

Contents

. . .............

Elementary Analytical Methods . . 3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometric Progressions; Arithmetic, Geometric, Harmonic and Generalized Means . . . , . . . . . . . . . . . 3.2. Inequalities . . . . . . . . . . . . . . . . . . . . . . 3.3. Rules for Differentiation and Integration . . . . . . . . . 3.4. Limits, Maxima and Minima . . . . . . . . . . . . . . 3.5. Absolute and Relative Errors . . . . . . , . . . . . . . 3.6. Infinite Series. . . . . , . . . . . . . . . , . . . . . 3.7. Complex Numbers and Functions . . . . . . . . . . . . 3.8. Algebraic Equations . . . . . . . . . . . , , . . . . . 3.9. Successive Approximation Methods . . . . . . . . . . . 3.10. Theorems on Continued Fractions . . . . . . , . . . . .

... . .

...,..........

Numerical Methods , , ., 3.11. Use and Extension of the Tables 3.12. Computing Techniques . . . . . References.

Page

10

10 10 11 13 14 14 16 17 18 19

. . . . . . , . , . . . . . . . . . . . . . . .

19 19 19

....,..,,.,,.......,...,...

23

Table 3.1. Powers and Roots . . . . , . . , n’, k = l ( l ) l O , 24, 1/2, 1/3, 1/4, 1/5 n=2(1)999, Exact or 10s

,

,

. . . .

, ,

. .

24

The author acknowledges the assistance of Peter J. O’Hara and Kermit C. Nelson in the preparation and checking of the table of powers and roots.

1

National Bureau of Standards. (Deceased.) 9

3. Elementary Analytical Methods 3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometric Progreseions; Arithmetic, Geometric, Harmonic and Generalized Means

3.1.9 Sum of Arithmetic Progression to n Terms

. . . +(a+(n--l)d) =nu+-1 n(n- l)d=zn (a+l) ,

a-t-(a+d)+(a+2d)+

Binomial Theorem

3.1.1

+

(a b) *=a"+

C) +G) an- b

2

last term in series=Z=a+(n-l)d

a" - 2b*

Sum of Geometric Progression to n Terms

+G)an-W+

. , . +bn

3.1.10 sn=a+ar+a13+.

(n a positive integer)

lim s,=a/(l-r)

(-l
n-m

n

*

1-r

Binomial Coefficients (see chapter 24)

3.1.2

. . +ar"-l=-a(1-rn)

n(n-1)

. . . (n-k+l)- k!

(k)'n61k=

3*1*3 c)=(nnk)=(-')k(

n! (n- k) !k!

Arithmetic Mean of n Quantities A

)

k-n-1 k

A= al+az+ n

3.1.11

a

. . +an

Geometric Mean of n Quantities G

n+l

(k

3.1.4

G=;(Ulaz. , .an)'/"

C)=C)=I

-=1 1

k=1,2,

. . .,n)

. . . +')

an

(ak>O, k=1 ,2,

. . .,n)

Generalized Mean

l-c)+c)-. . . +(-1)"@)=O Table of Binomial Coefficients

(-+-+ 1 1 az

H n al

l+C)+c)+.. . + C ) = 2 "

3.1.6

(Uk>O,

Harmonic Mean of n Quantities H

3.1.13

3.1.5

3.1.7

3.1.12

)'(nk)+(k!l)

3.1.14

M(t)=O(t
3.1.15

(9

3.1.16

3.1.8

lim M ( t )=max. l+-

3.1.17 limM(t)=min.

(al,a2, . . .,a,) =mtLx. a

(al,a2,. . .,a,)=min.

U

t+-m

3.1.18

limM(t)=G t 4

3.1.19

M(l)=A

3.1.20

M(-1)=H 3.2. Inequalities

Relation Between Arithmetic, Geometric, Harmonic and Generalized Means

3.2.1

A > G>H, equality if and only if al=a2=

For a more extensive table see chapter 24. *See page 11.

10

I

3.2.2

min. a<M(t)<max. a

. . . =an

11

ELEMENTARY ANALYTICAL METHODS

Minkowski’s Inequality for Sums

min. a
3.2.3

equality holds if all a k are equal, or t
3.2.4 M(t)<M(s) if t<s unless all ak are equal, or s
lull - la2l5 lal+a21 5 lall

3.2.5

+I4

If p > l and a,k, bk>O for all k , 3.2.12

(9 (ak+bk)’)l”<(& k=l

k=l

U$)”’+(&

k=l

b$)l”,

equality holds if and only if bk=cak (c=cons t an t >0) . Minkowski’s Inequality for Integrals

If P>l, Chebyshev’s Inequality

. . . >a, . . . >b,

If al>a22a,> blkb,>b,>

5 akbk>(&

3.2.7

kml

(& bk)

Hiilder’s Inequality for Sums

1 P

3.2.13

1 q

If -+-=1, p>l

equality holds if and only if g(x)=cf(x) (c=constant>O).

, q>l 3.3. Rules for Differentiation and Integration Derivatives

equality holds if and only if ]brl=cIQkIP-’ (c=constant>O). If p = q = 2 we get Cauchy’s Inequality

3.3.1

3.3.2

- (cu)=c

d dx

du -, c constant dx

d dx

d u dv dx dx

- (U+)=-+-

3.2.9 3.3.3

d dv - (uv)=u -+v dx dx

du dx

c constant). Hiilder’s Inequality for Integrals

1

1

If - + - = l , p > l , P q

q>l

3.3.5

d d u dv - u(v)=- dx dv d x

3.2.10

equality holds if and only if Ig(x)I=clflx)lP-‘ (c =constan t > O ) . If p = q = 2 we get Schwara’s Inequality

3.2.11

Leibnia’s Theorem for Differentiation of a n Integral

3.3.7

I

12 .

ELEMENTARY ANALYTICAL METHODS

s(u~$)~~c)n

The following - formulas are useful for evaluating

Leibniz's Theorem for Differentiation of a Product

3.3.8

where P ( x ) is a polynomial and

d"

n>l is an integer.

d"v

3.3.16

s

dx

2

(a?+ br +c) =(4ac-b 2 ) t

arctan

2ax+ b (4ac-bZ)t (b2- 4a~
3.3.9

-

3.3.17

lnl2 a x f b - ( b 2 - 4 ~ ) 4 (b2-4~)4 2ax+b+(b2-4~)*

_-

3.3.10

(b2-4~>0)

-2 2ax+b

(b2-4 a = ~ 0)

=-

3.3.18 3.3.19

1 In lax2+bx+c\--

Integration by Parts

J

3.3.20

J u d v =u v - vdu

3.3.12

dx bx

3.3.21

U

Integrals of Rational Algebraic Functions

(Integration constants are omitted) 3.3.14 S(ax+b)"dx= (ax+ b) n+'

(n#-1)

3*3*24

s

dx

(x2+

a2)

X X -_ arctan-+ 2-2a3 a 2a2(x2

+

3.3.15

Integrals of Irrational Algebraic Functions '

3.3.26 3.3.27

3.3.29 3.3.30

=-

-1

(- bd) '1'

arcsin

("bdiT+5+ ")

(b>O, d
a2)

13

ELEMENTARY ANALYTICAL METHODS

3.3.31 A

-X a

J-+=arcsin dx

dx 3.3.44 [(u+bx)(~+dx)]'/~

2-x2)f+- a2 arcsin X 2 (a 2 a

3.3.45 s(a2-x2)*dx=- X

3.3.32

dx [(a+bx)(c+d~)I'~~ 3.3.33

3.3.41

dx s ( a x 2 + bx+c)'/2 =a-'/2 In 12a'/2(ar2+ bx+~)'/~+2ax+bl (a>O)

3.3.48

-a-'/2

3.3.34

-

arcsinh (2az+ b) (4ac- b2)'I2 (a>(), 4w>b2)

=a-'/2 In j2ax+

3.3.35

--(-a)-lI2

3.3.36

bl (a>O, b2=4ac)

arcsin

Pax+ b) (b2-4ac)

. x-a dx +=arcsin S(2ux-x ) a

S ( 2 a x - x 2 ) + d x = m ( 2 a x - x 2 ) * +U2~arcsin x-a 2

U

3.3.49

s

dx (ax2+ b) (cx2+d)*

'I2

(a4w, 12ax+ bl< (b2-4ac)1/2) 3.3.50

3.3.37

I

1 [b(~~~+d)]++~(b~-ad)+ 2 [ b (bc -ad)]+ In [b(Cz2+d) I*- x (bc-ad)

I

J(ax2+bx+c)1/2dx=- 2ax+ b (ax2+bx+c)'/2 4a 4ac- b2 dx +8a ,,f(ax2+bx+c)1/2

(bc>ad)

3.3.38

dx s x (asz bx c) 'I2-

3.4. Limits, Maxima and Minima

dt

+ +

1/2

where t =1/x Indeterminate Forms (L'Hospital's Rule)

3.3.39

Let f ( x ) and g ( z ) be differentiable on an, interval a
XdX

If

s ( a x 2 + bx+c)1/2

limf(x)=O and lim g(x)=O

z+b-

z+b-

or if limf(x) = CO and lim g(x)= 03

z+b-

z+b-

and if 3.3.41 X

(z2f a2)idz=2 ( x z& a2)+ f

In Iz+

(z2fa2)*1

Both b and 1 may be finite or infinite.

14

ELEMENTARY ANALYTICAL METHODS

Maxima and Minima

3.4.2 ( 1 ) Functions of One Variable The function y=f(x) has a maximum at x=xo if f ’ ( x o ) = O and f”(xO)O. Points zo for which f’(xo) =O are called st,ationary points. 3.4.3 (2) Functions of Two Variables The function f ( 5 , y) has a maximum or minimum for those values of (xo, yo) for which

I

Approximate Values

If lel<
b<
1711<<1,

3.5.8

(a+ b)k =ak+ka”-’b

3.5.9

(1 +4 ( 1-to) = 1 +€+?

3.5.10

-lfe-l+€-V I+? 3.6. Infinite Series Taylor’s Formula for a Single Variable

3.6.1

+

(a) j ( x , y ) has a maximum

*

..

+-( nhn-I- l ) ! f (,-”(x)+Rn

(b) f (5, y) has a minimum if -2>0 ?f bX

w

and -2>0

at (xo, yo)

dY

=“-s‘

3.5. Absolute and Relative Errors

(n-l)!

(1-t)n-l f (“)(x+th)dt

( 1 ) If xo is an approximation to the true value of x , then

3.5.1 (a) the absolute error of xo is Ax=xo-x, x-xo is the correction to x. 3.5.2

(b) the re2ative error of zo is 6x=-

2

3.5.5 (3) If f(sl, x2, . . ., 5,) is a function of xl, x2, . . ,, x, and the absolute error in x f (i=l, 2, , . . n) is A q , then the absolute error infis

. . . +-df

1

f (,-l)(a) +Rn

50

3.5.4 (2) The absolute error of the sum or difference of several numbers is at most equal to the sum of the absolute errors of the individual numbers.

Ax2+

(5

=-

3.5.3 (c) the percentage error is 100 times the relative error.

Af =zl

-a) n. . . + (n-l)!

Ax Ax 3.6.5

(x-a), Rn=- n! f ( n )(€)

@<€<XI

Lagrange’s Expansion

3.6.6

3.6.7

Axn

axn

3.5.6 (4) The relative error of the product or quotient of several factors is at most equal to the sum of the relative errors of the individual factors. 3.5.7 (5) If y= j ( x ) , the relative error 6y=- AY =f’(’) -Ax Y f(x)

where g(x) is any function indefinitely differentiable. Binomial Series

3.6.8

15

ELEMENTARY ANALYTICAL ME!CHODS

I 3.6.14

3.6.9 (1+z)"= 1+ax+

1)

. ( a ~

2!

l)(a-2) 3!

. ( a x2+

x3+

. ., *

(1+2)-+=1--

3.6.10

...

(1+4-1=1-~+~2-2+~4-

91 -729

(-i<~
3.6.11

+...

A series

k=O

"+

Operation

I

cl

'

* *

akx-k is said to be an asymp-

n-1

for everyn=1,2,

154

(-l<x
f ( x ) - Ck -0 akx-k=O(x-n) as z+cu

3.6.13 ( l + x ) f = l + - 1 2-- 1 22+- 5 2-- 10 x4 3 9 81 243 22

...

totic expansion of a functionf(x) if

5 2 +--35x4 63x5 16 128 256

+-729 xb-- 6561

728 6561

xb+-x6-

m

3.6.15

x 3x2 --+--2 8

14 35 23- 2 4 81 243

Asymptotic Expansions

2 22 2 3 5x4 7x5 21xs (1+z)*= I+---+---+---2 8 16 128 256 1024

(1+x) -*=I

1 x+- 2 2,3 9

. . ,.

We write m

f(x) -r,a k x - k . k-0

I

(-'<'
I

I

Ca

3.6.16

-at

a: - al

3.6.17

-2al

3a: - 2ag

The series itself may be either convergent or divergent,

Ca

I

c4

3.6.18

1 1 1 3 5 --128' ' ~a4-~a~a~-p!+--a:a~ 16

3.6.19

paa+ga: 3 3 -3a4-3ak+-a: 1 1 5 35 128

3.6.20

nal

3.6.21 3.6.22 3.6.23 3.6.24

*See page I!.

I

1 T(n-l)clal+nal

*

I

16

ELEMENTARY ANALYTICAL METHODS,

3.6.25 Given y = ax b 9 + cx3+dx4+ ex5+fx6+ gx7

then x=Ay+By2+Cy3+Dy4+Ey5+Fy'+Gy7f where

z=TeiB=T(cos O+i sin

3.7.2

+..

+

Polar Form

I

Reversion of Series

Modulus:

3.7.3

121= (s2+y2)f=r

3.7.4 Argument: urg z=arctan (y/x)=e (other notations for arg z are am z and ph z ) .

.. .

3.7.5

aA= 1 a3B= -b a5C= 2 b2- ac a7D= Babc -a2d -5b3 a9E=6a2bd 3a2c2+ 14b4-a3e-2 lab2c

3.7.6

Real Part:

Z = ~ Z = T

cos e

Imaginary Part: y = Y z = ~ sin 0 Complex Conjugate of

-z=x-iy

3.7.7

+

5

PI= 14

3.7.8

a11F=7a3be+7a3cd +84ab3c-a4f -28a2bc2-42b6-28a2b2d

arg z=

3.789

G= 8a4bf +8a4ce+4a4d2+ 12Oa2b3d

a13

e)

-arg z

Multiplication and Division

+ 180a2b2c2+132

a5g-36a3b2e -72a3bcd- 12a3~3-330ab4c b6-

If Z i = X i f i Y i , z2=52+iy2, then zizz=~i52-yiy2

+i(xiy, +xzyi)

Kummer's Transformation of Series

3-7-10

3.6.26 Let 5 a k = s be agiven convcrgent series and

3*7*11

I 21z21= I 21I I 221

3.7.12

arg (zlzz)=arg zl+arg z2

..Ir=n" ~

m

k 10

ck=c be a given convergent series with known

3-7-13

sum c such that lim %=X#O. k - m ck Then

s=xc+g

22

(14

);

If

2

Powers

3.7.16

vergent series with sum s then

k

k

'=c 2ki-1

A k a o = E (-l)m(m)

k-0

arg (z)=&rg zl-arg z2

3.7.15

. . . is It con-

(-1)~ak=ao-al+a2-

rn=O

3.7.17

ak-,n

zn=Tnefne = r n cos n8+irn sin no

(n=O,&l,f2, 22=x2- Y2+ i (2xy)

3.7.18

Euler-Maclaurin Summation Formula

"

"

--

(0) ]

1 +30240 -[ f'

1

1209600

")

(n)-f

3.7.21

'"'(O)]

[f'""(n)-f'VN'(O)]+

* *

.)

2 =2 -3xy2+ i ( 3 2 y -y")

3.7.19

3.6.28

1 -720 - [ f '(n)-f '

d+$

1Z*l2

3.7.14

ak.

Euler's Transformation of Series

3.6.27

zl_zlz~=~l~z+~l~,+i(~*yl-~1~zZ

26=

2'-

10x3$+ 5xy4+i(5s4y- 10x2y"+ 9)

3.7.22

. . *

Z.=[P-@

x.-2y2+(3

e4y4-

. . .]

3.7. Complex Numbers and Functions Cartesian Form

3.7.1

z=x+iy

I

(n=1,2,.

. .)

17

ELEMENTARY ANALYTICAL METHODS

If zn=un+ivn, then zn+l=u,+l+iv,+l

where

3.8. Algebraic Equations

I

Solution of Quadratic Equations

3.8.1 Given a z 2 + b z + c = 0 , 3.7.24 21

3.7.25

-

zf=Jz=rfeftO=r* cos +e+irt sin $0

If -s<e5s this is the principal root. The other root has the opposite sign. The principal root is given by 3.7.27 zf=[$(r+x)]f fi[$ ( r - x ) ] f = u i i v where 2uv=y and where the ambiguous sign is taken to be the same as the sign of y. 3.7.28 zlin=r1lnetoln1(principal root if -?r<e<s). Other roots are r11nei(0+2nX'ln @ = I , 2 , 3 , . , ,, n-1). Inequalities

z2=

-b/a, z1zz= c/a

If p>O, two real roots, p=O, two equal roots, p
Roots

3.7.26

+

Solution of Cubic Equations

i

3.8.2

Given 29+a2Z2+a1z+a0=O, p=-

let

1 1 a1-4;r=-1 (ulu~-3u~)-- 1 a;. 3 9 6 27

If $+r2>0, one real root and a pair of complex conjugate roots, p3+r2=0, all roots real and at least two are equal, p3+r2<
+($ +r2)4]#, sa=[r- (p3+r') f]t

then Complex Functions, Cauchy-Riemann Equations

a2

z1= (s1Ss2) -3

f(z) =.f(x+iy)

= U (z, y) +iv(x, Y)where u b , y), v(x, y) are real, is analytic at those points z=x+iy a t which

3.7.30

If z=rele, 3.7.31

h l b v l --=-b u bv br r be r be br

If zl, zz, 23 are the roots of the cubic equation

+zz +z3= -a2 zizz + +

-E--)

21

zi 23

Laplace's Equation

Z ~ Z Z Z ~ -U0 =

The functions u(zl y) and v(2,y) are called harmonic functions and satisfy Laplace's equation: Cartesian Coordinates

Solution of Quartic Equations

3.8.3 Given z 4 + a 3 ~ + a 2 ~ * + a l ~ + a 0 =find 0 , the real root u1 of the cubic equation U3 -a&+

Polar Coordinates

Z223=al

(u1u3-4aO)u

- (a; +aoai -4uOuz) =0

and determine the four roots of the quartic as solutions of the two quadratic equations

18

ELEMENTARY ANALYTICAL METHODS

Method of Iteration (Successive Substitution)

If all roots of the cubic equation arc real, usc the valuc of u I which givcs rcal coefficiciits in tlic *quadratic cquation and select signs so that if

3.9.4 The iteration scheme converge to a zero of z=F(z) if

will

zk+l=F(zk)

(1) lF’(x)IIq
Newton’s Method of Successive Approximations

3.9.5

Newton’s Rule

If x = X k is an approximation to the solution z = t of f(x)=O then the sequence 3.9. Successive Approximation Methods General Comments

3.9.1 Let z=zl be an approximation to z=E where f(t)=O and both x1 and 5 are in the interval a l x l b . We define

xn+1=xn+CJ(xn)

(n=1, 2 ,

..

will converge quadratically to z = f : (if instead of the condition ( 2 ) above), (1) Monotonic convergence, f(xo)f” (xo)>O and f’(z), f”(z) do not change sign in the interval (z, E), or ( 2 ) Oscillatory convergence, f ( x o ) f ” (zo)
a).

Then, if f’(z)ZO and the constants cn are negative and bounded, the sequence zn converges monotonically to the root E. If cn=c=constant O , then the process converges but not necessarily monotonically.

Newton’s Method Applied to Real nth Roots

3.9.6 Given zn=N, if z k is an approximation x=N1lnthen the sequence

Degree of Convergence of an Approximution Process

3.9.2 Let xl, x2, x3, . . . be an infinite sequence of approximations to a number E. Then, if I~n+l-EI
(n=1, 2 ,

*)

will converge quadratically to 5.

where A and k are independent of n, the sequence is said to have convergence of a t most the kth degree (or order or index) to E. If k = l and A < l the convergence is linear; if k = 2 the convergence is quadratic. Regula Falsi (Falee Position) Aitken’s 6*-Process for Acceleration of Sequences

3.9.3 Given y=f(z) to find E such that f(E)=O, choose ro and z1 such that f(ro)and f(z,) have opposite signs and compute

Then continue with z2 and either of ro or zI for which f(ro) or f(zl) is of opposite sign to f(z2). Regula falsi is equivalent to inverse linear interpolation. *See page

11.

3.9.7 If x k , ; c ~ are + ~ three successive iterates in a. sequence converging with an error which is approximately in geometric progression, then

i

1

A2Ck=

xk-

2tk + 1

+

x k +2

is an improved estimate of z. In fact, if xk=z+* O(P)then Z=z+O(P), Ihl
19 *

ELEMENTARY ANALYTICAL METHODS

(5) For every n 2 0,

Definitions

3.10.1 a1

f =bo+=

(1) Let

If the number of terms is finite, .f is ciilled IL tenlliniiting continued friiction. Jf the nuniber of ternls is infinite,f is ciilled tin infinite continued frtiction und the terniintiting fraction

1

22

a , - +~aOalaz ~

* *

Xn

+

=- 1

aoala~. . . U,& a02

U12

ao+ a1-2+

az-x+

~

an-12

~

' * *

+an--2

A

is called the nth convergent ofJ. An (2) If lim - exists, the infinite continued frticn-s- Bn tion f is said to be convergent. If a,=l tind the bc are integers there is iilwuys convergence. Theorems

(1) If at iind br tire positive then

f*n<J2n+2,

2.0

-

1.8

-

1.6

-

1.4

-

1.2

-

1.0

-

.8 -

j i n - 1 >.f2n+l* .6 -

(2) Iffn=-J An

Bn An=bnAn- 1+anAn - 2

Bn= b n B n - 1+ a n B n - ~ 1, Ao=bo,B-l=O, Bo= 1

where

.4

-

.2

-

0

FIGURE 3.1. y=x". &tn=O,

Example 1. Compute xl' and using Table 3.1.

X47

(919.826)"' = 5.507144. B~ Newton's method for fourth roots with N=919.826,

for 2=29

1

~19=~'.$0

= (1.4507 1 4598. 1013)(4.20707 2333 * 1014)

=6.10326 1248. 102' (X")'/Z

x47=

= (1.25184 9008. 103')'/29

919'826 +3(5.507144) -5,50714 3845

I

Repetition yields the same result. Thus, x1/'=5.50714 3845/101= 1.74151 1796, ~ - ~ / ' = ~ t / ~ = . 1 8 905683. 33

=5.40388 2547 10ee Example 2.

Compute x-3/4 for x=9.19826.

1 1

5 -, 2 1, 2, 5 .

- 5

3.12. Computing Techniques Example 3. *See page

11.

Solve the quadratic equation

20

ELEMENTARY ANALYTICAL METHODS

I

.056 f.001. From 3.8.1 the solution is X=

3 (18.2f[ (18.2)'-4 ( .056)];)

To use Newton's method we first form the table of f ( ~ ) = 2 ~ - 1 8 . l r - 3 4 . 8

=3(18.21[331.01G]1)=3(18.2118.1930) =18.=9,

Solve the cubic equation x3- 18.12

Example 5.

-34.8 =O.

,002

Example 4.

Compute (-3

From 3.7.26, (- 3

+.0076i)?.

+.OO76i)+=

U+

We obtain by linear inverse interpolation:

iv where

Thus T=[

Using Newton's method, f'(2)=3x2-

(-3)'+

(.0076)']f=(9.00005776)~=3.000009627

-

Example 6.

Solve the quartic equntion

5.004- (-*07215 9936) =5.00526. 57.020048

Repetition yields x1=5.00526 5097. Dividing f(z) by x-5,00526 5097 gives ~ ~ f 5 . 0 0 5 2 50972 6 i-6.95267 869 the zeros of which are -2.50263 2549 & .83036 8OOi.

I

J~--2.377524922~'+6.07350 5 ? 4 1 ~ ' -11.17938 0 2 3 ~ + 9 . 0 5 2 6 55259=0.

We seek that value of p1 for whicli y(ql) = O . Inverse interpoltition in y(pl) gives y(pI) = O for pl -2.003. Then,

Resolution Into Quadratic Factors (x'+ PlZ Q1) (22 P2Z 42) by Inverse Interpolation

+

+ +

Sttirting with tlie triiil value pl= 1 we compute successively 2. 003

I -~ 9. 4. 053 526 4. 115

-1. 093

-2. 543 -3. 106

I

4. 520

1

-2. 550

I I ,

172

5: -2. 023

-2. 55259 257 -2. 55282 851 -2. 55306 447

~

17506 765 17530 358 17553 955

~

. 00078 . 00001

552 655 -. 00075 263

Inverse interpolution gives q1=2.00420 2152, nnd we get finnlly, 91

92

2. 00420 2152

4. 51683 7410

. 011

Inverse interpoltition between ql=2.2 and pl= 2.003 gives ql=2.0041, and thus,

i!i

- 1 . . 284 165 . 729

18.1 we get

21 =:so-f(~o)/f'(~o)

We note that the principal square root has been computed.

1 2 2. 2

fb) -43.2 - .3 72.6 181.5

4 5 6 7

The smaller root may be obtained more accurately from * .05f3/18.=9= .0031 ,0001.

PI -2. 55283 358

Y(qd

P2

. 17530

8659

-.

00000 0011

21

ELEMENTARY ANALYTICAL METHODS

Method (2)--If N and d are numbers each not more than 19 digits let N=N1+No109, d=dl+ dolog where No and do contain 10 digits and Nl and d, not more than 9 digits. Then

Double Precision Multiplication and Division o n a Desk Calculator

Example7. MultiplyM=20243 97459 71664 32102 by m=69732 82428 43662 95023 on a 10XlOX20 desk calculating machine. Let M0=20243 97459, i1f1=71664 32102, mo= 69732 82428, m,=43662 95023. Then M m = Momo1020 (Moml Mlmo)10"+ Mlm,. (1) Multiply .i%f1ml=3129O 75681 96300 28346 and record the digits 96300 28346 appearing in positions 1 to 10 of the product did. (2) Transfer the digits 31290 75681 from positions 11 to 20 of the product dial to positions 1 to 10 of the product dial. (3) Multiply cumulatively M,mo+ Moml$31290 75681=58812 67160 12663 25894 and record the digits 12663 25894 in positions 1 to 10. (4) Transfer the digits 58812 67160 from positions 11 to 20 to positions 1 to 10. (5) Multiply cumulatively Momo+58812 67160 =14116 69523 40138 17612. The results as obtained are shown below,

+

-= NOIOQ+N1 N =-

d

1 [ N - TNo ] di dolOe

Here

+

N=14116 69523 40138 1761, d=20243 97459 71664 3210 No=14116 69523, d0=20243 97459, d1=71664 3210 (1) Nodl=10116 63378 42188 8830 (productdial). (2) (Nodl)/do=49973 55504 (quotient dial). (3) N-(Nodl)/do= 14116 69522 90164 62106

(product dial). (4) [N- ( N ~ d , ) / d ~ ] / d ~ l.69732 O ~ = 82428=first 10 digits of quotient in quotient dial. Remainder =r=08839 11654, in positions 1 to 10 of product dial. (5) r/(dolog)=,43662 9502.1O-'O=next 9 digits of quotient. Nld= .69732 82428 43662 9502. This method may be modified to give the quotient of 20 digit numbers. Method (1) may be extended to quotients of numbers containing more than 20 digits by employing higher order interpolation.

9630028346 1266325894 14116695234013817612 141166952340138176121266325894963~?28346

If the product M m is wanted to 20 digits, only the result obtained in step 5 need be recorded. Further, if the allowable error in the 20th place is a unit, the operation Mlml may be omitted. When either of the factors M or m contains less than 20 digits it is convenient to position the numbers as if they both had 20 digits. This multiplication process may be extended to any higher accuracy desired.

Sum the series $=l-+++-t to 5D using the Euler transform. The sum of the first 8 terms is .634524 to 6D. If u n = l / n we get n Un Au, A%,, A%, A%,,

+

Example 9. , , ,

9

.llllll

10

.100000

-11111

Example 8. Divide N=14116 69523 40138 17612 by d=20243 97459 71664 32102. Method (lb-linearinterpolation.

2020 -9091

11

.090909

1515 -7576

12

NI20243 97459.10"= .69732 82430 90519 39054 NI20243 97460.10'0= .69732 82427 46057 26941 Difference=3 44462 12113.

.Of43333

-505 -349

156

1166 -6410

13

.076923

From 3.6.27 we then obtain

Difference X .71664 32102=24685 64402&10-20 (note this is an 1 1 x 1 0 multiplication).

.111111 (- .011111) 22

s= . 6 3 4 5 2 4 + 2 - -

-(-

Quotient = (69732 82430 90519 39054-246856 44028)-10-M =.69732 82428 43662 95025

There is an error of 3 units in the 20th place due to neglect of the contribution from second differences.

dolOg+dl

.002020

+ T

.000505) .000156 24 +,a

= .634524+ .055556+ ,0027784- ,000253

=.693148

I

(S=ln 2=.6931472 to 7D).

+.000032+.000005

22

ELEMENTARY ANALYTICAL METHODS

Example 10. Evaluate the integral a =- to 4D

2

lm 5k--'=F 10

dx

using the Euler transform.

k - 2 +k g= l (k+lo)-2

'=I

k=l

10

=E k-1 k - ' + S m 0

1 1 j(k)dk--2jo-E j; 1 +-720 j;"- . .

where j(k)=(k+10)-2. Evaluating the integrals in the last sum by numerical integration we get

m

k=l

kd2= 1.54976 7731

,

Thus,

+.1

- .005+.00016

6667- ,00000 0333

=1.64493 4065,

k

T2

as compared with -=1.64493 4067. 6 Example 12. Compute

1.85194 *

43379

.25661

arctanx=-

. 18260

A

. 14180 . 11593

-2587

A2

A3

A4

I

22

- --

..

,

to 5 D for x = . 2 . Here ul=x, un=(n-1)2x2 for n > l , bo=O, b,=2n-l1 A-l =l, B-l=O, Ao=O,

Bo= 1.

- 1788

.09805

-321 478

- 1000

For n > I

799

-1310 -08495

4x2 9x2 1+ 3+ 5+ 7 + 2

- 168

153

310

.07495

The sum to k = 3 is 1.49216. Applying the Euler transform to the remainder we obtain 1 1 1 (.14180)-,jj (-.02587)+2 (.00799) 2

-

1 1 -(- .00321)+5 (.00153) 24 = .07090+ .00647+ .00100+.00020

+.00005

= .07862

We obtain the value of the integral as 1.57018 as compared with 1.57080. m a2 Example 11. Sum the series k-2=T using k=l

the Euler-Maclaurin summation formula. From 3.6.28 we have for n = , Q)

["']-I =I]:[

B3

.6

.2

3.04

1

3.032

.6

15.36 3.04

1 ' /=I / I 7 )=I -16

.36

3.032 14=.197396 15.36

B3

21.440 /'4=.197396 108.6144 B 4

Note that in carrying out the recurrence method for computing continued fractions the numerators A , and the denominators B, must be used as originally computed. The numerators and denominators obtained by reducing AJB, to lower terms must not be used.

ELEMENTARY ANALYTICAL METHODS

23

References Texts [3.1] R. A. Buckingham, Sumerical methods (Pitman Publishing Corp., S e w York, Y.Y., 1957). [3.2] T. Fort, Finite differences (Clarendon Press, Oxford, England, 1948). [3.3] L. Fox, The use and construction of mathematical tables, Mathematical Tables, vol. 1, Kational Physical Laboratory (Her Majesty’s Stationery Office, London, England, 1956). [3.4] G. H. Hardy, A course of pure mathematics, 9th ed. (Cambridge Univ. Press, Cambridge, England, and The Macmillan Co., S e w York, N.Y., 1947). [3.5] D. R. Hartree, Numerical analysis (Clarendon Press, Oxford, England, 1952). [3.6] F. B. Hildebrand, Introduction to numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1956). [3.7] A. S. Householder, Principles of numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [3.8] L. V. Kantorowitsch and V. I. Krylow, Naherungsmethoden der Hoheren Analysis (VEB Deutscher Verlag der Wissenschaften, Berlin, Germany, 1956; translated from Russian, Moscow, U.S.S.R., 1952). [3.9] K. Knopp, Theory and application of infinite series (Blackie and Son, Ltd., London, England, 1951). [3.10] Z. Kopal, Numerical analysis (John Wiley & Sons, Inc., New York, S.Y., 1955). [3.11] G. Kowalewski, Interpolation und genaherte Quadratur (B. G. Teubner, Leipzig, Germany, 1932). [3.12] K. S. Kunz, Numerical analysis (McGraw-Hill Book Co., Inc., New York, X.Y., 1957). [3.13] C. Lanczos, Applied analysis (Prentice-Hall, Inc., Englewood Cliffs, S . J . , 1956). [3.14] I. M. Longman, S o t e on a method for computing infinite integrals of oscillatory functions, Proc. Cambridge Philos. Soc. 62, 764 (1956). [3.15] S. E. Mikeladze, Numerical methods of mathematical analysis (Russian) (Gos. Izdat. Tehn.Teor. Lit., Moscow, U.S.S.R. , 1953). [3.16] W. E. Milne, Numerical calculus (Princeton Univ. Press, Princeton, N.J., 1949). [3.17] L. M. Milne-Thomson, The calculus of finite differences (Macmillan and Co., Ltd., London, England, 1951).

[3.18] H. Mineur, Techniques de calcul numerique (Librairie Polytechnique Ch. BBranger, Paris, France, 1952). [3.19] Xationh Physical Laboratory, Modern computing methods, Notes on Applied Science No. 16 (Her Majesty’s Stationery Office, London, England, 1957). [3.20] J. B. Rosser, Transformations to speed the convergence of series, J. Research NBS 46, 56-64 (1951). [3.21] J. B. Scarborough, Numerical mathematical analysis, 3d ed. (The Johns Hopkins Press, Baltimore, Md.; Oxford Univ. Press, London, England, 1955). [3.22] J. F. Steffensen, Interpolation (Chelsea Publishing Co., New York, N.Y., 1950). [3.23] H. S. Wall, Analytic theory of continued fractions (D. Van Nostrand Co., Inc., New York, N.Y., 1948). [3.24] E. T. Whittaker and G. Robinson, The calculus of observations, 4th ed. (Blackie and Son, Ltd., London, England, 1944). [3.25] R . Zurmuhl, Praktische Mathematik (SpringerVerlag, Berlin, Germany, 1953).

Mathematical Tables and Collections of Formulas [3.26] E. P. Adams, Smithsonian mathematical formulae and tables of elliptic functions, 3d reprint (The Smithsonian Institution, Washington, D.C., 1957). 13.271 L. J. Comrie, Barlow’s tables of squares, cubes, square roots, cube roots and reciprocals of all integers up t o 12,500 (Chemical Publishing Co., Inc., New York, N.Y., 1954). [3.28] H. B. Dwight, Tables of integrals and other mathematical data, 3d ed. (The Macmillan Co., New York, N.Y., 1957). [3.29] Gt. Britain H.M. Nautical Almanac Office, Interpolation and allied tables (Her Majesty’s Stationery Office, London, England, 1956). [3.30] B. 0. Peirce, A short table of integrals, 4th ed. (Ginn and Co., Boston, Mass., 1956). [3.311 G. Schulz, Formelsarnmlung zur praktischen Mathematik (de Gruyter and Co., Berlin, Germany, 1945).

24

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS

Table 3.1

nk

k

1 2 3 4 5 6 7 8 9 10 24

4 5 6 7 8 9 10

24 1/2 1/3 1/4 115

d=

See Examples 1-5 for use of the table.

nf= no =

Floating decimal notation ; 910=34867 84401 =(9)3.4867 84401

3 19 97 (16)5.9604 2.2360 1.7099 1.4953 1.3797

5 25 125 625 3125 15625 78125 90625 53125 65625 64478 67977 75947 48781 29662

10000 1 00000

00000 00000 00000 00000

7.n6= = as, n9=

256 512 do= 1024 n24= 167 77216 nl:*= 1.4142 13562 n1/3= 1. 2599 21050 1.1892 07115 n1/5= 1.1486 98355

2 16 100 604 (18)4.7383 2.4494 1.8171 1.5650 1.4309

6 36 216 1296 7776 46656 79936 79616 77696 66176 81338 89743 20593 84580 69081

1 17 194 2143 ( 9)2.3579 (10)2.5937 (24)9.8497 3.3166 2.2239 1.8211 1.6153

11 121 1331 14641 61051 71561 87171 58881 47691 42460 32676 24790 80091 60287 94266 16 256 4096 65536 48576 77216 35456 67296 47674 11628 16251 00000 42100 00000 01127

10 100 1000

8 9 10 24 1/2 1/3 114 1/5

10 100 1000 ( 9 1.0000 (1011.0000 (24)1.0000 3.1622 2.1544 1.7782 1.5848

4 5 6 7 8 9 10 24 1/2 1/3 1/4 115

15 225 3375 50625 7 59375 113 90625 1708 59375 ( 9 2.5628 90625 ( 1 0 1 3.8443 35938 (11)5.7665 03906 (28)1.6834 11220 3.8729 83346 2.4662 12074 1.9679 89671 1.7187 71928

10 167 2684 f 9)4.2949 ( ioj 6;8719 (12)1.0995 (28)7.9228 4.0000 2.5198 2.0000 1.7411

20 400 8000 1 60000 32 00000 640 00000 ( 9)1.2800 00000 (ioj 2.5600 ooooo (11)5.1200 00000 (13)1.0240 00000 (31)1.6777 21600 4.4721 35955 2.7144 17617 2.1147 42527 1.8205 64203

21 441 9261 1 94481 40 84101 857 66121 ( 9 1.8010 88541 f 1013.7822 85936 (iij7;9428 00466 (13)1.6679 88098 (31)5.4108 19838 4.5825 75695 2.7589 24176 2.1406 95143 1.8384 16287

4 5 6

7 8 9 10 24 1/2 1/3 1/4 115

2 4 8 16 32 64

n2

n3=

00000

00000 77660 34690 79410 93192

(11)2. 8242 1.7320 1.4422 1.3160 1.2457

3 9 27 81 243 729 2187 6561 19683 59049 95365 50808 49570 74013 30940

1 8 57 403 2824 (20)1.9158 2.6457 1.9129 1.6265 1.4757

7 49 343 2401 16807 17649 23543 64801 53607 75249 12314 51311 31183 76562 73162

2 20 167 1342 ( 9)1.0737 (21)4.7223 2.8284 2.0000 1.6817 1.5157

2 29 358 4299 9)5.1597 10)6.1917 (25)7.9496 3.4641 2.2894 1.8612 1.6437

12 144 1728 20736 48832 85984 31808 81696 80352 36422 84720 01615 28485 09718 51830

13 169 2197 28561 3 71293 48 26809 627 48517 8157 30721 (10)1.0604 49937 ( 11)1.3785 84918 (26)5.4280 07704 3.6055 51275 2.3513 34688 1.8988 28922 1.6702 77652

14 241 4103 9)6.9157 11)1.1858 12)2.0159 (29)3.3944 4.1231 2.5712 2.0305 1.7623

17 289 4913 83521 19857 37569 38673 57441 78765 93900 86713 05626 81591 43185 40348

18 324 5832 1 04976 18 89568 340 12224 6122 20032 10)1.1019 96058

22 484 10648 2 34256 51 53632 1133 79904 12 1.2072 (1312.6559 ( 32)1.6525 4.6904 2.8020 2.1657 1.8556

69218 92279 10926 15760 39331 36771 00736

2 10 (14)2.8147 2.0000 1.5874 1.4142 1.3195

8 64 512

9

si

,"..,

AnQA

(30)l. 3382 4.2426 2.6207 2.0597 1.7826

32768 62144 97152 77216 17728 41824 66483 27125 00000 92831 16567

4 16 64 256 1024 4096 16384 65536 62144 48576 49767 00000 01052 13562 07911

5 47 430 3874 ( 9)3.4867 (22)7.9766 3.0000 2.0800 1.7320 1.5518

729 6561 59049 31441 82969 46721 20489 84401 44308 00000 83823 50808 45574

14 196 2744 38416 5 37824 75 29536

(27)3.2141 3.7416 2.4101 1.9343 1.6952

99700 57387 42264 36420 18203

19 361 6859 1 30321 24 76099 470 45881 8938 71739

58845 40687 41394 67144 02458

( 30)4.8987 62931

23 529 12167 2 79841 64 36343 1480 35889 9)3.4048 25447 10)7.8310 98528 12)1.8011 52661 (13j4.1426 51121 (32)4.8025 07640 4.7958 31523 2.8438 66980 2.1899 38703 1.8721 71231

24 576 13824 3 31776 19 62624 1911 02976 9)4.5864 71424 11)1.1007 53142 12 2.6418 07540 131 6.3403 38097 (33)l. 3337 35777 4.8989 79486 2.8844 99141 2.2133 63839 1. 8881 75023

4.3588 2.6684 2.0877 1.8019

98944 01649 97630 83127

25

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS k 1 2 3 4 5 6

25 625 15625 3 90625 97 65625 2441 40625 15625 78906 97266 43164 (33)3.5527 13679 5.0000 00000 2.9240 17738 2.2360 1.9036 67977 53939

24 1/2 1 1/5 ’4 1 2

30 900 27000 8 10000 243 00000 7290 00000 10)2.1870 00000 lll16.5610 00000 13 1.9683 00000 (14)5.9049 00000 (35)Z. 8242 95365 5.4772 25575 3.1072 32506 2.3403’ 1.9743 47319 50486

3

4 5 6 7 8

9 10 24 1/2

’I3 1 1/ ’45 1 2 3 4 5

35

1225 42875 00625 21875 65625 29688 75391 63867 47354 13124 79783 66310 99279 68005

15 525

( 37)1.1419

24

5.9160 3.2710 2.4322 2.0361

1 ’2 ’I3 1/5

40 1600 64000 25 60000 1024 00000

1 2 3 4

24

(38)2.8147 6.3245 3.4199 2.5148 2.0912

’” 1/5 lf4 1 2 3 4 5 6 7 8

9 10 24 1/2

49767 55320 51893 66859 79105

45 2025 91125 41 00625 1845 28125 ( 9)8. 3037 65625 (11)3.7366 94531 (13)1.6815 12539 (14 7. 5668 06426 (1613.4050 62892 (39)4.7544 50505 6.7082 03932 3.5568 93304 2.5900 20064 2.1411 27368

4

118

3089 918. 0318 1112.0882 12 5.4295 14 1.4116 (33)9.1066 5.0990 2.9624 2.2581 1.9186

\I

26 676 17576 56976 81376 15776 10176 70646 03679 70957 85770 19514 96068 00864 45192

31 961 29791 9 23521

27 729 19683 5 31441 143 48907 3874 20489

28 784 21952 6 14656 172 10368 4818 90304

( 34)2.2528 39954

( 34) 5.3925 32264

5.5677 3.1413 2.3596 1.9873

64363 80652 11062 40755

2.2795 07057 1.9331 82045

33 1089 35937 11 85921 391 35393 9 1.2914 67969 /10/4.2618 44298 12 1.4064 08618 13 4.6411 48440 (15)1.5315 78985

( 36)1.3292 27996

( 36)Z. 7818 55434

10 335 1.0737 3.4359 1.0995 3.5184

5.6568 3:1748 2.3784 2.0000

41 1681 68921 28 25761 1158 56201

42 1764 74088 31 11696 1306 91232

10945 24237 17240 39534 32478

46 2116 97336 44 77456 2059 62976

( 39)8.0572 70802

6.7823 3.5830 2.6042 2.1505

29983 47871 90687 60013

5.7445 3.2075 2.7967 2.0123

62647 34330 81727 46617

20 792 3.0109 1.1441 4.3477 1.6521 6.2782

38 1444 54872 85136 35168 36384 55826 92138 61013 11848

(37)8.2187 6.1644 3,3619 2.4828 2.0699

60383 14003 75407 23796 35054

54249 02104 14230 00000

37 1369 50653 18 74161 693 43957 9 2.5657 26409 10 9.4931 87713 12 3.5124 79454 14 1.2996 17398 15 4.8085 84372 ( 37) 4.3335 25711 6.0827 62530 3.3322 21852 2.4663 25715 2.0589 24137

(38)5.0911 6.4031 3.4482 2.5304 2.1016

02622 88972 26634 94361

32 1024 32768 48576 54432 41824 73837 11628 37209 99907

36 1296 46656 16 79616 604 66176 9 2.1767 82336 10 7.8364 16410 12 2.8211 09907 14 1.0155 99567 15)3.6561 58440 (37)2.2452 25771 6.0000 00000 3.3019 27249 2.4494 89743 2. 0476 72511

I

5.2915 3.0365 2.3003 1.9472

5.1961 52423 3.0000 00000

1.1258

( 35) 6.2041 26610

Talde 3.1

nk

( 9

11 12 14 15

29 841 24389 7 07281 205 11149 5948 23321

(35)l. 2518 5.3851 3.0723 2.3205 1.9610

34 1156 39304 13 36336 454 35424

(36)5. 6950 5.8309 3.2396 2.4147 2.0243

03680 51895 11801 36403 97459

23 902 3.5187 1.3723 5.3520 2.0872 8.1404

39 1521 59319 13441 24199 43761 10067 09260 83612 06085

(38)1.5330 6.2449 3.3912 2.4989 2.0807

29700 97998 11443 99399 16549

9

11 12 14 15

43 1849 79507 34 18801 1470 08443

44 1936 85184 37 48096 1649 16224

( 39)2.7724 53276

(38)9. 0778 6.4807 3.4760 2.5457 2.1117

49315 40698 26645 29895 85765

( 39) 1.5967 72093

38524 98060 49602 47461

6.6332 3.5303 2.5755 2.1315

49581 48335 09577 25513

1 48 2293 1.0779 5.0662 2.3811 1.1191 5.2599 (40)1.3500 6.8556 3.6088 2.6183 2.1598

47 2209 03823 79681 45007 21533 31205 28666 30473 13224 46075 54600 26080 30499 30012

48 2304 10592 08416 03968 59046 83423 28043 05461 06211 37322 03230 41186 48026 43542

1 57 2824 10 1.3841 lll\6.7822 (13 3.3232 1 5 1.6284 16 7.9792 (40)3.6703 7.0000 3.6593 2.6457 2.1779

49 2401 17649 64801 75249 28720 30728 93057 13598 26630 36822 00000 05710 51311 06425

6.5574 3.5033 2.5607 2.1217

1

53

2548 (10)1.2230 Ill)5.8706 13)2.8179 15 1.3526 (1616.4925 (40)2.2376 6.9282 3.6342 2.6321 2.1689

The numbers in square brackets a t the bottom of the page mean that the maximum error in n linear interpolate is a x 10-P ( p in parentheses), and that to interpolate to the full tabular accuracy W L p o p t s must be used in Lagrange’s and Aitkens methods for the respective functions Nr.

‘see page

11.

49008 64807 16826 95787 09057

26

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS

Table 3.1 k 1 2 3

s

9 10 24

1 2 3 4 5 6 7 8 9 10 24 1/2 1/3 114 1/ 5 1 2 3 d

5 6

50 2500 1 25000 b2 50000 00000 00000 00000 50000 25000 25000 (40)5.9604 64478 7. 0710 67812 3; 6840 31499 2.6591 47948 2.1867 24148 55 3025 1 66375 91 50625 5032 84375

( 41)5.8708 98173

7.4161 3.8029 2.7232 2.2288

98487 52461 69815 07384

60 3600 2 16000 129 60000 7776 00000

(40)9.5870 7.1414 3.7084 2. 6723 2.1954

51 2601 32651 65201 25251 28780 06779 94457 65173 24238 33090 28429 29769 45118 01897

56 3136 1 75616 98 34496 5507 31776

(41)9.0471 7.4833 3.8258 2.7355 2.2368

67858 14774 62366 64800 53829

61 372I 2 26981 138 45841 8445 96301

52 2704 1 40608 73 11616 3802 04032

(41)1.5278 7.2111 3.7325 2. 6853 2.2039

48342 02551 11157 49614 44575

57 3249 1 85193 105 56001 6016 92057

(42)1. 3835 7.5498 3.8485 2.7476 2.2447

55344 34435 01131 96205 86134

62 3844 2 30328 147 76336 9161 32832

1

s

9 10 24 1/2 1/ 3 114 1/5

(42)4.7383 7.7459 3.9148 2.7831 2.2679

81338 66692 67641 57684 33155 65 4225

1

-2 1. 4 b 2 5

178 50625 6 7 8 9 10 24 1/2 1/3 1/4 115 1 2 3 4 5 6 7 8 9 10 24 1/2 1/3 114 115

11017.5418 9 1.1602 (12 4.9022 14 3.1864 16 2.0711

90625 89063 27891 48129 91284

(43)3.2353 8.0622 4.0207 2.8394 2.3045

44710 57748 25759 11514 31620

(42)7.0455 7.8102 3.9364 2.7946 2.2754

(44)1.9158 8. 3666 4.1212 2.8925 2.3389

12314 00265 85300 07608 42837

( 43) 1. 0408 79722

66 4356

67 4489 3 00763 201 51121

2 07496

186 i4736

(43)4.6671 8. 1240 4.0412 2.8502 2.3115

70

4900 3 43000 240 10000

68477 49676 97183 82393 43032

78950 38405 40021 69883 79249 /I 5041 _. .

3 57911 254 11681

( 44)2. 6927 76876

8.4261 4.1408 2.9027 2.3455

49773 17749 83108 87669

7.8740 3.9578 2.8060 2.2828

(43)6.6956 8.1853 4.0615 2.8610 2.3185

07874 91610 66263 55056

88867 52772 48100 05553 41963

72 5184 3 73248 268 73856

16 5 1998 181 3: 7439 ( 44)3.7668 8.4852 4.1601 2.9129 2.3521

69781 06243 63772 81374 67646 50630 58045

nk

1 78 4181 10 2.2164 12 1.1747 13 6.2259 15 3.2997 17 1.7488 (41)2.4133 7.2801 3.7562 2.6981 2.2123

53 2809 48877 90481 95493 36113 11140 69041 63592 74704 53110 09889 85754 67876 56822

54 2916 1 57464 03056 65024 91130 25210 96134 05912 25193 (41)3.7796 38253 7.3484 69228 3; i797 63150 2.7108 06011 2.2206 43035

1 113 6563 10 3 8068 1212: 2079

58 3364 95112 16496 56768 69254 84168

59 3481 2 05379 121 17361 7149 24299

(42)2.1002 7.6157 3.8708 2.7596 2.2526

54121 73106 76641 69021 07878

2 157 9924 10 6 2523 121 3: 9389 14 2.4815 16 1.5633 17 9.8493 (43)1. 5281 7.9372 3.9790 2.8173 2.2901

63 3969 50047 52961 36543 50221 80639 57803 81416 02919 75339 53933 57208 13247 72049

!I

68 4624 3 14432 213 81376 9)1.4539 33568

(43)9.5546 8.2462 4.0816 2.8716 2.3254

30685 11251 55102 21711 22030 73 5329

3 sSOi7 283 98241

(44)5.2450 8.5440 4.1793 2.9230 2. 3586

38047 03745 39196 12786 55818

(42)3.1655 7. 6811 3.8929 2.7714 2.2603

43453 45748 96416 88002 22470

64 4096 2 62144 167 77216

(43)2.2300 8.0000 4.0000 2.8284 2.2973

74520 00000 00000 27125 96710

69 4761 3 28509 226 67121

(44)1. 3563 8.3066 4.1015 2.8821 2.3322

70007 23863 65930 21417 21626

74 5476 4 05224 299 86576

(44)7.2704 8.6023 4.1983 2.9329 2. 3650

49690 25267 36454 72088 82769

27

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS nk k 1 2 3 4

24 1/2 1/3 114 115 1

2

3 4 5 6 7 8

9 10 24 1/2 1/3 114 115

1 2 3

4

24 1/2 1/3 114 1/5

5 6 7 8

9 10 24

76 5776 4 38976 333 62176

75 5625 4 21875 316 40625

( 4 5 ) 1. 0033 8.6602 4.2171 2.9428 2.3714

91278 54038 63326 30956 40610 80

6400 5 12000 40960000 9 3,2768 00000 11 2.6214 40000 1 3 2.0971 52000 1 5 1.6777 21600 1 7 1.3421 77280 ( 1 9 1.0737 41824 ( 4 5 ) 4.7223 66483 8.9442 71910 4.3088 69380 2.9906 97562 2.4022 48868

I

85 7225 6 14125 522 00625

( 4 6 ) 2.0232 9.2195 4.3968 3.0363 2.4315

71747 44457 29672 70277 53252

90 8100 7 29000 656 10000 00000 10000 69000 72100 04890 84401 ( 4 6 ) 7.9766 44308 9.4868 32981 4.4814 04747 3.0800 70288 2.4595 09486 95 9025 8 57375 814 50625

( 4 5 ) 1. 3788 8.7177 4.2358 2.9525 2.3777

Ij

79182 97887 23584 91724 30992

77 5929 4 56533 351 53041

( 4 5 ) 1.8870 8.7749 4.2543 2.9622 2.3839

( 4 7 ) 2.9198 9.7467 4.5629 3.1219 2.4862

90243 94345 02635 85641 49570

78 6084 4,74552 370 15056

23915 64387 20865 56638 55503

( 45) 2.5719 97041

82 6724 5 51368

83 8889 5 71787

8.8317 4.2726 2.9718 2.3901

60866 58682 27866 15677

79 6241 4 93039 389 50081

( 4 5 ) 3.4918 8.8881 4.2908 2.9813 2.3962

06676 94417 40427 07501 12991

84 7056 5 92704

5 430 9 3.4867 11 2.8242 1 3 2.2876 1 5 1.8530 17 1.5009 ( 1 9 1.2157 ( 4 5 ) 6. 3626 9.0000 4.3267 3.0000 2.4082

81 6561 31441 46721 84401 95365 79245 20189 46353 66546 85441 00000 48711 00000 24685

9 11 13 15 17 19 ( 4 5 ) 8.5414 9.0553 4.3444 3.0092 2.4141

66801 85138 81486 16698 41771

6 547 4.7042 4.0456 3.4792 2.9921 2.5732 2.2130 (46)2. 6789 9.2736 4.4140 3.0452 2.4372

86 7396 36056 00816 70176 72351 78222 79271 74173 15789 39031 18495 04962 61646 47818

6 572 9)4.9842 lljl.3362 1 3 3.7725 1 5 3.2821 17 2.8554 19 2.4842 ( 4 6 ) 3.5355 9.3273 4.4310 3.0540 2.4428

87 7569 58503 89761 09207 62010 47949 16715 41542 34142 91351 79053 47622 75810 89656

( 9 111 13 15 17 19 (46)4.6514 9.3808 4.4479 3.0628 2.4484

04745 31520 60181 14314 79851

( 4 6 ) 6.1004 9.4339 4.4647 3.0714 2.4540

61181 04400 92014 41445 90619 50932

92 8464 7 78688 716 39296 15232 50013 66012 88731 13633 84542 ( 4 7 ) 1.3517 85726 9.5916 63047 4.5143 57435 3.0970 41015 2.4703 44749

8 748 9) 6.9568 11 6.4699 1 3 6.0170 15 5.5958 1 7 5.2041 1 9 4.8398 ( 4 7 ) 1. 7522 9.6436 4.5306 3.1054 2.4756

93 8649 04357 05201 83693 01834 08706 18097 10830 23072 28603 50761 54896 22799 91866

94 8836 8 30584 780 74896 40224 97811 75942 89385 48022 51141 ( 4 7 ) 2.2650 01461 9.6953 59715 4.5468 35944 3.1137 37258 2.4809 93182

96 9216 8 84736 849 34656

9464 9 12673 885 29281

9 922 ( 9) 9. 0392 11 8.8584 13 8.6812 1 5 8.5076 17 8.3374 19 8.1707 ( 4 7 ) 6.1578 9.8994 4.6104 3.1463 2.5017

98 9604 41192 36816 07968 23809 55332 30226 77621 28069 03365 94937 36292 46284 57527

99 9801 9 70299 960 59601

91 828I 7 53571 685 74961

19) 3.8941 ( 4 7 ) 1. 0399 9.5393 4.4979 3.0885 2.4649

( 4 7 ) 3.7541 9.7979 4.5788 3.1301 2.4914

32467 58971 56970 69160 61879

( 46) 1.1425 47375

9.1104 4.3620 3.0183 2.4200

( 4 7 ) 4.8141 9.8488 4.5947 3.1382 2.4966

72219 57802 00892 88993 30932

33579 70671 49479 01407 88

7744 6 81472 599 69536

91

9

10 24

Table 3.1

;i

( 4 6 ) 1.5230 9.1651 4.3795 3.0274 2.4258

10388 51390 19140 00104 04834

89 7921 7 04969 627 42241

( 4 7 ) 7.8567 9.9498 4.6260 3.1543 2.5068

25945 81132 45096 78656 19455

81408 74371 65009 42146 42442

28

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS nk

Table 3.1 k 1 2 3 4 5 6 7 8 9 10 24 1/2 1/3 1/4 1/5

24 1/2 113 114 1/5

1 2 3

4 5 6

100 10000 10 00000 1000 00000

(48)1.0000 00000 00000 4.6415 88834 3.1622 77660 2.5118 86432

( 1)l.OOOO

11 1215 1.2762 1.3400 1.4071 1.4774 1.5513 1.6288 (48)3.2250 ( 1)l. 0246 4.7176 3.2010 2.5365

105 11025 57625 50625 81563 95641 00423 55444 28216 94627 99944 95077 93980 85873 17482

110 12100 13 31000 1464 10000

101 10201 10 30301 1040 60401

(48)1.2697 ( 1)1.0049 4.6570 3.1701 2.5168

34649 87562 09508 53880 90229

11 1262 10 1.3382 12 1.4185 14 1.5036 16 1.5938 18 1.6894 20 1.7908 (48)4. 0489 ( 1) 1.0295 4.7326 3.2086 2.5413

106 11236 91016 47696 25578 19112 30259 48075 78959 47697 34641 63014 23491 80436 30642

111 12321 13 67631

102 10404 10 61208 1082 43216

(48)1.6084 ( 1)1.0099 4.6723 3.1779 2.5218

37249 50494 28728 71828 54548 107 11449

12 25043

(48)5.0723 ( 1) 1.0344

4.7474 3.2162 2.5461

79601 51731 30352 81476 86180 59212 51357 66953 08043 59398 21453 07613

112 12544 14 04928 1573 51936

103 10609 10 92727 1125 50881 74074 52297 73865 70081 73184 16379 (48)2.0327 94106 ( 1)l. 0148 89157 4.6875 48148 3.1857 32501 2.5267 80083

104 10816 11 24864 1169 85856

(48)2.5633 ( 1)1.0198 4.7026 3.1934 2.5316

108 11664 12 59712 1360 48896

(48)6.3411 ( 1)1.0392 4.7622 3.2237 2. 5508

80737 30485 03156 09795 49001

04165 03903 69375 36868 67508

109 11881 12 95029 1411 58161

(48)7.9110 83175 ( 1) 1.0440 30651

4.7768 56181 3.2311 46315 2.5555 55397 114 12996 14 81544 1688 96016

113 12769 14 42897 1630 47361

7

c

9 10 24 1/2 1/3 1/4 1/5

( 48) 9.8497 32676 ( 1)l. 0488 08848

4.7914 19857 3.2385 31840 2.5602 27376

(49)1.2239 ( 1)l. 0535 4.8058 3.2458 2.5648

15658 65375 95534 67180 65499

(49)1.5178 ( 1)l. 0583 4.8202 3.2531 2.5694

62893 00524 84528 53123 70314

1 2 3 4

115 13225 15 20875 1749 00625

116 13456 15 60896 1810 63936

117 13689 16 01613 1873 88721

24 1/2 1 0 1/4 115

( 49) 2.8625 17619 ( 1) 1.0723 80529

( 49) 3.5236 41704 ( 1)l. 0770 32961

( 1)1.0816 65383

120 14400 17 28000 2073 60000

121 14641 17 iiki 2143 58881

122 14884 18 15848 2215 33456

9 10 24 1/2 1/3 114 1/5

4.8629 44131 3. 2747 22171 2.5830 90178

( 20) 6.1917 (49)7.9496 ( 1)l. 0954 4.9324 3.3097 2.6051

36422 84720 45115 24149 50920 71085

4.8769 98961 3.2818 18035 2.5875,66964

(49)9.7017 ( 1)l. 1000 4.9460 3.3166 2.6094

23378 00000 87443 24790 98635

(49)4.3297 28675 4.8909 73246 3.2888 68168 2.5920 12982

(50)1.1820 ( 1)l. 1045 4.9596 3.3234 2.6137

50242 36102 15664 56186 97668

(20)3.3945 ( 49)1.8788 ( 1)l. 0630 4.8345 3.2603 2.5740

67390 09051 14581 88127 90439 42354

( 49)2.3212 20685 ( 1)l. 0677 07825

4.8488 07586 3.2675 79877 2.5785 82140

118 13924 16 43032 1938 77776

49) 5.3109 1)1.0862 4.9048 3.2958 2.5964

00627 78049 68131 73252 28703

123 15129 18 60867 2288 86641

(50)1.4378 ( 1)l. 1090 4.9731 3.3302 2.6180

80104 53651 89833 45713 68602

119 14161 16 85159 2005 33921

( 49) 6.5031 99444

( 1) 1.0908 71211

4.9186 84734 3.3028 33952 2.6008 14587

19 2364 10 2 9316 1213: 6352 14 4.5076 16 5.5895 18 6.9309 20 8.5944

!I

124 15376 06624 21376 25062 15077 66696 06703 88312 25506

(50)l. 7463 06393

( 1)l. 4.9866 1135 52873 30952

3.3369 93965 2.6223 11847

29

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS nk

Table 3.1

k

1 2 3 4 5 6 7 8 9 10 24

18) 7.4505 20)9.3132 ( 50) 2.1175 ( 1)1.1180 5.0000 3. 3437 2.6265

( 5 0 ) 5.4280 ( 1)1.1401 5.0657 3.3766 2.6472

1 2

24

5 6 7

Q

9 10 24

5 6 7 8 9 10 24

80597 25746 82368 33989 00000 01525 27804

130 16900 21 97000 2856 10000

4

5 6 7 8 9 10 24

126 15876 20 00376 2520 47376

125 15625 19 53125 2441 40625

07704 75425 97019 48375 11681

135 18225 24 60375 3321 50625

( 5 1 ) 1.3427 ( 1)l.1618 5.1299 3.4086 2.6672

97252 95004 27840 58099 68608

27 3841 ( 1 0 5 3782 121 7: 5295 /15\1.0541 17) 1.4757

140 19600 44000 60000 40000 36000 89056 35040

1 9 2.0661 04678 46550 (.5 1 .) 3.2141 99700 ( 1)l. 1832 15957 5.1924 94102 3.4397 90628 2.6867 39790

16) 6.3527 87975 18) 8.0045 12848 ( 2 1 ) 1.0085 68619 ( 5 0 ) 2.5638 52774 ( 1)1.1224 97216 5.0132 97935 3.3503 68959 2.6307 16865

( 5 0 ) 3.0994 ( 1)1.1269 5.0265 3.3569 2.6348

83316 42767 25695 96823 79413

22 2944 10 3.8579 12 5.0539

I 1

131 17161 48091 99921 48965 13144

132 17424 22 99968 3035 95776 10) 4.0074 64243

(50)6.5239 ( 1)l.1445 5.0787 3.3831 2.6512

57088 52314 53078 23282 71840

( 50) 7.8302 26935

136 18496 25 15456 3421 02016

137 18769 25 71353 3522 75361

( 51) 1.6030 ( 1)1.1661 5.1425 3.4149 2.6712

01028 90379 63181 52970 08461

141 19881 28 03221 3952 54161

128 16384 20 97152 2684 35456

127 16129 20 48383 2601 44641

( 50) 3.7414 ( 1)1.1313 5.0396 3.3635 2.6390

44192 70850 84200 85661 15822

23 3129 1 0 4 1615 121 5: 5349 1 4 7.3614 1 6 9.7906 19 1 , 3 0 2 1 21) 1.7318 ( 5 0 ) 9. 3851

133 17689 52637 00721 79589 00854 18136 86120 61254 74468 10346

( 5 1 ) 1.1233 50184

( 1 ) l . 1532 5.1044 3.3959 2.6593

56259 68722 62690 18337

( 1)1.1575 5.1172 3.4023 2.6633

I

( 1 ) l . 1489 12529

5.0916 43370 3.3895 61224 2.6553 07280

( 5 1 ) 1.9111 ( 1)1.1704 5.1551 3.4212 2.6751

44882 69991 36735 13222 25206

129 16641 2 1 46689 2769 22881

138 19044 26 28072 3626 73936

(5u2.2756 ( 1)1.1747 5.1676 3.4274 2.6790

11258 34012 49252 39296 19145

( 5 0 ) 4.5097 ( 1)1.1357 5.0527 3.3701 2.6431

56022 81669 74347 36005 26458

134 17956 24 06104 3224 17936

83690 29947 28159 05339

139 19321 26 85619 3733 01041

( 5 1 ) 2.7061 ( 1 ) l . 1789 5.1801 3.4336 2.6828

70815 82612 01467 31623 90577

143 20449 29 24207 4181 61601 10) 5,9797 10894

144 20736 29 85984 4299 81696

29930 37529 03446 10326 72696

( 5 1 ) 5.3464 42484 ( 1)1.1958 5.2293 21532 26074

( 5 1 ) 6.3197 48715 ( 1)l.ZOOO 5.2414 00000 82708

3.4580 71824 2.6981 56943

3.4641 01615 2.7019 20077

147 21609 31 76523 4669 48881

148 21904 32 41792

149 22201 33 07949 4928 84401

142 20164 28 63288 4065 86896

.21, 2.8925

30 4420 ( 1 0 6 4097 121 9: 2941 15) 1.3476 1 7 ) l . 9540 19 2 8334 (.2 1 1, 4: 1084 f. 51)7.4616 , ( 1)1. 2041 5.2535 3.4701 2.7056

i

145 21025 48625 50625 34063 14391 46587 87551 26948 69075 01544 59458 87872 00082 62363

( 5 1 ) 3.8129 28871 ( 1)1.1874 34209

5.2048 27863 3.4459 16727 2.6905 67070 146 21316 31 12136 4543 71856

(51)4.5177 ( 1)1.1916 5.2171 3.4520 2.6943

( 17) 2. 3019 38535 ( 1 9 3 4068 69032 (2115: 0421 66167

( 5 1 ) 8.7997 ( 1)l.2083 5.2656 3.4760 2.7093

13625 04597 37428 67602 84058

( 5 2 ) 1.0366 ( 1 ) 1.2124 5.2776 3.4820 2.7130

11527 35565 32088 04545 85417

( 5 2 ) l . 2197 ( 1)1.2165 5.2895 3.4879 2.7167

79049 52506 72473 11275 66686

( 5 2 ) 1.4337 ( 1 ) 1.2206 5.3014 3.4937 2.7204

40132 55562 59192 88147 28110

30

ELEMENTARY ANALYTICAL METHODS

Table 3.1 k 1 2 3

POWERS AND ROOTS nk 150 -~ ~

9 10 24 1/2 1/3 114 115

33 5062 110) 7.5937 13) 1.1390 1 5 1.7085 117 2. 5628 19 3.8443 21 5.7665 ( 5 2 ) 1. 6834 ( 1)1.2247 5.3132 3.4996 2.7240

22500 75000 50000 50000 62500 93750 90625 35938 03906 11220 44871 92846 35512 69927

10 24 1/2 1/3 114 1/5

37 5772 10 8 9466 1311: 3867 1 5 2 1494 171 3: 3316 19 5 1639 211 8: 0041 ( 5 2 ) 3.6979 ( 1)1.2449 5.3716 3.5284 2.7419

155 24025 23875 00625 09688 24502 22977 05615 88703 82490 47627 89960 85355 41525 92987

8

I

160 25600 40 96000 6553 60000

9 10 24 1/2 1/3 114 1/5

24 1/2 1/3 114 1/5 1 2 3

24 1/2 1/3 114 115

( 1 9 6.8719 (221 1.0995 (52)7.9228 ( 1)l. 2649 5.4288 3.5565 2.7594

47674 11628 16251 11064 35233 58820 59323

34 5198 1 0 7 8502 131 1: 1853

151 22801 42951 85601 72575 91159

94816 81203 79543 84905

( 5 2 ) 1.9744 52704

( 5 2 ) 2. 3133 75387 ( 1)1.2328 82801

156 24336 37 96416 5922 40896

157 24649 38 69893 6075 73201

5.3250 74022 3.5054 53712 2.7276 92374

( 5 2 ) 4.3150 94990 ( 1)1.2489 99600

5.3832 12612 3.5341 18843 2.7455 21947 161 25921 41 73281 6718 98241

5.3368 03297 3.5112 43086 2.7312 95679

( 5 2 ) 5.0302 ( 1)1.2529 5.3946 3.5397 2.7490

74186 96409 90712 68931 32856

162 26244 42- 51528 ..._. 6887 47536 11)1.1157 71008

153 23409 35 81577 5479 81281

( 52)Z. 7076 61312 ( 1 ) 1 . 2 3 6 9 31688

5.3484 81241 3.5170 03963 2.7348 80069

39 6232 ( 1 0 9.8465 1131 1.5557 1 5 2.4581 17 3.8837 1 9 6.1364 ( 2 1 ) 9.6955 ( 52) 5.8582 ( 1)1.2569 5.4061 3.5453 2.7525

158 24964 44312 01296 80477 59715 00350 98553 01714 14709 79483 80509 20176 92093 25920

163 26569 4 3 30747

154 23716 36 52264 5624 48656

( 5 2 ) 3.1659 ( 1)1.2409 5.3601 3.5227 2.7384

00782 67365 08411 36670 45765

159 25281 40 19679 6391 28961

( 5 2 ) 6.8160 22003 ( 1 ) 1 . 2 6 0 9 52021

5.4175 01515 3.5509 88625 2.7560 01343 164 26896 44 10944 7233 94816

( 5 2 ) 9 . 2 0 0 7 03274

( 5 3 ) 1.0674 81480

( 5 3 ) 1.2373 78329

( 5 3 ) 1.4330 20335

( 1)l.2688 57754

( 1)l.2727 92206

( 1)1.2767 14533

( 1)l.2806 24847

( 5 3 ) 2.2140 ( 1)1.2922 5.5068 3.5948 2.7831

5.4401 21825 3.5621 02966 2.7629 00056

( 5 3 ) 1.6581 15050

( 5 3 ) 1.9168 76411 ( 1 ) l . 2884 09873

170 28900 49 13000

171 29241 50 00211 8550 36081

5.4958 64660 3.5894 42676 2.1798 51635

( 5 3 ) 3.3944 86713

( 5 3 ) 3.9075 68945

( 1)1.3038 40481

( 1)1.3076 69683

5.5396 58257 3.6108 73137 2.7931 21220

iiios

( 1)1.2288 20573

( 1 ) 1.2845 23258

5.4848 06552 3.5840 24634 2.7764 94317

152 23104

35 5337 ( 1 0 8 1136 131 1: 2332 15) 1.8745

5.5504 99103 3.6161 71571 2.7963 99540

5.4513 61778 3.5676 21345 2.7663 23734

90189 84798 78446 36294 92813

172 29584 50 88440 8752 13056

( 5 3 ) 4.4945 ( 1)1.3114 5.5612 3.6214 2.7996

13878 87705 97766 46817 62559

5.4625 55571 3.5731 14235 2.7697 30547

5.4737 03675 3.5785 81908 2.7731 20681

( 5 3 ) 2 . 5551 87425

( 5 3 ) 2 . 9463 26763

( 1 ) l . 2961 48140

( 1)1.3000 00000

5.5178 48353 3.6002 05744 2.7865 18023

5.5287 74814 3.6055 51275 2.7898 27436

52 9166 11 1.5949 131 2.7752

174 30276 68024 36176 46946 07686

( 5 3 ) 5.9317 ( 1)1.3190 5.5827 3.6319 2.8061

37979 90596 70172 28683 43329

173

29429 51 77717 8957 45041

20 1 3880 221 2: 4013 ( 5 3 ) 5.1654 ( 1)1. 3152 5.5720 3.6266 2.8029

81379 80785 29935 94644 54656 99110 10436

31

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS k 1 2 3

175 30625 53 59375

176 30976 54 51776

24 1/2 11 3

( 53) 6.8063 32613 ( 1)1.3228 5.5934 75656 44710

3.6371 35763 2.8093 61392

(53)7.8037 62212 ( 1)5.6040 1.3266 49916 78661 3.6423 20574 2.8125 64777

1 2 3

180 32400 58 32000

181 32761 59 29741

24 1/2 1/3 1/4 115 1 23

24 1/2 1/3 114 115

i 5 6 7 8 9 10 24

1 2 3

24 1/2 113 114 1/5

(54)1.3382 ( 1)l. 3416 5.6462 3.6628 2.8252

58845 40786 16173 41501 34501

185 34225 63 31625

(54)2.5829 ( 1) 1.3601 5.6980 3.6880 2.8407

82606 47051 19215 17151 58702

190 36Iao 68 59000

(54)1.5285 ( 1) 1.3453 5.6566 3.6679 2.8283

71637 62405 52826 18217 66697

51775 18170 67413 90888 23174

191 36481 69 67871

178 31684

177 31329

(53)8.9404 ( 1)l. 3304 5.6146 3.6474 2.8157

29702 13470 72408 83337 53634

(54)1.0234 ( 1)l. 3341 5.6252 3.6526 2.8189

182 60 33124 28568

(54)l. 7446 ( 1)1.3490 5.6670 3.6729 2.8314

70074 73756 51108 73940 85080

(54)3.3434 ( 1)l. 3674 5. 7184 3.6979 2.8468

78670 79433 79065 44609 74493

179 32041 57 35339

81638 66406 26328 24271 28111

(54)1.1707 ( 1) 1.3379 5.6357 3.6577 2.8220

183 33489 61 28487

(54)1.9898 ( 1)l. 3527 5.6774 3.6780 2.8345

187 34969 65 39203

186 34596 64 34856

(54)2.9397 ( 1) 1.3638 5.7082 3.6929 2.8438

Table 3.1

nk

184 39856 62 29504

76639 74926 11371 08871 89786

( 1) 1.3564 5.6877 65997 33960

188 35344 66 44672

189 35721 67 51269

(54)3.8000 ( 1)l. 3711 5.7286 3.7028 2.8499

192 36864 70 77888 9 1.3589 54496 111 2.6091 92632

73122 08816 40794 43589 88352

41874 30920 54316 78502 12786

(54)2.2679 20111 3.6830 23210 2.8376 80950

(54)4.3160 ( 1)l. 3747 5.7387 3.7077 2.8529

18526 72708 93548 92751 38178

194 37636 73 01384

193 37249 71 89057

62931 04875 97079 87538 50791

( 54) 5.5564 . 93542 . ( 1) 1.3820 27496

(54)6.2983 89130

(54)7.1346 95065

(54)8.0768 40718

( 1) 1.3856 40646

( 1) 1.3892 44339

( 1) 1.3928 38828

5.7589 65220 3.7175 63041 2.8589 50746

5.7689 98281 3.7224 19436 2.8619 38162

5.7789 96565 3.7272 56899 2.8649 13156

5.7889 60372 3.7320 75599 2.8678 75844

195 38025 74 14875

196 38416 75 29536

197 38809 76 45373

198 39204 77 62392

199 39601 78 80599

( 55) 1.0331 07971 ( 1)1.4000 00000

( 55) 1.1673 18660 ( 1)1.4035 66885

( 55) 1.3181 49187 ( 1)1.4071 24728

(54)4.8987 ( 1) 1.3784 5.7488 3.7126 2.8559

(54)9.1375 ( 1)1.3964 5.7988 3.7368 2.8708

69069 24004 89998 75706 26340

5.8087 85734 3.7416 57387 2.8737 64756

5.8186 47867 3.7464 20805 2.8766 91203

1

1

ni[(-i)l]

5.8284 76683 3.7511 66123 2.8796 05790

;[(-:)5]

:[(-:I31

[

(55)1.4875 ( 1)1.4106 5.8382 3.7558 2.8825

ni (-46121

57746 73598 72461 93499 08624

32

ELEMENTARY ANALYTICAL METHODS

Table 3.1

POWERS AND ROOTS nk

k 200 40000 80 00000 6 7 8 9 10 24

1 2 3 4 5 6 7 8 9 10 24

16 1 2800 18)2: 5600 20 5 1200 231 1: 0240 (55) 1.6777 ( 1)1.4142 5.8480 3.7606 2.8853

00000 00000 00000 00000 21600 13562 35476 03093 99812

205 42025 86 15125

(55)3.0345 f 1) ~,1.4317 5.8963 3.7838 2.8996 ~

38594 82106 68540 89674 84668

201 -.40401 81 20601

(55)1.8910 ( 1) 1.4177 5.8577 3.7652 2.8882

202 40804 82 42408

60303 44688 66003 95059 79458

( 55)2.1302 61246 ( 1) 1.4212 67040

206 42436 87 41816

207 42849 88 69743

(55)3.4104 ( 1)1.4352 5.9059 3.7884 2.9025

5.8674 64308 3.7699 69549 2.8911 47666

62581 70009 40584 95756 08125

(55)3.8307 ( 1) 1.4387 5.9154 3.7930 2.9053

89523 49457 81700 85099 20638

203 41209 83 65427 9) 1.6981 81681 11) 3.4473 08812

23) 1.1883 (55) 2. 3983 ( 1)1.4247 5.8771 3.7746 2.8940

93805 07745 80685 30659 26716 04537

208 43264 89 98912 9)1.8717 73696

(55)4. 3005 ( 1)l. 4422 5.9249 3.7976 2.9081

10765 20510 92137 57844 22302

204 41616 84 89664

(23)1.2482 (55)Z.6985 ( 1)1.4282 5.8867 3. 7792 2.8968

50286 09916 85686 65317 66709 50171

91 91 1.9080 11 3.9877 13 8.3344 ( 16)1.7419

209 43681 29329 29761 82200 64799 03143

(23)1.5902 (55)4.8251 ( 1)1.4456 5.9344 3.8022 2.9109

40688 50531 83229 72140 14131 13212

1

1 2 3

210 44100 92 61000

211 44521 93 93931

212 44944 95 28128

24 1/2

( 5 5 ) 5.4108 19838 ( 1)1.4491 37675

( 5 5 ) 6.0642 75557 ( 1)1.4525 83905

(55)6.7929 85105

(55)7.6051 97251

(55)8.5100 19601

( 1)l. 4560 21978

5.9533 41813 3.8112 77876 2.9164 63134

5.9627 31958 3.8157 85604 2.9192 22328

( 1)1.4594 51952

( 1)1.4628 73884

'I3 1/4 1/5

5.9439 21953 3.8067 54096 2.9136 93459

5.9720 92620 3.8202 77414 2.9219 71130

5.9814 24030 3.8247 53435 2.9247 09627

217 47089 102 18313

218 47524 103 60232 30576 96656 44071 90075 60363 09359 80403 ( 56)1.3272 59512 ( 1)1.4764 82306 6.0184 61655 3.8425 02187 2.9355 62280

219 47961 105 03459

03342 87830 26415 13796 37906

216 46656 100 77696 82336 49846 99567 95064 81338 90369 39197 (56) 1.0638 73589 ( 1)1.4696 93846 6.0000 00000 3.8336 58625 2.9301 56052

10926 39697 10737 85107 28975

221 48841 107 93861 43281 29651 74353 14320 39641 65062 18787 (56)1.8425 30003 ( 1) 1.4866 06875 6.0459 43596 3.8556 54127 2.9435 97699

1

2

3 4 5 6 7 8 9 10 24

215 46225 99 38375

(55)9.5175 ( 1) 1.4662 5.9907 3.8292 2.9274

106 5 6 7 E 109 24

(56)1.6525 ( 1)1.4832 6.0368 3.8512 2.9409

(56)1.1885 ( 1)1.4730 6.0092 3.8380 2.9328

94216 91986 45007 88048 64149

222 49284 109 41048

213 45369 96 63597

223 49729

110

89567

( 56)2.0533 89736

( 56)2.2872 66205

I 1)1.4899 ~6. 0550 3.8600 2.9462

( 1)1.4933 18452

~

~I

66443 48941 08345 56780

6.0641 26994 3.8643 47878 2.9489 06295

214 45796 98 00344

16)2.4160 18)5.2911 21)1.1587 23)2. 5377 (56)1.4813 ( 1)1.4798 6.0276 3.8469 2.9382

66056 84663 69441 05076 53665 64859 50160 01167 50529

224 50176 112 39424 ( 9)2. 5176 30976 93386 46519 72201 65731 16324 88565 (56)2.5465 51362 ( 1)1.4966 62955 6.0731 77944 3.8686 72841 2.9515 46323

33

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS

Table 3.1

nk

k 225 50625 113 90625

226 51076 115 43176 ~~

~

227 51529 116 97083

228 51984 118 52352

229 52441 120 08989

(56)3.5044 55686 ( 1)1.5066 51917

( 56) 3.8943 62082 ( 1) 1.5099 66887

( 56) 4.3256 51988 ( 1)1.5132 74595

6.1001 70200 3.8815 61435 2.9594 10235

6.1091 14744 3.8858 29238 2.9620 13062

6.1180 33173 3.8900 83026 2.9646 06773

233 54289 126 49337

6.1534 49494 3.9069 60138 2.9748 91866

2 34 54756 128 12904 19536 33714 05089 89909 20386 00970 92271 (56)7.2640 79321 ( 1)1.5297 05854 6.1622 40148 3.9111 45426 2.9774 41049

238 56644 134 81272

239 57121 136 51919

5

6 7 8 9 10 24

(56)2.8338 I 1) 1.5000 -,-6.0822 3.8729 2.9541 \

1 2 3

121

4

5 6 7 8 9 10 24

(56)4.8025 ( 1)1.5165

6.1269 3.8943 2.9671

73334 00000 01996 83346 76939 230 52900 67000 10000 43000 58890 25447 98528 52661 51121 07640 75089 25675 22905 91438

235 55225 129 77875

9 10 24

10 24

(56)3.1521 ( 1) 1.5033 6.0911 3.8772 2.9567

18526 29638 99349 79507 98218 231 53361

124

(56)5.3295 12896 ( 1) 1.5198 68415

6..1357 92440 3.8985 48980 2.9697 67129 236 55696

131 44256

(56)5.9116 ( 1)1.5231 6.1446 3.9027 2.9723

89798 54621 33651 61357 33915

231 56169 133 12053

(56)6.5545 38287 ( 1)1.5264 33752

01671

(56)8.9102 12697

(56)9.8618 93410

(57)1.0910 55818

(57)1.2065 61943

( 1)1.5329 70972

( 1)1.5362 29150

( 1)1.5394 80432

( 1) 1.5427 24862

( 1)1.5459 62483

6.1971 54435

(. 56) . 8.0469

6.1710 05793 3.9153 17320 2.9799 81531

6.1797 46606 3.9194 75921 2.9825 13380

6.1884 62762 3.9236 21327 2.9850 36660

2.9875 3.9277 53635 51438

6.2058 21795 3.9318 72942 2.9900 57716

240 57600 138 24000

-241 ._ 58081 139 97521

242 .58564 141 72488

243 .59049 143 48907

244 59536 145 26784

(57)1.3337 ( 1)1.5491 6.2144 3.9359 2.9925

35777 93338 65012 79343 55740

245 60025 147 06125

(57)1.4736 ( 1) 1.5524 6.2230 3.9400 2.9950

99791 17470 84253 72930 45390

148

(57)1.6276 ( 1) 1.5556 6.2316 3.9441 2.9975

79087 34919 79684 53798 26790

247 61009 150 69223

(57)1.7970 ( 1)l. 5588 6.2402 3.9482 3.0000

152

5 6 7

e

9 10 24

( 57) 2.1876 91225 ( 1) 1.5652 47584

6.2573 24746 3.9563 20998 3.0049 220?4

(57)2.4123 62509 ( 1)1.5684 6.2658 3.9603 3.0073

( 57) 2.6590 52293 ( 1) 1.5716 23365

6.2743 05357 3.9643 70523 3.'0098 12147

(57)2.9298 ( 1)1.5748 6.2827 3.9683 3.0122

10300 45727 51469 22039 00000 248 61504 52992 42016 00200 37650 13371 13716 66016 91719 15956 01575 61305 76966 45305

(57)1.9831 ( 1)1.5620 6.2487 3.9522 3.0024

51223 49935 99770 77742 65081 249 62001

(57)3.2268 ( 1) 1.5779 6.2911 3.9723 3.0146

91257 73384 94552 71312 70627

34

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS

Table 3.1 k 1 2 3 4 5 6 7 8 9 10 24

250 _. 62500 156 25000

(57)3.5527 13679

(57)3.9099 33001

(57)4.3014 31179

( 1)1.5842 97952

( 1) 1.5874 50787

6.2996 05249 3.9163 53644 3.0170 88168

6.3079 93549 3.9803 24047 3. 0194 97986

255 65025 165 81375

256 65536 167 77216

(57)5.7143 ( 1)1.5968 6.3413 3.9960 3.0290

~

17018 71942 25705 88015 61117

(57) . . 6.2771 ( 1) 1.6000 6.3496 4.0000 3.0314

01735 00000 04208 00000 33133

253 64009 161 94277

252 63504 160 03008

( 1) 1.5811 38830

~~

8 9 10 24

251 63001

nk

6.3163 59598 3.9842 82604 3.0219 00136

(57)4.7303 ( 1)1.5905 6.3241 3.9882 3.0242

258. _. 66564 171 73512

257 66049 169 74593

(57)6.8927 ( 1) 1.6031 6.3578 4.0039 3.0337

88615 21954 61180 00541 97748

41643 97372 03543 29397 94671

(57)7.5661 ( 1) 1.6062 6.3660 4.0077 3.0361

254 64516 163 87064

(57)5.2000 70108 ( 1) 1.5937 37745

6.3330 25531 3.9921 64507 3.0266 81647

15089 37840 96760 89716 55014

259 67081 173 73979 60561 63886 . . 51463 48289 74507 24973 06068 ( 57)8.3022 21920 ( 1) 1.6093 47694 6.3743 11088 4.0116 67601 3.0385 04982 264 69696 183 99744

1 2 3

260 67600 175 76000

261 68121 111 79581

262 68644 179 84728

263 69169 181 91447

24 1/2 113 114

f. 57) . 9.1066 85770 ( 1)1.6124 51550

(57)9.9855 54265

( 58)1.0945 38372 ( 1) 1.6186 41406

(58)1.1993 27974

(58)1.3136 94086

( 1) 1.6155 49442

( 1)l. 6217 27474

( 1)l. 6248 07681

267 71289 34163 21521 26446 93611 92942 95915 65094 49380 40472 13464 76696 93240 47961

268 71824 48832 86976 28110 75334 69894 05132 29753 83974 68868 70554 05727 72854 34462

2 69 72361 194 65109

115

1 2 3

6.3825 04299 4.0155 34273 3.0408 47703

6.3906 76528 4.0193 89807 3.0431 83226

265 70225 186 09625

266 70756 188 21096

. . 24 1/2 114 1/5

1 2 3 4 5 6 7 8 9 10 24

70548 82060 58289 02045 54329

( 58) 1.5745 60235 ( 1) 1.6309 50643

270 72900 196 83000

271 73441 199 02511

(58)2.2528 39954

( 58) 2.4618 57897 ( 1)1.6462 07763

(58)1.4384 ( 1)l. 6278 6.4231 4.0347 3.0524

( 1) 1.6431 67673

6.4633 04070 4.0536 00464 3.0638 87063 1

n;[( L

6.4312 27591 4.0385 02994 3.0547 54599

6.4712 73627 4.0573 48596 3.0661 53254

6.3988 27910 4.0232 34278 3.0455 11602

190 5.0821 1.3569 3.6229 9.6733 2.5827 6.8960 1.8412 ( 58)1.7229 ( 1)l. 6340 6.4392 4.0422 3.0570

212

._.-.

201 23648

(58)2.6893 ( 1)1.6492 6.4792 4.0610 3.0684

89450 42250 23603 86370 127f5

6.4069 58577 4.0270 67760 3.0478 32879

192 9 5.1586 112 1.3825 14 3.7051 16 9.9298 19 2.6612 21 7.1320 24 1.9113 ( 58) 1.8846 ( 1)l. 6370 6.4413 4.0460 3.0593

I!

273 74529 203 46417 71841 98113 66847 56349 26834 42256 63236 . . (.5 8 .) 2.9369 97176 ( 1)1.6522 71164 6.4871 54117 4.0648 13851 3.0706 6564Q

6.4150 68660 4.0308 90325 3.0501 47105

9 I12 14 17 19 21 24 (58)2.0608 ( 1)l. 6401 6.4553 4.0498 3.0616

89564 21947 14811 41906 14147

274 75076 205 70824

(58)3.2063 69049 ( 1)1.6552 94536

6.4950 65288 4.0685 31106 3.0729 11923

~

~

~

35

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS

Table 3.1

nk

k

275 75625 207 96875 4

5 6 7 8 9 10 24

(58)3.4993 ( 1)1.6583 6.5029 4.0722 3.0751

28001 12395 57234 38199 51657

_280 __

212 53933 -- . .. ..

(58)4.1640 35828

(58)4.5402 01230

( 1) 1.6643 31698

( 1)1.6673 33200

6.5186 83915 4. 0796 22161 3.0796 11650

6.5265 18879 4.0832 99156 3.0818 31992

281 78961

78400 219 52000

221 88041

224

( 58)5.3925 32264 ( 1)1.6733 20053

( 58) 5.8742 39885 ( 1)1.6763 05461

( 1)l. 792

5 6 1

E

9 10 24

(58)6.3970 6.9576 4.0979 3.0906

282 79524 25768 66576 86774 50704 20498 81806 25669 68387 33126 85562 ... 72186 08689 49967 ~~

6.5421 32620 4.0906 23489 3.0862 53577

6.5499 11620 4.0942 70950 3.0884 54901

1 2 3

285 81225 231 49125

286 81796 233 93656

287 82369 236 39903

24 1/2 1/3

( 58) 8.2466 32480 ( 1)1.6881 94302

( 58) 8.9698 42039 ( 1)1.6911 53453

( 58)9.7536 13040 ( 1)1.6941 07435

6.5808 44365 4.1087 64171 3.0971 98013

6.5085 32275 4.1123 63618 3.0993 68441

1 2

290 84100

291 84681 246 42171

. . 24 1/2 1/ 3 114 115

( 59) 1.2518 49008 ( 1)l. 7029 38637

1 2 3 4 5 6 7 8 9 10 24 1/2 11 3 1/4 11 5

295 87025

6.6191 05948 4.1266 67707 3.1079 89906

I

9 7.5733 50625 12 2.2341 38434 14 6.5907 08381 17 1.9442 58973 19 5.7355 63969 22 1 6919 91371 2414: 9913 74544 25672375 ( 59) 1.8868 10930 ( 1)1.7175 56404 6.6569 30232 4.1443 41207 3.1186 33956 1

ni

(59)1.3596 ( 1) 1.7058 6.6267 4.1302 3.1101

64428 72211 05387 20588 30396

259 9 7 6765 1212: 2722 14 6.7258 17 1.9908 19 5.8929 22 1.7443 24 5.1631 ( 59) 2. 0464 ( 1)1.7204 6.6644 4.1478 3.1207

296 87616 34336 63456 62783 97838 65760 62649 16944 78155 49657 65053 43703 48904 45423

Ii

[(-46) 71

[(-:)

6.5962 02284 4.1159 53637 3.1015 32807

279 77841 217 17639 21281 22737 58437 19804 16253 25135 67126 ( 58)4.9488 11121 ( 1)1.6703 29309 6.5343 35077 4.0869 66245 3.0840 45954

278 77284 214 84952

277 76729

276 76176 210 24576 82976 68101 27960 10517 29027 52114 20383 (58)3.8178 42160 ( 1)1.6613 24773 6.5108 30071 4.0759 35196 3.0773 84885

283 -..

284

80089 226 65187

80656 229 06304

(58)6.9642 51599 ( 1)1.6822 60384

( 58) 7.5794 93086 ( 1)1.6852 29955

6.5654 14427 4.1015 36766 3.0928 38815

6.5731 38451 4.1051 55240 3.0950 21484

288 82944 238 87872

289 83521 241 37569

(59)1.0602 ( 1)1.6970 6.6038 4.1195 3.1036

77893 56275 54498 34288 91148

(59)1.1522 54005 ( 1)1.7000 00000

6.6114 89018 4.1231 05626 3.1058 43502

248 7.2699 2.1228 6.1986 h8100 5.2852 1.5432 4.5063

292 85264 97088 49696 25311 49909 05773 16858 83323 87302

I14129

251 7.3700 2.1594 6.3271 1.8538 5.4317 1.5915 4.6631

293 85849 53757 50801 24885 14912 44669 64881 07110 15833

294 86436 254 12184

(59)1.4763 ( 1)l. 7088 6.6342 4.1337 3.1122

46962 00749 87437 64325 65011

(59)1.6025 ( 1)l. 7117 6.6418 4.1372 3.1143

91698 24277 52195 98970 93785

( 59) 1.7391 45550 ( 1)l. 7146 42820

298 88804 264 63592

299 89401 267 30899

/I 17 19 22 24

297 88209 261 98073

(59)2.2189 ( 1)1.7233 6.6719 4.1513 3.1228

87131 68794 40272 47726 51191 1

21

2[

6.6493 99761 4.1408 24580 3.1165 16755

(59)2.4054 16789 ( 1)1.7262 67650

6.6794 20032 4.1548 37723 3.1249 51295

(-46) 11

(59)2.6068 ( 1)1.7291 6.6868 4.1583 3.1270

[(-47) 81

04847 61647 83077 18947 45768

36

ELEMENTARY, ANALYTICAL METHODS

POWERS AND ROOTS nk

Table 3.1 k 1 2 3 4 5 6 7 8 9 10 24 1/2 1/3 1/4 115

301 . ~90601 272 70901

91204 275 43608

95365 50808 29501 91450 34645

( 59) 3.0591 15639 ( 1)1.7349 35157

( 1)l. 7378 14720

6.7017 59395 4.165f- 55283 3.1312 17958

6.7091 72852 4.1687 10496 3.1332 95743

305 . ..

93025 283 72625

306 . . 93636 286 52616

( 59)4.1994 86063 ( 1)1.7464 24920

(59)4.5427 01868

(59)4.9127 08679

( 1)1.7492 85568

6.7313 15497 4.1790 24910 3.1394 96244 310 96iOo 297 91000

300

90000 270 00000

(59)2.8242 ( 1) 1.7320 6.6943 4.1617 3.1291

1

9 10 24 1/2

;$: 1/5 1 2 3

24 1/2 1/3 114 1/5

10 24 1/2

'I3 115 1 2

24 1/2 1/3 1/5

302 . ._

(59)3.3125 81949

303

704

91809

92416 280 94464

278 iiiii 9 8 4288 92481 121 2: 5539 54422

(59)3.5861 ( 1)1.7406 6.7165 4.1721 3.1353

(59)3.8811 ( 1)1.7435 6.7239 4.1755 3.1374

99856 59577 50814 95260 34853

3 08 94864 292 18112

95481 295 03629

( 1)1.7521 41547

( 59) 5.3115 00125 ( 1)1.7549 92877

( 59)5.7412 10972 ( 1)1.7578 39583

6.7386 64101 4.1824 46136 3.1415 52236

6.7459 96712 4.1858 58988 3.1436 02859

6.7533 13417 4.1892 63512 3.1456 48146

6.7606 14302 4.1926 59756 3.1476 88127

312 97344 303 71328

313 97969 306 64297

314 98596 309 59144

26610 81686 99452 47767 22833

311 96721 300 80231 51841 90023 02970 91124 12395 20355 50303 ( 59)6.7026 93132 ( 1)l. 7635 19209 6.7751 68952 4.1994 27591 3.1517 52295

315 99225 312 55875

316 . 99856 315 54496

(59)6.2041 ( 1)1.7606 6.7678 4.1960 3.1497

(59)9.1086 ( 1)1.7748 6.8040 4.2128 3.1598

34822 23935 92116 65931 18306

27996 54382 03787 85054 86385 1

[

~

9P7P9

289

(59)9.8285 ( 1) 1.7776 6.8112 4.2162 3.1618

62028 38883 84608 05502 21997

(60)1.4325 86248 6.8470 21278 4.2327 85474 3.1717 65030

n2 (-46)6]

Tns

28072 52173 22886 99273 76544

( 59) 7.8174 31800 ( 1)1.7691 80601

( 1)1.7720 04515

6.7896 61336 4.2061 62861 3.1557 95609

6.7968 84386 4.2095 18398 3.1578 09519

1 318 1.0098 3.2010 1.0147 3.2167 1.0197 3.2324 1.0246 (60)1.0602 ( 1) 1.7804 6.8184 4.2195 3.1638

317 00489 55013 03912 78401 41853 31615 03941 61493 90293 84208 49381 61941 37156 20622

318 1 oiiii 321 57432

319 1 01761 324 61759

321 1 03Oii 330 76161

( 1)1.7916 47287

34443

(59)7.2395 ( 1)1.7663 6.7824 4.2027 3.1537

~~

320 1 02400

(60)1.3292 ( 1)1.7888 6.8399 4.2294 3.1697

307

05682 89519 69962 57138 68030

(60)1.1435 38734

(60)1.2330 37808

( 1) 1.7832 55450

( 1)1.7860 57110

6.8256 24197 4.2228 60938 3.1658 14209

6.8327 71452 4.2261 76889 3.1678 02787

322 1 03684 333 86248

(60)1.5436 ( 1)1.7944 6.8541 4.2360 3.1737

[( 7 2 1

21862 35844 24002 78192 38749

(59)8.4393 99655

323 1 04329 336 98267

(60)1.6628 78568 ( 1)1.7972 20076

6.8612 12036 4.2393 63249 3.1757 07571

[(-46) 11

724

1 340 1.1019 3.5704 1.1568 3.7481 1.2143 3.9346 1.2748 (60)1.7909 ( 1)1.8000 6.8682 4.2426 3.1776

[(-1)71

04976 12224 96058 67227 31381 33676 95311 40808 23622 36736 00000 85455 40687 71523

37

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS nk 326 1 06276 346 45976

325 1 05625 343 28125

327 1 06929 349 65783

Table 3.1 328 07584 87552 31706 75994 11326 93150 48153 45942 47069 09169 77028 34481 75067 79164

1

352

8

9 10 24

(60)1.9284 ( 1) 1.8027 6.8753 4.2459 3.1796

15722 75638 44335 10547 30632

(~, 6012.0759

76350

( 1)1.8055 47009

6.8823 72871 88750 4.2491 3.1815 84924

(60)2.2343 ( 1)1.8083 6.8894 4.2524 3.1835

332 1 10224 365 94368

331 1 09561 362 64691

330 1 08900 359 37000

23554 14132 18774 27697 34426

( 60)2.4042

( 1)l.ellO

6.8964 4.2556 3.1854

333 1 10889 369 26037

1 356 1.1716 3.8546 1.2681 4.1722 1. 3726 4.5160 1.4857

329 08241 11289 11408 01533 63904 59245 73292 95129 95298

( 60)2.5864 34894

( 1)1.8138 35715

6.9034 35942 4.2589 15020 3.1874 19165 334 1 11556 372 59704

4 5 6 7 8

9 10 24

. (60)2.7818 ( 1) 1.8165 6.9104 4.2621 3.1893

55434 90212 23230 47595 54454

(. 60) . 2.9913

( 1)1.8193

6.9173 4.2653 3.1912

81825 40540 96417 72832 85058

.

(. 60) . 3.2159

6.9243 55573 4.2685 90770 3.1932 11001

336 1 12896 379 33056

335 1 12225 375 95375

84959

( 1)1.8220 86716

(60)3.4566 ( 1)l. 8248 6.9313 4.2718 3.1951

99320 28759 00768 01446 32308

338 1 14244 386 14472

1 382

4

9 10 24

.

(60)3.7146 26935 ( 1)1.8275 66688

6.9382 32074 4.2750 04899 3.1970 49006 339 1 14921 389 58219

10 1.3206 1214.4771 15 1.5177 17 5.1451 20 1.7442 22 5.9128 25 2.0044

.

( 60) 3.9909 41565 ( 1) 1.8303 00522

( 60)4.2868 93134 ( 1)1.8330 30278

6.9451 49558 4.2782 01166 3.1989 61118

4.2813 3.2008 90286 68669

340 1 15600 393 04000

341 1 16281 396 51821

6.9520 53290

12427 55975 43337 72295 71684

( 60)4.9431 16051 ( 1)1.8384 77631

342 1 16964 400 01688

343 1 17649 403 53607

(60)4.6038 ( 1)1.8357 6.9589 4.2845 3.2027

6.9658 19768 4.2877 47230 3.2046 70186

17486 83624 42828 48186 05235 55747 58098

(60)5.3063 11693 ( 1)1.8411 6.9726 95264 82649 4.2909 15128 3.2065 64201 344 1 18536 407 07584

A

8 9 10 24

1

. (60)5.6950 ( 1)1.8439 6.9795 4.2940 3.2084

03680 08891 32047 76026 53751

. .

.

(. 60) . 6.1108

( 1) 1.8466

6.9863 4.2972 3.2103

98859 18531 68028 29958 38860

346 1 19716 414 21736

345 1 19025 410 63625

6 7 8

9 10 24

(60)8.0845 ( 1)1.8574 7.0135 4.3097 3.2178

95243 17562 79083 76748 35355

(60)8.6661 53376 ( 1)1.8601 07524

7.0203 48952

4.3128 3.2196 96386 98608

-:)5]

(60) . . 6.5558 ( 1) 1.8493 6.9931 4.3003 3.2122

12822 24201 90657 76961 19552

76479 25918 00000 17071 95850

( 60)7.5405 43015 ( 1)1.8547 23699

348

349 1 21801 425 08549

( 60)9.9518 04932 ( 1)1.8654 75811

( 61) 1.0661 30203 ( 111.8681 54169

(60)7.0316 ( 1) 1.8520 7.0000 4.3035 3.2140

147

1 2oid9 81923 32728 19567 29090 79941 14939 91840 15168 ( 60)9.2876 83235 ( 1)1.8627 93601 7.0271 05788 4.3160 09269 3.2215 57557

;[(-46)2]

41 7

1

L

421

7.0338 49656 4.3191 15431 3.2234 12226

7.0067 96121 4.3066 50321 3.2159 67776

7.0405 80617 4.3222 14906 3.2252 62636

38

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS

Table 3.1

nk

k

1 2 3 4 5 6 7 8 9 10 24 1/2 1/3 1/4 115 1 2 3

1 428 10 1.5006 1215.2521 15 1.8382 17 6.4339 20 2.2518 22 7.8815 25 2. 7585 (61)l. 1419 ( 1)1.8708 7.0472 4.3253 3.2271

350 22500 75000 25000 87500 65625 29688 75391 63867 47354 13124 28693 98732 07727 08809

355 1 26025 447 38875

4

5 6 7 8 9 10 24 1/2 1/3 1/4 1/5

10 24 1/2 1/3 1/4 115

(61)1.6050 ( 1)1.8841 7.0806 4.3406 3.2362

20092 44368 98751 73183 76880

(61)1.7171 ( 1)l. 8867 7.0873 4. 3437 3.2380

17251 96226 41061 26771 98084

( 61)2.2452 25771 ( 1)1.8973 66596

( 61)2.3997 87825 ( 1) 1.9000 00000

7.1137 86609 4.3558 77175 3.2453 42223

7.1203 61359 4.3588 98944 3.2471 43191

165

366

1 332% 486 27125

1 33956 490 27896

'I3 114 1/5

7.1465 69499 4.3709 23607 3.2543 07394

v;

356 1 26736 451 18016 10)1.6062 01370

361 1 30321 470 45881

( 1)1.9104 97317

;$

351 23201 43551 48640 48727 04703 16508 64494 64375 84096 43263 99400 04063 93928 50768

360 1 29600 466 56000

1 2 3 4 5 6 7 8 9 10 24 1/2

1 2 3 4 5 6 7 8 9 10 24

1 432 10 1.5178 121 5.3276 15 1 8700 1716: 5637 20 2 3038 221 8: 0865 25) 2.8383 (61)1.2228 ( 1)1.8734 7.0540 4.3283 3.2289

(61)3.1262 86296

370 1 36900 506 53000

(61)3.3384 ( 1)1.9131 7.1530 4.3739 3.2560

59019 12647 90095 14319 88625

371 1 37641 510 64811

352 1 23904 436 14208 10)1.5352 20122

(61)1.3092 ( 1)1.8761 7.0606 4.3314 3.2307

54042 66304 96671 73541 88532

357 1 27449 454 99293

(61)1.8366 ( 1)1.8894 7.0939 4.3467 3.2399

95605 44363 70945 73933 15199 362

353 1 24609 439 86977

(61)1.4014 ( 1)1.8788 7.0673 4.3345 3.2326

99442 29423 76615 46600 22125

1 458 10 1.6426 12 5.8805 15 2.1052 17 7.5366 20 2.6981 22 9 6593 251 3: 4580 (61)1.9642 ( 1) 1.8920 7.1005 4.3498 3.2417

358 28ik4 82712 01090 11901 23260 99273 38340 35256 42022 31355 88793 88459 14700 28247

I'

354 1 25316 443 61864 10)1.5704 09986

22 8 7303 251 3: 0905 (61)1.4999 ( 1)1.8814 7.0740 4.3376 3.2344

03831 27556 55202 88772 43955 13137 51567

359 1 28881 462 68279

(61)2. 1002 ( 1)l. 8947 7.1011 4.3528 3.2435

29556 29532 93661 49104 37249

363 1 31769 478 32147

364 1 32496 482 28544

17652 29759 35967 14441 40172

( 61)2.7400 53237 ( 1)l. 9052 55888

( 1)l. 9078 78403

7.1334 92490 4.3649 23697 3.2507 33187

7.1400 36982 4.3679 26743 3.2525 22254

367 1 34689 494 30863

368 1 35424 498 36032

..

1 31044

. --

A7d, 1797R

(61)2.5645 ( 1) 1.9026 7.1269 4.3619 3.2489

I

(61)2.9270 70667

(61)3.5643 92671

(61)3.8049 38558

( 1)1.9157 24406

( 1)1.9183 32609

7.1595 98825 4.3768 98909 3.2578 65967

7.1660 95742 4.3798 77406 3.2596 39439

369 1 36161 502 43409 81792 92813 00148 36546 48485 44691 91910 ( 61)4.0609 98114 ( 1)1.9209 37271 7.1725 80900 4.3828 49839 3.2614 09059

372 1 38384 514 78848

373 1 39129 518 95117

374 1 39876 523 13624

( 61)4.9320 85051 ( 1)1.9287 30152

( 61)5.2603 17567

7.1919 66348 4.3917 31039 3.2666 95001

7.1984 04996 4.3946 79501 3.2684 49404

. . (61)4.3335 ( 1)1.9235 7.1790 4.3858 3.2631

25711 38406 54352 16237 74848 1

(61)4.6235 ( 1)1.9261 7.1855 4.3887 3.2649

31606 36028 16151 76627 36822

( 1)1.9313 20792

(61) . . 5.6094 ( 1)1.9339 7.2048 4.3976 3.2702

26383 07961 32147 22040 00047

39

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS nk

Table 3.1

k 375 1 40625 527 34375 4 5 6 7 8 9 10 24

10 24

6 7 8 9 10 24

1 2

6 7 8 9 10 24

376 1 41376 531 57376

377 1 42129 535 82633

( 6 1 ) 5.9806 78067 ( 61) 6.3754 12334 ( 1)1.9364 91673 , ( 1)1.9390 71943 7.2112 47852 4.4005 58684 3.2719 46950

7.2176 52160 4.4034 89461 3.2736 90130

( 61) 6.7950 46060 ( 1)1.9416 48784

( 61) 7.2410 77507 ( 1)1.9442 22210

( 1 ) 1 . 9 4 6 7 92233

380 1 44400 548 72000

381 1 45ibi 553 06341

382 1 45924 557 42968

383 1 46686 561 81887

384 1 47456 566 23104

( 6 1 ) 8.2187 60383

( 6 1 ) 8.7538 56362

( 1)1.9493 58869

( 1)1.9519 22130

( 61) 9.3222 49236 ( 1)1.9544 82029

( 61) 9.9259 15535 ( 1)1.9570 38579

( 62) 1.0566 94349 ( 1)1.9595 91794

7.2431 56443 4.4151 54436 3.2806 25976

7.2495 04524 4.4180 56280 3.2823 50807

385 1 48225 570 66625 10) 2.1970 65063

386 1 48996 575 12456

. . ( 6 2 ) 1.1247 ( 1)1.9621 7.2747 4.4296 3.2892

53901 41687 86349 06853 14120

(.6 2 .) 1.1970 ( 1)1.9646 7.2810 4.4324 3.2909

7.2558 41507 4.4209 52418 3.2840 72019 387 1 49769 579 60603

. 25). 7.5354 ( 6 2 ) 1.2736 ( 1)1.9672 7.2873 4.4353 3.2926

57941 88303 31557 61631 48416 24406

7.2304 26792 4.4093 33520 3.2771 65392

( 61) 7.7150 90756

7.2367 97216 4.4122 46858 3.2788 97510

7.2621 67440 4.4238 42876 3.2857 89631

7.2684 82371 4.4267 27679 3.2875 03659

388 1 50544 584 11072

389 1 51321 588 63869

(.6 2 .) 1. 3550 ( 1)1.9697 7.2936 4.4382 3.2943

69013 71560 33030 10856 24265

( 2 5 ) 7.9340 ( 6 2 ) 1.4414 ( 1)1.9723 7.2998 4.4410 3.2960

391 1 52881 597 76471

392 1 53664 602 36288

393 1 54449 606 98457

( 6 2 ) 1.5330 29700

( 62) 1.6302 04837

( 1 ) 1.9748 41766

( 1 ) l . 9773 7.3123 4.4467 3.2994

( 62) 1.7332 67559 ( 1)1.9798 98987

( 1)1.9824 22760

7.3248 29445 4.4524 40634 3.3027 71361

7.3310 36930 4.4552 70277 3.3044 50453

1 57609 625 70773

398 1 58404 630 44792

399 1 59201 635 21199 95840 63840 42722 42146 69163 43996 54554

( 62) 2.3494 82217 ( 1)1.9924 85885

( 62) 2.4957 07762 ( 1)1.9949 93734

71993 82812 65109 02898

395 1 56025 616 29875

396 1 56816 620 99136

( 62) 2.0812 78965 ( 1 ) 1.9874 60691

( 1)1.9899 74874

7.3186 11420 4.4496 05586 3.3010 88848

( 62) 1.8425 58176

1 55536 611 62984 10)2.4098 21570

( 62) 1.9584 35730 ( 1 ) 1.9849 43324

5 6 7

8

9 10 24

69734 19629 08292 93662 67768 20622

390 1 52100 593 19000

7.3061 43574 4.4439 19178 3.2977 13494

4

03202 88270 79420 80423 21030

7.2240 45124 4.4064 14397 3.2754 29605

379 1 43641 544 39939

1 540

7.3372 33921 4.4580 94538 3.3061 261?8

.i

( 6 2 ) 2.2114 87364

( 6 2 ) 2.6506 ( 1)1.9974 7.3619 4.4693 3. 3127

7.3495 96597 7.3434 20462 7.3557 62368 4.4637 27013 4.4609 13443 4.4665 35273 3.3094 67354 3.3111 32914 3.3077 98433 -46)4] .i[(-f)2] 1 "f(-i)8] &[(-25]

[(-

32365 98436 17821 38246 95131

40

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS nk

Table 3.1 k 1

2

24 1/2 1/3 114 1/5

mn 1 6OOiO

401 1 6osoI 644 81201

402 1 61604 649 64808

(62)2.8147 49767

( 62)2.9885 80393 ( 1)2.0024 98439

( 1)2.0049 93766

406 1 64836 669 23416

407 1 65649 674 19143

1 66464 679 17312

409 1 67281 684 17929

( 62)4.0236 92707 ( 1)2. 0149 44168

(62)4.2684 06980

( 62)4.5273 48373

( 62)4.8013 06073

( 1)2.0174 24100

( 1)2. 0199 00988

( 1)2.0223 74842

412 69744 34528 02554 96652 38207 25341 04405 84615 98461 06738 78313 18861 06108 06308

413 1 70569 704 44997

414 1 71396 709 57944 10)2.9376 58882

( 1)2.0000 00000

7.3680 62997 4.4721 35955 3.3144 54017

7.3741 97940 4.4749 28423 3.3161 09590

ArlC

1

2 3

1 640;; 664 30125

24 1/2 113 1/4 115

( 62)3.7924 56055

1 2

410 1 68100 689 21000 10 2.8257 bioob 131 1.1585 62010

( 1)2.0124 61180

7. 3986 36223 4.4860 46344 3.3226 99030

24 1/2 1/3 1/4 1/5

(6~95.0911 10945 ( 1)2. 0248 45673 7.4289 58841 4.4998 28522 3.3308 63008

1 2 3

415 1 72225 714 73375

24 1/2 1/3 114 115

6 7 8 9 10 24 1/2 113 114 115

7.4047 20630 4.4888 12948 3.3243 38251

7.4107 95055 4.4915 74446 3.3259 74245

1 699 10 2.8813 13 1.1870 15 4.8908 18)2.0150 20 8.3019 23 3.4203 26 1.4091 (62)5.7218 ( 1)2.0297 7.4410 4.5053 3.3341

37632 13493 93742 69814 86236

416 1 73056 719 91296

13045 54879 35926 85215 47722

1 740 10 3 1116 1311: 3069

420 76400 88000 96000 12320

(26)1.7080 ( 62)9.0778 ( 1)2.0493 7.4888 4.5270 3.3469

19812 49315 90153 72387 19056 54883

ni

7.3803 22692 4.4777 15674 3.3177 61862

I I

(62)5.3976 ( 1)2.0273 7.4349 4.5025 3.3324

(62)6.8101 ( 1)2.0371 7.4590 4.5134 3.3389

1

411 1 68921 694 26531

(62)3.1726 72718

(62)7.2150 ( 1)2.0396 7.4650 4.5162 3.3405

59801 07805 22314 01729 55305

(62)9.6110 38126 7.4948 11226 4.5297 11307 3.3485 47155

[(-46) 41

(62)3.3676 ( 1)2.0074 7.3864 4.4804 3.3194

(62)7.6430 ( 1)2.0420 7.4709 4.5189 3.3421

25690 57786 99115 13349 59799

7.4168 59539 4.4943 30860 3.3276 07026

16609 63858 40668 98767 36405 1

[(-46) 11

(62)6.0645 ( 1)2. 0322 7.4470 4.5080 3.3357

87127 40143 34238 37426 23237

418 1 74724 730 34632

(62)8. 0952 ( 1)2.0445 7.4769 4.5216 3.3437

59269 04830 66370 20097 61218

423 1 78929 756 86967

(62)3.5739 85306 ( 1)2.0099 75124

7. 3925 41792 4.4832 74611 3.3210 56568

7.4229 14120 4.4970 82211 3.3292 36609

(62)6. 4269 98328 ( 1)2.0346 98995

7.4530 39914 4.5107 63788 3.3373 37037 419 1 75561 735 60059

(62)8.5730 73581 ( 1)2.0469 48949

7.4829 24114 4.5243 21992 3.3453 59575 424 1 79776 767 25024

(63)1.0768 83734

(63)1.1396 73784

( 1)2.0566 96380

( 1)2.0591 7.5125 26028 71508

7.5066 60749 4.5350 81455 3.3517 22644

[(-1)71

ni

404 1 63216 659 39264

23 3 5727 53638 261 1: 4791 20006

422 1 78084 751 51448

(63)1.0174 ( 1)2.0542 7.5007 4.5323 3.3501

04703 85990 31295 97729 10850 on8

417 1 73889 725 11713

421 1 77241 746 18461

( 1)2.0518 28453

403 1 62409 654 50827

1

4.5377 59390 3.3533 05887

41

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS nk

24

425

426

427

1 806rg 767 65625

I 814% 773 00776

1 82329 778 54483

( 6 3 ) 1.2059 63938

( 63) 1.2759 40370 ( 1)2.0639 76744

( 1)2.0663 97832

( 1)2.0615 52813

7.5184 72981 4.5404 32593 3.3548 86145

1 2

430

431

795 n7nnn

1 85ibi 800 62991

1 84900

10) 3.4188 01000

10 24

1 2 3 4 5 6 7 8 9 10 24

7.5243 65204 4.5431 01082 3.3564 63431

( 6 3 ) 1.5967 ( 1 ) 2.0736 7.5478 4.5537 3.3627

72093 44135 42314 28292 43107

435 1 89225 823 12875

( 6 3 ) 2.1073 ( 1)2.0856 7.5769 4.5669 3.3705

76666 65361 04852 08540 27318

( 6 3 ) 1.6883 ( 1)2.0760 7.5536 4.5563 3.3643

436 1 90096 828 81856

( 63) 2.2267 71952 ( 1)2.0880 61302

7.5827 86527 4.5695 30941 3. 3720 75562

440

1 93bd0 851 84000

24

857 6 m i

(63)2. 7724 53276

( 6 3 ) 2.9276 97132 ( 1)2.1000 00000

445 1 98025 881 21125 1 0 3 9213 90063 131 1: 7450 18578

( 6 3 ) 3.6361 ( 1)2.1095 7.6346 4.5929 3.3858

37215 02311 06721 31864 83431 772

7.5302 48212 4.5457 64877 3.3580 37758 4 32 1 86624 806 21568

7.5595 26299 4.5590 14114 3.3658 65436

433 1 87409 82737 12512 87018 36787 45729 71901 59329 48890 ( 6 3 ) 1.8867 28946 ( 1)2.0808 65205 7.5653 54772 4.5616 50145 3.3674 22267

437 1 90969 834 53453 1 0 3.6469 15896 131 1.5937 02247 1 5 6 9644 78818 1813: 0434 77243 2 1 1.3299 99555 23 5.8120 98057 26 2.5398 86851 ( 63) 2.3526 34640 ( 1)2.0904 54496 7.5885 79338 4.5721 48834 3.3736 20969

438 1 91844 8421 27672 12034 20471 49662 64552 43274 99539 09998 ( 63) 2.4852 99040 ( 1)2.0928 44954 7.5943 63318 4.5747 62238 3.3751 63549

442 95364 50888 09250 85488 75858 62329 26950 33117 20038 52385 79604 11603 71321 05834

443 1 96249 869 38307 67000 55581 69224 13266 02777 31302 66867 ( 6 3 ) 3.2635 43677 ( 1 ) 2 . 1 0 4 7 56510 7.6231 51930 4.5877 62546 3.3828 34454

( 6 3 ) 1.7848 83700 ( 1)2.0784 60969

I

441 1 94481

( 1 ) 2.0976 17696

7.6059 04922 4.5799 75651 3.3782 40276

24

18906 53949 88825 73502 05720

( 6 3 ) 1.3497 98685

7.6116 62611 4.5825 75695 3.3797 74445

1 863 10 3.8167 1 3 1.6869 1 5 7.4564 18 3.2957 21 1.4567 2 3 6 4387 2612: 8459 ( 6 3 ) 3.0912 ( 1)2.1023 7.6174 4.5851 3.3813

446

447

1 98iib 887 16536

1 990d9 893 14623

( 6 3 ) 3.8373 ( 1)2.1118 7.6403 4.5955 3.3874

95917 71208 21250 09991 03811*

428 1 83184 784 02752 37786 12972 91521 12371 30495 10518 07702 ( 6 3 ) 1.4277 44370 ( 1)2.0688 16087 7.5361 22043 4.5484 23998 3.3596 09138

(63)4.0493 05610 ( 1)2.1142 37451

7.6460 27242 4.5980 03787 3. 3889 21465,

448 2 00704 899 15392 09562 37884 77719 80418 47227 59578 17891 (63)4.2724 04226 ( 1 ) 2 . 1 1 6 6 01049 7.6517 24731 4.6006 53268 3.3904 36406.

Table 3.1 429

1 84041

789 53589

( 6 3 ) 1.5099 93273 ( 1)2.0712 31518

7.5419 86732 4.5510 78463 3.3611 77583 434 1 88356 817 46504

( 6 3 ) 1.9941 30189 ( 1)2.0832 66666

7.5711 74278 4.5642 81614 3.3689 76223 419

1 92iZ 846 04519

26)2. 6585 52264 ( 63) 2.6251 15920 ( 1)2. 0952 32684

7.6001 38502 4.5773 71171 3. 3767 03314 444

1 97136 875 28384

( 63) 3.4450 16313 ( 1)2.1071 30751

7.6288 83626 4.5903 49388 3.3043 60316

449 2 01601 905 1864s

( 63) 4.5072 55570 ( 1)2.1189 62010

7. 6574 13748 4.6032 18450 3.3919 48644

42

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS nk

Table 3.1 L.

24 1/2 1/3 1/4 115

(63)4.7544 ( 1)2.1213 7.6630 4.6057 3.3934

1

50505 20344 94324 79352 58190

(63) 5.0146 ( 1)2.1236 7.6687 4.6083 3.3949

( 63) 6.1983 ( 1)2.1330 7.6913 4.6185 3.4009

24 1/2 113 114 115 1

2 3

24 1/2 1/3 1/4 115 1

2 3

24 1/2 113 1/4 1/5

1 2 3 0

5

6 7 8 9 10 24

13235 72901 71681 20218 65915

( 63) 6.5336 ( 1)2.1354 7.6970 4.6210 3.4024

460 2 11600 973 36000

( 63) 8.0572 ( 1)2.1447 7.7194 4.6311 3.4084

70802 61059 42629 56507 07924

09634 85865 10895 90198 85500

470 2 20900 23000 81000 50070 21533 31205 28666 30473 26) 5.2599 13224 ( 6 4 ) 1.3500 46075 ( 1)2.1679 48339 7.7749 80097 4.6561 23215 3.4230 99883

08183 76058 66491 35988 65055

(63) 5.2883 ( 1)2.1260 7.6744 4.6108 3.3964

77338 29163 30279 88377 69249

55383 15650 02263 55770 59532

( 6 3 ) 8.4883 ( 1)2.1470 7.7250 4.6336 3.4098

29103 91055 32380 71390 88554

69046 03314 60547 84795 53393

471 2 21841 1044 87111 42928 52519 55637 69048 61622 43924 03881 ( 64) 1.4206 98007 ( 1) 2.1702 53441 7.7804 90361 4.6585 97902 3.4245 55283 1

-46)31

453 05209 59677 73368 16236 01548 00201 13891 11927 99703

(63) 5.5764 ( 1) 2.1283 7.6800 4.6134 3.3979

37619 79665 85719 36534 70784

Ii

01000 27575 32843 80476 69669

45396 55833 24618 87171 50532

( 63) 7.2572 39774 ( 1)2.1400 7.7082 93456 38778 4.6261 14413 3.4054 38923

(63) 7. 6472 35292 ( 1)2.1424 28529 7.7138 4.6286 44772 37519

462 2 13444 986 11128

463 2 14369 992 52847

464 2 15296 998 97344

( 6 3 ) 9.4176 ( 1)2.1517 7.7361 4.6386 3.4128

38903 18526 14052 82186 66616

467 2 18089 1018 47563

( 64) 1.1577 ( 1)2.1610 7.7584 4.6486 3.4187

( 64) 1.4948 ( 1)2.1725 7.7859 4.6b10 3. 4260

76852 43479 87677 88909 42121

3.4069 24718

( 63) 9.9181 ( 1)2.1540 7.7417 4.6411 3.4143

468 2 19024 1025 03232

24259 18278 02264 75380 18768

( 64) 1.2187 ( 1)2.1633 7.7639 4.6511 3.4201

10278 30765 36077 61968 81635

07’3 2 23726 1058 23817

472 2 22784 1051 54048

-6)11

( 6 3 ) 5.8795 ( 1)2.1307 7.6857 4.6159 3.3994

459 2 10681 967 02579

( 63) 8.9414 ( 1)2.1494 7.7306 4.6361 3.4113

n3 [( - 4

454 2 06116 935 76664

458 2 09164 960 71912

(63) 6.8863 ( 1)2.1377 7.7026 4.6235 3.4039

466 2 17156 1 0 1 1 94696

( 64) 1.0996 ( 1)2.1587 7.7528 4.6461 3.4172

2 929 1 0 4.2110 131 1.9076 15 8.6415 1 8 3.9146 2 1 1.7733 23 8.0331 26 3.6389

457 2 08849 954 43993

461 2 12521 979 72181

465 2 16225 1005 44625

(64) 1.0444 ( 1)2.1563 7.7473 4.6436 3.4157

452 04304 45408 12442 53624 74379 08819 37986 15698 61895

456 2 07936 948 18816

455 2 07025 941 96375

2 3

2 923 4.1740 1.8866 8.5276 3.8545 1.7422 7.8749 3.5594

451 2 03401 917 33851

450 2 02500 9 1 1 25000

1 2 3

85630 56098 92832 68652 08213

( 6 4 ) 1.5727 ( 1)2.1748 7.7914 4. 6b35 3.4214

1

77826 56317 87536 35480 58683

469 2 19961 1031 61709

(64) 1.2827 ( 1)2.1656 7.7694 4.6536 3.4216

[

]

.;[(

68318 40783 62012 44575 42003

474 2 24676 96424 30498 19056 48832 65466 60231 24 1.2078 27949 26) 5.7251 04480 ( 64) 1.6545 51159 ( 1) 2.1771 54106 7.7969 74500 4.6659 98399 3.4289 06701

1

n4 (-7)5 4

69666 65923 53281 91574 15079

-?3] 3

43

ELEMENTARY ANALYTICAL METHODS

Table 3.1

POWERS AND ROOTS nk k 1 2

3 4 5 6 7 8 9 10 24 1/2 1/3 114 115

4 5 6 7

475 2 25625 1071 71875

476 2 26576

477 2 27529 1085 31333

478 2 28484 1092 15352

479 2 29441 1099 02239

( 6 4 ) 2.0242 75033

f 64) 2.1283 95451

( 1)2.1863 21111

( 1)2.1886 7.8242 4.6782 3.4361

1 8 ) 5.4557 60126

21 2 5914 86060 241 1: 2309 ( 2 6 ) 5.8470 ( 64) 1.7403 ( 1)2.1794 7.8024 4. 6684 3.4303

55878 40422 90207 49472 53753 57424 52278

( 64) 1.8304 87912

( 1 ) 2 . 1 8 1 7 42423

7.8079 25322 4.6709 12569 3. 4317 95422

24 1 2783 261 6: 0979 ( 64) 1.9250 ( 1)2.1840 7.8133 4. 6733 3.4332

96024 49036 45935 32967 89232 63849 36143

480 2 30400 1105 92000

481 2 31361 1112 84641

( 64) 2.2376 37322 ( 1 ) 2 . 1 9 0 8 90230

( 6 4 ) 2 . 3522 41094

( 6 4 ) 2 . 4724 57971

( 1 ) 2 . 1 9 3 1 71220

( 1)2.1954 49840

7.8297 35282 4. 6806 94639 3.4375 43855

7.8351 68827 4.6831 30598 3.4389 74973

7.8405 94846 4. 6855 62762 3.4404 03713

485 2 35225 1140 84125

486 2 36196 1147 91256

2 37169 1155 01303

~I

7.8188 45511 4.6758 11278 3.4346 74449

482 2 32324 1119 80168

483 2 33289 1126 78587

06863 94186 54870 10350

484 2 34256 1133 79904

8

9 10 24 1/2 1/3 1/4 115

4 5 6 7 8 9 10 24 1/2 1/3 114 115

( 2 6 ) 7.2014 ( 6 4 ) 2. 8694 ( 1)2.2022 7.8568 4.6928 3.4446

1

07489 70250 71555 28008 36620 75750

( 26) 7.3512 ( 6 4 ) 3.0148 ( 1)2.2045 7.8622 4. 6952 3.4460

A90

2

1176

75394 82996 40769 24183 53740 95065

1/2 113 1/4 115

7.8837 35163 4.7048 85081 3.4517 49066

1

8 9 10 24 1/2

;$:

n2

[(-:I

( 6 4 ) 3.3271 75643

(64)3.4947 21879

( 1)2.2090 72203

( 1)2.2113 34439

7.8729 94366 4.7000 76812 3.4489 26700

7.8783 68425 4.7024 82790 3.4503 39037

493

494

1 2 0 5 53784

26) 8.1435 ( 6 4 ) 3.8543 ( 1)2.2158 7.8890 4.7072 3.4531

71639 91376 51981 94604 83697 56794 496

(64)4.9154 ( 1)2.2271

7.9157 4.7192 3.4601

31

1169 30169

2 43649 1198 23157

495

06649 59546 59893 41683 64874 1

489

2 39121

497

1212

26) 8.8318 ( 6 4 ) 4.6830 ( 1 ) 2.2248 7.9104 4.7168 3.4587

48334 65139 51023 44480 42798 07649 12960 67133 12045

488 2 38144 1162 14272

92362 00000 24411 15760 54092

2 42db4 1190 95488 98010 73021 73526 97749 71692 18873 56854 (64)4. 0472 72689 ( 1 ) 2 . 2 1 8 1 07301 7.8944 46773 4.7096 78653 3.4545 62231

2 4

1 8 6.4968 211 3.1639 24) 1. 5408 26) 7.5039 ( 6 4 ) 3.1673 ( 1)2.2068 7. 8676 4.6976 3.4475

( 6 4 ) 2.7307 ( 1)2.2000 7.8514 4.6904 3.4432

491

8

( 64) 3.6703 36822 ( 1 ) 2 . 2 1 3 5 94362

AR7

50361 26098 13365 91145 30083

2 4ioii 1183 70771

5 6 7 9 10 24

( 6 4 ) 2.5985 ( 1)2.1977 7.8460 4.6879 3.4418

460i6 23936 87226 84064 84096 61115 39113 17000 08320 15513 05745 83219 22124 61227

497 2 47009 1227 63473 1 0 ) 6.1013 44608 68270 87030 22541 40603 52380 26) 9.1952 57326 ( 6 4 ) 5.1588 55098 ( 1)2.2293 49681 7.9210 99395 4. 7215 98967 3.4615 55329

[(-46) 11

(64)4.2493 ( 1 ) Z . 2203 7.8997 4.7120 3.4559

( 6 4 ) 4.4611 ( 1)2.2226 7.9051 4.7144 3.4573

49467 11077 29393 57633 66263

49R

499

2 48dii

2 490Oi 1242 51499 10)6.2001 49800 74750 43500 79067 85754 48691 60970 ( 6 4 ) 5.6808 47029 ( 1)2.2338 30790 7.9317 10391 4.7263 41916 3.4643 36816

(64)5.4138 ( 1)2.2315 7.9264 4.7239 3.4629

.f [(-1)51

84825 60331 91695 69960 65384

2 44d36

25162 91360 08444 72227 47190

[I(-37)3]

44

ELEMENTARY ANALYTICAL METHODS

Table 3.1

POWERS AND ROOTS

k 1 2 3 4 5 6 7 8 9 10 24 1/2 1/3 1/4 1/5

10 24 1/2 113 114 115

504 54016 24064 12826 16064

501 2 51001

502 2 52004 1265 06008

64478 67977 05260 08045 24216

( 64) 6.2532 44659

( 64)6.5597 79050

[ 64) 6.8806

( 1)2. 2383 02929

( 1)2.2405 35650

( 1)2.2427 66149

( 64)7.2166 04000 ( 1)2.2449 94432

505 2 55025 1287 87625 ( 10)6. 5037 75063

506 2 56036 1295 54216

508 2 58064 1310 96512

2 59081 1318 72229

500

2 1250 ( 10)6.2500 (13 3 1250 16) 1: 5625

50000 00000 00000 00000 00000

( 64)5.9604

[

503 2 53009

2 1280 10 6.4524 131 3.2520

27) 1.0163 35678 I

~

-

( 1)2.2360

7.9370 4.7287 3.4657

( 64)7.5682

507 2 57049 1303 23843.

7.9520 47628 4.7357 85203 3.4698 73139

96349

( 64)8.3212 97020

( 64)8.7242

( 1)2.2516 66050

7.9633 74242 4.7404 85740 3.4726 28104

7.9686 27129 4.7428 30775 3.4740 02314

7.9738 73099 4.7451 72336 3.4753 74353

( 1)2.2538 85534

510 2 60100 1326 51000

511 2 61121 1334 32831

512 2 62144 1342 17728

513 2 63169 1350 05697 ( 10)6.9257 92256

80873 72926 50848 30911 88272 04087 40954

( 65)1.0531 22917 ( 1)2.2627 41700

( 65)1.1036 12886

516 2 66256

517 2 67289 1381 88413

518 2 68324 1389 91832

2 3 4 5 6 7 8 ,

I

. 64), 9.5870 33090

(

( 1)2.2583 17958

7.9895 69740 4.7521 76299 3.4794 77522

515 2 65225 1365 90075

1

2 3 4 5 6 7

( 64)7,9361

7.9475 73855 4.7334 29676 3.4684 92370

84448

( 1)2.2494 44376

~I

08268

7.9422 93073 4. 7310 70628 3.4671 09398

. .

( 1)2.2472 20505

1

9 10 24 1/2 1/3 114 1/ 5

nk

24 2.3756 (2711.2139 (65)1.0048 ( 1)2.2605 7.9947 4.7545 3.4808

1373 88096

8.0000 00000 4.7568 28460 3.4822 02253

69942

7.9791 12176 4.7475 10436 3.4767 44229

( 1)2.2649 50331

8.0052 04946 4.7591 49431 3.4835 61427

7.9581 14416 4.7381 37221 3.4712 51715

(64)9.1459 ( 1)2.2561 7.9843 4.7498 3.4781

06897 02835 44383 45086 11950

514 2 64196 1357 96744

(65)1.1564 ( 1)2.2671 8.0104 4.7614 3.4849

18034 56810 03133 67011 10483 519

2 &bi 1397 98359

8

9 10 24 1/2 113 1/4 115

.

.

( 65)1.2116

39706

( 65)1.2693 83471

( 1)2.2693 61144

( 1)2.2715 63338

8.0155 94581 4.7637 81212 3.4862 73428

520 2 7040~ 1406 08000 10 7 3116 16000 131 3: 8020 40320

10 24 1/2

f I:

$5

(65)1. 5278 48342 ( 1)2.2803 50850

8.0414 51517 4. 7153 01928 3;4930 16754

8.0207 79314 4.7660 92045 3.4076 26271

(65)1.3297 ( 1)2.2737 8.0259 4.7683 3.4889

59294 63400 57353 99522 77017

(65)1. 3928 ( 1)2.2759 8.0311 4.7707 3.4903

81704 61335 28718 03654 25675

( 65)1.4588 69982 ( 1)2.2781 57150

8.0362 93433 4.7730 04452 3.4916 72252

521 2 71441 1414 20761

522 2 72484 1422 36640

523 2 73529 1430 55667

524 2 74576 1438 77824

( 65)1.5999 46126 ( 1)2.2825 42442

( 65)1.6752 98008 ( 1)2.2847 31932

( 65) 1.7540 44200 ( 1)2.2869 19325

( 1)2.2891 04628

8.0466 02993 4.7775 96092 3.4943 59190

8.0517 47881 4.7798 86957 3.4956 99566

8.0568 86203 4.7821 74532 3.4970 37889

(65)1.8363 30669 8.0620 17979 4.7844 58829 3.4903 74167

45

ELEMENTARY ANALYTICAL M[ETHODS

Table 3.1

POWER S AND ROOTS nk k 1 2 3 4 5 6 7 8 9 10 24

4

5 6 7 8 9 10 24

2 1447 (10 7 5969 131 3: 9883

( 65)1.9223 09365 ( 1) 2.2912 87847

528 2 78784 1471 97952

(65)2.0121 38448

(65)2. 1059 82534

( 65)2.2040 12944

(65)2.3064 07963

( 1) 2.2934 68988

( 1) 2.2956 48057

( 1)2.2978 25059

( 1)2.3000 00000

8.0671 43230 4.7867 39859 3.4997 08406

8,0722 61977 4.7890 17632 3.5010 40614

8.0773 74241 4.7912 92160 3.5023 70797

530 2 80900 1488 77000 10)7.8904 81000

531 2 81961 1497 21291

532 2 83024 1505 68768

(65)2.5250 ( 1)2.3043 8.0977 4.8003 3.5076

8.0926 72335 4.7980 96379 3.5063 49267

536 2 87296 1539 90656

535 2 86225 1531 . .-

41417 43724 58868 58033 71420

30375 . .

(27)1; si6o (65)2.6416 ( 1) 2.3065 8.1028 4.8026 3.5089

02002 73716 12519 39019 16494 91583

537 2 88369 1548 54153

8.0875 79399 4.7958 31523 3.5050 25117

533 84089 19437 55992 59644 84590 54187 48814 21518 27569 58943 79276 12808 71774 09762

534 2 85iS6 1522 73304

2 1514 10)8.0706 13)4.3016 16 2.2927 19 1.2220 21 6.5135 24 3.4717 271 1.8504 (65)2.7634 ( 1)2.3086 8.1079 4.8048 3.5103

538 2 89444 1557 20872

10) 8.1924 75063

24

(65)3.1619 49669

( 65)3.0233 66304 ( 1)2.3130 06701

( 1) 2.3151 67381

8,1180 41379 4.8093 72829 3.5129 40196

8.1230 96201 4.8116 18626 3.5142 52463

540 2 91600 1574 64000

541 2 92681 1583 40421

(65)3.3066 ( 1) 2.3173 8.1281 4.8138 3.5155

09101 26045 44739 61283 62774

542 2 93764 1592 20088

2 79841 1480 35889

8.0824 80041 4.7935 63454 3.5036 98962

I

( 65)2.4133 53110 ( 1)2. 3021 72887

529

527 2 77729

526 2 76676 1455 31576

525 75625 03125 14063 79883

(65)3.4575 ( 1) 2.3194 8.1331 4.8161 3.5168

98937 82701 87014 00810 71134

( 65)2.8906 14446 ( 1)2.3108 44002

8.1129 80255 4.8071 23882 3.5116 25964 539 2 90521 1565 90819 10) 8.4402 45144

(65)3.6151 83652 ( 1) 2.3216 37353

8.1382 23044 4.8183 37217 3.5181 77550

543 2 94849 1601 03007

2 95936 1609 89184

6 1

6 9 10 24

4 5 6 7 8 9 10 24

38253 90008 52850 70514 82029

( 65)3.9512 48669 ( 1)2.3259 40670

545 2 97025 1618 78625

546 2 98116 1627 71336

( 65)4.7153 73024 ( 1)2.3345 23506

( 65)4.9274 63602 ( 1)2.3366 64289

(65)3.7796 ( 1)2.3237 8.1432 4.8205 3.5194

8.1482 76449 4.8228 00711 3.5207 84576

8.1683 09170 4.8316 90704 3.5259 75582

L

8.1733 02026 4.8339 05553 3.5272 68570

(65)4.1303 ( 1)2.3280 8.1532 4.8250 3.5220

12169 89345 93862 27819 85199

2 99i09 1636 67323

(65)5.1486 ( 1)2.3388 8.1782 4.8361 3.5285

79188 03113 88788 17361 59664

(65)4.3171 ( 1) 2.3302 8.1583 4.8272 3.5233

37789 36040 05107 51847 83903

( 65)4.5120 46770 ( 1) 2.3323 80758

Elda

549 3 oiioi 1654 69149

8.1633 10204 4.8294 72806 3.5246 80696

3 005oi 1645 66592

(65)5.3793 ( 1)2.3409 8.1832 4.8383 3.5298

94612 39982 69477 26138 48871

(65)5.6199 ( 1) 2.3430 8.1882 4.8405 3.5311

-231

99369 74903 44110 31895 36198

46

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS nk

Table 3.1 L

552

551 3 03601

550 3 02500 1663 75000

1 2 3 4 5 6

3 04% 1681 96608

7

a

9 10 24

(65)5.8708 ( 1)2.3452 8.1932 4.8427 3.5324

1 2 3

555 3 08025 1709 53875

24

( 65)7.2951 05803 ( 1)2. 3558 43798

1 2 3 4 5 6 7 8 9 10 24

1 2 3 4 5 6 7 8 9 10 24

1 2

:

5 6 7 8 9 10 24

(65)9.0471 ( 1)2.3664 8.2425 4.8645 3.5451

67858 31913 70600 98558 74407

565 3 19225 1803 62125 46006 09935 49613 73032 54763 69410 onI7 (66)1.1198 57461 ( 1) 2.3769 72865 8.2670 29409 4.8754 20869 3.5514 82586 570 3 24900 1851 93000

(66)1.3835 55344 ( 1)2.3874 67277

8.2913 44342 4.8861 71586 3.5517 46263 1

ni[(

1700 31464

(65)6.9860 ( 1)2.3537 8.2130 4.8515 3.5375

92851 20459 27082 15700 44836

556 3 09136 1718 79616

557 3 10249 1728 08693

558 3 11364

559 3 12481 1746 76879

(65)7.9528 84664 ( 1)2.3600 84744

( 65)8. 3027 27311 ( 1)2. 3622 02362

( 65)8.6672 91224 ( 1)2.3643 18084

562 3 15844 1775 04328

563 3 16969 1784 53547

564 3 18096 1794 06144

8.2031 31859 4.8471 31136 3.5349 86956

93672 65225 98519 88409 95340

8.2278 25361 4.8580 70341 3.5413 67840

561 3 i4iZI 1765 58481

560 3 13600 1756 16000

554 3 06916

( 65)6.4052 76258 ( 1) 2.3494 68025

(65)7.6171 ( 1) 2.3579 8.2228 4.8558 3.5400

8.2179 65765 4.8537 03532 3.5388 21007

I:

553 05809 12377 14448 08690 99605 24482 30384 44203 53644 46227 95203 82453 24905 66821

11516 38919 75283 34384 05234

(65) . . 6.1325 ( 1)2.3473 8.1981 4.8449 3.5337

98173 07880 12706 34641 '21650

3 1691 10 9.3519 1315.1716 16 2.8598 19 1.5815 21 8.7458 24 4.8364 27 2.6745 ( 65)6.6896 ( 1) 2.3515 8.2080 4.8493 3.5362

(65)9.4429 ( 1)2.3685 8.2474 4.8667 3.5464

(65)9.8553 ( 1)2.3706 8.2523 4.8689 3i5477

71309 43856 73974 68801 39637

39138 53918 71525 36145 03064

8.2719 03838 4.8775 76704 3.5527 38859

567 3 21489 1822 84263 51771 38543 55254 02229 29264 59926 39578 (66)1.2189 71112 ( 1)2.3811 76180 8.2767 72529 4.8797 29685 3.5539 93358

571 3 26041 1861 69411 27337 86093 04959 31732 27119 54848 51718 (66)1.4430 00887 ( 1) 2.3895 60629 8.2961 90248 4.8883 13236 3.5589 93720

572 3 27184 49248 93699 23956 84103 20907 56759 72660 87161 (66)1.5048 89774 ( 1)2.3916 52149 8.3010 30501 4.8904 52074 3.5602 39430

566 3 20356 1813 21496

(66)1.1684 07534 ( 1)2.3790 75451

I

ni

[

8.2327 46311 4.8602 49331 3.5426 38514

(66)1.0284 ( 1) 2.3727 8.2572 4.8711 3.5489

93323 62104 63270 00598 64695

568 3 22624 1832 50432

(66)1.2716 ( 1)2. 3832 8.2816 4.8818 3.5552

27921 75058 35499 79820 46087

573 3 28329 1881 32517

22 1 1620 241 6: 6587 27)3.8154 (66)1.5693 ( 1)2.3937 8.3058 4.8925 3.5614

82539 32949 53980 17896 41841 65115 88109 83400

8.2376 61384 4.8624 25407 3.5439 07368

(66)1.0732 ( 1) 2.3748 8.2621 4.8732 3.5502

44065 68417 49226 62170 24533

569 3 23761 1842 20009 (11)1.0482 11851

(66)1.3264 ( 1)2.3853 8.2864 4.8840 3.5564

60719 72088 92764 27117 97054

574 3 29476 1891 19224

(66)l. 6363 ( 1)2.3958 8.3106 4.8947 3.5627

84728 29710 94107 21351 25633

47

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS k 1 2 3

575 3 30625 1901 09375

576 3 31776 1 9 1 1 02976

Table 3.1

nk

577 3 32929 1921 00033

578 3 34084 1931 00552

579 3 35241 1941 04539

~I

24 1/2 1/3

1 2 3

24 1/2 1/3 114 1/5 1 2 3 4 5 6 7 8 9 10 24

1 23

24 1/2 1/3 114 115

. .

f 66) 1.7061 93459

( 1)2.3979 8.3155 4.8968 3.5639

15762 17494 51807 66137

580 3 36400 1951 12000

( 6 6 ) 2.1002 ( 1)2.4083 8.3395 4.9074 3.5701

54121 18916 50915 62599 42892

585 3 42225

19 2 3447 221 1: 3716 24 8 0242 271 4: 6941 ( 66) 2.5807 ( 1) 2.4186 8.3634 4.9180 3.5762

20403 61436 19400 68349 19397 77324 46607 05007 77194 590

3 48100 2053 79000

( 6 6 ) 3.1655 ( 1)2.4289 8.3872 4.9284 3.5823

43453 91560 06527 80050 69695

595 3 54025 2106 44875

9 10 24

( 66) 3.8762 ( 1 ) 2.4392 8.4108 4.9388 3.5884

08928 62184 32585 88725 21030

( 6 6 ) 1.7788 ( 1)2.4000 8.3203 4.8989 3.5652

51122

( 66) 1.8544 68735

00000

( 1)2.4020 8.3251 4.9011 3.5664

35292 79486 04916

581 3 37561 1961 22941

( 6 6 ) 2.1889 ( 1)2.4103 8.3443 4.9095 3.5713

06331 94159 41009 76518 73127

586 3 43396 2012 30056

( 66) 2.6887 ( 1)2.4207 8.3682 4.9201 3.5774

02707 43687 09391 05372 99018

591 3 49281 2064 25071

( 6 6 ) 3,2968 52680 49156 42387 67063 83235

( 1)2.4310 8.3919 4.9305 3.5835

596 3 55216 2117 08736

(.6 6 .) 4.0356 ( 1)2.4413 8.4155 4.9409 3.5896

19703 11123 41899 62581 26411

82430 47517 04396 41976

( 66) 1.9331 ( 1)2.4041 8.3299 4.9032 3.5676

582 3 38724 1971 37368

( 66) 2.2811 ( 1)2.4124 8.3491 4.9116 3.5726

38380 67616 25609 87710 01670

44372 08521 08288 66760 03051 19175

( 66) 2.3770 ( 1)2.4145 8.3579 4.9137 3.5738

72793 05012 72908 51429 95134 597

3 56409 2127 76173

( 66) 4.2013 ( 1 ) 2.4433 8.4202 4.9430 3;5908

02448 58345 45948 33830 30176

88299 39294 04732 96184 28526

( 6 6 ) 2.9178 ( 1)2.4248 8.3777 4.9242 3.5799

02055 71131 18728 98052 37670

( 66) 2.4768 ( 1)2.4166 8.3586 4.9159 3.5750

01250 59132 98104 33156 05396

598 3 57604 2138 47192 11 1.2788 06208 13 7.6472 61125 16 4.5730.62153 19 2.7346 91167 22 1.6353 45318 24 9.7793 65002 27 5.8480 60271 ( 6 6 ) 4.3734 92798 ( 1)2.4454 03852 8.4249 44747 4.9451 02478 3.5920 3232,9

I!

99188 09195 78393 01946 53698 589

3 466% 2043 36469

( 6 6 ) 3.0392 ( 1)2.4269 8.3824 4.9263 3.5811

593 3 51649 2085 27857

( 6 6 ) 3.5753 ( 1 ) 2.4351 8.4013 4.9347 3.5860

48620 41883 55313 45944 10958

584 3 41056 1991 76704

588 3 45744 2032 97472

592 3 50464 2074 74688

( 6 6 ) 3.4333 ( 1)2.4331 8. 3966 4.9326 3.5847

( 66) 2.0150 ( 1)2.4062 8.3347 4.9053 3.5689

583 3 39889 1981 55287

587 3 44569 2022 62003

( 27) 4.8571 ( 66) 2.8010 ( 1)2.4228 8.3729 4.9222 3.5787

61432 63056 54185 26546 77321

54545 32220 65312 90382 54508

594 3 52836 2095 84584

(66)3.7228 ( 1) 2.4372 8.4061 4.9368 3.5872

42640 11521 17992 12252 14026

599 3 58801 2149 21799

( 66) 4. 5524 ( 1 ) 2.4474 8.4296 4.9471 3.5932

3

34829 47650 38310 68534 32875

48

ELEMENTARY ANALYTICAL METHODS

POW’ERS AND ROOTS nk

Table 3.1 k

1 23

24 1/2 1/3 114 1/5 1 2

601 3 61201

600 3 60000 2160 00000

(66)4.7383 ( 1)2.4494 8.4343 4.9492 3.5944

81338 89743 26653 32004 31819

( 66)4.9315 94142 ( 1)2.4515 30134

8.4390 09789 4.9512 92896 3.5956 29165

605 3 66025

(66)5.7826 ( 1)2.4596 8.4576 4.9595 3.6004

77757 74775 90558 10838 02669

(66)5.1323 44384 ( 1)2.4535 68829

8.4436 87734 4.9533 51218 3.5968 24918

603 3 63609 2192 56227

(66)5.3409 ( 1)2.4556 8.4483 4.9554 3.5980

12849 05832 60500 06978 19083

604 3 64816

2203 48864

(66)5.5575 90288 ( 1)2.4576 8.4530 41145 28104 4.9574 60182 3.5992 11665

607

608

609

3 67236

3 68449

3 69664

2225 45016

2236 48543

2247 55712

3 70881

606

3

24 1/2 1/3 114 1/5

602 3 62494 2181 67208

(66)6.0164 86963 ( 1)2.4617 06725

8.4623 47878 4.9615 58954 3.6015 92098

(66)6.2593 ( 1)2.4637 8.4670 4.9636 3.6027

40623 36999 00076 04536 79959

(66)6.5115 ( 1)2.4657 8.4716 4.9656 3.6039

72833 65601 47168 47592 66255

(66)6.7735 ( 1)2.4677 8.4762 4.9676 3.6051

29447 92536 89168 88130 50991

1 2 3

72100 22693 81000

611 3 73321 2280 99131

612 3 74544 2292 20928

3 75769 2303 46397

614 3 76996 2314 75544

24 1/2 1/3 114 1/5

(66)7.0455 68477 ( 1)2.4698 8.4809 17807 26088 4.9697 26156 3.6063 34171

(66)7.3280 60494 ( 1)2.4718 41419 8.4855 57944 4.9717 61679 3.6075 15802

(66)7.6213 89047 ( 1)2.4738 63375 8.4901 84749 4.9737 94704 3.6086 95885

(66)7.9259 51097 ( 1)2.4758 8.4948 83681 06516 4.9758 25239 3.6098 74428

(66)8.2421 57465 ( 1)2.4779 02339 8.4994 23260 4.9718 53291 3.6110 51433

1

615 3 78225

616 3 79456 2337 44896

617 3 80689 2348 85113

618 3 81924

619 3 83161 2371 76659

24 1/2 1/3 1/4 115

( 66)8.5704 33286 ( 1)2.4799 19354

( 66)8.9112 18488 ( 1)2. 4819 34729

( 66)9.2649 68280 ( 1)2.4839 48470

( 66)9.6321 53659 ( 1)2.4859 60579

( 67)1.0013 26192 ( 1)2.4879 71061

1

620 3 84400 2383 28000

622

3 85641

623 3 88129

624 3 89376

2

2

3

24 1/2 1/3 114 1/5

610

8.5040 34993 4.9798 78868 3.6122 26906

(67)1.0408 ( 1)2.4899 8.5270 4.9899 3.6180

79722 79920 18983 69859 81437

8.5086 41730 4.9819 01975 3.6134 00850

8.5132 43484 4.9839 22621 3.6145 73271

621

3 86884

613

2258 66529

8.5178 40269 4.9859 40813 3.6157 44173

8.5224 32097 4.9879 56556 3.6169 13560

2394 83061

(67)1. 0819 28109

(67)1.1245 25305

( 1)2.4919 87159

( 1)2.4939 92783

8.5316 00940 4.9919 80728 3.6192 47808

8.5361 77980 4.9939 89170 3.6204 12677 1

ni[(-!)2]

i[c‘-1”7]

(67)1.1687 ( 1)2.4959 8.5407 4.9959 3.6215

27115 96795 50116 95191 76049

(67)l. 2145 ( 1)2.4979 8.5453 4.9979 3.6227

91262 99199 17363 98799 37928

49

ELEMENTARY ANALYTICAL METHODS

Table 3.1

POWERS AND ROOTS nk k 625 3 90625 2441 40625 6 7 8 9 10 24

1 2 3

4 5 6 7 8 9 10 24

626 3 91876 2453 14376

--.

629 3 95641 2488 58189

A77

,._-.

47419 99201 37239 98801 57224

3 93129 2464 91883 04106 10747 24838 42174 82943 41505 12238 (67)1.3627 65028 ( 1) 8.5589 2.5039 96805 89894 5.0039 95209 3.6262 14650

630 3 96900 2500 47000

631 3 98161 2512 39591

632 3 99424 2524 35968

633 4 00689 2536 36137

(67)1.5281 75339

( 67)1.5874 66692 ( 1)2.5119 71337

( 67)1.6489 59081 ( 1)2.5139 61018

(67)1.7127 30535

(67)1.7788 61719

( 1)2.5159 49125

( 1)2.5179 35662

8.5726 18882 5.0099 70139 3.6296 78090

8. 5771 52262 5.0119 57040 3.6308 29638

8.5816 80854 5.0139 41581 3.6319 79727

635 4 03225 2560 47875

636 4 04496 2572 59456

637 4 osiks 2584 74853

36020 20634 38034 81108 21280

( 67)1.9185 39634

--,_. ( 67) 61654 - 1.9922 -( 1)2.5238 8.6042 85893 52449

( 67)2.0686 94164

(67)2.1479 32334

( 1)2.5258 66188 8.6087 52582

( 1)2.5278 44932 8.6132 48015

5.0238 29110 3.6377 08430

5.0257 99626 3.6388 49851

5.0277 67827 3.6399 89842

640 4 09600 2621 44000

642 4 12164 2646 09288

643 4 13449 2658 41707 00756 44686 00332 02713 50945 78757 19041 (67)2.4949 58638 ( 1)2.5357 44467 8.6311 82992 5.0356 17605 3.6445 35581

644 4 14736

74520 22128 38760 33719 28406

641 4 10881 2633 74721 31962 56687 24366 76219 27156 31507 63096 ( 67)2.3152 22362 ( 1)2.5317 97780 8.6222 24830 5.0316 97308 3.6422 65548

645 4 16025 2683 36125

646 4 17316 2695 86136

. . (. 67)l. .

2621 ( 1) 2.5000 8.5498 5.0000 3.6238

77448 00000 79733 00000 98318

( 1) 2.5099 80080

(67)l. 3115 ( 1)2.5019 8.5544 5.0019 3.6250

~

2476 73152

. . (67)1.4158 ( 1)2. 5059 8.5635 5.0059 3.6273

96309 92817 37711 89230 70600

(671 . . 1.4710 09545 ( 1)2.5079 87241 8.5680 80703

5.0079 80871 3.6285 25079 634

4 Olsi6 2548 40104

8.5862 04672 5.0159 23768 3.6331 28361

8. 5907 23728 5.0179 03608 3.6342 75544

638 4 07044 94072

639 4 08321 2609 17119

7596

4

i

9 10 24

~

(67)1.8474 ( 1) 2.5199 8.5952 5.0198 3.6354

10 24

6 7 8 9 10 24

(67)2.2300 ( 1) 2.5298 8.6177 5.0297 3.6411

( 67)2.6880 24057 ( 1)2.5396 85020

8.6401 22598 5.0395 28767 3.6467 99973

( 1)2.5219 04043

8.5997 47604 5.0218 56273 3.6365 65574

(67)2. 7898 ( 1)2.5416 8.6445 5.0414 3.6479

47292 53005 85472 80939 30063

I

I

.

(. 67) . 2.4034

( 1)2.5337

8.6267 5.0336 3.6434

80891 71892 06237 58602 01272

647 4 18609

(67)2.8953 ( 1)2.5436 8.6490 5.0434 3.6490

61105 19468 43742 30845 58755

28)1.1173 95163

648 4 19904 2720 97792

(67)3.0046 ( 1)Z. 5455 8.6534 5.0453 3.6501

93247 84412 97422 78492 86051

. . (67) . . 2.5897 ( 1)2.5377 8.6356 5.0375 3.6456

67740 15508 55108 74325 68481 649

4 21IOi 2733 59449

(67)3.1179 ( 1) 2.5475 8.6579 5.0473 3.6513

75679 47841 46522 23886 11957

50

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS

Table 3.1 k 1

2 3

24 1/2 113 114 1/5

1 2 3

650 4 22500 2746 25000

(67)3.2353 ( 1)2.5495 8.6623 5.0492 3.6524

44710 09757 91053 67033 36476

651 4 23801 2758 94451 72876 43442 74809 65401 77776 46432 30227 (67)3.3569 41134 ( 1)2.5514 70164 8.6668 31029 5.0512 07939 3.6535 59612 656 4 30336

655 4 29025 2810 11375

i

652 4 25104 2771 67808

(67)3.4829 ( 1)2.5534 8.6712 5.0531 3.6546

nk 653 4 26409 2784 45077

10364 29067 66460 46611 81368

(67)3.6134 ( 1)2.5553 8.6756 5.0550 3.6558

657 4 31649 2835 93393

A

2848

' 19 5.2278 95232 ' 22 3.4294 99272

25 2.2497 51522 ' 28 1.4758 36999

24 1/2 1/3 114 115

( 67) 3.8885 81447 ( 1)2.5592 96778

1

660 4 35600 2874 96000

2

3 4 5 6 7 8 9 10 24 1/2 1/3 1/4 1/5

6 7 8 9 10 24 1/2 1/3 114 1/5 1 2 3 4 5 6 7 8 9 10 24 1/2 1/3

'I4 1/5

8.6845 45603 5.0589 49277 3.6580 38399

(67)4.6671 ( 1)2.5690 8.7065 5.0685 3.6636

78950 46516 87691 76246 06215

( 67)4.0335 93654 ( 1)2.5612 49695

8.6889 62971 5.0608 79069 3.6591 54676

4 2888 11 1 9089 141 1: 2618 16 8.3408 19 5.5132 !22 3. 6442 25 2.4088 28 1.5922 (67)4.8398 ( 1)2.5709 8.7109 5.0704 3.6647

'I

661 36921 04781 99602 48737 20153 82121 79482 68738 62236 84834 92026 82739 95071 15727

(67)4.1837 80288 ( 1)2.5632 01124

8.6933 75853 5.0628 06656 3.6602 69592 662 4 38244 2901 17528

671 4 50241 3021 11711

612 4 51584 3034 64448

8.7503 40i23 5.0876 67266 3.6746 41374

664

4 4086b 2927 54944

669 4 47561 2994 18309

(67)6.0120 14426

96605 66769 91362 64588 3762:

8.7021 88202 5.0666 55239 3.6624 95358

660

( 1)2.5826 34314

(67)6.9396 ( 1)2.5903 8.7546 5. 0895 3. 6757

663 4 39569

( 67)4.5003 87920 ( 1)2.5670 99531

4 46224 2980 77632

(67)5.7993 79113

( 67)6.6956 88867 ( 1)2.5884 35821

659 4 34281 2861 91179

bb7

( 1)2.5806 97580

670 48900 63000 12100 25107 38217

658 32964 90312 78253 72490 48987 91834 43626 40706 54385 17689 51068 84260 32044 83152

5.0570 17274 3.6569 20758

4 44809 2967 40963

( 67)5.5939 61683 ( 1)2.5787 59392

4 3007 11)2. 0151 14 1 3501 l6]9: 0458

( 67) 3.7485 72888 ( 1)2.5573 8.6801 42371 23736

( 67)5.3955 27431 ( 1)2.5768 19745

05901 36066 13356 11720 23896

666 4 43556 2954 08296

8.7328 91741 5.0800 56673 3.6702 43226

02582 86468 97359 83054 01749

48947 78638 59553 26200 30727

(67)5.0187 ( 1)2.5729 8.7153 5.0724 3.6658

665 4 42225 2940 79625

8.7285 18735 5.0781 48670 3.6691 40389

(67)4.3393 ( 1)2.5651 8.6977 5. 0647 3.6613

654 4 27716 26264 09767 33987 78277 39593 40094 68021 54286

7747

8.7372 60372 5.0819 62528 3.6713 44740

(67)5.2038 ( 1)2.5748 8.7197 5.0743 3.6669

(67)6.2321 ( 1)2.5845 8.7416 5.0838 3.6724

09844 69597 24639 66242 44934

673 4 52929 3048 21217

8.7241 41343 5.0762 38514 3.6680 36224

(67)6.4599 15340 03431 8.7459 84552 5.0857 67819 3.6735 43810

( 1)2.5865

674 4 54276 3061 82024

(67)7.1922 13208

(67)7.4535 22063

(67)7.7239 15552

( 1)2.5922 96279

( 1)2.5942 24354

( 1)2.5961 50997

8.7590 38280 5.0914 59790 3.6768 32575

8.7633 80887 5.0933 52878 3.6779 2621?

8.7677 19196 5.0952 43858 3.6790 18565

51

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS

5 6 7

8 9

10 24

4 3075 (11)2.0759 ( 14)1.4012 (16)9.4585 191 6.3844 (22 4. 3095 25 2.9089 28 1.9635 (67)8.0036 ( 1) 2.5980 8.7720 5.0971 3.6801

i

675 55625 46875 41406 60449 08032 92922 32722 34587 30847 95322 76211 53215 32735 09614

677 4 58329 3102 88733

676 4 56976 3089 15776

Table 3.1

nk 678

4 59684 3116 65752 (11)2. 1130 93799

(28j 2 ; 0525 65092 ( 67)8.2931 72571 ( 1)2.6000 00000

8.7763 82955 5.0990 19514 3.6811 99371

680 4 62400

(28)2.0830 (67)9.2230 ( 1)2.6057 8.7893 5.1046 3.6844

40639 50418 62844 46612 67319 60923

68325 22366 08428 04200 87840

( 67)8.9025 13744

682 4 65124

683 4 66489 3186 11987

684 4 67856 3200 13504

(67)8.5926 ( 1)2.6019 8.7807 5.1009 3.6822

681 4 63761

679 4 61041 3130 46839 (11)2.1255 88037

( 1)2.6038 43313

8.7850 29644 5.1027 86801 3.6833 75023

4

38701

( 68)1.0619 32441

( 68)1.0998 82878

( 1)2.6076 80962

( 1)2. 6095 97670

( 1)2.6115 12971

( 1)2.6134 26869

( 1)2.6153 39366

5.1065 3.6855 45762 45546

5.1084 22134 3.6866 28893

5.1102 96441 3.6877 10968

685 4 69225 3214 19125

686 4 70596 3228 28856

687 4 71969 3242 42703

688 73344 60672 45423 95251 48733

689 4 74721 3270 82769

(, 67) . 9.5546

(. 67) . 9.8976

30685

8.7936 59344

17949

8.7979 67850

(. 68) . 1. 0252

8.8022 72141

17 1 0331 02539 191 7: 0767 52393 10 24

( 68)1.1391 31118 ( 1)2.6172 50466

8.8151 59819 5.1159 07022 3.6909 49595

(28) 2 ; 3080 (68)1.1797 ( 1)2.6191 8.8194 5.1177 3.6920

690 4 76ioo 3285 09000

28446 19551 60171 47349 73120 26615

(68)1.2216 ( 1)2.6210 8.8237 5.1196 3.6931

691 4 77481 3299 39371

4 5 6 7

91886 68484 30714 37179 02381

8.8065 72225 5.1121 68688 3.6887 91774

4 3256 11)2.2405 14 1.5414 171 1.0605

(68)l. 2650 93189 ( 1)2.6229 75410

8.8280 09925 5.1214 99204 3.6941 76894 693 4 80249 3328 12557

692 4 78864 3313 73888

8

9 10 24

(68)1.3563 ( 1)2.6267 8.8365 5.1252 3.6963

70007 85107 55922 17173 22179

(68)1.4043 42816

(68)1.4539 39271

( 1)2. 6286 87886

( 1) 2.6305 89288

8.8408 22729 5.1270 73128 3.6973 92956 696 4 84416 3371 53536

695 4 83025 3357 02375 31506 26397 60846 77878 02625 10 24

( 68)1.6130 03502 ( 1)2.6362 85265

8.8578 48911 5.1344 76863 3.7016 63707 1

[

8.8450 85422 5.1289 27069 3.6984 62494

(68)1.6696 ( 1)2.6381 8.8620 5.1363 3.7027

.2 (-36121

35809 81192 95243 22801 28321

(68)1.5052 ( 1)2.6324 8.8493 5.1307 3.6995

698 d 87204 3400 68392 77376 26809 65112 26484 44286 72312 75074

697 4 85809 3386 08873

(68)1.7281 70846 ( 1)2. 6400 75756

8.8663 37511 5.1381 66751 3.7037 91713

.:[(-1)51

11857 89316 44010 79001 30796

( 68) 1.7886 69670 ( 1)2.6419 68963

8.8705 75722 5.1400 08719 3.7048 53884

.a[(-37) "3

8.8108 68115 5.1140 38880 3.6898 71315

(68)l. 3099 ( 1)2.6248 8.8322 5.1233 3.6952

69927 80950 84991 59200 50159

694 4 81636 3342 55384 32365 94261 66617 30324 58245 23822 59533 ( 68)1.5582 14678 ( 1) 2.6343 87974 8.8535 98503 5.1326 28931 3.7005 97866 699 1

3415 11 21: 3873 141 6687

72960 57313 ( 68)1.8511 95210 ( 1)2.6438 8.8748 09888 60813

5.1418 48708 3.7059 14839

52

ELEMENTARY ANALYTICAL METHODS

Table 3.1

POWERS AND ROOTS nk

k 1

7nn

2 3

4 9006i 3430 00000

701 4 91401 3444 72101

24 1/2

( 68)1.9158 12314 ( 1)2.6457 51311

( 1)2.6476 40459

114 115

8.8790 40017 5.1436 86724 3.7069 74581

8.8832 66120 5.1455 22771 3.7080 33112

705 4 97025 3504 02625

706 4 98436 3518 95816

1 2 3 4 5 6 7 8 9 10 24

(68)1.9825 87808

8.9001 30453 5.1528 47377 3.7122 55193 1

2 3

710 5 04100 3579 11000

1 2 3 4 5 6 7

8.9211 21404 5.1619 59433 3.7175 05928

1

(68)2.1965 63787 ( 1)2.6532 99832

8.8959 20362 5.1510 19154 3.7112 01473

( 68)2.4325 73275

( 1)2.6589 47160

( 68)2.5165 07242 ( 1)2.6608 26939

8.9085 38706 5.1564 979% 3.7143 59051

8.9127 36887 5.1583 20404 3.7154 09195

( 1)2.6627 05391

711 5 05521 3594 25431

712 5 06944 3609 44128 22191 84600 06635 83244 00070 04050 11683 ( 68)2.8808 44702 ( 1)2.6683 32813 8.9294 90191 5.1655 90782 3.7195 97942

713 5 08369 3624 67097 90402 70356 23964 64864 73748 79582 34042 (68)2.9795 36544 ( 1)2.6702 05985 8.9336 68708 5.1674 03588 3.7206 42186

714 5 09796 3639 94344 19616 28606 18825 20408 83171 29584 57523 (68)3.0814 63889 ( 1)2.6720 77843 8.9378 43321 5.1692 14489 3.1216 85260

717 5 14089 3686 01813

718 5 15524 3701 46232

(68)3.4076 87302 ( 1)2.6776 85568

( 68)3.5236 00491 ( 1)2.6795 52201

8.9503 43817 5.1746 35801 3.7248 07483

8.9545 02899 5.1764 39125 3.7258 45902

719 5 16961 3716 94959 86755 17977 71425 98549 85457 31343 31336 ( 68) 3.6432 86875 ( 1)2.6814 17536 8.9586 58122 5.1782 40566 3.7268 83164

122

729 5 227% 3779 33067

720 5 24iid 3795 03424

(68)2.7852 ( 1)2.6664 8.9253 5.1637 3.7185

89985 58325 07760 76065 52523

(68)2.6032 12640 8.9169 31117 5.1601 40881 3.7164 58153

8.9420 14037 5.1710 23488 3.7227 27165

716 5 12656 3670 61696 61743 63808 42887 75069 34149 79651 35030 ( 68)3.2954 33372 ( 1)2.6758 17632 8.9461 80866 5.1728 30591 3.7237 67905

720 5 18400 3732 48000

721 5 19841 3748 05361

( 68)3.7668 63772 ( 1)2.6832 81573

( 68)3.8944 51981

(. 68) . 4.0261

( 1)2.6851 44316

( 1)2.6870 05769

( 1)2.6888 65932

( 1)2.6907 24809

8.9628 09493 5.1800 40128 3.7279 19273

8.9669 57022 5.1818 37817 3.7289 54232

8.9711 00718 5.1836 33637 3.7299 88042

8.9752 40590 5.1854 27593 3.7310 20708

8.9793 76646 5.1872 19688 3.7320 52232

715 5 11225 3655 25875

8

9 10 24 1/2 1/3 114 1/5

91511 14717 06283 88981 46554

25887 66051 36564 73657 07718

8

(68)2.6927 76876

(68)2.1228 ( 1)2.6514 8.8917 5.1491 3.7101

709 5 02681 3564 00829

4

( 1)2.6645 82519

90555 28260 88205 56856 90435

708 5 01264 3548 94912

5 6 7 9 10 24 1/2 113 11 4 115

(68)2. 0515 ( 1)2.6495 8.8874 5.1473 3.7090

707 4 99849 3533 93243

(68)2.3513 ( 1) 2.6570 8,9043 5.1546 3.7133

( 68)2.2726 82709 ( 1)2.6551 83609

iiiii

704 4 95616 3489 13664

703 4 94209 3474 28927

702 A 97RnA

3459

( 68) 3.1867 28051 ( 1)2.6739 48391

5 21284 3763 67048

i

9 10 24 1/2 1/3 1/4 115

1

[(-36) 21

,a [(-251

75870

( 68)4.1621

[(-371 21

63488

. .

( 68)4.3025

[(-3"3

46659

53

ELEMENTARY ANALYTICAL METHODS

Table 3.1

POWERS AND ROOTS nk k

5 6 7 i3

9 10 24

5 3810 ill 2.7628 14 2.0030 17 1.4522 20 1.0528 22 7.6331 25 5.5340 28 4.0121 (68)4.4474 ( 1)2.6925 8.9835 5.1890 3.7330

ii

725 25b25 78125 16406 41895 05374 48896 54495 37009 76831 61095 82404 08896 09928 82616

726 5 27076 3826 57176

(68)4.5970 46501 ( 1)2.6944 38717

8.9876 37347 5.1907 98317 3.7341 11864

730 5 32900 3890 17000 4 5 6 7 8 9 10 24

731 5 3436I 3906 17891

f. 68). 5.4202 ( 1)2.7037 9.0082 5.1997 3.7392

735 5 40225 3970 65375

740 5 47600 4052 24000

1 2 3 4 5 6 7 8 9 10 24

46686 93753 62009 84860 39979

( 68)4.9108 09683 ( 1)2.6981 47513

7 32 5 35824 3922 23168

733 5 37289 3938 32837

( 68)5.6010 04807

( 68)5.7875 58467 ( 1)2.7073 97274

(68)4.7514 ( 1)2.6962 8.9917 5.1925 3.7351

21655 01167 22937 12653 41158

736 5 41696 3986 86256

( 1)2.7055 49852

9.0123 28782 5.2014 90029 3.7402 63647 717

5 43164 4003 15553

. . ( 68)6.1786 86185 ( 1) 2.7110 88342

9.0246 23926 5.2068 11253 3.7433 24423

10 24

728 5 29984 3858 28352

8.9958 82891 5.1943 69560 3.7361 66963

5 3874 ( 11)2.8242 (14)2. 0589 (17)l. 5009

729 31441 20489 95365 11321 46353

( 68)5.0752 87861

( 1)2.7000 00000 9.0000 00000

5.1961 52423 3.7371 92819 770

5 3954 11 2 9025 1412: 1304

38ii6 46904 80275 93922

~,

( 68)5.2450 38047 ( 1)2.7018 51217

9.0041 13346 5.1979 33452 3.7382 17550

5 6 7 8 9 10 24

727 5 28529 3842 40583

(68) . . 6.3836 ( 1)2.7129 9.0287 5.2085 3.7443

27605 31993 14871 81374 42461 7dl

5 496ii

4068 . .- - 69021 . .- -

(2ej4;9909 36907 (68)7.2704 ( I) 2.7202 9.0450 5.2156 3;7484

49690 94102 41696 43874 03580

( 68)7.5099 49065 ( 1)2.7221 31518

9.0491 14206 5.2174 05023 3.7494 16115

7A5

7 AA

5 5502s 4134 93625

5 56% 4151 60936

(68)8.5457 ( 1)2.7294 9.0653 5.2244 3.7534

57129 68813 67701 31847 55355 1

ni[(

(68)8.8253 48404 ( 1)2.7313 00057

9.0694 21981 5.2261 84131 3.7544 62453

(68)6.5950 74542 ( 112.7147 74392

9.0328 02112 5.2103 49693 3.7453 59393 742 5 50564 4085 18488 07181 55728 73550 04174 16974 56995 01490 (68)7.7569 98844 ( 1)2.7239 67694 9.0531 83053 5.2191 64391 3.7504 27557 747 5 58009 4168 32723 40441 64109 95190 08907 79533 48511 09038

(68)5.9800 58576 ( 1)2.7092 43437

9.0164 30890 5.2032 65584 3.7412 85019

9.0205 29268 5.2050 39324 3.7423 05277

738 5 44644 4019 47272 70867 81700 16095 24678 56123 24819 16516 (68)6.8132 24254 ( 1)2.7166 15541 9.0368 85658 5.2121 16213 3.7463 75222

139 5 46121 4035 83419

743 5 52049 4101 72401

( 68)7.0382 79698 ( 1)2.7184 55444

9.0409 65517 5.2138 80938 3.7473 89950 744 5 53536 4118 30784

9.0572 48245 5.2209 21982 3.7514 37909

(68)8.2146 ( 1)2.7276 9.0613 5.2226 3.7524

748 5 59504 4185 08992

4201

( 68)8,0118 26396 ( 1)2.7258 02634

65623 36339 09792 77799 47174 749

5 6iooi 89749

( 68)9.1136 94019 ( 1)2.7331 30074

(6Q9.4110 55807

(68)9.7177 03069

( 1)2.7349 58866

( 1)2.7367 86437

9.0734 72639 5.2279 34653 3.7554 68472

9.0775 19683 5.2296 83419 3.7564 7341.5

9.0815 63122 5.2314 30432 3.7574 77282

54

ELEMENTARY ANALYTICAL METHODS

POWERS ARD ROOTS nk

Table 3.1 k 750 62500 75000 62500 46875 85156 38867 29150 68628 51471 91278 12788 02964 75697 80079

751 5 64001 4235 64751 71280 09431 70983 47308 57828 52291 88271 ( 6 9 ) 1.0359 96977 ( 1) 2.7404 37921 9.0896 39217 5.2349 19217 3.7594 81806

I

755

2 3

5 70625 4303 68875 11) 3.2492 85006

756 5 71536 4320 81216

1 2

5 4218 11) 3.1640 14) 2.3730

3

4 5 6 7 8 9 10 24

( 6 9 ) 1.0033 ( 1 ) 2.7386 9.0856 5.2331 3.7584

A

5 6 7 8 9 10 24

( 6 9 ) l . 2148 ( 1)2.7495 9.1097 5.2436 3.7644

51214 45417 66916 10795 74495

760 5 77600 4389 76000 11) 3.3362 17600 25376 99286 19457 34787 64385 88932 ( 6 9 ) 1.3788 79182 ( 1)2.7568 09750 9.1258 05271 5.2505 33069 3. 7684 49662

5 4407 11 3 3538 141 2: 5522 17 1 9422 201 1: 4780 23 1.1248 25 8.5597 28 6.5139 ( 6 9 ) 1.4230 ( 1 ) 2.7586 9.1298 5.2522 3.7694

761 79121 11081 11326 50419 62569 61815 05041 66364 82203 88020 22845 06063 59366 40838

I

1 2 3

8 9 10 24

1 2 3

8 9 10 24

1 2 3 4 5 6 7 8 9 10 24

765 5 85225 4476 97125

( 2 8 ) 6.8645 ( 6 9 ) 1.6138 ( 1 ) 2.7658 9.1451 5.2591 3.7733

86020 91907 63337 74274 47590 95151

770 5 92900 4565 33000

( 6 9 ) 1.8870 ( 1) 2.7748 9.1656 5.2677 3.7783

23915 87385 56454 19986 14849

( 6 9 ) 1.0696 ( 1)2.7422 9.0936 5.2366 3. 7604

16698 61840 71888 60997 82467

758 5 74564 4355 19512

5 76601 4372 45479

753 5 67009 4269

757

5 73049

4337 98093 11 3.2838 51564 1 4 2.4858 75634

I1

65520 26333 48491 75936 78075

( 6 9 ) 1.1768 i 112.7477 -,-9.1057 5.2418 3.7634

38685 03656 ( 6 9 ) 1.1042 80565 ( 1) 2.7440 84547 9.0977 00985 5.2384 01041 3.7614 82064

754 5 68516 4286 61064 04423 06735 03078 77321 49900 60245 01825 ( 69) 1.1400 19555 ( 1) 2.7459 06044 9.1017 26517 5.2401 39353 3.7624 80599

752 5 65504 4252 59008

I

766 5 86756 4494 55096 11) 3.4428 26035

( 69) 1.6652

( 6 9 ) 1.2540 ( 1 ) 2.7513 9.1137 5.2453 3.7654

10313 63298 81798 43934 69862

759

17 1 9118 36089 201 1: 4510 83592

( 6 9 ) 1.2943 ( 1 ) 2.7531 9.1177 5.2470 3.7664

77441 79980 93146 75356 64176

( 6 9 ) 1.3359 ( 1 ) 2.7549 9.1218 5.2488 3.7674

88198 95463 00968 05067 57442

53390 34748 03351 83963 30972

763 5 82169 A A A 1 9A947 11 3.3895 07446 1 4 2.5859 65281 1 7 1.9730 91509 20 1.5054 68822 23 1.1486 72711 25 8.7643 72784 ( 2 8 ) 6.6872 16435 ( 6 9 ) 1.5156 15056 ( 1) 2.7622 45463 9.1377 97144 5.2557 06863 3.7714 20068

764 5 83696 4459 43744 10204 55796 58228 34886 71853 -... 96958 78876 ( 6 9 ) 1.5640 13890 ( 1 ) 2.7640 54992 9.1417 87449 5.2574 28071 3.7724 08126

767 5 88289 4512 17663 11) 3.4608 39475

768 5 89824 4529 84832 ( 1 1 ) 3.4789 23510

769 5 91361 4547 56609

762 5 80644 4424 50728 11) 3.3714 74547

( 6 9 ) 1.4686 ( 1)2.7604 9.1338 5.2539 3.7704

( 1 ) 2.7676 9.1497 5.2608 3.7743

92289 70501 57625 65424 81144

( 6 9 ) 1.7182 ( 1)2.7694 9.1537 5.2625 3.7753

59425 76485 37512 81576 66108

5 4583 11 3.5336 1 4 2.7244 1 7 2.1005 20 1.6194 23 1.2486 25 9.6269 28 7.4223 1.9467 2.7766 9.1696 5. 2694 3. 7192

771 94441 14011 01025 06390 17327 98859 33620 65212 90179 27094 88675 22555 29452 95720

772 5 95984 4600 99648 69283 20286 16861 59817 48578 27025 23664 ( 69) 2. 0082 38127 ( 1 ) 2.7784 88798 9.1735 85227 5. 2711 37257 3.7802 75573

I /I

( 6 9 ) 1.7728 ( 1) 2.7712 9.1577 5.2642 3.7763

38934 81292 13940 96052 50045

773 5 97529 4618 89917

(69)2. 0716 ( 1) 2.7802 9.1775 5.2728 3.7812

09310 87755 44479 43403 54412

( 6 9 ) 1.8290 ( 1)2.7730 9.1616 5,2660 3.7773

77701 84925 86919 08854 32958

774 5 99076 4636 84824

( 69) 2.1368 ( 1)2. 7820 9.1815 5.2745 3.7822

94378 85549 00317 47894 32239

55

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS

Table 3.1

nk

k

10 24

6 4654 11 3.6075 14 2.7958 17 2.1667 20 1.6792 23 1.3014 26 1,0085 28 7.8165 ( 69)2.2041 ( 1) 2.7838 9.1654 5.2762 3.7832

I!

775 00625 84375 03906 15527 57034 36701 08443 91544 84463 48547 82181 52750 50735 09055

780 6 08400 4745 52000 . ._ ..

776 6 02176 4612 88576

( 69)2.2734 28553 ( 1) 2.7856 77655

9.1894 01784 5.2779 51928 3.7841 84864

777 6 03729 4690 91433

(69)2.3447 ( 1)2.7874 9.1933 5.2796

92689 71973 47428 51478

778 6 05284 4109 10952

(69)2.4183 ( 1)2.7892 9.1972 5.2813 317861

00846 65136 89687 49388 33467

779 6 06641 4727 29139

(69)2.4940 ( 1)2.7910 9.2012 5.2830 3.7871

14558 57147 28569 45663 06266 784

782 6 11524 4782 11768

783 6 13089 4800 48687

13239 37722 96233 33318 48871

( 69)2. 7350 29868 ( 1)2.7964 26291

( 1)2.7982 13716

9.2130 25029 5.2881 24706 3.7900 18681

9.2169 50477 5.2898 14473 3.7909 87500

9.2208 72584 5.2915 02622 3.7919 55329

786 6 17796 4855 87656

787 6 19369 4874 43403

788 6 20944 4893 03872

789 6 22521 4911 69069

93218 46695 99652 69154 06804 74081 88029

( 69) 3.1870 84488 ( 1)2. 8053 52028

(69)3.2857 09926 ( 1)2.8071 33770

( 69)3.3872 56439 ( 1)2.8089 14381

791 6 25681 4949 i367i

792 6 27264 4967 93088

781 6 09961 4763 79541

6 14656 4818 90304 11) 3.7780 19983

5

6 7 8 9 10 24

(69)2.5719 ( 1) 2.7928 9.2051 5.2847 3.7880

97041 48009 64083 40305 78066 7A5

4

6 16225 4837 36625

(69)2.6523 ( 1)2.7946 9.2090 5.2864 3.7890

(69)Z. 8202 15463

( 69)2. 9079 40422 ( 1)2.8000 00000

5

6 7 8 9 10 24 1/2 1/3

'115 I4

( 69)2.9982 77060 ( 1)2.8017 85145

9.2247 91357 5.2931 89157 3.7929 22172 ,

6 7 8 9 10 24

790 6 24100 4930 39000

(69)3.4918 06676 ( 1) 2.8106 93865

9.2443 35465 5.3015 97745 3.7977 41656 1 2 3

4 5 6 7 8 9 10 24 1/2 1/3

795 6 32025 5024 59875

26 1,1449 281 8.9996 ( 69)3. 0912 ( 1)2.8035 9.2287 5.2948 3.7938

(69)3.5994 ( 1) 2.8124 9.2482 5.3032 3.7987

45514 72222 34384 74670 02623

796 6 33bi6 5043 58336

9.2326 18931 5.2965 57399 3.7948 52904

(69)3.7102 60118 ( 1)2.8142 49456

9.2521 30018 5.3049 50005 3.7996 62619 797

6 35264 5062 61573

9.2365 27746 5.2982 39113 3.7958 16799

6 4986 11 3 9545 141 3: 1359

793 28849 77257 10648 26944

(69)3.8243 ( 1)2.8160 9.2560 5.3066 3.8006

39997 25568 22375 23755 21646

9.2404 33255 5.2999 19227 3.7967 79716 19P

6 30436 5005 66184

(69)3.9417 ( 1)2.8178 9.2599 5.3082 3.8015

77065 00561 11460 95923 79705

798 6 36804 5081 69592

799 6 38401 5100 82399

( 69)4.4470 23172 ( 1)2.8248 89378

( 69)4.5827 13463 ( 1)2.8266 58805

(29)1.0603 95298 (69)4.0626 ( 1)2.8195 9.2637 5.3099 3.8025

65702 74436 97282 66512 36800

( 69)4.1871 02820 ( 1)2.8213 47196

9.2676 79846 5.3116 35526 3.8034 92932

(69)4.3151 ( 1)2.8231 9.2715 5.3133 3.8044

87922 18843 59160 02968 48104

9.2754 35230 5.3149 68841 3.8054 02317

9.2793 08064 5.3166 33150 3.8063 55574

56

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS nk

Table 3.1 k 1 2 3 4 5 6 7

800 6 40000

801

h 41601

6 5158 11 4.1371 1 4 3.3179 ( 1 7 ) 2.6610

I1

803 6 44809 5177 81627 11) 4.1577 86465

802 43204 49608 13856 65313 08181

a

9 10 24 1/2 1/3 114 1/5 1

2 3 4 5 6 7 8 9 10 24 1/2 1/3 114 1/5 1 2

8 9 10

24 1/2 1/3 114 115

( 2 6 1.4037 97416 (2911.1286 53123 ( 6 9 ) 4.7223 ( 1) 2.8284 9.2831 5.3182 3.8073

66483 27125 77667 95897 07877

805 6 48025 5216 60125

( 6 9 ) 5.4840 ( 1)2. 8372 9.3024 5.3265 3.8120

46503 52192 77468 86329 55159

810 6 56100 5314 41000

20 2 2876 231 1: 8530 26 1 5009 291 1: 2157 ( 69) 6.3626 ( 1)2. 8460 9.3216 5.3348 3.8167

79245 20189 46353 66546 85441 49894 97518 38230 78910

815 6 64225 5413 43375 6 1

8 9 10 24 1/2 1/3 114 115

5 6 7 8 9 10 24 1/2 1/3 114 1/5

804 6 464i6 5197 18464

(69) 7. 3753 ( 1 ) 2. 8548 9.3408 5.3430 3.8214

49576 20485 38634 52016 79391

6 5513 11)4.5212 14) 3.7073

820 72400 68000 17600 98432

26 1 6761 291 1: 3744 (69) 8.5414 ( 1)2. 8635 9.3599 5.3512 3.8261

95504 80313 66801 64213 01623 28095 56858 1

(69) 4.8660 ( 1 ) 2.8301 9.2870 5.3199 3.8082

92789 94340 44047 57086 59229

6 5236 ( 11)4.2202 ( 1 4 3 4015 1712: 7416

806 49636 06616 69325 37076 38883

( 69) 5. 6499 ( 1)2. 8390 9.3063 5. 3282 3.8130

03151 13913 27832 39778 01783

6 5334 11 4 3259 1413: 5083 17 2.8452 20 2.3075 23 1.8714 26 1.5177 29 1.2308

811 57721 11731 69138 60971 80748 22686 00899 06129 59670

( 69) 6.5539 ( 1)2. 8478 9.3255 5.3364 3.8177

10420 06173 32030 84023 20859

6 5433 11 4.4336 1 4 3.6178 17 2.9521 20 2.4089 23 1.9651 26 1 6040 291 1: 3088

816 65856 38496 42127 51976 67212 68445 18251 26093 85292

( 6 9 ) 7. 5956 ( 1) 2.8565 9.3446 5.3446 3.8224

30157 71371 57457 90236 16717

821 6 74041 5533 87661

(69) 5.0140 ( 1)2.8319 9.2909 5.3216 3.8092

05879 60452 07211 16720 09631

(69) 5.1662 ( 1)2. 8337 9.2947 5. 3232 3.8101

807 6 51249 5255 57943

( b9) 5.8205 ( 1)2. 8407 9.3101 5.3298 3.8139

60843 74542 75012 91690 47468

808 6 52864 5275 14112 14025 49732 11384 30798 32085 19524 78976

809 6 54481 5294 75129

( 1)2.8354 9.2986 89376 23915 5.3249 31338 3.8111 07593

52346 34081 19016 42067 92216

( 6 9 ) 6.1768 13927 ( 1 ) 9.3178 2.8442 92531 59849

813 6 60969 5373 67797

814 6 62596 5393 53144 11)4.3903 34592

5.3331 90912 3.8158 36029

26 1 5689 88079 (291 1: 2771 56297

21222 36166 61370 63391 28295 61880

( 69) 6.9530 ( 1)2.8513 9.3331 5.3397 3.8196

13847 15486 91608 71049 01974

818 6 69124 5473 43432

817 6 67489 5453 38513

( 69) 7.8222 ( 1)2. 8583 9.3484 5.3463 3.8233

( 69) 5.3228 61548

( 6 9 ) 5.9961 ( 1)2. 8425 9.3140 5. 3315 3.8148

812 6 59344 5353 07328

f 29) 1.2461 (69) 6.7506 ( 1 ) 2.8495 9.3293 5. 3381 3.8186

22264 25463 67164 74803 59085

07941 21186 73160 26950 53125

( 69) 8.0552 ( 1)2. 8600 9.3522 5.3479 3.8242

54907 69929 85752 62163 88616

(69) 7.1611 ( 1)2. 8530 9.3370 5.3414 3.8205

98588 68524 16687 12288 41144

6 5493 11 4 4992 141 3: 684&

819 70761 53259 03191 47414

( 69)8.2949 ( 1 ) 2.8618 9.3560 5.3495 3.8252

47511 17604 95231 95877 23193

A24 6 7867b 5594 76224

823 6 77329 5574 41767

822 6 75684 5554 12248 ( 11)4.5654 88679

(29) 1.3913 34555 ( 69) 8.7949 ( 1)2. 8653 9.3637 5.3528 3.8270

98523 09756 04916 58822 89612

(69)9.0557 ( 1)2. 8670 9.3675 5.3544 3.8280

33244 54237 05121 88059 21458 1

[

( 69) 9.3238 ( 1)2. 8687 9.3713 5.3561 3.8289

n4 (-37121

66467 97658 02245 15810 52397

(69) 9.5995 ( 1)2. 8705 9.3750 5.3577 3.8298

f [‘-911

98755 40019 96295 42079 82432

57

ELEMENTARY ANALYTICAL METHODS

Table 3.1

POWERS AND ROOTS nk k

24

825 6 80625 15625 03906 15723 97971 23326 09244 57626 29) 1.4606 27542 ( 69) 9.8831 35853 ( 1)2.8722 81323 9.3788 87277 5.3593 66869 3.8308 11564 830 6 88900 5717 87000 32100 40643 03734 05099 92232 02553 04119 ( 7 0 ) 1.1425 ( 1)2.8809 72058 9.3977 96375 5.3674 68731 3.8354 43756

b 7 8

9 10 24

1 2

24

1 2 3 4 5 6 7 8 9 10 24

835 6 97225 5821 82875 27006 24550 68999 23115 52801 32589 49211 ( 70) 1.3197 00592 ( 1)2.8896 36655 9.4166 29685 5.3755 34071 3.8400 53677 840 7 05600 5927 04000 13600 19424 80316 03466 58911 57485 12288 ( 7 0 ) 1.5230 10388 ( 1 )2.8982 75349 9.4353 87961 5.3835 63271 3.8446 41568

__

845

7 14025 6033 51125

( 7 0 ) 1.7561 ( 1 ) 2.9068 9. 4540 5. 3915 3.8492

47601 88371 71946 56705 07664

826 6 82216 5635 59976

( 7 0 ) 1.0174 ( 1)2.8740 9.3826 5.3609 3.8317

68882 21573 75196 90182 39795

827 6 83929 5656 09283

( 7 0 ) 1. 0474 ( 1)2.8757 9.3864 5.3626 3.8326

831 6 90561 5738 56191

47415 60769 60060 12021 67128

832 6 92224 5759 30368

6 5676 11)4.7002 14) 3.8918

828 85584 bj552 54211 10486

( 7 0 ) 1.0782 ( 1 ) 2 . 8774 9.3902 5.3642 3.8335

71392 98914 41873 32391 93565

833 6 93889 5780 09537

29) 1.5894 00198 ( 1)2.8827 07061

( 1)2. 8844 41020

9.4053 38751 5. 3706 99229 3.8372 90383

( 1)2.8861 73938

9.4015 69076 5.3690 84709 3.8363 67514 836 b 98896 5842 77056

837 7 00569 5863 76253

838 7 02244 5884 80472

il ( 7 0 ) 1. 3581 ( 1)2.8913 9.4203 5. 3771 3.8409

59133 66459 87319 42790 73010

841 7 07281 5948 23321 ( 1 1 ) 5.0024 64130

( 7 0 ) 1.5671 ( 1 ) 2 . 9000 9.4391 5.3851 3.8455

25939 00000 30677 64807 56523

846 7 isiib 6054 95736 93927

20 2.8779 23 2.4088 26 2.0161 ( 2 9 ) 1.6875 ( 7 0 ) 1. 3976 ( 1 ) 2 . 8930 9.4241 5.3787 3.8418

16611 16204 79163 41959 90431 95228 41957 50067 91464 R42

7 08Gdi 5969 47688

( 7 0 ) 1.6124 64626 ( 1)2. 9017 23626

9.4428 70428 5. 3867 64916 3. 8464 70609

847 7 17409 6076 45423 ( 11)5.1467 56733

9.4091 05407 5.3723 12294 3.8382 12366

26 2 0379 2911: 7078 ( 7 0 ) l . 4383 ( 1 ) 2 . 8948 9.4278 5.3803 3.8428

829 6 87241 5697 22789

( 7 0 ) 1.1099 ( 1)2.8792 9.3940 5.3658 3.8345

63591 36010 20643 51293 19107

834 6 95556 5800 93704

26 1 9520 291 1: 6280 ( 70) 1.2822 ( 1)2.8879 9.4128 5.3739 3.8391

65814 22889 86929 05816 69049 23907 33463

839 7 03921 5905 89719

62403 12493 23072 22965 93606 55904 09040

( 1)2. 8965 49672

843 7 10649 5990 77107

844 7 12336 6012 11584

( 7 0 ) 1.6590 ( 1)2.9034 9.4466 5.3883 3.8473

58848 46228 07220 63600 83826

( 7 0 ) 1.4800 86372 9.4316 42272 5.3819 60304 3.8437 25741

( 7 0 ) 1.7069 ( 1)2.9051 9.4503 5.3899 3.8482

41821 67809 41057 60862 96177

848 7 19104 6098 00192

849 7 20801 6119 60049

( 7 0 ) l . 8067 11101

( 7 0 ) l . 8586 68111

( 7 0 ) 1.9120 55324

( 7 0 ) 1.9669 10351

( 1)2.9086 07914

( 1)2.9103 26442

( 1)2.9120 43956

( 1)2.9137 60457

9.4577 99893 5.3931 51133 3.8501 18288

9.4615 24903 5. 3947 44148 3.8510 28051

9.4652 46982 5.3963 35753 3.8519 36956

9.4689 66137 5.3979 25951 3.8528 45003

58

ELEMENTARY ANALYTICAL METHODS

Table 3.1

POWERS AND ROOTS

nk

k 1 2 3 4 5 6 7 8 9 10

850 22500 25000 62500 53125 95156 70883 05250 69463 44043

7 6184 111’5. 2693 14) 4.4894 17 3 8250 (2013: 2589 12312.7766 26 2.3656 29 2.0155

852 25904 70208 66172 99979 53982 45993 21986 81932 61006

(70) 2.1406 ( 1)2.9189 9.4801 5.4026 3.8555

( 70) 2.0232 71747

7 6162 11 5.2446 141 4.4632 17 3 7 9 8 1 20) 3: 2322 2332.7506 26) 2.3408 29) 1.9920 ( 7 0 ) 2.0811

851 24201 9505I 70884 14922 95899 64710 57268 09335 28744 79034

( 1)2.9154 9.4726 5.3995 3.8537

75947 82372 14744 52195

( 1)2.9171 9.4763 5.4011 3.8546

90429 95693 02137 58534

7 6250 ( 1 1 5.3439 ( 1 4 1 4. 5690 17) 3.9065 20) 3. 3401 23) 2.8558 26) 2.4417 ( 29) 2. 0876 ( 7 0 ) 2.3290 ( 1)2. 9240 9.4912 5.4074 3.8582

855 31025 26375 75506 99058 79694 25639 07421 15345 66620 92589 38303 19958 37751 75391

7 6272 11 5.3690 14 4.5958 17 3.9340 20 3.3675 23 2.8826 26 2.4675 29)2.1122

856 32736 22016 20457 81511 74574 67835 38067 38185 12686

857 7 34449 6294 22793 ( 1 1 ) 5. 3941 53336

(7O)Z. 3953 ( 1 ) 2 . 9257 9.4949 5.4090 3.8591

57569 47768 18797 18180 77490

(7w2.4634 ( 1)2.9274 9.4986 5.4105 3.8600

860 39600 56000 81600 70176 72351 78222 79271 74173 15789

7 6382 11115.4955 1 4 4.7316 17 4.0739

861 41321 77381 68250 84264 80151

8 9 10

7 6360 11) 5.4700 14) 4.7042 17) 4.0456 20) 3.4792 23) 2.9921 26)Z. 5732 29)2.2130

24 1/2 113 114 115

( 7 0 ) 2.6789 ( 1)2.9325 9.5096 5.4153 3.8627

39031 75660 85413 26084 77475

29) 2.2388 (7O)Z. 7547 ( 1)2.9342 9.5133 5.4168 3.8636

83597 08410 80150 69910 99621 75378

24 1/2 113 114 115 1 2 3 4

5 6 7 8 9 10 24 1/2 1/3 1/4 115

5 6 7

7 6141 11)5.2200 14) 4.4370 117) 3.7714 20) 3.2057 23) 2.7249 ( 2 6 2 3161 (291 1: 9687

I

865 7 48225 6472 14625

I I

866 7 49956 6494 61896

5 6 7

8 9 10 24 1/2 1/3 114 1/5

( 7 0 ) 3.0788 ( 1)2.9410 9.5280 5.4231 3.8672

36164 88234 79435 80095 58668

7 6585 11 5 7289 141 4: 9842

870 56900 03000 76100 09207

26 2.8554 291 2.4842

41542 34142

( 7 0 ) 3.5355 ( 1)2.9495 9.5464 5.4310 3.8717

91351 76241 02709 00130 1918?

1

2 3 4 5 6 7

8 9 10 24 1/2 1/3 114 1/5

( 7 0 ) 3.1654 ( 1 ) 2.9427 9.5317 5.4247 3.8681

05907 87794 49727 46809 52418

7 6607 11 5.7553 141 5.0129 1 7 4.3662 20 3.8030 23 3.3124 26 2.8851 29 2.5129 ( 7 0 ) 3.6344 ( 1)2. 9512 9.5500 5.4325 3.8726

5864i 76311 61669 20014 53332 06652 18794 16769 36706 25075 70913 58934 60090 088q7

R71

72719

7 6206 11)5.2941 14) 4. 5159 ( 1 7 ) 3.8520 ( 2 0 3 2858 ( 2 3 1 2: 8028 ( 2 6 ) 2. 3907 ( 2 9 ) 2.0393 ( 7 0 ) 2.2017

853 27609 50477 48569 08729 70146 15835 00907 89174 43165 94325

03904 06107 88131 64021

( 1)2.9206 9.4838 5.4042 3.8564

16373 13619 72729 68659

I

7 6316 11) 5.4193 14)4. 6498 1 7 3.9895 20 3.4230 23 2.9369 26 2.5199 29)2.1620 ( 7 0 ) 2.5333 ( 1) 2.9291 9.5023 5.4121 3.8609

i

27165 56234 14756 97225 78746

862 7 43044 6405 03928

( 7 0 ) 2.8325 ( 1)2.9359 9.5170 5.4184 3.8645

29097 83651 51555 71787 72447

7 6517 11 5 6503 1414: 8988 17 4.2473 20 3.6824 23 3.1926 26 2.7680 29 2.3998 ( 7 0 ) 3.2543 ( 1)2.9444 9.5354 5.4263 3.8690

A67 51689 14363 63527 65178 16109 23067 60799 36913 88003 05644 86373 17196 12167 45344

872 7 60% 6630 54848

( 7 0 ) 3.7359 ( 1)2.9529 9.5537 5. 4341 3.8734

03403 64612 12362 18707 9765:

858 36164

28712

74349 23191 48298 32440 61833 13253 85571 48329 63703 07842 74889 79161

863 7 44769 6427 35647 .. ..

( 2 9 ) 2; 2914 ( 7 0 ) 2.9124 ( 1)2.9376 9.5207 5.4200 3.8654

37189 54150 86164 30354 42587 68684

868 7 53424 6539 72032

( 7 0 ) 3.3455 ( 1 ) 2.9461 9.5390 5.4278 3.8699

95291 83973 81845 76171 37445

7 6653 11 5 8084 141 5: 0707 17 4.4267 20 3.8645 23 3.3737 26 2.9452 29 2.5712 ( 7 0 ) 3.8400 ( 1 ) 2.9546 9.5573 5.4356 3.8743

877 62126 38617 06126 38548 54753 56899 58173 90885 38943 93943 57341 62998 75984 8566l

7 6228 ( 1 1 ) 5.3190 ( 1 4 4.5424 17 3.8792 l Z23) l j 32.8291 .3128

854 29316 35864 18279 41610 45135 75345 95545

26 2 4161 32995 ( 2 9 ) 2: 0633 77578 ( 7 0 ) 2.2645 86409

( 1)2.9223 9.4875 5.4058 3.8573

27839 18234 55935 72448

859 7 37881 6338 39779

(7W2.6051 ( 1 ) 2.9308 9.5059 5.4137 3.8618

69182 70178 98059 51174 78737

864 7 46496 6449 72544

(70)Z. 9945 ( 1)2.9393 9.5244 5.4216 3.8663

37938 87691 06312 12022 64090

869 7 55161 6562 34909

(29)Z. 4558 ( 7 0 ) 3.4393 ( 1)2.9478 9.5427 5.4294 3.8708

26970 36231 80595 43681 38824 28725

U74 7 63876 6676 27624

( 7 0 ) 3.9470 ( 1)2.9563 9.5610 5.4372 3.8752

65953 49100 10846 31924 72857

59

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS

Table 3.1

nk

k 7 6699 11 5 8618 14) 5: 1290

10 24

h

5 6 7 8 9 10 24

875 65625 21875 16406 89355

. . (70) . . 4.0568 ( 1)2.9580 9.5646 5.4387 3.8761

90376 39892 55914 86530 59242

880 7 74400 6814 72000 11) 5.9969 53600

(70)4.6514 04745 ( 1)2.9664 79395

9.5828 39714 5.4465 39631 3.8805 79047 885 7 83225 6931 54125 4

8 9 10 24

~I

(70) . , 5.3289 ( 1)2.9748 9.6009 5.4542 3.8849

11365 94956 54766 59763 78808

5 6 7 8 9 10 24

(70)4.1696 ( 1)2.9597 9.5682 5.4403 3.8770

39882 29717 98205 39803 44816

881 7 76161 6837 97841 11 6 0242 58979 141 5: 3073 72161 94874 75284 69625 98440 19925 ( 70) 4.7799 32920 ( 1) 2.9681 64416 9.5864 68204 5.4480 86284 3.8814 60596 7 6955 11)6.1621 14)5.4596 17 4.8372 20 4.2858 23 3.7972 26 3.3643 29 2.9808 (70)5.4753 ( 1)2.9765 9.6045 5.4557 3.8858

886 84996 06456 87200 97859 92303 40981 55109 68027 30072 17719 75213 69584 99862 56373

7 92100

891 7 93881 7073 47971

(70) . . 6.1004 25945 ( 1)2.9832 86778 9.6190 01716

( 1)2.9849 62311

890

10 24

U76 7 67376 6722 21376 11)5.8886 59254

(70)6.2670 75070

5.4619 47252 3.8893 58728

9.6226 02990 5.4634 80860 3.8902 32348

895 8 01025 7169 17375 10506 87403 05226 36177 32378 43979 45861 (70)6.9783 51604 ( 1)2.9916 55060 9.6369 81200 5.4696 02417 3.8937 19006

896 R 02816 7193 23136 35299 41228 57740 34935 76902 63304 79120 (70)7.1679 04854 ( 1)2.9933 25909 9.6405 69057 5.4711 29599 3.8945 88722

7 6745 11)5.9155 14)5.1879 17 4.5498 201 3.9902 23) 3.4994 26 3.0689 ( 29) 2.6915 (70)4.2853 ( 1)2.9614 9.5719 5.4418 3.8779

I

877 69129 26133 94186 76101 55041 22871 25458 96127 09603 88904 18579 37725 91747 29583

882 7 77924 6861 28968 11) 6. 0516 57498

(70)4.9118 ( 1)2.9698 9.5900 5.4496 3.8823

60716 48481 93948 31621 41346

887 7 86769

A978 - . . - 64103 . - ..

7 6768 (11)5.9426 (14 5 2176

878 70884 36152 21415 21602

120 1714: 4.0221 5810 23{ 3.5314 26 3 1006 (29)2: 7223

71767 81011 74928 34987 57518

(70)4.4042 ( 1)2.9631 9.5755 5.4434 3.8788

13682 06478 74480 42365 13542

7 6884 11) 6.0791 14)5.3678 (17 4 7398 ( 201 4: 1852

883 79689 65387 49367 88891 45891 83922

( 70) 5.0472 74047

( 1)2.9715 31592

9.5937 16954 5.4511 75645 3.8832 21296 7 8Gii 7002 27072

879 7 72641 6791 51439 11)5.9697 41149

(70)4.5261 92303 ( 1)2.9647 93416

9.5792 08475 5.4449 91658 3.8796 96696 884 7 81456 6908 07104 34799

(70)5.1862 ( 1)2.9732 9.5973 5.4527 3.8841

53563 44549 75782 20991 31356 22119 60897 13749 -. 37224 18358 00450

7 90321 7025 95369

29)3.0146 45144 ( 70) 5.6255 74442

( 1)2.9782 54522

9.6081 81682 5.4573 38658 3.8867 33146

892 7 95664 7091 32288 11) 6.3308 12009

(70)6. 4380 ( 1)2.9866 9.6262 5.4650 3.8911 8

(70)7.3623 ( 1)2.9949 9.6441 5.4726 3.8954

(70)5.7797 ( 1)2.9799 9.6117 5.4588 3.8876

78281 32885 91067 76153 09128 893

7 97449

(70)5.9380 ( 1)2.9816 9.6153 5.4604 3.8884

28303 10303 97744 12350 84321

894 7 99236 7145 16984

82017 36905 01570 13179 05185

(70)6.6135 55666 ( 1)2.9883 9.6297 10559 97462

( 1)2.9899 9.6333 83278 90671

(70)b. 7936 07487

5.4665 44210 3.8919 77239

5.4680 73955 3.8928 48512

897 04609 34273 56429 38917 03608 76237 11184 16432 86240 86846 95826 54244 55504 57662

898 8 06404 50792 74112 80953 43696 61439 37172 05980 70570 (70)7.5619 20026 ( 1)2.9966 64813 9.6477 36769 5.4741 80133 3.8963 25828

899 8 08201 7265 72699 88564 67819 78869 91904 56821 34582 35490 (70)7.7666 29743 ( 1) 2.9983 32870 9.6513 16634 5.4757 03489 3.8971 93220

60

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS nk

Table 3.1 k 1

2 3

900 8 10000 7290 00000

901 8 iisoi 7314 32701

903

902 8 13604 7338 70808

904 8 i7ii6 7387 63264

8 15406

20 4.8956 93857

24 1/2 113 114 1/5 1 2 3

(70)7.9766 ( 1)3.0000 9.6548 5.4772 3.8980

44308 00000 93846 25575 59841

(70)8.1920 ( 1)3.0016 9.6584 5.4787 3.8989

905

906

7412 17625

( 70)9.1109 96943 ( 1)3.0083 21791

1 2 3

910 8 28100 7535 71000

9.6727 40271 5.4848 17035 3.9023 81426

(70)8.4131 ( 1)3.0033 9.6620 5.4802 3.8997

13 - -201336 ____

8 19025

24 1/2 1/3 114 1/5

95066 66204 68409 46393 25692

7436 77416

(70)9.3557 ( 1)3.0099 9.6763 5.4863 3.9032

09844 83389 01663 31551 43449

907

(70)9.8641 ( 1)3.0133 9.6834 5.4893 3.9049

912 8 31744 7585 50528

4

29) 3.6448 88981 ( 70) 8.8724 24888 ( 1)3.0066 59276

9.6691 76254 5.4833 01264 3.9015 18640

-R

8

746I 42643

14616 44069 60436 44813 04712

92832 11553 69527 46120 95840 09608 84235 55089

908 24464 7486 i33iz

a 22649

(70)9.6067 ( 1)3.0116 9.6798 5.4878 3.9041

911 8 29921 7560 58031

16465 31484 40328 65946 90774

12314,4208 26 3.9919 29 3.6047 ( 70) 8.6398 ( 1)3.0049 9.6656 5.4817 3.9006

65825 03835 16593 56824 65216

7510 11)6.8274 14)6.2061 17)5.6413 20) 5.1279 23 4.6613 26 4.2371 29 3.8515 ( 71)1.0128 ( 1)3.0149 9.6869 5.4908 3.9058

I

909

7h7AI _____

89429 02910 09245 53304 90153 43049 60832 79196 22166 62686 70141 67587 24962

914 8 35396 7635 51944 ._. . .

913 8 33569 7610 48497

5 6 7 8

9 10 24 1/2 1/3 114 1/5

(71)l. 0399 ( 1)3.0166 9.6905 5.4923 3.9066

04400 20626 21083 77104 83951

(71)1.0676 ( 1)3.0182 9.6940 5.4938 3.9075

915 8 37225 7660 60875

29) 3.9805 (71)l. 0961 ( 1)3.0199 9.6976 5.4953 3.9083

79852 77655 69425 85378 42186

916 8 39056 7685 75296

98576 65476 33774 15172 92410 99668

(29)4. 0244 (71)1.1253 ( 1)3.0215 9.7011 5.4968 3.9092

61484 78622 88986 58327 98203 56397

(26)4.4515 (29)4; 0687 (71)1.1553 ( 1)3.0232 9.7046 5.4984 3.9101

919 8 44561 7761 51559

918 8 42724 7736 20632

917 8 40889 7710 95213

96186 58914 37042 43292 98896 02760 12356

6 7

8

9 10 24 1/2 1/3 1/4 1/5

(71)l. 1860 ( 1)3.0248 9.7082 5.4999 3.9109

58902 96692 36884 06083 67606

(71)1.2175 ( 1)3.0265 9.7117 5.5014 3.9118

920 a 4641~ 778i 88000 6 7 8 9 10 24 1/2 1/3 114 115

(71)1.3517 ( 1)3.0331 9.7258 5.5074 3.9152

85726 50178 88262 04268 32576 1

n2

[(-:I

62793 49190 72294 08174 22089

(71)1.2498 ( 1)3.0282 9.7153 5.5029 3.9126

921 8 48241 7812 29961

(71)l. 3874 ( 1)3.0347 9.7294 5.5089 3.9160

11

94035 98181 10859 00236 83344

67732 00786 05133 09036 75826

(71)1.2829 ( 1)3.0298 9.7188 5.5044 3.9135

922 8 50084 7837 77448

.f[

(71)1.4241 ( 1)3.0364 9.7329 5.5103 3.9169

(-:I31

05308 45290 30906 94986 33373

93183 51482 35404 08671 28819

(71)1.3169 59057 ( 1)3.0315 01278

9.7223 63112 5.5059 07081 3.9143 81068 924 8 53776 7888 89024

923 8 51929 7863 30467

(71)l. 4616 41363

(71)l. 5001 24518

( 1)3.0380 91506

( 1)3.0397 36831

9.7364 48410 5.5118 88520 3.9177 82664

9.7399 63373 5.5133 80842 3.9186 31220

.'[ (-:)2]

I!-([$

11

61

ELEMENTARY ANALYTICAL METHODS

Table 3.1

POWERS AND ROOTS nk 8 7914 11)7.3209 14)6.7718

925 55625 53125 41406 70801

26 4 9576 291 4: 5858 (71)1.5395 ( 1)3. 0413 9.7434 5.5148 3.9194

46934 23414 77607 81265 75802 71952 79042

1 2

6 7 8 9 10

24 1/2 1/3 1/4 1/ 5

927

928

59129

8 61184

7965 97983

7991 78752

( 71) 1,5800 23988 ( 1)3.0430 24811

( 71) 1.6214 87554 ( 1)3.0446 67470

( 1)3.0463 09242

911

932 68624 8095 57568

926 8 57476

-R

7940 22776 (11 7. 3526 50906 (1416.8085 54739

29)4. 6356 41763 9.7469 85700 5.5163 61854 3.9203 26131

8 8043

9.7504 93072 5.5178 50550 3.9211 72488

20 6.1081 1231 5.6928 26)5.3057 29)4.9449 (71)1.8449 ( 1) 3.0528 9.7679 5.5252 3.9253

7

i

(71)1.7980 ( 1)3. 0512 9.7644 5.5237 3.9245

(71)1.7522 28603 ( 1) 3.0495 90136

9.7610 00077 5.5223 09423 3.9237 07185

1 2

8

9 10 24

i

9 10 24 1/2 1/3 114 1/5 1 23

24 1/2 1/3 1/4 1/5

(71)2. 0977 ( 1) 3.0610 9.7854 5. 5326 3.9295

( 71) 2.0446 56558 ( 1)3.0594 11708

9.7819 46493 5.5311 94905 3.9287 57017

32860 45573 28852 71663 96137

8 72356 8147 80504

812I 66237

(11)7.5775 (14 7.0698 (1716.5961 20) 6.1541 1231 5.7418 26 5.3571 29 4.9982 (71)1.8930 ( 1) 3.0545 9.7714 5.5267 3.9262

10991 17755 39965 98588 67202 62174 32309 38514 04870 84510 57521 35348

938 8 79844 8252 93672

29) 5.2726 (71)2.1521 ( 1) 3.0626 9.7889 5.5341 3.9304

43202 28115 78566 08735 47239 34540

947 8 89244 8385 61807

942 8 87364 8358 96888

941 8 85481 8332 37621

9.7575 00256 5.5208 24332 3.9228 63013

933

8200 25856

(71)l.9423 ( 1)3.0561 9.7749 5.5282 3.9270

38996 41358 74326 37837 76625 979

8 8iiii

8279 (11)7.7743 (14)7.3000 (17 6 8547 (2016: 4366 (23 6.0440 (26 5.6753 (29 5.3291 (71)2.2078 ( 1) 3.0643 9.7923 5.5356 3.9312

i

36019 19218 85746 80516 38904 03931 19691 25190 73640 10689 86145 21636 72229

944 8 91136 8412 32384 11)7.9412 33705

ii

26 6.3063 23 5.9531 19276 65396

(. 71)2.3235 .

~1

( 1)3. 0675

9.7993 5.5385 3.9329

5.5370 94855 3.9321 09204

29) 5.5018 53574 (71)2. 3835 35733

44320 72330 33566 66899 45467

( 1)3.0692 01851

9.8028 03585 5.5400 37771 3.9337 81020

946 8 94916 8465 90536

945 8 93025 8439 08625

. 47511 85230 98931 43371 83427 I

73809 17990 06367 18334 38512 67504 92199 76015 93351

( 71)1.7075 64573 ( 1)3. 0479 50131

8 70489

937 8 77969

8 76096

940 8 83600 8305 84000 89600 40224 97811 75942 89385 48022 29) 5.3861 51141 (7112.2650 01461 ( 1)3. 0659 41943 9.7958 61087

. . (71)2.5725 ( 1)3.0740 9.8131 5.5444 3.9362

10997 29260 97390 93317 50630 936

935

8 74225 8174 00375 11)7.6426 93506 18428 33731 40538 76403 06437 15018 (71)1.9928 68584 ( 1) 3.0577 76970 9.7784 61652 5.5297 16964 3.9279 17180

9.7539 97922 5.5193 38042 3.9220 18115

8

8 667bi 8069 54491

5 6 9 10 24

(71)1.6639 92748

929 63041 65089 97677 61442 72579 65226 62795 64537 33055

8 8017 (11)7.4483 (14 6.9195 ( 171 6.4282 (20 5.9718 (2315.5478 (26)5.1539 (29)4.7880

947 8 96809 8492 78123

39464 09921 30507 71149 07472 15863

29 5.6197 88134 (71)2.5080 01911

( 1)3.0724 58299

9.8097 36263 5.5429 76005 3.9354 49998

948

949

8 98704

9 00601

8519 71392

8546 70349

.

(71) , . 2.6386 ( 1)3.0757 9.8166 5.5459 3.9371

[ -36) 11

nZ (

(29)5.5605 (71)2.4450 ( 1)3.0708 9.8062 5.5415 3.9346

83331 11300 59156 09574 16151

(71)2.7064 46809 ( 1)3.0773 36511

9.8201 16944 5.5473 74614 3.9379 48170

[(-37) 31

(71)2.7758 ( 1)3.0789 9.8235 5.5488 3.9387

ni[(-37) 21

76218 60864 72299 38494 79487

( 71)2.8470 10693 ( 1)3.0805 84360

9.8270 25224 5.5503 01217 3.9396 10103

62

ELEMENTARY ANALYTICAL METHODS

Table 3.1 k 1 2 3 4 5 6 7

POWERS AND ROOTS

9 8573 (11)8.1450 1 4 7.7378 171 7.3509

950 02500 75000 62500 09375 18906

( 7 1 ) 2.9198 ( 1)3.0822 9.8304 5.5517 3.9404

90243 07001 75725 62784 40019

8

9 10 24 1/2

G 1/5

1 2 3

h

5 6

955 9 12025 8709 83875 11)8.3178 96006

9 8600 (11 8 1794 (1417: 7786 1171 7.3974 20 7.0349 2 3 6.6902 ( 2 6 6 3624 (291 6: 0506 ( 7 1 ) 2.9945 ( 1)3.0838 9.8339 5.5532 3.9412

951 04401 85351 11688 20515 68110 92173 77556 53956 93712 55775 28789 23805 23198 69236

956 9 13936 8737 22816

1 2

24 1/2 1/3 114 1/5 1 2 3 A

5 6 7 8 9 10 24

( 7 1 ) 3.3119 28238

( 7 1 ) 3.3961 69948

( 1)3.0903 07428

( 1)3.0919 24967

9.8476 92005 5.5590 53362 3.9445 79145

9 8847 8.4934 8.1537 7.8275 7.5144 7.2138 6.9253 6.6483 ( 7 1 ) 3.7541 ( 1)3.0983 9.8648 5.5663 3.9487

960 21600 36000 65600 26976 77897 74781 95790 39958 26360 32467 86677 48297 15367 00972

965 9 31225 8986 32125

( 71) 4.2526 09649 ( 1)3.1064 44913

9.8819 45122 5.5735 49061 3.9528 05659

5 6 7 8 9 10 24 112'

( 7 1 ) 3.0710 49109 ( 1)3.0854 49724

9.8373 69469 5.5546 82461 3.9420 97756

9.8511 28046 5.5605 08040 3.9454 04889

9 8875 11 8.5289 1418.1962 17 7.8766 20 7.5694 23 7.2742 26 6.9905 ( 2 9 6.7179 ( 7 1 ) 3.8491 ( 1)3.1000 9.8682 5.5677 3.9495

961 23521 03681 10374 82870 27838 39352 31217 36200 05288 18699 00000 72403 64363 23275

9 33ilb 9014 28696

( 7 1 ) 4.3596 ( 1 ) 3.1080 9.8853 5.5749 3.9536

44069 54054 57396 92425 24554

9711

971

9 40666 9126 73000

9 428ii 9154 98611

(704.8141 ( 1)3.1144 9.8989 5. 5807 3.9568

72219 82300 82992 54698 93368

(71)4.9347 ( 1)3.1160 9.9023 5.5821 3.9577

08664 87290 83537 92482 08886

9 8655 11 8.2484 14 7.8607 1 7 7.4913 20 7.1392 23 6.8036 26 6 4838 2916: 1791 ( 7 1 ) 3.1494 ( 1)3.0870 9.8408 5.5561 3.9429

953 08209 23177 35877 59391 03699 12425 69441 96978 53820 12996 69808 12721 40574 25580

9 8792 11 8.4229 141 8.0691 17 7 7302 2017: 4055 23) 7.0945 26 6 7965 2916: 5111 ( 7 1 ) 3.5708 ( 1)3.0951 9.8579 5.5634 3.9470

958 17764 in12 07597 45478 41368 71230 37239 66675 10874 55021 57508 92945 13977 54307

i

957

9 8764 11 8.3877 1 4 8.0271 17 7.6819 20 7.3516 23 7.0355 26 6.7329 ( 2 9 j 6; 4434 ( 7 1 ) 3.4824 ( 1)3.0935 9.8545 5.5619 3.9462

1 i

7

8 9 10 24 1/2 1/3 114 115

952 9 06304 0628 01408

nk

15849 67493 93908 18770 52663 28698 08664 81792 63575 62966 41660 61691 61578 29943

962 9 25444

963 9 27369

95A

9 ioiid 8682 50664 (11)8.2831 11335 (14)7.9020 88213

( 7 1 ) 3.2296 91146 ( 1)3.0886 89042

9.8442 53565 5.5575 97541 3.9437 52709 959

9 8819 (11 8.4581 (141 8.1113

i968i 74079 31418 48029

26 6 8606 291 6: 5793 ( 71) 3.6613 ( 1)3.0967 9.8614 5.5648 3.9478

84761 96686 94899 72513 91813 65240 77983

964 9 29296

(11

( 7 1 ) 3.9464 ( 1)3.1016 9.8716 5.5692 3.9503

05693 12484 94135 12228 44894

967 9 35089 9042 31063 (11)8.7439 14379

( 7 1 ) 4.4692 ( 1)3.1096 9.8887 5.5764 3.9544

57504 62361 67316 34668 42771

972 9 44784 9183 30048 68067 35361 00771 68349 47635 53501 06003 ( 7 1 ) 5.0581 34323 ( 1)3.1176 91454 9.9057 81747 5.5836 29155 3.9585 23732

( 7 1 ) 4 . 0460 46699

(71)4.1480 96142

( 1)3.1032 24130

( 1 ) 3 . 1 0 4 8 34939

9bR

969 9 38961 9098 53209 11)8.8164 77595

9.8751 13495 5.5706 58964 3.9511 65831 9 37054 9070 39232

( 7 1 ) 4.5815 ( 1 ) 3.1112 9.8921 5.5778 3.9552

09331 691137 i4886 75794 60312

973 9 46729 9211 67317 57994 58129 92259 83968 61601 58138 11068 ( 7 1 ) 5.1845 15371 ( 1)3.1192 94792 9.9091 77627 5.5850 64719 3.9593 37908

9.8785 30490 5.5721 04575 3.9519 86085

( 7 1 ) 4.6964 60232 ( 1)3.1128 76483

9.8955 80110 5.5793 15803 3.9560 77177

974 9 48676 9240 10424 61530 65130 52637 65868 50755 57236 39148 ( 7 1 ) 5 . 3139 19427 ( 1)3.1208 97307 9.9125 71181 5.5864 99178 3.9601 51415

63

ELEMENTARY ANALYTICAL METHODS

POWERS AND ROOTS nk

Table 3.1

k

975 9 50625 9268 59375

1

2 3 4 5 6 7 8 9 10 24

26 7 9623 291 7: 7632 ( 71) 5.4464 ( 1) 3.1224 9.9159 5.5879 3.9609

55086 96209 15584 98999 62413 32533 64254

9

mon

94ii 11)9.2236 14)9.0392 17 8 8584 2018: 6812

81600 07968 23809 55332

1 2

24 1/2 113 1/4 115

I

(71)5.5820 ( 1)3.1240 9.9193 5.5893 3.9617

~1

03365 95168 83884 82813 18474

70383 99154 22218 25992 98776 9R3

9 662i9 9498 62087

29) 8.0877 (71)6.0087 ( 1)3.1288 9.9295 5.5936 3.9642

35096 56477 97569 04202 54950 08956

984 9 68256 9527 63904

(71)6. 3103 89657

(71)6.4665 95666

(71)6.6265 03443

(71)6.7901 96812

( 1)3.1336 87923

( 1) 3.1352 83081

( 1) 3.1368 77428

986 9 72196

987 9 74169 9615 04803

988 9 76144 9644 30272

989 9 78121 9673 61669

9.9362 61267 5.5965 09584 3.9658 27331

985 9 70225

(71)6.9577 ( 1)3.1384 9.9497 5.6022 3.9690

a

( 71)7.8567 ( 1) 3.1464

9.9665 5.6093 3.9730

61406 70965 47896 05785 56179

(71)7.1292 ( 1)3.1400 9.9531 5.6036 3.9698

00499 01494 53479 46944 72475 20750 81408 26545 54934 01690 77521

20)9.3867 57134

(71)8. 0494 77813 ( 1)3.1480 15248

9.9699 09547 5.6107 17644 3.9738 79839 9 92oib 9880 47936

74075 95006 87531 25094

(71)8.8665 ( 1)3.1543 9.9833 5.6163 3.9770

35105 62059 05478 70767 82648 I

9.9396 36356 5.5979 35265 3.9666 35529

(29)9.6071 (71)9.0828 ( 1)3.1559 9.9866 5.6177 3.9778

[(-36)

11

23735 91413 46768 48849 81384 81740

9.9430 09155 5.5993 59857 3.9674 43069

9.9463 79667 5.6007 83363 3.9682 49952

(71)7.3048 56083

(71)7.4845 66822

(7U7.6685 10178

( 1)3.1416 55614

( 1)3.1432 46729

( 1)3.1448 37039

9.9564 77521 5.6050 47381 3.9706 66671

992 9 84064 9761 91488 19561 49004 98212 62227 36129 15040 94120 (71)8.2466 98779 ( 1) 3.1496 03150 9.9732 61904 5.6121 32527 3.9746 81509

9.9598 38925 5.6064 66560 3.9714 70939

994

(71)8.4485 ( 1) 3.1511 9.9766 5.6135 3.9754

45822 90251 12009 46340 82534 998

9 9AlMA

0, O,-"", Ann9

9910 26973 ( 11)9.8805 38921 ( 14)9.8508 97304

9940

iiSSi

( 71) 9.3043 02025 ( 1)3.1575 30681

(71)9.5308 ( 1)3.1591 9.9933 5.6205 3.9794

79767 13800 28884 99434 78001

[(-37)

9.9899 89983 5.6191 90939 3.9786 80191

31

9.9631 98061 5.6078 84662 3.9722 74555 9 woid 9821 07784

9 9791

997

991.

995

9850 (11)9. 8014 (14 9 7524 1719: 7037

i.

84708 63694 13846 27123 61752

991 9 82081 9732 42271

9 90025

115

982 9 64324 9469 66168

(71)5.8631 ( 1) 3.1272 9.9261 5.5922 3.9633

( 1) 3.1320 91953

990

'I3

979 9 58441 9383 13739

~I

(71)6.1578 ( 1) 3.1304 9.9328 5.5950 3.9650

9

5 6 7 8 9 10 24 1/2

74443 99870 51328 64785 76427

981 9 62361

~iooo

9702

9 10 24

97R

9 56484 9354 41352 ( 11) 9.1486 16423

[

980

4 5 6 7 8 9 10 24 1/2 1/3 114 1/5

977 9 54529 9325 74833 (11)9.1112 56118 (14)8.9016 97228 (17)8.6969 58191 (20 8.4969 28153 231 8.3014 98806 26 8.1105 64333 (2917.9240 21353 (71)5.7209 68141 ( 1)3.1256 99922 9.9227 37928 5,5907 95938 3.9625 87934

976 9 52576 9297 i4i76

ni [(-g11

1

(71)8.6551 22630 ( 1) 3.1527 76554

9.9799 59866 5.6149 59086 3.9762 82913 999 9 98001 9970 0299s

( 71)9.7627 39866 ( 1)3.1606 96126

9.9966 65555 5.6220 06871 3.9802 75173

4. Elementary Transcendental Functions Logarithmic, Exponential, Circular and Hyperbolic Functions RUTHZUCKER

Contents Mathematical Properties , . . . . . 4.1. Logarithmic Function , . , , 4.2. Exponential Function , . . . 4.3. Circular Functions . . . , , 4.4. Inverse Circular Functions . . 4.5. Hyperbolic Functions , , . . 4.6. Inverse Hyperbolic Functions

. . , , . . . . . . . . . . . . . . . . , , , , . , . . . . . . , . . . . . . . . . . , . . , ,

Numerical Methods . . , . . , . . . 4.7. Use and Extension of the Tables References

,

. . . . . . .

,

. . . .

Table 4.1. Common Logarithms (100 log,, 5 , ~=100(1)1350, 10D

,

. . .

. . , . . . . . . . . . , .

.

Page I

. . . . . . . . . .

67 67 69

71

. .

. . . . . , . , . . . .. . . . . . . . . . . , . . . . .

89 89

.

,

.

. . . . . .

. .

93

.

95

,

.

,

5 1350) . . . . . . .

,

, ,

,

Table 4.3. Radix Table of Natural Logarithms In (l+z), --In (1-z), r=lO-"(lO-n)lO-n+l,

,

,

1z=O(.OO1)1, 18D, x=0(.1)5, ~=5(.1)10, 12D, -z=O(.I)IO, ~ z = O ( l ) l O O , 19s

'

.

,

,

. . .

,

.

100

,

. . .

.

. . .

.

.

114

. . . . . . . . . . .

116

. . . ,

n=lO(-l)l,

Table 4.4. Exponential Function (0 5 1x1 5 100) .

25D

15D 20D

Table 4.5. Radix Table of the Exponential Function ez, e-r,

,

. , . . . . , . . . . . . . . . . . . . .

79 83 86

Table 4.2. Natural Logarithms (0 5 ~ 5 2 ~ 1. ). . In z, s=0(.001)2.1, 16D

ez,

,

~=10-~(10-")10-"+~,n=lO(-l)l,

,

. . . . . . . .

140

.

142

. . . . . .

174

25D

Table 4.6. Circular Sines and Cosines for Radian Arguments (0
Table 4.8. Circular Sines and Cosines for Large Radian Arguments

.

.

.

( O ~ z ~ 1 0 0 0 .) . . . . . . . . , . . . . sin 2, cos 2, z=0(1)100, 23D, z=100(1)1000,

1

. .. . . . . . ,

175

8D

National Bureau of Standards.

66

ELEMENTARY TRANSCENDENTAL FUNCTIONS

66

Table 4.9. Circular Tangents, Cotankents, Secants and Cosecants for Page Radian Arguments (0 I z 21.6) . . . . . . . . . . . . . . . . . . 186 ttin z, cot z, sec 5 , csc r, z=0(.01)1.6, 7 to 9D z-'-cot 2 , csc x-x-', ~=0(.01).5, 8D Table 4.10. Circular Sines and Cosines to Tenths of a Degree (Oo 28590") sin 8, cos 8, 8=Oo(.lo)900, 15D

,

189

Table 4.11. Circular Tangents, Cotangents, Secants and Cosecants to Five Tenths of a Degree (O"I8290") . . . . . . . . . . . . . . 198 tan 8, cot 8, 8=0°(.50)900, 15D sec 8, csc 8, 8=0°(.50)900, 8D Table 4.12. Circular Functions for the Argument f z ir

ir

ir

2

2

2

sin -2, cos -2, tan

-2,

7r

7r

?r

2

2

2

cot -2, sec -2, csc -2,

Table 4.13. Harmonic Analysis. . . . . 21rr 2nr sin -t cos -I r=l(l)[s/2], s even S

(OSz 2 1) . . . .

200

z=O(.O1)1, 20D

. . . . . . . . . . . . . .

202

. . . . . .

203

S

r=l(l)[(s-l)/2], s=3(1)25, 10D

s odd

Table 4.14. Inverse Circular Sines and Tangents ( O I z < l ) . arcsin z, arctan z, z=O(.OOl)l, 12D f(z)=[2(1-z)]-i[3n-arcsin z], z=.95(.001)1, 12D

Table 4.15. Hyperbolic Functions (0 2 z 5 10) . . . . . . . . sinh 2 , cosh z, z=0(.01)2, 9D, 2=2(.1)10, 9D tanh z, coth z, 2=0(.01)2, 8D, 7D, 2=2(.1)10, 10D

. . . .

21 3

Table 4.16. Exponential and Hyperbolic Functions for the Argument 7rx (05221). . . . . . . . . . . . . . . . . . . . . . . . . . ,219 e*rz, sinh T X , cosh m, tanh 7rz, z=O(.Ol)l, 10D Table 4.17. Inverse Hyperbolic Functions (0 2 ~ s) .~. ). . arcsinh z, arctanh z, z=O(.O1)1, 9D arcsinh 2, (arccosh x)/(zz-l)+, z= 1(.01)2, 9D, 8D arcsinh x-ln z, arccosh z-ln z, ~ - ~ = . 5 ( - - . 0 1 ) 0 , 10D

. . . . .

221

7D

. . . . . .

223

l<X<m)

. . . . .

224

. . . . . . . . . . .

225

Table 4.18. Roots z, of cos z, cosh zn= & 1, n=1(1)5, Table4.19. Roots z, of tan z n = h n --X=0(.05)1, n=1(1)9, 5D X-'=-l(.O5)1, n=1(1)9, 5D

(-a

<X
Table 4.20. Roots 2 , of cot z n = h , . (O<X<m) X=0(.05)1, n=1(1)9, 5D X-'=1(-.05)0, n=1(1)9, 5D

The author acknowledges the assistance of Lois K. Cherwinski and Elizabeth F.Godefroy in the preparation and checking of the tables.

4. Elementary Transcendental Functions Logarithmic, Exponential, Circular and Hyperbolic Functions Mathematical Properties 4.1. Logarithmic Function

I

Logarithmic Identities

I I

Ln (zlzz)=Ln zl+Ln z z .

4.1.6

Integral Representation

(i.e., every value of Ln ( z l z z )is one of the values of Ln zl+Ln z z . )

4.1.1

4.1.7

In (zlzz)=ln zl+ln zz (--n<arg zl+arg z d a )

4.1.8

Z Ln J=Ln z,-Ln zz

4.1.9

In a = l n zl-ln z z

zz

zz

(-a<arg

I f.

where the path of integration does not pass through the origin or cross the negative real axis. In z is a single-valued function, regular in the z-plane cut along the negative real axis, real when z is positive.

In z=ln r+i8

4.1.3 r=(s2+y2)i,

X=T

In 1=0

4.1.12

(-a<ela).

(n integer)

(see chapter 1)

Special Values

z = x +iy =reto.

zzsa)

In x"=nln z (n integer, -a
4.1.11

( a not an integer or zero.)

4.1.2

Ln zn=n Ln z

4.1.10

FIGURE4.1. Branch c u t j o r In z and

2,-arg

4.1.13

In 0=-

4.1.14

In (- 1)=ai

4.1.15

In (hi)=&+&

4.1.16 In e=1,

e is the real number such that

cos 8, y=r sin 8, @=arctanY

=2.71828 18284.. .

-a

2

n-t

m

(see 4.2.21)

The general logarithmic function is the manyvalued function Ln z defined by

Logarithms to General Barn

4.1.18

log, z=ln z/ln a

4.1.4 ~

4.1.19

where the path does not pass through the origin. 4.1.5 Ln (re'B)=ln (refo)+2kA=1n r+i(e+2ka),

k being an arbitrary integer. In z is said to be the

I

4.1.20 4.1.21

log, z=ln z

491.22 log,, Z=ln Z/ln 1O=lOgio e In z

68

ELEMENTARY TRANSCENDENTAL FUNCTIONS

4.1.23 In z=ln 10 log,, z= (2.30258 50929 . . .) log,, z

(log, z=ln z, called natural, Napieriun, or hyperbolic logarithms; log,, z, called common or Briggs logarithms.) Series Expansions

In

4.1.24

(l+2)=~-~2~++2~-.

,

4.1.35 4.1.36 111 t _ < z - l 4.1.37 In x l n ( x l / n - l ) for any positive n

.

(X>O)

(X>O)

( l z J I 1and z#-1)

4.1.38

/In (l+z)lI--In

4.1.25

(1-IzO

(lzl
Continued Fractions

4.1.39

2 3)

(38

4.1.26 In

2=(2-1)-3(2-1)~++(2-1)3-.

,

(l2-1lS1,

( z in the plane cut from

. ZZO)

In(*)=----.

4.1.27

22 z2 4z2 9z2 1-3-5-7-

1-2

J+,(

In 2=2 [ (%)+j

1 z

(my+.. .]

.,

( z in the cut plane of Figure 4.7.)

1 2-1

1

- 1 to - w )

4.1.40

Polynomial Approximations 2

4.1.28

(:+$+@+. 1 .)

In (“+‘)=2 2-1

4.1.41

,

+

logl, x=alt +a3t3 ~ ( zI ) t= (z- l)/(z+ 1) 4.1.29

IE (2) I 5 6X 10-4 ai=.86304

a3=.36415

4.1.42

log,, ~ = a ~ ~ + a ~ t ~ + a ~ ~ ~ + a ~ t ~ + u ~ ~ ~ + ~ ( z ) t= (x-l)/(Z+l)

Limiting Values

lim x - a In z=O

4.1.30

z-m

(aconstant,

1€(2)(510-’

3a>O)

lim za In x=O

4.1.31

z+o

(aconstant,

9a>O)

11

.

Inequalities

1+x

(l-z)<

O~X,
In (1 +z) = a l z + a 2 x ~ + a 3 ~ 3 + a 4 z 4 + a ~ z 6 + ~ ( ~ ) lt(z)lIlXlO-~ i

.99949 = 556

az= - ,49190 896

-
x<-In

~g=.191337714

~

X

(x>-1,

4.1.34

U3=.28933 5524

4.1.43

)

m = y (Euler’s constant) =.57721 56649. . (see chapters 1, 6 and 23)

4.1.33

a,= .09437 6476

a6= ,17752 2071

4.1.32

% ( Z F l

a,= .86859 1718

ZZO)

X l-x

@
ZZO)

a4= - .13606 275

a,=.03215 845

aa= .28947 478 2 The approximations 4.1.41 t o 4.1.44 ere from C.Hastings, Jr., Approximations for digital computers. Princeton Univ. Press, Princetoil, N.J., 1955 (with permission).

69

ELEMENTARY TRANSCENDENTAL FUNCTIONS

4.1.44

Olxll

In ( 1 + z ) = a l r + a ~ r ~ + ~ ~ 3 + a ~ r 4 + a 5 r 5 + a ~ r 6 +a,z7+a939+e(2)

le(z>lS3 x 10-8 .99999 64239 a 2 = - . 49987 41238 %= .33179 90258 a4=-. 24073 38084 uI=

~

5

=

. 16765 40711

a6=-. 09532 93897 ~

7

=

Us=-.

.03608 84937 00645 35442

Approximation in Terms of Chebyshev Polynomials a

4.1.45

O l X S l

Tn*(z)=cosn8, cos b 2 ~ - 1 (see chapter 22) n-0

4.1.56

n

An

n 0

.37645 2813 1 .34314 5750 2 -. 02943 7252 3 .00336 7089 4 -. 00043 3276 5 .00005 9471

An -. 00000 8503 .OOOOO 1250 -. 00000 0188 ,00000 0029

6 7 8 9 10 -. 00000 0004 11 ,00000 0001

4.1.47

l'

fo'

dt=-r2/12

&=Zi(z)

(see 5.1.3)

4.2. Exponential Function Series Expansion

z 22 l! 2!

z3

e'=exp z=1+-+-+-+

d 1 z=zln 2 d" -In d z"

4.1.57

dt =-r2/6

4.2.1

Differentiation Formulas

4.1.46

4.1.55

A,T,*(z)

In ( i + z ) = g

so'

Definite Integrals

3!

.- .

(z=z+iy)

where e is the real number defined in 4.1.16 Fundamental Properties

z=(-l)n-l(n-l)!z-n 4.2.2

Ln (exp z)=z+2kri

4.2.3

In (exp z ) = z

(k any integer)

Integration Formulas

J %=In

4.1.48 4.1.49

z

S

In z dz=z In z - z

4.2.4

exp (In z ) =exp (Ln z ) =z

4.2.5

- exp z=exp z

4.1.50

S

4.2.6

If N=ae, then z=Log, N

4.2.7

a'=exp ( z In a)

4.2.8

4.1.51

(In z)" dz=

d dz

Definition of General Powers

(n# - 1, n integer)

Z"

(-r<$zIn)

"+'(In '1" n+l

n+l

Jzn(ln z)J?l-ldz

4.2.9 4.2.10

(n#-1)

If a=laJ exp (i arg a) (-r<arg

a<*)

la*l=lalze-Yarga

arg (a')=y In lal+z arg a

4.2.11

Ln a'=z In a for one of the values of Ln a' The approximation 4.1.45 is from C. W. Clenshaw, Polynomial approximations to elementary functions, Math. Tables Aids Comp. 8, 143-147 (1964) (with permission). 3

4.2.12 4.2.13

In az=z In a

(a real and positive)

le'l=ez

4.2.14

arg (ez) =y

4.2.15

GZIQ,22 =a 21+22

4.2.16

a%'= ( ~ b ) ~

Special Values (see chapter 1)

(--n
Y

1

4.2.22

e=2.71828 1 8 2 8 4 . .

4.2.23

eo= 1

4.2.24

em=

4.2.25

e-"=O

4.2.26

e*t"i=

a0

-1

*t*'

4.2.27

e

4.2.28

e2*kI

2 = f i

(k any integer)

=1

Exponential Inequalities

If z is real and different from zero

--

X

4.2.29

e l-x
4.2.30

eZ>l+x

(x< )

1

FIGURE 4.2. Logarithmic and exponentialfunctions.

4-2-31

Periodic Property

4.2.32

4.2.17

er+2nkt

=e z

(k any integer)

4.2.33

eZ<- 1-x 2

-<(l-e-")
x<

(ex- 1)

(x>-1)

<-1-x

Exponential Identities

4.2.18 4.2.19

=ezlz2

(- T < Y z2~T )

The restriction (-a
4.2.20

lim zue-'=O

(larg zI < + x - e < g ~ ,

121-'

4.2.21 m+

-

4.2.40

(For e,(z) see 6.5.11)

(x<1)

2

4.2.34

e'le'z=e'~+za (e'l)za

(x<1)

402.35

l+z>e'+z

(x>-I)

(n>o,

eZ>1+$

Z>O)

ZII

I

(x>O, y>O)

'>e=

4.2.37

e-z
4.2.38

I,1 zl
a constant)

4.2.39

X

2 1<\I,

(O<sS 1.5936) 7

zI

(O
~ e ' - l ~ _ < e i ' i - l ~ ( z ( e ' * ~(all z )

71

ELEMENTARY TRANSCENDENTAL FUNCTION8

I

4.2.42

2a

1

2-1+---

e2a 'cm

Z-a+

u2+1 u2+4 u2+9 3 Z + 5Z+ 7Z+

4.2.48 * ' *

( z in the cut plane of Figure 4.4.) Polynomial Approximations4

O l z l l n 2=.693.

4.2.43

le(z) I 5 3 x 10-3

u2=.3536

OlxSln 2

4.2.44

+e

eV2= 1+ulz+u2$ +u3x3+u4z4

ez=xA,TE(x) m

n=O

&=--.I5953 32 ,0293641

d

4.2.51 (2)

4.2.52

Ub=-.OO83O 13598 .00132 98820 u7= - .00014 13161

uB=

Integration Formulae

e"dz =ea2Ja

7 x 10-4 u3=.20800 30 ar=.1268089

4.2.55

eaz znea*dz=a,+l [(uz)"-n(az)n-'+n(n-l)(uz)"-2

S

+

,

. . +(-l)"-ln!(uz)+(-l)"n!]

(n20)

4.2.56

05x51

4.2.47

+

+

1Oz= (1 alz+ u2x2 u3z3+a4x4+a5;cb

+ +a7z7)2+ a6Z6

6 (2)

(See chapters 5, 7 and 29 for other integrals involving exponential functions.) 4.3. Circular Functions

1e(2)1<5X10-8 ~ l = l 15129 . 277603 ~ 2 =

.66273 088429

~ a = .25439 357484 ~ 4 =

~ 5 = .

uB=. ~ 7 = .

01742 111988 00255 491796 00093 264267

Definitions

4.3.1

.07295 173666

~-

The a proximations 4.2.43 to 4.2.45 are from B. Carlson, M. &oldstein, Rational approximation of functions, Los Alamos Scientific Lahoratory LA-1943, Los Alanios, N. Mex., 1955 (with permission). 6 The Bpproximations 4.2.46 to 4.2.47 are from C. Hastings, Jr., Approximations for digital computers. Princeton Univ. Press, Princeton, N.J., 1955 (with permission). 4

S

4.2.54

10z= (1+ a 1 2 + u 2 ~ 2 + u 3 ~ + u 4 ~ 4 ) 2 + ~ ( ~ )

ul=1.14991 96 a2= .6774323

eg=ez

d - z2= (1 +In z)zl dz

4.2.53

ogx11 le(z) I

An .64503 5270 -. 31284 1606 ,038704116 -. 00320 8683 .00019 9919 -. 00000 9975 .OOOOO 0415 -. 00000 0015

d" ,pz=anp dz" d - a'=uz In a dz d p=aza--l dz

4.2.50

~ 4 =

Ie (2)I p 2 x 10-'O

4.2.46

n

0 1 2 3 4 5 6 7

4.2.49

+a6x6+a7s7+

~1=--.9999999995 ,49999 99206 a3= - ,16666 53019 U(= ,04165 73475

n-0

An 1. 75338 7654 ,85039 1654 . 10520 8694 .00872 2105 .00054 3437 .00002 7115 . 00000 1128 . 00000 0040 . 00000 0001

e -*= 1+ ~ ~ x + a ~ x ~ + u ~ ~ + a ~ x ~ + a ~ ~ ~

~ 2 =

m

e - z = C A,T$(z)

Differentiation Formulas

O
4.2.45

I

(2)

(e(z)l53XlO-b ~1=--.9998684 .4982926

Olxll

TE(z)=cos ne, cos e = 2 ~ - 1 (see chapter 22)

I

n 0 1 2 3 4 5 6 7 8

..

e-z=l +u~z+u2z2++(z)

al=-.9664

Approximations in Terms of Chebyshev Polynomials 8

sin z=

-

e"-

I2

(2= 2

2i

+iy)

eLz+e-'c

4.3.2

cos z=

2

6 The approximations 4.2.48 are from C. W. Clenshaw, Polynomial approximations to elementary functions, Math. Tables Aids Comp. 8, 143-147 (1954) (with permission).

72 4.3.3

ELEMENTARY TRANSCENDENTAL FUNCTIONS

4.3.4

csc z=-

4.3.5

sec z=-

4.3.6

sin z cos z

tan z=-

1

4.3.17

cos (zl+z2)=cos z1 cos z2-sin z1 sin z2

4.3.18

tan zl+tan 22 tan (zl4-z2)=1 -tan z1 tan z2

sin z 1 cos z

Half-Angle Formulas

1

cot z=-

tan z

Periodic Properties

4.3.7

(k any integer)

sin (z+2kr)=sin z

4.3.8

cos (z+2kr)=cos

4.3.9

tan (z+kr)=tan z

2

Z 1-cos z t-1-cos zsin z 4.3.22 tan-=& ---sin -z 2 (l+cos z ) l+cos z

-

The ambiguity in sign may be resolved with the aid of a diagram.

Y

Transformation of Trigonometric Integrals

If tan

i=z

then

4.3.23 sin U=-

+22

1

22’

cosu=-

1-22

1+22’

au=-

l+ZZ

dz

Multiple-Angle Formulas

2 tan z 1 tan2z

+

4.3.24

sin 22=2 sin z cos z=

4.3.25

cos22=2 cos2z-1=1-2

sinZz 1-tan2 z =cos2 z-sin2 z= 1 tan2z

+

sinx __---cos x

-CSC x ........

tanx

cot x

sec x

FIQURE 4.3. Circular junctions. Relations Between Circular Functions

4.3.10

sin2 z+cos2 z = 1

4.3.11

sec2 z-tan2 z= 1

4.3.12

csc2 z-cot2 z = 1

4.3.27

sin 32=3 sin 2-4 sin3z

4.3.28

COS

4.3.29

sin 42=8 cos3 z sin

4.3.30

COS

3~=-3

4z=8

COS 2+4

COS* 2-8

cos3 Z

2-4

cos z sin z

cos2 z + l

Producte of Sines and Cosines

4.3.31

2 sin z1 sin zz=cos (z1-z2)-cos

(z1-l-z~)

Negative Angle Formulas

4.3.13

sin (--)=-sin

4.3.14

cos (-z)=cos

4.3.15

tan (--)=-tan

z

4.3.32 2 COS 21 COS Z ~ = C O S(Zl-ZJ+COS 4.3.33 2 sin z1 cos z2=sin (zl-z2)+sin

z z

Addition Formulas

4.3.16 sin (zl+z2)=sin z1 cos z2+cos z1 sin zz

(zI+ZZ)

(zl+zJ

Addition and Subtraction of Two Circular Functions

73

ELEMENTARY TRANSCENDENTAL FUNCTIONS

I

4.3.35

sin zl-sin zn=2 cos(?) 4.3.36

cos z'fcos

22=2

cos

4.3.41

-Z?7 sin ( )cos2 z1-cos2 z,=-sin 21

(

) cos (y) 4.3.42cos2 zl-sin2

Zl+Z2

(zl+zz> sin (zl-z2)

z2=cos (zl+z2) cos ( ~ 1 - 2 ~ )

4.3.37

cos zl-cos z2=-2

s i na (+ z72 ) sin(?) 4.3.43

Signs of the Circular Functions in the Four Quadrants

4.3.38

sin (zl fz 2 ) tan zlf tan z2= cos 2' cos 22

Quadrant

4.3.39

cot

Z l i C C O t z2=

sin ( z 2 k z 1 ) sin zl sin zz

Relations Between Squares of Sines and Cosines

4.3.40

i

sin2 zl-sin2 z2=sin (zl+z2) sin (zl-zz)

?T

-e

Fsin 0 icot e +sec e FCSC

- 2i e

2k7rke

e e e e ~ C S C e

e

COS

COS

-

31r

Tke

-sin 8

e -tan 8 -CSC e sec e --cot e 4.3.45

+e

-il + +

I I1 I11 IV

isin 8 +cos 8 ftan 8

-COS

isin wot -sec

e

Ftan 0

e

~ C S C

+sec e &cot e

'Ftan 0

Relations Between Circular (or Inverse Circular) Functions cos x=a

sin x=a

I

tanx=a

I

csc x=a

cot x=a

sec x=a

a

+ ( 1 +a') - 4

U- '( 1 -aZ)4

a

(a'- l ) - t

(a2- I)+

a-'

( 1 -a2)-*

a-'(l+a2),

a

a ( d - I)-*

(1+a2)*

a-'

(1+a2)4

a(a2- I)-+

a

a-l(1 +a2)'

a( 1 -a2)-*

a-'

(a2- 1)'

(a2- I ) - +

a

(1-a2)*

a(1 d )- f

If sin x=a, cot x=a-'(l-u2)+ (O<x<- -3 Illustration: arcsec a=arccot (a2- 1)-+

a-'

a-'(a2- 1)'

( 1 +a2) - $

a-'(aZ- I ) *

a-'

a(1+a2)-*

I

74

ELEMENTARY TRANSCENDENTAL FUNCTIONS

4.3.46

Euler's Formula

Circular Functions for Certain Angles 4.3.47

ez=ez+iv-

-e

(cos y + i sin y)

De Moivre's Theorem

1-

-I

-1-1

4.3.48

(cos z+i sin

vz+i sin

Z)"=COS

(- a < 9 z

YZ

5 a unless Y is an integer)

Relation to Hyperbolic Functions (see 4.5.7 to 4.5.12)

1

51/12 75'

tan

2+&

csc

dZ(J3-1)

l$l Q)

1

JZ(&+l)

cot

Q)

7a/12 105'

3a/4 135'

sin z= -i sinh iz

4.3.50

COS

4.3.51

tan z=-i

4.3.52

csc z=i csch iz

4.3.53

sec z=sech iz

4.3.54

cot z = i coth iz

Z=cosh iz tanh iz

Circular Functions i n Terms of Real andImaginary Parts

4.3.55

sin z=sin x cosh y + i cos x sinh y

4.3.56

cos z=cos x cosh y--i sin x sinh y

4.3.57

sin 2x+i sinh 2y tan z= cos 2x+cosh 2y

4.3.58

sin 22-i sinh 2y cot z= cosh ~ Y - C O S 22

--(2+4)

-4

Modulus and Phase (Argument) of Circular Functions

JZ(&-1)

24313

4.3.59

-@(J3+1)

-2

42-4)

-&3

0

2-&

1

2113 120'

4.3.49

5716 150'

111112 165'

Isin zj=(sin2 x+sinh2 y)i =[$ (cosh

arg sin z=arctan (cot x tanh y)

4.3.61

Icos zJ=(cos2 z+sinh2 y)' =[$ (cosh

4.3.62

~ Y + C O S 22)]*

arg cos z=-arctan

(tan x tanh y)

4.3.63

~ Y - C C O S 2s)) I tan z 1 =(cosh cosh 2y+cos 22

4.3.64

arg tan z=arctan

c223

Series Expansions

tan

-1

-4/3

-(2--J3)

0

4.3.65

sin z=z--+---+ 3! 51 71

z6 z7

...

z4 2' 21 4! 6!

...

23

csc

Jz

2

Jz(&+1)

sec

-&

-2413

-4(8-1) -1

COt

-1

-&

-(2+J3)

OD

4.3.66 '

Q)

2~)]*

4.3.60

a

180'

~Y-COS

22

cos z=l--+---+

75

ELEMENTARY TRANSCENDENTAL FUNCTIONS

I

4.3.67 23

tan z=z+-+-+-+ 3

+

225 1727 15 315

Z2n

-

->sin x

4.3.79

, , ,

( -1)n- 12y22n- 1)Bz, (2n) !

Inequalities

+

...

X

B 2.

(-;<.<;)

(lz'<%)4.3.80

sin x 5 x S t a n x

4.3.81

sin x cos XS-< 1

(01x17)

4.3.82

sin T X B
(O<X<1)

(01x2;)

4.3.68 7

1

2

z

6 360

csc z=-+-+-

z3+-

31 z6+ 15120

...

1)B2, + (- 1)"- 12(22%-1(2n) ! +... 2n-

z

(IZJ
4.3.69 z2 5 z 4 61ze sec z=1+-+-+-+

2

24

.. .

720

x -

4.3.83

lsinh y(1Isin zl Scosh y

4.3.84

J s i n h y j I l c o sz(
4.3.85

Jcsc zI Icschly)

4.3.86

(cos z 12cosh(zI

4.3.87

/sin zI SsinhIzI

(lzI<;)

4.3.88

6

Jcoszl< 2,

/sin zl<- / z / -5

(lzl<1)

Infinite Products

4.3.89

s i n z = z II k-1 "

4.3.90

cos

2

4 k=l

(1-- kri2)

(1-

(2k-1)2~2

Expansion in Partial Fractions

I

4.3.73

1 cot z=;+2z

4.3.91

" 1 E k = l zZ-k2r2

(z#O,f*,k27r, "

(lzl<$T)

4.3.92

where B, and E,, are the Bernoulli and Euler numbers (see chapter 23). sin x liin -=I z-to

4.3.75

tan xlim --I z-to

4.3.76 4.3.77 4.3.78

1 csc z=-+22

4.3.93

Z

I I

2

x

k=-m

1 (z-kn)'

(z#O,k?r,k27r,

Limiting Values

4.3.74

csc2 z=

"

(-1)k Z2-k2?r2

(z#O,&a,k2a,. Continued Fractions

X

n

X

lim n tan -=z n+ m n X

lini cos -=1 n-tm

12

4.3.95

tan az=-

a tan z (l-az) tan2 z 1+

3 f

. . .)

~

k-1

lirn n sin -=x n-t m

. . .)

(&-a9

tan2 z

5 f

. .)

76

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Polynomial Approximations 7

4.3.101

012
o
4.3.96

2

sin --x X

- 1+a222 +a4z4

+e (x)

1 c (2)1 12 x 10 - 4 a2=-.

a4= .00761

16605

4.3.97

lc(5)/12xl0-8

01x1;

sin --x - 1+a2~*+a4z4+a6z6+a8x8+aIox10+ e(z>

a2= .33333 14036

as= .02456 50893

~4=.1333923995

~10=.0029005250

aa= .05337 40603

~12=.0095168091

4.3.102

01x5;

2

*

1€(2)/12Xlo-Q UZ=

- -16666 66664

a4=

1 € (2)I5SX 10-6

- ,00000 00239

u~O=

o<x<;

4.3.103

0125; cos 2 = 1+a2x2+a~z'+

e (2)

z cot z= 1+a2~~+a4x4+agz6+aszg+a,ox10+ e (2)

/e(x)/14X10-'0

Ie(x)l19X 10-4

a2 = - .49670

a4= .03705

0125;

4.3.99

a?= - .33333 33410

- .00020 78504

- .02222 20287

ale=-.00002 62619

a,=-.00211

cos z= 1+a& +a4x' +a& +aszs+alozlO+E @ )

77168

Approximations in Terms of Chebyshev Polynomials *

4.3.104

Jt(x)112X10-9

a2=- ,49999 99963

as=

.00002 47609

.04166 66418

ale=

- .OOOOO 02605

a,=

a4=- ,024369

a2= - .332867

aa= - ,00019 84090

4.3.98

+

x= 1 a2~2+a,x4+t (2)

.OOOOO 27526

as=

,00833 33315

z cot

-11x11

1

(see chapter 22)

T ~ ( x ) = c o s cos8=2z-1 ~~,

a8= - .00138 88397 An

n

4.3.100

O l X C i

0

-4

1.27627 8962

1 -.28526

x

ls(z) 1 s 1 10-3 ad= .31755

a(= ,20330

7 The a proximations 4.3.96 to 4.3.103 are from B. Carlson, M. &oldstein, Rational approximation of functions, Los Alamos Scientific Laboratory LA-1943, Los Alamos, N. Mex., 1955 (with permission).

1569

n 0

An .47200 1216

1 -.49940 3258

2

.00911 8016

2

.02799 2080

3

-,00013 6587

3

-.00059 6695

4 5

.OOOOO 1185 -.OOOOO 0007

4 5

.OOOOO 6704 -,OOOOO 0047

8 The spproxilnations 4.3.104 are from C.W, clenshslv, Polyliomial approsiniations to elementary functlons, Math. Tablvs Aids Comp. 8, 143-147 (1954) (with permission).

77

ELEMENTARY TRANSCENDENTAL FUNCTIONS

I

Differentiation Formulas

4.3.122 -2

d

4.3.105

dz --

4.3.106

d -cos dz

4.3.107

d - tan z=sec2 z dz

4.3.108

- csc z=-csc

4.3.109

d - sec z=sec z tan z dz

4.3.110

d cot z=-csc2 dz

sin z=cos z z=-sin

d dz

sin" z-(n-1) z

I

2

sinn-l z

- (n-I)

1 -, (n-2) sin"+ z

4.3.123

I1

cot z

cos z

Jz" cos zdz=zn sin z-n

S

zn-l

sin zdz

4.3.124

Y y d Z=

I

-(n-1)P-l cos z

s4

4.3.125

cos2 2

- L s c z d z n-1 zn-l

(n>l)

dz=z tan z+ln cos z

z

(n>2)

j I

I

1

Integration Formulas

4.3.127

S

sinmz

S S

sin zdz=-cosz

4.3.113 4.3.114

COS"

z dz=

z

COS"-'

z

m+n + H S s i n " z cos"-2 z dz

cos zdz=sinz

s?an z dz= -1n cos z=ln sec z

4.3.115 4.3.116

S

Z

csc zdz=ln tan --ln 2-

1 1-cos2 (csc z-cot z)=- 2 ln- l+cos z

(m#-n) 4.3.128

4.3.117 7 r z

s s e c z dz =In (sec z +tan z ) =In tan (-+-)=gd-' 4 2

(z)

=Inverse Gudermannian Function gd 2=2 arctan ez-4.3.118

7r

2

S c o t zdz=ln sin z=-ln

-

csc z

-1 (m-1) sinm-l z cos"-' z

4.3.119

S

znsinzdz=-z" cos z+n

S

2"-'

cos zdz

4.3.120 I

4.3.121

S sin2 l dz=- z

cot z+ln sin z

4.3.129

S

tan" z dz=-

tann-' z n- 1

z d z (n # 1)

78

ELEMENTARY TRANSCENDENTAL FUNCTIONS

4.3.131

sa+,":in

z

=-

2

=-

1

a tan ( d + b

(a2-b2)*

(a2-b2)t

?r

l s i n 2nt d t = l c o s 2 nt dt=- 2

4.3.141

>

(a2 b2) (n an integer, n#O)

[

a tan (i)+b-(b2-llli]

4.3.142

sin mt

7r

dt=g

(b2-a2)f In a t'an (i)+b+(b2-a2)i

cos at-cos bt

4.3.143 4.3.133

sa+f!os z

arctan (a2-b2)*

(a-b) tan 2 (a2-b2)1 (a2>b2)

(b-a) tan -i- (bz-a2)f

J&z-

--cot

2

4.3.145

1

*I2

In sin t d t = l

*I2

7r

In cos t dt=-- 2 In 2

4.3.146

J k z = L a n -z 2

4.3.134

dt=ln (b/a)

2 4.3.144 l - s i n t2 d t = i m c o s t 2 dt=l$

(b-a) tan g+(b2-a2)+ z =-In[ 1 (b'-a2)*

4.3.135

t

z

=---

(m=O)

=O

(:Tij

J&=Ftan

4.3.132

(m>O>

(See chapters 5 and 7 for other integrals involving circular functions.) (See [5.3]for Fourier transforms.)

3z

4.3.147 4.3.136

ear (a sin bz-b cos b z ) ear sin b z d z = m 2

S

I

Formulas for Solution of Plane Right Triangles

I

4.3.137

S

ear eaz cos b z dz=- a$+ b2 (a cos bz+b sin b z )

4.3.138

I

ear &n-1

S

ear sinn b z dz=

b z (a sin bz-nb cos b z ) a2fn2b2

sin A=%-

4.3.139

S

If A, B and C are the vertices (C the right angle), and a, b and c the sides oppositerespectively,

eaz cos" b z dz=

ea* cos"-l b z (a cos bz+nb sin b z ) a2+n2b2

c

1

csc A

Definite Integrals

4.3.140

l s i n mt sin nt dt=O

versine A=vers A=l-cos

A

coversine A=covers A=l-sin (m Zn, m and n integers) l c o s mt cos nt dt=O

haversine A=hav A=$ vers A exsecant A=exsecA=secA-1

A

79

ELEMENTARY TRANSCENDENTAL FUNCTIONS

4.3.148 Formulas for Solution of Plane Triangles

I

4.4. Inverse Circular Functions Definitions

4.4.1

arcsin

.=I* &

(2=x

+i y )

4.4.2 7r

arcsin z

4.4.3 b

A

C

arctan z=J In a triangle with angles A, B and C and sides opposite a, b and c respectively, a --

sin A COS

--=-b

A= a=

C

sin B sin C

c2+

9

b cos C+c cos B

tan $(A+B) a- b-tan +(A- B) bc sin A area=-=[s(s-a) 2

+(a+b

The path of integration must not cross the real axis in the case of 4.4.1 and 4.4.2 and the imaginary axis in the case of 4.4.3 except possibly inside the unit circle. Each function is singlevalued and regular in the z-plane cut along the in the ca?e real axis from - to - 1 and 1 to of 4.4.1 and 4.4.2 and along the imaginar axis from i to i w and --i to - - i w in the case o 4.4.3. Inverse circular functions are also written arcsin z=sin-' z, arccos z=cos-' z, arctan z =tan-' z, . . . . When - 1< x < 1, arcsin x and arccos x are real and

+ +

b"a2 2bc

a+b --

S=

dt lSt2

(6-

b) (s-c>]i

+c)

4.3.149 Formulas for Solution of Spherical Triangles

0 0

A

If A, B and C are the three angles and a, b and c the opposite sides,

4.4.4

- 35 . arcsin x < $1r,

0 < i1rccos x 5 7r

4.4.5

arctan zfarccot

RzlQ Rz
z=&+7r

4.4.6

nrccsc z=urcsiii 11."

4.4.7

arcsec z=arccos l / z

4.4.8

arccot z=arctan l / z

4.4.9

arcsec z+arccsc z=$r

(see 4.3.45)

++ -I

0 +I

arcsin z and arccorz

sin A --= sin -- B sin C sin a sin b sin c

i

arcton z

liY

cos a=cos b cos c+sin b sin c cos A -cos b cos (the) COS

e

where tan O=tan b cos A cos A=-cos

B cos C+sin 13 sin C cos a

arccsc z and arcsec z

arccat z

FIGURE4.4. Branch cuts f o r incerse circular junctions.

80

ELEMENTARY TRANSCENDENTAL FUNCTIONS

I

Fundamental Property

The general solutions of the equations sin t = z

4.4.23

Arccsc z = i Arccsch iz

4.4.24

Arcsec z= fi Arcsech z

4.4.25

Arccot z = i Arccoth iz

cos t = z

Logarithmic Representations

tan t=z

4.4.26

Arcsin s = - i L n

4.4.27

Arccos x=--i

are respectively 4.4.10

t=Arcsin z= (- l)* arcsin z + k a

4.4.11

t=Arccos z= farccos z+2kn

4.4.28 Arctan x=-

[(1-x2)*+ix]

(z2<1)

Ln [z+i(l-x2)+]

i Ln -=1-ix 2 1+ax

(~"1)

i Ln ifs 2 a-x

4.4.12

t=Arctan z=arctan z + k A

(x real)

(z2#-1)

where k is an arbitrary integer. 4.4.13

4.4.29

Arccsc x=-i

4.4.31

Arccot

Ln

[

]

(z2- I )'+i

( x 2 2 1)

Interval containing principal value Y x positive x negative or zero

arcsin x and arctan x 0 5 y 5 7r/2 -n/2 5 y
n/2< y 5 A

(x real)

arccot x and arccsc x 0 5 y 5 s/2 - 712 5 y
Addition and Subtraction of Two Inverse Circular Functions

Y

.

4.4.32

'I

Arcsin z1fArcsin z2 =Arcsin [ zl( 1-zl)+sfr z2( 1-z!)+] 4.4.33

Arccos z1fArccos zz =Arccos { z1z2i [ (1 -22) (1 -231i} 1.20

4.4.34

IICICC. I

- - 0:lCoI

Arctan zlfArctan z2=Arctan

FIGURE 4.5. Inverse circular junctions.

4.4.35

Functions of Negative Arguments

4.4.14

arcsin (- z ) = -arcsin z

4.4.15

arccos (- z ) =A- arccos z

4.4.16

arctan (--)=-arctan

4.4.17

arccsc (- z ) = -arccsc z

4.4.18

arcsec

(-2)

=A-arcsec

4.4.19

arccot

(-2)

=- arccot z

Arcsin z1fArccos z2 =Arcsin{ zrzaf [(l-z;) (l-4)]+} =Arccos [z2(1-zf)+F 21 (1- 23*]

z

4.4.36

Arctan zlfArccot z 2 = ~ r tan c ( s ) = ~ r c c ot

z

Arcsin z= -i Arcsinh iz

4.4.21

A~CCOS Z = fi Arccosh z

4.4.22

Arctan z=-i

Arctanh iz

(--)

zziz1

ZlZZfl

Inverse Circular Functions in Terms of Red and Imaginary Parts

Relation to Inverse Hyperbolic Functions (see 4.6.14 t o 4.6.19)

4.4.20

(Z13

4.4.37

Arcsin ~ = l c ? r + ( - - l ) ~arcsin6 +(-I)%

In [a+ (2- l)t]

4.4.38

(z2Z-1)

I

Arccos z=2k?rf {arccos6-iIn[ar+(a2-ll)*Ij

I

81

ELEMENTARY TRANSCENDENTAL FUNCTIONS

4.4.46

4.4.39

Olxll I

2.c Arctari z=ka++ arctan (1 -xz- y J

7r

arcsin x=-- (1 -x)~(uO+ulx+u2x2+a3x3 2 +u424+u525+u6x6+u,57)

I

where k is an integer or zero uiid y'li

ar=$l(2+1)~+~2]t+$[(Z-l)'L+

/3=$ [ ( 2 + 1)2+ y']+-$ [ ( 2 -

1)2

f y"]"

Series Expansions

4.4.40 arcsin z=z+-+-+-

I

+.

z 3 1 . 3 ~1.3.52' ~ 2.3 2.4.5 2.4-6.7

*

e(5)

&) 1 1 2 x 10-8 u0= 1.57079 63050

u4=

al= --.21459 SS016

~ 5 =

- ,01708 81256

a2=

uB=

.00667 00901

~

(lz/
+

3

,03897 89874 -.05017 43046

4.4.47

.03089 lSSl0

a7=-.00126

=

I

24911

-l<sll

4.4.41

+

arct an z= ulz u3z3+u5x5+u7z7+ u9s9+ ~ ( 5 ) ie(z)I 5 10-5

(I I <2)

4.4.42

ul=

+. . .

arctan z=z--+---z 3 z 5 z7 3 5 7

=---+--a 1 1 2

((z/<1andz2#-1)

~3=-.33029 95 as=

+ . . .(lz/>landz2#-1)

1 323 525

2

a,= - .08513 30

.99986 60

u9=

.02083 51

.18014 10

-1lzll

4.4.48 l0 arctan -=x

+

5

+

1 .28x2

e (2)

(z2f-1) Continued Fractions

z z2 4z2 92' 162' - -- -. I + 3+ 5+ 7+ 9+ ( z in the cut plane of Figure 4.4.)

4.4.43 arctan z=-

1

Olxll

4.4.49" 8

arctan "=1 X

( z in the cut plane of Figure 4.4.)

u2= - .33333 1452s

u~O=

- .07528 96400

aI2= .04290 96138

Polynomial Approximations 9

u4=

.19993 55085

Olxll

a6=

- ,14208 89944

U14=

- ,01616 57367

.10656 26393

alB=

,00286 62257

4.4.45

-

arcsin x= - ( 1-s)t(uo+alz +u2x2+u3x3) 2 7r

+(

e 5)

1.57072 88

u1= -.21211 44

~ 2 =

US=

I

le(~)l _<5X10-5 uO=

+r,u2k22"+e(5) k=l

.07426 10

a3= -.01872 93

9 The approximations 4.4.45 to 4.4.47 are from C. Hastings, Jr., Approximations for digital computers. Princeton Univ. Press, Princeton, N.J.. 1955 (with permission).

10 The approximation 4.4.48 is from C. Hastings, Jr., Note 143. Math. Tables Aids Comp. 6, 68 (1953) (with permission). 11 The approximation 4.4.49 is from B. Carlson, M. Goldstein, Rational approximatio11 of functions, Los Alamos Scientific Laboratory LA-1943, Los Alamos, N . Mex., 1955 (with permission).

I

82

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Approximations in Terms of Chebyshev Polynomials 12

4.4.50

d - arccsc z=dz

4.4.57

-15x11

T:(x)=cos no,

Integration Formulas

(see chapter 22)

cos 8 = 2 ~ - 1 0

arctan x=x

1 z (22- 1)'

C A,T:(x2)

4.4.58

Sarcsin z dz=z arcsin z+(l-zZ)*

4.4.59

sarccos z dz=z arccos z-(1-~2)+

n-0

n 0

n 6 7 8 9 10

An ,88137 3587 1 -. 10589 2925 2 .01113 5843 3 -.00138 1195 4 .00018 5743 5 -.00002 6215

An .OOOOO 3821 -. 00000 0570 .OOOOO 0086 -. 00000 0013 . 00000 0002

S

4.4,60

arctan z dz=z arctan z-iln

(l+z2)

4.4.61

sarccsc z dz=z arccsc zrfln [~+(22-1)f]

For x >1, use arctan x=+arctan

(1/x)

-3Jz5XS3J2

4.4.51

OD

arcsin x=x

AnT:(2x2) n-0

4.4.62

OSxS3Jz

d.&rcsec z dz=z arcsec z F l n [z+(z2-l)f]

m

arccos x=+x

n 0 1 2 3 4

An 1.05123 1959 .05494 6487 .00408 0631 .00040 7890 .00004 6985

AnTz(2x2)

O<arcsec z<E

n=0

n 5 6 7 8 9

An .OOOOO .OOOOO .OOOOO ,00000 * 00000

For ? j @ s x < l , use arcsin x=arccos(l-x2)t, cos x=arcsin (1-9)+.

5881 0777 0107 0015 0002

4,4,63

Sarccot z dz=z arccot z++ In (1+z2)

arc-

4.4.64

S

z arcsin z dz=

Differentiation Formulas

4.4.52

d - arcsin z=(l-zZ)-f dz

4.4.53

d - arccos z=-(l-z2)-1

4.4.54

d 1 - arctan z=dz 1+22

4.4.55

d -1 - arcco t z =dz 1+ 2 2

I

IS

zn arcsin z dz=-

The approximations 4.4.50 t o 4.4.51 are from C. W. Clenshaw, Polynomial approximations to elementary functions, Math. Tables Aids Comp. 8, 143-147 (1954)

arcsin z+z (1-z2)4 4

p + l

n+l (n#-1)

dz

d 1 arcsec z= dz z (z*- 1)'

---

4.4.65

4.4.66

4.4.56

(z :)

I s

z arccos z dz=

(z :) ---

Z

arccos z-- 4 (1-zz)t

4.4.67

S

zn arccos z dz=-

2n+1

n+l

arccos (n#-1)

4-44-68 z arctan z dz=-

1

(1+z2) arctan z--

Z

83

ELEMENTARY TRANSCENDENTAL FUNCTIONS

I

4.4.69

S

cosh

4.5.8

zn arctan z dz=-

n+l

arctan

tanh z=-i (n#-1)

4.4.70

S

1

iz tan iz

4.5.10

csch z = i csc iz

4.5.11

sech z=sec iz

4.5.12

coth z=i cot iz

Z

z arccot z dz=- 2 (1+z2) arccot z+- 2

Periodic Properties

4.4.71

S

zn arccot z dz=-

Zn+l

n+l

sinh (z+2kai) =sinh z

4.5.13

arccot zf-

(k any integer) (n#-1)

4.5. Hyperbolic Functions

4.5.14

cosh (z+2kai)=cosh z

4.5.15

tanh (z+klri)=tanh z

Definitions

4.5.1

,?.=COS

Zn+l

Relations Between Hyperbolic Functions

eZ-e-2 sinh z=2

(2=5

+i y )

4.5.2

4.5.16

cosh2 z -sinh2 z =1

4.5.17

tanh2 z+sech2 z= 1

4.5.18

cotha z-csch2 ~ = 1

4.5.3

tanh z=sinh z/cosh z

4.5.19

cosh z+sinh z=ez

4.5.4

csch z=l/sinh z

4.5.20

cosh z-sinh z=e-'

4.5.5

sech z=l/cosh z

4.5.6

coth z=l/tanh z

Negative Angle Formulas

Y

t

4.5.21

sinh (-z)=-sinh

4.5.22

cosh (-

4.5.23

tanh (- z ) = -tanh z

z

Z ) =cosh

z

Addition Formulas

4.5.24

sinh (zl+z2)=sinh z1 cosh

a2

+cosh z1 sinh 4.5.25

-tinh -2.3

I

I

21

cosh 2.2 +sinh z1 sinh z2

I

--- csch x sech .'.--'.' colh

cosh (z1+Zz)=COSh

Za

4.5.26

tanh (z,+zJ=(tanh zl+tanh zz)/ (l+tanh z1 tanh z2)

I FIGURE 4.6. Hyperbolic functions.

4.5.27

00th ( ~ 1 + ~ 2 ) = ( ~ 021t coth h ~z+l)/ (coth ~ 2 + ~ 0 21) th

Relation to Circular Functions (see 4.3.49 t o 4.3.58)

Hyperbolic formulas can be derived from trigonometric identities by replacing z by iz sinh z=-i

sin iz

Half-Angle Formulas

4,5.28

sinhE=( 2

coshz-1

)4

84 4.5.29

ELEMENTARY TRANSCENDENTAL FUNCTIONS

4.5.44

)

cosh?=( c o s h z + l 3 2 4.5.45

4.5.30

cosh 2-1 #-cosh 2-1tanhE-( 2- cosh z + l - sinh z

)

tanh zl+tanh zz=

sinh 2 cosh z + 1

-

4.5.46

coth z,+coth

Multiple-Angle Formulas

4.5.31

sinh 22=2 sinh z cosh z=

2 tanh z 1-tanh2 z

4.5.32

cosh 22=2 cosh2 2-1=2

sinh' z + l

sinh (z1+z2) cosh 21 cosh 22 sinh (z,+z2) sinh z1 sinh zz

~ 2 =

Relations Between Squares of Hyperbolic Sines and Cosines

4.5.47

sinh2 zl-sinh2 z2=sinh (zl+zz) sinh ( z l - z z ) =Cosh2 zl-cOsh2 22

=cosh2 z+sinh2 z

4.5.48

sinh2 z,+cosh2 zz=cosh (zl+z2) cosh (zl-zz) =cosh2 zl+sinh2 zz

4.5.33

2 tanh z tanh 22=1+tanh2

4.5.34

sinh 32=3 sinh z t 4 sinh3 z

4.5.35

cosh 3 Z z - 3 cosh Z + 4 cosh3 z

4.5.36

sinh 4z=4 sinha z cosh zi-4 cosh3 z sinh z

4.5.49 sinh z=sinh x cos y + i cosh x sin y

4.5.37

cosh 4z=cosh4 z+6 sinh2z cosh2 z+sinh4 z

4.5.50 cosh z=cosh x cos y + i sinh x sin y

Hyperbolic Functions in Terms of Real and Imaginary Parts (2=x

Products of Hyperbolic Sines and Cosines

4.5.38

4.5.51 tanh z=

sinh 2x+i sin 2y cosh 2xfcos 2y

4.5.52 coth

sinh 2 x 4 sin 2y cosh 22-COS 2y

2 sinh z1 sinh z2=cosh (z1+z2) -cosh

(21

- ZJ

+iy)

Z=

De Moivre's Theorem

4.5.53 4.5.40

(cosh z+sinh z)"=cosh nz+sinh nz

Modulus and Phase (Argument) of Hyperbolic Functions

2 sinh z1 cosh z2=sinh (zl+z2) +sinh (z1-z2)

4.5.54

lsinh zI = (sinh2 x+sin2 y)' =[$(cosh ~ Z - C O S 2y)l'

4.5.55

arg sinh z=arctan (coth x tan y)

Addition and Subtraction of Two Hyperbolic Functions

4.5.41

sinh zl+sinh z2=2 sinh 4.5.42

('9 ( ) ' 9 ) cosh

(zlzzz)

sinh zl-sinh z2=2 cosh - sinh

(y)

4.5.56

/cosh z/=(sinh2 x+cos2 y)* =[+(cosh ~ X + C O S 2y)]i

4.5.57

arg cosh z=arctan (tanh x tan y)

4.5.58

tanh

21

=(cosh 2x+cos 2y

I 4.5.59

arg tanh z=arctan ( SE2z)

85

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Relations Between Hyperbolic (or Inverse Hyperbolic) Functions

4.5.60

sinh x=a

tanh z = u

cosh x=a

csch x=a

U

(a*- l ) t

a ( 1-a2)-4

a-'

(1+a')'

U

(1-d)-'

a- 1( 1

a(1+ a y '

a-'(a*- 1)'

U

U-'

(a*- 1)-+

( 1 a*)-4 a-'(a*+ l ) t

+

sech X = J

coth

X=U

-'

a-'(l-d)'

((32-

U-'

a(a2- 1)-'

(1+aZ)-t

(1-a"'

a-'

a- '( 1-a*) 4

U

a ( 1-a2)

(a*- 1)+

U-'

(1-a*)'

a ( 1 a2)-f

U

a-'(al- l)+

a(a2- 1)-'

a-'

(1 +a34

(1-d )- 4

U

+u2)'

-'

+

,

1)

Illustration: If sinh z=a, coth z=a-l(a2+ l)* arcsech a=arccoth (l--a*)-* 4.5.61 Special Values of the Hyperbolic Functions

sinh z

22

_______

5

sech z=l--+-2 24 24-,

Io

z

4.5.66 61 Z S + . 720

,

. +!Z E2n

2n

(24

(I

+ .. <;)

*

ZI

4.5.67

0

cosh z _ _ _ _ _ _ _ 1

(I zl
tanh z _ - - _ - _ 0 csch z

_______

where B, and E, are the nth Bernoulli and Euler numbers, see chapter 23.

0)

sech z _ - _ _ - _1_ coth

_______

2

Infinite Products

Q)

4.5.68

Series Expansions 23

26

4.5.62

sinh z=z+-+-+-+

4.5.63

cosh ~=1+-+-+-+

4.5.64

4.5.69

27

31 5 ! 7!

tanh z=z--+- 2 3 3

*

22

24

26

2!

4!

6!

,

. . (l4< Q))

,

..

2 17 z6--2' 15 315 2**(22"(27b)!

+. .+ *

(lzl< Q))

1 2 2=---+2

7 23,31 2 5 + . 6 360 15120

Z2n-1+,

..

..

-2 (2'"- '- 1)B z n 22"- + . . . (2n) ! (I4 <..I *See page 11.

Continued Fraction

4.5.70

tanh z=-

2

22 23 22 --

1+ 3+ 5+ 74-

* * '

Differentiation Formulas

4.5.65

csch

1

cosh z = i l [l+(2k-l)2T2 4z2

4.5.71

d sinh z =cosh z dz -

4-5-72

d cosh z=sinh dZ

4.5.73

d - tanh z=sechz z dz

4.5.74

d a c h Z=-aCh dz

z

z mth 2

86

ELEMENTARY TRANSCENDENTAL FUNCTIONS,

d - sech z=-sech dz

4.5.75

4.5.81

z tanh z

Integration Formulas

S S S S

sinh zdz=cosh z

4.5.77 4.5.18 4.5.79

cosh z dz=sinh z

(See chapters 5 and 7 for other integrals involving hyperbolic functions.)

tanh zdz=ln cosh z

4.6. Inverse Hyperbolic Functions Definitions

csch zdz=ln tanh

4.5.80

-Z2

S

sech zdz=arctan (sinh z )

4.5.81

S

sinh z dz=z" cosh z-n

4.5.84

S

Z"

4.6.2

arccosh

cosh z dz=z" sinh z-n

S S

zn-l cosh z dz

1

m+n

to

1)'

=-

m+n

- w to + 1 axis from - CO to -1

zdz

arccsch z=arcsinh l / z arcsech z = arccosh l / z arccoth z=arctanh l / z

4.6.4 4.6.5 4.6.6

z cosh" z dz

(m+n#o>

dz - -1 1 -sinh" z cosh" z m-1 sinh"-' z cosh"-l z dz -m+n-2S m- 1 sinhm-2z cosh" z

and +l to

Inverse hyperbolic functions are also written sinh-l z , arsinh z, d r sinh z, etc.

sinhm-l z coshnC1z

s

to -i and i

iw

4.6.3 real

sinhm+lz coshn-l z

--Ssinhm-2 m-1 m+n

dt

4.6.2 real axis from

z"-' sinh z dz

+*m+n Ssinh" z

(z=z+iy)

( dtt

4.6.1 imaginary axis from -iw

+=

sinh" z cosh" z dz=-

4.5.86

dt

The paths of integration must not cross the following cuts,

4.5.85

S

z = l m, z=l'

arctanh z=J

4.6.3

4.5.83 2"

arcsinh

2-

s c o t h z dz=ln sinh z

4.5.82

4.6.1

(m#U

- 1 1 -n-1 sinh"-l z cosh"-l z

I

-k.inh

I

x

z

X

arccosh z

orctonh z

-qo idl I

[i'

iy

o+

-=

x -1

orccsch L

+I

t;

orcsech z

-I

iy

x orccoth z

FIGURE 4.7. Branch cuts f o r inverse hyperbolic functions.

87

ELEMENTARY TRANSCENDENTAL FUNCTIONS

arctanh z=arccoth zf $+i

4.6.7

(see 4.5.60)

4.6.25

(according as Y z ' ( 0 )

1 x+l arccoth x=- In 2 2-1

Fundamental Property

The general solutions of the equations z=sinh t z=cosh t z=tanh t are respectively arcsinh z+k+i

4.6.8

t=Arcsinh z=(-l)&

4.6.9

t=Arccosh z= karccosh z+2k+i

4.6.10

t=Arctanh z=arctanh z+kxi

(k,integer)

0rctO"h

I

Functiono of Negative Argument0

4.6.11 *4.6.12 4.6.13

arcsinh (-

2)

1;

FIGURE 4.8. Inverse hyperbolic Junctions.

= -nrcsinh z

arccosh (- z ) =+i- arccosh z Addition and Subtraction of Two Inverse Hyperbolic Functions

arctanh (- z ) = -arctanh z

Relation to Inverse Circular Functiono (see 4.4.20 to 4.4.25)

Hyperbolic identities can be derived from trigonometric identities by replacing z by i z .

\

I)

- - ohcalk I

4.6.14

Arcsinh z= -i Arcsin iz

4.6.15

Arccosh z= fi Arccos z

4.6.16

Arctanh

4.6.17

Arccsch z =i Arccsc i z

4.6.18

Arcsech z= f i Arcsec z

4.6.19

Arccoth z=i Arccot i z

2=

4.6.26

Arcsinh z1kArcsinh z 2 =Arcsinh [z,(l+zi)if z2(l +z3+1 4.6.27

Arccosh z1fArccosh z2

-i Arctan i z

=Arccosh { ~ ~ ~ ~ f [ ( ~ ~ - l ) ( z ~ - l ) J ~ } 4.6.28

Arctanh zlfArctanh z2=Arctanh

Logarithmic Repreeentationo

4.6.29

4.6.20

arcsinh x=ln [x+(x2+1)+]

Arcsinh z,kArccosh z 2

4.6.21

arccosh x=ln [x+ (x2- 1)4]

4.6.22

l+x arctauh x=$ In 1-x

4.6.23

arccsch t = l n

[:: -+(t2 -+1

4.6.24

arcsech s=ln

[:-+ 2-:(

(l2::2)

=Arcsinh{ z 1 z 2[(1+23(z:-l)]+} ~

(221)

=Arccosh [z2(1+z:)if z,(zZ,- 1)+] 4.6.30

>'I 1

>1

(x#O)

Arctanh z,fArccoth z2=Arctanh =Arccoth

(0<~51)

(zz:) ~

("-)

2 z t 21

ZIZ2h *Sec page 11.

1

88

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Series Expansions

arcsinh z=z--

1 z3+- 1 . 3 2.3 2'4.5

=In 22+--2

1 *

z5

Integration Formulas

1.3 2 * 4 * 4z4

222

d dz - arccoth z=(l-z2)-1

4.6.42

4.6.31

4.6.43

Sarcsinh z dz=z arcsinh z-(l+z2)+

4.6.44

Sarccosh z d z = z arccosh z--(z2-1)+

4.6.45

Sarctanhz dz=z arctanh z+31n (1-z2)

4.6.46

Sarccsch z dz=z arccsch zfaxcsinh z

*

(according as WzZO) 1 1.3 arccosh z=ln 22--- 2 * 222 2 * 4 * 4 z 4

4.6.47

Sarcsech z dz=z arcsech zfarcsin z

4.6.48

sarccoth z dz=z arccoth z++ In (z2-1>

*

4.6.49

S

z arcsinh z dz=- 222+1 arcsinh 4

4.6.33 4.6.34

z5 z7

arctanh z=z+-+-+-+ 3

1

5

1

...

7 1

...

72

S

S

2 - 5

.\11+22 -1+

-

.

.

1 2z2 1 222 3 4z2 3 4 9 ---3+ 5+ 7+ 9+ ' . 9

d

d 2 arcsinh -

*

Zntl

S

(n#-1)

4.6.53

d 2 arccosh z=(z2-

4.6.39

d d z arctanh z=(1-z2)-l

4.6.40

d 2 arccsch z = F -

S

22-

1 arctanh z+-

I

Z

2

2

I

4.6.54 p + 1

S -

4.6.38

nfl

z arctanh z dz=-

z=(1+z2)-+

d

4

znarccosh zdz=-

Differentiation Formulas

4.6.37

arccosh z dz= 222-1 -arccosh z--(z2-1)+ 4

4.6.52

( z in the cut plane of Figure 4.7.)

arcsinh

Z

2

arctanli z=- 1- -3- 5- 7-

4.6.36

n+l

4.6.51

422 922

22

,

(n#-1) (121>1)

2

(z2+l)+

Zntl

Continued Fractions

4.6.35

Z

4.6.50 zn arcsinh zdz=-

1

arccoth z=-+-+-+-,+ 2 323 525

(1z1<1)

2-2

zn arctanh z dz=-

ntl

arctanh (n#-1)

l)-+

4.6.55

d

S

z arccsch z dz=- 22 arccsh

1 2 (1+ Z Z )

+

(according as 9zZO) d 1 - arcsech z=F2(1-z2), dz

4.6.41 'See page

11.

2

zf51 (1+z2)+

(according as W z SO) 4.6.56

S

z"arccsch z dz=-

p + 1

n+ 1

1

arccsch z f-

(n# -1)

*

89

ELEMENTARY TRANSCENDENTAL FUNCTIONS

*

4h*57

s

z arcsech z dz=-

1 arcsech Z T(1-z2)+ ~ 2 (according as 9zZO)

4.6.58

S

4.6.59

22

p + I

arcsech z f-S nfl n+l

zn arcsech z dz =-

L

(1-22)'

d

z

(nZ-1)

S

22-

z arccoth z dz=-

I 4.6.60

1

s z n arccot,h z dz=-

1

2

arccoth z+-

2

2

n+ 1

zn+l

nfl

arccoth z+-

1 n+l

1

S L

2 '

(n#

- 1)

Numerical Methods

I

4.7. Use and Extension of the Tables

NOTE:In

the examples given it is nssumed thnt the nrguinents nre exact.

Example 1. Coniput,ation of Comnion Logarithms. To comput,e common lognrithnis, the number must be espressed in the form x l o g , (1 sz
-

2

2.lOQ

.009836 9.836. 10-3

log,, x. 100

-3,99281 85=(-2,00718

Example 2. Compute X - ~ I I for x=9.19826 to lOD using t,he Table of Common Logarithnis. From Table 4.1, four-point Lagrangian inter olation gives loglo(9.19826) = .96370 56812. d e n , --43 log10 (x)= -.72277 92609=9.27722 07391-10. Linear inverse interpolation in Table 4.1 yields nntilog (7.27722)= ,18933. For 10 place nccurncy subt,nbulation with 4- oint Lngrnnginn intcrpolniits produces the tn le

g

N

logio N ,27721 94350 .27724 23729 .27726 53095

.18933 ,18934 .18935

15)

,9836

- 13

9.836.10-'

-1.9928185=(-0.00718

9.836.10'

0.99281 85

9.836.10'

1.9928185

Convert loglox to In z for z=.009836. Using 4.1.23 nnd Table 4.1, In (.009836)= In 10 log,, (.009836)=2.30258 5093 (-2.00718 15) = -4.62170 62.

9.836.10'

2.99281 85

Example 4.

15)

Interpolat,ion in Table 4.1 between 983 and 984 gives .99281 85 ns the innntissa of 9836. Note that 5.99281 85=-3+.99281 85. When q is negative the coinnion logarithm can be expressed in the alternative forins

Compute In z for s=.OO278 to 6D. Using 4.1.7, 4.1.11 niid Table 4.2, In (.00278)= In (.278.10-2)=ln (.27S)-2 ln 10=-5.885304. Linenr intcrpoltition between z= .002 nnd ln(.00278)= -5.808. To z=.003 would obtain 5 decitnn plwc accurncy with linear interpolation it is neccssnrp thnt x>.175.

fvc

log,, (.009836) =3.99281 85=7.99281 85-10 = -2.00718 15.

Example 5.

The last form is convenient for conversion from common logarithms to natural logarithiiis. The inverse of log,, x is called the ant,ilogarithm of x, and is written antilog x or 10 x. The logarithm of the reciprocal of a nun1%-Icr is called the cologarithm, written colog.

(1131.718 1131) In 1131.718=111 1131

*See page 11.

29379 29366

~-~/'=.1893305685.

Example 3.

983.6

A?

By linear inverse interpolation

.09836 9.836. 10-2 3.99281 85=(--1.00718 15) 9.836 98.36

A

Compute In z for z=1131.718 t.o 8D. Using 4.1.7, 4.1.11 and Table 4.2

=In 1131.718+1n 1.131 +In 103 1131 =111(1.00063 4836)+1n 1.1314-3 lu 10.

90

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Then from 4.1.24 In 1131.718= (.00063 4836) -$(.00063 4836)2 +In 1.131+3 In 10=.00063 4836-.OOOOO 0202 +.12310 2197+6.90775 5279=7.03149 211.

Example 9.

Example 6. Compute In x working with 16D for

Linear interpolation with an error of 1 X 10-

x=1.38967 12458 179231.

Compute er.wm O9 to 75. Using 4.2.18 and Table 4.4, e 4 . e e ~e9,e4.9e.0872a ~

89,

e.087u) 6e=1.10217 67

,

e1g8728"= (134.28978) (1.10217 67) = 148.01 11.

X

Since -- 1.00048 32583 282384= 1+a, using 1.3894.1.24 and Table 4.2 we compute successively

a=

.00048 32583 282384

a2 --=2

'

aa -= 3

.

a' --=4

In (l+a)=

1167 693059

a2 -2!-

136

315 88140 97019

*

aa --

.00048 31415 965388

3!-

In 1.389 = ,32858 40637 722067 In x= .32906 72053 687455.

2646 54842

'

a4

a=

16630

*

ea=1.00025

Example 7. Compute the principal value of In ( f 2 f3i). From 4.1.2, 4.1.3 and Tables 4.2 and 4.14. 3 In (2+3i)=z 1 In (22+32)+i arctan 2 =1.282475+{(.982794) 1 In (-2+3i)=In 13+i r-arctan 2

(\

-

23,

1 In 13+i 2

(-+arctan

23,

=1.282475- i(2.158799) 1

(

In (2-3i)=- 2 In 13+i -arctan =1.282475-i(.982794).

Example 8. Compute (.227).08to 7D. Using 4.2.7 and Tables 4.2 and 4.4, (,227).69,e.09 in 6227) ,e.08(-1.488280 6zaz) =e-*.OB13 ua1= .35946 60.

e.8e7=2.37976 08513 29496 863 from Table 4.4 - 2--2.38035 90768 39006 089. -e

Example 11. Compute eW to 7s. X

In 10

and d= the decimal part of

X -a

In 10

Then exp x=exp

-

13805 15472 81184

eae .SO1

Let n=-

=1.282475+i (2.158799)

In (-2-3i)=-

Compute ez to 18D for x=.86725 13489 24685 12693. Let a=x-.867. Using 4.2.1, compute successively 1.00000 00000 00000 00000

a= -00025 13489 24685 12693

376199

.

Example 10.

)

-In 10 =exp [(n+d) In 101 (lnxlo

=exp (In 10") exp (d In 10) =10" exp (d In 10)

From Table 4.4

(E0h

ee48=exp

lO)=exp (281.42282 42 In 10)

=1OzE1 exp (.42282 42 In 10)=102s1exp .97358 87 =102*l(2.647428) = (281)2.647428.

Example 12. Com ute e-z for x=.75 using the expansion in ChebysYlev polynomials.

91

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Following the procedure in [4.3] we have from

4.2.118

Compute sin x and cos 2 for 2=2.317 to 7D. From 4.3.4 and Table 4.6

7

e-'=xAkn

Example 15.

(2)

k=O

*

c(x)

where are the Chebyshev polynomials defined in chapter 22. Assuming bs=b9=0 we generate br, k = 7 , 6 , 5 , . . , 0 from the recurrence relation b k = (4s-2)bk+l - b L + 2 +At

COS

(2.317)=COS (U-2 .3 1 7 )~--COS (-82459 2654) --.67885 60.

Compute sin z for z= 12.867 to 8D. From 4.3.16 and Tables 4.6 and 4.8 sin (12.867)=sin 12 cos .867+cos 12 sin ,867 =.29612 142. The method of reduction to an angle in the first quadrant which was given in Example 15 may also be used.

since f ( z ) = b o - (22- l ) b l ,

Example 17.

e-.76=.33520 2828- (.5)(-.27432 =.47236 6553.

7449)

Example 13. in radians to 6D.

1°=.01745 32925 19943 29577r 1'=.00029 08882 08665 72159 62 r 1" =.OOOOO 48481 36811 09535 9936 r

Compute sin 2 to 19D for ~=.86725 13489 24685 12693. Let a=.867, 8=z-a0. From 4.3.16 Table 4.6

and

sin (a+@)=sin a cos @+cos a sin @ sin a=.76239 10208 07866 22598 COS a=.64711 66288 94312 75010 With the series expansions for sin @ and cos 8 we compute successively

Therefore 3S0=.66322 51 r 42'=.01221 73 r 32"=.00015 51 r 3S042'32''= -675598r.

1.00000 00000 00000 00000

---@ . 218'. 41-

Example 14.

315 88140 97019 16630 ~

Express x= 1.6789 radians in degrees, minutes and seconds to the nearest tenth of a second. From Table 1.1 giving the mathematical constants we have 1800 1 r=-=57.29577 U

1.6789 r=96.1938S0 .19388O X 60= 11.633' ,633' X 60=38.O' 1.6789 r =96°11'38.0". 11.

=.73427 12

Example 16.

.00018 9959 -.00300 9164 .03550 4993 -.27432 7449 .33520 2828

*See page

(32459 2654)

Linear interpolation for 21.82459 2654 gives an error of 9 X 10-8.

bk

-.OOOOO 0015 .OOOOO 0400 -,OOOOO 9560

Express 38'42'32''

sin (2.317)=sin (*--2.317)=sin

95130'.

..

COS

8=

.99999 99684 11859 19611 @= ,00025 13489 24685 12693

-88,-. 31

$=

2646 54842

.

1

sin 8= .WO25 13489 22038 57852 a n Q! cos 8=.76239 09967 25351 31308 cos U sin 8=.00016 26520 67105 82436 Sin ~=.76255 36487 92457 1374

92

ELEMENTARY TRANSCENDENTAL FUNCTIONS

This procedure is equivalent to interpolation with Taylor's formula 3.6.4. Example 18.

Example 22.

Compute arctan 20 and arccot 20 to 9D. * Using 4.4.5, 4.4.8, and Table 4.14

In the plane triangle ABC, a=123, B=29'16', ~ = 3 2 1 ;find A, b.

arctan 2O=--arctan 2

b2=a2+c2-2~ cos B= (123)2+(321)* -2(123)(321) COS 29'16'

arccot 20=--arctan 2

b=221.99934 00

Example 23.

sin A=-= a sin B (123)(.48887 50196)=.27086 39918 b 221.99934 00 A=15'42'56.469''.

U

U

1/20=1.52083 7931

2O=arctan .05=.04995 8396.

Express z=3+9z' in polar form. z =z+ iy=refe, where r=

(2' +y2)+,

O=arctan Y -+27rk, k is an integer.

Example 19.

For k=O,

X

In the plane triangle ABC, a=4, b=7, c=9, find A, B, and C. c*+b2-a2 COS A= 2bc

-814-49-16 2.7.9

114-.90476 1w5 =-126

r= (32+g2) +=Ji%= 9.486833 O=arctan 9/3=arctan 3=1.24904 58. Thus 3+93=9.486833 exp (1.24904 584. Example 24.

A=.43997 5954=25'12'31.6''

sin A=.425917709

Compute arctan x for.x=1/3 to 12D. From 4.4.34 and 4.4.42 we have arctan x=arctan (xo+h) =48'11'22.9''

9(.42591 7709) sin C= , 4

c= 1.86054 803

=arctan xo+arctan

h 1+xdL+z:

=106'36'5.6"

where the sup lementary an le must be chosen for C. As a chec we get A+ I7+C=lSOoO0'.l''.

We have

Example 20.

x=3= .33333 33333 33 so that h=.00033

R

Compute cot x for x=.4589 to 6D. Since x<.5, using Table 4.9 with interpolation in (x-'-cot x), we find -cot(.4589)= ,4589 .155159. Therefore cot (.4589) =2.179124.155159=2.023965.

and, from Table 4.14, arctan xo=arctan ,333 Since = .32145 05244 03. .00030 00300 03 we get arctan x=.32145 05244 03+.00030 00300 03

-.ooooo

Example 21.

Compute arcsin 5 for x=.99511. For x>.95, using Table 4.14 with interpolation in the auxiliary functionf(x) we find U

arcsin x=z- [2(1-x)]+f(x)

33333 33

00000 09

=.32175 05543 97.

If x is given in the form b/a it is convenient to use 4.4.34 in the form b b -axo arctan -=arctan s,+arctan -. a a+bxO In the present example we get

=1.57079 6327-(.09889 388252) (1.O0040 7951) =1.47186 2100.

1

arctan --arctan 3-

.333+arctan

1 -3333

93

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Example 25. Compute arcsec 2.8 to 5D. Using 4.3.45 and Table 4.14 arcsec z=arcsin arcsec 2.8=arcsin

arctanh .96035=3 In

=3

In 49.44136 191

=3(3.90078 7359) = 1.950394.

(Z”l)* Z

Example 27.

[(2.8)2--1]* 2.8

Compute arccosh x for x=1.5368 to 6D. Using Table 4.17

=arcsin .93404 97735

arccosh z-arccosh 1.5368(z’- 1)*- [ (1.5368)2-1]*- ,852346

= 1.20559

or using 4.3.45 and Table 4.14

arccosh 1.5368= (.852346)(1.361754)*

arcsec z=arctan (z2-1)t

= (.852346)(1.166942)

arcsec 2.8=arctan 2.61533 9366

= ,994638.

T --arctan .38235 95564,

2

+

1 .96035 =& ln- 1.96035 1- .96035 .03965

from 4.4.3 and 4.4.8

= 1.570796-.365207

Example 28. Compute arccosh 2 for x=31.2 to 5D. Using Tables 4.2 and 4.17 with 1/~=1/31.2 = .03205 128205

= 1.20559.

Example 26.

arccosh 31.2 -1n 31.2 =,692886

Compute arctanh x for x=.96035 to 6D. From 4.6.22 and Table 4.2

+

arccosh 31.2=,692886 3.440418=4.13330.

References Texts [4.1] B. Carlson, M. Goldstein, Rational approximation of functions, Los Alamos Scientific Laboratory LA-1943 (Los Alamos, N. Mex., 1955). [4.2] C. W. Clenshaw, Polynomial approximations to elementary functions, Math. Tables Aids Comp. 8, 143-147 (1954). [4.3] C. W. Clenshaw, A note on the summation of Chebyshev series, Math. Tables Aids Comp. 9, 118-120 (1955). [4.4] G. H. Hardy, A course of pure mathematics 9th ed. (Cambridge Univ. Press, Cambrid e h g l a n d , and The Macmillan Co., New York, #.k., 1947). [4.5] C. Hastin s, Jr., Approximations for digital computers (gprinceton Univ. Press, Princeton, N.J., 1955). [4.6] C. Hastings, Jr., Note 143, Math. Tables Aids Comp. 6, 68 (1953). [4.7] E. W. Hobson, A treatise on plane tri onometry, 4th ed. (Cambridge Univ. Press, barnbridge, England, 1918). [4.8] H. S. Wall, Analytic theory of continued fractions (D. Van Nostrand Co., Inc., New York, N.Y., 1948). Tables [4.9] E. P. Adams, Smithsonian mathematical formulae and tables of elliptic functions, 3d reprint (The Smithsonian Institution, Washington, D.C., 1957). [4.10] H. Andoyer, Nouvelles tables tri onometriques fondamentales (Hermann et fils, baris, France, 1916).

[4.11] British Association for the Advancement of Science, Mathematical Tables, vol. I. Circular and hyperbolic functions, exponential, sine and cosine integrals, factorial function and allied functions, Hermitian probabilit functions, 3d ed. (Cambridge Univ. Press, ambridge, England, 1951). [4.12] Chemical Rubber Com any, Standard mathematical tables, 12th ed. (Zhemical Rubber Publ. Co., Cleveland, Ohio, 1959). [4.13] L. J. Comrie, Chambers’ six-figure mathematical tables, vol. 2 (W. R. Chambers, Ltd., London, England, 1949). [4.14] H. B. Dwight, Tables of integrals and other mathematical data, 3d rev. ed. (The Macmillan Co., New York, N.Y., 1957). [4.15] W. Grobner and N. Hofreiter, Inte raltafel, unbestimmte und bestimmte Integrale (jpringerVerlag, Wien und Innsbruck, Austria, 1949-1950). [4.16] Harvard Computation Laboratory, Tables of the function arcsin z (Harvard Univ. Press, Cam0 521475, bridge, Mass., 1956). z=z+iy, 0 < y 5475, 6D, varying intervals. [4.17] Harvard Computation Laboratory, Tables of inverse hyperbolic functions (Harvard Univ. Press, Cambridge, Mass., 1949). arctanh x , 0
8

94

ELEMENTARY TRANSCENDENTAL FUNCTIONS

[4.19] National Bureau of Standards, Table of natural logarithms for arguments between zero and five t o sixteen decimal places, 2d ed., Applied Math. Series 31 (U.S. Government Printing office, Washin ton, D.C., 1953). z=O(.OOOl)5, 16 D. [4.20] National Bureau of Standards, Tables of the exponential function ez, 3d ed., Applied Math. Series 14 (U.S. Government Printing Office, Washington, D.C., 1951). x= -2.4999(.0001) .9999, 18D, z= 1(.0001) 2.4999, 15D, ~=2.5(.001)4.999, 15D, ~=5(.01)9.99, 12D, Z= -.000099(.000001) .000099, 18D, ~=-100(1)100, 19S, z = - 9 X 10-n(10-")9X lO-n, n = 10, 9, 8, 7, 18D; values of e and l/e, 2556D. [4.21] National Bureau of Standards, Table of the descending exponential z=2.5 t o z=10, Applied Math. Series 46 &.S. Government Printing Office, Washington, D.C., 1955). z=2.5(.001)10, 20D. [4.22] National Bureau of Standards, Tables of sines and cosines for radian arguments, 2d ed., Applied Math. Series 43 (U.S. Government Printing Office, Washington, D.C., 1955). sin z, cos z, z=0(.001)25.2, 0(1)100, 8D, Z= 10-"(10-")9X l O - n , n=5, 4, 3, 2, 1 15D, x=O(.OOOOl) .01, 12D. [4.23] National Bureau of handards, Tables of circular and hyperbolic sines and cosines for radian arguments, 2d ed., Applied hlath. Series 36 (US. Government Printing Office, Washington, D.C., 1953). sin 2, cos z, sinh x, cosh 2, z=O(.OOOl) 1.9999, O(.l)lO, 9D. [4.24] National Bureau of Standards, Table of circular and hyperbolic tangents and cot,angents for radian arguments, 2 d p n t i n g (Columbia Univ. Press, New York, N. ., 1947). tan z, cot z, tanh 2, coth x, 2=0(.0001)2, 8D or 8S, z=O(.l)lO, 10D. [4.25] National Bureau of Standards, Table of sines and cosines to fifteen decimal places a t hundredths of a degree, Applied Math. Series 5 (U.S. Government Printing Office, Washington, D.C., 1949). sin x, cos x, ~=Oo(.Ol0)9Oo,15D. supplementary table of sin z, cos z, z=lo(l0)89", 30 D. [4.26] National Bureau of Standards, Table of secants and cosecants to nine significant figures a t hundredths of a degree, Applied Math. Series 40 (U.S. Government Printing Office, Washington, D.C., 1954). [4.27] National Bureau of Standards, Tables of functions and of zeros of functions, Collected short tables of the Computation Laboratory, Applied Math. Series 37 (U.S. Government Printing Office, Washington, D.C., 1954).

[4.28] National Bureau of Standards, Table of arcsin z (Columbia Univ. Press, New York, N.Y., 1945). arcsin z, z=0(.0001).989(.00001) 1, 12D; a.\iary table of f(v)=[)a-arcsin (l-u)]/(2v) , v=0(.00001).0005, 13D. [4.29] National Bureau of Standards, Tables of arctan 2, 2d ed., Applied Math. Series 26 (U.S. Government Printing Office, Washington D.C., 1953). z= O( .001) 7 (.01) 50(. 1)300( 1)2000 ( 10) 10000, 12D. [4.30] National Bureau of Standards, Table of hyperbolic sines and cosines, z = 2 to z=10, Applied Math. Series 45 (U.S. Government Printing Office, Washington, D.C., 1955). x=2(.001)10, 9s. [4.31] B. 0. Peirce, A short table of integrals, 4th ed. (Ginn and Co., Boston, Mass., 1956). [4.32] J. Peters, Ten- lace logarithm table, vols. 1, 2 (together witg an appendix of mathematical tables) (Berlin, 1922; rev. ed., Frederick Ungar Publ. bo., New York, N.Y., 1957). [4.33] J. Peters, Seven-Dlace values of trigonometric functions for eGery thousandth ofVa de ree (Berlin-Friedenau, 1918; D. Van Nostrand Inc., New York, N.Y., 1942). [4.34] L. W. Pollak, Rechentafeln zur harmonischen Analyse (Johann Ambrosius Barth, Leipzig, Germany, 1926). [4.35] A. J. Thompson, Standard table of logarithms to twenty decimal places, Tracts for Computers, No. 22 (Cambridge Univ. Press, Cambridge, England, and New Ynrk , N 1952) - .Y - . ) -""-,. [4.361 J. Todd, -Table of arctangents of rational numbers, NBS Applied Math. Series 11 (U.S. Government Printing Office, Washington, D.C., 1951). arctan mln and arccot m/n, O<m
80.,

I

95

ELEMENTARY TRANSCENDENTAL FUNCTIONS

COMMON LOGARITHMS X

log10 x

X

log10 x

X

log10 x

X

7nn

250 251 252 253 254

39794 no087

39967 372i5 40140 05408 40312 05212 40483 37166

300. _

202 203 204

30102 99957 30?19 60574 30535 13694 30749 60379 30963 01674

16982 45984 96524 70870 71243

205 206 207 208 209

31175 31386 31597 31806 32014

38611 72204 03455 33350 62861

255 256 257 258 259

40654 40823 40993 41161 41329

20411 20682 20951 21218 21484

99827 58i60 50145 76044 38480

7111

165 166 167 168 169

21748 22010 22271 22530 22788

170 171

X

151 152 153 154

12591 69471 35879 14308 07208

92991 58653 37777 37555 64979

155 156 157 158 159

19033 19312 19589 19865 20139

04139 04532 04921 05307 05690

26852 29788 80227 84435 48513

161 162 163 164

160 ---

115 116 117 118 119

06069 06445 06818 07188 07554

78404 79892 58617 20073 69614

120 121 122 123 124

07918 12460 08278 53703 08635 98307 OSSSO 5Iii4 09342 16852

125 126 127 128 129

09691 10037 10380 10720 11058

130 131 ~. 132 133 134

00000 00432 00860 01283 01703

00000 13738 01718 72247 33393

105 106 107 108 109

02118 02530 02938 03342 03742

110 111 112 113 114

150 _.

log10 x 17609 17897 18184 18469 18752

100 101 102 103 104

Table 4.1

iOi

log10 z

301 302 303 304

47712 47856 48000 48144 48287

12547 64956 69430 26285 35836

01804 99653 31233 97060 97641

305 306 307 308 309

48429 48572 48713 48855 48995

98393 14265 83755 07165 84794

iii 52428 24553

32221 92947

_260 _

41497 33480

212 213 214

32633 58609 32837 96034 33041 37733

261 262 263 264

41664 41830 41995 42160

OS073 12913 57485 39269

310 311 312 313 314

49136 49276 49415 49554 49692

16938 03890 45940 43375 96481

39442 80880 64711 92817 67046

215 216 217 2i8 219

33243 33445 33645 33845 34044

84599 37512 97338 64936 41148

265 266 267 268 269

42324 42488 42651 42813 42975

58739 16366 12614 47940 22800

315 316 317 318 319

49831 49968 50105 50242 50379

05538 70826 92622 71200 06831

89214 61104 84469 bioli 92483

220 221 224

34242 34439 34635 54830 35024

26808 22737 29745 48630 80183

270 271 272 273 274

43136 37642 43296 92909

174

23044 23299 23552 23804 24054

4 K b 26470 43775 05628

320 321 322 323 324

50514 50650 50785 50920 51054

99783 50324 58717 25223 50102

00130 05451 37210 99696 97103

175 176 177 178 179

24303 24551 24797 25042 25285

80487 26678 32664 00023 30310

225 226 227 228 229

35218 35410 35602 35793 35983

25181 84391 58572 48470 54823

275 276 277 278 279

43933 44090 44243 44404 44560

26938 90821 97691 47959 42033

325 326 327 328 329

51188 51321 51454 51587 51719

33610 76001 77527 38437 58979

11394 11727 12057 12385 12710

33523 12957 %3I2 16410 47984

180

25051 85749 13880 10897 78230

230 233 234

2%

36172 78360 36361 19799 56548 59849 36735 59210 36921 58574

280 281 282 283 284

44715 44870 45024 45178 45331

80313 63199

182 183 184

25527 25767 26007 26245 26481

64355 83400

330 331 332 333 334

51851 51982 52113 52244 52374

39399 79938 80837 42335 64668

135 136 137 138 139

13033 13353 13672 13987 14301

37685 89084 05672 90864 48003

185 186 187 188 189

26717 26951 27184 27415 27646

17284 29442 16065 78493 18042

235 236 237 238 239

37106 37291 37474 37657 37839

78623 20030 83460 69571 79009

285 286 287 288 289

45484 45636 45788 45939 46089

48600 60331 18967 24878 78428

335 336 337 338 339

52504 52633 52762 52891 53019

48070 92774 99009 67003 96982

140 I41 142 143 144

14612 14921 15228 15533 15836

80357 9iiZ7 83444 60375 24921

190

36010 33672 12287 73090 17299

240

192 193 194

27875 28103 28330 28555 28780

242 243 244

38021 12417 70426 38381 53660 38560 62736 38738 98263

290 291 292 293 294

46239 46389 46538 46686 46834

79979 29890 28514 76204 73304

340 341 342 343 344

53147 53275 53402 53529 53655

89170 43790 61061 41200 84426

145 146 147 148 149

16136 16435 16731 17026 17318

80022 28558 73347 17154 62684

195 196 197 198 199

29003 29225 29446 29666 29885

46114 60714 62262 51903 30764

245 246 247 248 249

38916 39093 39269 39445 39619

60844 51071 69533 16808 93471

295 296 297 298 299

46982 47129 47275 47421 47567

20160 17111 64493 62641 11883

345 346 347 348 349

53781 53907 54032 54157 54282

90951 60988 94748 92439 54270

150

17609 12591

200

30102 99957

250

39794 00087

300

47712 12547

350

54406 80444

~

172

iij

181 ---

i9i

777

2%

271

i4i 3iioi

43456 89040

SiOm

For use of common logarithms see Examples 1-3. For 100<~<135 interpolate in the range 1000
I

96

ELEMENTARY TRANSCENDENTAL FUNC!TIONS

Table 4.1 2'

COMMON LOGARITHMS

log10 z

X

log10 2

X

log10 x

x

log10 x

X

log10 x

350 351 352 353 354

54406 54530 54654 54777 54900

80444 71165 26635 47054 32620

400 401 402 403 404

60205 60314 60422 60530 60638

99913 43726 60531 50461 13651

450 451 452 453 454

65321 65417 65513 65609 65705

25138 65419 84348 82020 58529

500 501 502 503 504

69897 69983 70070 70156 70243

00043 77259 37171 79851 05364

550 551 552 553 554

74036 74115 74193 74272 74350

26895 15989 90777 51313 97647

355 356 357 358 359

55022 55144 55266 55388 55509

83531 99980 82161 30266 44486

405 406 407 408 409

60745 60852 60959 61066 61172

50232 60336 44092 01631 33080

455 456 457 458 459

65801 65896 65991 66086 66181

13967 48427 62001 54780 26855

505 506 507 508 509

70329 70415 70500 70586 70671

13781 05168 79593 37123 77823

555 556 557 558 559

74429 74507 74585 74663 74741

29831 47916 51952 41989 18079

360 361 362 363 364

55630 55750 55870 55990 56110

25008 72019 85705 66250 13836

410 411 412 413 414

61278 61384 61489 61595 61700

38567 18219 72160 00517 03411

460 461 462 463 464

66275 66370 66464 66558 66651

78317 09254 19756 09910 79806

510 511 512 513 514

70757 70842 70926 71011 71096

01761 09001 99610 73651 31190

560 561 562 563 564

74818 74896 74973 75050 75127

80270 28613 63156 83949 91040

365 366 367 368 369

56229 56348 56466 56584 56702

28645 10854 60643 78187 63662

415 416 417 418 419

61804, 61909 62013 62117 62221

80967 33306 60550 62818 40230

465 466 467 468 469

66745 66838 66931 67024 67117

29529 59167 68806 58531 28427

515 516 517 518 519

71180 71264 71349 71432 71516

72290 97016 05431 97597 73578

565 566 567 568 569

75204 75281 75358 75434 75511

84478 64312 30589 83357 22664

370 371 372 373 374

56820 56937 57054 57170 57287

17241 39096 29399 88318 16022

420 421 422 423 424

62324 62428 62531 62634 62736

92904 20958 24510 03674 58566

470 471 472 473 474

67209 67302 67394 67486 67577

78579 09071 19986 11407 83417

520 521 522 523 524

71600 71683 71767 71850 71933

33436 77233 05030 16889 12870

570 571 572 573 574

75587 75663 75739 75815 75891

48557 61082 60288 46220 18924

375 3 76 377 378 379

57403 57518 57634 57749 57863

12677 78449 13502 17998 92100

425 426 427 428 429

62838 62940 63042 63144 63245

89301 95991 78750 37690 72922

475 476 477 478 479

67669 67760 67851 67942 68033

36096 69527 83790 78966 55134

525 526 527 528 529

72015 72098 72181 72263 72345

93034 57442 06152 39225 56720

575 576 577 578 579

75966 76042 76117 76192 76267

78447 24834 58132 78384 85637

380 381 382 383 384

57978 58092 58206 58x9 58433

35966 49757 33629 87740 12244

430 431 432 433 434

63346 63447 63548 63648 63748

84556 72702 37468 78964 97295

480 481 482 483 484

68124 68214 68304 68394 68484

12374 50764 70382 71308 53616

530 531 532 533 534

72427 72509 72591 72672 72754

58696 45211 16323 72090 12570

580 581 582 583 584

76342 76417 76492 76566 76641

79936 61324 29846 85548 28471

385 386 387 388 389

58546 58658 58771 58883 58994

07295 73047 09650 17256 96013

435 436 437 438 439

63848 63948 64048 64147 64246

92570 64893 14370 41105 45202

485 486 487 488 489

68574 68663 68752 68841 68930

17386 62693 89612 98220 88591

535 536 537 538 539

72835 72916 72997 73078 73158

37820 47897 42857 22757 87652

585 586 587 588 589

76715 76789 76863 76937 77011

58661 76160 81012 73261 52948

390 391 392 393 394

59106 59217 59328 59439 59549

46070 67574 60670 25504 62218

440 441 442 443 444

64345 64443 64542 64640 64738

26765 85895 22693 37262 29701

490 49 1 492 493 494

69019 69108 69196 69284 69372

60800 14921 51028 69193 69489

540 541 542 543 544

73239 73319 73399 73479 73559

37598 72651 92865 98296 88997

590 591 592 593 594

77085 77158 77232 77305 77378

20116 74809 17067 46934 64450

395 396 397 398 399

59659 59769 59879 59988 60097

70956 51859 05068 30721 28957

445 446 447 448 449

64836 64933 65030 65127 65224

00110 48587 75231 80140 63410

495 496 497 498 499

69460 69548 69635 69722 69810

51989 16765 63887 93428 05456

545 546 547 548 549

73639 73719 73798 73878 73957

65023 26427 73263 05585 23445

595 596 597 598 599

77451 77524 77597 77670 77742

69657 62597 43311 11840 68224

400

6D205 99913

450

65321 25138

500

69897 00043

550

74036 26895

600

77815 12504

I

[(-:PI

[(-:)31

97

ELEMENTARY TRANSCENDENTAL FUNCTIONS

COMMON LOGARITHMS

600 601 602 603 604

log10 x 77815 12504 77887 44720 77959 64913 78031 73121 78103 69386

log10 X 650 81291 33566 651 81358 09886 652 81424 75957 653 81491 31813 654 81557 77483

X log10 x 700 84509 80400 701 84571 80180 702 84633 71121 703 84695 53250 704 84757 26591

605 606 607 608 609

78175 78247 78318 78390 78461

53747 26242 86911 35793 72926

655 656 657 658 659

81624 81690 81756 81822 81888

13000 38394 53696 58936 54146

705 706 707 708 709

84818 84880 84941 85003 85064

610 611 612 613 614

78532 78604 78675 78746 78816

98350 12102 14221 04745 83711

660 661 662 663 664

81954 82020 82085 82151 82216

39355 14595 79894 35284 80794

710 711 712 713 714

615 616 617 618 619

78887 78958 79028 79098 79169

51158 07122 51640 84751 06490

665 666 667 668 669

82282 82347 82412 82477 82542

16453 42292 58339 64625 61178

620 621 622 623 624

79239 79309 79379 79448 79518

16895 16002 03847 80467 45897

670 671 672 673 674

82607 82672 82736 82801 82865

625 626 627 628 629

79588 00173 79657 43332 79726 75408 79795 96437 79865 06454

675 676 617 678 679

630 631 632 633 634

79934 80002 80071 80140 80208

05495 93592 70783 37100 92579

635 636 637 638 639

80277 80345 80413 80482 80550

37253 71156 94323 06787 08582

640 641 642 643 644 645 646 647 648 649

2

.

X

Table 4.1

750 751 752 753 754

log10 X 87506 12634 87563 99370 87621 78406 87679 49762 87737 13459

800 801 802 803 804

log10 x 90308 99870 90363 25161 90417 43683 90471 55453 90525 60487

91170 47011 94138 32577 62352

755 756 757 758 759

87794 87852 87909 87966 88024

69516 17955 58795 92056 17759

805 806 807 808 809

90579 58804 90633 50418 90687 35347 90741 13608 90794 85216

85125 85186 85247 85308 85369

83487 96007 99936 95299 82118

760 88081 35923 761 88138 46568 762 88195 49713 763 88252 45380 764 88309 33586

810 811 812 813 814

90848 90902 90955 91009 91062

715 716 7L7 718 719

85430 85491 85551 85612 85672

60418 30223 91557 44442 88904

765 88366 14352 766 88422 87696 767 88479 53639 768 88536 12200 769 88592 63398

815 816 817 818 819

91115 76087 91169 01588 91222 20565 91275 33037 91328 39018

48027 25202 92731 50642 98965

720 721 722 723 724

85733 85793 85853 85913 85973

24964 52647 71976 82973 85662

770 771 772 773 774

88649 88705 88761 88817 88874

07252 43781 73003 94939 09607

820 821 822 823 824

91381 38524 91434 31571 91487 18175 91539 98352 91592 72117

82930 82994 83058 83122 83186

37728 66959 86687 96939 97743

725 726 727 728 729

86033 86093 86153 86213 86272

80066 66207 44109 13793 75283

775 776 777 778 779

88930 88986 89042 89097 89153

17025 17213 10188 95970 74577

a25 826 827 828 829

91645 39485 91698 00473 91750 55096 91803 03368 91855 45306

680 681 682 683 684

83250 83314 83378 83442 83505

89127 71119 43747 07037 61017

730 731 732 733 734

86332 86391 86451 86510 86569

28601 73770 10811 39746 60599

780 89209 46027 781 89265 10339 782 89320 67531 783 89376 17621 784 89431 60627

830 831 832 833 834

91907 80924 91960 10238 92012 33263 92064 50014 92116 60506

685 686 687 688 689

83569 83632 83695 83758 83821

05715 41157 67371 84382 92219

735 736 737 738 739

86628 73391 86687 78143 86746 74879 86805 63618 86864 44384

785 786 787 788 789

89486 89542 89597 89652 89707

96567 25460 47324 62175 70032

835 836 837 838 839

92168 64755 92220 62774 92272 54580 92324 40186 92376 19608

80617 80685 80753 80821 80888

99740 690 83884 90907 80295 691 83947 80474 50281 692 84010 60945 09729 693 84073 32346 58674 694 84135 94705

740 741 742 743 744

86923 86981 87040 87098 87157

17197 82080 39053 88138 29355

790 791 792 793 794

89762 89817 89872 89927 89982

70913 64835 51816 31873 05024

840 841 842 843 844

92427 92479 92531 92582 92634

92861 59958 20915 75746 24466

80955 81023 81090 81157 81224

97146 25180 42807 50059 46968

695 696 697 698 699

84198 84260 84323 84385 84447

745 746 747 748 749

87215 87273 87332 87390 87448

62727 88275 06018 15979 18177

795 796 797 798 799

90036 71287 90091 30677 90145 83214 90200 28914 90254 67793

845 846 847 848 849

92685 92737 92788 92839 92890

67089 03630 34103 58523 76902

650 81291 33566

700

84509 80400

750 87506 12634

800

90308 99870 850

48046 92396 27781 54226 71757

[(-l) ‘1

X

[(-47111

X

50189 08542 60292 05456 44049

92941 89251

[(-f’”3

98

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.1 z

COMMON LOGARITHMS

log10 2

log10 x

X

log10

x

X

log10 x

X

log10 x

850 851 852 853 854

92941 92992 93043 93094 93145

89257 95601 95948 90312 78707

900 901 902 903 904

95424 95472 95520 95568 95616

25094 47910 65375 77503 84305

950 951 952 953 954

97772 97818 97863 97909 97954

36053 05169 69484 29006 83747

1000 1001 1002 1003 1004

00000 00043 00086 00130 00173

00000 40775 77215 09330 37128

1050 1051 1052 1053 1054

02118 02160 02201 02242 02284

92991 27160 57398 83712 06109

855 856 857 858 859

93196 93247 93298 93348 93399

61147 37647 08219 72878 31638

905 906 907 908 909

95664 95712 95260 95808 95856

85792 81977 72871 58485 38832

955 956 957 958 9 59

98000 98045 98091 98136 98181

33716 78923 19378 55091 86072

1005 1006 1007 1008 I009

00216 00259 00302 00346 00389

60618 79807 94706 05321 11662

1055 1056 1057 1058 1059

02325 02366 02407 02448 02489

24596 39182 49873 56677 59601

860 861 862 363 364

93449 93500 93550 93601 93651

84512 31515 72658 07957 37425

910 911 912 913 914

95904 95951 95999 96047 96094

13923 83770 48383 07775 61957

960 961 962 963 964

98227 98272 98317 98362 98407

12330 33877 50720 62871 70339

1010 1011 1012 1013 1014

00432 00475 00518 00560 00603

13738 11556 05125 94454 79550

1060 1061 1062 1063 1064

02530 02571 02612 02653 02694

58653 53839 45167 32645 16280

865 866 867 868 869

93701 93751 93801 93851 93901

61075 78920 90975 97252 97764

915 916 917 918 919

96142 96189 96236 96284 96331

10941 54737 93357 26812 55114

965 966 967 968 9 69

98452 98497 98542 98587 98632

73133 71264 64741 53573 37771

1015 1016 1017 1018 1019

00646 00689 00732 00774 00817

60422 37079 09529 77780 41840

1065 1066 1067 1068 1069

02734 02775 02816 02857 02897

96078 72047 44194 12527 77052

870 871 872 873 874

93951 94001 94051 94101 94151

92526 81550 64849 42437 14326

920 921 922 923 924

96378 96425 96473 96520 96567

78273 96302 09211 17010 19712

970 971 972 973 9 74

98677 98721 98766 98811 98855

17343 92299 62649 28403 89569

1020 1021 1022 1023 1024

00860 00902 00945 00987 01029

01718 57421 08958 56337 99566

1070 1071 1072 1073 1074

02938 02978 03019 03059 03100

37777 94708 47854 97220 42814

87 5 876 877 878 879

94200 94250 94299 94349 94398

80530 41062 95934 45159 88751

925 926 927 928 929

96614 96661 96707 96754 96801

17327 09867 97341 79762 57140

975 976 977 978 979

98900 98944 98989 99033 99078

46157 98177 45637 88548 26918

1025 1026 1027 1028 1029

01072 01114 01157 oiiS9 01241

38654 73608 04436 31147 53748

1075 1076 1077 1078 1079

03140 03181 03221 03261 03302

84643 22713 57033 87609 14447

880 881 882 883 884

94448 94497 94546 94596 94645

26722 59084 85851 07036 22650

930 9 31 932 933 934

96848 96894 96941 96988 97034

29486 96810 59124 16437 68762

980 981 982 983 984

99122 99166 99211 99255 99299

60757 90074 14878 35178 50984

1030 1031 1032 1033 1034

01283 01325 01367 01410 01452

72247 86653 96973 03215 05388

1080 1081 1082 1083 1084

03342 03382 03422 03462 03502

37555 56940 72608 84566 92822

885 886 887 888 889

94694 94743 94792 94841 94890

32707 37219 36198 29658 17610

9 35 936 9 37 9 38 939

97081 97127 97173 97220 97266

16109 58487 95909 28384 55923

985 986 9 87 988 989

99343 99387 99431 99475 99519

62305 69149 71527 69446 62916

1035 1036 1037 1038 1039

01494 01535 01577 01619 01661

03498 97554 87564 73535 55476

1085 1086 1087 1088 1089

03542 03582 03622 03662 03702

97382 98253 95441 88954 78798

89 0 891 892 893 894

94939 94987 95036 95085 95133

00066 77040 48544 14589 75188

940 941 942 943 944

97312 97358 97405 97451 97497

78536 96234 09028 16927 19943

990 991 992 993 994

99563 99607 99651 99694 99738

51946 36545 16722 92485 63844

1040 1041 1042 1043 1044

01703 01745 01786 01828 01870

33393 07295 77190 43084 04987

1090 1091 1092 1093 1094

03742 03782 03822 03862 03901

64979 47506 26384 01619 73220

895 896 897 898 899

95182 95230 95279 95327 95375

30353 80097 24430 63367 96917

945 946 947 948 949

97543 97589 97634 97680 97726

18085 11364 99790 83373 62124

995 996 997 998 999

99782 99825 99869 99913 99956

30807 93384 51583 05415 54882

1045 1046

62904 16845 66817 12826 54882

1095 1096

1048 1049

01911 01953 01994 02036 02077

i09S 1099

03941 03981 04020 04060 04099

41192 05541 66276 23401 76924

900

95424 25094

950

97772 36053

1000

00000 00000

1050

02118 92991

1100

04139 26852

[(-:I

71

“-38’61

1047

[(-38)

51

1097

[(-pi

I

99

ELEMENTARY TRANSCENDENTAL FUNCTIONS

COMMON LOGARITHMS

1100 1101 1102 1103 1104

log10 1 04139 26852 04178 73190 04218 15945 04257 55124 04296 90734

1105 1106 1107 1108 1109

04336 04375 04414 04453 04493

1110 1111 1112 1113 1114

1150 1151 ii52 1153 1154

log10 x 06069 78404 06107 53236 06145 24791 06182 93073 06220 58088

22780 51270 76209 97604 15461

1155 1156 1157 1158 1159

06258 06295 06333 06370 06408

04532 04571 04610 04649 04688

29788 40589 47872 51643 51908

1160 1161 1162 1163 1164

1115 1116 1117 1118 1119

04727 04766 04805 04844 04883

48674 41946 31731 18036 00865

1120 1121 1122 1123 1124

04921 04960 04999 05037 05076

1125 1126 1127 1128 1129

x

1200 1201 1202 1203 1204

log10 5 07918 12460 07954 30074 07990 44677 08026 56273 08062 64869

19842 78341 33590 85594 34360

1205 1206 1207 1208 1209

06445 06483 06520 06557 06595

79892 22197 61281 97147 29803

1165 1166 1167 1168 1169

06632 06669 06707 06744 06781

80227 56126 28569 97563 63112

1170 1171 1172 1173 1174

05115 05153 05192 05230 05269

25224 83905 39160 90996 39419

1130 1131 1132 1133 1134

05307 05346 05384 05422 05461

1135 1136 1137 1138 1139

Table 4.1

1250 1251 1252 1253 1254

log10 2 09691 00130 09725 73097 09760 43289 09795 10710 09829 75365

1300 1301 1302 1303 1304

log10 2 11394 33523 11427 72966 11461 09842 11494 44157 11527 75914

08098 70469 081.3473078 08170 72701 08206 69343 08242 63009

1255 1256 1257 1258 1259

09864 09898 09933 09968 10002

37258 96394 52777 06411 57301

1305 1306 1307 1308 1309

11561 11594 11627 11660 11693

05117 31769 55876 77440 96466

1210 1211 1212 1213 1214

08278 08314 08350 08386 08421

53703 41431 26198 08009 86867

1260 1261 1262 1263 1264

10037 10071 10105 10140 10174

05451 50866 93549 33506 70739

1310 1311 1312 1313 1314

11727 11760 11793 11826 11859

12957 26917 38350 47261 53652

59254 85504 08560 28428 45112

1215 1216 1217 1218 1219

08457 08493 08529 08564 08600

62779 35749 05782 72883 37056

1265 1266 1267 1268 1269

10209 10243 10277 10311 10346

05255 37057 66149 92535 16221

1315 1316 1317 1318 1319

11892 11925 11958 11991 12024

57528 58893 57750 54103 47955

06818 06855 06892 06929 06966

58617 68951 76117 80121 80969

1220 1221 1222 1223 1224

08635 08671 08707 08742 08778

98307 56639 12059 64570 14178

1270 1271 1272 1273 1274

10380 10414 10448 10482 10516

37210 55506 71113 84037 94280

1320 1321 1322 1323 1324

12057 12090 12123 12155 12188

39312 28176 14551 98442 79851

1175 1176 1177 1178 1179

07003 07040 07077 07114 07151

78666 73217 64628 52905 38051

1225 1226 1227 1228 1229

08813 08849 08884 08919 08955

60887 04702 45627 83668 18829

1275 1276 1277 1278 1279

10551 10585 10619 10653 10687

01848 06744 08973 08538 05445

1325 1326 1327 1328 1329

12221 12254 12287 12319 12352

58783 35241 09229 80750 49809

84435 26049 64269 99099 30546

1180 1181 1182 1183 1184

07188 07224 07261 07298 07335

20073 98976 74765 47446 17024

1230 1231 1232 1233 1234

08990 09025 09061 09096 09131

51114 80529 07078 30766 51597

1280 1281 1282 1283 1284

10720 10754 10788 10822 10856

99696 91297 80252 66564 50237

1330 1331 1332 1333 1334

12385 12417 12450 12483 12515

16410 80555 42248 01494 58296

05499 05537 05576 05614 05652

58615 83314 04647 22621 37241

1185 1186 1187 1188 1189

07371 07408 07445 07481 07518

83503 46890 07190 64406 18546

1235 1236 1237 1238 1239

09166 09201 09236 09272 09307

69576 84708 96996 06447 13064

1285 1286 1287 1288 1289

10890 10924 10957 10991 11025

31277 09686 85469 58630 29174

1335 1336 1337 1338 1339

12548 12580 12613 12645 12678

12657 64581 14073 61134 05770

1140 1141 1142 1143 1144

05690 05728 05766 05804 05842

48513 56444 61039 62304 60245

1190 1191 1192 1193 1194

07554 69614 07591 17615 07627 62554 07664 04437 07700 43268

1240 1241 1242 1243 1244

09342 09377 09412 09447 09482

16852 17815 15958 11286 03804

1290 1291 1292 1293 1294

11058 97103 11092 62423 11126 25137 11159 85249 11193 42763

1340 1341 1342 1343 1344

12710 12742 12775 12807 12839

47984 87779 25158 60127 92687

1145 1146 1147 1148 1149

05880 05918 05956 05994 06032

54867 46176 34179 18881 00287

1195 1196 1197 1198 1199

07736 07773 07809 07845 07881

79053 11797 41504 68181 91831

1245 1246 1247 1248 1249

09516 09551 09586 09621 09656

93514 80423 64535 45853 24384

1295 1296 1297 1298 1299

11226 11260 11293 11327 11360

97684 50015 99761 46925 91511

1345 1346 1347 1348 1349

12872 12904 12936 12968 13001

22843 50599 75957 98922 19497

1150

06069 78404

1200

07918 12460

1250

09691 00130

1300

11394 33523

1350

13033 37685

z

5

[(-;I41

2

2

I

100

ELEMENTARY TRANSCENDENTAL FUNCTIONS

NATURAL LOGARITHMS

Table 4.2

In x

7

0.000 0.001

In

2

- Q )

-6.90775 52789 821371 0.002 -6.21460 80984 221917 0.003 -5.80914 29903 140274 0.004 -5.52146 09178 622464

In x 50929 940457 47621 408776 24656 978660 62907 525013 43798 407644

X

z

0.050 0.051 0.052 0.053 0.054

-2.99573 -2.97592 -2.95651 -2.93746 -2.91877

22735 96462 15604 33654 12324

539910 578113 007097 300152 178627

-2.30258 0. i o i -2.29263 0.102 -2.28278 0.103 -2.27302 0.104 -2.26336 0.100

0. 005 3.006 0.007 0.008 0.009

-5.29831 -5.11599 -4.96184 -4.82831 -4.71053

73665 58097 51299 37373 07016

480367 540821 268237 023011 459177

0.055 0.056 0.057 0.058 0.059

-2.90042 -2.88240 -2.86470 -2.84731 -2.83021

20937 35882 40111 22684 78350

496661 469878 475869 357177 764176

0.105 0.106 0.107 0.108 0.109

-2.25379 -2.24431 -2.23492 -2.22562 -2.21640

49288 61848 64445 40518 73967

246137 700699 202309 579174 529934

0,010

-4.60517 -4.50986 -4.42284 -4.34280 -4.26869

01859 00061 86291 59215 79493

880914 837665 941367 206003 668784

0.060 0.061 0.062 0.063 0.064

-2.81341 -2.79688 -2.78062 -2.76462 -2.74887

07167 14148 08939 05525 21956

600364 088258 370455 906044 224652

0.110

0.011 0.012 0.013 0.014

0.111 0.112 0.113 0.114

-2.20727 -2.19822 -2.18925 -2.18036 -2.17155

49131 50776 64076 74602 68305

897208 698029 870425 697965 876416

0.015 0.016 0. 017 0.018 0. 019

-4.19970 -4.13516 -4.07454 -4.01738 -3.96331

50778 65567 19349 35210 62998

799270 423558 259210 859724 156966

0.065 0.066 0.067 0.068 0.069

-2.73336 -2.71810 -2.70306 -2.68824 -2.67364

80090 05369 26595 75738 87743

864999 557115 911710 060304 848777

0.115 0.116 0.117 0.118 0.119

-2.16282 -2.15416 -2.14558 -2.13707 -2.12863

31506 50878 13441 06545 17858

188870 757724 843809 164723 706077

0.020 0.021 0.022 0.023 0.024

-3.91202 -3.86323 -3.81671 -3.77226 -3.72970

30054 28412 28256 10630 14486

281461 587141 238212 529874 341914

0.070 0.071 0.072 0.073 0.074

-2.65926 -2.64507 -2.63108 -2.61729 -2.60369

00369 54019 91599 58378 01857

327781 408216 660817 337459 779673

0.120 0.121 0.122 0.123 0.124

-2.12026 -2.11196 -2.10373 -2.09557 -2.08747

35362 47333 42342 09236 37133

000911 853960 488805 097196 771002

0. 025 0.026 0. 027 0. 028 0.029

-3.68887 -3.64965 -3.61191 -3.57555 -3.54045

94541 87409 84129 07688 94489

139363 606550 778080 069331 956630

0.075 0.076 0.077 0.078 0.079

-2.59026 -2.57702 -2.56394 -2.55104 -2.53830

71654 19386 98571 64522 74265

458266 958060 284532 925453 151156

0.125 0.126 0.127 OI i28 0.129

-2.07944 -2.07147 -2.06356 -2.05572 -2.04794

15416 33720 81925 50150 28746

798359 306591 235458 625199 204649

0.030 0.031 0.032 0.033 0.034

-3.50655 78973 -3.47376 80744 -3.44201 93761 -3.41124 77175 -3.38139 47543

199817 969908 824105 156568 659757

0.080 0.081 0.082 0.083 0.084

-2.52572 -2.51330 -2.50103 -2.48891 -2.47693

86443 61243 60317 46711 84801

082554 096983 178839 855391 388234

0.130 0.131 0.132 0.133 0.134

-2.04022 -2.03255 -2.02495 -2.01740 -2.00991

08285 79557 33563 61507 54790

265546 809855 957662 603833 312257

0.035 0.036 0.037 0.038 0.039

-3.35240 72174 927234 -3.32423 63405 260271 -3.29683 73663 379126 -3.27016 91192 557513 -3.24419 36328 524906

0.085 0.086 0.087 0.088 0.089

-2.46510 -2.45340 -2.44184 -2.43041 -2.41911

40224 918206 79827 286293 71603 275533 84645 039306 89092 499972

0.135 0.136 0.137 0.138 0.139

-2.00248 -1.99510 -1.98777 -1.98050 -1.97328

05005 03932 43531 15938 13458

437076 460850 540121 249324 514453

0.040 0.041 0.042 0.043 0.044

-3.21887 -3.19418 -3.17008 -3.14655 -3.12356

58248 32122 56606 51632 56450

682007 778292 987687 885746 638759

0.090 0.091 0.092 0.093 0.094

-2.40794 -2.39689 -2.38596 -2.37515 -2.36446

56086 57724 67019 57858 04967

518720 652870 330967 288811 121332

0.140 0.141 0.142 0.143 0.144

-1.96611 -1.95899 -1.95192 -1.94491 -1: 93794

28563 53886 82213 06487 19794

728328 039688 808763 222298 061364

0.045 0.046 0.047 0.048 0.049

-3.10109 27892 -3.07911 38824 -3.05760 76772 -3.03655 42680 -3.01593 49808

118173 930421 720785 742461 715104

0.095 0.096 0.097 0.098 0.099

-2.35387 -2.34340 -2.33304 -2.32278 -2.31263

83873 70875 43004 78003 54288

815962 143008 787542 115651 475471

0.145 0.146 0.147 0.148 0.149

-1.93102 -1.92414 -1.91732 -1.91054 -1.90380

15365 86572 26922 30052 89730

615627 738006 034008 180220 366779

0.050 -2.99573 22735 539910

0.100

-2. 30258 50929 940457

0.150

-1.89711 99848 858813

For use of natural logarithms see Examples 4-7.

In 10= 2.30258 50929 940457

[‘-95’11

101

ELEALENTARY TRANSCENDENTAL FZMCTIONS

NATURAL LOGARITHMS

In x

In x

Table 4.2

In x

x 0.150 0.151 0.152 0.153 0.154

-1.89711 -1.89047 -1.88387 -1.87731 -1.87080

99848 54421 47581 73575 26765

858813 672127 358607 897016 685079

0.200 0.201 0.202 0.203 0.204

-1.60943 -1.60445 -1.59948 -1.59454 -1.58963

79124 03709 75815 92999 52851

341004 230613 809323 403497 379207

0.250 0.251 0.252 0.253 0.254

-1.38629 -1.38230 -1.37832 -1.37436 -1.37042

43611 23398 61914 57902 10119

198906 503532 707137 546168 636005

0.155 0.156 0.157 0.158 0.159

-1.86433 -1.85789 -1.85150 -1.84516 -1.83885

01620 92717 94736 02459 10767

628904 326000 338290 551702 619055

0.205 0.206 0.207 0.208 0.209

-1.58474 -1.57987 -1.57503 -1.57021 -1.56542

52998 91101 64857 71992 10270

437289 925560 167680 808191 173260

0.255 0.256 0.257 0.258 0.259

-1.36649 -1.36257 -1.35867 -1.35479 -1.35092

17338 78345 91940 56940 72172

237109 025746 869173 605196 825993

0.160 -1.83258 0.161 -1.82635 0.162 -1.82015 0.163 -1.81400 0.164 -1.80788

14637 09139 89437 50781 88511

483101 976741 497530 753747 579386

0.210 0.211 0.212 0.213 0.214

-1.56064 -1.55589 -1.55116 -1.54646 -1.54177

77482 71455 90043 31132 92639

646684 060706 101246 727119 602856

0.260 0.261 0.262 0.263 0.264

-1.34707 -1.34323 -1.33941 -1.33560 -1.33180

36479 48716 07752 12468 61758

666093 594436 210402 043725 358209

0.165 0.166 0.167 0.168 0.169

-1.80180 -1.79576 -1.78976 -1.78379 -1.77785

98050 74906 14665 12995 65640

815564 255938 653819 788781 590636

0.215 0.216 0.217 0.218 0.219

-1.53711 -1.53247 -1.52785 -1.52326 -1.51868

72508 68712 79254 02161 35491

544743 979720 416775 930480 656362

0.265 0.266 0.267 0.268 0.269

-1.32802 -1.32425 -1.32050 -1.31676 -1.31304

54529 89702 66205 82984 38993

959148 004380 818875 712804 802979

0.170 0.171 0.172 0.173 0.174

-1.77195 -1.76609 -1.76026 -1.75446 -1.74869

68419 17224 08021 36844 99797

318753 794772 686840 843581 676080

0.220 -1.51412 77326 297755 0.221 -1.50959 25774 643842 0.222 -1.50507 78971 098576 0.223 -1.50058 35075 220183 0.224 -1.49610 92271 270972

0.270 0.271 0.272 0.273 0.274

-1.30933 -1.30563 -1.30195 -1.29828 -1.29462

33199 64581 32126 34837 71725

837623 024362 861397 971773 940668

0.175 0.176 0.177 0.178 0.179

-1.74296 -1.73727 -1.73160 -1.72597 -1: 72036

93050 12839 55464 17286 94731

586230 439853 083079 900519 413821

0.225 0.226 0.227 0.228 0.229

-1.49165 -1.48722 -1.48280 -1.47840 -1.47403

48767 02797 52615 96500 32754

777169 098512 007344 276963 278974

0.275 0.276 0.277 0.278 0.279

-1.29098 -1.28735 -1.28373 -1.28013 -1.27654

41813 44132 77727 41652 34971

155658 649871 947986 915000 607714

0.180 0.181 0.182 0.183 0.184

-1.71479 -1.70925 -1: 70374 -1.69826 -1.69281

84280 82477 85919 91261 95213

919267 163113 053417 407161 731514

0.230 0.231 0.232 0.233 0.234

-1.46967 -1.46533 -1.46101 -1.45671 -1.45243

59700 75684 79073 68254 41636

589417 603435 158271 164365 244356

0.280 0.281 0.282 0.283 0.284

-1.27296 -1.26940 -1.26584 -1.26230 -1.25878

56758 06096 82080 83813 10408

128874 483913 440235 388994 209310

0.185 0.186 0.187 0.188 0.189

-1.68739 -1.68200 -1.67664 -1.67131 -1.66600

94539 86052 66621 33161 82639

038122 689358 275504 521878 224947

0.235 6.236 0.237 0.238 0.239

-1.44816 -1.44392 -1.43969 -1.43548 -1.43129

97648 34739 51378 46053 17270

379781 565270 470059 106624 506264

0.285 0.286 0.287 0.288 0.289

-1.25526 -1.25176 -1.24827 -1.24479 -1.24132

60987 34681 30632 47988 85908

134865 622845 225159 461911 697049

0.190 0.191 0.192 0.193 0.194

-1.66073 -1.65548 -1.65025 -1.64506 -1.63989

12068 18509 99069 50900 71199

216509 355072 543555 772515 188089

0.240 0.241 0.242 0.243 0.244

-1.42711 -1.42295 -1.41881 -1.41469 -1.41058

63556 83454 75528 38356 70536

401457 914821 254507 415886 889352

0.290 0.291 0.292 0.293 0.294

-1.23787 -1.23443 -1.23100 -1.22758 -1.22417

43560 20118 14767 26699 55116

016173 106445 138553 650697 434554

0.195 0.196 0.197 0.198 0.199

-1.63475 -1.62964 -1.62455 -1.61948 -1.61445

57204 06197 15502 82482 04542

183903 516198 441485 876018 576447

0.245 0.246 0.247 0.248 0.249

-1.40649 -1.40242 -1.39836 -1.39432 -1.39030

70684 37430 69423 65328 23825

374101 497742 541599 171549 174294

0.295 0.296 0.297 0.298 0.299

-1.22077 -1.21739 -1.21402 -1.21066 -1.20731

99226 58246 31401 17924 17055

423172 580767 794374 767326 914506

0.200

-1.60943 79124 341004

0.250

-1.38629 43611 198906

0.300

-1.20397 28043 259360

X

5

I

~

I

1 I

In 10 = 2.30258 50929 940457

~

I

1

I ~

I

I

I

I I

Ii I

1

i

102

ELEMENTARY TRANSCENDENTAL FUNCTIONS

NATURAL LOGARITHMS

Table 4.2

In

2'

In ns

5

2.

0.300 0.301 0.302 0.303 0.304

-1.20397 -1.20064 -1.19732 -1.19402 -1.19072

28043 50142 82616 24734 75775

259360 332613 072674 727679 759154

0.350 0.351 0.352 0.353 0.354

-1.04982 -1.04696 -1.04412 -1.04128 -1.03845

21244 90555 41033 72220 83658

986777 162712 840400 488403 483626

0.400 0.401 0.402 0.403 0.404

-0.91629 -0.91379 -0.91130 -0.90881 -0.90634

07318 38516 31903 87170 04010

741551 755679 631160 354541 209870

0.305 0.306 0.307 0.308 0.309

-1.18744 -1.18417 -1.18090 -1.17765 -1.17441

35023 01770 75313 54960 40020

147254 297563 949399 085626 843916

0.355 0.356 0.357 0.358 0.359

-1.03563 -1.03282 -1.03001 -1.02722 -1. 02443

74895 45481 94972 22925 28904

067213 301066 024980 814367 938582

0.405 0.406 0.407 0.408 0.409

-0.90386 -0.90140 -0.89894 -0.89648 -0.89404

82118 21193 20935 81045 01229

755979 804044 395421 779754 393353

0.310 0.311 0.312 0.313 0.314

-1.17118 -1.16796 -1.16475 -1.16155 -1.15836

29815 23668 20911 20884 22930

029451 029029 726547 419838 738837

0.360 0.361 0.362 0.363 0.364

-1.02165 -1. 01887 -1.01611 -1.01335 -1.01060

12475 73206 10671 24447 14113

319814 492561 563660 172863 453964

0.410 0.411 0.412 0.413 0.414

-0.89159 -0.88916 -0.88673 -0.88430 -0.88188

81192 20644 19296 76860 93051

837836 859024 326107 211043 568227

0.315 0.316 0.317 0.318 0.319

-1.15518 -1.15201 -1.14885 -1.14570 -1.14256

26401 30653 35051 38962 41761

565040 952249 048564 019602 972925

0. 365 0.366 0.367 9.368 0. 369

-1.00785 -1.00512 -1.00239 -0.99967 -0.99695

79253 19455 34309 23408 86349

996455 807708 275668 132061 416099

0.415 0.416 0.417 0.418 0.419

-0.87947 -0.87707 -0.87466 -0.87227 -0.86988

67587 00187 90571 38464 43590

514388 208738 833356 573807 599993

0.320 0.321 0.322 0.323 0.324

-1.13943 -1.13631 -1.13320 -1.13010 -1.12701

42831 41558 37334 29557 17631

883648 521212 377287 594805 898077

0.370 0.371 0.372 0.373 0.374

-0.99425 -0.99155 -0.98886 -0.98617 -0.98349

22733 32163 14247 68593 94815

438669 747019 089905 383215 676051

0.420 0.421 0.422 0.423 0.424

-0.86750 -0.86512 -0.86274 -0.86038 -0.85802

05677 24452 99649 30999 18237

047231 997556 461252 358591 501793

0.325 0.326 0.327 0.328 0.329

-1.12393 -1.12085 -1.11779 -1.11474 -1.11169

00966 78976 51080 16705 75282

523996 154294 848837 979933 167652

0.375 0.376 0.377 0.378 0.379

-0.98082 -0.97816 -0.97551 -0.97286 -0.97021

92530 61355 00915 10833 90738

117262 922425 341263 625494 997107

0.425 0.426 0.427 0.428 0.429

-0.85566 -0.85331 -0.85097 -0.84863 -0.84629

61100 59327 12657 20834 83600

577202 127666 535125 003403 541201

0.330 0.331 0.332 0.333 0.334

-1.10866 -1.10563 -1.10262 -1.09961 -1.09661

26245 69036 03100 27890 42860

216111 050742 656485 016932 054366

0.380 0.381 0.382 0.383 0.384

-0.96758. -0.96495 -0.96233 -0.95972 -0.95711

40262 59038 46703 02898 27263

617056 554361 755619 014911 944102

0.430 0.431 0.432 0.433 0.434

-0.84397 -0.84164 -0.83932 -0.83701 -0.83471

00702 71888 96907 75509 07448

945289 783893 380267 796472 817322

0.335 0.336 0.337 0.338 0.339

-1.09362 -1.09064 -1.08767 -1. 08470 -1.08175

47471 41190 23486 93834 51716

570706 189328 297753 991183 016868

0. 385 0.386 0.387 0.388 0.389

-0.95451 -0.95191 -0.94933 -0.94674 -0.94417

19446 79095 05859 99393 59353

943528 173062 523552 588636 636908

0.435 0.436 0.437 0.438 0.439

-0.83240 -0.83011 -0.82782 -0.82553 -0.82325

92478 30356 20838 63686 58659

934530 331027 865469 056909 069657

0.340 0.341 0.342 0.343 0.344

-1.07880 -1.07587 -1.07294 -1.07002 -1.06711

96613 28016 45419 48318 36216

719300 986203 195319 161971 087387

0.390 0.391 0.392 0.393 0.394

-0.94160 -0.93904 -0.93649 -0.93394 -0.93140

85398 77189 34391 56671 43696

584449 967713 916745 128758 842032

0.440 0.441 0.442 0.443 0.444

-0.82098 -0.81871 -0.81644 -0.81418 -0.81193

05520 04035 53969 55089 07165

698302 352911 044389 370014 499123

0.345 0.346 0.347 0.348 0.349

-1.06421 -1.06131 -1.05843 -1.05555 -1.05268

08619 65039 04990 27992 33567

507773 244128 352779 076627 797099

0.395 0.396 0.397 0.398 0.399

-0.92886 -0.92634 -0.92381 -0.92130 -0.91879

95140 10677 89982 32736 38620

810152 276565 949466 976993 922736

0.445 0.446 0.447 0.448 0.449

-0.80968 -0.80743 -0.80519 -0.80296 -0.80073

09968 63269 66843 20465 23912

158968 620730 685682 671519 398828

0.350

-1.04982 21244 986777

0.400

-0.91629 07318 741551

0.450

-0.79850 76962 177716

*

In 10 = 2.30258 50929 940457 *See pabge U.

r(-n8i

1'

7'

J

ELEMENTARY TRANSCENDENTAL FUNCTIONS

NATURAL LOGARITHMS

In x

2

In x

X

Table 4.2

In x

X

0.450 0.451 0.452 0.453 0.454

-0.79850 -0.79628 -0.79407 -0.79186 -0.78965

76962 79394 30991 31534 80809

177716 794587 499059 991030 407891

0.500 0.501 0.502 0.503 0.504

-0.69314 -0.69114 -0.68915 -0.68716 -0.68517

71805 91778 51592 51088 90109

599453 972723 904079 823978 107684

0.550 0.551 0.552 0.553 0.554

-0.59783 -0.59602 -0.59420 -0.59239 -0.59059

70007 04698 72327 72774 05922

556204 292226 050417 598023 348532

0.455 0.456 0.457 0.458 0.459

-0.78745 -0.78526 -0.78307 -0.78088 -0.77870

78600 24694 18880 60948 50689

311866 677510 879324 679521 215919

0.505 0.506 0.507 0.508 0.509

-0.68319 -0.68121 -0.67924 -0.67727 -0.67530

68497 86096 42753 38314 72624

067772 946715 909539 036552 316143

0.555 0.556 0.557 0.558 0.559

-0. 58878 -0.58698 -0.58519 -0; 58339 -0.58160

71652 69847 00390 63166 58058

357025 315547 548530 008261 270379

0.460 0.461 0.462 0.463 0.464

-0.77652 -0.77435 -0.77219 -0.77002 -0.76787

87894 72359 03879 82248 07267

989964 854885 003982 959030 558818

0.510 0.511 0.512 0.513 0.514

-0.67334 -0.67138 -0.66943 -0.66747 -0.66553

45532 56887 06539 94338 20135

637656 784326 426293 113675 269719

0.560 0.561 0.562 0.563 0.564

-0.57981 -0.57803 -0.57625 -0.57447 -0.57270

84952 43734 34290 56508 10274

529421 594407 884460 424467 840782

0.465 0.466 0.467 0; 468 0.469

-0.76571 -0.76356 -0.76142 -0; 75928 -0.75715

78733 96448 60213 69830 25105

947807 564912 132397 644903 358577

0.515 ol5I6 0.517 0.518 0.519

-0.66358 -0; 66164 -0.65971 -0.65778 -0.65585

83783 85135 24044 00367 13958

184009 005743 737079 226540 162484

0.565 0.566 0.567 0.568 0.569

-0.57092 -0.56916 -0.56739 -0.56563 -0.56387

95478 12007 59752 38602 48448

356961 789541 543850 609857 558061

0.470 0; 471 0.472 0.473 0.474

-0.75502 -0.75289 -0.75077 -0.74865 -0.74654

25842 71849 62933 98904 79572

780328 657193 965817 902041 870606

0.520 0.521 0.522 0.523 0.524

-0.65392 -0.65200 -0.65008 -0.64817 -0.64626

64674 52372 76910 38149 35946

066640 287701 994983 172142 610949

0.570 0.571 0.572 0.573 0.574

-0.56211 -0.56036 -0; 55861 -0.55686

89181 60693 62876 95622

535412 261268 023392 673975

’ i

-0; 55512 58826 625706

i

0.475 0.476 0.477 0.478 0.479

-0.74444 -0.74233 -0.74023 -0.73814 -0.73605

04749 74247 87880 45464 46815

474958 507170 937958 906811 712218

0.525 0.526 0; 527 0.528 0.529-

-0.64435 -0.64245 -0; 64055 -0.63865 -0.63676

70163 40662 47304 89952 68471

905133 444272 407747 758756 238377

0.575 0.576 0.577 0.578 0.579

-0.55338 -0.55164 -0.54991 -0.54818 -0.54645

52381 76182 30124 14103 28014

847866 862458 740375 097596 091418

0.480 0.481 0.482 0.483 0.484

-0.73396 -0.73188 -0.72981 -0.72773 -0.72567

91750 80088 11649 86253 03722

802004 763759 315367 295644 655053

0.530 0.531 0.532 0.533 0.P34

-0.63487 -0.63299 -0.63111 -0.62923 -0.62735

82724 32577 17896 38548 94400

359695 401982 404927 162925 219422

0.580 0.581 0.582 0.583 0.584

-0.54472 -0.54300 -0.54128 -0.53956 -0.53785

71754 45221 48312 80926 42961

416720 302258 506992 316447 539100

0.485 0.486 0.487 0.488 0.489

-0.72360 -0.72154 -0.71949 -0.71743 -0.71539

63880 66550 11558 98731 27895

446539 816433 995473 289899 072650

0.535 0.536 0.537 0.538 0.539

-0.62548 85320 861305 -0.62362 11179 113351 -0.62175 71844 732724 -0.61989 67188 203526 -0.61803 97080 731399

0.585 0.586 0.587 0.588 0.589

-0.53614 -0.53443 -0.53273 -0.53102 -0.52932

34317 54894 04591 83310 90953

502806 051244 540406 835101 305503

0.490 0.491 0.492 0.493 0.494

-0.71334 -0.71131 -0.70927 -0.70724 -0.70521

98878 11511 65624 61049 97617

774648 876165 898289 394469 942145

0.540 0.541 0; 542 0.543 0.544

-0.61618 -0.61433 -0; 61248 -0.61064 -0.60880

61394 60001 92775 59590 60321

238170 356555 424908 482016 261944

0.590 0.591 0.592 0.593 0.594

-0.52763 -0.52593 -0.52424 -0.52256 -0.52087

27420 92615 86440 08799 59596

823719 760389 981314 844116 194921

0.495 0.496 0.497 0.498 0.499

-0.70319 -0.70117 -0.69916 -0.69715 -0.69514

75164 93522 52528 52019 91832

134468 572096 855083 574841 306184

0.545 0.546 0.547 0.548 0.549

-0.60696 -0.60513 -0.60330 -0.60147 -0.59965

94843 63032 64765 99920 68374

188930 372320 601558 341215 726064

0.595 0.596 0.597 0.598 0.599

-0.51919 -0.51751 -0.51583 -0.51416 -0.51249

38734 46119 81655 45250 36808

365073 167873 895350 315053 666877

0.500

-0.69314 71805 599453

0.550

-0.59783 70007 556204

0.600

-0.51082 56237 659907

In 10 = 2.30258 50929 940457

1

I

I

104

ELEMENTARY TRANSCENDENTAL FUNCTIONS

NATURAL LOGARITHMS

Table 4.2

In x

2

In x

X

29160 56367 07170 81497 79275

In x

0.600 0.601 0.602 0.603 0.604

-0.51082 -0.50916 -0.50749 -0.50583 -0.50418

56237 03444 78336 80822 10810

659907 469295 733160 549516 473221

0.650 0: 6 5 1 0.652 0.653 0.654

-0.43078 -0; 42924 -0.42771 -0.42617 -0.42464

924543 735678 554841 057060 249384

0.700 0.701 0.702 0.703 0.704

-0.35667 -0: 35524 -0.35382 -0.35239 -0.35097

49439 73919 18749 83871 69228

387324 475470 563259 714721 240947

0.605 0.606 0.607 0.608 0.609

-0.50252 -0.50087 -0.49922 -0.49758 -0.49593

68209 52929 64879 03970 70112

512956 128226 226388 159700 722400

0.655 0.656 0.657 0.658 0.659

-0.42312 00433 468851 -0.42159 44900 380480 -0.42007 12604 975265 -0.41855 03476 568199 -0.41703 17444 796298

0.705 0.706 0.707 0.708 0.709

-0.34955 -0.34814 -0.34672 -0.34531 -0.34389

74761 00414 46130 11852 97524

698684 888950 855643 884173 500096

0.610 0.611 0.612 0.613 0.614

-0.49429 -0.49265 -0.49102 -0.48939 -0.48776

63218 83198 29964 03430 03508

147801 105417 698110 459257 349946

0.660 0.661 0.662 0.663 0.664

-0.41551 -0.41400 -0.41248 -0.41098 -0.40947

54439 14391 97230 02887 31295

616658 304508 451288 962745 057032

0.710 0.711 0.712 0.713 0.714

-0.34249 -0.34108 -0.33967 -0.33827 -0: 33687

03089 28491 73675 38585 23166

467759 788962 701613 678411 425527

0.615 0.616 0.617 0.618 0.619

-0.48613 -0.48450 -0.48288 -0.48126 -0.47965

30111 83154 62550 68215 00062

756192 486173 767492 244463 975409

0.665 0.666 0.667 0.668 0.669

-0.40796 -0.40646 -0.40496 -0.40346 -0.40197

82383 56084 52330 71054 12188

262829 417479 665133 454913 539086

0.715 0.716 0.717 0.718 0.719

-0.33547 -0. 33407 -0.33267 -0.33128 -0.32989

27362 51120 94383 57099 39212

881294 214914 825167 339129 610904

0.620 0.621 0.622 0.623 0.624

-0.47803 -0.47642 -0.47481 -0.47320 -0.47160

58009 41970 51862 87601 49106

429998 486583 429576 946839 127094

0.670 0.671 0.672 0.673 0: 674

-0.40047 -0.39898 -0.39749 -0.39600 -0.39452

75665 61420 69384 99493 51680

971253 104553 589875 374092 698300

0.720 0.721 0.722 0.723 0; 724

-0.32850 -0.32711 -0.32573 -0.32434 -0.32296

40669 61416 01400 60568 38865

720361 971880 893108 233724 964207

0.625 0.626 0.627 0.628 0.629

-0.47000 -0.46840 -0.46680 -0.46521 -0.46362

36292 49078 87383 51125 40222

457356 820385 492164 139384 816965

0.675 0.676 0.677 0.678 0.679

-0.39304 -0.39156 -0.39008 -0.38860 -0.38713

25881 22029 40060 79910 41514

096072 391730 698621 417415 234409

0.725 0.726 0.727 0.728 0.729

-0.32158 -0.32020 -0.31882 -0.31745 -0.31608

36241 52641 88014 42307 15469

274623 573410 486177 854511 734789

0.630 0; 631 0.632 0.633 0.634

-0.46203 -0; 46044 -0.45886 -0.45728 -0.45570

54595 94164 58848 48568 63245

965587 409239 352796 379609 449111

0.680 0.681 0.682 0.683 0.684

-0.38566 -0.38419 -0.38272 -0.38126 -0.37979

24808 29728 56211 04194 73613

119847 326247 386750 113470 595866

0.730 0.731 0.732 0.733 0.734

-0.31471

07448 397002

0.635 0.636 0.637 0.638 0.639

-0.45413 -0.45255 -0.45098 -0.44941 -0.44785

02800 67156 56234 69956 08246

894454 420149 099737 373472 046022

0.685 0.686 0.687 0.688 0.689

-0.37833 -0.37687 -0.37542 -0.37396 -0.37251

64407 76512 09867 64410 40079

199118 562518 597877 487934 684785

0.640 0.641 0.642 0.643 0.644

-0.44628 71026 284195 -0.44472 58220 614670 -0.44316 69752 921759 -0.44161 05547 445177 -0.44005 65528 777834

0.690 0.691 0.692 0.693 0.694

-0.37106 -0.36961 -0: 36816 -0.36672 -0.36528

36813 54552 93233 52797 33184

0.645 0. 646 0.647 0.648 0.649

-0.43850 -0.43695 -0.43540 -0.43386 -0.43232

49621 57751 89844 45826 25622

863646 995352 812365 298624 780471

0.650

-0.43078

29160 924543

-0; 31334 18192 323585 -0.31197

47650 208255

-0: 31060 95770 954856 -0.30924

62503 676215

0.735 0.736 0.737 0.738 0.739

-0.30788 -0.30652 -0.30516 -0.30381 -0.30245

47797 51602 73867 14543 73580

693004 532608 928004 816646 339353

908320 144672 644675 922338 753326

0.740 0.741 0.742 0.743 0.744

-0.30110 -0.29975 -0.29840 -0.29705 -0: 29571

50927 46536 60358 92342 42441

839216 860502 147566 643779 490452

0.695 0.696 0.697 0.698 0.699

-0.36384 34334 173449 -0.36240 56186 477174 -0.36096 98682 216132 -0.35953 61762 197646 -0.35810 45367 483268

0.745 0.746 0.747 0.748 0.749

-0.29437 -0.29302 -0: 29169 -0.29035 -0.28901

10606 96787 00938 23010 62954

025775 783762 493197 076598 649176

0.700

-0.35667

0.750

-0.28768

20724 517809

49439 387324

In 10=2.30258 50929 940457

105

ELEMENTARY TRANSCENDENTAL FUNCTIONS

NATURAL LOGARITHMS

In x

X

In x

X

0.750 0.751 0.752 0.753 0.754

-0.28768 -0.28634 -0.28501 -0.28369 -0.28236

20724 96272 89550 00511 29109

517809 180023 322973 822435 741810

0.755 0.756 0.757 0.758 0.759

-0.28103 -0.27971 -0.27839 -0.27707 -0.27575

75297 39028 20255 18933 35015

0.760 0.761 0.762 0.763 0.764

-0.27443 -0.27312 -0.27180 -0.27049 -0.26918

0.765 0.766 0.767 0.768 0.769

Table 4.2

In x

X

0.800 0.801 0.802 0.803 0.804

-0.22314 -0.22189 -0.22064 -0; 21940 -0.21815

35513 43319 66711 05650 60098

142098 137778 156226 353754 031707

0.850 0.851 0.852 0.853 0.854

-0.16251 -0.16134 -0.16016 -0.15899 -0.15782

89294 31504 87521 57314 40851

977749 087629 528213 904579 935672

331123 026041 446883 397654 865071

0.805 0.806 0.807 0.808 0.809

-0.21691 -0.21567 -0.21443 -0.21319 -0.21195

30015 15364 16107 32204 63619

635737 755088 121883 610417 236454

0.855 0.856 0.857 0.858 0.859

-0.15665 -0.15548 -0.15431 -0.15315 -0.15198

38100 49028 73603 11794 63569

453768 403950 843573 941748 978817

68457 19211 87232 72476 74898

017603 204512 954908 976800 156166

0.810 0.811 0.812 0.813 0.814

-0.21072 -0.20948 -0.20825 -0.20702 -0.20579

10313 72248 49388 41694 49129

156526 667241 204591 343265 795968

0.860 0.861 0.862 0.863 0.864

-0.15082 -0.14966 -0.14850 -0.14734 -0.14618

28897 07745 00083 05878 25101

345836 544063 184440 987091 780814

-0.26787 -0.26657 -0.26526 -0.26396 -0.26266

94451 31092 84776 55458 43094

556012 415458 148809 344649 764931

0.815 0.816 0.817 0.818 0.819

-0.20456 -0.20334 -0.20211 -0.20089 -0.19967

71657 09240 61841 29423 11951

412743 180300 221342 793900 290676

0.865 0.866 0.867 0.868 0.869

-0.14502 -0.14387 -0.14271 -0; 14156 -0.14041

57720 03704 63022 35643 21537

502577 197019 015952 217869 167450

0.770 0.771 0.772 0.773 0.774

-0.26136 -0.26006 -0.25877 -0.25747 -0.25618

47641 69054 07289 62303 34053

344075 188076 573609 947151 924099

0.820 0.821 0.822 0.823 0.824

-0.19845 -0.19723 -0.19601 -0.19479 -0.19358

09387 21695 48839 90783 47490

238383 297088 259571 050672 726654

0.870 0: 871 0.872 0.873 0.874

-0.13926 -0; 13811 -0.13696 -0.13581 -0.13467

20673 33021 58550 97231 49033

335076 296343 731574 425348 266016

0.775 0.776 0.777 0.778 0.779

-0.25489 -0.25360 -0.25231 -0.25102 -0.24974

22496 27587 49286 87548 42331

287901 989183 144896 037454 113888

0.825 0.826 0.827 0.828 0.829

-0.19237 -0.19116 -0.18995 -0.18874 -0.18753

18926 05054 05839 21245 51238

474561 611590 584457 968774 468421

0.875 0.876 0.877 0.878 0.879

-0.13353 -0.13238 -0.13124 -0.13010 -0.12897

13926 91880 82866 86853 03812

245226 457456 099540 470204 969601

0.780 0.781 0.782 0.783 0.784

-0.24846 -0.24718 -0.24590 -0.24462 -0.24334

13592 01291 05384 25829 62586

984996 424511 368260 913340 317292

0.830 0.831 0.832 0.833 0.834

-0.18632 -0.18512 -0.18392 -0.18272 -0.18152

95781 54841 28381 16368 18766

914934 266889 609285 152944 233903

0.880 0.881 0.882 0.883 0.884

-0.12783 -0.12669 -0.12556 -0.12443 -0.12329

33715 76530 32229 00783 82163

098849 459575 753457 781770 444936

0.785 0.786 0.787 0.788 0.789

-0.24207 -0.24079 -0.23952 -0.23825 -0.23698

15611 84865 70305 71891 89581

997286 529305 647338 242579 362628

0.835 0.836 0.837 0.838 0.839

-0.18032 -0.17912 -0.17793 -0.17673 -0.17554

35541 66658 12084 71785 45725

312816 974354 926617 000540 149309

0.885 0.886 0.887 0.888 0.889

-0.12216 -0.12103 -0.11991 -0.11878 -0.11765

76339 83283 02966 35359 80434

742075 770561 725576 899670 682325

0.790 0.791 0.792 0.793 0.794

-0.23572 -0.23445 -0.23319 -0.23193 -0.23067

23335 73112 38871 20573 18177

210699 144832 677112 472891 350013

0.840 0.841 0.842 0.843 0.844

-0.17435 -0.17316 -0.17197 -0.17078 -0.16960

33871 36190 52647 83209 27843

447778 091890 398103 802816 861799

0.890 0.891 0.892 0.893 0.894

-0.11653 -0.11541 -0.11428 -0.11316 -0.11204

38162 08515 91464 86981 95038

559515 113277 021277 056380 086229

0.795 0.796 0.797 0.798 0.799

-0.22941 -0.22815 -0.22690 -0.22564 -0.22439

31643 60931 06001 66815 43332

278052 377540 919220 323283 158624

0.845 0.846 0.847 0.848 0.849

-0.16841 -0.16723 -0.16605 -0.16487 -0.16369

86516 59193 45843 46431 60926

249632 759138 300827 902340 707897

0.895 0.896 0; 897 0.898 0.899

-0.11093 -0.10981 -0; 10869 -0.10758 -0.10647

15607 48660 94169 52106 22445

072817 072066 233409 799374 105168

0.800

-0.22314

35513 142098

0.850

-0.16251

0.900

-0.10536

89294 977749

[(-;I21

In 10= 2.30258 50929 940457

05156 578263

[(-;I21

106

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.2

NATURAL LOGARITHMS

In x

2

In x

X

In x

2

0.900 0: 901 0.902 0.903 0.904

-0.10536 -0: 10425 -0.10314 -0.10203 -0.10092

05156 00213 07589 27255 59185

578263 737991 195134 651516 899606

0.950 0.951 0.952 0.953 0.954

-0.05129 -0.05024 -0.04919 -0.04814 -0.04709

32943 12164 02441 03753 16075

875505 367467 907717 279349 338505

1.000 1.001 1.002 1.003 1.004

0.00000 0.00099 0.00199 0.00299 0.00399

00000 95003 80026 55089 20212

000000 330835 626731 797985 695375

0.905 0.906 0.907 0.908 0.909

-0.09982 -0.09871 -0.09761 -0.09651 -0.09541

03352 59729 28288 09003 01848

822109 391577 670004 808438 046582

0.955 0.956 0.957 0.958 0.959

-0.04604 -0.04499 -0.04395 -0.04290 -0.04186

39385 73659 18875 75010 42040

014068 307358 291828 112765 986988

1.005 1.006 1.007 1.008 1.009

0.00498 0.00598 0.00697 0.00796 0.00895

75415 20716 56137 81696 97413

110391 775475 364252 491769 714719

0.910 0.911 0.912 0.913 0.914

-0.09431 -0.09321 -0.09211 -0.09101 -0.08992

06794 23817 52889 93983 47075

712413 221787 078057 871686 279870

0.960 0.961 0.962 0.963 0.964

-0.04082 -0.03978 -0.03874 -0.03770 -0.03666

19945 08700 08283 18671 39843

202551 118446 164306 840115 715914

1.010 1.011 1.012 1.013 1.014

0.00995 0.01093 0.01192 0.01291 0.01390

03308 99400 85708 62252 29051

531681 383344 652738 665463 689914

0.915 0.916 0.917 0.918 0.919

-0.08883 -0.08773 -0.08664 -0.08555 -0.08446

12137 89143 78067 78883 91566

066157 080068 256722 616466 264500

0.965 0.966 0.967 0.968 0.969

-0.03562 -0.03459 -0.03355 -0.03252 -0.03149

71776 14447 67835 31917 06670

431511 696191 288427 055600 913708

1.015 1.016 1.017 1.018 1.019

0.01488 0.01587 0.01685 0.01783 0.01882

86124 33491 71170 99181 17542

937507 562901 664229 283310 405878

0.920 0.921 0.922 0.923 0.924

-0.08338 -0.08229 -0.08121 -0,08012 -0.07904

16089 52427 00554 60444 32073

390511 268302 255432 792849 404529

0.970 0.971 0.972 0.973 0.974

-0.03045 -0.02942 -0.02839 -0.02737 -0.02634

92074 88106 94745 11967 39753

847085 908121 216980 961320 396020

1.020 1.021 1.022 1.023 1.024

0.01980 0.02078 0.02176 0.02273 0.02371

26272 25391 14917 94869 65266

961797 825285 815127 694894 173160

0.925 0.926 0.927 0.928 0.929

-0.07796 -0.07688 -0.07580 -0.07472 -0.07364

15414 10443 17134 35461 65401

697119 359577 162819 959365 682985

0.975 0.976 0.977 0.978 0.979

-0.02531 -0.02429 -0.02326 -0.02224 -0.02122

78079 26925 86269 56089 36364

842899 690446 393543 473197 516267

1.025 1.026 1.027 1.028 1.029

0.02469 0.02566 0.02664 0.02761 0.02858

26125 77467 19309 51670 74568

903715 485778 464212 329734 519126

0.930 0.931 0.932 0.933 0.934

-0.07257 -0. 07149 -0.07042 -0; 06935 -0.06827

06928 60017 24642 00781 88407

348354 050700 965459 347932 532944

0.980 0.981 0.982 0.983 0.984

-0.02020 -0.01918 -0.01816 -0.01714 -0.01612

27073 28194 39706 61588 93819

175194 167740 276712 349705 298836

1.030 1.031 1.032 1.033 1.034

0.02955 0; 03052 0.03149 0.03246 0.03343

88022 92050 86670 71901 47760

415444 348229 593710 375015 862374

0.935 0.936 0.937 0.938 0.939

-0.06720 -0.06613 -0.06507 -0.06400 -0.06293

87496 98025 19967 53299 97997

934501 045450 437149 759124 738741

0.985 0.986 0.987 0.988 0.989

-0.01511 -0.01409 -0.01308 -0.01207 -0.01106

36378 89243 52395 25812 09473

100482 795016 486555 342692 594249

1.035 1.036 1.037 1.038 1.039

0.03440 0.03536 0.03633 0.03729 0.03825

14267 71438 19292 57847 87121

173324 372913 473903 436969 170903

0.940 0.941 0.942 0.943 0.944

-0.06187 -0.06081 -0.05975 -0.05868 -0.05762

54037 21393 00044 89963 91128

180875 967574 057740 486796 366364

0.990 0.991 0.992 0.993 0.994

-0.01005 -0.00904 -0.00803 -0.00702 -0.00601

03358 07446 21716 46149 80723

535014 521491 972643 369645 255630

1.040 1.041 1.042 1.043 1.044

0.03922 0.04018 0.04114 0.04210 0.04305

07131 17896 19433 11760 94894

532813 328318 311752 186354 604470

0.945 0.946 0.947 0.948 0.949

-0.05657 -0.05551 -0.05445 -0.05340 -0.05234

03514 27099 61857 07767 64803

883943 302588 960588 271152 722092

0.995 0.996 0.997 0.998 0.999

-0.00501 -0.00400 -0.00300 -0.00200 -0.00100

25418 80213 45090 20026 05003

235443 975388 202987 706731 335835

1.045 1.046 1.047 1.048 1.049

0.04401 0.04497 0.04592 0.04688 0.04783

68854 33656 89318 35858 73294

167743 427312 883998 988504 141601

0.950

-0.05129 32943 875505

1.000

0.00000 00000 000000

1.050

0.04879 01641 694320

[(-;'"I

In 1.O= 2.30258 50929 940457

[(-67)

1'

107

ELEMENTARY TRANSCENDENTAL FUNCTIONS

NATURAL LOGARITHMS

In x

2

In x

2

Table 4.2

In x

2

1. 050 1.051 1.052 1.053 1.054

0.04879 0.04974 0.05069 0.05164 0.05259

01641 20918 31143 32331 24501

694320 948141 155181 518384 191706

1.100 1.101 1.102 1.103 1.104

0.09531 0.09621 0.09712 0.09803 0.09893

01798 88577 67107 37402 99478

043249 405429 307227 713654 549036

1.150 1.151 1.152 1.153 1.154

0.13976 0.14063 0.14149 0.14236 0.14323

19423 11297 95622 72412 41680

751587 397456 736995 869220 859078

1.055 1.056 1.057 1.058 1.059

0.05354 0.05448 0.05543 0.05638 0.05732

07669 81852 47068 03334 50666

280298 840697 881006 361076 192694

1.105 1.106 1.107 1.108 1.109

0.09984 0.10074 0.10165 0.10255 0.10345

53349 99031 36537 65883 87083

697161 001431 264998 250921 682300

1.155 1.156 1.157 1.158 1.159

0.14410 0.14496 0.14583 0.14669 0.14755

03439 57702 04482 43791 75643

737569 501857 115395 508035 576147

1.060 1.061 1. 062 1.063 1.064

0.05826 0.05921 0.06015 0.06109 0: 06203

89081 18596 39228 50993 53909

239758 318461 197471 598109 194526

1.110 1.111 1.112 1.113 1.114

0.10436 0.10526 0.10616 0.10705 0.10795

00153 05106 01958 90722 71415

242428 574929 283906 934078 050923

1.160 1.161 1.162 1.163 1.164

0.14842 0.14928 0.15014 0.15100 0.15186

00051 17027 26584 28735 23493

182733 157544 297195 365274 092461

1.065 1.066 1.067 1.068 1.069

0.06297 0.06391 0.06485 0.06578 0: 06672

47991 33257 09723 77405 36320

613884 436528 196163 380031 429082

1.115 1.116 1.117 1.118 1.119

0.10885 0.10975 0.11064 0.11154 0.11243

44049 08639 65200 13747 54293

120821 591192 870637 329074 297882

1.165 1.166 1.167 1.168 1.169

0.15272 0.15357 0.15443 0.15529 0.15614

10870 90879 63533 28844 86824

176639 283006 044189 060353 899314

1.070 1.071 1.072 1.073 1.074

0.06765 0.06859 0.06952 0.07045 0.07138

86484 27914 60626 84636 99960

738148 656117 486102 485614 866729

1.120 1.121 1.122 1.123 1.124

0.11332 0.11422 0.11511 0.11600 0.11689

86853 11440 28071 36757 37514

070032 900229 005046 563061 714993

1.170 1.171 1.172 1.173 1.174

0.15700 0.15785 0.15871 0.15956 0.16041

37488 80846 16911 45696 67214

096648 155803 548209 713384 059047

1.075 1.076 1.077 1.078 1.079

0.07232 0.07325 0.07417 0.07510 0.07603

06615 04617 93981 74724 46862

796261 395927 742515 868054 759976

1.125 1.126 1.127 1.128 1.129

0.11778 0.11867 0.11955 0.12044 0.12133

30356 15297 92350 61530 22851

563835 174986 576392 758672 675250

1.175 1.176 1.177 1.178 1.179

0.16126 0.16211 0.16296 0; i638i 0.16466

81475 88494 88282 80852 66215

961223 764352 781397 293950 552339

1.080 1.081 1.082 1: 083 1.084

0.07696 0.07788 0.07881 0.07973 0.08065

10411 65386 11804 49680 79030

361283 570712 242898 188536 174545

1.130 1.131 1.132 1.133 1.134

0.12221 0.12310 0.12398 0.12486 0.12575

76327 21971 59797 89820 12053

242492 339834 809912 458693 055603

1.180 1.181 1.182 1.183 1.184

0.16551 0: 16636 0.16720 0.16805 0.16889

44384 15372 79189 35849 85364

775734 152253 839065 962497 618139

1. 085 1.086 1.087 1.088 1.089

0.08157 0.08250 0.08342 0.08434 0.08525

99869 12215 16081 11484 98439

924229 117437 390724 337509 508234

1.135 1.136 1.137 1.138 1.139

0.12663 0.12751 0.12839 0.12927 0.13015

26509 33202 32147 23357 06844

333660 989596 683990 041392 650451

1.185 1.186 1.187 1.188 1.189

0.16974 0.17058 0.17142 0.17227 0.17311

27745 63005 91156 12209 26177

870945 755337 275310 404532 086448

1. 090 1.091 1.092 1. 093 1.094

0.08617 0.08709 0.08801 0.08892 0.08984

76962 47068 08773 62091 07039

410523 509338 227133 944015 997895

1.140 1.141 1.142 1.143 1.144

0.13102 0.13190 0.13278 0.13365 0.13453

82624 50708 11112 63848 08929

064041 799386 338185 126736 576062

1.190 1.191 1.192 1.193 1.194

0.17395 0.17479 0.17563 0.17647 0.17730

33071 32903 25686 11431 90149

234380 731631 431580 157791 704103

1.095 1.096 1.097 1.098 1.099

0.09075 0.09166 0.09257 0.09349 0.09440

43632 71885 91812 03430 06754

684641 258238 930932 873389 214843

1.145 1.146 1.147 1.148 1.149

0.13540 0.13627 0.13714 0.13802 0.13889

46370 76182 98381 12978 19988

062030 925478 472336 973747 666186

1.195 1.196 1.197 1.198 1.199

0.17814 0.17898 0; i n 8 1 0.18065 0; 18148

61853 26555 84265 34996 78760

834740 284400 758361 932576 453772

1.100

0.09531 01798 043249

1.150

0.13976 19423 751587

1.200

0.18232 15567 939546

[(-;I11

I;-([

'3

In 10 = 2.30258 50929 940457

108

ELEMENTARY TRANSCENDENTAL FUNCTIONS

NATURAL LOGARITHMS

Table 4.2

In x

X

In x

X

In x

X

1.200 1.201 1.202 1.203 1.204

0.18232 0.18315 0.18398 0.18481 0.18564

15567 45430 68361 84369 93468

939546 978465 130158 925418 866293

1.250 1.251 1.252 1.253 1.254

0.22314 0.22394 0.22474 0.22554 0.22633

35513 32314 22726 06759 84422

142098 847741 779068 139312 107290

1.300 1.301 1.302 1.303 1.304

0.26236 0.26313 0.26390 0.26466 0.26543

42644 31995 15437 92981 64635

674911 303682 863775 427081 044612

1.205 1.206 1.207 1.208 1.209

0.18647 0.18730 0.18813 0; 18896 0.18979

95669 90983 79421 60995 35716

426183 049937 153944 126232 326556

1.255 1.256 1.257 1.258 1.259

0.22713 0.22793 0.22872 0.22952 0.23031

55725 20680 79296 31582 77550

837472 460069 081104 782488 622101

1.305 1.306 1.307 1.308 1.309

0.26620 0.26696 0.26773 0.26849 0.26926

30407 90308 44346 92530 34869

746567 542393 420849 350070 277629

1.210 1.211 1.212 1.213 1.214

0.19062 0.19144 0.19227 0.19309 0.19392

03596 64645 18876 66299 06926

086497 709552 471227 619131 373065

1.260 1.261 1.262 1.263 1.264

0.23111 0.23190 0.23269 0.23348 0.23428

17209 50569 77641 98433 12957

633866 827825 190214 683541 246657

1.310 1.311 1.312 1.313 1.314

0.27002 0.27079 0.27155 0.27231 0.27307

71372 02047 26905 45953 59200

130602 815628 218973 206591 624188

1.215 1.216 1.217 1.218 1.219

0.19474 0.19556 0.19638 a. 19721 0.19803

40767 67835 88140 01692 08504

925118 439753 053901 877053 991345

1.265 1.266 1.267 1.268 1.269

0.23507 0.23586 0.23665 0.23744 0.23822

21221 23237 19013 08560 91887

794836 219844 390020 150342 322506

1.315 1.316 1.317 1.318 1.319

0.27383 0.27459 0.27535 0.27611 0.27687

66656 68329 64227 54360 38737

297279 031255 611440 803155 351775

1.220 1.221 1.222 1.223 1.224

0.19885 0.19967 0.20048 0.20130 0.20212

08587 01951 88607 68567 41840

451652 285676 494036 050353 901343

1.270 1.271 1.272 1.273 1.274

0.23901 0.23980 0.24059 0.24137 0.24216

69004 39922 04649 63195 15571

704999 073170 179304 752695 499716

1.320 1.321 1.322 1.323 1.324

0.27763 0.27838 0.27914 0.27990 0.28065

17365 90255 57414 18851 74575

982795 401883 294945 328186 148165

1.225 1.226 1.227 1.228 1.229

0.20294 0.20375 0.20457 0.20538 0.20620

08439 68375 21657 68297 08305

966903 140197 287744 249507 838978

1.275 1.276 1.277 1.278 1.279

0.24294 0.24373 0.24451 0.24529 0.24607

61786 01849 35770 63559 85225

103895 225981 504022 553431 967056

1.325 1.326 1.327 1.328 1.329

0.28141 0.28216 0.28292 0.28367 0.28442

24594 68917 07553 40510 67797

381855 636708 500705 542421 311083

1.230 1.231 1.232 1; 233 1.234

0.20701 0.20782 0.20863 0.20945 0.21026

41693 68472 88651 02241 09254

843261 023165 113280 822072 831961

1.280 1.281 1.282 1.283 1.284

0.24686 0.24764 0.24842 0.24920 0.24998

00779 10229 13584 10856 02052

315258 145972 984783 334994 677694

1.330 1.331 1.332 1.333 1.334

0.28517 0.28593 0.28668 0.28743 0,28818

89422 05394 15721 20411 19474

336624 129746 181974 965716 934320

1.235 1.236 1.237 1.238 1.239

0.21107 0.21188 0.21268 0.21349 0.21430

09700 03590 90934 71742 46026

799405 354990 103508 624044 470054

1.285 1.286 1.287 1.288 1.289

0.25075 0.25153 0.25231 0.25309 0.25386

87183 66258 39286 06276 67239

471831 154276 139896 821619 570503

1.335 1.336 1.337 1.338 1.339

0.28893 0.28968 0.29042 0.29117 0.29192

12918 00751 82981 59617 30667

522129 144540 198061 060367 090355

1.240 1.241 1.242 1.243 1.244

0.21511 0.21591 0;21672 0.21752 0: 21833

13796 75062 29835 78125 19943

169455 224702 112870 285741 169877

1.290 1.291 1.292 1.293 1.294

0.25464 0.25541 0.25619 0.25696 0.25773

22183 71118 14053 50997 81960

735807 645054 604101 897204 787088

1.340 1.341 1.342 1.343 1.344

0.29266 0.29341 0.29416 0.29490 0.29565

96139 56042 10385 59175 02421

628200 995415 494901 411005 009578

1.245 1.246 1.247 1.248 1.249

0.21913 0.21993 0.22074 0.22154 0.22234

55299 84203 06666 22699 32311

166709 652614 978994 472359 434406

1.295 1.296 1.297 1.298 1.299

0.25851 0.25928 0.26005 0.26082 0.26159

06951 25979 39053 46182 47376

515011 300830 343068 818983 884625

1.345 1.346 1.347 1.348 1.349

0.29639 0.29713 0.29787 0.29862 0.29936

40130 72312 98974 20124 35772

538024 225361 282269 901153 256188

1.250

0.22314 35513 142098

1. 300

0.26236 42644 674911

1.350

0.30010 45924 503381

['-;I83

In 1(3 = 2.30258 50929 940457

109

E L E M E N T A R Y TRANSCENDENTAL F U N C T I O N S

NATURAL LOGARITHMS

In x

X

In x

2

Table 4.2

In x

2

1.350 1.351 1.352 1.353 1.354

0.30010 0.30084 0.30158 0.30232 0.30306

45924 50589 49776 43491 31744

503381 780618 207723 886510 900833

1.400 1.401 1.402 1.403 1.404

0.33647 0.33718 0.33789 0.33861 0.33932

22366 62673 97886 28011 53056

212129 548700 123983 203239 036194

1.450 1.451 1.452 1.453 1.454

0.37156 0.37225 0.37294 0.37363 0.37431

35564 29739 19164 03845 83791

324830 020508 026043 881459 113276

1.355 1.356 1.357 1.358 1.359

0.30380 0.30453 0.30527 0.30601 0.30674

14543 91895 63808 30291 91351

316642 182038 527321 365044 690067

1.405 1.406 1.407 1.408 1.409

0.34003 0.34074 0.34145 0.34217 0.34288

73027 87933 97781 02577 02329

857091 884732 322520 358507 165432

1.455 1.456 1.457 1.458 1.459

0.37500 0.37569 0.37637 0.37706 0.37775

59006 29497 95272 56335 12695

234558 744942 130678 864664 406486

1.360 1.361 1.362 1.363 1.364

0.30748 0.30821 0.30895 0.30968 0.31042

46997 97236 42077 81527 15594

479606 693290 273206 143956 212704

1.410 1.411 1.412 1.413 1.414

0.34358 0.34429 0.34500 0.34571 0.34642

97043 86728 71390 51037 25674

900769 706770 710503 023904 743810

1.460 1.461 1.462 1.463 1.464

0.37843 0.37912 0.37980 0.38048 0.38117

64357 11327 53613 91220 24155

202451 685624 275868 379873 391198

1.365 1.366 1. 367 1.368 1.369

0.31115 0.31188 0.31261 0.31334 0.31408

44286 67611 85577 98192 05463

369231 485983 418125 003587 063118

1.415 1.416 1.417 1.418 1.419

0.34712 0. 34783 0.34854 0.34924 0.34995

95310 59952 19607 74281 23981

952009 715280 085434 099358 779056

1.465 1.466 1.467 1.468 1.469

0.38185 0.38253 0.38321 0.38390 0.38458

52424 76034 94991 09301 18971

690306 644597 608447 923238 917403

1.370 1.371 1.372 1.373 1.374

0.31481 0.31554 0.31626 0.31699 0.31772

07398 04005 95293 81267 61938

400335 801773 036935 858340 001576

1.420 1.421 1.422 1.423 1.424

0.35065 0.35136 0.35206 0.35276 0.35346

68716 08491 43313 73191 98129

131694 149636 810491 077153 897840

1.470 1.471 1.472 i. 473 1.474

0.38526 0.38594 0.38662 0: 38730 0.38797

24007 24416 20203 11374 97937

906449 193005 066845 804932 671449

1.375 1.376 1.377 1.378 1.379

0.31845 0.31918 0.31990 0.32063 0.32135

37311 07395 72197 31725 85988

185346 111519 465178 914668 111648

1.425 1.426 1.427 1.428 1.429

0.35417 0.35487 0.35557 0. 35627 0.35697

18137 33219 43384 48639 48989

206138 921042 946994 173926 477304

1.475 1.476 1.477 1.478 1.479

0.38933 0.39001 0.39068 0.39136

57261 30035 98225 61837

782808 492427 260100 286627

1.380 1.381 1.382 1.383 1.384

0.32208 0.32280 0.32353 0.32425 0.32497

34991 78744 17253 50526 78571

691133 271551 454782 826212 954778

1.430 1.431 1.432 1.433 1.434

0.35767 0.35837 0.35907 0.35977 0.36046

44442 35005 20685 01488 77421

718159 743139 384539 460348 774286

1.480 1.481 1.482 1.483 1.484

0.39204 0.39271 0.39339 0.39406 0.39474

20877 75352 25268 70631 11447

760237 856617 738951 557950 451887

1.385 1.386 1.387 1.388 1.389

0.32570 0.32642 0.32714 0.32786 0.32858

01396 f9007 31413 38620 40637

393018 677115 326945 846128 722067

1.435 1.436 1.437 1.436 1.439

0.36116 0.36186 0.36255 0.36325 0.36394

48492 14706 76070 32592 84279

115844 260324 968879 988549 052308

1.485 1.486 1.487 1.488 1.489

0.39541 0.39608 0.39676 0.39743 0.39810

47722 79462 06674 29364 47537

546629 955674 780180 109001 018719

0.32930 37471 426004

0.33217 73123 383321

1.440 1.441 1.442 1.443 1.444

0.36464 0.36533 0.36603 0.36672 0.36741

31135 73170 10388 42797 70404

879093 173850 627573 917338 706345

1.490 1.491 1.492 1.493 1.494

0.39877 0.39944 0.40011 0.40078 0.40145

61199 70357 75017 75185 70867

573678 826014 815691 570533 106256

1.395 1.396 1.397 1.398 1.399

0.33289 ol3336i 0.33432 0.33504 0.33575

1.445 1I446 1.447 1.448 1.449

0.36810 OI36880 0.36949 0.37018 0.37087

93215 11237 24476 32939 36633

643955 365729 493468 635246 385453

1.495 1.496 1.497 1.498 1.499

0.40212 0.40279 0.40346 0.40413 0.40479

62068 48795 31054 08850 82191

426497 522855 374913 950277 204607

1.400

0.33647 22366 212129

1.450

0.37156 35564 324830

1.500

0.40546 51081 081644

1.390 1.391 1.392 1.393 1.394

0: 33002 29129 413059

0.33074 15619 122279 0: 33145 96947 976686

44152 10043 70802 26438 76956

733290 401807 748248 116185 833441

In 10 = 2.30258 50929 940457

[(-;I61

110

ELEMENTARY TRANSCENDENTAL FUNCTIONS

NATURAL LOGARITHMS

Table 4.2

In x

2

In x

L

In x

2

1.500 1.501 1.502 1.503 1.504

0.40546 0.40613 0.40679 0.40746 0.40812

51081 15526 75533 31107 82255

081644 513249 419430 708374 276481

1.550 1.551 1.552 1.553 1.554

0.43825 0.43889 0.43954 0.44018 0.44083

49309 98841 44217 85441 22519

311553 944018 610270 665500 454557

1.600 1.601 1.602 1.603 1.604

0.47000 0.47062 0.47125 0.47187 0.47250

36292 84340 28486 68736 05094

457356 145776 461675 274159 443228

1.505 1.506 1.507 1.508 1.509

0.40879 0.40945 0.41012 0.41078 0.41144

28982 71293 09196 42695 71797

008391 777018 443584 857643 857118

1.555 1.556 1.557 1.558 1.559

0.44147 0.44211 0.44276 0.44340 0.44404

55456 84257 08928 29474 45900

311975 561999 518613 485565 756395

1.605 1.606 1.607 1.608 1.609

0.47312 0.47374 0.47436 0.47499 0.47561

37565 66155 90867 11707 28680

819792 245699 553755 567746 102462

1.510 1.511 1.512 1.513 1.514

0.41210 0.41277 0.41343 0.41409 0.41475

96508 16832 32777 44348 51550

268330 906025 573413 062189 152570

1.560 1.561 1.562 1.563 1.564

0.44468 0.44532 0.44596 0.44660 0.44724

58212 66415 70514 70514 66421

614457 332950 174942 393396 231193

1.610 1.611 1.612 1.613 1.614

0.47623 0.47685 0.47747 0.47809 0.47871

41789 51041 56440 57991 55698

963716 948373 844365 430718 477571

1.515 1.516 1.517 1.518 1.519

0.41541 0.41607 0.41673 0.41739 0.41805

54389 52872 47003 36789 22236

613325 201799 663952 734382 136358

1.565 1.566 1.567 1.568 1.569

0.44788 0.44852 0.44916 0.44980 0.45043

58239 45975 29633 09219 84737

921165 686114 738838 282161 508955

1.615 1.616 1.617 1.618 1.619

0.47933 0.47995 0.48057 0.48119 0.48180

49566 39600 25805 08186 86746

746199 989036 949698 362999 954981

1.520 1.521 1.522 1.523 1.524

0.41871 0.41936 0.42002 0.42068 0.42133

03348 80132 52594 20739 84572

581850 771558 394941 130248 644545

1.570 1.571 1.572 1.573 1.574

0.45107 0.45171 0.45234 0.45298 0.45362

56193 23592 86940 46240 01499

602167 734841 070148 761408 952115

1.620 1.621 1.622 1.623 1.624

0.48242 0.48304 0.48365 0.48427 0.48489

61492 32427 99556 62885 22417

442927 535391 932212 324542 394862

1.525 1.526 1.527 1.528 1.529

0.42199 0.42264 0.42330 0.42395 0.42461

44100 99328 50262 96907 39269

593749 622653 364954 443287 469252

1.575 1.576 1.577 1,578 1.579

0.45425 0.45488 0.45552 0.45615 0.45679

52722 99914 43079 82224 17352

775964 356874 809013 236825 735050

1.625 1.626 1.627 1.628 1.629

0.48550 0.48612 0.48673 0.48735 0.48796

78157 30111 78282 22675 63296

817008 256188 369007 803486 199081

1.530 1.531 1.532 1.533 1.534

0.42526 77354 043441 0.42592 11166 755467 0.42657 40713 183996 0.42722 65998 896771 0.42787 87029 450644

1.580 1.581 1.582 1.583 1.584

0.45742 0.45805 0.45868 0.45932 0.45995

48470 75582 98693 17808 32933

388754 273350 454621 988751 922341

1.630 1.631 1.632 1.633 1.634

0.48858 0.48919 0.48980 0.49041 0.49103

00148 33236 62565 88139 09964

186710 388768 419153 883281 378111

1.535 1.536 1.537 1.538 1.539

0.42853 0.42918 0.42983 0.43048 0.43113

03810 16347 24645 28710 28548

391605 254804 564588 834522 567422

1.585 1.586 1.587 1.588 1.589

0.46058 0.46121 0.46184 0.46247 0.46310

44073 51232 54415 53628 48875

292439 126562 442720 249440 545789

1.635 1.636 1.631 1.638 1.639

0.49164 0.49225 0.49286 0.49347 0.49408

28043 42381 52983 59054 62997

492167 805553 889979 308777 616926

1.540 1.541 1.542 1.543 1.544

0.43178 0.43243 0.43308 0.43372 0.43437

24164 15563 02751 85733 64516

255378 379787 411377 810238 025844

1.590 1.591 1.592 1.593 1.594

0.46373 0.46436 0.46499 0.46561 0.46624

40162 27493 10874 90309 65803

321402 556498 221913 279115 680233

1.640 1.641 1.642 1.643 1,644

0.49469 0.49530 0.49591 0.49652 0.49713

62418 58121 50110 38390 22966

361071 079538 302365 551310 339882

1.545 1.546 1.547 1.548 1.549

0.43502 0.43567 0.43631 0.43696 0.43760

39103 09501 75715 37751 95614

497088 652302 909291 675354 347316

1.595 1.596 1.597 1.598 1.599

0.46687 0.46750 0.46812 0.46875 0.46937

37362 04990 68692 20473 84338

368079 276170 328754 440829 518172

1.645 1.646 1.647 1.648 1.649

0.49774 0.49834 0.49895 0.49956 0.50016

03842 81022 54511 24314 90435

173352 548781 955033 872800 774619

1.550

0.43825 49309 311553

1.600

0.47000

36292 457356

1.650

0.50077 52879 124892

[(-58)6]

In 10 = 2.30258 50929 940457

111

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.2

NATURAL LOGARITHMS

In x

X

In 2

X

In x

2

1.650 1.651 1.652 1.653 1.654

0.50077 0.50138 0.50198 0.50259 0.50319

52879 11649 66750 18188 65966

124892 379910 987863 388871 014996

1.700 1.701 1.702 1.703 1.704

0.53062 0.53121 0.53180 0.53239 0.53297

82510 63134 40301 14016 84284

621704 137247 511824 805512 071240

1.750 1.751 1.752 1.753 1.754

0.55961 0.56018 0.56075 0.56132 0.56189

57879 70533 79925 86059 88939

354227 037148 141997 390974 499913

1.655 1.656 1.657 1.658 1.659

0.50380 0.50440 0.50500 0.50561 0.50621

10088 50559 87384 20567 50112

290262 630679 444259 131032 083074

1.705 1.706 1.707 1.708 1.709

0.53356 0.53415 0.53473 0.53532 0.53590

51107 14490 74438 30953 84041

354801 694874 123036 663781 334538

1.755 1.756 1.757 1.758 1.759

0.56246 0.56303 0.56360 0.56417 0.56474

88569 84952 78092 67992 54657

178291 129249 049601 629853 554211

1.660 1.661 1.662 1.663 1.664

0.50681 0.50741 0.50802 0.50862 0.50922

76023 98306 16964 32002 43423

684519 311578 332564 107906 990168

1.710 1.711 1.712 1.713 1.714

0.53649 oI53707 0.53766 0.53824 0.53882

33705 79949 22777 62193 98201

145685 ioo564 195504 419829 755880

1.760 1.761 1.762 1: 763 1.764

0.56531 0.56588 0.56644 0; 56701 0.56758

38090 18295 95275 69034 39575

500604 140691 139878 157332 845996

1.665 1.666 1.667 1.668 1.669

0.50982 0.51042 0.51102 0.51162 0.51222

51234 55437 56037 53039 46446

324071 446509 686569 365550 796980

1.715 1I716 1.717 1.718 1.719

0.53941 0.53999 0.54057 0.54116 0.54174

30806 60010 85819 08235 27264

179032 657705 153385 620636 007122

1.765 1.766 1.767 1.768 1.769

0.56815 0.56871 0.56928 0.56984 0.57041

06903 71021 31933 89642 44151

852601 817683 375593 154517 776482

1.670 1.671 1.672 1.673 1.674

0.51282 0.51342 0.51402 0.51461 0.51521

36264 22496 05146 84220 59720

286637 132567 625099 046869 672836

1.720 1.721 1.722 1.723 1.724

0.54232 0.54290 0.54348 0.54406 0.54464

42908 55172 64060 69575 71722

253617 294024 055391 457926 415014

1.770 1.771 1.772 1.773 1.774

0.57097 0.57154 0.57210 0.57267 0.57323

95465 43588 88521 30270 68838

857378 006965 828892 920708 873877

1.675 1.676 1.677 1.678 1.679

0.51581 0.51641 0.51700 0.51760 0.51819

31652 00020 64828 26080 83780

770298 598913 410718 450144 954038

1.725 1.726 1.727 1.728 1.729

0.54522 0.54580 0.54638 0.54696 0.54754

70504 65926 57991 46703 32067

833231 612362 645415 818639 011534

1.775 1.776 1.777 1.778 1.779

0.57380 0.57436 0.57492 0.57548 0.57605

04229 36445 65491 91370 14086

273791 699783 725143 917128 836981

1.680 1.681 1.682 1.683 1.684

0.51879 0.51938 0.51998 0.52057 0.52117

37934 88544 35615 79152 19158

151676 264786 507563 086690 201350

1.730 1.731 1.732 1.733 1.734

0.54812 0.54869 0.54927 0.54985 0.55043

14085 92761 68101 40107 08783

096876 940722 402434 334690 583501

1.780 1.781 1.782 1.783 1.784

0.57661 0.57717 0.57773 0.57829 0.57885

33643 50043 63290 73388 80341

039938 075246 486176 810034 578176

1.685 1.686 1.687 1.688 1.689

0.52176 0.52235 0.52295 0.52354 0.52413

55638 88595 18035 43961 66378

043250 796637 638312 737654 256630

1.735 1.736 1.737 1.738 1.739

0.55100 0.55158 0.55215 0.55273 0.55331

74133 36162 94872 50268 02353

988225 381584 589679 432003 721460

1.785 1.786 1.787 1.788 1.789

0.57941 0.57997 0.58053 0.58109 0.58165

84152 84824 82361 76767 68045

316024 543073 772910 513224 265821

1.690 1.691 1.692 1.693 1.694

0.52472 0.52532 0.52591 0.52650 0.52709

85289 00699 12611 21031 25962

349821 164432 840315 509983 298627

1.740 1.741 1.742 1.743 1.744

0.55388 0.55445 0.55503 0.55560 0.55616

51132 96607 38784 77665 13254

264377 860520 303111 378839 867879

1.790 1.791 1.792 1.793 1.794

0.58221 0.58277 0.58333 0.58389 0.58444

56198 41230 23145 01946 77636

526636 785747 527387 229958 366044

1.695 1.696 1.697 1.698 1.699

0.52768 0.52827 0.52886 0.52945 0.53003

27408 25373 19862 10878 98426

324136 697113 520893 891556 897950

1.745 1.746 1.747 1.748 1.749

0.55675 0.55732 0.55790 0.55847 0.55904

45556 74574 00311 22772 41960

543905 174105 519195 333437 364650

1.795 1.796 1.797 1.798 1.799

0.58500 0.58556 0.58611 0.58667 0.58723

50219 19698 86078 49360 09549

402422 800079 014220 494285 683961

1.700

0.53062 82510 621704

1.750

0.55961 57879 354227

1.800

0.58778 66649 021190

In 10 = 2.30258 50929 940457

112

ELEMENTARY TRANSCENDENTAL FLTNCTIONS

NATURAL LOGARITHMS

Table 4.2

In x

X

In x

X

In x

2

1.800 1.801 1.802 1.803 1.804

0.58778 0.58834 0.58889 0.58945 0.59000

66649 20661 71591 19442 64216

021190 938190 861462 211802 404319

1.850 1.851 1.852 1.853 1.854

0.61518 0.61572 0.61626 0.61680 0.61734

56390 60335 61362 59473 54671

902335 913605 239876 032227 436634

1.900 1.901 1.902 1.903 1.904

0.64185 0.64238 0.64290 0.64343 0.64395

38861 00635 59641 15883 69363

723948 062921 231986 140124 691736

1.805 1.806 1.807 1.808 1.809

0.59056 0.59111 0.59166 0.59222 0.59277

05917 44549 80116 12619 42064

848442 947937 100914 699848 131581

1.855 1.856 1.857 1.858 1.859

0.61788 0.61842 0.61896 0.61950 0.62003

46960 36343 22823 06403 87087

593985 640088 705687 916468 393070

1.905 1.906 1.907 1.908 1.909

0.64448 0.64500 0.64553 0.64605 0.64657

20085 68052 13266 55730 95447

786643 320104 182820 260948 436106

1.810 1.811 1.812 1.813 1.814

0.59332 0.59387 0.59443 0.59498 0.59553

68452 91789 12076 29317 43516

777344 012763 207876 727140 929449

1.860 1.861 1.862 1.863 1.864

0.62057 0.62111 0.62165 0.62218 0.62272

64877 39776 11788 80916 47162

251099 601137 548753 194514 633994

1.910 1.911 1.912 1.913 1.914

0.64710 0.64762 0.64814 0.64867 0.64919

32420 66652 98146 26904 52930

585385 581360 292095 581158 307625

1.815 1.816 1.817 1.818 1.819

0.59608 0.59663 0.59718 0.59773 0.59828

54677 62801 67894 69957 68995

168141 791016 140341 552871 359852

1.865 1.866 1.867 1.868 1.869

0.62326 0.62379 0.62433 0.62486 0.62540

10530 71024 28645 83398 35284

957789 251521 595856 066509 734258

1.915 1.916 1.917 1.918 1.919

0.64971 0.65023 0.65076 0.65128 0.65180

76226 96795 14640 29764 42170

326093 486688 635074 612465 255629

1.820 1.821 1.822 1.823 1.824

0.59883 0.59938 0.59993 0.60048 0.60103

65010 58007 47988 34956 18916

887040 454709 377666 965260 521396

1.870 1.871 1.872 1.873 1.874

0.62593 0.62647 0.62700 0.62754 0.62807

84308 30472 73780 14234 51838

664953 919526 554003 619515 162304

1.920 1.921 1.922 1.923 1.924

0.65232 0.65284 0.65336 0.65388 0.65440

51860 58837 63105 64666 63522

396902 864196 481007 066427 435147

1.825 1.826 1.827 1.828 1.829

0.60157 0: 60212 0.60267 0.60322 0.60376

99870 77821 52773 24730 93694

344548 727767 958697 319583 087286

1.875 1.876 1.877 1.878 1.879

0.62860 0.62914 0.62967 0.63020 0.63073

86594 18505 47576 73807 97204

223741 840329 043718 860712 313283

1.925 1.926 1.927 1.928 1.929

0.65492 0.65544 0.65596 0.65648 0.65700

59677 53133 43894 31961 17339

397475 759338 322293 883539 235920

1.830 1.831 1.832 1.833 1.834

0.60431 0.60486 0.60540 0.60595 0.60649

59668 22656 82662 39688 93738

533296 923737 519385 575680 342731

1.880 1.881 1.882 1.883 1.884

0.63127 0.63180 0.63233 0.63286 0.63339

17768 35503 50411 62496 71761

418578 188933 631879 750154 541713

1.930 1.931 1.932 1.933 1.934

0.65752 0.65802 0.65855 0.65907 0.65959

00029 80034 57357 32002 03970

167942 463774 903263 261938 311026

1.835 1.836 1.837 1.838 1.839

0.60704 0.60758 0.60813 0.60867 0.60922

44815 92921 38062 80239 19456

065336 982987 329886 334953 221840

1.885 1.886 1.887 1.888 1.889

0.63392 0.63445 0.63498 0.63551 0.63604

78208 81842 82663 80677 75885

999741 112658 864132 233089 193725

1.935 1.936 1.937 1.938 1.939

0.66010 0.66062 0.66114 0.66165 0.66217

73264 39888 03844 65134 23762

817451 543853 248588 685745 605148

1.840 1.841 1.842 1.843 1.844

0.60976 0.61030 0.61085 0.61139 0.61193

55716 89022 19378 46786 71251

208943 509408 331151 876862 344021

1.890 1.891 1.892 1.893 1.894

0.63657 0.63710 0.63763 0.63816 0.63869

68290 57896 44706 28722 09947

715510 763204 296865 271858 638865

1.940 1.941 1.942 1.943 1.944

0.66268 0; 66320 0.66371 0.66423 0.66474

79730 33041 83698 31703 77060

752368 868732 691332 953030 382473

1.845 1.846 1.847 1.848 1.849

0.61247 0: 61302 0.61356 0: 61410 0.61464

92774 11360 27012 39732 49524

924905 806604 171029 194924 049878

1.895 1.896 1.897 1.898 1.899

0.63921 Ol63974 0.64027 0.64080 0.64132

88385 64038 36909 07001 74318

343897 328301 528772 877361 301488

1.945 1: 946 1.947 1.948 1.949

0.66526 0.66577 0.66628 0.66680 0.66731

19770 59837 97263 32052 64205

704096 638133 900626 203434 254238

1.850

0.61518 56390 902335

1.900

0.64185 38861 723948

1.950

0.66782 93725 756554

In 10= 2.30258 50929 940457

113

E L E M E:NTARY T R A N S C E N D E N T A L FUNCTIONS

Table 4.2

NATURAL LOGARITHMS

In x

X

In x

2

In x

X

1.950 1.951 1.952 1.953 1.954

0.66782 0.66834 0.66885 0.66936 0.66987

93725 20616 44879 66518 85536

756554 409742 909007 945419 205910

2.000 2,001 2,002 2.003 2.004

0.69314 0.69364 0.69414 0.69464 0.69514

71805 70556 66808 60566 51832

599453 015964 930288 836812 226184

2.050 2.051 2.052 2.053 2.054

0.71783 0.71832 0.71881 0.71930 0.71978

97931 74790 49273 21380 91115

503168 902436 085231 367965 063665

1.955 1.956 1.957 1.958 1: 959

0.67039 0.67090 0.67141 0.67192 0.67243

01934 15716 26884 35441 41389

373291 126256 139392 083186 624037

2.005 2.006 2.007 2.008 2.009

0.69564 0.69614 0.69664 0.69713 0.69763

40607 26895 10698 92018 70858

585325 397438 142011 294828 327974

2.055 2.056 2.057 2.058 2.059

0.72027 0.72076 0.72124 0.72173 0.72222

58479 23475 86106 46374 04280

481979 929187 708201 118579 456524

1.960 1.961 1.962 1.963 1.964

0.67294 0.67345 0.67396 0.67447 0.67498

44732 45472 43611 39152 32099

424259 142092 431713 943240 322741

2.010 2.011 2.012 2.013 2.014

0.69813 0.69863 0.69912 0.69962 0.70012

47220 21107 92522 61466 27942

709844 905150 374928 576544 963706

2.060 2.061 2.062 2.063 2.064

0.72270 0.72319 0.72367 0.72416 0.72464

59828 13019 63855 12340 58476

014897 083220 947682 891148 193163

1.965 1.966 1.967 1.968 1.969

0.67549 0.67600 0.67650 0.67701 0.67752

22453 10217 95394 77986 57996

212246 249748 069220 300617 569885

2.015 2.016 2.017 2.018 2.019

0.70061 0.70111 0.70161 0.70210 0.70260

91953 53502 12589 69219 23393

986463 091222 720747 314172 307004

2.065 2.066 2.067 2.068 2.069

0.72513 0.72561 0.72609 0.72658 0.72706

02264 43706 82806 19566 53987

129961 974468 996312 461827 634060

1.970 1.971 1.972 1.973 1I974

0.67803 35427 0.67854 10281 0;67904 82561 0.67955 52270 0.68006 19410

498971 705832 804437 404783 112898

2.020 2.021 2,022 2.023 2.024

0.70309 0.70359 0.70408 0.70458 0.70507

75114 24384 71205 15581 57514

131134 214840 982797 856084 252191

2.070 2.071 2.072 2.073 2.074

0.72754 0.72803 0.72851 0.72899 0.72947

86072 15824 43243 68334 91098

772777 134471 972366 536425 073356

1.975 1.976 1.977 1.978 1.979

0.68056 0.68107 0.68158 0.68208 0.68259

83983 45993 05441 62332 16666

530852 256761 884799 005204 204287

2,025 2.026 2.027 2.028 21 029

0.70556 0.70606 0.70655 0.70705 0: 70754

97005 34058 68674 00857 30608

585025 264916 698630 289367 436777

2.075 2.076 2.077 2.078 2.079

0.72996 0.73044 0.73092 0.73140 0.73188

11536 29653 45448 58926 70088

826616 036422 939753 770357 758759

1.980 1.981 1.982 1.983 1.984

0.68309 0.68360 0.68410 0.68461 0.68511

68447 17677 64359 08495 50088

064439 164139 077962 376589 626811

2.030 2.031 21 032 2.033 2,034

0.70803 0.70852 01 70902 0.70951 0.71000

57930 82825 05297 25346 42976

536960 982476 162355 462096 263682

2.080 2.081 2.082 2.083 2.084

0.73236 0.73284 0.73332 0.73380 0.73428

78937 85474 89701 91622 91238

132266 114974 927771 788349 911205

1.985 1.986 1.987 1.988 1.989

0.68561 0.68612 0.68662 0.68712 0.68763

89141 25656 59635 91082 19998

391537 229808 696798 343823 718351

2.035 2.036 2.037 2.038 2I039

0.71049 0.71098 0.71147 0.71196 0.71245

58188 70986 81372 89348 94915

945583 882763 446688 005331 923181

2.085 2.086 2.087 2.088 2.089

0.73476 0.73524 0.73572 0.73620 0.73668

88552 83565 76280 66700 54825

507648 785807 950637 203923 744287

1.990 1.991 1.992 1.993 1.994

0.68813 0.68863 0.68913 0.68964 0.69014

46387 70250 91591 10412 26715

364010 820592 624065 306577 396466

2.040 2.041 2.042 2.043 2.044

0.71294 0.71343 0.71392 0.71441 0.71490

98078 98838 97197 93158 86723

561250 277077 424738 354850 414580

2. 090 2.091 2.092 2.093 2.094

0.73716 0.73764 0.73812 0.73859 0.73907

40659 24204 05462 84434 61124

767196 464965 026765 638627 483451

1.995 1.996 1.997 1.998 1.999

0.69064 0.69114 0.69164 0.69214 0.69264

40503 51778 60544 66802 70555

418268 892722 336782 263618 182630

2.045 2.046 2.047 2,048 2.049

0.71539 0.71588 0.71637 0.71686 0.71735

77894 66675 53066 37071 18692

947651 294347 791525 772614 567627

2.095 2.096 2.097 2.098 2.099

0.73955 0; 74003 0.74050 0; 74098 0.74146

35533 07664 77519 45099 10408

741011 587957 197829 741054 384959

2.000

0.69314 71805 599453

2.050

0.71783 97931 503168

2.100

0.74193 73447 293773

[(-58’31

For 2>2.1 see Example 5 .

In 10 = 2.30258 50929 940457

114

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.3 X

n 10

10 10 10 10 10 10 10 10

9

1 2

In (1fzlO-n) 0.00000 00000 99999 99999 0.00000 00001 99999 99998 0.00000 00002 99999 99995 0.00000 00003 99999 99992 0.00000 00004 99999 99987 0.00000 00005 99999 99982 0.00000 00006 99999 99975 0.00000 00007 99999 99968 0.00000 00008 99999 99959

-In (1 -210-n)

50000 00000 50000 00000 50000 00000 50000 00000 50000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

00001 00002 00003 00004 00005 00006 00007 00008 00009

00000 00000 00000 00000 00000 00000 00000 00000 00000

00000 00002 00004 00008 00012 00018 00024 00032 00040

50000 00000 50000 00000 50000 00000 50000 00000 50000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

00009 00019 00029 00039 00049 00059 00069 00079 00089

99999 99999 99999 99999 99999 99999 99999 99999 99999

99950 99800 99550 99200 98750 98200 97550 96800 95950

00000 00000 00000 00000 00000 00001 00001 00002 00002

0.00000 0.00000 0,00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

00010 00020 00030 00040 00050 00060 00070 00080 00090

00000 00000 00000 00000 00000 00000 00000 00000 00000

00050 00200 00450 00800 01250 01800 02450 03200 04050

00000 00000 00000 00000 00000 00001 CtO001 00002 00002

8 8 8 8 8 8 8 8 8

0.00000 0.00000 0.00000 0.00000 0.00000 0,00000 0.00000 0.00000 0.00000

00099 00199 00299 00399 00499 00599 00699 00799 00899

99999 99999 99999 99999 99998 99998 99997 99996 99995

95000 80000 55000 20000 75000 20000 55000 80000 95000

00003 00027 00090 00213 00417 00720 01143 01707 02430

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

00100 00200 00300 00400 00500 00600 00700 00800 00900

00000 00000 00000 00000 00001 00001 00002 00003 00004

05000 20000 45000 80000 25000 80000 45000 20000 05000

00003 00027 00090 00213 00417 00720 01143 01707 02430

7 7 7 7 7 7 7 7 7

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

00999 01999 02999 03999 04999 05999 06999 07999 08999

99995 99980 99955 99920 99875 99820 99755 99680 99595

00000 00000 00000 00002 00004 00007 00011 00017 00024

03333 26667 90000 13333 16667 20000 43333 06666 29998

0.00000 0.00000 0.00000 0,00000 0,00000 0.00000 0.00000 0.00000 0.00000

01000 02000 03000 04000 05000 06000 07000 08000 09000

00005 00020 00045 00080 00125 00180 00245 00320 00405

00000 00000 00000 00002 00004 00007 00011 00017 00024

03333 26667 90000 13333 16667 20000 43334 06668 30002

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

09999 19999 29999 39999 49999 59999 69999 79999 89999

99500 98000 95500 92000 87500 82000 75500 68000 59500

00033 00266 00899 02133 04166 07199 11433 17066 24299

33331 66627 99798 32693 65104 96760 27331 56427 83598

0.00000 0.00000 0.00000 0,00000 0.00000 0.00000 0.00000 0,00000 0,00000

10000 20000 30000 40000 50000 60000 70000 80000 90000

00500 02000 04500 08000 12500 18000 24500 32000 40500

00033 00266 00900 02133 04166 07200 11433 17066 24300

33336 66707 00203 33973 68229 03240 39336 76907 16403

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8

RADIX TABLE OF NATURAL LOGARITHMS

1

For n>10, In (lkz10-n) =fzlO-n--z210-2n 2

to 25 D.

ELEMENTARY TRANSCENDENTAL FUNCTIONS

115

RADIX TABLE OF NATURAL LOGARITHMS

Table 4.3

-In (1 -zIO-n)

In ( l + z l O - n )

X

n

1 2 3 4 5 6 7 8 9

5 5 5 5 5 5 5 5 5

0.00000 0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 0.00007 0.00008

99999 99998 99995 99992 99987 99982 99975 99968 99959

50000 00002 50008 00021 50041 00071 50114 00170 50242

33333 66662 99979 33269 66510 99676 32733 65642 98359

08334 66673 75049 33538 42292 01555 11695 73220 86809

0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 0.00007 0.00008 0.00009

00000 00002 00004 00008 00012 00018 00024 00032 00040

50000 00002 50009 00021 50041 00072 50114 00170 50243

33333 66670 00020 33397 66822 00324 33933 67690 01640

58334 66673 25049 33538 92292 01555 61695 73221 36811

1 2 3 4 5 6 7 8 9

4 4 4 4 4 4 4 4 4

0.00009 0.00019 0.00029 0,00039 0.00049 0.00059 0.00069 0.00079 o.00089

99950 99800 99550 99200 98750 98200 97551 96801 95952

00333 02666 08997 21326 41651 71967 14273 70564 42836

30833 26673 97548 93538 04791 61554 34192 33215 09300

53332 06560 58785 06509 40636 42280 77369 90059 94948

0,00010 0.00020 0.00030 0.00040 0.00050 0.00060 0.00070 0.00080 0.00090

00050 00200 00450 00800 01250 01800 02451 03201 04052

00333 02667 09002 21339 41682 72032 14393 70769 43164

35833 06673 02548 73538 29791 41555 39196 13224 14318

53335 06773 61215 20162 92719 97800 69533 63873 66419

1 2 3 4 5 6 7 8 9

3 3 3 3 3 3 3 3 3

0.00099 0.00199 0.00299 0.00399 0.00498 0.00598 0.00697 0.00796 0.00895

95003 80026 55089 20212 75415 20716 56137 81696 97413

33083 62673 79798 69537 11039 77547 36425 49176 71471

53316 05601 47881 45299 07361 46378 24209 87351 90444

68094 82538 16106 90751 21022 20189 95222 07973 31465

0.00100 0.00200 0.00300 0.004oo 0.00501 0.00601 0.00702 0.00803 0.00904

05003 20026 45090 80213 25418 80723 46149 21716 07446

33583 70673 20298 97538 23544 25563 36964 97264 52149

53350 07735 72181 81834 28204 01620 45987 25903 06220

01430 16511 32509 87927 30937 19350 41123 86494 55241

1 2 3 4 5 6 7 8 9

2 2 2 2 2 2 2 2 2

0.00995 0.01980 0.02955 0.03922 0.04879 0.05826 0.06765 0.07696 0.08617

03308 26272 88022 07131 01641 89081 86484 10411 76962

53168 96179 41544 53281 69432 23975 73814 36128 41052

08284 71302 40273 29626 00306 77552 80526 32498 33234

82154 60291 26194 92009 53744 57184 84159 42170 13335

0.01005 0.02020 0.03045 0.04082 0.05129 0.06187 0.07257 0.08338 0.09431

03358 27073 92074 19945 32943 54037 06928 16089 06794

53501 17519 84708 20255 87550 18087 34835 39051 71241

44118 44840 54591 12955 53342 47179 43071 05839 32687

35489 80453 92613 45771 61961 78001 15733 47658 71427

1 2 3 4 5 6 7 8 9

1 1 1 1 1 1 1 1 1

0.09531 0.18232 0.26236 0.33647 0.40546 0.47000 0.53062 0.58778 0.64185

01798 15567 42644 22366 51081 36292 82510 66649 38861

04324 93954 67491 21212 08164 45735 62170 02119 72394

86004 62621 05203 93050 38197 55365 39623 00818 77599

39521 17180 54960 45934 80131 09370 15432 97311 10360

0.10536 0.22314 0.35667 0.51082 0.69314 0.91629 1.20397 1.60943 2.30258

05156 35513 49439 56237 71805 07318 28043 79124 50929

57826 14209 38732 65990 59945 74155 25935 34100 94045

30122 75576 37891 68320 30941 06518 99262 37460 68401

75010 62951 26387 55141 72321 35272 27462 07593 79915

1

0

0.69314 71805 59945 30941 72321

a)

116

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.4

EXPONENTIAL FUNCTION ez

z

e-z

0.000 0.001 0.002 0.003 0.004

1.00000 1,00100 1.00200 1100300 1.00400

00000 05001 20013 45045 80106

00000 66708 34000 03377 77341

000 342 267 026 872

1.00000 0,99900 0.99800 0.99700 0.99600

00000 04998 19986 44955 79893

00000 33374 67333 03372 43991

000 992 067 976 472

0. 005 0.006 0.007 0,008 0.009

1.00501 1.00601 1.00702 1.00803 1.00904

25208 80360 45572 20855 06217

59401 54064 66848 04273 73867

063 865 555 431 814

0.99501 0.99401 0.99302 0.99203 0.99104

24791 79640 44429 19148 03787

92682 53935 33235 37060 72883

313 265 105 630 662

0.010 0.011 0.012 0.013 0.014

1.01005 1.01106 1.01207 1.01308 1.01409

01670 07224 22888 48673 84589

84168 44719 66077 59809 38492

058 556 754 158 345

0.99004 0.98906 0.98807 0.98708 0.98609

1d337 02787 17128 41350 75442

49168 75368 61930 20287 62861

054 698 540 583 903

0.015 0.016 0.017 0.018 0.019

1.01511 1.01612 1.01714 1.01816 1.01918

30646 86854 53223 29763 16486

15718 06094 25240 89793 17408

979 822 748 761 011

0.98511 0.98412 0.98314 OI98216 0.98117

19396 73200 36846 10323 93622

03062 55285 34909 58300 42806

661 115 635 718 006

0.020 0.021 0.022 0.023 0.024

1.02020 1.02122 1.02224 1.02326 1.02429

13400 20516 37844 65395 03178

26755 37528 70438 47217 90621

810 653 235 475 534

0.98019 0.97921 0.97824 0.97726 0.97628

86733 89645 02350 24837 57097

06755 69459 51210 73277 57909

302 588 045 073 314

0.025 0.026 0.027 0.028 0.029

1.02531 1.02634 1.02736 1.02839 1.02942

51205 09484 78027 56844 45944

24428 73442 63489 21425 75130

841 115 392 045 820

0.97530 0.97433 0.97336 0.97238 0.97141

99120 50896 12415 83668 64644

28332 08749 24336 01246 66604

669 328 791 891 825

0.030 0: 031 0.032 0.033 0.034

1.03045 1.03148 1.03251 1.03355 1.03458

45339 55038 75053 05392 46067

53516 86522 05118 41305 28117

856 716 420 472 894

0.97044 0.96947 0.96850 0.96753 0.96657

55335 55730 65820 85595 15046

48508 76025 79197 89032 37506

177 948 585 009 651

0.035 0.036 01 037 0.038 0.039

1.03561 1.03665 1.03769 1.03873 1.03977

97087 58464 30208 12328 04836

99623 90923 38157 78497 50157

260 727 074 733 831

0.96560 0.96464 0.96367 0.96271 0.96175

54162 02934 61353 29408 07091

57566 83123 49053 91199 46366

478 030 452 529 723

0.040 0.041 0.042 0. 043 0.044

1.04081 1.04185 1.04289 1.04393 1.04498

07741 21055 44787 78948 23548

92388 45479 50763 50612 88443

227 549 238 586 779

0.96078 0.95982 0.95886 0195791 0.95695

94391 91299 97805 13900 39574

52323 47798 72484 67030 73046

209 914 552 669 678

0.045 0.046 0.047 0.048 0.049

1.04602 1.04707 1.04812 1.04917 1.05022

78599 44109 20090 06553 03507

08716 56937 79655 24470 40028

943 184 638 516 148

0.95599 0.95504 0.95408 0.95313 0.95218

74818 19621 73975 37870 11296

33099 90714 90371 77504 98504

907 635 141 745 853

0.050

1.05127 10963 76024 040

I;-([

'3

0.95122 94245 00714 009

For use and extension of the table see Examples 8-1 1. z? 2 -22 1 -See Table 7.1 for values of - e and Table 26.1 for - e 2 . J;; Jz;

117

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.4

EXPONENTIAL FUNCTION e-=

e2

X

0.050 0.051 0.052 0.053 0.054

1.05127 1.05232 1.05337 1.05442 1.05548

10963 28932 57425 96451 46021

76024 83203 13364 19355 55080

040 913 763 907 041

0.95122 0.95027 0.94932 0.94838 0.94743

94245 86705 88668 00124 21065

00714 32426 42889 82298 01798

009 935 583 184 300

0.055 0.056 0.057 0.058 0.059

1.05654 1.05759 1.05865 1.05971 1.06077

06146 76837 58103 49957 52407

75494 36611 95500 10287 40159

286 252 087 540 012

0.94648 0.94553 0.94459 0,94364 0.94270

51479 91358 40693 99474 67691

53483 90396 66523 36798 57099

869 267 349 514 754

0.060 0.061 0.062 0.063 0.064

1.06183 1.06289 1.06396 1.06502 1.06609

65465 89141 23447 68392 23987

45359 87195 28033 31305 61505

622 264 669 464 244

0.94176 0.94082 0.93988 0.93894 0.93800

45335 32397 28867 34736 49995

84248 76009 91088 89133 30729

710 730 928 241 488

0.065 0.066 0.067 0.068 0.069

1.06715 1.06822 1.06929 1.07036 1.07143

90243 67171 54781 53084 62091

84192 65993 74600 78774 48346

625 321 202 366 205

0.93706 Oi93613 0.93519 0.93426 0.93332

74633 08642 52013 04735 66800

77403 91618 36776 77213 78201

433 844 558 542 958

0.070 0.071 0.072 0.073 0.074

1.07250 1.07358 1.07465 1.07573 1.07680

81812 12258 53440 05369 68054

54216 68357 63813 14703 96219

479 383 620 476 891

0.93239 oi93146 0.93053 0.92960 0.92867

38199 18921 08958 08300 16938

05948 27592 11205 25792 41287

229 106 732 713 187

0.075 0.076 0.077 0.078 0.079

1.07788 1.07896 1.08004 1.08112 1.08220

41508 25741 20763 26586 43220

84631 5728> 92600 70083 70314

536 889 313 133 717

0.92774 0.92681 0.92588 0.92496 0.92403

34863 62065 98536 44265 99244

28552 59382 06495 43539 45086

892 237 377 280 807

0,080 0.081 0.082 0.083 0.084

1.08328 1.08437 1.08545 1.08654 1.08762

70676 08965 58098 18085 88938

74958 66760 29549 48238 08826

554 341 059 061 156

0.92311 0.92219 Oi92127 0.92035 Oi91943

63463 36914 19586 11472 12560

86635 44608 96348 20124 95124

783 072 654 706 674

0.085 0.086 0.087 0.088 0.089

1.08871 1.08980 1.09089 1.09198 1.09308

70666 63283 66797 81220 06563

98398 05128 18277 28197 26330

696 660 747 460 201

0.91851 0.91759 Oi91667 0.91576 0.91484

22844 42312 70956 08767 55735

01457 20150 33152 23325 74452

356 982 295 631 003

0.090 0.091 0.092 0.093 0.094

1'.09417 1.09526 1.09636 1.09746 1.09855

42837 90052 48220 17352 97459

05210 58465 80816 68081 17173

358 401 975 994 736

0.91393 0.91301 0.91210 0.91119 0.91028

11852 77108 51495 35002 27622

71228 99265 45090 96140 40766

187 803 403 557 940

0.095 0.096 0.097 0.098 0.099

1.09965 1.10075 1.10186 1.10296 1.10406

88551 90639 03736 27851 62995

26102 93978 21010 08507 58881

942 912 606 743 902

0.90937 0.90846 0.90755 0.90664 0.90574

29344 40160 60061 89037 27080

68231 68706 33272 53920 23548

420 150 654 921 496

0.100

1.10517 09180 75647 625

0.90483 74180 35959 573

118

ELEMENTARY TRANSCENDENTAL FUNCTIONS

EXPONENTIAL FUNCTION

Table 4.4 ez

5

e-z

0.100 0.101 0.102 0.103 0.104

1.10517 1.10627 1.10738 1.10849 1.10960

09180 66417 34717 14090 04549

75647 63423 27933 76007 15582

625 521 371 230 540

0.90483 0.90393 0.90302 0.90212 0.90122

74180 30328 95516 69734 52974

35959 85864 68876 81516 21204

573 089 819 470 780

0.105 0.106 0.107 0.108 0.109

1.11071 1.11182 1.11293 1.11404 1.11516

06103 18765 42544 77453 23503

55705 06530 79325 86467 41447

232 839 605 594 807

0.90032 0.89942 0.89852 0.89762 0.89673

45225 46480 56729 75964 04174

86265 75924 90305 30434 98235

613 059 534 876 450

0.110 0,111 0.112 0.113 0.114

1.11627 1.11739 1.11851 1.11963 1.12075

80704 49068 28606 19329 21248

58871 54458 45045 48585 84153

292 258 196 987 031

0.89583 0.89493 0.89404 0.89315 0.89225

41352 87489 42575 06601 79558

96528 29031 00357 16015 82408

251 000 257 519 325

0.115 0.116 0.117 0.118 0.119

1.12187 1.12299 1.12411 1.12524 1.12636

34375 58721 94296 41113 99182

71938 33254 90536 67342 88352

354 738 839 307 913

0.89136 0.89047 0.88958 0.88869 0.88780

61439 52232 51931 60526 78007

06831 97472 63411 14617 61950

368 599 334 364 067

0.120 0.121 0.122 0.123 0.124

1.12749 1.12862 1.12975 1.13088 1.13201

68515 49123 41017 44209 58709

79375 67343 80318 47489 99175

671 967 682 324 153

0.88692 0.88603 0.88514 0.88426 0.88337

04367 39595 83685 36625 98408

17157 92875 02627 60820 82750

516 591 096 866 886

0.125 0.126 0.127 0.128 0.129

1.13314 1.13428 1.13541 1.13655 1.13769

84530 21682 70177 30026 01241

66826 83024 81486 97060 65731

317 976 442 307 582

0.88249 0.88161 0.88073 0.87985 0.87897

69025 48467 36725 33791 39655

84595 83416 97156 44643 45583

403 046 940 827 178

0.130 0.131 0.132 0.133 0.134

1.13882 1.13996 1.14110 1.14224 1.14339

83833 77813 83192 99983 28196

24621 11990 67235 30894 44646

831 306 091 235 898

0.87809 OI87721 0.87634 0.87546 0.87459

54309 77743 09950 50921 00646

20561 91043 79373 08771 03334

324 564 297 138 043

0.135 0.136 0.137 0.138 0.139

1.14453 1114568 1.14682 1.14797 1.14912

67843 18935 81485 55502 41000

51314 94861 20398 74178 03605

488 807 195 672 088

0.87371 0.87284 0.87197 0.87109 0.87022

59116 26324 02261 86917 80284

88034 88719 32109 45798 58251

434 322 436 347 595

0.140 0.141 0.142 0.143 0.144

1.15027 1.15142 1.15257 1.15372 1.15488

37988 46479 66485 98016 41085

57227 84744 37004 66010 24913

268 161 992 407 632

0.86935 0.86848 0.86762 0.86675 0.86588

82353 93116 12564 40688 77480

98805 820 9.7667 890 85914 032 95488 962 59205 017

0.145 0.146 0.147 0.148 0.149

1.15603 1.15719 1.15835 1.15951 1.16067

95702 61880 39630 28963 29892

68021 50796 29855 62973 09085

623 218 297 936 563

0.86502 0.86415 0.86329 0.86243 0.86156

22931 77031 39774 11149 91148

10741 84642 16319 42045 98958

0.150

1.16183 42427 28283 123

288 755 421 443 277

0.86070 79764 25057 807

“-l’lI

119

ELEMENTARY TRANSCENDENTAL FUNCTIONS EXPONENTIAL FUNCTION

Table 4.4

0.150 0.151 0.152 0.153 0.154

1.16183 1.16299 1.16416 1.16532 1.16649

ez 42427 28283 66580 81820 02364 32112 49789 42737 08867 78439

0.155 0.156 0.157 0.158 0.159

1.16765 1.16882 1.16999 1.17116 1.17233

79611 62030 56139 61947 79466

05125 89869 00913 07669 80717

080 080 572 465 662

0.85641 0.85555 0.85470 0.85384 0.85299

51774 91903 40588 97819 63589

83613 71018 17685 68481 69131

531 473 083 735 511

0.160 0.161 0.162 0.163 0.164

1.17351 1.17468 1.17586 1.17703 1.17821

08709 49688 02413 66896 43150

91810 13871 20999 88467 92722

235 592 654 025 171

0.85214 0.85129 0.85044 0.84959 0.84874

37889 20711 12045 11884 20218

66211 07151 40232 14590 80206

338 144 998 263 741

0.165 0.166 0.167 0.168 0.169

1.17939 1.18057 1.18175 1.18293 1.18412

31187 31017 42653 66106 01389

11390 23276 08361 47810 23969

594 011 533 843 378

0.84789 0.84704 0.84619 0.84535 0.84450

37040 62341 96113 38346 89033

87915 89399 37188 84658 86034

828 660 270 733 326

0.170 0.171 0.172 0.173 0.174

1.18530 1.18649 1.18767 1.18886 1.19005

48513 07490 78332 61050 55658

20365 21711 13905 84032 20362

514 746 874 188 660

0.84366 0.84282 0.84197 0.84113 0.84029

48165 15734 91731 76148 68976

96383 71619 68499 44623 58431

682 939 904 201 438

0.175 0.176 0.177 0.178 0.179

1.19124 1.19243 1.19363 1.19482 1.19602

62166 80586 10931 53212 07441

12358 50669 27138 34800 67883

122 468 834 796 563

0.83945 0.83861 0.83777 0.83694 0.83610

70207 79833 97845 24234 58993

69207 37074 22993 88768 97035

358 003 869 073 511

0.180 0.181 0.182 0.183 0.184

1.19721 1.19841 1.19961 1.20081 1.20201

73631 51792 41938 44080 58230

21810 93199 79868 80830 96301

165 657 311 812 462

0.83527 0.83443 0.83360 0.83276 0.83193

02114 53586 13404 81557 58038

11272 95789 15735 37090 26671

021 549 309 951 728

0.185 0.186 0.187 0.188 0.189

1.20321 1.20442 1.20562 1.20683 1.20804

84401 22603 72850 35153 09524

27695 77629 49924 49605 82901

376 686 742 317 811

0.83110 0.83027 0.82944 0.82861 0.82778

42838 35949 37363 47072 65066

52125 81932 85403 32680 94733

659 701 915 634 637

0.190 0.191 0.192 0.193 0.194

1.20924 1.21045 1.21167 1.21288 1.21409

95976 94520 05169 27935 62829

57251 81299 64900 19119 56233

458 533 562 527 085

0.82695 0.82613 0.82530 0.82448 0.82365

91339 25881 68684 19741 79042

43362 51193 91682 39108 68576

318 854 387 186 832

0.195 0.196 0.197 0.198 0.199

1.21531 1.21652 1.21774 1.21896 1.22018

09864 69053 40407 23938 19658

89730 34316 05908 21642 99872

774 229 396 747 499

0.82283 0.82201 0.82119 0.82036 0.81954

46580 22346 06333 98531 98933

56018 78186 12657 37831 32925

384 562 919 021 626

0.200

1.22140 27581 60169 834

5

e-2

123 230 335 886 490

0.86070 0.85984 0.85898 0.85812 0.85727

79764 76986 82807 97218 20210

25057 59205 41123 11393 11457

807 488 482 800 440

0.81873 07530 77981 859 L

"

J

i

120

ELEMENTARY TRANSCENDENTAL FUNCTIONS

EXPONENTIAL FUNCTION

Table 4.4

e-z

ez

X

0.200 0.201 0.202 0.203 0.204

1.22140 1.22262 1.22384 1.22507 1.22629

27581 47718 80081 24682 81534

60169 23327 11358 47499 56210

834 112 099 185 607

0.81873 0.81791 0.81709 0.81627 0.81546

07530 24315 49279 82414 23711

77981 53859 42236 25609 87292

859 397 649 934 668

0.205 0.206 0.207 0.208 0.209

1.22752 1.22875 1.22998 1.23121 1.23244

50649 32039 25717 31695 49985

63177 95312 80752 48867 30254

678 005 723 721 869

0.81464 0.81383 0.81301 0,81220 0.81139

73164 30762 96499 70367 52356

11414 82920 87570 11939 43411

545 720 998 015 427

0.210 0.211 0.212 0.213 0.214

1.23367 1.23491 1.23614 1.23738 1.23862

80599 23550 78850 46512 26547

56743 61394 78503 43600 93452

251 396 512 719 285

0.81058 0.80977 0.80896 0.80815 0.80734

42459 40668 46975 61372 83850

70187 81276 66499 16488 22681

100 291 845 379 475

0.215 0: 216 0.217 0.218 0.219

1.23986 1.24110 1.24234 1.24358 1.24483

18969 23790 41021 70676 12766

66061 00671 37764 19061 87531

862 728 020 978 187

0.80654 0.80573 0.80492 0.80412 0.80332

14401 53018 99693 54416 17181

77326 73479 05001 66559 53626

874 662 467 655 521

0.220 0.221 0.222 0.223 0.224

1.24607 1.24732 1.24857 1.24982 1.25107

67305 34305 13778 05737 10194

87380 64064 64283 35983 28362

820 879 447 926 294

0.80251 0.80171 0.80091 0.80011 0.79931

87979 66802 53643 48492 51343

62478 90195 34659 94554 69365

483 284 186 165 114

0.225 0.226 0.227 0.228 0.229

1.25232 1.25357 1.25482 1.25608 1.25734

27161 56652 98679 53254 20390

91864 78186 40279 32344 09839

345 948 295 151 113

0.79851 0.79771 0.79692 0.79612 0.79532

62187 81016 07822 42598 85335

59377 65674 90139 35453 05093

043 274 647 721 973

0.230 0.231 0.232 0.233 0.234

1.25860 1.25985 1.26111 1.26238 1.26364

00099 92394 97288 14793 44922

29477 49231 28329 27261 07777

863 426 426 349 797

0.79453 0.79373 0.79294 0.79215 0.79136

36025 94660 61233 35735 18158

03334 35242 06683 24314 95583

008 758 687 003 855

0.235 0.236 0.237 0.238 0.239

1.26490 1.26617 1.26744 1.26870 1.26997

87687 43101 11177 91928 85365

32891 66879 75283 24910 83836

756 857 640 818 547

0,79057 08496 0.78978 06739 0.78899 12880 0.78820 26910 0.78741 48823

28735 32802 17609 93770 72687

550 754 706 426 922

0.240 0.241 0.242 0.243 0.244

1.27124 1.27252 1.27379 1.27506 1.27634

91503 10353 41928 86241 43304

21404 08229 16194 18459 89454

692 095 849 570 665

0.78662 0.78584 0.78505 0.78427 0.78348

78610 16263 61775 15137 76342

66553 88345 51829 71556 62862

409 515 496 451 532

0.245 0.246 0.247 0.248 0.249

1.27762 1.27889 1.28017 1.28145 1.28274

13132 95735 91127 99321 20330

04886 41738 78269 94021 69811

611 230 966 162 341

0.78270 0.78192 0.78114 OI78035 0.77957

45382 22249 06935 99432 99733

41868 25477 31376 78034 84700

168 270 458 273 396

0.250

1.28402 54166 87741 484

4

0.77880 07830 71404 868

I I

121

ELEMENTARY TRANSCENDENTAL FUNCTIONS

EXPONENTIAL FUNCTION

Table 4.4 e--5

ez

X

0.250 0.251 0.252 0.253 0.254

1.28402 1.28531 1.28659 1.28788 1.28917

54166 00843 60372 32768 18042

87741 31195 84840 34630 67804

484 317 591 366 299

0.77880 0.77802 0.77724 0.77646 0.77569

07830 23715 47380 78818 18020

71404 58957 68946 23737 46476

868 312 150 828 034

0.255 0.256 0.257 0.258 0.259

1.29046 1.29175 1.29304 1.29433 1.29563

16208 27279 51267 88186 38048

72889 39703 59353 24237 28048

931 974 603 745 373

0.77491 OI77414 0.77336 OI77259 0.77182

64979 19687 82137 52321 30230

61080 92248 65449 06928 43703

928 360 096 045 483

0.260 0.261 0.262 0.263 0.264

1.29693 1.29822 1.29952 1.30082 1.30212

00866 76654 65424 67189 81963

65771 33689 29381 51724 00894

798 967 755 266 131

0.77105 0,77028 0.76951 0.76874 0.76797

15858 09196 10237 18973 35396

03566 15079 07575 11160 56706

284 142 806 303 173

0.265 0.266 0.267 0.268 0.269

1.30343 1.30473 1.30604 1.30734 1.30865

09757 50586 04463 71400 51410

78368 86927 30654 14936 46466

808 883 372 028 646

0.76720 0.76643 0.76567 0.76490 0.76414

59499 91275 30714 77811 32556

75855 01019 65373 02864 48198

698 133 938 015 937

0.270 0.271 0.272 0.273 0.274

1.30996 1.31127 1.31258 1.31390 1.31521

44507 50703 70013 02448 48022

33247 84587 11108 24739 38724

364 979 252 218 500

0.76337 0.76261 0.76185 0.76109 0.76033

94943 64964 42610 27876 20752

36853 05065 89837 28934 60882

186 386 543 278 066

0.275 0.276 0.277 0.278 0.279

1.31653 1.31784 1.31916 1.32048 1.32180

06748 78640 63710 61972 73438

67621 27303 34958 09095 69539

623 324 873 387 151

0.75957 0.75881 0.75805 0.75729 0,75653

21232 29307 44971 68215 99032

24968 61241 10508 14335 15047

476 409 337 547 380

0.280 0.281 0.282 0.283 0.284

1.32312 1.32445 1.32577 1.32710 1.32843

98123 36039 87199 51618 29307

37436 35257 86792 17157 52794

936 318 007 164 731

0.75578 0.75502 0.75427 0.75351 0.75276

37414 83354 36845 97878 66447

55725 80208 33088 59717 06196

472 002 932 250 222

0.285 0.286 0.287 0.288 0.289

1.32976 1.33109 1.33242 1.33375 1.33509

20281 24552 42134 73041 17285

21473 52291 75675 23384 28508

753 710 843 488 403

0.75201 0.75126 0.75051 0.74976 0.74901

42543 26159 17288 15922 22054

19382 46886 37067 39041 02669

630 026 974 301 348

0.290 0.291 0.292 0.293 0.294

1.33642 1.33776 1.33910 1.34044 1.34178

74880 45839 30176 27904 39036

25472 50035 39293 31681 66971

103 199 724 481 373

0.74826 0.74751 0.74676 0.74602 0.74527

35675 56780 85359 21406 64914

78565 18091 73357 97221 43288

215 016 128 444 626

0.295 0.296 0; 297 0.298 0.299

1.34312 1.34447 1.34581 1.34716 1.34850

63586 01568 52994 17878 96234

86276 32052 48097 79554 72910

747 735 594 052 654

0.74453 0.74378 0.74304 0.74230 0.74155

15874 74280 40123 13397 94094

65909 20179 61939 47774 35010

357 599 843 369 502

0.300

1.34985 88075 76003 104

"-:I

0.74081 82206 81717 866

~

122

ELEMENTARY TRANSCENDENTAL FUNCTIONS

EXPONENTIAL FUNCTION

Table 4.4 2

e -2

er

0.300 0.301 0.302 0.303 0.304

1.34985 1.35120 1.35256 1.35391 1.35526

88075 93415 12267 44644 90560

76003 38015 09482 42288 89671

104 618 272 348 692

0.74081 0.74007 0.73933 0.73859 0.73786

82206 77727 80648 90963 08664

81717 46707 89531 70482 50591

866 647 848 549 171

0.305 0.306 0.307 0.308 0.309

1.35662 1.35798 1.35934 1.36070 1.36206

50030 23065 09680 09889 23705

06224 47892 71980 37150 03421

066 497 642 137 961

0.73712 0.73638 0.73565 0.73491 0.73418

33743 66194 06009 53180 07700

91627 56100 07253 09068 26263

732 112 313 726 391

0.310 0.311 0.312 0.313 0.314

1.36342 1.36478 1.36615 1.36752 1.36888

51141 92211 46930 15310 97365

32177 86161 29479 27605 47375

794 378 880 258 624

0.73344 Oi73271 0.73198 0.73124 0.73051

69562 38758 15282 99125 90281

24289 69332 28312 68882 59424

264 482 628 001 881

0.315 0.316 0.317 0.318 0.319

1.37025 1.37163 1.37300 1.37437 1.37575

93109 02556 25719 62612 13249

56996 26042 25458 27561 06039

611 743 804 208 370

0.72978 0.72905 0.72833 0.72760 0.72687

88742 94501 07551 27884 55493

69056 67623 25701 14595 06338

797 797 720 463 254

0.320 0.321 0.322 0.323 0.324

1.37712 1. 37850 1.37988 1.38126 1.38264

77643 55808 47759 53509 73071

35957 93753 57246 05630 19479

085 895 476 003 542

0.72614 0.72542 0.72469 0.72397 0.72325

90370 32509 81903 38543 02423

73690 90141 29902 67915 79842

925 181 880 300 419

0.325 0.326 0.327 0.328 0.329

1.38403 1.38541 1.38680 1.38818 1.38957

06459 53688 14771 89722 78555

80751 72784 80302 89412 87610

421 617 136 403 642

0.72252 0.72180 0.72108 0.72036 0.71964

73536 51874 37430 30197 30167

42072 31715 26607 05301 47075

189 812 016 338 395

0.330 0.331 0.332 0.333 0.334

1.39096 1.39235 1.39375 1.39514 1.39654

81284 97923 28485 72984 31435

63780 08194 12516 69803 74505

266 268 609 608 339

0.71892 0.71820 0.71748 0.71677 0.71605

37334 51690 73228 01941 37822

31926 40570 54443 55698 27208

170 286 294 947 486

0.335 0.336 0.337 0.338 0.339

1.39794 1.39933 1.40073 1.40214 1.40354

03852 90248 90637 05034 33452

22467 10930 38535 05320 12726

023 424 249 540 081

0.71533 0.71462 0.71390 0.71319 0.71248

80863 31058 88399 52878 24490

52559 16057 02720 98282 89191

924 326 095 260 756

0.340 0.341 0.342 0.343 0.344

1.40494 1.40635 iI40776 1.40916 1.41057

75905 32408 02975 87619 86355

63593 62169 14102 26450 07678

797 155 572 817 418

0.71177 0.71105 0.71034 0.70963 0.70892

03227 89082 82047 82115 89280

62609 06409 09177 60208 49510

715 751 248 649 748

0.345 0.346 0.347 0.348 0.349

1.41198 1.41340 1.41481 1.41623 1.41764

99196 26158 67253 22497 91903

67659 17677 70428 40023 41986

075 066 658 522 146

0.70822 0.70751 0.70680 0.70609 0.70539

03534 24871 53282 88762 31302

67799 06501 57749 14384 69954

973 685 463 398 390

0.350

1.41906 75485 93257 248

r(-m L' 6 ' I

0.70468 80897 18713 434

I

123

ELEMENTARY TRANSCENDENTAL FUNCTIONS Table 4.4

EXPONENTIAL FUNCTION e -=

e2

X

0.350 0.351 0. 352 0.353 0.354

1.41906 1.42048 1.42190 1.42333 1.42475

75485 73259 85237 11434 51864

93257 12195 18577 33601 79888

248 200 438 886 380

0.70468 0.70398 0.70328 0.70257 0.70187

80897 37538 01219 71933 49673

18713 55620 76340 77241 55394

434 921 929 521 037

0.355 0.356 0.357 0.358 0.359

1.42618 1.42760 1.42903 1.43046 1.43189

06542 75482 58698 56204 68015

81480 63844 53876 79897 71658

082 915 979 983 672

0.70117 0.70047 0.69977 0.69907 0.69837

34432 26202 24977 30750 43513

08572 35252 34611 06525 51573

398 399 008 666 587

0.360 0.361 0.362 0.363 0.364

1.43332 1.43476 1.43619 1.43763 1.43907

94145 34608 89419 58592 42141

60340 78555 60351 41209 58046

258 a48 880 556 276

0.69767 0.69697 0.69628 0.69558 0.69489

63260 89984 23678 64334 11947

71031 66872 41770 99095 42910

057 738 967 062 621

0.365 0.366 0.367 0. 368 0.369

1.44051 1.44195 1.44339 1.44484 1.44628

40081 52426 79191 20389 76036

49217 54516 15177 73879 74739

078 071 881 090 677

0.69419 0.69350 0.69280 0.69211 0.69142

66508 28012 96450 71816 54104

77978 09755 44391 88730 50308

831 768 707 425 508

0.370 0.371 0.372 0.373 0.374

1.44773 1I44918 1.45063 1.45208 1.45353

46146 30733 29812 43398 71504

63324 86644 93158 32775 56852

462 554 799 223 487

0.69073 0.69004 0.68935 0.68866 0.68797

43306 39415 42425 52328 69118

37354 58789 24222 43955 28979

660 010 423 806 422

0.375 0.376 0.377 0.378 0.379

1.45499 1.45644 1.45790 1.45936 1.46082

14146 71337 43093 29428 30357

18201 71086 71225 75796 43431

336 052 910 632 842

0.68728 0.66660 0.68591 0.68523 0,68454

92787 23330 60738 05006 56126

90972 42301 96020 65870 66278

199 040 141 297 222

0. 380 0.381 0.382 0.383 0.384

1.46228 1.46374 1.46521 1.46667 1.46814

45894 76054 20851 80300 54416

34224 09728 32959 68398 81989

532 512 881 485 380

0.68386 0.68317 0.68249 0.68181 0.68113

14092 78896 50532 28992 14271

12355 19899 05390 85990 79547

858 696 084 553 125

0.385 0.386 0.387 0.388 0.389

1,46961 1.47108 1.47255 1.47402 1.47550

43214 46708 64912 97842 45513

41144 14743 73135 88141 33054

302 133 370 592 939

0.68045 0.67977 0.67909 0.67841 0.67773

06362 05256 10949 23432 42700

04587 80321 26636 64104 13971

638 060 810 077 142

0.390 0.391 0.392 0.393 0.394

1.47698 1.47845 1.47993 1.48141 1.48290

07938 85134 77114 83893 05486

82642 13147 02288 29264 74753

577 180 401 352 084

0.67705 0.67638 0.67570 0.67502 0.67435

68744 01560 41139 87475 40562

98164 39289 60626 86133 40444

700 177 058 209 198

0.395 0.396 0.397 0.398 0.399

1.48438 1.48586 1.48735 1.48884 1.49033

41909 93175 59300 40299 36186

20914 51389 51306 07277 07402

066 667 642 615 565

0.67368 0.67300 0.67233 Oi67166 0.67099

00392 66959 40256 20276 07013

48867 37386 32657 62009 53445

624 438 274 771 901

0.400

1.49182 46976 41270 318 r (- 7 ~ 7

L'

6'

J

0.67032 00460 35639 301

124

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.4

EXPONENTIAL FUNCTION ez

5

e-z

0.400 0.401 0.402 0.403 0.404

1.49182 1.49331 1.49481 1.49630 1.49780

46976 72684 13326 68916 39469

41270 99960 76042 63582 58138

318 030 686 585 840

0.67032 0.66965 0.66898 0.66831 0.66764

00460 00610 07456 20993 41212

35639 37934 90346 23560 68928

301 596 733 309 902

0.405 0.406 0.407 0.408 0.409

1.49930 1.50080 1.50230 1.50380 1.50531

25000 25524 41056 71611 17204

56766 58019 61950 70111 85559

870 898 452 860 754

0.66697 0.66631 0.66564 0,66497 0.66431

68108 01674 41903 88788 42323

58474 24886 01521 22401 22216

400 338 227 888 786

0.410 0.411 0.412 0.413 0.414

1.50681 1.50832 1.50983 1.51134 1.51285

77851 53565 44363 50259 71268

12853 58058 28744 33993 84394

578 082 838 742 526

0.66365 0.66298 0.66232 0.66166 0.66100

02501 69316 42760 22828 09512

36319 00727 52122 27848 65912

366 386 256 372 454

0.415 0.416 0.417 0.418 0.419

1.51437 1.51588 1.51740 1.51892 1.52044

07406 58688 25129 06744 03547

92048 70568 35084 02239 90196

265 894 718 927 115

0.66034 0.65968 0.65902 0.65836 0.65770

02807 02704 09199 22284 41952

04982 84389 44120 24827 67816

886 050 673 158 932

0.420 0.421 0.422 0.423 0.424

1.52196 1.52348 1.52500 1.52653 1.52806

15556 42784 85246 42959 15937

18633 08753 83279 66456 84057

796 926 422 685 126

0.65704 0.65639 0.65573 0.65507 0.65442

68198 01014 40393 86331 38819

15056 09171 93441 11806 08858

782 201 728 293 560

0.425 0.426 0.427 0.428 0.429

1.52959 1.53112 1.53265 1.53418 1.53572

04196 07751 26617 60809 10343

63378 33247 24018 67579 97349

690 382 802 666 347

0.65376 0.65311 0.65246 0.65181 0.65115

97851 63421 35522 14147 99291

29847 20675 27900 98731 81032

271 593 462 930 515

0.430 0.431 0.432 0.433 0.434

1.53725 1.53879 1.54033 ii54187 1.54341

75235 55499 51151 62207 88681

48281 56865 61127 00632 16487

402 110 008 428 038

0.65050 0.64985 0.64920 0.64856 0.64791

90947 89107 93766 04918 22554

23316 74749 85147 04976 85350

545 506 398 075 604

0.435 0.436 0,437 0.438 0.439

1.54496 1.54650 1.54805 1.54960 1.55115

30589 87947 60770 49074 52873

51338 49377 56340 19508 87714

384 427 096 826 108

0.64726 0.64661 0.64597 0.64532 0.64468

46670 77259 14314 57828 07796

78034 35439 10624 57294 29801

611 635 479 565 285

0.440 0.441 0.442 0.443 0.444

1.55270 1.55426 1.55581 1.55737 1.55893

72185 07023 57404 23343 04856

11336 42305 34107 41779 21915

042 879 580 367 277

0.64403 0.64339 0.64274 0.64210 0.64146

64210 27065 96354 72070 54208

83141 72956 55531 87795 27319

359 185 200 233 863

0.445 0.446 0.447 0.448 0.449

1.56049 1.56205 1.56361 1.56517 1.56674

01958 14665 42992 86956 46571

32666 33744 86418 53521 99451

719 035 055 663 356

0.64082 0.64018 0.63954 0.63890 0.63826

42760 37720 39082 46840 60986

32318 61647 74800 31916 93768

776 123 880 208 809

0.450

1.56831 21854 90168 811

,

0.63762 81516 21773 293

125

ELEMENTARY TRANSCENDENTAL FUNCTIONS

EXPONENTIAL FUNCTION e -I

ez

X

Table 4.4

0.450 0.451 0.452 0.453 0.454

1.56831 1.56988 1.57145 1.57302 1.57459

21854 12820 19485 41865 79974

90168 93202 77649 14175 75018

811 449 003 089 775

0.63762 0.63699 0.63635 0.63571 0.63508

81516 08421 41697 81336 27332

21773 77982 25087 26414 45928

293 535 037 293 153

0.455 0.456 0.457 0.458 0.459

1.57617 1.57775 1.57932 1.58090 1.58249

33830 03447 88842 90030 07027

33991 66477 49440 61419 82533

152 911 916 781 449

0.63444 0.63381 0.63318 0.63254 0.63191

79679 38370 03400 74762 52448

48228 98549 62759 07363 99495

182 030 794 387 898

0.460 0.461 0.462 0.463 0.464

1.58407 1.58565 1.58724 1.58883 1.59042

39849 88512 53032 33424 29704

94481 80547 25595 16080 40039

775 101 846 087 147

0.63128 0.63065 0.63002 0.62939 0.62876

36455 26773 23399 26325 35544

06925 98054 41912 08162 67098

969 154 291 872 411

0.465 0.466 0.467 0.468 0.469

1.59201 1.59360 1.59520 1.59679 1.59839

41888 69993 14034 74026 49987

87101 48484 17000 87052 54640

182 772 511 601 444

0.62813 Oi62750 0.62688 0.62625 0.62562

51051 72840 00904 35236 75831

89640 47340 12377 57555 56310

814 750 027 956 730

0.470 0.471 0.472 0.473 0.474

1.59999 1.60159 1.60319 1.60480 1.60640

41932 49876 73837 13829 69870

17360 74406 26574 76258 27460

241 589 077 891 416

0.62500 Oi62437 0.62375 0.62313 0.62250

22682 75784 35129 00711 72525

82700 11411 17752 77657 67686

796 229 104 876 754

0.475 0.476 0.477 0.478 0.479

1.60801 1.60962 1.61123 1.61284 1.61445

41974 30159 34440 54833 91355

85782 58436 54240 83623 58623

835 741 740 064 174

0.62188 Oi62126 0.62064 0.62002 0.61940

50564 34822 25292 21969 24846

65020 47461 93437 81993 92799

075 685 314 957 250

0.480 0.481 0.482 0.483 0.484

1.61607 1.61769 1.61930 1.62092 1.62255

44021 12849 97853 99050 16456

92893 01700 01927 12074 52261

382 456 238 265 382

0.61878 0.61816 0.61754 0.61692 0.61631

33918 49177 70617 98233 32019

06140 02925 64680 73547 12289

853 827 018 436 639

0.485 0.486 0.487 0.488 0.489

1.62417 1.62579 1.62742 1.62905 1.63068

50088 99962 66093 48499 47196

44229 11341 78585 72574 21548

364 538 406 272 865

0.61569 0.61508 0.61446 0.61385 0.61323

71967 18073 70329 28730 93269

64285 13528 44630 42817 93927

113 659 776 043 508

0.490 0.491 0.492 0.493 0.494

1.63231 1.63394 1.63558 1.63722 1.63885

62199 93526 41192 05213 85607

55378 05565 05239 89170 93758

970 057 912 270 453

0.61262 0.61201 0.61140 0.61079 0,61018

63941 40740 23658 12690 07830

84416 01349 32408 65884 90679

069 867 668 251 799

0.495 0.496 0.497 0.498 0.499

1.64049 1.64213 1.64378 1.64542 1.64707

82390 95578 25187 71234 33735

57044 18705 20061 04072 15345

002 315 292 971 173

0.60957 0.60896 0.60835 0.60774 0.60713

09072 16410 29838 49349 74937

96309 72896 11176 02490 38789

287 868 269 178 634

0.500

1.64872 12707 00128 147

0.60653 06597 12633 424

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.4

EXPONENTIAL FUNCTION e-=

ez

X

0.500 0.501 0.502 0.503 0.504

1.64872 1.65037 1.65202 1.65367 1.65532

12707 08166 20128 48611 93631

00128 06319 83464 82760 57054

147 214 418 175 920

0.60653 0.60592 0.60531 0.60471 0.60410

06597 44322 88106 37943 93828

12633 17187 46224 94122 55864

424 470 228 075 709

0.505 0.506 0.507 0.508 0.509

1.65698 1.65864 1.66030 1.66196 1.66362

55204 33347 28076 39409 67361

60850 50305 83232 19105 19058

766 156 516 918 736

0.60350 0.60290 0.60229 0.60169 0.60109

55754 23715 97704 77717 63747

27040 03842 83065 62109 38975

541 093 390 362 237

0.510 0.511 0.512 0.513 0.514

1.66529 1.66695 1I66862 1.67029 1167196

11949 73190 51101 45698 56998

45886 64047 39666 40535 36112

308 601 871 333 826

0.60049 0.59989 0.59929 0.59869 0.59809

55788 53833 57878 67916 83940

12265 81185 45538 05729 62761

943 502 434 153 369

0.515 0.516 0.517 0.518 0.519

1.67363 1.67531 1.67698 1.67866 1.68034

85017 29773 91283 69562 64627

97529 97587 10762 13205 82744

486 414 348 342 439

0.59750 0.59690 0.59630 0.59571 0.59511

05946 33926 67876 07789 53658

18237 74358 33920 00321 77550

489 019 965 238 053

0.520 0.521 0.522 0.523 0.524

1.68202 1.68371 1.68539 1.68708 1.68876

76496 05186 50712 13093 92344

98886 42818 97408 47211 78463

347 123 851 326 738

0.59452 0.59392 0.59333 0.59273 0.59214

05479 63245 26951 96589 72156

70194 83436 23052 95412 07481

339 138 015 460 294

0.525 0.526 0.527 0.528 0.529

1.69045 1.69215 li69384 1.69553 1.69723

88483 01527 31492 78396 42254

79091 38708 48618 01819 93001

359 232 855 881 803

0.59155 0.59096 0.59037 0.58978 0.58919

53643 41046 34359 33576 38690

66815 81562 60463 12850 48643

082 533 912 450 749

0.530 0.531 0.532 0.533 0.534

1.69893 1.70063 1.70233 1.70403 1.70574

23086 20906 35733 67583 16474

18550 76549 66781 90727 51574

654 702 146 817 883

0.58860 0.58801 0.58742 0.58684 0.58625

49696 66589 89361 18008 52523

78355 13085 64523 44946 67219

196 372 463 670 626

0.535 0.536 0.537 0.538 0.539

1.70744 1.70915 1.71086 1.71257 1.71429

82422 65445 65559 82781 17130

54211 05232 12940 87347 40175

545 748 887 510 036

0.58566 0.58508 0.58449 0.58391 0.58333

92901 39135 91221 49151 12920

44793 91706 22582 52628 97638

803 932 409 716 836

0.540 0.541 0.542 0.543 0.544

1.71600 1.71772 1.71944 1.72116 1.72288

68621 37273 23102 26125 46360

84858 36547 12106 30118 10887

460 069 159 747 296

0.58274 0.58216 0.58158 0.58100 0.58042

82523 57953 39205 26273 19151

73989 98641 89137 63601 40742

665 430 107 839 351

0.545 0.546 0.547 0.548 0.549

1.72460 1.72633 1.72806 1.72978 1.73152

83823 38533 10506 99760 06311

76435 50509 58581 27847 87233

429 656 095 197 477

0.57984 0.57926 0.57868 0.57810 0.57752

17833 22313 32586 48646 70487

39846 80782 83997 70519 61954

373 055 389 631 718

0.550

1.73325 30178 67395 237

0.57694 98103 80486 695

[(-:I8]

127

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.4

EXPONENTIAL FUNCTION e -=

ez

X

0.550 0.551 0.552 0.553 0.554

1.73325 1.73498 1.73672 1.73846 1.74019

30178 71378 29927 05843 99144

67395 00719 21325 65069 69542

237 302 750 647 780

0.57694 0.57637 0.57579 0.57522 0.57464

98103 31489 70638 15546 66205

80486 48877 90464 29163 89465

695 132 548 839 693

0.555 0.556 0.557 0.558 0.559

1.74194 1.74368 1.74542 1.74717 1.74892

09847 37970 83529 46543 27028

74075 19737 49342 07446 40349

399 955 837 121 310

0.57407 0.57349 0.57292 0.57235 0.57178

22611 84758 52640 26251 05586

96436 75715 53518 56633 12420

024 391 425 257 941

0.560 0.561 0.562 0.563 0.564

1.75067 1.75242 1.75417 1.75593 1.75768

25002 40484 73489 24037 92143

96101 24499 77091 07179 69816

083 041 459 036 648

0.57120 0.57063 0.57006 0.56949 0.56892

90638 81402 77873 80045 87911

48814 94320 78013 29541 79121

886 280 522 648 761

0.565 0.566 0.567 0.568 0.569

1.75944 1.76120 1.76297 1.76473 1.76649

77827 81105 01995 40515 96682

21815 21742 29927 08459 21189

104 902 989 520 621

0.56836 0.56779 0.56722 0.56665 0.56609

01467 20706 45624 76213 12469

57540 96153 26884 82224 95233

464 288 125 657 792

0.570 0.571 0; 572 0.573 0. 574

1.76826 1.77003 1.77180 1.77357 1.77535

70514 62029 71244 98177 42846

33735 13479 29574 52941 56273

152 471 208 024 392

0.56552 0.56496 0.56439 0.56383 0,56326

54386 01959 55181 14047 78551

99537 29326 19358 04955 22004

097 229 370 664 648

0.575 0.576 0.577 0.578 0.579

1.77713 1.77890 1.78068 1.78246 1.78425

05269 85463 83445 99235 32850

14038 02478 99612 85240 40940

362 341 864 377 016

0.56270 0.56214 0.56158 0.56101 0.56045

48680 24451 05837 92838 85449

06955 96822 29181 42170 74490

693 437 224 538 445

0.580 0.581 0.582 0.583 0.584

1.78603 1.78782 1.78961 1.79140 1.79319

84307 53624 40820 45912 68918

50073 97786 71010 58466 50662

382 336 772 414 599

0.55989 0.55933 0.55877 0.55822 0.55766

83665 87480 96888 11884 32463

65402 54726 82846 90701 19791

033 843 320 245 179

0.585 0: 586 0.587 0.588 0.589

1.79499 1.79678 1.79858 1.80038 1.80218

09856 68744 45599 40441 53286

39900 20272 87669 39776 76077

067 757 600 313 198

0.55710 0,55654 0.55599 0.55543 0.55488

58618 90344 27635 70486 18892

12173 10464 57836 98018 75294

905 868 621 264 892

0.590 0;591 0.592 0.593 0.594

1.80398 1.80579 1.80760 1.80940 1.81121

84153 33061 00026 85067 88202

97856 08202 12004 15959 28572

940 413 477 787 596

0.55432 0.55377 0.55321 0.55266 0.55211

72847 32345 97380 67948 44043

34507 21050 80873 60481 06930

035 107 848 771 610

0.595 0.596 0.597 0.598 0.599

1.81303 1.81484 1.81666 1.81847 1.82029

09449 48827 06353 82045 75923

60156 22836 30550 99051 45908

569 588 566 264 101

0.55156 0.55101 0.55046 0.54991 0.54936

25658 12789 05431 03577 07222

67829 91340 26176 21601 27429

766 753 649 542 984

0.600

1.82211 88003 90508 975 2 ,[,]

0.54881 16360 94026 433

r ( -a 7 7 L'

6'

1

ELEMENTARY TRANSCENDENTAL FUNCTIONS

EXPONENTIAL FUNCTION

Table 4.4

e-"

ez

X

0.600 0.601 0.602 0.603 0.604

1.82211 1.82394 1.82576 1.82759 1.82942

88003 18305 66846 33645 18719

90508 54062 59597 31970 97859

975 083 740 203 499

0.54881 0.54826 0.54771 0.54716 0.54662

16360 30987 51097 76683 07741

94026 72304 13727 70305 94597

433 710 448 543 605

0.605 0.606 0.607 0.608 0.609

1.83125 1.83308 1.83491 1.83675 1.83859

22088 43770 83782 42143 18872

85773 26048 50853 94189 91893

244 479 497 676 312

0.54607 0.54552 0.54498 0.54443 0.54389

44266 86251 33692 86582 44917

39709 59293 07547 39217 09589

413 368 943 140 946

0.610 0.611 0.612 0.613 0.614

1.84043 1.84227 1.84411 1.84596 1.84780

13987 27507 59448 09832 78674

81637 02933 97134 07433 78869

455 750 270 364 496

0.54335 0.54280 0.54226 0.54172 0.54118

08690 77897 52533 32591 18066

74499 90323 13983 02940 15202

787 981 200 922 890

0.615 0.616 0.617 0.618 0.619

1.84965 1.85150 1.85335 1.85521 1.85707

65995 71812 96145 39011 00429

58327 94538 38085 41401 58772

090 381 258 120 725

0.54064 0.54010 0.53956 0.53902 0.53848

08953 05246 06940 14030 26510

09316 44370 79994 76357 94167

571 616 313 053 789

0.620 0.621 0.622 0.623 0.624

1.85892 1.86078 1.86264 1.86451 1.86637

80418 78996 96182 31995 86452

46342 62108 65928 19523 86472

044 121 925 215 402

0.53794 0.53740 0.53686 0.53633 0.53579

44375 67620 96238 30226 69576

94674 39663 91459 12924 67456

492 618 568 149 037

0.625 0.626 0.627 0.628 0.629

1.86824 1.87011 1.87198 1.87385 1.87573

59574 51378 61883 91108 39071

32222 24085 31242 24743 77511

407 530 321 442 543

0.53526 0.53472 0.53419 0.53365 0.53312

14285 64346 19754 80505 46591

18990 31997 71484 02990 92592

242 571 093 602 086

0.630 0.631 0.632 0.633 0.634

1.87761 1.87948 1.88136 1.88325 1.88513

05792 91289 95581 18687 60625

64343 61910 48763 05331 13924

132 454 361 198 678

0.53259 0.53205 0.53152 0.53099 0.53046

18010 94754 76818 64198 56888

06897 13047 78717 72114 61974

190 683 927 344 883

0.635 0.636 0.637 0.638 0.639

1.88702 1.88891 1.89079 1.89269 1.89458

21414 01074 99623 17079 53463

58737 25849 03226 80722 50084

766 565 199 703 912

0.52993 0.52940 0.52887 0.52834 0.52781

54883 58177 66765 80641 99802

17568 08694 05682 79390 01207

489 574 485 975 673

0.640 0.641 0.642 0.643 0.644

1.89648 1.89837 1.90027 1.90217 1.90408

08793 83087 76365 88646 19949

04951 40855 55225 47391 18580

353 140 865 502 301

0.52729 0.52676 0.52623 0.52571 0.52518

24240 53951 88930 29172 74670

43048 77357 77105 15790 67436

557 426 369 242 140

0.645 0.646 0.647 0.648 0.649

1.90598 1.90789 1.90980 1.91171 1.91362

70292 39696 28178 35758 62456

71922 12453 47112 84748 36119

692 188 287 384 674

0.52466 0.52413 0.52361 0.52309 0.52256

25421 81418 42656 09131 80836

06592 08335 48263 02500 47694

872 432 478 807 830

0.650

1.91554 08290 13896 070

[(-p]

0.52204 57767 61016 048

129

ELEMENTARY TRANSCENDENTAL FUNCTIONS

EXPONENTIAL FUNCTION 5

Table 4.4 e -2

ez

0.650 0.651 0.652 0.653 0.654

1.91554 1.91745 1.91937 1.92129 1.92321

08290 73279 57443 60800 83371

13896 32661 08913 61070 09468

070 108 867 883 067

0.52204 0.52152 0.52100 0.52048 0.51996

57767 39919 27286 19862 17644

61016 20157 03334 89283 57261

048 530 394 277 823

0. 655 0.656 0.657 0.658 0.659

1.92514 1.92706 1.92899 1.93092 1.93285

25173 86227 66552 66167 85091

76362 85934 64290 39462 41411

630 997 740 496 902

0.51944 0.51892 0.51840 0.51788 0.51736

20625 28801 42166 60715 84443

87048 58940 53755 52831 38021

156 364 974 438 612

0.660 0.661 0.662 0.663 0.664

1.93479 1.93672 1.93866 1.94060 1.94254

23344 80944 57912 54266 70027

02031 55146 36517 83841 36754

522 776 879 774 070

0.51685 0.51633 0.51581 0.51530 0.51478

13344 47414 86648 31039 80584

91699 96754 36594 95141 56836

238 426 140 674 146

0.665 0.666 0.667 0.668 0.669

1.94449 1.94643 1.94838 1.95033 1.95228

05213 59844 33939 27518 40601

36830 27591 54498 64961 08339

982 272 192 432 065

0.51427 0.51375 0.51324 0.51273 0.51222

35277 95112 60085 30190 05423

06631 29998 12918 41890 03924

974 365 798 516 002

0.670 0.671 0.672 0.673 0.674

1.95423 1.95619 1.95814 1,96010 1.96206

73206 25354 97063 88354 99246

35939 01023 58805 66457 83108

496 417 754 630 314

0.51170 olsiii9 0.51068 0.51017 0.50966

85777 71249 61833 57524 58316

86542 77781 66187 40820 91247

470 383 865 271 632

0.675 0.676 0.677 0; 678 0.679

1.96403 1.96599 1.96796 1.96993 1.97190

29759 79912 49726 39218 48411

69847 89725 07759 90929 08185

187 700 335 575 868

0.50915 0.50864 0.50813 0.50763 0.50712

64206 75186 91254 12402 38627

07549 80313 00639 60132 50908

157 718 348 723 661

0.680 0.681 0.682 0.683 0.684

1.97387 1.97585 1.97782 1.97980 1.98178

77322 25972 94380 82567 90552

30447 30606 83526 66049 56994

594 040 371 605 589

0.50661 0.50611 0.50560 0.50509 0.50459

69923 06285 47709 94188 45719

65589 97305 39691 86890 33551

610 142 448 827 185

0.685 0.686 0.687 0.688 0.689

1.98377 1.98575 1.98774 1.98973 1.99172

18355 65995 33493 20869 28142

37159 89326 98257 50703 35403

979 220 531 885 001

0.50409 0.50358 0.50308 0.50258 0.5U207

02295 63913 30566 "2250 ,3960

74825 06371 24350 25428 06773

526 449 644 387 037

0.690 0.691 0.692 0.693 0.694

1.99371 1.99571 1.99770 1.99970 2.00170

55332 02459 69544 56605 63663

43082 66461 00252 41163 87902

329 043 033 899 948

0,50157 0.50107 0.50057 0.50007 0.49957

60690 47437 39194 35956 37720

66055 01448 11627 95767 53544

534 895 713 658 971

0.695 0.696 0.697 0.698 0.699

2.00370 2.00571 2.00772 2.00972 2.01173

90739 37852 05021 92268 99613

41175 03688 80153 77288 03818

193 356 865 865 219

0.49907 0.49857 OI49807 0.49757 0.49708

44479 56229 72965 94682 21374

85'135 91216 72961 32044 70637

969 541 653 844 732

0.700

2.01375 27074 70476 522

0.49658 53037 91409 515

130

ELEMENTARY TRANSCENDENTAL FTINCTIONS

EXPONENTIAL FUNCTION

Table 4.4

ez

X

0.700 0.701 0.702 0.703 0.704

2.01375 2.01576 2:01778 2.01980 2.02182

27074 74673 42430 30365 38498

70476 90010 77179 48759 23544

522 108 065 247 296

0.49658 0.49608 0.49559 0.49509 0.49460

53037 89666 31256 77802 29299

91409 97526 92651 80943 67056

515 471 465 451 976

0.705 0.706 0.707 0.708 0.709

2.02384 2.02587 2.02789 2.02992 2.03195

66849 15438 84286 73414 82840

22347 68004 85374 01341 44819

653 586 210 511 374

0.49410 0.49361 0.49312 0.49262 0.49213

85742 47126 13446 84698 60875

56141 53841 66295 00135 62485

685 826 756 445 987

0.710 0.711 0.712 0.713 0.714

2.03399 2.03602 2.03806 2.04010 2.04214

12586 62672 33118 23945 35173

46750 40109 59906 43184 29026

612 996 288 280 822

0.49164 0.49115 0.49066 0.49017 0.48968

41974 27990 18916 14750 15485

60965 03682 99240 56730 85736

102 649 129 197 169

0.715 0.716 0.717 0.718 0.719

2.04418 2.04623 2.04827 2.05032 2.05237

66822 18913 91467 84503 98043

58556 74939 23384 51146 07529

873 531 083 049 226

0.48919 0.48870 0.48821 0.48772 0.48723

21117 31641 47053 67346 92516

96331 99079 05032 25731 73205

534 460 312 153 263

0.720 0.721 0: 722 0.723 0.724

2.05443 2.05648 2.05854 2.06060 2.06266

32106 86714 61886 57644 74008

43887 13628 72211 77154 88034

743 106 257 626 i89

0.48675 0.48626 0.48577 0.48529 0.48480

22559 57469 97243 41873 91357

59971 99034 03884 88500 67343

650 560 990 207 253

0.725 0.126 0.727 0.728 0.729

2.06473 2.06679 2.06886 2.07093 2.07300

10999 68637 46943 45938 65642

66486 76210 82971 54598 60992

$29 896 273 438 036

0.48432 0.48384 0.48335 0.48287 0.48239

45689 04864 68878 37725 11401

55362 67990 21146 31229 15125

467 997 315 734 923

0.730 0.731 0.732 0.733 0.734

2.07508 2.07715 2.07923 2.08131 2.08339

06076 67261 49218 51967 75529

74122 68033 18844 04749 06025

645 852 323 882 589

0.48190 0.48142 0.48094 0.48046 0.47998

89900 73219 61352 54295 52042

90202 74309 85778 43422 66536

427 180 027 238 031

0.735 0.736 0.737 0.738 0.739

2.08548 2.08756 2.08965 2.09174 2.09384

19925 85175 71302 78325 06265

05027 86196 36056 43220 98392

819 344 419 868 173

0.47950 0.47902 0.47854 OI47806 0.47759

54589 61931 74064 90982 12680

74894 88751 28841 16377 73052

090 082 182 589 052

0.740 0.741 0.742 0.743 0.744

2.09593 2.09803 2.10013 2.10223 2I10433

55144 24983 15801 27621 60464

94364 26026 90360 86450 15478

563 109 816 725 007

0.47711 0.47663 0.47616 0.47568 0.47520

39155 70400 06412 47186 92716

21034 82972 81989 41687 86144

388 004 423 803 466

0.745 0.746 0.747 0.748 0.749

2.10644 2.10854 2.11065 2.11277 2I11488

14349 89299 85335 02477 40747

80727 87586 43551 58226 43325

065 641 917 625 155

0.47473 0,47425 0.47378 0.47331 0.47283

42999 98029 57801 22312 91555

39912 28019 75969 09739 55779

416 867 767 326 537

0.750

2.11700 00166 12674 669

0.47236 65527 41014 707

[(-p3]

131

ELEMENTARY TRANSCENDENTAL FUNCTIONS

EXPONENTIAL FUNCTION

Table 4.4

ez

X

e-=

0.750 0.751 0.752 0.753 0.754

2.11700 2.11911 2.12123 2.12336 2.12548

00166 80754 82534 05526 49752

12674 82217 70011 96236 83191

669 212 830 688 190

0.47236 0.47189 0.47142 0.47095 0.47048

65527 44222 27637 15766 08604

41014 92841 39130 08222 28930

707 982 875 791 562

0.755 0.756 0.757 0.758 0.759

2.12761 2.12974 2.13187 2.13400 2.13613

15233 01990 10044 39417 90130

55298 39105 63289 58655 58141

098 663 745 946 739

0.47001 0.46954 0.46907 0.46860 0.46813

06147 08390 15328 26958 43273

30537 42799 95938 20650 48096

969 274 749 211 543

0. 760 0.761 0.762 0.763 0.764

2.13827 2.14041 2.14255 2.14470 2.14684

62204 55662 70523 06810 64544

96818 11894 42714 30765 19676

602 152 282 301 075

0.46766 OI46719 0.46673 0.46626 0.46579

64270 89943 20288 55301 94976

09909 38187 65499 24879 49828

234 907 852 557 242

0.765 0.766 0.767 0.768 0.769

2.14899 2.15114 2.15329 2.15545 2.15760

43746 44438 66642 10379 75670

55220 85318 60038 31603 54385

173 010 993 678 916

0.46533 0.46486 0.46440 0.46394 0.46347

39309 88296 41931 00210 63129

74313 32768 60091 91646 63261

393 297 573 708 598

0.770 0.771 0.772 0.773 0.774

2.15976 2.16192 2.16409 2.16625 2.16842

62537 71002 01087 52812 26199

84915 81877 06121 20653 90647

008 866 167 514 604

0.46301 0.46255 0.46208 0.46162 0.46116

30683 02866 79675 61105 47152

11228 72301 83700 83104 08658

073 444 034 714 446

:0.77

0.778 0.779

2.17059 2.17276 2.17493 2.17711 2.17929

21271 38049 76555 36810 18836

83442 68545 17634 04559 05347

386 234 114 757 830

0.46070 0.46024 0.45978 0.45932 0.45886

37809 33074 32942 37407 46465

98965 93092 30565 51370 95954

818 580 189 344 527

0.780 0; 781 0.782 0.783 0.784

2.18147 2.18365 2.18583 2.18802 2.19021

22654 48288 95758 65087 56296

98201 63501 83813 43882 30643

117 691 099 545 070

0.45840 0.45794 0.45749 0.45703 0.45657

60113 78344 01154 28540 60496

05223 20542 83733 37077 23314

545 069 175 890 727

0.785 0.786 0.787 0.788 0.789

2.19240 2.19460 2.19679 2.19899 2.20119

69407 04442 61423 40372 41311

33215 42911 53235 59883 60752

744 852 086 740 903

0.45611 0.45566 0.45520 0.45475 0.45429

97017 38100 83740 33931 88670

85639 67703 13615 67940 75695

236 540 885 176 532

0.790 0.791 0.792 0.793 0.794

2.20339 2.20560 2.20780 2.21001 2.21222

64262 09247 76288 65407 76626

55936 47730 40632 41347 58787

659 288 465 466 377

0.45384 0.45339 0.45293 0.45248 0.45203

47952 11773 80127 53011 30420

82355 33849 76557 57316 23414

822 215 724 754 649

0.795 0.796 0.797 0.798 0.799

2.21444 2.21665 2.21887 2.22109 2.22331

09968 65453 43106 42947 64999

04074 90542 33740 51434 63608

299 561 936 850 607

0.45158 0.45112 Oi45067 0.45022 Oi44977

12349 98794 89750 85213 85178

22592 03042 13409 02789 20727

237 379 518 227 758

0.800

2.22554 09284 92467 605

;;;

0.44932 89641 17221 591

I

132

ELEMENTARY TRANSCENDENTAL FUNCTIONS

EXPONENTIAL FUNCTION

Table 4.4 X

e-2

ez

0.800 0; 801 0.802 0: 803 0.804

2.22554 2.22776 2.22999 2.23222 2.23446

09284 75825 64644 75762 09202

92467 62440 00181 34573 96726

605 556 717 111 759

0.44932 0.44887 0.44843 0.44798 0.44753

89641 98597 12042 29971 52381

17221 42716 48109 84743 04412

591 986 530 691 369

0.805 0.806 0.807 0.808 0.809

2.23669 2.23893 2.24117 2.24341 2.24566

64988 43140 43681 66635 12022

19986 39932 94378 23379 69230

909 270 249 186 599

0.44708 0.44664 0.44619 0.44574 0.44530

79265 10621 46442 86726 31467

59356 02264 86271 64960 92358

447 340 556 242 738

0.810 0.812 0.813 0.814

2.24790 2.25015 2.25240 2.25466 2.25691

79866 70189 83014 18363 76258

76471 91886 64507 45618 88752

419 242 569 061 788

0.44485 0.44441 0.44396 0.44352 0.44308

80662 34305 92392 54918 21880

22941 11626 13780 85209 82167

134 826 063 512 806

0.815 0.816 0.817 0.818 0.819

2.25917 2.26143 2.26369 2.26596 2.26823

56723 59719 85450 33758 04725

49701 86510 59486 31195 66470

480 786 532 979 087

0.44263 0.44219 0.44175 0.44131 0.44087

93273 69092 49333 33992 23064

61351 79898 95392 65856 49756

106 654 332 218 146

0.820 0.821 0.822 0.823 0.824

2.27049 2.27277 2.27504 2.27732 2.27960

98375 14729 53812 15645 00251

32405 98368 35993 19188 24138

781 215 046 700 650

0.44043 0.43999 0.43955 0.43911 0.43867

16545 14429 16714 23395 34466

05999 93933 73347 04469 47967

263 588 574 662 847

0.825 0.826 0.827 0.828 0.829

2.28188 2.28416 2.28644 2.28873 2.29102

07653 37874 90936 66863 65678

29303 15424 65522 64904 01164

690 217 506 998 583

0.43823 0.43779 0.43735 0.43692 0.43648

49924 69765 93983 22575 55537

64949 16959 65982 74441 05193

237 611 985 171 342

0.830 0.831 0.832 0.833 0. a34

2.29331 2.29561 2.29790 2.30020 2.30251

87402 32060 99674 90267 03862

64182 46132 41479 46985 61709

888 567 593 553 945

0.43604 0.43561 0.43517 0.43474 0.43430

92863 34549 80592 30987 85729

21535 87200 66356 23608 23995

593 502 699 428 109

0. a35

0.836 0.837 0.838 0.839

2.30481 2.30712 2.30942 2.31173 2.31405

40482 00151 82890 88724 17676

87012 26555 86305 74537 01834

474 358 628 437 366

0.43387 0.43344 0.43300 0.43257 0.43214

44814 08238 75996 48084 24498

32990 16504 40877 72886 79740

906 293 616 664 233

0.840 0.841 0.842 0.843 0.844

2.31636 2.31868 2.32100 2.32332 2.32565

69767 45023 43465 65117 10003

81091 27518 58641 94304 56672

734 913 644 351 462

0.43171 0.43127 0.43084 0.43041 0.42998

05234 90286 79652 73326 71304

29079 88978 27942 14906 19239

693 558 052 679 788

0. a45 0.846 0.847 0.848 0.849

2.32797 2.33030 2.33263 2.33497 2.33730

78145 69567 84292 22343 83745

70234 61805 60527 97872 07647

734 575 370 812 233

0.42955 0.42912 0.42869 0.42827 0.42784

73582 80155 91020 06172 25606

10739 59632 36577 12659 59395

148 516 204 654 005

0.850

2.33964 68519 25990 937

0: 811

0.42741 49319 48726 670

133

ELEMENTARY TRANSCENDENTAL FUNCTIONS

EXPONENTIAL FUNCTION

Table 4.4 e-=

ez

2

0.850 0.851 0.852 0.853 0.854

2.33964 2.34198 2.34433 2.34667 2.34902

68519 76689 08280 63314 41814

25990 91381 44636 28914 89719

937 538 295 459 607

0.42741 0.42698 0.42656 0.42613 0.42570

49319 77306 09563 46085 86869

48726 53025 45091 98148 85850

670 901 367 726 193

0.855 0.856 0.857 0.858 0.859

2.35137 2.35372 2.35608 2.35843 2.36079

43805 69310 18352 90954 87141

74901 34660 21547 90464 98674

997 911 002 656 336

0.42528 0.42485 0.42443 0.42400 0.42358

31910 81204 34746 92533 54560

82274 61924 99730 71047 51652

123 574 893 281 373

0.860 0.861 0.862 0.863 0.864

2.36316 2.36552 2.36789 2.37026 2.37263

06937 50363 17445 08206 22669

05794 73806 67050 52237 98442

948 196 946 586 400

0.42316 0.42273 0.42231 0.42189 0.42147

20823 91317 66039 44983 28147

17748 45962 13343 97363 75917

817 841 840 945 606

0.865 0.866 0.867 0.868 0.869

2.37500 2.37738 2.37976 2.38214 2.38452

60859 22799 08513 18024 51357

77111 62065 29496 57978 28460

933 359 863 010 126

0.42105 0.42063 0.42021 0.41979 0.41937

15526 07115 02910 02908 07103

27321 30312 64050 08114 42503

165 439 296 234 963

0.870 0.871 0.872 0.873 0.874

2.38691 2.38929 2.39168 2.39408 2.39647

08535 89582 94522 23379 76177

24276 31145 37171 32849 11065

682 671 999 872 184

0.41895 0.41853 0.41811 0.41769 0.41727

15492 28071 44834 65779 90901

47638 04358 93919 97998 98691

983 162 324 822 126

0.875 0.876 0.877 0.878 0.879

2.39887 2.40127 2.40367 2.40608 2.40849

52939 53690 78455 27255 00117

67097 98624 05720 90861 58929

915 518 327 947 666

0.41686 0141644 0.41602 0.41561 0.41519

20196 53660 91288 33076 79020

78508 20380 07652 24088 53866

403 096 513 408 560

0,880 0.881 0,882 0.883 0.884

2.41089 2.41331 2.41572 2.41814 2.42056

97064 18119 63308 32654 26181

17209 75397 45597 42330 82530

851 361 956 708 413

0.41478 0.41436 0.41395 0.41354 0.41312

29116 83360 41748 04276 70938

81581 92242 71274 04515 78218

367 420 097 140 250

0.885 0.886 0.887 0.888 0.889

2.42298 2.42540 2.42783 2.43026 2.43269

43914 85877 52094 42590 57387

85550 73163 69565 01380 97656

015 018 911 593 799

0.41271 0.41230 0.41188 0.41147 0.41106

41732 16653 95698 78861 66139

79049 94088 10827 17170 01433

666 753 593 568 949

0.890 0.891 0.892 0.893 0.894

2.43512 2.43756 2.44000 2.44244 2.44488

96512 59989 47841 60092 96769

89874 11946 00220 93481 32956

527 472 460 882 134

0.41065 0.41024 0.40983 0.40942 0.40901

57527 53022 52620 56315 64106

52345 59044 11078 98410 11406

488 001 959 082 922

0.895 0.896 0.897 0.898 0.899

2.44733 2.44978 2.45223 2.45468 2.45714

57894 43493 53589 88208 47374

62311 27659 77561 63026 37516

060 394 203 343 904

0.40860 0.40819 0.40779 0.40738 0.40697

75986 91952 12001 36127 64327

40848 77922 14226 41763 52948

458 685 207 826 135

0.900

2.45960 31111 56949 664

0.40656 96597 40599 112 [() : -

51

134

ELEMENTARY TRANSCENDENTAL FUNCTIONS

EXPONENTIAL FUNCTION

Table 4.4

e -=

ez

X

0.900 0.901 0.902 0.903 0.904

2.45960 2.46206 2.46452 2.46699 2.46946

31111 39444 72398 29997 12266

56949 79698 66597 80940 88490

664 548 083 863 006

0.40656 0.40616 0.40575 0.40535 0.40494

96597 32932 73330 17784 66293

40599 97943 18615 96654 26504

112 710 453 028 879

0.905 0.906 0.907 0.908 0.909

2.47193 2.47440 2.47688 2.47935 2.48183

19230 50913 07340 88536 94525

57471 58582 64990 52339 98748

626 298 529 232 200

0.40454 0.40413 0.40373 0.40333 0.40292

18851 75454 36098 00780 69495

03018 21451 77463 67118 86885

802 540 377 736 773

0.910 0.911 0.912 0.913 0.914

2.48432 2.48680 2.48929 2.49178 2.49427

25333 80984 61504 66916 97246

84816 93625 10739 24212 24583

587 386 912 291 942

0.40252 0.40212 0.40171 0.40131 0.40091

42240 19010 99800 84609 73430

33635 04643 97586 10541 41992

975 753 047 915 136

0.915 0.916 0.917 0.918 0.919

2.49677 2.49927 2.50177 2.50427 2.50678

52519 32759 37992 68243 23537

04888 60652 89900 93156 73445

075 177 513 620 810

0.40051 0.40011 0.39971 0.39931 0.39891

66260 63096 63933 68767 77594

90818 56304 38134 36389 51555

809 950 089 877 677

0.920 0.921 0.922 0.923 0.924

2.50929 2.51180 2.51431 2.51682 2.51934

03899 09353 39926 95642 76526

36297 89748 44344 13141 11713

671 577 189 971 703

0.39851 0.39812 0.39772 0.39732 0.39692

90410 07212 27995 52755 81488

84514 36546 09334 04954 25882

173 962 165 021 492

0.925 0.926 0.927 0.928 0.929

2.52186 2.52439 2.52691 2.52944 2.53197

82603 13899 70439 52249 59352

58147 73052 79557 03317 72513

991 794 936 633 022

0.39653 0.39613 0.39573 0.39534 0.39494

14190 50858 91487 36074 84614

74992 55555 71236 26099 24603

866 360 720 830 311

0.930 0.931 0.932 0.933 0.934

2.53450 2.53704 2.53958 2.54212 2.54466

91776 49544 32683 41218 75174

17854 72585 72481 55857 63568

680 166 544 927 010

0.39455 0.39415 0.39376 0.39337 0.39297

37103 93538 53915 18229 86477

71601 72342 32469 58022 55429

130 199 987 122 996

0.935 0.936 0.937 0.938 0.939

2.54721 2.54976 2.55231 2.55486 2.55742

34577 19452 29824 65720 27164

39007 28117 79384 43847 75094

611 220 537 026 464

0.39258 0.39219 0.39180 0.39140 0.39101

58655 34758 14784 98728 86585

31518 93504 49000 06006 72918

373 997 198 497 221

0.940 0.941 0.942 0.943 0.944

2.55998 2.56254 2.56510 2.56767 2.57024

14183 26801 65045 28940 18511

29271 65080 43782 29203 87732

496 189 593 299 007

0.39062 0.39023 0.38984 0.38945 0.38906

78353 74027 73604 77079 84448

58521 71991 22897 21196 77236

102 894 977 971 341

0.945 0.946 0.947 0.948 0.949

2.57281 2.57538 2.57796 2.58054 2.58312

33785 74788 41544 34079 52420

88326 02513 04393 70643 80516

089 161 651 376 117

0.38867 0.38829 0.38790 0.38751 0.38712

95709 10856 29886 52794 79579

01753 05872 01110 99369 12940

010 971 896 747 390

0.950

2.58570 96593 15846 199

0.38674 10234 54501 207 [(-:)5]

[(-231

135

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.4

EXPONENTIAL FUNCTION e-z

ez

X

0.950 0.951 0.952 0.953 0.954

2.58570 2.58829 2.59088 2.59347 2.59607

96593 66622 62535 84356 32112

15846 61051 03133 31686 38890

199 072 898 135 126

0.38674 0.38635 0.38596 0.38558 0.38519

10234 44757 83143 25389 71491

54501 37117 74242 79713 67755

207 707 140 111 194

0.955 0.956 0.957 0.958 0.959

2.59867 2.60127 2.60387 2.60647 2.60908

05829 05532 31248 83003 60823

19521 70952 93153 88696 62757

695 740 828 799 366

0.38481 0.38442 0.38404 0.38365 0.38327

21445 75247 32893 94380 59703

52978 50378 75335 43613 71361

545 516 273 409 560

0.960 0.961 0.962 0.963 0.964

2.61169 2.61430 2.61692 2.61954 2.62216

64734 94761 50932 33272 41807

23117 80169 46914 38971 74573

718 136 592 373 688

0.38289 0.38251 0.38212 0.38174 0.38136

28859 01844 78654 59286 43734

75112 71780 78665 13447 94189

023 368 061 076 517

0.965 0.966 0.967 0.968 0.969

2.62478 2.62741 2.63004 2.63267 2.63530

76564 37569 24848 38428 78334

74575 62452 64304 08861 27480

291 101 825 583 539

0.38098 0.38060 0.38022 0.37984 0.37946

31997 24069 19947 19628 23107

39337 67716 98534 51378 46217

233 437 325 697 574

0.970 0.971 0.972 0.973 0.974

2.63794 2.64058 2.64322 2.64587 2.64851

44593 37232 56276 01753 73689

54152 25503 80798 61940 13478

532 708 158 558 808

0.37908 0.37870 0.37832 0.37794 0.37756

30381 41445 56296 74931 97345

03398 43649 88076 58165 75777

818 757 798 054 964

0.975 0.976 0.977 0.978 0.979

2.65116 2.65381 2.65647 2.65913 2.66179

72109 97042 48512 26548 31174

82606 19166 75651 07209 71643

682 470 628 434 642

0.37719 0.37681 0.37643 0.37606 0.37568

23535 53497 87227 24721 65976

63156 42920 38065 71965 68367

913 859 949 147 855

0.980 0.981 0.982 0.983 0.984

2.66445 2.66712 2.66979 2.67246 2.67513

62419 20308 04868 16127 54109

29417 43654 80145 07345 96380

138 602 169 099 441

0.37531 0.37493 0.37456 0.37418 0.37381

10988 59753 12267 68527 28529

51399 45561 75729 67156 45466

539 350 751 142 482

0.985 0.986 0.987 0.988 0.989

2.67781 2.68049 2.68317 2;68585 2.68854

18844 10356 28673 73822 45830

21049 57826 85862 86989 45722

708 547 418 272 235

0.37343 0.37306 0.37269 0.37232 0.37194

92269 59743 30948 05880 84535

36660 918 67113 412 63571 361 53155 231 63358 181

0.990 0.991 0.992 0.993 0.994

2.69123 2.69392 2.69662 2.69932 2.70202

44723 70528 23273 02984 09688

49262 87498 53013 41079 49668

289 962 016 142 652

0.37157 0.37120 OI37083 0.37046 0.37009

66910 53000 42802 36313 33529

22045 57455 98195 73247 11961

691 187 674 362 296

0.995 0.996 0.997 0.998 0.999

2.70472 2.70743 2.71013 2.71285 2.71556

43412 04184 92030 06977 49053

79452 33802 18796 43219 18566

181 382 637 755 687

0.36972 0.36935 0.36898 0.36861 0.36824

34445 39058 47366 59363 75046

44058 99632 09141 03418 13662

983 024 744 822 921

1.000

2.71828 18284 59045 235 [(-231

0.36787 94411 71442 322

r (-8)51

L

6

J

136

ELEMENTARY TRANSCENDENTAL FWCTIONS

Table 4.4

EXPONENTIAL FUNCTION ez

5

e-2

0.0 0.1 0.2 0.3 0. 4

1.00000 1.10517 1.22140 1.34985 1.49182

00000 09180 27581 88075 46976

00000 75648 60170 76003 41270

1.00000 0.90483 0.81873 0.74081 0.67032

00000 74180 07530 82206 00460

00000 35959 77981 81717 35639

00000 57316 85867 86607 30074

0.5 0. 6 0. 7 0. 8 0. 9

1.64872 1.82211 2.01375 2.22554 2.45960

12707 88003 27074 09284 31111

00128 90509 70477 92468 56950

0.60653 0.54881 0.49658 0.44932 0.40656

06597 16360 53037 89641 96597

12633 94026 91409 17221 40599

42360 43263 51470 59143 11188

1. 0 1.1 1. 2 1. 3 1.4

2.71828 3.00416 3.32011 3.66929 4.05519

18284 60239 69227 66676 99668

59045 46433 36547 19244 44675

0.36787 0.33287 0.30119 0.27253 0.24659

94411 10836 42119 17930 69639

71442 98079 12202 34012 41606

32160 55329 09664 60312 47694

1.5 1. 6 1.7 1.8 1.9

4.48168 4.95303 5.47394 6.04964 6.68589

90703 24243 73917 74644 44422

38065 95115 27200 12946 79269

0.22313 0.20189 0.18268 0.16529 0.14956

01601 65179 35240 88882 86192

48429 94655 52734 21586 22635

82893 40849 65022 53830 05264

2.0 2.1 2. 2 2.3 2.4

7.38905 8.16616 9.02501 9.97418 11.02317

60989 99125 34994 24548 63806

30650 67650 34121 14721 41602

0.13533 0.12245 0.11080 0.10025 0.09071

52832 64282 31583 88437 79532

36612 52981 62333 22803 89412

69189 91022 88333 73373 50338

2.5 2. 6 2.7 2.8 2.9

12.18249 13.46373 14.87973 16.44464 18.17414

39607 80350 17248 67710 53694

03473 01690 72834 97050 43061

0.08208 0.07427 0.06720 0.06081 0.05502

49986 35782 55127 00626 32200

23898 14333 39749 25217 56407

79517 88043 76513 96500 22903

3. 0 3.1 3.2 3. 3 3. 4

20.08553 22.19795 24.53253 27.11263 29.96410

69231 12814 01971 89206 00473

87668 41633 09349 57887 97013

0.04978 0.04504 0.04076 0.03688 0.03337

70683 92023 22039 31674 32699

67863 93557 78366 01240 60326

94298 80607 21517 00545 07948

3.5 3.6 3.7 3.8 3.9

33.11545 36.59823 40.44730 44.70118 49.40244

19586 44436 43600 44933 91055

92314 77988 67391 00823 30174

0.03019 0.02732 0.02472 0.02237 0.02024

73834 37224 35264 07718 19114

22318 47292 70339 56165 45804

50074 56080 39120 59578 38847

4. 0 4.1 4.2 4. 3 4. 4

54.59815 60.34028 66.68633 73.69979 81.45086

00331 75973 10409 36995 86649

44239 61969 25142 95797 68117

0.01831 0.01657 0.01499 0.01356 0.01227

56388 26754 55768 85590 73399

88734 01761 20477 12200 03068

18029 24754 70621 93176 44118

4.5 4.6 4.7 4.8 4.9

90.01713 99.48431 109.94717 121.51041 134.28977

13005 56419 24521 75187 96849

21814 33809 23499 34881 35485

0.01110 0.01005 0.00909 0.00822 0.00744

89965 18357 52771 97470 65830

38242 44633 01695 49020 70924

30650 58164 81709 02884 34052

5. 0 148.41315 91025 76603 0.00673 79469 99085 46710 From C. E. Van Orstrand, Tables of the exponential function and of the circular sine and cosine to radian arguments, Memoirs of the National Academy of Sciences, vol. 14, Fifth Memoir. US. Government Printing Office, Washington, D.C., 1921 (with permission) for e-z, z 12.4.

137

E L E M E N T A R Y TRANSCENDENTAL FUNCTIONS EXPONENTIAL FUNCTION X

Table 4.4 e -2

ez

5.0 5.1 5.2 5.3 5.4

148.41315 164.02190 181.27224 200.33680 221.40641

91025 72999 18751 99747 62041

77 02 51 92 87

0.00673 0.00609 0.00551 0.00499 0.00451

79469 67465 65644 15939 65809

99085 65515 20760 06910 42612

46710 63611 77242 21621 66798

5.5 5. 6 5.7 5.8 5.9

244.69193 270.42640 298.86740 330.29955 365.03746

22642 74261 09670 99096 78653

20 53 60 49 29

0.00408 0.00369 0.00334 0.00302 0,00273

67714 78637 59654 75547 94448

38464 16482 57471 45375 18768

06699 93082 27277 81475 36923

6. 0 6.1 6.2 6.3 6.4

403.42879 445.85777 492.74904 544.57191 601.84503

34927 00825 10932 01259 78720

35

56 29 82

0,00247 0.00224 0.00202 0.00183 0.00166

87521 28677 94306 63047 15572

76666 19485 36295 77028 73173

35842 80247 73436 90683 93450

6. 5 6. 6 6.7 6. 8 6. 9

665.14163 735.09518 812.40582 897.84729 992.27471

30443 92419 51675 16504 56050

62 73 43 18 26

0.00150 0.00136 0.00123 o.ooiii 0.00100

34391 03680 09119 37751 77854

92977 37547 02673 47844 29048

57245 89342 48118 80308 51076

7. 0 7.1 7.2 7.3 7.4

1096.63315 1211.96707 1339.43076 1480.29992 1635.98442

84284 44925 43944 75845 99959

59 77 18 45 27

0.00091 0.00082 0.00074 0.00067 0.00061

18819 51049 65858 55387 12527

65554 23265 08376 75193 61129

51621 90427 67937 84424 57256

7. 5 7.6 7.7 7.8 7.9

1808.04241 1998.19589 2208.34799 2440.60197 2697.28232

44560 51041 18872 76244 82685

63 18 09 99 09

0.00055 0.00050 0.00045 0.00040 0,00037

30843 04514 28271 97349 07435

70147 33440 82886 78979 40459

83358 61070 79706 78671 08837

8. 0 8.1 8.2 8.3 8.4

2980.95798 3294.46807 3640.95030 4023.87239 4447.06674

70417 52838 73323 38223 76998

28 41 55 10 56

0.00033 0.00030 0.00027 0.00024 0.00022

54626 35391 46535 85168 48673

27902 38078 69972 27107 24178

51184 86666 14233 95202 84827

8.5 8.6 8.7 8.8 8.9

4914.76884 5431.65959 6002.91221 6634.24400 7331.97353

02991 13629 72610 62778 91559

34 80 22 85 93

0.00020 0.00018 0.00016 0.00015 0.00013

34683 41057 65858 07330 63889

69010 93667 10987 75095 26482

64417 57912 63341 47660 01145

9.0 9.1 9.2 9. 3 9.4

8103.08392 8955.29270 9897.12905 10938.01920 12088.38073

75753 34825 87439 81651 02169

84 12 16 84 84

0.00012 0.00011 0,00010 0.00009 0.00008

34098 16658 10394 14242 27240

04086 08490 01837 31478 65556

67955 11474 09335 17334 63226

9. 5 9. 6 9.7 9.8 9.9

13359.72682 14764.78156 16317.60719 18033.74492 19930.37043

96618 55772 80154 78285 82302

72 73 32 11 89

0.00007 0.00006 0.00006 0.00005 0.00005

48518 77287 12834 54515 01746

29887 36490 95053 99432 P2056

70059 85387 22210 17698 17530

10.0

ii

22026.46579 48067 17

0.00004 53999 29762 48485

i I

138

ELEMENTARY TRANSCENDENTAL FUNCTIONS

EXPONENTIAL FUNCTION

Table 4.4

ez

5

e-=

1.00000 2.71828 7.38905 2.00855 5.45981

00000 18284 60989 36923 50033

00000 59045 30650 18766 14423

000 235 227 774 908

1.00000 3.67879 1.35335 4.97870 1.83156

00000 44117 28323 68367 38888

00000 14423 66126 86394 73418

000 216 919 298 029

1.48413 4.03428 1.09663 2.98095 8.10308

15910 79349 31584 79870 39275

25766 27351 28458 41728 75384

034 226 599 275 008

6.73794 2.47875 9.11881 3.35462 1.23409

69990 21766 96555 62790 80408

85467 66358 45162 25118 66795

097 423 080 388 495

10 11 12 13 14

2.20264 5.98741 1.62754 4.42413 1.20260

65794 41715 79141 39200 42841

80671 19781 90039 89205 64776

652 846 208 033 778

4.53999 1.67017 6.14421 2.26032 8.31528

29762 00790 23533 94069 71910

48485 24565 28209 81054 35678

154 931 759 326 841

15 16 17 18 19

3.26901 8.88611 2.41549 6.56599 1.78482

73724 05205 52753 69137 30096

72110 07872 57529 33051 31872

639 637 821 114 608

3.05902 1.12535 4.13993 1.52299 5.60279

32050 17471 77187 79744 64375

18257 92591 85166 71262 37267

884 145 660 844 540

20 21 22 23 24

4.85165 1.31881 3.58491 9.74480 2.64891

19540 57344 28461 34462 22129

97902 83214 31591 48902 84347

780 697 562 600 229

2.06115 7.58256 2.78946 1.02618 3.77513

36224 04279 80928 79631 45442

38557 11906 68924 70189 79097

828 728 808 030 752

25 26 27 28 29

7.20048 1.95729 5.32048 1.44625 3.93133

99337 60942 24060 70642 42971

38587 88387 17986 91475 44042

252 643 167 174 074

1.38879 5.10908 1.87952 6.91440 2.54366

43864 90280 88165 01069 56473

96402 63324 39083 40203 76922

059 720 295 009 910

30 31 32 33 34

1.06864 2.90488 7.89629 2.14643 5.83461

74581 49665 60182 57978 74252

52446 24742 68069 59160 74548

215 523 516 646 814

9.35762 3.44247 1.26641 4.65888 1.71390

29688 71084 65549 61451 84315

40174 69976 09417 03397 42012

605 458 572 364 966

35 36 37 38 39

1.58601 4.31123 1.17191 3.18559 8.65934

34523 15471 42372 31757 00423

13430 15195 80261 11375 99374

728 227 131 622 695

6.30511 2.31952 8.53304 3.13913 1.15482

67601 28302 76257 27920 24173

46989 43569 44065 48029 01578

386 388 794 629 599

40 41 42 43 44

17)2.35385 6.39843 1.73927 4.72783 1.28516

26683 49353 49415 94682 00114

70199 00549 20501 29346 35930

854 492 047 561 828

4.24835 1.56288 5.74952 2.11513 7.78113

42552 21893 22642 10375 22411

91588 34988 93559 91080 33796

995 768 807 487 516

71057 94206 28861 59120 65724

48509 02448 90067 97631 95099

535 875 396 739 691

2.86251 1.05306 3.87399 1.42516 5.24288

85805 17357 76286 40827 56633

49393 55381 87187 40935 63463

644 238 113 106 937

45 46 47 48 49 50

I1 19 19 20 20 21

3.49342 9.49611 2.58131 7.01673 1.90734

( 21) 5.18470 55285 87072 464

(-22) 1.92874 98479 63917 783

139

ELEMENTARY TRANSCENDENTAL FUNCTIONS

EXPONEXTIAL FUNCTION

Table 4.4

ez

5

e-z

50 51 52 53 54

5.18470 1.40934 3.83100 1.04137 2.83075

55285 90824 80007 59433 33032

87072 26938 16576 02908 74693

464 796 849 780 900

1.92874 7.09547 2.61027 9.60268 3.53262

98479 41622 90696 00545 85722

63917 84704 67704 08676 00807

783 139 805 030 030

55 56 57 58 59

7.69478 2.09165 5.68571 1.54553 4.20121

52651 94960 99993 89355 04037

42017 12996 35932 90103 90514

138 154 223 930 255

1.29958 4.78089 1.75879 6.47023 2.38026

14250 28838 22024 49256 64086

07503 85469 24311 45460 94400

074 081 649 326 606

60 61 62 63 64

1.14200 3.10429 8.43835 2.29378 6.23514

73898 79357 66687 31594 90808

15684 01919 41454 69609 11616

284 909 489 879 883

8.75651 3.22134 1.18506 4.35961 1.60381

07626 02859 48642 00000 08905

96520 92516 33981 63080 48637

338 089 006 974 853

65 66 67 68 69

1.69488 4.60718 1.25236 3.40427 9.25378

92444 66343 31708 60499 17255

10333 31291 42213 31740 87787

714 543 781 521 600

70 71 72 73 74

2.51543 6.83767 1.85867 5.05239 1.37338

86709 12297 17452 36302 29795

19167 62743 84127 76104 40176

006 867 980 195 188

3.97544 1.46248 5.38018 1.97925 7.28129

97359 62272 61600 98779 01783

08646 51230 21138 46904 21643

808 947 414 554 834

75 76 77 78 79

3,73324 1.01480 2.75851 7.49841 2.03828

19967 03881 34545 69969 10665

99901 13888 23170 90120 12668

640 728 206 435 767

2.67863 9,85415 3.62514 1.33361 4.90609

69618 46861 09191 48155 47306

08077 11258 43559 02261 49280

944 029 224 341 566

80 81 82 83 84

5.54062 1.50609 4.09399 1.11286 3.02507

23843 73145 69621 37547 73222

93510 85030 27454 91759 01142

053 548 697 412 338

85 86 87 88 89

8.22301 2.23524 6.07603 1.65163 4.48961

27146 66037 02250 62549 28191

22913 34715 56872 94001 74345

510 047 150 856 246

1.21609 4.47377 1.64581 6.05460 2.22736

92992 93061 14310 18954 35617

52825 81120 82273 01185 95743

564 735 651 885 739

1.22040 32943 17840 802 I 39391 9i01762 3.31740 00983 35742 626 84050 34298 931 2.45124 55429 20085 786 6.66317 62164 10895 834

8.19401 3.01440 1.10893 4.07955 1.50078

26239 87850 90193 86671 57627

90515 65374 12136 77560 07394

430 553 379 158 888

1.81123 4.92345 1.33833 3.63797 9.88903

5.52108 2.03109 7.47197 2.74878 1.01122

22770 26627 23373 50079 14926

28532 34810 42990 10214 10448

732 926 161 930 530

90 91 92 93 94 95 96 97 98 99

90828 82860 47192 09476 03193

89023 12058 04269 08804 46946

282 400 500 579 771

100 ( 43) 2.68811 71418 16135 448 For 1z1>100 see Example 11.

5.90009 05415 97061 391 I -29 -29 1 2.17052 20113 03639 412 7.98490 42456 86978 808 2.93748 21117 10802 947 1.08063 92777 07278 495

-35 1.80485 13878 45415 172 I -36 1 2,44260 6.63967 71995 80734 401 07377 40527 679 ,

8.98582 59440 49380 670 3.30570 06267 60734 298

(-44) 3.72007 59760 20835 963

140

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.5

RADIX TABLE OF THE EXPONENTIAL FUNCTION eZl0-n

e--210-n

x

n

1 2 3 4 5 6 7 8 9

10 10 10 10 10 10 10 10 10

1,00000 00001 00000 00000 50000 1.00000 00002 00000 00002 00000 1.00000 00003 00000 00004 50000 1.00000 00004 00000 00008 00000 1,00000 00005 00000 00012 50000 1.00000 00006 00000 00018 00000 1.00000 00007 00000 00024 50000 1.00000 00008 00000 00032 00000 1,00000 00009 00000 00040 50000

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99999 00000 99998 00000 99997 00000 99996 00000 99995 00000 99994 00000 99993 00000 99992 00000 99991 00000

1 2 3 4 5 6 7 8 9

9 9 9 9 9 9 9 9 9

1.00000 00010 00000 00050 00000 1.00000 00020 00000 00200 00000 I.00000 00030 00000 00450 00000 1,00000 00040 00000 00800 00000 1.00000 00050 00000 01250 00000 1.00000 00060 00000 01800 00000 1.00000 00070 00000 02450 00001 1.00000 00080 00000 03200 00001 1.00000 00090 00000 04050 00001

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99990 00000 00050 99980 00000 00200 99970 00000 00450 99960 00000 00800 99950 00000 01250 99940 00000 01800 99930 00000 02449 99920 00000 03199 99910 00000 04049

00000 00000 00000 00000 00000 00000 99999 99999 99999

1 2 3 4 5 6 7 8 9

8 8 8 8 8 8 8 8 8

1.00000 00100 1,00000 00200 1.00000 00300 1.00000 00400 1.00000 00500 1.00000 00600 1,00000 00700 1.00000 00800 1.00000 00900

00000 00000 00000 00000 00001 00001 00002 00003 00004

05000 00002 20000 00013 45000 00045 80000 00107 25000 00208 80000 00360 45000 00572 20000 00853 05000 01215

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99900 00000 99800 00000 99700 00000 99600 00000 99500 00001 99400 00001 99300 00002 99200 00003 99100 00004

04999 19999 44999 79999 24999 79999 44999 19999 04999

99998 99987 99955 99893 99792 99640 99428 99147 98785

1 2 3 4 5 6 7 8 9

7 7 7 7 7 7 7 7 7

1.00000 01000 00005 1,00000 02000 00020 1.00000 03000 00045 1.00000 04000 00080 1.00000 05000 00125 1.00000 06000 00180 1.00000 07000 00245 1.00000 08000 00320 1.00000 09000 00405

00000 01667 00000 13333 00000 45000 00001 06667 00002 08333 00003 60000 00005 71667 00008 53334 00012 15000

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99000 00004 99999 98333 98000 00019 99999 86667 97000 00044 99999 55000 96000 00079 99998 93333 95000 00124 99997 91667 94000 00179 99996 40000 93000 00244 99994 28333 92000 00319 99991 46667 91000 00404 99987 85000

1 2

6 6 6 6 6 6 6 6 6

1.00000 10000 00500 1.00000 20000 02000 1.00000 30000 04500 1.00000 40000 08000 1,00000 50000 12500 1.00000 60000 18000 1.00000 70000 24500 1.00000 80000 32000 1.00000 90000 40500

00016 00133 00450 01066 02083 03600 05716 08533 12150

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

90000 00499 99983 33334 80000 01999 99866 66673 70000 04499 99550 00034 60000 07999 98933 33440 50000 12499 97916 66927 40000 17999 96400 00540 30000 24499 94283 34334 20000 31999 91466 68373 10000 40499 87850 02734

3

4 5 6 7 8 9

66667 33340 00034 66773 33594 00540 67667 35040 02734

00000 50000 00002 00000 00004 50000 00008 00000 00012 50000 00018 00000 00024 50000 00032 00000 00040 50000

For n>10, e*zlo-n=l i ~ l O - ~ + ;z210-2nt o 25D. Compiled from C. E. Van Orstrand, Tables of the exponential function and of the circular sine and cosine to radian arguments, Memoirs of the National Academy of Sciences, vol. 14, Fifth Memoir. U.S.Government Printing Office, Washington, D.C., 1921 (with permission).

141

ELEMENTARY TRANSCENDENTAL FUNCTIONS

RADIX TABLE OF THE EXPONENTIAL FUNCTION @O-

Table 4.5

e-zio-n

n

2

n

1 2 3 4 5 6 7 8 9

5 5 5 5 5 5 5 5 5

1.00001 1.00002 1.00003 1.00004 1.00005 1.00006 1.00007 1.00008 1.00009

00000 00002 00004 00008 00012 00018 00024 00032 00040

50000 00001 50004 00010 50020 00036 50057 00085 50121

1 2 3 4 5 6 7 8 9

4 4 4 4 4 4 4 4 4

1.00010 1.00020 1.00030 1.00040 1.00050 1.00060 1.00070 1.00080 1.00090

00050 00200 00450 00800 01250 01800 02450 03200 04051

01333 04500 10667 20835 36005 57176 85350 21527

40000 33752 73341 93776 40064 67223 40273 34242

1 2 3 4 5 6 7 8 9

3 3 3 3 3 3 3 3 3

1.00100 1.00200 1.00300 1.00400 1.00501 1.00601 1.00702 1,00803 1.00904

05001 20013 45045 80106 25208 80360 45572 20855 06217

66708 34000 03377 77341 59401 54064 66848 04273 73867

1 2 3 4 5 6 7 8 9

2 2 2 2 2 2 2 2 2

1.01005 1.02020 1.03045 1.04081 1.05127 1.06183 1.07250 1.08328 1.09417

01670 13400 45339 07741 10963 65465 81812 70676 42837

1 2 3 4 5 6 7 8 9

1 1 1 1 1 1 1 1 1

1.10517 1.22140 1.34985 1.49182 1.64872 1.82211 2.01375 2.22554 2.45960

09180 27581 88075 46976 12707 88003 27074 09284 31111

1

0

2.71828 18284 59045 23536 02875

70833 00000 37502 33342 37526 00065 70973 00273 37992

0.99999 0.99998 0.99997 0.99996 0.99995 0.99994 0.99993 0.99992 0.99991

00000 00001 00004 00007 00012 00017 00024 00031 00040

49999 99998 49995 99989 49979 99964 49942 99914 49878

83333 66667 50003 33343 16692 00053 83433 66837 50273

37500 33333 37498 99991 70807 99935 37360 33060 37008

00155 67083 34167 26668 02510 86724 04384 80648 40801 10308 14882

0.99990 0.99980 0.99970 0.99960 0.99950 0.99940 0.99930 0.99920 0.99910

00049 30199 00449 00799 01249 01799 02449 03199 04048

99833 98666 95500 89334 79169 64005 42843 14683 78527

33749 73333 33747 39991 27057 39935 33609 73060 33257

99167 06668 97512 46724 29384 20648 95801 30307 99880

34166 26675 02601 87235 06338 86485 55523 43117 81406

80558 55810 29341 88080 35662 55845 16000 20736 25705

0.99900 0.99800 0.99700 0.99600 0.99501 0.99401 0.99302 0.99203 0.99104

04998 19986 44955 79893 24791 79640 44429 19148 03787

33374 67333 03372 43991 92682 53935 33235 37060 72883

99166 06675 97601 47235 31335 26474 10490 63033 66216

80554 55302 20662 23064 25642 44988 47970 98697 45648

84168 26755 53516 92388 76024 45359 54216 74958 05210

05754 81016 85561 22675 03969 62222 47905 55443 35787

21655 01439 24400 70448 75176 46849 31039 59878 28976

0.99004 0.98019 0.97044 0.96078 0.95122 0.94176 0.93239 0.92311 0.91393

98337 86733 55335 94391 94245 45335 38199 63463 11852

49168 06755 48508 52323 00714 84248 05948 86635 71228

05357 30222 17693 20943 00909 70953 22885 78291 18674

39060 08141 25284 92107 14253 71528 79726 07598 73535

75647 60169 76003 41270 00128 90508 70476 92467 56949

62481 83392 10398 31782 14684 97487 52162 60457 66380

17078 10720 37443 48530 86508 53677 45494 95375 01266

0.90483 0.81873 0.74081 0.67032 0.60653 0.54881 0.49658 0.44932 0.40656

74180 07530 82206 00460 06597 16360 53037 89641 96597

35959 77981 81717 35639 12633 94026 91409 17221 40599

57316 85866 86606 30074 42360 43262 51470 59143 11188

42491 99355 68738 44329 37995 84589 48001 01024 34542

16666 33334 50003 66677 83359 00054 16766 33504 50273

0.36787 94411 71442 32159 55238

142

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS

sin x

X

cos x

0.000 0,001 0.002 0.003 0.004

0.00000 0.00099 0.00199 0.00299 0.00399

00000 99998 99986 99955 99893

00000 33333 66666 00002 33341

00000 34166 93333 02499 86666

000 667 331 957 342

1.00000 0,99999 0.99999 0.99999 0.99999

00000 95000 80000 55000 20000

00000 00041 00666 03374 10666

00000 66666 66657 99898 66097

000 528 778 750 778

0.005 0.006 0.007 0.008 0.009

0.00499 0.00599 0.00699 0.00799 0.00899

99791 99640 99428 99146 98785

66692 00064 33473 66939 00492

70831 79994 39150 73291 07405

783 446 327 723 100

0.99998 0.99998 0.99997 0.99996 0.99995

75000 20000 55001 80001 95002

26041 53999 00041 70666 73374

64496 93520 50326 30257 26188

529 004 542 819 857

0.010 0.011 0.012 0.013 0.014

0.00999 0.01099 0.01199 0.01299 0.01399

98333 97781 97120 96338 95426

34166 68008 02073 36427 71148

66468 75446 59289 42921 51241

254 684 053 659 801

0.99995 0.99993 0.99992 0.99991 0.99990

00004 95006 80008 55011 20016

16665 10039 63995 90034 00656

27778 20617 85281 96278 20901

026 059 066 551 438

0.015 0.016 0.017 0.018 0.019

0.01499 0.01599 0.01699 0.01799 0.01899

94375 93173 91811 90280 88568

06328 42071 78498 15746 53967

09109 41340 72691 27852 31431

944 585 726 832 205

0.99988 0.99987 0.99985 0.99983 0.99981

75021 20027 55034 80043 95054

09359 30643 80008 73952 29976

17975 36508 14243 76107 32558

106 430 829 331 650

0.020 0.021 0.022 0.023 0.024

0.01999 0.02099 0.02199 0.02299 0.02399

86666 84565 82253 79722 76960

93333 34033 76279 20302 66354

07936 81764 77175 18277 28999

649 335 771 769 311

0.99980 0.99977 0.99975 0.99973 0.99971

00066 95081 80097 55116 20138

66577 03255 60509 59836 23734

77841 88132 19593 06320 58193

270 556 878 750 002

0.025 0.026 0.027 0.028 0.029

0.02499 0.02599 0.02699 0.02799 0,02899

73959 70707 67196 63414 59353

14712 65676 19572 76750 37589

33066 53973 14955 38952 48577

217 517 411 746 881

0.99968 0.99966 0.99963 0.99960 0.99957

75162 20190 55221 80256 95294

75702 40237 42836 09997 69215

58624 62215 92299 38394 53557

967 698 214 779 207

0.030 0.031 0.032 0.033 0.034

0.02999 0.03099 0.03199 0.03299 0.03399

55002 50350 45389 40108 34497

02495 71904 46280 26119 11951

66076 13288 11602 81908 44553

853 752 188 762 435

0.99955 0.99951 0.99948 0.99945 0.99942

00337 95384 80436 55494 20556

48987 78809 89175 11581 78521

51627 04381 38584 32936 14926

216 810 710 824 773

0.035 0.036 0.037 0.038 0.039

0.03499 0.03599 0.03699 0.03799 0.03899

28546 22245 15584 08553 01142

04336 03869 11180 26937 51841

19281 25183 80633 03228 09720

702 461 489 414 085

0.99938 0.99935 0.99931 0.99927 0.99923

75625 20699 55780 80868 95963

23488 80976 86478 76484 88487

57581 76116 24487 91840 98862

460 700 902 819 358

0.040 0.041 0.042 0.043 0.044

0.03998 0.04098 0.04198 0.04298 0.04398

93341 85141 76530 67500 58040

86634 32096 89047 58349 40905

15945 36751 85918 76078 18626

255 449 946 755 492

0.99920 0.99915 0.99911 0.99907 0.99903

01066 96177 81296 56424 21561

60977 33444 46376 41262 60588

94031 49770 58494 28564 80138

457 040 043 524 853

0.045 0.046 0.047 0.048 0.049

0.04498 0.04598 0.04698 0.04798 0.04898

48140 37790 26980 15701 03941

37660 49604 77774 23249 87159

23632 99745 54095 92191 17808

066 054 689 340 403

0.99898 0.99894 0.99889 0.99884 0.99879

76708 21865 57033 82211 97401

47842 47508 05071 67013 80818

40921 41817 12480 76767 48087

992 869 849 299 272

0.050

0.04997 91692 70678 32879 487

0.99875 02603 94966 24656 287

For conversion from degrees to radians see Example 13. For use and extension of the table see Examples 15-17. From C. E. Van Orstrand, Tables of the exponential functionand of thecircular sine and cosine to radian arguments,Memoirs of the National Academy of Sciences, vol. 14,Fifth Memoir. US.Government Printing Office, Washington, D.C., 1921 (with permission). Known errors have been corrected.

I

143

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS

sin x

X

CO8

Table 4.6 x

0.050 0.051 0.052 0.053 0.054

0.04997 0.05097 0.05197 0.05297 0.05397

91692 78943 65685 51906 37598

70678 75032 01496 51396 26109

32879 37375 29184 03981 55099

487 800 649 925 505

0.99875 0.99869 0.99864 0.99859 0.99854

02603 97818 83046 58287 23542

94966 58936 23208 39259 59564

24656 84647 81242 37585 41634

287 237 407 623 531

0.055 0.056 0.057 0.058 OIO59

0.05497 0.05597 0.05696 0.05796 Oi05896

22750 07352 91395 74868 57761

27067 55755 13712 02534 23875

73387 47070 61601 99503 40214

446 891 567 794 896

0.99848 0.99843 0.99837 0.99831 0.99826

78812 24097 59397 84714 00048

37598 27834 85743 67796 31461

40913 37163 80900 65862 23365

005 704 770 676 235

0.060 0.061 0.062 0.063 0.064

0.05996 0.06096 0.06196 0.06295 0.06395

40064 21768 02863 83337 63182

79444 71012 00408 69523 80309

59919 31380 23757 02430 28803

909 500 982 343 166

0.99820 0,99814 0.99807 0.99801 0.99795

05399 00768 86156 61562 26989

35204 38490 01782 86542 55229

16554 34561 86552 95687 92968

766 437 769 334 628

0.065 0.066 0.067 0.068 0.069

0.06495 0.06595 0.06694 0.06794 0.06894

42388 20944 98840 76067 52615

34782 35022 83173 81445 32117

60114 49232 44449 89264 22165

361 601 361 458 004

0.99788 0.99782 0.99775 0.99768 0.99762

82436 27904 63395 88907 04443

71301 99211 04415 53362 13501

10999 77634 09538 05636 40472

144 635 592 926 866

0.070 0.071 0.072 0.073 0,074

0.06994 0.07094 0.07193 0.07293 0.07393

28473 03632 78081 51811 24811

37532 00106 22323 06738 55977

76397 79734 54229 15974 74838

655 071 480 250 360

0.99755 0.99748 0.99740 0.99733 0.99726

10002 05586 91195 66830 32492

53279 42140 50526 49875 12624

57462 62048 14757 24157 39707

091 084 726 139 777

0.075 0.076 0.077 0.078 0.079

0.07492 0.07592 0.07692 0.07792 0.07891

97072 68584 39337 09320 78524

72742 59805 20017 56301 71660

34208 90718 33972 46257 02252

684 980 485 015 478

0.99718 0.99711 0.99703 0.99695 0.99688

88181 33898 69644 95419 11225

12207 23055 20596 81256 82457

44522 48023 78496 75551 82476

774 568 785 417 279

0,080 0.081 0.082 0.083 0.084

0.07991 0.08091 0.08190 0.08290 0.08390

46939 14555 81362 47349 12508

69172 51998 23374 86621 45140

68730 04247 58826 73635 80655

688 389 394 718 638

0.99680 0.99672 0.99663 0.99655 0.99647

17063 12932 98834 74769 40739

02619 21157 18485 76013 76147

38497 70937 87272 67091 53953

771 933 823 212 598

0.085 0.086 0.087 0.088 0.089

0.08489 0.08589 0.08689 0.08788 0.08888

76828 40298 02910 64653 25516

02416 62015 27592 02885 91720

02338 51260 29764 29594 31524

544 514 492 973 112

0.99638 0.99630 0.99621 0.99613 0.99604

96745 42786 78864 04980 21135

02290 38841 71197 85750 69887

47151 93367 78234 17797 49872

570 506 626 412 388

0.090 0.091 0.092 0.093 0. 094

0.08987 0.09087 0.09187 0.09286 0.09386

85491 44568 02735 59984 16304

98011 25760 79059 62093 79136

04969 07600 84943 69966 82662

125 919 819 323 751

0.99595 0.99586 0.99577 0.99567 0.99558

27330 23565 09841 86159 52521

11994 01450 28634 84916 62665

25309 99152 21703 29482 36090

284 586 483 217 844

0.095 0.096 0.097 0.098 0.099

0.09485 0.09585 0.09684 0.09784 0.09883

71686 26119 79593 32099 83627

34557 32817 78472 76177 30679

29625 03609 83083 31775 98210

724 347 006 683 683

0.99549 0.99539 0.99529 0.99520 0.99510

08927 55378 91875 18419 35011

55245 57015 63330 70541 75992

22976 30094 46473 00679 51179

426 649 881 686 796

0.100

0.09983 34166 46828 15230 681

“-7Y

0.99500 41652 78025 76609 556

I

144

ELEMENTARY TRANSCENDENTAL FUNCTIONS Table 4.6

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS sin x

X

cos X

0.100 0,101 0.102 0.103 0.104

0.09983 0.10082 0.10182 0.10281 0.10381

34166 83707 32239 79754 26240

46828 29567 83945 15107 28302

15230 99512 51074 52769 69768

681 975 864 040 897

0.99500 0.99490 0.99480 0.99470 0.99459

41652 38343 25085 01879 68726

78025 75976 70176 61949 53618

76609 65937 08533 84132 52703

556 840 469 117 737

0.105 0.106 0.107 0.108 0.109

0.10480 0.10580 0.10679 0.10779 0.10878

71688 16088 59430 01704 42900

28882 22302 14121 10007 15731

49043 18823 88052 45835 60869

655 209 588 941 939

0.99449 0.99438 0.99428 0.99417 0.99406

25627 72583 09595 36665 53792

48497 50896 66120 00466 61230

44220 48325 03900 88538 07909

501 268 596 307 607

0.110 0.111 0.112 0.113 0,114

0.10977 0.11077 0.11176 0.11275 0.11375

83008 22018 59921 96706 32364

37174 80326 51285 56261 01575

80866 31964 18131 20553 97013

495 714 952 909 636

0.99395 0.99384 0.99373 0.99362 0.99350

60979 58226 45535 22907 90342

56696 96148 89860 49101 86134

85035 49459 26316 25308 29576

784 483 578 652 080

0.115 0.116 0.117 0.118 0.119

0.11474 0.11574 0.11673 0.11772 0.11871

66883 00256 32471 63519 93389

93663 39072 44465 16621 62434

81259 82361 84055 44080 93496

372 097 722 790 613

0.99339 0.99327 0.99316 0.99304 0.99292

47843 95409 33043 60744 78516

14215 47595 01517 92218 36926

84471 86235 70568 01110 57814

755 439 768 921 950

0.120 0.121 0.122 0.123 0.124

0.11971 0.12070 0.12169 0.12269 0.12368

22072 49559 75838 00900 24735

88919 03206 12547 24315 46003

35996 47206 73970 33626 13267

735 615 447 003 407

0.99280 0.99268 0.99256 0.99244 0.99232

86358 84272 72259 50321 18458

53866 62252 82294 35193 43142

25224 80653 82259 57029 88655

810 067 329 382 070

0.125 0.126 0.127 0.128 0.129

0.12467 0.12566 0.12665 0.12765 0.12864

47333 68685 88780 07608 25160

85227 49729 47372 86148 74174

68995 25157 73569 72735 47043

744 389 978 909 273

0.99219 0.99207 0.99194 0.99181 0.99169

76672 24964 63335 91787 10320

29329 17930 34118 04055 54896

05314 67355 54873 55198 50278

910 462 474 803 123

0.130 0.131 0.132 0.133 0.134

0.12963 0.13062 0.13161 0.13260 0.13359

41426 56395 70058 82404 93425

19694 31083 16843 85608 46144

85954 43179 35844 43632 07929

121 968 433 907 171

0.99156 0.99143 0.99130 0.99116 0.99103

18937 17638 06424 85298 54260

14788 12868 79267 45107 42499

03959 49177 75039 13813 27814

451 481 751 659 325

0.135 0.136 0.137 0.138 0.139

0.13459 0.13558 0.13657 0.13756 0.13855

03110 11448 18431 24048 28290

07348 78252 68023 85962 41508

30938 74799 60677 67852 32784

844 575 867 453 107

0.99090 0.99076 0.99063 0.99049 0.99035

13312 62454 01689 31018 50441

04547 65348 59985 24535 96067

96193 01628 16913 91451 37644

339 375 714 667 937

0.140 0.141 0.142 0.143 0.144

0.13954 0.14053 0.14152 0.14251 0.14350

31146 32607 32662 31302 28517

44236 03861 30237 33359 23362

48171 61995 76542 47427 82584

799 092 691 025 791

0.99021 0.99007 0.98993 0.98979 0.98964

59962 59580 49297 29115 99035

12637 13293 38073 28007 25111

17189 27270 86655 21689 52197

895 829 145 546 214

0.145 0.146 0.147 0.148 0.149

0.14449 0.14548 0.14647 0.14746 0.14844

24297 18632 11512 02927 92868

10526 05272 18167 59922 41398

41263 32992 16543 98870 34041

332 773 800 997 627

0.98950 0.98936 0.98921 0.98906 0.98892

59058 09187 49421 79764 00216

72394 13854 94478 60241 58111

77275 60997 18007 99027 76256

984 551 704 617 193

0.150

0.14943 81324 73599 22149 773

0.98877 10779 36042 28673 498

145

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS sin x

X

cos

Table 4.6

x

0.150 0.151 0.152 0.153 0.154

0.14943 0.15042 0.15141 0.15240 0.15339

81324 68286 53744 37687 20107

73599 67680 34944 86847 34994

22149 08215 81070 72225 54727

773 725 532 604 267

0.98877 0.98862 0.98847 0.98831 0.98816

10779 11454 02243 83147 54168

36042 42977 28849 44579 42076

28673 27245 20028 17178 75856

498 283 611 614 382

0.155 0.156 0.157 0.158 0.159

0.15438 0.15536 0.15635 0.15734 0.15833

00992 80334 58122 34347 08998

91143 67205 75247 27490 36311

41996 86651 79319 47428 53983

190 555 902 529 354

0.98801 0.98785 0.98770 0.98754 0.98738

15307 66566 07947 39451 61079

74239 94954 59094 22522 42087

85038 50224 78054 60814 60855

006 794 663 736 150

0.160 0.161 0.162 0.163 0.164

0.15931 0.16030 0.16129 0.16227 0.16326

82066 53540 23412 91670 58306

14245 73987 28387 90460 73379

96331 04906 41960 00278 01876

146 020 095 226 705

0.98722 0.98706 0.98690 0.98674 0.98658

72833 74715 66727 48869 21144

75626 81965 20914 53272 40826

94904 18284 09029 51905 22328

095 099 574 638 234

0.165 0.166 0.167 0.168 0.169

0.16425 0.16523 0.16622 0.16721 0.16819

23309 86670 48378 08424 66798

90480 55265 81396 82704 73183

96685 61216 97208 30268 08481

825 228 916 843 981

0.98641 0.98625 0.98608 0.98592 0.98575

83553 36098 78780 11602 34564

46347 33596 67316 13241 38088

70185 03560 72356 51818 25966

554 791 233 712 434

0.170 0.171 0.172 0.173 0.174

0.16918 0.17016 0.17115 0.17213 0.17312

23490 78490 31789 83376 33241

66996 78473 22117 12595 64750

01015 96702 02607 42577 55773

762 805 812 560 865

0.98558 0.98541 0.98524 0.98507 0.98490

47669 50917 44312 27854 01546

09560 96348 68126 95555 50280

70917 38117 37476 20391 62691

193 998 124 598 158

0.175 0.176 0.177 0.178 0.179

0.17410 0.17509 0.17607 0.17706 0.17804

81375 27769 72411 15292 56403

93595 14318 42278 93011 82230

95189 26146 24778 76492 74417

433 505 176 317 975

0.98472 0.98455 0.98437 0.98419 0.98402

65389 19384 63534 97840 22304

04933 33129 09469 09537 09903

47463 47797 09416 33225 57745

670 052 699 443 046

0.180 0.181 0.182 0.183 0.184

0.17902 0.18001 0.18099 0.18198 0.18296

95734 33274 69014 02944 35054

25824 39859 40581 44417 67974

17834 10581 59452 72574 57756

180 029 980 233 116

0.98384 0.98366 0.98348 0.98330 0.98311

36927 41713 36661 21775 97056

88121 22728 93246 80179 65017

41459 45058 13586 58485 39552

272 522 083 974 448

0.185 0.186 0.187 0.188 0.189

0.18394 0.18492 0.18591 0.18689 0.18787

65335 93776 20368 45100 67964

28041 41589 25775 97940 75611

20836 64000 84083 70855 05288

370 231 224 554 013

0.98293 0198275 0.98256 0.98237 0.98219

62506 18126 63919 99886 26029

30231 59276 36591 47595 78693

46781 82121 41132 94537 69683

122 799 959 971 022

0.190 0.191 0.192 0.193 0.194

0.18885 0.18984 0.19082 0.19180 0.19278

88949 08046 25244 40533 53905

76500 18510 19732 98445 73120

57799 86484 35325 32380 87958

285 571 424 691 485

0.98200 0,98181 0.98162 0.98143 0.98124

42351 48852 45535 32402 09455

17270 51693 71313 66461 28451

31896 65751 56228 69777 35290

788 875 034 178 214

0.195 0.196 0.197 0.198 0.199

0.19376 0.19474 0.19572 0.19670 0.19768

65349 74855 82414 88016 91650

62421 85204 60517 07604 45907

92769 16058 03723 76404 27565

058 510 204 820 917

0.98104 0.98085 0.98065 0.98046 0.98026

76695 34125 81746 19561 47571

49577 23115 43322 05437 05677

24965 35080 66661 06062 05434

723 479 867 170 796

0.200

0.19866 93307 95061 21545 941

0.98006 65778 41241 63112 420

I

146

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS sin x

X

COB 2

0.200 0.201 0.202 0.203 0.204

0.19866 0.19964 0.20062 0.20160 0.20258

93307 92978 90653 86321 79972

95061 74900 05459 06969 99863

21545 91597 37903 25571 82615

941 545 151 640 083

0.98006 0.97986 0.97966 0.97946 0.97926

65778 74185 72793 61604 40621

41241 10310 12041 46575 15030

63112 03887 59192 47187 52742

420 090 306 084 047

0.205 0.206 0.207 0.208 0.209

0.20356 0.20454 0.20552 0.20650 0.20748

71599 61189 48734 34224 17648

04777 42549 34218 01031 64439

97905 19110 50612 51399 32944

397 856 330 175 665

0.97906 0.97885 0.97865 0.97844 0.97823

09845 69278 18923 58781 88855

19505 63076 49802 84716 73834

07327 68803 01113 53874 41879

536 784 156 491 553

0.210 0.211 0.212 0.213 0.214

0.20845 0.20943 0.21041 0.21139 0.21237

98998 78263 55434 30501 03453

46099 67877 51846 20289 95699

57060 33732 18932 12409 55467

871 895 346 982 398

0.97803 0.97782 0.97761 0.97740 0.97718

09147 19658 20391 11348 92531

24148 43628 41225 26863 11448

24491 84946 09554 66806 86380

614 201 014 039 882

0.215 0.216 0.217 0.218 0.219

0.21334 0.21432 0.21530 0.21627 0.21725

74283 42978 09530 73930 36166

00782 58454 91846 24303 79385

28707 49764 71012 77249 83368

677 905 439 851 434

0.97697 0.97676 0.97654 0.97633 0.97611

63942 25583 77456 19564 51909

06862 25963 82586 91546 68630

38054 10511 90059 39246 75378

344 247 555 782 736

0.220 0.221 0.222 0.223 0.224

0.21822 0.21920 0.22018 0.22115 0.22213

96230 54112 09802 63290 14566

80869 52747 19233 04757 33970

31995 91115 51671 25146 41115

179 124 977 920 484

0.97589 0.97567 0.97545 oI97523 0.97501

74493 87317 90385 83699 67259

30605 95212 81168 08I67 96877

48940 21920 46034 40857 71849

602 392 788 388 392

0.225 0.226 0.227 0.228 0.229

0.22310 0.22408 0.22505 0.22602 0.22700

63621 10445 55028 97360 37433

31745 23176 33582 88504 13708

44782 94494 59230 16071 47642

417 428 720 214 363

0.97479 0.97457 0.97434 0.97412 0.97389

41070 05133 59450 04024 38856

68943 46983 54590 16334 57756

28292 01125 60681 34326 84008

737 708 052 607 477

0.230 0.231 0.232 0.233 0.234

0.22797 0.22895 0.22992 0.23089 0.23187

75235 10757 43990 74924 03549

35188 79163 72082 40621 11686

39540 77732 45933 22962 80075

462 354 437 869 884

0.97366 0.97343 0.97320 0.97297 0.97274

63950 79306 84929 80819 66980

05374 86678 30133 65176 22218

83696 96733 53085 26494 11536

773 940 695 602 294

0.235 0.236 0.237 0.238 0.239

0.23284 0.23381 0.23478 0.23575 0.23673

29855 53832 75472 94763 11697

12416 70180 12580 67453 62868

78273 65586 74343 18405 90384

112 809 904 752 520

0.97251 0.97228 0.97204 0.97181 0.97157

43413 10121 67106 14371 51917

32643 28807 44041 12644 69892

00578 60642 10166 95675 68349

389 091 529 843 034

0.240 0: 241 0.242 0.243 0.244

0.23770 0.23867 0.23964 0.24061 0.24158

26264 38453 48256 55663 60663

27134 88793 76627 19655 47136

58836 65429 22091 08131 67335

079 334 869 828 933

0.97133 0.97109 0.97086 0.97062 0.97037

79748 97865 06272 04970 93961

52029 96272 40809 24800 88375

60492 61916 96210 96928 83670

618 095 262 391 294

0.245 0.246 0.247 0.248 0.249

0.24255 0.24352 0.24449 0.24546 0.24643

63247 63406 61130 56408 49232

88572 73702 32513 95231 92328

05043 85196 27365 03750 36159

522 546 389 445 337

0.97013 0.96989 0.96965 0.96940 0.96915

73249 42836 02723 52914 93411

72635 19650 72463 75084 72494

38069 79682 41782 47054 83195

313 233 166 425 397

0.250

0.24740 39592 54522 92959 685 L

'

J

0.96891 24217 10644 78414 459

147

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR RADIAN ARGUlMENTS sin

X

Table 4.6

cos x

1:

0.250 0.251 0.252 0.253 0.254

0.24740 0.24837 0.24934 0.25030 0.25127

39592 27478 12879 95788 76193

54522 12778 98307 42569 77272

92959 86007 67549 27105 88317

685 332 922 742 722

0.96891 0.96866 0.96841 0.96816 0.96791

24217 45333 56762 58508 50572

10644 36453 97810 43570 23561

78414 76838 13822 91154 52178

459 955 250 897 941

0.255 0.256 0.257 0.258 0.259

0.25224 0.25321 0.25418 0.25514 0.25611

54086 29456 02294 72590 40335

34378 46095 44888 63473 34820

05782 61854 63424 38674 33804

506 486 714 587 209

0.96766 0.96741 0.96715 0.96690 0.96664

32956 05664 68698 22061 65754

88575 56805 90374 56434 81687 68781 16211 52599 48609 82314

375 780 180 126 035

0.260 0.261 0.262 0.263 0.264

0.25708 0.25804 0.25901 0.25997 0.26094

05518 68131 28163 85606 40448

92155 68959 98972 16189 54868

09735 38788 01336 82426 68386

339 820 401 844 239

0.96638 0.96613 0.96587 0.96561 0.96535

99781 24144 38845 43888 39275

34513 30519 94190 84058 59618

22555 02595 90687 68308 04309

822 835 131 107 520

0.265 0.266 0.267 0.268 0.269

0.26190 0.26287 0.26383 0.26480 0.26576

92681 42295 89280 33627 75325

49524 34933 46135 18431 87386

43392 86023 65779 39579 48230

399 278 278 372 942

0.96509 0.96483 0.96456 0.96430 0.96403

25008 01091 67525 24313 71458

81330 10622 09885 42476 72716

28964 07924 16072 11288 08109

923 537 584 118 368

0.270 0.271 0.272 0.273 0.274

0.26673 0.26769 0.26865 0.26962 0.27058

14366 50740 84437 15447 43761

88831 58861 33839 50396 45431

12873 31394 74821 83684 64354

229 301 451 915 828

0.96377 0.96350 0.96323 0.96296 0.96269

08963 36830 55063 63662 62633

65890 88248 07004 90334 07377

51301 89328 47727 02389 52736

623 696 972 084 246

0.275 0.276 0.277 0.278 0.279

0.27154 0.27250 0.27347 0.27443 0.27539

69369 92262 12429 29862 44551

56112 19879 74443 57786 08166

85351 73627 10825 29507 09350

302 557 981 043 952

0.96242 0.96215 0.96188 0.96160 0.96133

51976 31695 01792 62271 13133

28237 23980 66634 29189 85596

94814 94278 59286 13299 67778

248 169 807 879 997

0.280 0.281 0.282 0.283 0.284

0.27635 0.27731 0.27827 0.27923 0.28019

56485 65656 72054 75669 76492

64113 64435 48215 54812 23866

73331 83865 38926 68142 28856

967 270 293 411 909

0.96105 OI96077 0.96050 0.96022 0.95994

54383 86021 08052 20478 23302

10770 80586 71880 62449 31050

94792 99523 92684 62830 48581

459 878 682 504 495

0.285 0.286 0.287 0.288 0.289

0.28115 0.28211 0.28307 0.28403 0.28499

74512 69722 62110 51667 38384

95294 09293 06345 27208 12929

02165 88922 05725 80861 50237

110 591 374 997 384

0.95966 0.95938 0.95909 0.95881 0.95852

16526 00154 74188 38630 93485

57401 22179 07021 94525 68245

10746 04351 50572 08568 47227

590 746 193 713 984

0.290 0.291 0.292 0.293 0.294

0.28595 0.28691 0.28786 0.28882 0.28978

22251 03258 81396 56656 29027

04835 44540 73943 35230 70875

53268 28750 10698 24153 80965

394 981 841 475 551

0.95824 0.95795 0.95767 0.95738 0.95709

38755 74442 00549 17080 24037

12697 13353 56644 29961 21649

16807 20481 85799 36036 61457

013 688 478 308 636

0.295 0.296 0.297 0.298 0.299

0.29073 0.29169 0.29265 0.29360 0.29456

98501 65067 28716 89439 47225

23642 36583 53042 16653 71345

75547 80597 42792 78457 69198

489 155 582 616 389

0.95680 0.95651 0.95621 0.95592 0.95563

21423 09241 87494 56184 15316

21013 18315 04772 72560 14809

90483 60759 90127 47507 23678

768 429 632 858 590

0.300

0.29552 02066 61339 57510 532

py41

0.95533 64891 25606 01964 231

I

148

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS sin x

0.300 0.301 0.302 0.303 0.304

0.29552 0.29647 0.29743 0.29838 0.29933

02066 53952 02873 48819 91782

61339 31151 25592 89771 69093

57510 42357 74716 53102 19051

532 025 586 518 897

0.95533 0.95504 0.95474 0.95444 0.95414

64891 04912 35384 56308 67687

x 25606 99993 33968 24485 69450

0.305 0.306 0.307 0.308 0.309

0.30029 0.30124 0.30220 OI30315 0.30410

31752 68718 02672 33604 61505

09261 56279 56451 56380 02974

52585 67635 07447 39950 53093

026 045 613 549 365

0.95384 0.95354 0.95324 0.95294 0.95263

69525 61825 44589 17820 81523

67727 19130 24430 85350 04568

06164 11990 12121 63513 47552

084 559 945 878 001

0.310 0.311 0.312 0.313 0.314

0.30505 0.30601 0.30696 0.30791 0.30886

86364 08173 26921 42601 55200

43443 25301 96367 04767 98932

50156 45030 57464 08284 14579

564 632 615 189 138

0.95233 0.95202 0.95172 0.95141 0.95110

35698 80351 15483 41098 57199

85713 33367 53066 51295 35494

39784 79558 39561 95271 94302

281 038 711 383 111

0.315 0.316 0.317 0.318 0.319

0.30981 0.31076 0.31171 0.31266 0.31361

64712 71125 74430 74619 71680

27602 39828 84966 12688 72974

84860 14184 79252 33468 01977

120 658 234 402 833

0.95079 0.95048 0.95017 0.94986 0.94954

63789 60870 48447 26523 95099

14053 96311 92562 14047 72959

25664 88923 63269 76481 73811

080 617 094 749 467

0.320 0.321 0.322 0. 323 0.324

0.31456 0.31551 0.31646 0.31741 0.31836

65606 56385 44010 28470 09756

16117 92727 53724 50346 34148

76666 11130 15619 51938 28330

176 659 332 844 674

0.94923 0.94892 0.94860 0.94828 0.94796

54180 03769 43869 74482 95613

82440 56583 10427 59963 22130

86757 01754 28762 69764 87164

531 395 501 173 613

0. 325 0. 326 0.327 0.328 0.329

0.31930 0.32025 0.32120 0.32215 0.32309

87858 62767 34474 02968 68241

57000 71094 28937 83360 87512

94315 35507 68391 35077 98012

718 128 319 048 460

0.94765 0.94733 0.94701 0.94668 0.94636

07264 09438 02139 85370 59134

14815 56853 68025 69063 81642

72098 12639 61918 06147 32541

048 034 976 877 351

0.330 0.331 0.332 0.333 0.334

0.32404 0.32498 0.32693 0.32687 0.32782

30283 89085 44637 96929 45953

94868 59222 34694 75730 37100

34670 32199 82047 74545 93468

020 224 011 756 777

0.94604 0.94571 0.94539 0.94506 0.94473

23435 78275 23658 59587 86065

28386 32866 19598 14042 42606

97152 92611 15765 35228 58837

941 768 535 939 502

0.335 0.336 0.337 0.338 0.339

0.32876 0.32971 0.33065 oi33160 0.33254

91698 73903 10553 241 34156 41562 79990 386 73316 95834 32882 957 oSiio 92801 71669 766 41708 88879 64517 288

0.94441 0.94408 0.94375 oI94341 0.94308

03096 10683 08829 97537 76811

32643 12448 11264 59275 87612

01006 49997 35085 93637 38092

864 577 413 243 499

0.340 0.341 0.342 0.343 0.344

0.33348 OI33442 0.33537 0.33631 0.33725

70921 96799 19332 38512 54328

40814 05684 40903 04216 53706

39678 79816 16300 23460 12813

177 635 519 104 399

0.94275 0.94242 0.94208 0.94174 0.94141

46655 07071 58062 99633 31786

28346 14493 80011 59801 89707

22850 11062 41330 94311 59229

264 025 105 834 468

0.345 0.346 0.347 0.348 0.349

0.33819 0.33913 0.34007 0.34101 0.34195

66772 75834 81505 83774 82634

47791 45227 05108 86866 50276

27257 35228 24823 97891 64093

928 880 531 850 188

0.94107 0.94073 0.94039 0.94005 0.93971

54526 67854 71775 66292 51409

06513 47944 52668 60293 11367

00285 22986 40365 39119 45650

905 218 059 944 473

0.350

0.34289 78074 55451 34918 963

X

L

'

cos

J

01964 28826 84359 52692 92289

231 414 763 116 242

0.93937 27128 47378 92003 503

I

I 149

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS sin x

z

cos

Table 4.6

x

0.350 0.351 0.352 0: 353 0.354

0.34289 0.34383 0.34477 0.34571 0.34665

78074 70085 58658 43783 25451

55451 62847 33263 27841 98071

34918 17681 09467 91058 20819

963 237 102 778 319

0.93937 0.93902 0.93868 0.93833 0.93799

27128 93454 50389 97937 36103

47378 10755 44865 94014 03447

92003 81724 55613 57391 99266

503 321 841 869 461

0.355 0.356 0.357 0.358 0.359

0.34759 0.34852 0.34946 0.35040 0.35133

03652 78377 49617 17363 81604

35784 73161 82729 27364 70292

28543 09276 17091 58840 87868

852 237 064 891 632

0.93764 0.93729 0.93694 0.93659 0.93624

64888 84296 94332 94998 86299

19349 88839 59978 81762 04124

27409 87337 89202 72980 73578

412 915 418 716 312

0.360 0.361 0.362 0.303 0.364

0.35227 0.35320 0.35414 0.35508 0.35601

42332 99538 53211 03343 49923

75089 05683 26351 01729 96801

97684 15610 96384 15734 63913

991 866 608 065 294

0.93589 0.93554 0.93519 0.93483 0.93448

68236 40815 04038 57910 02433

77934 54999 88060 30795 37816

85835 29438 13742 02492 78462

091 322 042 855 165

0.365 0.366 0.367 0.368 0.369

0.35694 0.35788 0.35881 0.35975 0.36068

92944 32396 68268 00552 29239

76911 07756 55391 86229 67042

39203 41380 65142 93504 91160

863 647 021 354 721

0.93412 0.93376 0.93340 0.93304 0.93268

37611 63448 79948 87113 84948

64673 67846 04751 33740 14096

07984 05404 97425 87371 19348

897 739 922 606 871

0.370 0.371 0.372 0.373 0.374

0.36161 0.36254 0.36347 0.36441 0.36534

54319 75783 93621 07825 18384

64961 47479 82448 38085 82970

97803 21412 31502 52343 56131

729 373 813 006 067

0.93232 0.93196 0.93160 0.93123 0.93087

73456 52640 22505 83054 34291

06034 70704 70188 67499 26582

42320 74082 65151 62553 73524

381 737 560 347 125

0.375 0.376 0.377 0.378 0.379

0.36627 0.36720 0.36813 0.36906 0.36999

25290 28534 28105 23995 16194

86047 16625 44381 39357 71964

56137 99809 61843 37211 34164

291 733 251 926 758

0.93050 0.93014 0.92977 0.92940 0.92903

76219 08841 32163 46186 50916

12314 90501 27881 92123 51824

29114 47704 98417 64451 06312

948 265 211 836 328

0.380 0.381 0.382 0.383 0.384

0.37092 0.37184 0.37277 0.37370 0.37463

04694 89484 70556 47899 21506

12982 33562 05224 99862 89741

67184 49909 88020 72083 70366

549 881 096 184 479

0.92866 0.92829 0.92792 0.92754 0.92717

46355 32508 09378 76968 35283

76510 36638 03592 49686 48161

24949 24806 76777 81063 29943

253 858 471 030 792

0.385 0.386 0.387 0.388 0.389

0.37555 0.37648 0.37741 0.37833 0.37926

91367 57472 19812 78378 33161

47501 46155 59093 60081 23263

21610 27762 46681 84790 89706

089 945 397 240 110

0.92679 0.92642 0.92604 0.92566 0.92528

84326 24101 54613 75863 87857

73184 99852 04187 63138 54580

70454 66966 63679 47019 07941

235 223 438 143 297

0.390 0.391 0.392 0.393 0.394

0.38018 0.38111 0.38203 0.38296 0.38388

84151 31339 74716 14272 49999

23161 34675 33087 94059 93636

42823 51860 43373 55222 29011

118 671 349 774 366

0.92490 0.92452 0.92414 0.92376 0.92338

90598 84090 68337 43342 09109

57313 51063 16481 35142 89547

04145 22192 39537 86457 07898

068 776 314 070 401

0.395 0.396 0.397 0.398 0.399

0.38480 0.38573 0.38665 0.38757 0.38849

81888 09928 34110 54427 70867

08245 14697 90188 12300 59002

02477 01854 34186 79611 83601

888 707 658 426 363

0.92299 0.92261 0.92222 0.92183 0.92144

65643 12947 51025 79880 99517

63117 40199 06064 46904 49832

25225 97879 84939 06602 05558

693 040 589 584 150

0.400

0.38941 83423 08650 49166 631

0.92106 09940 02885 08279 853

150

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SINES AND COSINES FOR RADIAN ARGUiMENTS cos x

sin x

X

0.400 0.401 0.402 0.403 0.404

0.38941 0.39033 0.39125 0.39217 0.39309

83423 92084 96842 97687 94611

08650 39988 32150 64660 17434

49166 29019 17700 43663 61324

631 595 358 363 955

0.92106 0.92067 0.92028 0.91988 0.91949

09940 11151 03157 85959 59563

02885 95020 16118 56976 09315

08279 86221 16919 45007 43137

853 075 248 979 110

0.405 0.406 0.407 0.408 0.409

0.39401 Or39493 0.39585 0.39677 0.39769

87603 76656 61759 42903 20080

70780 05398 02384 43226 09812

43071 71230 29995 97324 36782

820 202 816 356 508

0.91910 0.91870 0.91831 0.91791 0.91751

23971 79189 25219 62067 89735

65774 19913 66209 00060 17781

72800 45073 81253 73416 44875

745 295 568 956 737

0.410 0.411 0.412 0: 413 0.414

0.39860 0.39952 0.40044 0.40135 0.40227

93279 62493 27711 88925 46126

84422 49738 88838 85200 22702

89359 65238 35528 23958 98524

380 251 558 010 766

0.91712 0.91672 0.91632 0.91592 0.91551

08228 17549 17704 08695 90527

16605 94682 51081 85785 99696

10547 37232 03796 61266 92832

564 150 202 649 194

0.415

0.419

0.40318 0.40410 0.40501 0.40593 0.40684

99303 48449 93554 34608 71603

85626 58653 26869 75762 91229

63109 49047 06660 96747 82037

550 645 654 939 655

0.91511 0.91471 0.91430 0.91390 0.91349

63204 26730 81109 26344 62441

94631 73322 39416 97475 52975

73753 31180 03880 01872 65972

232 180 251 722 725

0.420 0.421 0.422 0.423 0.424

0.40776 0.40867 0.40958 0.41049 0.41140

04530 33379 58142 78808 95370

59570 67491 02108 50946 01936

18597 47203 84671 15143 81337

279 546 703 980 201

0.91308 Oi91268 0.91227 0.91186 0.91145

89403 07233 15937 15518 05981

12308 82776 72597 90901 47728

27243 66357 72866 04379 45647

609 915 996 332 576

0.425 0.426 0.427 0.428 0.429

0.41232 OI41323 0.41414 0.41505 0.41596

07817 16141 20333 20384 16283

43424 64165 53326 00488 95646

75749 31825 15081 14189 32014

435 593 889 067 301

0.91103 OI91062 0.91021 0.90979 0.90938

87329 59567 22698 76727 21659

54033 21681 63449 93022 24998

67564 86066 20950 54591 90577

373 990 808 701 360

0.430 0.431 0.432 0.433 0.434

0.41687 oi41777 0.41868 oi41959 0.42050

08024 95595 78989 58196 33207

29210 92007 75279 70687 70310

76621 52231 50136 39579 58584

692 243 257 028 774

0.90896 0.90854 0.90813 Or90771 0.90729

57496 84244 01906 10488 09991

74885 59097 94960 00709 95484

12247 41143 95366 47844 84510

591 638 563 729 435

0.435 0.436 0.437 0.438 0.439

0.42141 0,42231 0.42322 0.42412 0.42503

04013 70605 32974 91110 45005

66648 52619 21565 67248 83856

04753 26011 11315 81323 79016

684 018 146 456 027

0.90687 0.90644 0.90602 0.90560 0.90517

00422 81785 54083 17320 71502

99336 33221 19003 79452 38245

62385 67577 73181 97096 59747

731 465 601 848 647

0.440 0.441 0.442 0.443 0.444

0.42593 0.42684 0.42774 0.42865 0,42955

94650 40036 81153 17992 50545

65999 08712 07458 58123 57025

60276 84433 04751 58891 59317

972 381 750 823 745

0.90475 0.90432 0.90389 0.90346 0.90304

16632 52714 79753 97753 06718

19963 50093 55027 62061 99394

41716 41286 31889 19473 99766

554 061 904 892 305

0.445 0.446 0.447 0.448 0.449

0.43045 0.43136 0.43226 0.43316 0.43406

78803 02755 22395 37711 48696

00908 86947 12746 76342 76203

83666 65073 82453 50745 11100

443 141 917 219 244

0.90261 0.90217 0.90174 0.90131 0.90088

06653 97562 79449 52319 16175

96132 82279 88745 47341 90780

15457 13291 01061 04523 24210

899 573 718 319 832

0.450

0.43496 55341 11230 21042 084

0: 416

0.417 0; 418

151

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS sin x

X

cos

Table 4.6

x

0.450 0.451 0.452 0.453 0.454

0.43496 0.43586 0.43676 0.43766 0.43856

55341 57635 55571 49140 38331

11230 80759 84561 22842 96246

21042 44573 42243 61170 25020

084 567 681 507 568

0.90044 0.90001 0.89957 0.89913 0.89870

71023 16866 53709 81556 00412

52676 67546 70803 98765 88646

92166 28580 98337 67474 59552

884 847 319 569 965

0.455 0.456 0.457 0.458 0.459

0.43946 0.44036 0.44125 0.44215 0.44305

23138 03549 79557 51152 18326

05853 53183 40194 69287 43301

23944 04468 59344 17350 33053

492 918 542 215 008

0.89826 0.89782 0.89738 0.89693 0.89649

10281 11168 03076 86010 59975

78561 07522 15441 43127 32287

11933 31966 53089 90836 98759

463 167 030 721 714

0.460 0.461 0.462 0.463 0.464

0.44394 0.44484 0.44573 0.44663 0.44752

81069 39373 93228 42626 87558

65519 39668 69916 60878 17615

76524 23010 42563 89618 92537

151 752 218 275 506

0.89605 0.89560 0.89516 0.89471 0.89426

24975 81014 28097 66229 95414

25525 66339 99127 69179 22683

24253 64298 21110 57699 53342

639 937 867 908 602

0.465 0.466 0.467 0.468 0.469

0.44842 0.44931 0.45020 0.45110 0.45199

28014 63986 95465 22442 44907

45634 50888 39782 19166 96343

43101 85958 08029 27868 84976

319 244 479 603 342

0.89382 0.89337 0.89292 0.89247 0.89202

15656 26959 29329 22770 07286

06720 69266 59190 26256 21120

58962 52423 93730 80142 01196

873 883 459 134 857

0.470 0.471 0.472 0.473 0.474

0.45288 0.45377 0.45466 0.45555 0.45644

62853 76270 85149 89482 89259

79068 75545 94432 44843 36343

29070 09309 63474 07100 22566

327 736 735 635 671

0.89156 0.89111 0.89066 0.89020 0.88974

82881 49562 07330 56193 96153

95328 01323 92437 22891 47800

93645 96296 04773 26178 33674

402 541 005 292 367

0.475 0.476 0.477 0.478 0.479

0.45733 0.45822 0.45911 0.46000 0.46089

84471 75110 61167 42633 19498

78955 83158 59888 20540 76967

48139 66969 96047 75103 55473

307 994 279 180 739

0.88929 0.88883 0.88837 0.88791 0.88745

27216 49386 62667 67065 62583

23168 05888 53744 25407 80438

20970 56721 38842 48723 05369

288 822 074 197 212

0.480 0.481 0.482 0.483 0.484

0.46177 0.46266 0.46355 0.46443 0.46532

91755 59394 22406 80783 34515

41482 26861 46338 13613 42849

88913 16364 56679 95295 72867

664 968 522 430 132

0.88699 0.88653 0.88606 0.88560 0.88514

49227 27001 95910 55958 07150

79284 83281 54652 56506 52837

19439 47206 44417 20075 90129

995 469 051 401 517

0.485 0.486 0.487 0.488 0.489

0.46620 0.46709 0.46797 0.46886 0.46974

83594 28011 67757 02823 33201

48672 46175 50915 78918 46678

73849 15033 34040 77761 90760

162 451 104 558 024

0.88467 0.88420 0.88374 0.88327 0.88280

49491 82984 07636 23450 30432

08528 89343 61933 93833 53462

31072 33453 55301 75463 46844

223 094 874 416 214

0.490 0.491 0.492 0.493 0.494

0.47062 0.47150 0.47238 0.47327 0.47415

58881 79855 96114 07649 14451

71158 69788 60472 61583 91970

03618 21242 11121 91533 19709

136 715 556 149 261

0.88233 0.88166 0.88138 0.88091 0.88044

28586 10121 17916 33995 98427 96151 70125 68537 33014 23984

49570 44058 23994 69230 98588

547 307 541 763 075

0.495 0; 496 0.497 0; 498 0.499

0.47503 0.47591 0.47679 OI47766 0.47854

16512 13823 06374 94157 77164

70950 18319 54345 99774 75827

79950 71693 97532 51191 05452

264 150 118 668 099

0.87996 0.87949 0.87901 0.87853 0.87806

87098 32382 68872 96571 15485

22574 96012 70581 47270 28743

157 154 529 917 023

0.500

0.47942 55386 04203 00027 329

36204 79786 30204 63808 57828

0.87758 25618 90372 71611 628

152

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS sin x

X

cos 2

0.500 0.501 0.502 0.503 0.504

0.47942 0.48030 0.48117 0.48205 0.48293

55386 28813 97437 61249 20240

04203 07080 07116 27448 91696

00027 29394 30578 70881 35573

329 947 414 314 583

0.87758 0.87710 0.87662 0.87614 0.87565

25618 26976 19562 03383 78441

90372 40428 87859 13407 98689

71611 38630 50795 39357 97748

628 733 903 847 295

0.505 0.506 0.507 0.508 0.509

0.48380 0.48468 0.48555 0.48643 0.48730

74403 23727 68204 07826 42584

23960 48823 91355 77106 32116

15529 94818 38243 78840 05316

617 170 967 928 931

0.87517 0.87469 0.87420 0.87371 0.87323

44744 02294 51098 91160 22484

26201 79311 42264 00180 39053

33418 19588 46912 75052 84166

203 355 391 318 561

0.510 0.511 0.512 0.513 0.514

0.48817 0.48904 0.48992 0.49079 0.49166

72468 97471 17583 32796 43101

82907 56492 80371 82532 91455

49450 73435 57187 85582 35667

013 934 006 104 778

0.87274 0.87225 0.87176 0.87127 0.87078

45076 58941 64083 60507 48219

45751 08013 14454 54560 18687

26310 76750 85187 26898 53787

581 129 176 565 441

0.515 0.516 0.517 0.518 0.519

0.49253 0.49340 0.49427 0.49514 0.49601

48490 48953 44482 35068 20703

36108 45953 50944 81528 68647

63810 92799 98899 98859 36861

364 025 617 309 855

0.87029 0.86979 0.86930 0.86881 0.86831

27222 97523 59126 12036 56258

98065 84793 71841 53049 23126

45347 59540 83584 84660 60524

504 132 429 240 189

0.520 0.521 0.522 0.523 0.524

0.49688 0.49774 0.49861 0.49948 0.50034

01378 77084 47812 13555 74302

43736 38729 86055 18641 69914

71433 62299 57189 78596 10484

446 043 109 658 518

0.86781 0.86732 0.86682 0.86632 0.86582

91796 18657 36844 46363 47218

77649 13065 26688 16698 82144

90038 83614 33565 64378 82893

785 647 898 779 524

0.525 0.526 0.527 0.528 0.529

0.50121 0.50207 0.50294 0.50380 0.50467

30046 80778 26489 67171 02815

73797 64718 77603 47881 11483

84942 68796 50161 24954 83349

748 092 411 981 596

0.86532 0.86482 0.86431 0.86381 0.86331

39416 22960 97856 64109 21723

22941 39868 34571 09560 68210

28399 22644 19753 56071 99902

561 077 996 436 671

0.530 0.531 0.532 0.533 0.534

0.50553 0.50639 0.50725 0.50811 0.50898

33412 58953 79431 94836 05160

04846 64911 29121 35431 22300

96181 01306 89905 92741 66364

366 143 473 999 220

0.86280 0.86230 0.86179 0.86128 0.86077

70705 11058 42788 65901 80400

14761 54312 92829 37140 94932

01180 41041 81312 13920 10201

670 248 894 311 726

0.535 0.536 0.537 0.538 0.539

0.50984 0.51070 0.51156 0.51241 0.51327

10394 10529 05558 95471 80260

28695 94093 58481 62356 46726

79260 97962 73096 25387 31605

534 456 946 754 686

0.86026 0.85975 0.85924 0.85873 0.85822

86292 83581 72273 52372 23884

74755 86021 39001 44824 15482

70140 71507 18926 92837 98393

025 760 068 581 339

0.540 0.541 0.542 0.543 0.544

0.51413 0.51499 0.51585 0.51670 0.51756

59916 34431 03796 68002 27041

53113 23551 00588 27290 47234

10467 08484 85758 01726 00855

728 914 874 969 920

0.85770 0.85719 0.85667 0.85616 0.85564

86813 41166 86946 24160 52812

63824 03555 49241 16304 21022

14253 41303 51282 35326 52425

797 947 623 032 567

0.545 0.546 0.547 0.548 0.549

0.51841 0.51927 0.52012 0.52098 0.52183

80905 29584 73071 11356 44432

04516 43752 10073 49129 07094

98283 65410 15436 88849 38858

861 714 812 675 868

0.85512 0.85460 0.85408 0.85356 0.85304

72907 84452 87450 81907 67829

80530 12819 36734 71975 39096

77799 51181 25018 12587 36027

957 787 472 703 442

0.550

0.52268 72289 30659 16778 838

0.85252

45220 59505 74280 498

ELEMENTARY TRANSCENDENTAL FUNCTIONS

153

CIRCULAR SIXES AND COSINES FOR RADIAN ARGUMENTS cos x

sin x

X

Table 4.6

0.550 0.551 0.552 0.553 0.554

0.52268 0.52353 0.52439 0.52524 0.52609

72289 94919 12314 24465 31364

30659 67038 63969 69712 33053

16778 57359 64065 94301 44585

838 653 565 297 976

0.85252 0.85200 0.85147 0.85095 0.85042

45220 14086 74432 26263 69585

59505 55464 50084 67333 32026

74280 10953 82092 23867 20180

498 761 114 110 431

0.555 0.556 0.557 0.558 0.559

0.52694 0.52779 0.52864 0.52949 0.53033

33002 29370 20460 06264 86773

03301 30292 64391 56488 58002

35674 97627 54824 10933 33815

635 180 757 415 002

0.84990 0.84937 0.84884 0.84831 0.84778

04402 30721 48545 57881 58734

69831 07267 71701 91352 95285

50182 35704 88608 58049 77652

218 287 318 047 517

0.560 0.561 0.562 0.563 0.564

0.53118 0.53203 0.53287 0.53372 0.53457

61979 31872 96446 55691 09598

20883 97610 41195 05179 43639

40385 81418 26300 47726 06347

187 533 543 585 607

0.84725 0.84672 0.84619 0.84565 0.84512

51110 35012 10448 77421 35938

13416 76506 16165 64850 55863

12609 06683 29136 21564 44654

452 799 481 438 991

0.565 0.566 0.567 0.568 0.569

0.53541 0.53626 0.53710 0.53794 0.53878

58160 01367 39212 71686 98780

11183 62956 54637 42441 83121

35362 25057 07291 39926 91211

572 521 168 969 553

0.84458 0.84405 0.84351 0.84297 0.84244

86004 27624 60803 85547 01861

23353 02313 28580 38838 70611

24855 00958 70603 36691 53715

579 945 796 011 445

0.570 0.571 0.572 0.573 0.574

0.53963 0.54047 0.54131 0.54215 0.54299

20487 36797 47702 53195 53266

33969 52812 98021 28505 03714

24099 80524 65614 31859 63213

446 005 465 028 905

0.84190 0.84136 0.84082 0.84027 0.83973

09751 09222 00279 82928 57175

62268 53020 82920 92863 24582

74013 93925 99876 14368 41893

376 658 632 839 605

0.575 0.576 0.577 0.578 0.579

0.54383 0.54467 0.54551 0.54634 0.54718

47906 37109 20865 99165 72002

83642 28825 00342 59818 69423

59158 18694 24296 25797 24232

222 718 136 231 321

0.83919 0.83864 0.83810 0.83755 0.83701

23024 80481 29551 70241 02555

20654 24493 80354 33330 29351

14757 38825 39176 05683 38499

543 019 658 918 807

0.580 0.581 0.582 0.583 0.584

0.54802 0.54886 0.54969 0.55053 0.55136

39367 01252 57649 08548 53942

91873 90432 28912 71672 83624

55618 74682 38538 90300 42652

270 851 382 563 424

0.83646 0.83591 0.83536 0.83481 0.83426

26499 42078 49298 48164 38683

15186 38442 47559 91816 21326

93465 27434 43511 36205 36508

789 927 337 988 907

0.585 0.586 0.587 0.588 0.589

0.55219 0.55303 0.55386 0.55469 0.55552

93823 28181 57009 80299 98041

30227 71494 91989 40829 91685

61353 48692 26889 21434 44380

309 799 504 637 278

0.83371 0.83315 0.83260 0.83205 0.83149

20858 94697 60204 17385 66245

87037 40732 35026 23370 60044

56877 36143 84331 27399 51895

861 543 337 720 332

0.590 0.591 0.592 0.593 0.594

0.55636 0.55719 0.55802 0.55885 0.55968

10229 16852 17904 13375 03258

12783 72905 41388 88127 83575

77572 55827 50056 50327 48880

254 556 192 409 201

0.83094 0.83038 0.82982 0.82926 0.82870

06791 39026 62959 78593 85934

00163 99672 15348 04797 26455

49524 61643 23660 09361 75147

800 346 255 243 786

0.595 0.596 0.597 0.598 0.599

0.56050 0.56133 0.56216 0.56299 0.56381

87544 66226 39293 06739 68555

98744 05205 75090 81092 96468

23078 18307 30821 90525 43709

004 516 541 792 545

0.82814 0.82758 0.82702 0.82646 0.82589

84988 75761 58257 32484 98446

39590 04294 81491 32932 21193

04193 50517 82974 29164 19254

468 407 799 660 799

0.600

0.56464 24733 95035 35720 095

0.82533 56149 09678 29724 095

I

154

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS

sin x

X

cos 2

0.400 0.601 0.602 0.603 0.604

0.56464 0.56546 0.56629 0.56711 0.56793

24733 75265 20142 59356 92899

95035 51175 39837 36531 17336

35720 93580 08553 18642 91043

095 897 336 028 574

0.82533 0.82477 0.82420 0.82363 0.82307

56149 05598 46800 79760 04483

09678 62617 45065 22901 62830

29724 27022 11146 59135 68484

095 123 193 858 934

0.605 0.606 0.607 0.608 0.609

0.56876 0.56958 0.57040 0.57122 0.57204

20762 42938 59418 70194 75257

58900 38434 33722 23115 85537

04538 31827 21808 81800 59705

687 607 719 299 300

0.82250 0.82193 0.82136 0.82079 0.82022

20976 29243 29292 21127 04753

32380 99900 34564 06368 86127

00471 23403 55786 09403 32317

116 216 102 380 893

0.610 0.611 0.612 0.613 0.614

0.57286 0.57368 0.57450 0.57532 0.57614

74601 68215 56093 38225 14604

00481 48012 08770 63966 95387

26119 56380 12563 25415 76236

098 111 221 904 989

0.81964 0.81907 0.81850 0.81792 0.81734

80178 47406 06443 57296 99969

45479 56882 93612 29766 40259

51790 17114 42372 49108 08915

075 225 770 549 198

0.615 0.616 0.617 0.618 0.619

0.57695 0.57777 0.57859 0.57940 0.58022

85222 50071 09141 62426 09916

85396 16931 73507 39217 98732

78697 60606 45614 34861 88572

975 809 047 330 073

0.81677 0.81619 0.81561 0.81503 0.81445

34469 60800 78970 88984 90847

00822 88007 79180 52524 87037

85945 79339 65565 40689 62551

685 051 411 288 318

0.620 0.621 0.622 0.623 0.624

0.58103 0.58184 0.58266 0.58347 0.58428

51605 87483 17542 41775 60174

37305 40765 95525 88579 07505

07584 14825 36729 84595 35888

296 522 641 681 387

0.81387 0.81329 0.81271 0.81213 0.81154

84566 70146 47593 16913 78111

62533 59641 59801 45270 99116

92868 39252 97147 91684 19458

400 335 027 290 331

0.625 0.626 0.627 0.628 0.629

0.58509 0.58590 0.58671 0.58752 0.58833

72729 79433 80279 75257 64359

40462 76194 04032 13891 96274

15480 76836 83139 88356 18246

540 923 861 252 006

0.81096 0.81037 0.80979 0.80920 0.80861

31195 76168 13038 41809 62489

05217 48267 13768 88032 58182

90218 68483 15067 28536 86569

953 556 973 214 178

0.630 0.631 0.632 0.633 0.634

0.58914 0.58995 0.59075 0.59156 0.59237

47579 24907 96335 61856 21462

42269 43555 92400 81661 04785

51311 99690 89983 44033 59635

811 151 484 509 440

0.80802 0.80743 0.80684 0.80625 0.80566

75083 79596 76035 64405 44713

12151 38679 27315 68414 53140

87252 90282 58094 96904 97676

371 722 522 569 566

0.635 0.636 0.637 0.638 0.639

0.59317 0.59398 0.59478 0.59559 0.59639

75143 22893 64703 00565 30471

55812 29375 20697 25599 40494

91193 30315 86352 66873 58084

198 454 425 364 641

0.80507 0.80447 0.80388 0.80328 0.80269

16964 81165 37320 85437 25521

73462 22155 92798 79775 78276

77004 17917 10598 93030 91556

837 411 548 752 338

0.640 0.641 0.642 0.643 0.644

0.59719 0.59799 0.59879 0.59959 0.60039

54413 72383 84374 90376 90382

62392 88897 18215 49145 81087

05188 92681 24594 04673 16496

355 375 757 426 070

0.80209 0.80149 0.80089 0.80030 0.79970

57578 81614 97636 05648 05657

84292 94617 06847 19380 31415

61358 26862 22056 30729 26635

611 715 216 469 842

0.645 0.646 0.647 0.648 0.649

0.60119 0.60199 0.60279 0.60359 0.60439

84385 72375 54345 30288 00194

14041 48606 85984 27978 76993

03535 49156 56561 28662 47908

151 949 576 868 070

0.79909 0.79849 0.79789 0.79729 0.79668

97669 81690 57726 25783 85868

42951 54786 68519 86546 12061

13571 65377 65855 48612 36819

848 243 159 327 444

0.650

0.60518 64057 36039 56037 252

0.79608 37985 49055 82891 760

155

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SJNES .4ND COSINES FOR RADIAN ARGUMENTS sin x

X

cos

Table 4.6

1:

0.650 0.651 0.652 0.653 0.654

0.60518 0.60598 0.60677 0.60757 0.60836

64057 21868 73618 19302 58909

36039 08730 99284 12527 53891

56037 33782 80505 93778 48897

252 358 818 646 929

0.79608 0.79547 Oi79487 0.79426 0.79365

37985 82142 18343 46596 66907

49055 02318 77432 80778 19531

82891 08089 42041 62180 33114

760 927 183 929 757

0.655 0.656 0.657 0.658 0.659

0.60915 0.60995 0.61074 0.61153 0.61232

92433 19865 41198 56423 65533

29414 45745 10140 30466 15201

78343 51174 52364 62074 34867

652 755 359 073 307

0.79304 0.79243 0.79182 0.79121 0.79060

79281 83724 80243 68843 49532

01659 35925 31885 99886 51069

45900 57253 28666 65458 55734

987 785 909 154 550

0.660 0.661 0.662 0.663 0.664

0.61311 0.61390 0.61469 0.61548 0.61627

68519 65375 56091 40660 19075

73433 14865 49810 89197 44570

78861 34819 55178 83019 30974

515 272 137 186 165

0.78999 0.78937 0.78876 0.78814 0.78753

22314 87197 44186 93287 34506

97365 51494 26970 38093 99953

09278 96354 86436 86857 81380

382 080 061 558 523

0.665 0.666 0.667 0.668 0.669

0.61705 0.61784 0.61863 0.61941 0.62020

91327 57408 17311 71027 18550

28086 52521 31267 78333 08348

60071 58518 20428 24475 12498

171 785 576 901 919

0.78691 0.78629 0.78568 0.78506 0.78444

67851 93326 10938 20693 22598

28428 40184 52672 a4132 53587

68686 00789 21368 04022 90446

643 551 279 017 244

0.670 0.671 0.672 0.673 0.674

0.62098 0.62176 0.62255 0.62333 Oi62411

59870 94980 23873 46540 62974

36559 78835 51665 72160 58052

68035 94799 95092 48154 88456

744 654 281 700 349

0.78382 0.78320 0.78257 0.78195 0.78133

16658 02880 81270 51835 14579

80849 86510 91948 19324 91581

28530 10376 10240 22393 98907

294 414 374 698 578

0.675 0.676 0.677 0.678 0.679

0.62489 0.62567 OI62645 0.62723 0.62801

73167 77111 74797 66220 51370

27699 00082 94805 32101 32827

83921 14094 48239 23383 22288

682 496 849 477 658

0.78070 0.78008 0.77945 0.77882 0.77820

69511 16635 55959 87488 11228

32446 66425 18805 15655 83820

87358 68455 93590 22308 59699

526 830 877 414 786

0.680 0.681 0.682 0.683 0.684

0.62879 0.62957 0.63034 0.63112 OI63189

30240 02822 69108 29091 82762

18468 11138 33578 09159 61884

51370 18547 11028 73043 83499

418 018 644 207 197

0.77757 0.77694 0.77631 0.77568 0.77505

27187 35370 35783 28434 13327

50927 45381 96362 33829 88518

93718 32416 41105 79438 38411

239 339 566 156 247

0.685 0.686 0.687 0.688 0.689

0.63267 0.63344 0.63422 0.63499 0.63576

30115 71140 05832 34181 56180

16386 97929 32410 46361 66947

33585 04308 43963 45549 24110

498 084 542 306 566

0.77441 0.77378 0,77315 0.77251 0.77188

90470 59869 21530 75460 21664

91938 76376 74891 21318 50263

77293 60473 94232 63436 68154

390 500 293 286 418

0.690 0.691 0.692 0.693 0.694

0.63653 0.63730 0.63807 0.63884 0.63961

71822 81098 84001 80523 70657

21967 39859 49694 81182 64670

94023 46216 25323 06781 73855

743 467 984 899 200

0.77124 0.77060 0.76997 OI76933 0.76869

60149 90922 13989 29357 37030

97106 97998 89862 10392 98049

60197 79579 90904 19670 88505

354 541 069 418 132

0.695 0.696 0.697 0.698 0.699

0.64038 0.64115 0.64192 0.64268 0.64345

54395 31729 02651 67154 25230

31146 12236 40207 47966 69063

94603 98782 54600 45892 48031

464 185 136 698 063

0.76805 Oi76741 0.76677 OI76612 0.76548

37017 29324 13956 90921 60224

92068 32449 59961 16142 43294

53315 39366 77279 38958 73431

502 321 757 434 759

0.700

0.64421 76872 37691 05367 261

0.76484 21872 84488 42625 586

156

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SIXES AND COSINES FOR RADIAN ARGUMENTS sin x

X

cos x

0.700 0.701 0.702 0.703 0.704

0.64421 0.64498 0.64574 0.64650 0.64727

76872 22071 60821 93113 18940

37691 88685 57525 80337 93892

05367 07414 65445 88940 61979

261 902 583 870 783

0.76484 0.76419 0.76355 0.76290 0.76225

21872 75872 22230 60953 92046

84488 83558 85105 34492 77847

42625 57055 11442 20253 53166

586 252 075 368 023

0.705 0.706 OI707 0.708 0.709

0.64803 0.64879 OI64955 0.65031 0.65107

38295 51169 57555 57446 50833

35607 43546 56422 13597 55081

19561 23864 40438 14335 46169

705 641 747 062 354

0.76161 0.76096 0.76031 0.75966 0.75901

15517 31372 39617 40259 33304

62061 34787 44439 40193 71984

70453 58298 64022 31253 34997

752 030 815 107 406

0.710 0.711 0.712 0.713 0.714

0.65183 0.65259 0.65334 0.65410 0.65486

37710 18068 91900 59199 19957

21536 54275 95261 87111 73096

68121 19866 24450 64083 55888

013 915 173 709 565

0.75836 0.75770 0.75705 0.75640 0.75574

18759 96631 66925 29649 84809

90508 47219 94330 84811 72391

16654 18942 20755 71940 28003

146 159 235 852 128

0.715 0.716 0.717 0.718 0.719

0.65561 0.65637 0.65712 0.65787 0.65863

74166 21820 62909 97427 25366

97140 03821 38376 46694 75324

27566 93009 27837 44880 69585

883 463 851 853 417

0.75509 0.75443 0.75378 0.75312 0.75246

32412 72463 04970 29939 47378

11552 57536 66335 94701 00135

84730 12745 91983 46092 76755

074 203 563 263 558

0. 720

0.721 0.722 0: 723 0.724

0.65938 0.66013 0.66088 0.66163 0.66238

46719 61478 69636 71185 66117

71473 83004 58443 46973 98439

15361 58862 15198 13079 69907

800 952 027 967 065

0.75180 0.75114 0.75048 0.74982 0.74916

57291 59686 54570 41949 21830

40894 75987 65174 68966 48626

97944 70091 34189 45814 09078

549 576 363 983 707

0.725 0.726 0.727 0.728 0.729

0.66313 OI66388 0.66463 0.66537 0.66612

54426 36103 11142 79534 41272

63349 92872 38839 53748 90759

66778 23443 73184 37633 01524

441 354 280 666 309

0.74849 0.74783 0.74717 0.74650 0.74584

94219 59123 16549 66503 08992

66165 84344 66673 77410 81559

10497 52795 88624 54215 02955

806 369 209 910 103

0.730 0.731 0.732 0.733 0.734

0.66686 0.66761 0.66835 0.66910 0.66984

96350 44758 86490 21539 49897

03697 47057 75996 46342 14589

87373 30099 51573 35102 99849

259 195 181 739 159

0.74517 0.74450 0,74383 0.74317 0.74250

44023 71602 91736 04431 09695

44870 33841 15714 58475 30855

38879 50102 42167 71321 77713

013 364 693 153 862

0.735 0.736 0.737 0.738 0.739

0.67058 0.67132 0.67206 0.67280 0.67354

71556 86509 94749 96269 91060

37903 74117 81736 19936 48565

75177 74942 71700 70863 84779

973 523 537 650 796

0.74183 0.74115 0.74048 0.73981 0.73914

07534 97954 80963 56567 24772

02328 43109 24156 17168 94586

18528 01033 15559 68402 14660

866 791 237 998 158

0.740 0.741 0.742 0.743 0.744

0.67428 0.67502 0.67576 0.67650 0.67723

79116 60429 34991 02796 63836

28145 19868 85605 87900 89971

06748 84968 96417 20669 13633

388 216 996 485 096

0.73846 0.73779 0.73711 0.73644 0.73576

85587 39016 85068 23749 55065

29587 96092 68756 22975 34881

90979 48243 84181 75897 12335

142 787 492 532 582

0.745 0.746 0.747 0.748 0.749

0.67797 0.67870 0.67944 0.68017 0.68090

18104 65592 06293 40199 67304

55714 49704 37191 84105 57056

81235 53032 55745 86745 87450

936 193 803 313 880

0.73508 0.73440 0.73373 0.73305 0.73237

79023 95631 04894 06821 01416

81341 39960 89077 07766 75833

26664 28591 36602 10125 81627

537 681 285 695 975

0.750

0.68163 87600 23334 16673 324

0.73168 88688 73820 88631 184

ELEMENTARY TRANSCENDENTAL FUNCTIONS

157

CIRCULAR SINES AND COSINES FOR RADJ.44 ARGUMENTS sin x

X

cos

Table 4.6

x

0.750 0.751 0.752 0.753 0.754

0.68163 0.68237 0.68310 0.68383 0.68456

87600 01079 07735 07559 00545

23334 50908 08431 65237 91345

16673 23885 22423 62625 04892

324 163 554 080 285

0.73168 0.73100 0.73032 0,72964 0.72895

88688 68643 41288 06630 64676

73820 83000 85375 63683 01388

88631 05659 76111 44059 85978

184 342 160 608 367

0.755 0: 756 0.757 0.758 0.759

0.68528 0: 68601 0.68674 0.68747 0.68819

86686 65974 38401 03962 62647

57454 34953 95911 13086 59922

92691 25484 31587 40963 57950

917 772 089 419 885

0.72827 0.72758 0.72689 0.72621 0.72552

15431 58904 95102 24030 45696

82687 92503 16489 41026 53220

42395 49472 70515 27404 31961

268 750 436 867 494

0.760 0; 7 6 1 0.762 0.763 0.764

0.68892 0: 68964 0.69036 0.69109 0.69181

14451 59365 97383 28497 52700

10551 39792 23154 36835 57724

33914 39835 38826 58582 63761

776 383 030 200 700

0.72483 0.72414 0.72345 0.72276 0.72207

60107 67269 67190 59877 45336

40905 92639 97707 46116 28598

17233 68715 55489 61298 15545

969 814 548 318 123

0.765 0.766 0.767 0.768 0.769

0.69253 0.69325 0.69397 0.69469 0.69541

69985 80345 83772 80259 69800

63401 32137 42896 75335 09807

28295 07631 10903 73038 26789

794 223 039 195 802

0.72138 0.72068 0.71999 0.71930 0.71860

23574 94598 58415 15033 64457

36606 62317 98627 39157 78243

24219 00753 96800 32949 29362

693 084 072 410 010

0.770 0.771 0.772 0.773 0.774

0.69613 0.69685 0.69756 0.69828 0.69900

52386 28011 96667 58347 13045

27356 09725 39351 99368 73609

74701 61005 43442 65024 25718

988 296 524 972 983

0.71791 0.71721 0.71651 0.71581 0.71512

06696 41755 69642 90364 03928

10943 33033 41008 32078 04171

36337 64806 16757 15581 36356

129 626 355 770 807

0.775 0.776 0.777 0.778 0.779

0.69971 0.70043 0.70114 0.70185 0.70256

60753 01464 35170 61865 81541

46603 03580 30469 13900 41203

54062 78713 99923 60948 19385

747 256 379 949 818

0.71442 0.71372 0.71302 0.71231 0.71161

10340 09608 01739 86740 64618

55931 86716 96600 86370 57525

36051 83660 90273 39059 15198

117 709 093 972 564

0.780 0.781 0.782 0.783 0.784

0.70327 0.70398 0.70469 0.70540 0.70611

94192 99809 98387 89918 74395

00410 80256 70180 60324 41535

18436 58108 66337 70046 66131

790 374 280 581 480

0.71091 0.71020 0.70950 0.70880 0.70809

35380 99032 55582 05038 47405

12277 53550 84980 10910 36395

35721 79296 15931 36614 82877

626 239 435 737 671

0.785 0.786 0.787 0.788 0.789

0.70682 0.70753 0.70823 0.70894 0.70964

51811 22158 85430 41620 90720

05365 44073 50625 18692 42656

92374 98290 15901 30436 50970

614 801 193 730 857

0.70738 0.70668 0.70597 0.70526 0.70455

82691 10904 32049 46135 53168

67199 09793 71355 59771 83632

76290 47885 67509 73107 99934

330 059 330 880 173

0.790 0.791 0.792 0.793 0.794

0.71035 0.71105 0.71175 0.71246 0.71316

32724 67624 95414 16086 29633

17607 39345 04380 09933 53937

80981 88841 78239 58529 15005

403 574 979 620 776

0.70384 0.70313 0.70242 0.70171 0.70099

53156 46105 32023 10917 82793

52236 75582 64376 30026 84643

09691 19602 31409 60306 63792

278 208 812 275 314

0.795 0.796 0.797 0.798 0.799

0.71386 0.71456 0.71526 0.71596 0.71665

36049 35326 27458 12436 90256

35036 52590 06672 98066 28277

79112 98579 07482 96241 81536

713 148 391 109 630

0.70028 0.69957 0.69885 0.69814 0.69742

47660 05524 56392 00271 37169

41039 12728 13922 59535 65179

70466 08742 35499 64661 95703

123 151 779 971 964

0.800

0.71735 60908 99522 76162 718

0.69670 67093 47165 42092 075

I

158

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS cos 2

sin x

X

0.800 0.801 0.802 0.803 OI804

0.71735 0.71805 OI71874 0.71944 Oi72013

60908 24388 80686 29797 71714

99522 14736 77571 92397 64303

76162 58803 43741 50488 73354

718 753 255 651 263

0.69670 0.69598 0.69527 0.69455 0.69383

67093 90050 06047 15091 17189

47165 22499 08886 24727 89116

42092 59652 74871 13123 26831

075 695 538 218 236

0.805 0.806 0.807 0.808 0.809

0.72083 0.72152 0.72221 0.72290 0.72359

06429 33937 54228 67298 73139

99098 03310 84188 49704 08550

50932 35522 62476 19472 15721

396 503 322 935 677

0.69311 0.69239 0.69166 0.69094 0.69022

12350 00579 81884 56273 23752

21844 43394 74945 38365 56214

23558 94027 40074 02528 89026

425 956 951 784 151

0.810 0,811 0.812 0.813 0.814

0.72428 0.72497 0.72566 0.72635 0.72703

71743 63105 47217 24072 93664

70142 44620 42849 76416 57636

51092 85175 06266 00277 19583

818 959 069 085 027

0.68949 0.68877 0.68804 0.68732 0.68659

84329 38011 84805 24719 57759

51747 48903 72316 47306 99881

01754 65129 53394 18165 15892

964 158 472 280 545

0.815 0.816 0.817 0.818 0.819

0.72772 0.72841 0.72909 0.72977 0.73046

55985 11030 58790 99259 32430

99550 15926 21260 30776 60427

51786 8841,4 93542 72343 39565

534 775 651 223 302

0.68586 0.68514 0.68441 0.68368 0.68295

83934 03250 15714 21335 20118

56737 45257 93509 30246 84907

35262 24529 18772 67094 59742

969 414 652 544 692

0.820 0.821 0.822 0.823 0.824

0.73114 0.73182 0.73250 0.73318 0.73386

58297 76852 88089 92001 88581

26895 47595 40670 24998 20187

87938 56503 98872 51414 01366

131 084 320 329 283

0.68222 0.68148 0.68075 0.68002 0.67929

12072 97204 75521 47030 11740

87613 69169 61060 95457 05207

55166 07005 91008 31885 30088

656 802 857 232 213

0.825 0.826 0.827 0.828 0.829

0.73454 0.73522 0.73590 0.73658 0.73725

77822 59718 34261 01446 61264

46578 25249 78008 27404 96715

54873 04953 99391 08557 93150

150 477 793 557 579

0.67855 0.67782 0.67708 0.67635 0.67561

69656 20786 65139 02720 33538

23839 85563 25264 78508 81536

88530 39229 69888 50409 59331

058 106 949 750 781

0.830 0.831 0.832 0.833 0.834

0.73793 0.73860 0.73927 0.73995 0.74062

13711 58777 96458 26746 49635

09962 91899 68020 64557 08481

71872 89026 82039 48913 15603

858 752 434 544 989

0.67487 0.67413 0.67339 0.67265 0.67191

57600 74913 85485 89323 86434

71267 85293 61885 39984 59207

10211 77928 24928 27394 01352

246 481 580 537 983

0.835 0.836 0.837 0.838 0.839

0.74129 0.74196 0.74263 0.74330 0.74397

65117 73186 73836 67059 52849

27503 50074 05390 23383 34732

03320 95758 06248 44844 85324

808 049 576 755 932

0.67117 0.67043 0.66969 0.66895 0.66820

76826 60506 37482 07761 71351

59842 82850 69864 63185 05786

28712 E3235 56439 83438 68708

570 098 445 385 357

0.840 0.841 0.842 OI843 0.844

0.74464 0.74531 0.74597 OI74664 0.74730

31199 02103 65554 21545 70070

70859 63927 46848 53275 17609

32125 87199 16798 18184 86260

657 577 923 539 385

0.66746 0.66671 0.66597 0.66522 0.66447

28258 78491 22056 58962 89216

41308 14059 69016 51824 08791

11792 32935 98654 47240 14192

267 396 482 065 152

0.845 OI846 0.847 0.848 0.849

0.74797 OI74863 0.74929 0.74995 0.75062

11121 44693 70779 89371 00464

74999 61339 13273 68190 64233

80133 89598 01550 66317 63922

429 886 724 368 547

0.66373 0.66298 0.66223 0.66148 0.66073

12824 29796 40137 43857 40961

86891 33764 97713 27703 73363

57589 83391 70674 96802 62530

286

0.850

0.75128 04051 40292 70271 207

r(-mi L 7 1

100 409 946 783

0.65998 31458 84982 17039 542

159

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS sin x

X

cos

Table 4.6

x

0.850 0.851 0.852 0.853 0.854

0.75128 0.75194 0.75259 0.75325 0.75391

04051 00125 88679 69708 43204

40292 36009 91775 48737 48790

70271 23260 88815 26849 57151

207 432 295 594 380

0.65998 0.65923 0.65847 0.65772 0.65697

31458 15356 92661 63381 27524

84982 13509 10556 28392 19944

17039 82909 81024 55410 98010

542 449 321 547 152

0.855 0.856 0.857 0.858 0.859

0.75457 0.75522 0.75588 0.75653 0.75718

09161 67572 18431 61731 97466

34586 49528 37776 44244 14602

25193 67867 79144 75659 62217

237 227 450 143 260

0.65621 0.65546 0.65470 0.65395 0.65319

85097 36108 80564 18474 49843

38799 39199 76042 04884 81933

73388 43373 91635 48193 13861

013 300 218 134 148

0.860 0.861 0.862 0.863 0.864

0.75784 0.75849 0.75914 0.75979 0.76044

25628 46213 59212 64620 62430

95276 33451 77068 74826 76186

97229 58068 06350 53141 24087

459 441 566 684 122

0.65243 0.65167 0.65092 0.65016 0.64940

74681 92995 04791 10079 08865

64051 08756 74216 19251 03332

84627 75966 47091 25131 29254

203 794 357 418 574

0.865 0.866 0.867 0.868 0.869

0.76109 0.76174 0.76239 0.76303 0.76368

52636 35230 10208 77561 37284

31366 91346 07866 33429 21299

24465 04168 22598 13500 49706

750 073 272 144 858

0.64864 0.64787 0.64711 0.64635 0.64559

01156 86962 66288 39144 05536

86580 29767 94312 42282 36391

94718 96858 75010 56369 79782

373 196 176 276 561

0.870 0.871 0.872 0.873 0.874

0.76432 0.76497 0.76561 0.76625 0.76690

89370 33813 70606 99742 21217

25505 00837 02852 87869 12977

07814 32779 02438 91953 38182

480 191 134 834 114

0.64482 0.64406 0.64329 0.64253 0.64176

65472 18960 66007 06621 40810

40001 17117 32390 51117 39234

19477 08727 63447 05733 87329

766 234 280 091 202

0.875 0.876 0.877 0.878 0.879

0.76754 0.76818 0.76882 0.76946 0.77010

35022 41152 39600 30359 13424

36027 15638 11198 82862 91555

03963 42337 60682 84773 22769

458 736 252 027 271

0.64099 0.64022 0.63946 0.63869 0.63792

68581 89942 04901 13466 15643

63325 90610 88955 26862 73477

13035 64049 21244 88380 15258

656 903 528 872 639

0.880 0.881 0.882 0.883 0.884

0.77073 0.77137 0.77201 0.77264 0.77328

88788 56445 16388 68611 13107

98969 67568 60587 42032 76680

29120 68399 79051 37074 19618

965 506 337 497 049

0.63715 0.63638 0.63560 0.63483 0.63406

11441 00868 83931 60638 30997

98580 72592 66570 52208 01835

20801 16079 27264 18529 14874

550 131 710 695 218

0.885 0.886 0.887 0.888 0.889

0,77391 0.77454 0.77518 ol7758i 0.77644

49871 78895 00174 13701 19470

30081 68560 59214 69915 69310

68504 53673 36552 33343 78237

290 706 600 321 045

0.63328 0.63251 0.63174 0.63096 0.63018

95014 52699 04059 49102 87834

88415 85546 67460 09021 85724

24894 63485 74481 53235 69127

213 020 571 256 530

0.890 0.891 0.892 0; 893 0.894

0.77707 Oi77770 0.77832 0.77895 0.77958

17475 07709 90165 64839 31723

26823 12654 97778 53950 53704

86549 17776 38577 85676 28683

033 316 722 211 432

0.62941 0.62863 0.62785 0.62707 0.62629

20265 46402 66252 79824 87125

73696 49694 91105 75942 82849

88020 94643 14919 38222 39581

355 540 057 428 242

0.895 0; 896 0.897 0.898 0.899

0.78020 0.78083 0.78145 0.78208 0.78270

90811 42097 85575 21238 49080

70350 77980 51465 66458 99392

32846 21716 39740 14771 20508

443 548 163 667 171

0.62551 0.62473 0.62395 0.62317 0.62239

88163 82946 71482 53778 29842

91096 80578 31818 25961 44179

01810 37587 11458 61790 22654

880 545 656 683 524

0.900

0.78332 69096 27483 38846 138

0.62160 99682 70664 45648 472 L

'

J

,

160

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SINES AKD COSINES FOR RADIAN ARGUMENTS sin x

X

cos

x

0.900 0.901 0.902 0.903 0.904

0.78332 0.78394 0.78456 0.78518 0.78580

69096 81278 85620 82117 70762

27483 28730 81914 66602 63143

38846 22159 55501 18722 48518

138 796 279 439 260

0.62160 0.62082 0.62004 0.61925 0.61847

99682 63306 20722 71938 16961

70664 86633 76323 23992 14519

45648 21658 02558 22842 21204

472 870 530 983 658

0.905 0.906 0.907 0.908 0.909

0.78642 0.78704 0.78765 0.78827 0.78888

51549 24472 89524 46700 95992

52674 17115 39174 02347 90915

00391 10540 57664 24696 60447

817 713 940 094 888

0.61768 0.61689 0.61611 0.61532 0.61453

55799 88460 14953 35284 49462

33401 66755 01314 24430 24068

62045 56924 85952 19111 37523

040 921 792 466 020

0.910 0.911 0.912 0.913 0.914

0.78950 0.79011 0.79072 0.79134 0.79195

37396 70905 96513 14214 24001

89950 85311 63647 12398 19793

41187 32130 48850 18619 41660

896 474 789 897 812

0.61374 0.61295 0.61216 0.61137 0.61058

57494 59390 55155 44799 28329

88811 07856 71013 68705 91968

54652 37447 27423 61677 93848

118 803 839 674 110

0.915 0.916 0.917 0.918 0.919

0.79256 0.79317 0.79378 0.79438 0.79499

25868 19810 05820 83893 54021

74854 67394 88020 28129 79915

52325 80192 11086 48016 72036

499 738 785 785 860

0.60979 0.60899 0.60820 0.60741 0.60661

05754 77080 42317 01471 54552

32450 82406 34706 82824 20845

15011 74518 00764 21909 86522

758 350 999 476 589

0.920 0.921 0.922 0.923 0.924

0.79560 0.79620 0.79681 0.79741 0.79801

16200 70422 16683 54975 85293

36366 91262 39183 75501 96389

03026 60393 23692 93169 50226

828 471 319 858 129

0.60582 0.60502 0.60422 0.60343 0.60263

01566 42522 77428 06291 29120

43462 45973 24282 74899 94936

84179 65991 65074 16960 79945

741 745 984 980 468

0.925 0.926 0.927 0.928 0.929

0.79862 0.79922 0.79982 0.80042 0.80102

07631 21983 28343 26704 17061

98814 80542 40138 76967 91191

17797 20660 45653 01823 80485

639 537 978 638 294

0.60183 0.60103 0.60023 0.59943 0.59863

45923 56708 61482 60254 53031

82112 34746 51758 32673 77612

55377 043 07885 466 85549 703 40005 791 46494 584

0.930 0.931 0.932 0.933 0.934

0.80161 0.80221 0.80281 0.80340 0.80400

99408 73739 40048 98328 48574

83777 56488 11892 53358 85059

15208 41719 57726 82661 17341

432 806 899 218 371

0.59783 0.59703 0.59622 0.59542 0.59462

39822 20635 95478 64358 27284

87298 63051 06791 21032 08887

23849 54424 03960 41397 58618

491 260 905 846 345

0.935 0.936 0.937 0.938 0.939

0.80459 0.80519 0.80578 0.80637 0.80696

90781 24941 51049 69100 79086

11969 39867 75339 25773 99364

03555 83565 59525 52827 63359

863 545 671 488 313

0.59381 0.59301 0.59220 0.59140 0.59059

84263 35305 80416 19605 52881

74063 20863 54181 79507 02922

90139 32740 65034 66977 39319

324 634 867 785 443

0.940 0.941 0.942 0.943 0.944

0.80755 0.80814 0.80873 0.80932 0.80991

81004 74845 60605 38278 07857

05114 52830 53130 17436 57982

28687 83153 16899 34799 15321

022 915 872 758 017

0.58978 0.58898 0.58817 0.58736 0.58655

80250 01721 17303 27003 30829

31098 71298 31375 19770 45514

22996 18462 04967 59766 77276

099 976 973 388 748

0.945 0.946 0.947 0.948 0.949

0.81049 0.81108 0.81166 0.81225 0.81283

69337 22713 67977 05125 34150

87809 20770 71528 55555 89138

69300 98639 54920 97938 54154

383 669 560 351 591

0.58574 0.58493 0.58412 0.58330 0.58249

28790 20893 07147 87560 62139

18224 48104 45944 23117 91583

88177 78446 08339 31310 12874

827 913 436 012 994

0.950

0.81341 55047 89373 75068 542

0.58168 30894 63883 49416 618

161

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS cos x

sin x

X

Table 4.6

0.950 0.951 0.952 0.953 0.954

0.81341 0.81399 0.81457 0.81515 0.81573

55047 67810 72433 68910 57236

89373 74171 62256 73166 27252

75068 95507 91835 40081 73984

542 433 411 165 145

0.58168 0.58086 0.58005 0.57924 0.57842

30894 93832 50961 02290 47826

63883 53142 73067 37944 62640

49416 86928 39704 08966 01435

618 810 748 253 096

0.955 0,956 0.957 0.958 0.959

0.81631 0.81689 0.81746 0.81804 0.81861

37404 09409 73245 28907 76388

45683 50441 64327 10956 14762

42959 69980 09381 04577 45701

322 433 654 644 891

0.57760 0.57679 0.57597 0.57515 0.57433

87578 21554 49762 72210 88907

62601 53853 52997 77213 44256

47846 21403 56176 65441 59961

300 511 536 113 007

0.960 0.961 0.962 0.963 0.964

0.81919 0.81976 0.82033 0.82090 0.82147

15683 46785 69691 84393 90886

00998 95734 25859 19084 03938

27163 05121 54877 28189 10495

322 101 569 263 962

0.57351 0.57270 0.57188 0.57105 0.57023

99860 05078 04569 98342 86403

72456 80718 88520 15912 83518

66212 44551 07322 36911 03741

505 395 513 940 923

0.965 0.966 0.967 0.968 0.969

0.82204 0.82261 0.82318 0.82375 0.82432

89164 79221 61053 34652 00014

09771 66757 05889 58985 58683

78067 55069 70544 15315 98799

694 656 986 328 136

0.56941 0.56859 0.56777 0.56694 0.56612

68763 45428 16407 81708 41340

12530 24714 42403 88498 86469

84208 78562 28733 36093 79171

614 699 004 162 417

0.970 0.971 0.972 0.973 0.974

0.82488 0.82545 0.82601 0.82657 0.82714

57133 06003 46618 78974 03063

38450 32571 76161 05158 56326

05747 52898 45547 34034 70155

662 564 087 750 495

0.56529 0.56447 0.56364 0.56282 0.56199

95311 43629 86302 23338 54747

60354 34754 34839 86340 15554

31303 78229 35633 66624 99167

653 727 190 480 663

0.975 0.976 0.977 0.978 0.979

0.82770 0.82826 0.82882 0.82938 0.82993

18881 26422 25681 16651 99327

67257 76369 22907 46947 89390

63479 37592 86257 29486 69534

226 699 689 397 022

0.56116 0.56034 0.55951 0.55868 0.55785

80535 00712 15285 24263 27654

49341 15121 40876 55149 87042

43450 09200 22937 45183 87601

813 110 736 654 358

0.980 0.981 0.982 0.983 0.984

0.83049 0.83105 0.83160 0.83216 0.83271

73704 39776 97538 46983 88107

91970 97248 48619 90304 67360

46808 95697 00310 50142 95650

453 028 290 703 254

0.55702 0.55619 0.55536 0.55452 0.55369

25467 17710 04390 85517 61099

66217 22891 87840 92397 68448

30087 37806 78167 37748 39160

666 645 757 295 207

0.985 0.986 0.987 0.988 0.989

0.83327 0.83382 0.83437 0.83492 0.83547

20904 45368 61493 69275 68708

25676 11970 73796 59543 18432

03744 13205 90007 82563 76889

902 801 262 379 279

0.55286 0.55202 0.55119 0.55036 0.54952

31144 95660 54656 08140 56120

48435 65354 52753 44732 75943

57861 38911 13672 16453 01100

376 453 322 272 969

0.990 0.991 0.992 0.993 0.994

0.83602 0.83657 0.83712 0.83766 0.83821

59786 42503 16855 82835 40439

00520 56699 38697 99079 91248

51678 33299 50701 90248 50455

926 444 883 385 694

0.54868 0.54785 0.54701 0.54617 0.54534

98605 35603 67123 93173 13760

81587 97417 59732 05380 71756

57534 28224 24618 43512 83362

313 252 647 268 006

0.995 0.996 0.997 0.998 0.999

0.83875 0.83930 0.83984 0.84038 0.84093

89661 30495 62937 86979 02618

69442 88741 05059 75154 56621

96654 15567 69798 52241 40408

953 733 245 668 555

0.54450 0.54366 0.54282 0.54198 0.54114

28894 38584 42836 41661 35065

96802 19004 77392 11542 61572

60547 25576 79237 88693 03531

375 412 026 907 067

1.000

0.84147 09848 07896 50665 250

0.54030 23058 68139 71740 094

162

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SIXES AND COSINES FOR RADIAN ARGUMENTS sin

2

cos x

2:

1.000 1.001 1: 002 1.003 1.004

0.84147 0.84201 0.84254 0.84308 0.84362

09848 08662 99057 81026 54565

07896 88256 57821 77549 09246

50665 92390 22046 97169 30271

250 268 578 747 873

0.54030 0.53946 0.53861 0.53777 0.53693

23058 05648 82844 54653 21084

68139 72446 16233 41780 91907

71740 55654 47828 86864 73184

094 214 237 465 669

1. 005 1. 006 1.007 1.008 1.009

0.84416 0.84469 0.84523 0.84576 0.84629

19667 76327 24541 64302 95605

15556 59970 06821 21289 69397

42661 18177 56844 28431 25943

273 851 116 774 853

0.53608 0.53524 0.53439 0.53355 0.53270

82147 37848 88197 33202 72871

09970 39863 26016 13394 47496

84748 92716 77062 42130 32136

188 262 668 747 904

1.010 1.011 1.012 1.013 1.014

0.84683 0.84736 0.84789 0.84842 0.84895

18446 32818 38716 36136 25071

18015 34859 88491 48323 84612

19012 07211 73284 36290 04660

310 051 331 466 810

0.53186 0.53101 0.53016 0.52931 0.52846

07213 36237 59950 78362 91481

74355 40537 93140 79791 48651

46620 55841 16121 85137 37156

673 426 808 984 798

1.015 1.016 iI 017 1.018 1.019

0.84948 0.85000 0.85053 0.85105 0.85158

05517 77468 40919 95865 42300

68464 71835 67530 29204 31363

29173 55845 78730 92646 45804

940 003 164 111 549

0.52761 0.52677 0.52591 0.52506 0.52421

99315 01873 99163 91194 77974

48406 28274 37999 27850 48627

78219 61274 01253 90098 11734

896 932 921 832 503

1.020 1.021 1.022 1.023 1; 024

0.85210 0.85263 0.85315 0.85367 0.85419

80219 09617 30489 42829 46633

49362 59411 38569 64749 16717

92361 44882 26719 24308 39374

655 415 808 778 945

0.52336 0.52251 0.52166 0.52080 0.51995

59512 35816 06896 72758 33413

51649 88764 12341 75271 30969

56988 38461 05336 58150 63497

961 245 792 502 542

1.025 1.026 1.027 1.028 1.029

0.85471 0.85523 0.85575 0.85626 0.85678

41894 28609 06771 76375 37417

74093 17351 27819 87681 79977

41057 17949 30046 60616 67982

997 715 586 931 525

0.51909 0.51824 0.51738 0.51653 0.51567

88868 39132 84213 24121 58864

33369 36926 96612 67920 06859

68691 16373 59061 73657 75899

985 373 276 956 186

1.030 1.031 1.032 1.033 1.034

0.85729 0.85781 0.85832 0.85883 0.85935

89891 33792 69115 95855 14006

88603 98311 94711 64271 94317

37214 31744 44887 51283 58248

627 398 626 734 998

0.51481 0.51396 0.51310 0.51224 0.51138

88449 12887 32184 46351 55396

69955 14248 97296 77168 12447

34753 86768 50370 40101 80821

350 878 116 715 625

1.035 1.036 1.037 1.038 1.039

0.85986 0.86037 0.86088 0.86139 0.86189

23564 24523 16879 00625 75758

73034 89466 33518 95953 68397

57043 938 7,4054 819 21889 224 50385 634 97536 975

0.51052 0.50966 0.50880 0.50794 0.50708

59326 58151 51880 40520 24082

62230 86122 44242 97216 06180

21842 51023 08807 02209 18757

776 535 028 404 138

1.040 1.041 1.042 1.043 1.044

0.86240 0.86291 0.86341 0.86391 0.86442

42272 00162 49422 90049 22036

43338 14123 74964 20934 47972

40328 45486 20150 62441 11963

079 997 131 124 456

0.50622 0.50535 0.50449 0.50363 0.50276

02572 76000 44374 07704 65997

32778 39161 87986 42416 66117

40373 57213 81451 61010 93250

447 919 427 426 711

1.045 1.046 1.047 1.048 1.049

0.86492 0.86542 0.86592 0.86642 0.86692

45379 60073 66112 63493 52209

52878 33318 87822 15788 17477

00206 00866 80077 46561 01685

699 385 424 037 140

0.50190 0.50103 0.50017 0.49930 0.49843

19263 67509 10745 48980 82221

23261 78520 97070 44586 87247

38600 34140 07134 88513 26307

728 520 396 415 756

1.050

0.86742 32255 94016 89438 141

0.49757 10478 91726 99029 085

I

163

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS sin x

X

Table 4.6

cos x

1.050 1.051 1. 052 1,053 1.054

0.86742 0.86792 0.86841 0.86891 0.86940

32255 03628 66321 20330 65651

94016 47403 80499 97035 01611

89438 46316 51123 74685 29477

141 092 146 276 198

0.49757 0.49670 0.49583 0.49496 0.49409

10478 33760 52074 65430 73836

91726 25200 55338 50311 78782

99029 29002 95651 48726 21490

085 975 499 051 510

1.055 1.056 1.057 1.058 1.059

0.86990 0.87039 0.87088 0.87137 0.87186

02276 30203 49427 59941 61741

99694 97621 02601 22711 66899

19162 88046 70443 39954 58660

460 624 529 543 794

0.49322 0.49235 0.49148 0.49061 0.48974

77302 75835 69444 58139 41927

09910 13349 59246 18239 61459

43854 55459 18707 31756 41446

806 008 979 732 534

1.060 1.061 1.062 1.063 1.064

0.87235 0.87284 0.87333 0.87381 0.87430

54823 39181 14811 81707 39866

44986 67663 46494 93918 23243

26228 28925 88556 11299 36468

295 947 345 356 402

0.48887 0.48799 0.48712 0.48625 0.48537

20818 94820 63943 28194 87582

60527 87554 15140 16372 64825

56191 58818 19351 07757 06632

864 317 528 202 362

1.065 1.066 1,067 1.068 1.069

0.87478 0.87527 0.87575 0.87623 0.87671

89281 29948 61863 85020 99415

48654 85211 48845 56366 25458

85179 08932 38105 30362 18969

424 453 753 492 874

0.48450 0.48362 0.48275 0.48187 0.48100

42117 91807 36660 76686 11893

34560 00124 36547 19345 24514

23847 05142 46667 07484 22014

867 311 387 800 811

1.070 1.071 1.072 1.073 1.074

0.87720 0.87768 0.87815 0.87863 0.87911

05042 01898 89976 69274 39784

74681 23473 92149 01900 74797

61030 85627 41877 46904 33716

706 336 919 963 111

0.48012 0.47924 0.47836 0.47749 0.47661

42290 67886 88689 04709 15953

28534 08365 41447 05700 79522

12436 01039 22529 36282 38551

509 904 904 289 762

1.075 1.076 1.077 1.078 1.079

0.87959 0.88006 0.88053 0,88101 0.88148

01504 54428 98551 33868 60376

33788 02703 06248 70011 20461

98997 50816 56244 88879 76291

101 869 731 619 297

0.47573 0.47485 0.47397 0.47309 0.47221

22432 24153 21126 13359 00861

41788 71851 49538 55152 69469

74632 50968 47223 28292 56273

160 911 840 396 392

1.080 1.081 1.082 1.083 1.084

0.88195 0.88242 0.88289 0.88336 0.88383

78068 86941 86990 78210 60596

84947 91699 69831 49337 61096

47373 79609 46247 63390 36998

533 169 031 660 790

0.47132 0.47044 0.46956 0.46868 0.46779

83641 61708 35070 03737 67717

73740 49685 79499 45845 31856

02391 58871 50767 47743 75803

353 547 810 217 727

1.085 1.086 1.087 1.088 1.089

0.88430 0.88476 0.88523 0.88570 0.88616

34144 98849 54706 01710 39858

36869 09301 11921 79145 46272

09797 08104 88562 84791 53940

534 243 972 522 000

0.46691 0.46602 0.46514 0.46425 0.46337

27019 81651 31624 76945 17623

21135 97750 46239 51605 99315

28984 80991 96791 44159 05181

862 522 014 401 235

1.090 1.091 1.092 1.093 1.094

0.88662 0.88708 0.88755 0.88801 0.88846

69144 89564 01113 03786 97579

49487 25861 13352 50807 77956

23160 35990 98641 26207 88779

860 371 470 951 948

0.46248 0.46159 0.46071 0.45982 0.45893

53668 85088 11892 34089 51687

75300 65958 58145 39181 96847

87702 36738 45833 68372 28855

790 852 190 764 783

1.095 1.096 1.097 1.098 1.099

0.88892 0.88938 0.88984 0.89029 0.89075

82488 58507 25633 83860 33184

35422 64713 08227 09252 11966

57470 50354 78315 90807 21527

660 274 047 488 609

0.45804 0.45715 0.45626 0.45537 0.45448

64697 73125 76983 76277 71018

19382 95485 14314 65483 39062

34113 84487 84956 56224 45757

686 142 158 382 688

1,100

0.89120 73600 61435 33995 180

0.45359 61214 25577 38777 137

[(-78161

164

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SIXES AND COSINES FOR RADIAN ARGUMENTS sin x

X

cos

x

1.100 1.101 1.102 1.103 1.104

0.89120 0.89166 0.89211 0.89256 0.89301

73600 05105 27692 41359 46100

61435 03618 85365 54417 59408

33995 67046 80240 99171 60693

180 971 901 080 678

0.45359 0.45270 0.45181 0.45092 0.45002

61214 46874 28007 04621 76727

25577 16008 01790 74808 27402

38777 69206 30573 86868 83352

137 400 730 576 928

1.105 1.106 1.107 1.108 1.109

0.89346 0.89391 0.89436 0.89480 0.89525

41911 28787 06724 75718 35763

49863 76201 89735 42671 88110

58063 85981 85553 89157 65223

585 812 594 146 027

0.44913 0.44824 0.44734 0.44645 0.44555

44332 07446 66077 20235 69929

52361 42924 92780 96065 47363

57327 48852 11424 22607 94616

478 689 866 305 628

1.110 1: 111 1.112 1.113 1.114

0.89569 0.89614 0.89658 0.89702 0.89747

86856 28992 62167 86375 01614

80047 73373 23874 88234 24030

62924 56775 91147 24683 74633

063 801 427 120 785

0.44466 0.44376 0.44286 0.44197 0.44107

15167 55958 92312 24237 51742

41706 74570 41874 39984 65707

84864 06453 38633 37201 44874

374 951 030 474 890

1.115 1.116 1.117 1.118 1.119

0.89791 0.89835 0.89878 0.89922 0.89966

07877 05162 93463 72776 43097

89740 44737 49293 64577 52658

61099 51180 03041 09884 43829

138 079 321 230 826

0.44017 0.43927 0.43838 0.43748 0.43658

74837 93529 07829 17745 23287

16293 89431 83253 96329 27666

01603 54849 69812 39623 95482

891 166 438 410 777

1.120 1.121 1.122 1.123 1.124

0.90010 Oi90053 0.90097 0.90140 0.90183

04421 56744 00062 34371 59665

76504 99984 87864 05813 20399

99711 38780 32313 05144 79088

910 263 880 201 276

0.43568 0.43478 0.43388 0.43298 0.43207

24462 21281 13752 01884 85686

76712 43347 27890 31095 54146

16761 41055 74199 00232 91323

399 736 612 420 845

1.125 1.126 1.127 1.128 1.129

0.90226 0.90269 0.90312 0.90355 0.90398

75940 83194 81420 70615 50774

99095 10271 23203 08069 35948

16291 62482 90131 41527 71758

842 258 256 464 658

0.43117 0.43027 0.42937 0.42846 0.42756

65167 40337 11204 77777 40066

98666 66704 60745 83700 38914

17655 57257 05806 86372 59134

197 452 078 749 030

1.130 1.131 1.132 1.133 1.134

0.90441 0.90483 0.90526 0.90568 0.90611

21893 83969 36996 80970 15887

78825 09589 02030 30848 71644

91603 10334 78425 30177 26245

708 160 425 523 348

0.42665 0.42575 0.42485 0.42394 0.42303

98079 51825 01314 46554 87555

30157 61627 37950 64178 45785

31037 65422 91605 14410 23669

122 763 376 540 902

1.135 1.136 1.137 1.138 1.139

0.90653 0.90695 0.90737 0.90779 0.90821

41744 58534 66256 64903 54473

00926 96110 35516 98372 64813

96078 80269 72815 63281 78880

401 960 632 260 126

0.42213 0.42122 0.42031 0.41941 0.41850

24325 56874 85211 09345 29285

88672 99161 83998 50349 05800

03673 42580 41784 25243 48758

585 219 656 478 379

1.140 1.141 1.142 1.143 1.144

0.90863 0.90905 0.90946 0.90988 0.91029

34961 06362 68673 21889 66006

15883 33532 00620 00918 19102

26459 34395 94400 03234 04326

422 940 939 153 885

0.41759 0.41668 0.41577 0.41486 0.41395

45039 56618 64029 67283 66389

58358 16446 88907 85000 14400

09217 53794 89108 90333 10281

519 933 094 707 852

1.145 1.146 1.147 1.148 1.149

0.91071 0.91112 0.91153 0.91194 0.91235

01020 26927 43723 51403 49965

40761 52394 41410 96130 05786

29314 39475 67087 56676 06195

164 912 073 684 821

0.41304 0.41213 0.41122 0.41031 0.40940

61354 52190 38904 21505 00004

87194 13888 05397 73050 28587

88428 59906 64456 55331 08169

529 732 120 381 395

1.150

165

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS cos x

sin x

X

Table 4.6

1.150 1.151 1.152 1.153 1.154

0.91276 0.91317 0.91357 0.91398 0.91439

39402 19712 90890 52933 05835

60521 51391 70367 10330 65075

08094 90306 57146 30107 88579

403 792 165 602 865

0.40848 0.40757 0.40666 0.40574 0.40483

74408 44728 10972 73149 31269

84157 52320 46045 78706 64086

29815 67107 15621 28372 24481

258 284 071 706 224

1.155 1.156 1.157 1.158 1.159

0.91479 0.91519 0.91560 0.91600 0.91640

49594 84204 09663 25966 33109

29314 98669 69679 39800 07401

10465 12711 91743 63815 05261

816 431 383 143 556

0.40391 0.40300 0.40208 0.40117 0.40025

85341 35373 81375 23357 61326

16372 50159 80441 22620 92496

97790 25449 76456 20152 34689

397 945 266 779 958

1.160 1.161 1.162 1.163 1.164

0.91680 0.91720 0.91759 0.91799 0.91839

31087 19898 99536 69999 31282

71766 33100 92520 52063 14682

92661 42911 53200 40902 83374

866 136 023 883 147

0.39933 0.39842 0.39750 0.39658 0.39566

95294 25267 51257 73271 91320

06273 80553 32340 79035 38435

15445 83402 93491 42889 79278

164 355 775 706 377

1.165 1.166 1.167 1.168 1.169

0.91878 0.91918 0.91957 0.91996 0.920.35

83380 26291 60010 84533 99857

84250 65556 64310 87139 41592

57652 80075 45798 68222 18336

941 906 178 492 360

0.39475 0.39383 0.39291 0.39199 0.39107

05412 15556 21762 24039 22396

28737 68530 76800 72926 76682

09066 05567 17146 75312 02789

125 898 187 486 366

1.170 1.171 1.172 1.173 1.174

0.92075 0.92114 0.92152 0.92191 0.92230

05977 02889 90590 69076 38343

36135 80158 83968 58796 16793

63957 08886 31967 26061 36915

301 071 851 369 902

0.39015 0.38923 0.38830 0.38738 0.38646

16843 07387 94040 76809 55705

08230 88126 37316 77135 29304

21533 60718 64679 00821 67479

266 072 599 054 575

1.175 1.176 1.177 1.178 1.179

0.92268 0.92307 0.92345 0.92384 0.92422

98386 49203 90789 23140 46253

71033 35513 25145 55777 44173

01956 88974 34733 83468 25312

127 783 097 944 701

0.38554 0.38462 0.38369 0.38277 0.38184

30736 01911 69240 32733 92397

15936 59525 82956 09495 62792

01753 87293 62048 25982 48743

942 547 718 487 902

1.180 1.181 1.182 1.183 1.184

0.92460 0.92498 0.92536 0.92574 0.92612

60124 64748 60123 46244 23108

08020 65932 37446 43024 04056

34610 08156 03329 76141 19188

754 619 642 242 645

0.38092 0.38000 0.37907 0.37814 0.37722

48243 00280 48517 92963 33627

66881 46178 25478 29958 85174

77302 43547 71840 86542 19493

960 271 534 917 444

1.185 1.186 1.187 1.188 1.189

0.92649 0.92687 0.92724 0.92762 0.92799

90710 49047 98116 37912 68432

42853 82657 47634 62876 54404

99516 96383 38942 43819 52606

095 480 352 290 588

0.37629 0.37537 0.37444 0.37351 0.37258

70520 03649 33025 58656 80552

17058 51921 16451 37709 43133

17454 49518 14476 48149 30684

471 342 334 962 752

1.190 1.191 1.192 1.193 1.194

0.92836 0.92874 0.92911 0.92947 0.92984

89672 01628 04297 97675 81758

49166 75038 60825 36260 32004

69260 97404 77546 24192 62877

202 950 899 928 403

0.37165 0.37073 0.36980 0.36887 0.36794

98722 13176 23922 30970 34330

60532 18091 44362 68273 19116

93806 28040 89893 08995 95213

568 589 026 672 382

1.195 1.196 1.197 1.198 1.199

0.93021 0.93058 0.93094 0.93131 0.93167

56542 22025 78201 25068 62622

79650 11719 61664 63866 53638

67095 95146 26876 00337 48349

956 303 083 679 974

0.36701 0.36608 0.36515 0.36422 0.36328

34010 30020 22369 11066 96122

26558 20629 31729 90622 28438

45714 52004 06923 11604 82399

570 819 698 876 631

1.200

0.93203 90859 67226 34967 013

0.36235 77544 76673 57763 837

166

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS sin x

X

cos 2

1.200 1.201 1.202 1.203 1.204

0.93203 0.93240 0.93276 0.93312 0.93348

90859 09776 19369 19634 10568

67226 41805 15485 27305 17240

34967 91853 54567 98748 76215

013 542 367 519 175

0.36235 0.36142 0.36049 0.35956 0.35862

77544 55343 29528 00108 67092

76673 67184 32190 04273 16376

57763 05108 27614 71008 30309

837 539 189 651 065

1.205 1.206 1.207 1.208 1.209

0.93383 0.93419 0.93455 0.93490 0.93526

92167 64427 27346 80919 25144

26196 96013 69465 90260 03041

50966 35090 24584 35070 37431

302 992 444 567 162

0.35769 0.35675 0.35582 0.35488 0.35395

30490 90310 46564 99259 48405

01799 94203 27606 36382 55261

56527 63341 33727 26557 83165

660 607 018 166 039

1.210 1.211 1.212 1.213 1.214

0.93561 0.93596 0.93632 0.93667 0.93702

60015 85530 01686 08478 05904

53385 87806 53752 99608 74693

93341 90713 79041 04663 45913

646 291 926 095 598

0.35301 0.35208 0.35114 0.35021 0.34927

94012 36088 74644 09688 41230

19330 64027 25144 38826 41568

33870 04470 22698 24640 61124

301 775 521 616 730

1.215 1.216 1.217 1.218 1.219

0.93736 0.93771 0.93806 OI93841 0.93875

93960 72642 41946 01870 52410

29266 14521 82590 86543 80386

48199 58969 62598 15169 79170

416 959 617 574 848

0.34833 0.34739 0.34646 0.34552 0.34458

69279 93845 14937 32564 46736

70217 61966 54360 85289 92990

04069 52800 40329 39601 69704

578 358 260 140 455

1.220 1I221 1.222 1I223 1.224

0.93909 Or93944 0.93978 0.94012 0.94046

93563 25324 47691 60660 64228

19067 58470 55418 67676 53946

58093 30937 86621 58302 57600

524 151 257 957 622

0.34364 0.34270 0.34176 0.34082 0.33988

57463 64752 68615 69060 66097

16047 93385 64277 68336 45517

02047 66500 57501 40132 56153

552 405 890 702 996

1.225 1.226 1.227 1.228 1.229

0.94080 0.94114 0.94148 0.94181 0.94215

58391 43146 18490 84419 40930

73872 88036 57965 46123 15917

08723 82507 30357 18091 59701

559 685 157 912 104

0.33894 0.33800 0.33706 0.33612 0.33518

59735 49983 36852 20349 00486

36117 80771 20455 96483 50503

30011 74807 98234 08479 20093

855 668 533 750 523

1.230 1.231 1.232 1.233 1.234

0.94248 0.94282 0.94315 0.94348 0.94381

88019 25683 53919 72724 82093

31697 58754 63320 12574 74633

51002 03206 76390 12870 70486

382 998 684 299 175

0.33423 0.33329 OI33235 0.33140 0.33046

77271 50713 20823 87608 51080

24502 60802 02059 91261 71729

59823 72418 26391 19759 85740

955 427 462 164 328

1.235 1.236 1.237 1.238 1.239

0.94414 0.94447 0.94480 0.94513 0.94545

82025 72515 53560 25157 87303

18562 14367 32999 46354 27272

55790 57139 77695 68328 60431

164 322 223 851 046

0.32952 0.32857 0.32763 0.32668 0.32574

11247 68119 21705 72015 19058

87117 81408 98914 84277 82466

98424 78405 98386 88743 43066

316 786 387 487 054

1.240 1.241 1.242 1.243 1.244

0.94578 0.94610 0.94643 0.94675 0.94707

39994 83227 17000 41308 56148

49538 87884 17986 16467 60895

98628 73405 53628 18984 92311

471 063 942 738 309

0.32479 0.32385 0.32290 0.32195 0.32101

62844 03381 40681 74751 05601

38776 98828 08569 14269 62521

23657 67007 89227 91456 65238

769 475 042 764 364

1.245 1.246 1.247 1.248 1.249

0.94739 0.94771 0.94803 0.94835 0.94866

61518 57414 43832 20770 88225

29788 02608 59766 82619 53474

71844 63367 12259 35461 53335

815 118 472 479 262

0.32006 0.31911 0.31816 0.31721 0.31627

33242 57681 78930 96997 11891

00239 74660 33339 24152 95292

97855 77643 99262 68947 09714

712 341 871 423 116

1.250

0.94898 46193 55586 21434 849 L

'

J

0.31532 23623 95268 66544 754 L

'

J

167

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS sin x

X

cos

Table 4.6

x

1.250 1.251 1.252 1.253 1.254

0.94898 0.94929 0.94961 0.94992 0.95023

46193 94671 33656 63145 83135

55586 73157 91340 96237 74899

21434 62180 96439 75008 10006

849 713 444 528 196

0.31532 0.31437 0.31342 0.31247 0.31152

23623 32202 37637 39938 39114

95268 72909 77355 58064 64805

66544 11534 49010 20615 10363

754 791 665 601 979

1.255 1.256 1.257 1.258 1.259

0.95054 0.95085 0,95116 0.95147 0.95178

93623 94605 86078 68040 40486

15326 06469 38232 01466 87975

06166 92038 51091 52726 83188

303 225 729 783 287

0.31057 0.30962 0.30867 0.30772 0.30676

35175 28130 17989 04761 88456

47660 57024 43600 58403 52756

49664 22311 69445 94485 68021

355 242 729 052 196

1.260 1.261 1.262 1.263 1.264

0.95209 0.95239 0.95270 0.95300 0.95330

03415 56824 00708 35065 59892

90515 02793 19468 36151 49407

76385 44617 09200 31003 40886

682 416 227 222 709

0.30581 0.30486 0.30391 0.30295 0.30200

69083 46652 21173 92654 61106

78289 86939 30948 62866 35544

32688 08001 95158 81822 46859

634 291 833 373 693

1.265 1.266 1.267 1.268 1.269

0.95360 0.95390 0.95420 0.95450 0.95480

75186 80944 77163 63840 40972

56753 56660 48552 32808 10759

70045 80258 94039 24694 06289

767 512 032 963 671

0.30105 0.30009 0.29914 0.29819 0.29723

26538 88959 48379 04808 58254

02136 16100 31192 01472 81295

65060 11818 67791 23518 84019

070 814 595 675 121

1.270 1.271 1.272 1.273 1.274

0.95510 0.95539 0.95569 0.95598 0.95627

08555 66588 15067 53989 83351

84692 57849 34427 19578 19409

23509 41432 35209 19640 78657

018 673 944 104 170

0.29628 0.29532 0.29437 0.29341 0.29245

08729 56240 00799 42413 81094

25318 88493 26068 93588 46891

73355 39166 57175 35661 19906

114 425 182 000 579

1.275 1.276 1.277 1.278 1.279

0.95657 0.95686 0.95715 0.95744 0.95772

03150 13383 14048 05142 86661

40985 92326 82408 21166 19488

94719 78101 96095 02109 64678

118 497 419 886 437

0.29150 0.29054 0.28958 0.28863 0.28767

16850 49691 79626 06666 30819

42108 35665 84278 44951 74982

96613 98290 07609 61732 56616

869 890 308 860 726

1.280 1.281 1.282 1.283 1.284

0.95801 0.95830 0.95858 0.95887 0.95915

58602 20964 73742 16935 50539

89224 43180 95120 59764 52796

96370 82604 10371 96853 17957

075 453 286 962 320

0.28671 0.28575 0.28479 0.28383 0.28288

52096 70505 86057 98761 08626

31955 73742 58503 44681 91007

51277 72036 16730 58895 51923

939 934 332 050 831

1.285 1.286 1.287 1.288 1.289

0.95943 0.95971 0.95999 0.96027 0.96055

74551 88969 93790 89011 74629

90853 91535 73400 55966 59710

36739 31748 25260 11427 84322

577 357 814 805 094

0.28192 0.28096 0.28000 0.27904 0.27808

15663 19881 21288 19896 15714

56494 00438 82417 62291 00198

33192 28157 54428 25809 56310

303 651 993 577 871

1.290 1.291 1.292 1.293 1.294

0.96083 0.96111 0.96138 0.96166 0.96193

50642 17046 73839 21018 58580

06072 17450 17203 29652 80080

65890 33810 49249 84528 50693

556 354 056 675 590

0.27712 0.27615 0.27519 0.27423 0.27327

08750 99015 86519 71271 53280

56557 92064 67693 44692 84588

64138 75651 29289 79480 00512

661 234 769 997 263

1.295 1.296 1.297 1.298 1.299

0.96220 0.96248 0.96275 0.96302 0.96329

86523 04845 13541 12610 02048

94730 00807 26481 00880 54098

24982 78203 02013 36103 95282

339 231 782 915 920

0.27231 0.27135 0.27038 0.26942 0.26846

32557 09111 82951 54087 22529

49177 00534 01003 13198 00008

90379 74605 10035 88600 41057

053 108 206 711 992

1.300

0.96355 81854 17192 96470 135

/

0.26749 88286 24587 40699 798 p-78141

168

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SINES .4ND COSINES FOR RADIAN ARGUMENTS sin x

X

cos 2

1.300 1.301 1. 302 1.303 1.304

0.96355 0.96382 0.96409 0.96435 0.96462

81854 52024 12556 63446 04694

17192 22181 02048 90740 23167

96470 85589 64366 17032 36927

135 331 761 855 537

0.26749 0.26653 0.26557 0.26460 0.26364

88286 51368 11785 69546 24661

24587 50360 41018 60519 73088

40699 07039 09469 70890 71318

798 695 650 877 016

1.305 1.306 1.307 1.308 1.309

0.96488 0.96514 0.96540 0.96566 0.96592

36295 58247 70548 73195 66185

35205 63694 46439 22209 30740

53009 56266 26036 56221 81411

126 806 635 061 924

0.26267 0.26171 0.26074 0.25978 0.25881

77140 26992 74227 18854 60883

43213 35646 15401 47755 98246

51456 16255 38427 61955 05556

761 031 774 494 626

1. 310 1.311 1.312 1. 313 1.314

0.96618 0.96644 0.96669 0.96695 0.96720

49516 23185 87189 41527 86195

12734 09856 64740 20986 23159

02916 14689 29162 02983 62656

926 520 218 276 736

0.25785 0.25688 0.25591 0.25495 0.25398

00325 37188 71482 03217 32402

32669 17082 17797 01385 34673

66133 22194 37244 63156 43517

818 242 030 911 173

1.315 1.316 1.317 1.318 1.319

0.96746 0.96771 0.96796 0.96821 0.96846

21191 46512 62156 68121 64403

16794 48390 65416 16306 50465

30085 48019 05402 62628 76697

794 478 607 991 879

0.25301 0.25204 0.25108 0.25011 0.24914

59047 83163 04758 23842 40425

84742 18927 04816 10251 03323

16937 20348 92269 76046 23067

022 457 738 556 996

1.320 1.321 1.322 1.323 1.324

0.96871 0.96896 0.96920 0.96945 0.96970

51001 27911 95132 52661 00495

18265 71045 61115 41752 67204

26273 36648 04608 23202 06414

590 340 211 252 685

0.24817 0.24720 0.24623 0.24526 0.24429

54516 66126 75263 81939 86161

52372 25991 93018 22539 83886

95957 71738 44974 30889 68450

398 199 865 004 760

1.325 1.326 1.327 1.328 1.329

0.96994 0.97018 0.97042 0.97066 0.97090

38632 67070 85806 94838 94163

92687 74387 69462 36036 33208

13740 74662 13034 71365 35004

188 236 465 051 060

0.24332 0.24235 0.24138 0.24041 0.23944

87941 87287 84210 78719 70824

46638 80615 55885 42753 11769

23445 91516 01181 16828 41682

582 463 759 662 448

1.330 1.331 1.332 1.333 1.334

0.97114 0.97138 0.97162 0.97185 0.97209

83779 63683 33874 94348 45104

21044 60583 13835 43780 14372

56233 78261 59117 95451 46235

768 900 786 405 282

0.23847 0.23750 0.23653 0.23556 0.23458

60534 47859 32810 15395 95624

33723 79643 20797 28690 75063

20751 43748 47988 21258 04672

578 768 097 288 221

1.335 1.336 1.337 1.338 1.339

0.97232 0.97256 0.97279 0.97302 0.97325

86138 17450 39036 50894 53021

90534 38163 24129 16271 83406

56369 80187 04871 73757 09557

230 900 129 066 931

0.23361 0.23264 0.23167 0.23069 0.22972

73508 49055 22276 93180 61778

31892 71391 66003 88407 11512

95492 805 49935 286 85946 099 85958 358 99624 085

1.340 1.341 1.342 1.343 1.344

0.97348 0.97371 0.97394 0.97416 0.97439

45416 28077 01000 64184 17626

95319 22772 37498 12205 20575

37478 08238 2.0994 46167 48173

787 616 365 522 349

0.22875 0.22777 0.22680 0.22583 0.22485

28078 92090 53825 13291 70500

08459 52617 17584 77188 05482

46523 18849 84074 87585 55305

264 831 691 859 819

1.345 1.346 1.347 1.348 1.349

0.97461 0.97483 0.97506 0.97528 0.97550

61324 95276 19479 33932 38633

37264 37901 99092 98416 14428

08052 46006 43832 67265 88217

713 501 603 423 916

0.22388 0.22290 0.22193 0.22095 0.21998

25459 78180 28672 76944 23007

76744 65480 46415 94502 84913

96286 05279 65290 50100 26774

212 929 729 463 007

1.350

0.97572 33578 26659 06926 111

0.21900 66870 93041 58142 002

169

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS sin z

z

Table 4.6

cos z

1.350 1.351 1.352 1.353 1.354

0.97572 0.97594 0.97615 0.97637 0.97659

33578 18766 94194 59861 15764

26659 15612 62771 50591 62506

06926 73996 12353 39095 87244

111 110 536 407 418

0.21900 0.21803 0.21705 0.21607 0.21510

66870 08543 48036 85358 20520

93041 94501 65124 80961 18281

58142 05261 29854 96725 76154

002 504 627 291 163

1.355 1.356 1.357 1.358 1.359

0.97680 0.97701 0.97723 0.97744 0.97765

61901 98270 24869 41696 48748

82927 97238 91804 53965 72037

27405 89325 83352 21803 40225

609 386 894 706 805

0.21412 0.21314 0.21217 0.21119 0.21021

53530 84399 13137 39753 64257

53567 63517 25046 15278 11553

46271 95410 24434 49048 02083

899 772 790 406 908

1.360 1.361 1.362 1.363 1.364

0.97786 0.97807 0.97828 0.97848 0.97869

46024 33521 11237 79171 37319

35316 34074 59561 04006 60615

18567 02249 23135 20406 61343

849 690 125 864 685

0.20923 0.20826 0.20728 0.20630 0.20532

86658 06968 25195 41349 55440

91419 32637 13175 11211 05130

35767 23964 64404 80880 25435

598 842 112 089 952

1.365 1.366 1.367 1.368 1.369

0.97889 0.97910 0.97930 0.97950 0.97970

85681 24253 53035 72024 81217

23574 88047 50175 07082 56868

61999 07786 73954 45982 39862

774 196 516 521 027

0.20434 0.20336 0.20238 0.20140 0.20042

67477 77471 85432 91369 95291

73521 95182 49113 14517 70801

80524 61151 16990 34489 38946

932 240 457 495 217

1.370

1.372 1.373 1.374

0.97990 0.98010 0.98030 0.98050 0.98069

80613 70211 50007 20000 80189

98614 32380 59206 81114 01103

22288 30754 93540 49613 68424

769 328 094 233 652

0.19944 0.19846 0.19748 0.19650 0.19552

97209 97133 95072 91036 85036

97572 74640 82010 99890 08682

96568 16515 52911 06852 28380

820 079 545 798 853

1.375 1.376 1.377 1.378 1.379

0.98089 0.98108 0.98128 0.98147 0.98166

30570 71142 01903 22852 33986

23155 52232 94276 56212 45944

69608 42586 66065 27452 42153

920 155 826 479 343

0.19454 0.19356 0.19258 0.19160 0.19062

77079 67178 55340 41577 25898

88987 21600 87511 67905 44156

18444 30840 74132 13553 72884

822 918 912 129 094

1.380 1.381 1.382 1.383 1.384

0.98185 0.98204 0.98223 0.98241 0.98260

35303 26802 08480 80336 42368

72359 45326 75694 75296 56947

72787 48298 82965 95320 26961

813 791 850 221 571

0.18964 0.18865 0.18767 0.18669 0.18571

08312 88831 67462 44217 19105

97834 10696 64691 41955 24813

36320 50314 25395 37980 32156

915 508 757 715 930

1.385 1.386 1.387 1.388 1.389

0.98278 0.98297 0.98315 0.98333 0.98352

94574 36952 69500 92216 05100

34442 22562 37068 94707 13205

61276 42059 92032 31273 95537

561 162 708 673 148

0.18472 0.18374 0.18276 0.18178 0.18079

92135 63319 32665 00183 65884

95776 37540 32988 65185 17379

21451 90577 97169 73489 28124

016 542 360 451 404

1.390 1.391 1.392 1.393 1.394

0.98370 0.98388 0.98405 0.98423 0.98441

08148 01359 84731 58262 21952

11276 08614 25898 84790 07939

54484 29809 13274 84637 29485

004 722 870 207 405

0.17981 0.17882 0.17784 0.17686 0.17587

29776 91871 52177 10704 67464

72999 15656 29142 97424 04651

47659 98336 27690 66173 28756

616 311 484 860 976

1. 39.5 1.396 1.397 1.398 1.399

0.98458 0.98476 0.98493 0.98510 0.98527

75797 19796 53948 78250 92701

18974 42512 04152 30479 49063

56974 17462 20048 50013 86162

360 083 145 670 846

0.17489 0.17390 0.17292 0.17193 0.17095

22464 75715 27228 77011 25074

35146 73409 04115 12112 82423

16514 18192 11759 65937 41718

467 681 690 830 833

1.400

0.98544 97299 88460 18065 947

1. 371

"

);-

'I

0.16996 71429 00240 93861 675

170

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENgS cos

sin x

X

x

1.400 1.401 1.402 1.403 1.404

0.98544 0.98561 0.98578 0.98595 0.98612

97299 92043 76931 51961 17131

88460 78208 48834 31850 59751

18065 63203 84013 04837 28769

947 840 966 776 609

0.16996 0.16898 0.16799 0.16701 0.16602

71429 16083 59048 00332 39947

00240 50929 20024 93227 56412

93861 675 72373 233 23971 842 93533 ~ 8 5 4 25523 303

1.405 1.406 1.407 1.408 1.409

0.98628 0,98645 0.98661 0.98677 Oi98693

72440 17886 53468 79184 95031

66021 85129 52531 04669 78970

54406 92502 82515 09070 18307

982 294 912 631 486

0.16503 0.16405 0.16306 0.16207 0.16109

77901 14205 48869 81902 13314

95615 97042 47062 32209 39179

65404 61039 64065 31258 25882

770 544 184 571 568

1.410 1.411 1.412 1.413 1.414

0.98710 0.98725 0.98741 0.98757 0.98773

01010 97117 83352 59712 26197

13850 48711 23943 80922 62012

34142 74427 67004 65672 66048

909 198 304 895 706

0.16010 0.15911 0.15812 0.15714 0.15615

43115 71315 97924 22952 46408

54831 66184 60420 24876 47050

19016 90869 32080 44997 44945

356 577 359 336 751

1.415 1.416 1.417 1.418 1.419

0.98788 0.98804 0.98819 0.98834 0.98850

82805 29533 66381 93348 10430

10565 70919 88402 09330 81005

21328 57953 91177 40532 45199

142 120 144 586 170

0.15516 0.15417 0.15319 oii5220 0.15121

68303 88646 07447 24716 40463

14596 15325 37202 68347 97033

61477 39606 41027 45317 51126

752 967 471 231 135

1.420 1.421 1.422 1.423 1.424

0.98865 0.98880 0.98895 0.98909 0.98924

17628 14939 02362 79896 47539

51719 70753 88375 55844 25405

79273 66940 97544 40562 60478

627 521 222 021 351

0.15022 oIi4923 0.14824 0.14725 0.14626

54699 67432 78672 88430 96716

11685 00880 53344 57953 03732

77348 64281 74765 95314 37224

698 559 840 499 747

1.425 1.426 1.427 1.428 1.429

0.98939 0.98953 0.98967 0.98982 0.98996

05289 53145 91106 19171 37337

50295 84738 83949 04132 02480

31560 52533 61159 48716 74376

129 174 714 941 619

0.14528 0.14429 0.14330 0.14231 0.14132

03538 08908 12835 15329 16400

79851 75628 80526 84153 76259

37675 60810 98807 72928 34563

648 986 514 666 848

1.430 1.431 1.432 1.433 1,434

0.99010 0.99024 0.99038 0.99052 0.99065

45603 43968 32431 10990 79644

37177 67397 53301 56046 37773

79485 01748 89307 14729 88889

729 121 176 460 346

0.14033 0.13934 0.13835 0.13736 0.13637

16058 14312 11173 06651 00755

46736 85619 83083 29440 15144

66253 82699 31761 95441 90849

390 275 733 799 940

1.435 1.436 1.437 1.438 1.439

0.99079 0.99092 0.99106 0.99119 0.99132

38391 87230 26160 55180 74287

61619 91709 93157 32073 75552

74754 01072 75959 00385 81565

605 941 459 060 735

0.13537 0.13438 0.13339 0.13240 0.13141

93495 84881 74924 63632 51017

30784 67086 14910 65254 09245

71160 26554 85143 13887 19491

849 495 546 244 852

1.440 1.441 1.442 1.443 1.444

0.99145 0.99158 0.99171 0.99184 0.99197

83481 82761 72125 51571 21099

91686 49554 19229 71773 79243

46252 53923 09874 78212 94748

760 766 676 505 990

0.13042 0.12943 0.12844 0.12744 0.12645

37087 21853 05325 87512 68425

38145 43347 16375 48881 32647

49297 92153 79275 85098 28105

752 306 576 002 135

1.445 1.446 1.447 1.448 1.449

0.99209 Oi99222 0.99234 0.99247 0.99259

80708 30395 70160 00002 19919

14686 52141 66639 34203 31850

79795 50856 35228 82494 76923

055 088 024 216 086

0.12546 0712447 0.12348 0.12248 0.12149

48073 26467 03616 79530 54219

59580 21717 11217 20366 41572

71654 24785 43017 29130 33939

525 871 513 391 548

1.450

0.99271 29910 37588 49766 535

0.12050 27693 67366 57053 287 L

'

J

171

ELEMENTARY TRANSCENDENTAL FTJNCTIONS

CIRCULAR SINES .4ND COSINES FOR RADIAN ARGUMENTS sin x

X

cos

Table 4.6

x

1.450 1.451 1.452 1.453 1.454

0.99271 0.99283 0.99295 0.99307 0.99318

29910 29974 20109 00315 70591

37588 30417 90332 98319 36356

49766 91459 63717 11543 75120

535 118 946 325 114

0.12050 0.11950 0.11851 0.11752 0.11653

27693 99962 71037 40925 09639

67366 90401 03450 99404 71276

57053 47620 05063 79804 73971

287 080 327 068 735

1.455 1.456 1.457 1.458 1.459

0.99330 0.99341 0.99353 0.99364 0.99375

30934 81345 21821 52362 72967

87418 35468 65467 63366 16112

01619 56903 37123 80232 77380

777 143 830 355 893

0.11553 0.11454 0.11355 0.11255 0.11156

77188 43581 08828 72940 35927

12194 15402 74262 82249 32951

42103 91829 84551 36104 17410

061 237 407 618 313

1.460 1.461 1.462 1.463 1.464

0.99386 0.99397 0.99408 0.99419 0.99430

83634 84362 75150 55998 26904

11644 38896 87794 49260 15209

84228 32148 39331 21797 04300

683 075 194 223 286

0.11056 0.10957 0.10858 0.10758 0.10659

97798 58563 18232 76816 34324

20069 37417 78917 38604 10617

55117 32232 88737 22199 88365

465 463 835 915 556

1.465 1.466 1.467 1.468 1.469

0.99440 0.99451 0.99461 0.99472 0.99482

87866 38885 79958 11085 32265

78550 33187 74019 96938 98831

31137 76860 56879 37979 48727

923 141 043 012 437

0.10559 0.10460 0.10361 0.10261 0.10162

90765 46151 00491 53795 06072

89208 68730 43646 08521 58026

01747 36201 25487 63826 06440

983 884 846 230 584

1.470 1.471 1.472 11473 1.474

0.99492 0.99502 0.99512 0.99522 0.99531

43497 44780 36112 17493 88922

77580 32063 62150 68709 53602

89785 44122 87122 96604 62729

993 430 898 762 932

0.10062 0.09963 0.09863 0.09764 0.09664

57333 07588 56847 05119 52415

86931 90112 62542 99295 95545

70090 33595 38345 88804 53005

698 391 147 678 525

1.475 1.476 1.477 1.478 1.479

0.99541 0.99551 0.99560 0.99569 0.99578

50398 01919 43486 75096 96749

19685 70812 11829 48581 87906

97818 46063 93145 75747 90969

664 854 787 356 720

0.09564 0.09465 0.09365 0.09266 0.09166

98745 44118 88544 32034 74598

46561 47711 94456 82355 07058

63028 15478 71943 59452 70920

806 186 189 948 484

1.480 1.481 1.482 1.483 1.484

0.99588 0.99597 0.99606 0.99614 0.99623

08445 10182 01959 83775 55630

37640 06611 04648 42571 32200

05648 65569 04588 53643 49677

408 851 337 374 461

0.09067 0.08967 0.08867 0.08768 0.08668

16244 56984 96827 35783 73863

64309 49943 59886 90154 36851

65577 69400 75526 44661 05477

623 641 752 519 303

1.485 1.486 1.487 1.488 1.489

0.99632 0.99640 0.99649 0.99657 0.99665

17522 69452 11417 43417 65452

86349 18829 44446 79005 39305

44454 13277 63607 43586 50450

246 079 933 693 815

0.08569 0.08469 0.08369 0.08270 0.08170

11075 47431 82940 17612 51456

96168 64385 37866 13060 86499

55002 59004 52356 39407 94334

845 070 240 518 076

1.490 1.491 1.492 1.493 1.494

0.99673 0.99681 0.99689 0.99697 0.99705

77520 79621 71753 53917 26110

43143 09312 57602 08799 84688

38855 29093 15215 73054 68141

320 143 811 448 099

0.08070 0.07971 0.07871 0.07771 0.07672

84484 16705 48128 78764 08624

54800 14659 62854 96243 11762

61486 55729 62770 39483 14220

832 907 926 234 152

1.495 1.496 1.497 1.498 1.499

0.99712 0.99720 0.99727 0.99735 0.99742

88334 40586 82865 15173 37506

08049 02660 93295 05727 66724

63530 27521 41279 06360 52131

364 334 821 877 595

0.07572 0.07472 0.07372 0.07273 0.07173

37716 66050 93638 20488 46611

06424 77322 21620 36561 19459

87121 30411 88691 79219 92192

354 478 060 898 943

172

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.6

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS sin x

X

cos

2

1.500 1.501 1.502 1.503 1.504

0.99749 0.99756 0.99763 0.99770 0.99776

49866 52250 44659 27091 99547

04054 46480 23765 66667 06942

43094 86109 37519 10173 80349

172 251 509 501 750

0.07073 0.06973 0.06874 0.06774 0.06674

72016 96714 20715 44028 66664

67702 78750 50131 79447 64365

91008 12531 67342 39990 89231

819 065 208 761 245

1.505 1.506 1.507 1.508 1.509

0.99783 0.99790 0.99796 0.99802 0.99809

62024 14524 57044 89585 12145

77346 11631 44547 11841 50260

94581 76379 32859 61264 55394

063 092 104 976 397

0.06574 0.06475 0.06375 0.06275 0.06175

88633 09943 30607 50633 70031

02623 92023 30434 15789 46086

48257 24928 01988 37280 63952

343 268 470 758 953

1.510 1.511 1.512 1.513 1.514

0.99815 0.99821 0.99827 0.99833 0.99838

24724 27322 19938 02571 75221

97548 92446 74695 85033 65198

11924 36636 SO542 95912 42198

274 332 912 947 118

0.06075 0.05976 0.05876 0.05776 0.05676

88812 06985 24560 41548 57959

19385 33809 87538 78816 05945

90658 01748 57464 94113 24248

160 769 281 053 072

1.515 1.516 1.517 1.518 1.519

0.99844 0.99849 0.99855 0.99860 0.99865

37887 90569 33265 65976 88701

57923 06943 56990 53793 44081

91859 86092 10456 00399 46683

188 495 612 163 784

0.05576 0.05476 0.05377 0.05277 0.05177

73801 89086 03823 18023 31695

67282 61243 86301 40981 23862

36836 97425 48297 08625 74620

851 545 399 609 716

1.520 1.521 1.522 1.523 1.524

0.99871 0.99876 0.99880 0.99885 0.99890

01439 04190 96954 79730 52517

75583 97023 58128 09621 03224

00717 79776 72136 42098 34913

231 634 872 089 328

0,05077 0.04977 0.04877 0.04777 0.04677

44849 57495 69644 81305 92488

33579 68814 28305 10835 15238

19672 94487 27218 23593 67036

613 284 360 598 388

1.525 1;526 1.527 1.528 1,529

0.99895 OI99899 0.99904 0.99908 0.99912

15314 68123 10941 43769 66606

91658 28645 68902 68148 83100

81616 03749 17990 40684 92265

285 180 729 234 762

0.04578 0.04478 0.04378 0.04278 0.04178

03203 13460 23270 32642 41586

40397 85239 48738 29915 27830

18782 17991 81854 05695 63073

371 291 166 871 262

1.530 1.531 1.532 1.533 1.534

0.99916 0.99920 0.99924 0.99928 0.99932

79452 82306 75169 58038 30915

71476 91989 04354 69286 48498

01592 10170 76285 79026 22220

427 755 152 436 463

0.04078 0.03978 0.03878 0.03778 0.03678

50112 58230 65951 73283 80238

41591 70343 13276 69617 38633

05868 64380 47406 42326 15178

899 513 277 008 390

1.535 1.536 1.537 1.538 1.539

OL99935 0.99939 0.99942 0.99946 0.99949

93799 46689 89585 22486 45393

04701 01607 03928 77374 88654

38256 91817 83506 53376 84360

819 592 202 306 752

0.03578 0.03478 0.03378 0.03279 0.03179

86825 93054 98935 04478 09693

19628 11943 14956 28078 50755

10734 52566 43115 63750 74831

312 435 073 505 796

1.540 1.541 1.542 1.543 1.544

0.99952 0.99955 0.99958 0.99961 0.99964

58306 61222 54144 37069 09999

05479 96555 31593 81300 17383

05600 95674 85726 62497 71251

596 180 242 095 832

0.03079 0.02979 0.02879 0.02779 0.02679

14590 19180 23471 27475 31200

82466 22720 71058 27051 90300

15762 05041 40314 98418 35423

248 568 858 526 217

1.545 1,546 1.547 1.548 1.549

0.99966 0.99969 0.99971 0.99974 0.99976

72932 25868 68807 01749 24694

12550 40506 75959 94615 73179

18609 75272 78656 35418 23886

586 821 660 249 150

0.02579 0.02479 0.02379 0.02279 0.02179

34658 37858 40810 43524 46010

60430 37097 19980 08784 03238

86673 66826 69885 69229 17647

867 971 184 328 934

1.550

0.99978 37641 89356 96389 761

0.02079 48278 03092 47364 391

ELEMENTARY TRANSCENDENTAL FUNCTIONS

173

CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS Table 4.6

sin

1.550 1.551 1.552 1.553 1.554

0.99978 0.99980 0.99982 0.99984 0.99985

37641 40591 33542 16495 89450

89356 21853 50374 55624 19308

96389 81488 86102 97539 85428

761 767 606 966 298

0.02079 0.01979 0.01879 0.01779 0.01679

48278 50338 52200 53874 55370

cos x 03092 08120 18116 32894 52286

47364 70061 76905 38564 05229

391 827 802 929 507

1.555 1.556 1.557 1.558 1.559

0.99987 0.99989 0.99990 0.99991 0.99993

52406 05363 48321 81281 04241

24131 53795 93 07 2470 43888

03543 91538 76575 74851 93030

342 676 277 093 623

0.01579 0.01479 0.01379 0.01279 0.01176

56698 57869 58891 59775 60532

76142 04329 36731 73245 13782

06628 52043 30323 09896 38778

284 433 849 874 533

1.560 1.561 1.562 1.563 1.564

0.99994 0.99995 0.99996 0.99996 0.99997

17202 20163 13125 96087 69050

29966 74406 66914 98192 59945

29574 75969 17856 36062 07529

517 172 344 758 731

0.01079 0.00979 0.00879 0.00779 0.00679

61170 61701 62133 62478 62744

58267 06636 58835 14822 74562

44582 34527 95443 93777 75597

392 146 014 062 546

1.565 1.566 1.567 1.568 1.569

0.99998 0.99998 0.99999 0.99999 0.99999

32013 84976 27939 60902 83866

44876 46689 60087 80775 05456

06142 03461 6934.8 72499 80873

794 318 142 201 162

0.00579 0.00479 0.00379 0.00279 0.00179

62943 63084 63176 63231 63258

38028 05200 76064 50611 28835

66597 72096 77045 46023 23243

372 784 359 436 059

1.570 1.571 1.572 1.573 1.574

0.99999 0.99999 0.99999 0.99999 0.99999

96829 99792 92755 75719 48682

31834 58612 85495 13185 43386

62021 83315 12082 15626 61164

053 895 337 285 539

+0.00079 -0.00020 -0.00120 -0.00220 -0.00320

63267 36732 36729 36714 36677

10733 03695 14450 21533 24944

32548 22583 59042 14087 45343

541 254 804 901 613

1.575 1.576 1.577 1.578 1.579

0.99999 0.99998 0.99998 0.99997 0.99996

11645 64609 07572 40536 63500

78803 23138 81096 58379 61693

15654 45523 16298 92137 35254

423 419 798 261 568

-0.00420 -0.00520 -0.00620 -0.00720 -0.00820

36608 36497 36334 36109 35811

24688 20771 13205 02006 87197

30802 68822 78129 97812 87324

109 280 029 142 647

1.580 1.581 1.582 1.583 1.584

0.99995 0.99994 0.99993 0.99992 0.99991

76464 79429 72395 55361 28327

98740 78223 09847 04315 73330

05255 58361 46545 16554 08844

179 895 499 408 324

-0.00920 -0.01020. -0.01120 -0.01220 -0.01320

35432 34961 34388 33702 32895

68808 46876 21448 92583 60348

26480 15451 74764 45294 88260

539 796 568 454 743

1.585 1.586 1.587 1.588 1.589

0.99989 0.99988 0.99986 0.99985 0.99983

91295 44263 87233 20204 43177

29595 86814 59691 63927 16226

56407 83504 04289 21344 24106

893 374 313 232 322

-0.01420 -0.01520 -0.01620 -0.01720 -0.01820

31956 30874 29641 28245 26678

24825 86108 44304 99538 51948

85219 38055 68973 20485 55400

553 737 475 440 452

1.590 1.591 1.592 1.593 1.594

0.99981 0.99979 0.99977 0.99975 0.99973

56151 59127 52105 35085 08068

34290 36823 43527 75103 53254

87198 68657 08066 24582 14867

158 422 646 972 933

-0.01920 -0.02020 -0.02120 -0.02220 -0.02320

24929 22987 20843 18488 15910

01692 48945 93900 36773 77799

56809 28070 92788 94801 98151

503 065 583 039 502

1.595 1.596 1.597 1.598 1.599

0.99970 0.99968 0.99965 0.99963 0.99960

71054 24042 67033 00029 23027

00681 41086 99171 00635 72179

50917 77790 11241 35248 99440

259 702 891 219 759

-0.02420 -0.02520 -0.02620 -0.02720 -0.02819

13101 10049 06745 03180 99342

17236 55365 92491 28945 65082

87068 65939 59282 11714 87922

552 492 234 764 093

1.600

0.99957 36030 41505 16434 211

r

E

[(-77111

For r>l..6 see Example 16. "-1.57079 63267 94896 61923 132 2-

-0.02919 95223 01288 72620 577

[(-79) 31

~=3.1415926535 89793 23846 264

174

ELEMENTARY TRANSCENDENTAL FUNCTIONS RADIX TABLE OF CIRCUL,AR SINES AND COSINES

Table 4.7 r

1

sin ~ 1 0 - n

11

cos 110-n

0.00000 00001 00000 0 0 0 0 0 00000

10 10 10 10

0.00000 00002 0.00000 00003 0.00000 00004 0.00000 00005 0.00000 00006

00000 00000 00000 0 0 0 0 0 0 0 0 0 0 00000 00000 0 0 0 0 0 00000 0 0 0 0 0 0.00000 00007 00000 00000 0.00000 00008 0 0 0 0 0 00000 0.00000 00009 0 0 0 0 0 00000

00000 00000 00000 00000 00000 00000 00000 00000

0.99999 0.99999 0.99999 0.99999 0.99999 0,99999 0.99999 0.99999 0.99999

99999 99999 99999 99999 99999 99999 99999 99999 99999

99999 99999 99999 99999 99999 99999 99999 99999 99999

99999 99998 99995 99992 99987 99982 99975 99968 99959

50000 00000 50000 00000 50000 00000 50000

50000

99999 99999 99999 99999 99999 99999 99999 99999 99999

99999 99999 99999 99999 99999 99999 99999 99999 99999

99950 99800 99550 99200 98750 98200 97550 96800 95950

00000 00000 00000 00000 00000 00000 00000 00000 00000

00000

2 3 4 5 6 7 8 9

10 10 10 10

1 2 3 4 5 6 7 8 9

9 9 9 9 9 9 9 9 9

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

00040 00050 00060 00069 00079 00089

00000 00000 00000 99999 99999 99999

00000 00000 00000 99999 99999 99999

00000 00000 00000 99999 99999 99999

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

1

8 8 8 8 8 8 8 8 8

0.00000 0.00000 0.00000 0.00000 0,00000 0.00000 0.00000 0.00000 0.00000

00099 00199 00299 00399 00499 00599 00699 00799 00899

99999 99999 99999 99999 99999 99999 99999 99999 99999

99999 99999 99999 99999 99999 99999 99999 99999 99999

99998 99987 99955 99893 99792 99640 99428 99147 98785

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99999 99999 99999 99999 99999 99999 99999 99999 99999

99999 99999 99999 99999 99998 99998 99997 99996 99995

95000 80000 55000 20000 75000 20000 55000 80000 95000

00000 00000 00000 00000 00000 00000 00000 00000 00000

7 7 7 7 7 7 7 7 7

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

00999 01999 02999 03999 04999 05999 06999 07999 08999

99999 99999 99999 99999 99999 99999 99999 99999 99999

99999 99999 99999 99998 99997 99996 99994 99991 99987

98333 86667 55000 93333 91667 40000 28333 46667 85000

0.99999 0.99999 0.99999 0.99999 0,99999 0.99999 0.99999 0.99999 0.99999

99999 99999 99999 99999 99999 99999 99999 99999 99999

99995 99980 99955 99920 99875 99820 99755 99680 99595

00000 00000 00000 00000 00000 00000 00000 00000 00000

00000 00000 00000 00000 00000 00000 00000 00000 00000

6 6 6 6 6 6 6 6 6

0.00000 0,00000 0.00000 0.00000 0,00000 0.00000 0.00000 0.00000 0,00000

09999 19999 29999 39999 49999 59999 69999 79999 89999

99999 99999 99999 99999 99999 99999 99999 99999 99999

99903 99866 99550 98933 97916 96400 94283 91466 87850

33333 66667 00000

99999 99999 99999 99999 99999 99999 99999 99999 99999

99500 98000 95500 92000 87500 82000 75500 68000 59500

00000 00000 00000 00000 00000 00000 00000 00000 00000

00000 00007 00034 00107 00260 00540

66667 00000

0.99999 0.99999 0,99999 0.99999 0.99999 0.99999 0,99999 0,99999 0,99999

5 5 5 5

0.00000 99999 0.00001 99999 0.00002 99999 o.oooo3 99999 0.00004 99999 0.00005 99999 0.00006 99999 0.00007 99999 0.00008 99999

99999 99998 99995 99989 99979 99964 99942 99914 99878

83333 33333 66666 66667 50000 00002 33333 33342 16666 66693 00000 00065 83333 33473 66666 66940 50000 00492

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99999 99998 19995 99992 99987 99982 99975 99968 99959

50000 00000 50000 00000 50000 00000 50000 00000 50000

00000 04167 00000 66667 00003 37500 00010 66667 00026 04167 00054 00000 00100 04167 00170 66667 00273 37500

0.00009 0.00019 0.00029 0.00039 0.00049 0.00059 0.00069 0.00079 0.00089

99833 98666 95500 89333 79166 64000 42833 14666 78500

33333 66666 00002 33341 66692 00064 33473 66939 00492

0.99999 0,99999 0.99999 0.99999 0.79999 0.99999 0.99999 0.99999 0.99999

99950 99800 99550 99200 9 m o 98200 97550 96800 95950

00000 00000 00000 00001

00416 06666 33749 06666 60416 39999 00416 06666 33749

2

3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9 1 2

3

4 5 6 7 8 9

10

5

6 7 8 9

5 5 5

5

1

2

3

4

1

3

0.00000 00010 00000 00000 00000 0.00000 00020 00000 00000 00000 0.00000 00030 00000 00000 00000

99999 99999 99999 99999 99999 99999 99999 99999 99998

33333

66667 00000

33333

34167 93333 02500 86667 70833 80000 39167 73333 07499

0.00099 99998 33333 34166 66665 = slO-n; cos T1O-n = 1 - 1,x~lO-zn; to

For >1O, sin 11O-n

oooaz

00005

OOOlD

00017 00027

01000 01707 02734

66667 66666 99990 66610 66450 99352 65033 63026 92619

0.99999 95000 00041 66666 52778

25~.

2

From C. E. Van Orstrand, Tables of tiie exponential function and of the circular sine and cosine to radian arguments, Memoirs of the National Academy of Sciences, vol. 14, Fifth Memoir. US. Government Printing Office, Washington, D.C., 1921 (with permission).

175

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS

Table 4.8

0 1 2 3 4

0.00000 +0.84147 +0.90929 +0.14112 -0.75680

00000 09848 74268 00080 24953

sin x 00000 07896 25681 59867 07928

00000 50665 69539 22210 25137

000 250 602 074 264

1.00000 +0.54030 -0.41614 -0,98999 -0,65364

00000 23058 68365 24966 36208

cos x 00000 68139 47142 00445 63611

00000 71740 38699 45727 91463

000 094 757 157 917

5 6 7 8 9

-0.95892 -0.27941 +0.65698 +0.98935 +0.41211

42746 54981 65987 82466 84852

63138 98925 18789 23381 41756

46889 87281 09039 77780 56975

315 156 700 812 627

~0.28366 21854 +0.96017 02866 +0.75390 22543 -0.14550 00338 -0.91113 02618

63226 50366 43304 08613 84676

26446 02054 63814 52586 98836

664 565 120 884 829

10 11 12 13 14

-0.54402 -0.99999 -0,53657 +0.42016 +0.99060

11108 02065 29180 70368 73556

89369 50703 00434 26640 94870

81340 45705 97166 92186 30787

475 156 537 896 535

-0.83907 +0.00442 +0.84385 +0.90744 +0.13673

15290 56979 39587 67814 72182

76452 88050 32492 50196 07833

45225 78574 10465 21385 59424

886 836 396 269 893

15 16 17 18 19

+0.65028 -0.28790 -0.96139 -0.75098 +0.14987

78401 33166 74918 72467 72096

57116 65065 79556 71676 62952

86582 29478 85726 10375 32975

974 446 164 016 424

-0.75968 -0.95765 -0.27516 +0.66031 +0.98870

79128 94803 33380 67082 46181

58821 23384 51596 44080 86669

27384 64189 92222 14481 25289

815 964 034 610 835

20 21 22 23 24

+0.91294 +0:83665 -0.00885 -0.84622 -0.90557

52507 56385 13092 04041 83620

27627 36056 90403 75170 06623

65437 03186 87592 63524 84513

610 648 169 133 579

+0.40808 -0,54772 -0.99996 -0.53283 +0.42417

20618 92602 08263 30203 90073

13391 24268 94637 33397 36996

98606 42138 12645 55521 97593

227 427 417 576 705

25 26 27 28 29

-0,13235 +0.76255 +0.95637 +0.27090 -0.66363

17500 84504 59284 57883 38842

97773 79602 04503 07869 12967

02890 73751 01343 01998 50215

201 582 254 634 117

+0.99120 +0.64691 -0.29213 -0.96260 -0,74805

28118 93223 88087 58663 75296

63473 28640 33836 13566 89000

59808 34272 19337 60197 35176

329 138 140 545 519

30 31 32 33 34

-0.98803 -0.40403 +0.55142 +0.99991 +0.52908

16240 76453 66812 18601 26861

92861 23065 41690 07267 20023

78998 00604 55066 14572 82083

775 877 156 808 249

+0.15425 +0.91474 +0.83422 -0.01327 -0.84857

14498 23578 33605 67472 02747

87584 04531 06510 23059 84605

05071 27896 27221 47891 18659

866 244 553 522 997

35 36 37 38 39

-0,42818 -0.99177 -0i64353 +0.29636 +0:96379

26694 88534 81333 85787 53862

96151 43115 56999 09385 84087

00440 73683 46068 31739 75326

675 529 567 230 066

-0.90369 -0.12796 +0.76541 +0.95507 +0.26664

22050 36896 40519 36440 29323

91506 27404 45343 47294 59937

75984 68102 35649 85758 25152

730 833 108 654 683

40 41 42 43 44

+0.74511 -0,15862 -0.91652 -0.83177 +0.01770

31604 26688 15479 47426 19251

79348 04708 15633 28598 05413

78698 98710 78589 28820 57780

771 332 899 958 795

-0.66693 -0.98733 -0.39998 +0.55511 +0.99984

80616 92775 53149 33015 33086

52261 23826 88351 20625 47691

84438 45822 29395 67704 22006

409 883 471 483 901

45 46 47 48 49

+0.85090 +0.90178 +0.12357 -0.76825 -0,95375

35245 83476 31227 46613 26527

34118 48809 45224 23666 59471

42486 18503 00406 79904 81836

238 329 153 497 042

+0.52532 -0.43217 -0.99233 -0,64014 +0.30059

19888 79448 54691 43394 25437

17729 84778 50928 69199 43637

69604 29495 71827 73131 08368

746 278 975 294 703

X

50 -0.26237 48537 03928 78591 439 +0.96496 60284 92113 27406 896 From C. E. Van Orstrand, Tables of the exponential function and of the circular sine and cosine to radian arguments, Memoirs of the National Academy of Sciences, vol. 14, Fifth Memoir. U.S. chwnment Printing Office, Washington, D.C., 1921 (with permission) for ~1100.

176.

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.8

CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS cos x

sin x

X

50 51 52 53 54

-0.26237 +0.67022 +0;98662 +0.39592 -0.55876

48537 91758 75920 51501 90488

03928 43374 40485 81834 51616

78591 73449 29658 18150 24581

439 435 757 339 787

+0.96496 +0.74215 -Oil6299 -0.91828 -0.62930

60284 41968 07807 27862 98328

92113 13782 95705 12118 63150

27406 53946 48100 89119 14772

896 738 333 973 785

55 56 57 58 59

-0.99975 -0,52155 +0.43616 +0.99287 +0.63673

51733 10020 47552 26480 80071

58619 86911 47824 84537 39137

83659 88018 95908 11816 88077

863 741 053 509 123

+0.02212 +0.85322 +0.89986 +0.11918 -0.77106

67562 01077 68269 01354 02229

61955 22584 69193 48819 75845

73456 11396 78650 28543 22938

356 968 300 584 744

60 61 62 63 64

-0.30481 -0.96611 -0.73918 +0.16735 +0.92002

06211 77700 06966 57003 60381

02216 08392 49222 02806 96790

70562 94701 86727 92152 68335

565 829 602 784 154

-0.95241 -0.25610 +0.67350 +0:98589 +0.39185

29804 16359 71623 65815 72304

15156 38267 23586 82549 29550

29269 44570 25288 69743 00516

382 121 783 864 171

65 66 67 68 69

+0.82682 -0i02655 -0.85551 -0.89792 -0.11476

86794 11540 99789 76806 48137

90103 23966 75322 89291 83187

46771 79446 25899 26040 22054

021 384 683 073 507

-0.56245 -0.99964 -0.51776 +0.44014 +0.99339

38512 74559 97997 30224 03797

38172 66349 89505 96040 22271

03106 96483 06565 70593 63756

212 045 339 105 155

70 71 72 73 74

+0.77389 +0:95105 +0.25382 -0.67677 -0.98514

06815 46532 33627 19568 62604

57889 54374 62036 87307 68247

09778 63665 27306 62215 37085

733 657 903 498 189

+0.63331 -0.30902 -0.96725 -0i73619 +0.17171

92030 27281 05882 27182 73418

86299 66070 73882 27315 30777

83233 70291 48729 96016 55609

201 749 171 815 845

75 76 77 78 79

-0.38776 +0.56610 +0.99952 +0.51397 -0.44411

16354 76368 01585 84559 26687

09430 98180 80731 87535 07508

43773 32361 24386 21169 36850

094 028 610 609 760

+0.92175 +0.82433 -0.03097 -0.85780 -0,89597

12697 13311 50317 30932 09467

24749 07557 31216 44987 90963

31639 75991 45752 85540 14833

230 501 196 835 703

80 81 82 83 84

-0.99368 -0.62988 +0.31322 +0.96836 +0.73319

86539 79942 87824 44611 03200

23375 74453 33085 00185 73292

18973 87856 15263 40435 16636

081 521 353 015 321

-0.11036 +0.77668 +0.94967 +0.24954 -0.68002

72438 59820 76978 01179 34955

39847 21631 82543 73338 87338

55811 15768 20471 12437 79542

787 342 326 735 720

85 86 87 88 89

-0.17607 -0.92345 -0.82161 +0.03539 +0.86006

56199 84470 78366 83027 94058

48587 04059 30822 33660 12453

07696 80260 54487 68362 22683

212 163 211 543 685

-0.98437 -0.38369 +0.56975 +0.99937 +0.51017

66433 84449 03342 32836 70449

94041 49741 65311 95124 41668

89491 84477 92000 65698 89902

821 893 851 442 379

90 91 92 93 94

+0.89399 +Oil0598 -0.77946 -0i94828 -0.24525

66636 75117 60696 21412 19854

00557 51156 15804 69947 67654

89051 85002 68855 23213 32522

827 021 400 104 044

-0.44807 -0i99436 -0,62644 +0.31742 +0.96945

36161 74609 44479 87015 93666

29170 28201 10339 19701 69987

15236 52610 06880 64974 60380

548 672 027 551 439

95 96 97 98 99

+0.68326 +0.98358 +0.37960 -0.57338 -0.99920

17147 77454 77390 18719 68341

36120 34344 27521 90422 86353

98369 85760 69648 88494 69443

958 773 192 922 272

+0.73017 -0.18043 -0.92514 -0.81926 +0.03982

35609 04492 75365 82452 08803

94819 91083 96413 91459 93138

66479 95011 89170 25267 89816

352 850 475 566 180

100

-0.50636 56411 09758 79365 656

+0.86231 88722 87683 93410 194

177

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS 2

sin z

cos 2

5

sin z

Table 4.8 cos z

100 101 102 103 104

-0.50636 +0.45202 +O. 99482 -0,62298 -0.32162

564 579 679 863 240

+0.86231 +0.89200 +O. 10158 -0.78223 -0.94686

887 487 570 089 801

150 151 152 153 154

-0.71487 +0.20214 +0.93332 +0.80640 -0.06192

643 988 052 058 034

+0.69925 +0.97935 +Or35904 -0.59136 -0.99808

081 460 429 968 109

105 106 107 108 109

-0.97053 -0.72714 +0.18478 +0.92681 +0.81674

528 250 174 851 261

-0.24095 +0.68648 +0.98277 +0.37550 -0.57700

905 655 958 960 218

155 156 157 158 159

-0.87331 -0.88178 -0.07954 +0.79582 +0.93951

198 462 854 410 973

-0.48716 +0.47165 +0.99683 +0.60552 -0,34249

135 229 099 787 478

110 111 112 113 114

-0.04424 -0.86455 -0.88999 -0.09718 +OI78498

268 145 560 191 039

-0.99902 -0.50254 +0.45596 +0.99526 +Or61952

081 432 910 664 061

160 161 162 163 164

+0.21942 -0.70240 -0.97845 -0.35491 +0.59493

526 779 035 018 278

-0.97562 -0.71177 +0.20648 +0.93490 +0.80377

931 476 223 040 546

115 116 117 118 119

~0.94543 533 +0.23666 139 -0.68969 794 -0.98195 217 -0.37140 410

-0.32580 -0.97159 -0.72409 +O. 18912 +0.92847

981 219 720 942 132

165 166 167 168 169

+0.99779 +0.48329 -0.47555 -0.99717 -0,60199

728 156 019 329 987

-0.06633 -0.87545 -0i87968 -0.07513 +Or79849

694 946 859 609 619

120 121 122 123 124

+0.58061 +0.99881 +0.49871 -0.45990 -0.99568

118 522 315 349 699

+0.81418 -0.04866 -0,86676 -0,88796 -0.09277

097 361 709 891 620

170 171 172 173 174

+0.34664 +0.97659 +0.70865 -0.21081 -0.93646

946 087 914 053 197

+0.93799 +0.21510 -0I70555 -0.97752 -0.35076

475 527 101 694 911

125 126 127 128 129

-0.61604 +0.32999 +O. 97263 +0.72103 -0.19347

046 083 007 771 339

+O. 78771 +0.94398 +O. 23235 -0.69289 -0.98110

451 414 910 582 552

175 176 177 178 179

-0.80113 +0.07075 +0.87758 +0.87757 +0.07072

460 224 979 534 217

+0.59848 +0.99749 +0.47941 -0.47943 -0.99749

422 392 231 877 605

130 131 132 133 134

-0.93010 -0.81160 +0.05308 +0.86896 +0.88592

595 339 359 576 482

-0.36729 +0.58420 +0.99859 +0.49487 -0.46382

133 882 007 222 887

180 181 182 183 184

-0.80115 -0,93645 -0.21078 +0.70868 +0.97658

264 140 107 041 438

-0.59846 +0.35079 +0.97753 +0.70552 -0.21513

007 734 329 964 471

135 136 137 138 139

+0.08836 -0.79043 -0.94251 -0.22805 +0.69608

869 321 445 226 013

-0.99608 -0.61254 +0.33416 +0.97364 +0.71796

784 824 538 889 410

185 186 187 188 189

+0.34662 -0.60202 -0.99717 -0,47552 +0.48331

118 394 102 367 795

-0.93800 -0.79847 +O. 07516 +0.87970 +0.87544

520 804 615 293 489

140 141 142 143 144

+0.98023 +0.36317 -0I58779 -0.99834 -0.49102

966 137 501 536 159

-0.19781 -0.93172 -0.80900 +0.05750 +0.87114

357 236 991 253 740

190 191 192 193 194

+0.99779 +0.59490 -0.35493 -0.97845 -0.70238

928 855 836 657 633

+0.06630 -0.80379 -0.93488 -0.20645 +0.71179

686 339 971 273 593

145 146 i47 148 149

+0.46774 +0.99646 +0160904 -0.33833 -0.97464

516 917 402 339 865

+0.88386 +0.08395 -0.79313 -0.94102 -0,22374

337 944 642 631 095

195 196 197 198 199

+0.21945 +0.93953 +0.79580 -0.07957 -0.88179

467 006 584 859 884

+0.97562 +O. 34246 -0.60555 -0.99682 -0.47162

270 646 186 859 571

150

-0.71487 643

+0.69925 081

200

-0,87329 730

+0.48718 768

178

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.8 X

CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS sin x

cos z

X

sin z

CO8

x

200 201 202 203 204

-0.87329 -0.06189 +0.80641 +O. 93330 +0.20212

730 025 841 970 036

+0.48718 +0.99808 +0.59134 -0.35907 -0,97936

768 296 538 242 069

250 251 252 253 254

-0.97052 -0.32159 +O. 62301 +0.99482 +0.45199

802 386 221 373 890

+On24098 +0.94687 +O. 78221 -0.10161 -0.89201

831 771 211 569 850

205 206 207 208 209

-0.71489 -0,97464 -0.33830 +0.60906 +0.99646

751 190 503 793 664

-0.69922 +0.22377 +0.94103 +0.79311 -0.08398

926 033 651 806 947

255 256 257 258 259

-0.50639 -0.99920 -0.57335 +0.37963 +0.98359

163 803 717 563 318

-0.86230 -0.03979 +0.81930 +0.92513 +0.18040

361 076 553 609 080

210 211 212 213 214

+0.46771 -0,49104 -0.99834 -0.56777 +0.36319

852 785 709 062 945

-0.88387 -0.87113 -0.05747 +0.80902 +0.93171

747 260 243 763 141

260 261 262 263 264

+0.68323 -0.24528 -0.94829 -0.77944 +0.10601

970 121 171 719 749

-0.73019 -0.96945 -0.31740 +0.62646 +0.99436

416 197 012 794 427

215 216 217 2 18 219

+0.98024 +0.69605 -0.22808 -0,94252 -0.79041

562 849 161 453 474

+0.19778 -0,71798 -0.97364 -0.33413 +0.61257

403 508 202 697 207

265 266 267 268 269

+0.89401 +0.86005 +0.03536 -0.82183 -0.92344

017 403 818 501 688

+0.44804 -0.51020 -0.99937 -0.56972 +0.38372

667 297 435 556 628

220 22 1 222 223 224

+0.08839 +0,88593 +0.86895 +0.05305 -0.81162

871 880 084 349 100

+0.99608 +Q.46380 -0.49489 -0.99859 -0.58416

517 216 841 167 435

270 271 272 273 274

-0.17604 +0.73321 +0.96835 +0.31320 -0.62991

595 082 694 015 141

+0.98438 tO.68000 -0.24956 -0.94968 -0,77666

195 139 931 714 699

225 226 227 228 229

-0.93009 -0.19344 +0.72105 +0.97262 +0.32996

488 382 860 306 237

+O. 36731 +0.98111 +0.69287 -0,23238 -0,94399

937 135 409 842 409

275 276 277 278 279

-0.99388 -0.44408 +0.51400 +0.99952 +0.56608

533 566 431 109 279

+0.11041 +0.89598 +0.85778 +0.03094 -0,82434

720 433 760 490 840

230 231 232 233 234

-0.61606 -0.99568 -0.45987 +0.49873 +O. 99881

420 419 672 928 669

-0,78769 +0.09280 +0.88798 +0.86675 +O. 04863

594 622 277 206 350

280 281 282 283 284

-0.38780 -0.98515 -0.67674 +0.25385 +0.95106

942 144 976 252 397

-0.92173 -0.17168 +0.73621 +0.96724 +0.30899

958 765 312 294 406

235 236 237 238 239

+O. 58058 -0.37143 -0.98195 -0.68967 +0.23669

664 209 787 611 068

-0.81419 -0.92846 -0.18909 +0.72411 +0.97158

847 012 982 799 506

285 286 287 288 289

+0.77387 -0.11481 -0.89794 -0.85550 -0.02652

159 476 095 437 102

-0,63334 -0,99338 -0,44011 +0.51779 +0.99964

253 692 595 559 826

240 241 242 243 244

+0.94544 +0.78496 -0.09721 -0I89000 -0.86453

515 171 191 935 630

+0.32578 -0.61954 -0.99526 -0.45594 +0.50257

131 428 371 228 038

290 291 292 293 294

+0.82684 +0.92001 +0.16732 -0.73920 -0.96610

563 423 598 100 999

+0.56242 -0.39188 -0.98590 -0i67348 +0.25813

893 496 163 488 076

245 246 247 248 249

-0.04421 +0.81676 +O. 92680 +0.18475 -0.72716

256 000 719 212 319

+0.99902 +0.57697 -0.37553 -0,98278 -0,68646

215 756 754 515 463

295 296 297 298 299

-0.30478 +0.63676 +0.99286 +0.43613 -0.52157

191 125 906 763 672

+0.95242 +O,77106 -0.11921 -0.89987 -0.85320

217 103 006 997 439

250

-0.97052 802

+0.24098 831

300

-0.99975 584

-0.02209 662

I

I

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS

x

sin z

cos x

z

sin z

179 Table 4.8 cos 5

300 301 302 303 304

-0.99975 -0.55876 +0.39595 +O. 98663 +0.67020

584 405 283 250 680

-0.02209 +0.82932 +0.91827 +O. 16296 -0.74217

662 668 085 104 440

350 351 352 353 354

-0.95893 -0.75678 +O. 14114 +0.90930 +0.84145

283 279 985 997 470

-0.28363 +0.65366 +O. 98998 +0.41611 -0.54032

328 643 824 943 767

305 306 307 308 309

-0.26240 -0.95376 -0.76823 +0.12360 +0.90180

394 171 536 304 137

-0.96495 -0.30056 +0.64016 +0.99233 +0.43215

812 379 750 174 076

355 356 357 358 359

-0.00003 -0.84148 -0.90928 -0.14109 +0.75682

014 727 488 017 220

-1.00000 -0.54027 +0.41617 +0.98999 +0.65362

000 694 425 675 081

310 311 312 313 314

+O. 85088 +0.01767 -0,83179 -0.91650 -0.15859

769 179 148 949 291

-0.52534 -0.99984 -0.55508 +0.40001 +O. 98734

764 384 823 294 406

360 361 362 363 364

+0.95891 +0.27938 -0.65700 -0.98935 -0.41209

572 655 932 386 102

-0.28369 -0.96017 -0.75388 +0.14552 +0.91114

109 871 245 986 268

315 316 317 318 319

+0.74513 +0.96378 +0.29633 -0.64356 -0.99177

326 735 979 121 500

+0.66691 -0.26667 -0.95508 -0.76539 +0.12799

560 199 258 465 359

365 366 367 368 369

+0.54404 +0.99999 +0.53654 -0.42019 -0.99061

640 007 748 439 148

+0.83905 -0.00445 -0.84387 -0.90743 -0.13670

513 584 013 412 736

320 321 322 323 324

-0.42815 tO.52910 +0.99991 +0.55140 -0.40406

543 827 226 153 522

+0.90370 +0.84855 +0.01324 -0.83423 -0.91473

511 433 661 998 018

370 371 372 373 374

-0.65026 +0.28793 +0.96140 +0.75096 -0,14990

494 218 579 734 701

+O. 75970 +0.95765 +0.27513 -0.66033 -0.98670

752 080 436 935 010

325 326 327 328 329

-0.98803 -0.66361 +0.27093 +0.95638 +0.76253

627 133 481 473 895

-0.15422 +0.74807 +0.96259 +0.29210 -0.64694

167 753 770 998 231

375 376 377 378 379

-0.91295 755 -0.83663 913 +0.00888 145 +0.84623 647 ~0.90556 557

-0.40805 +0.54775 +0.99996 +0.53280 -0,42420

454 448 056 751 631

330 331 332 333 334

-0.13238 -0.90559 -0.84620 -0.00882 +0.83667

163 115 434 117 215

-0.99119 -0.42415 +0.53285 +OS99996 +0.54770

882 171 853 109 404

380 381 382 383 384

+0.13232 -0,76257 -0,95636 -0,27087 +0.66365

187 795 712 677 643

-0.99120 -0,64689 +0.29216 +0.96261 +0.74803

680 634 764 403 752

335 336 337 338 339

+0.91293 +O. 14984 -0.75100 -0,96138 -0.28787

295 741 715 920 445

-0.40810 -0.98870 -0.66029 +0.27519 +O. 95766

958 914 407 232 816

385 386 387 388 389

+0.98802 +0.40401 -0.55145 -0.99991 -0,52905

697 007 183 146 711

-0.15428 -0,91475 -0.83420 +0.01330 +0.84858

123 454 674 689 622

340 341 342 343 344

+0.65031 +O. 99060 +0.42013 -0.53659 -0.99999

074 323 968 836 034

+0.75966 -0.13676 -0.90745 -0.84383 -0.00439

831 708 945 778 555

390 391 392 393 394

+0.42820 +0.99178 +0.64351 -0.29639 -0.96380

991 271 506 737 342

+0.90367 +0.12793 -0.76543 -0.95506 -0.26661

930 379 345 471 388

345 346 347 348 349

-0.54399 +O. 41214 +O. 98936 +0.65696 -0.27944

582 595 263 387 444

+0.83908 +O. 91111 +O. 14547 -0.75392 -0.96016

793 784 021 206 186

395 396 397 398 399

-0.74509 +0.15865 +0:91653 +0.83175 -0.01773

306 243 361 801 206

+0.66696 +0.98733 +0.39995 -0.55513 -0,99984

052 450 769 837 277

350

-0.95893 283

-0.28363 328

400

-0,85091 936

-0.52529 634

180

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.8 X

CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS cos x

sin x

X

sin x

cos 2

400 401 402 403 404

-0.85091 -0,90177 -0.12354 +0.76827 +0.95374

936 532 321 396 359

-0.52529 +0.43220 +0.99233 +0.64012 -0.30062

634 513 919 118 129

450 451 452 453 454

-0.68328 -0.98358 -0.37957 +0.57340 +0.99920

373 231 985 657 563

-0.73015 +0.18046 +0.92515 +0.81927 -0.03985

296 010 898 096 100

405 406 407 408 409

+0.26234 -0.67025 -0.98662 -0.39589 +0.55881

577 155 268 747 405

-0.96497 -0.74213 +0.16302 +0.91829 +0.82929

394 399 052 472 299

455 456 457 458 459

+0.50633 -0,45205 -0.99482 -0.62296 +0.32165

965 268 985 505 095

-0.86233 -0.89199 -0.10155 +0.78224 +0.94685

414 124 572 967 832

410 411 412 413 414

+0.99975 +0.52152 -0.43619 -0.99287 -0.63671

451 528 188 624 476

-0.02215 -0.85323 -0.89985 -0.11915 +0.77109

689 583 368 021 942

460 461 462 463 464

+0.97054 +0.72712 -0.18481 -0.92682 -0.81672

255 181 137 982 521

+0.24092 -0.68650 -0.98277 -0.37548 +0.57702

979 847 401 166 680

415 416 417 418 419

+0.30483 +0.96612 +0.73916 -0.16738 -0.92003

933 555 039 542 785

+0.95240 +0.25807 -0.67352 -0.98589 -0,39182

379 251 944 154 950

465 466 467 468 469

+0.04427 +0.86456 +0.88998 +0.09715 -0.78499

279 660 186 190 906

+0.99901 +0.50251 -0.45599 -0.99526 -0.61949

948 826 593 957 695

420 421 422 423 424

-0,82681 +0.02658 +0.85553 +0.89791 +0.11475

172 129 559 441 487

+0.56247 +0.99964 +0.51774 -0.44017 -0.99339

878 666 401 009 384

470 471 472 473 474

-0.94542 -0I23663 +0.68971 +0.98194 +0.37137

551 211 977 647 611

+0.32583 +OI97159 +0.72407 -0.18915 -0.92848

830 932 641 902 252

425 426 427 428 429

-0.77390 -0.95104 -0.25379 +0.67679 +0.98514

977 534 421 415 108

-0.63329 +0.30905 +0.96725 +0.73617 -0.17174

587 140 824 232 704

475 476 477 478 479

-0.58063 -0.99881 -0.49868 +0.45993 +0.99568

573 376 703 026 978

-0.81416 +0.04869 +0.86678 +0.88795 +0.09274

347 372 212 504 619

430 431 432 433 434

+0.38775 -0i56613 -0.99951 -0.51395 +0.44413

385 249 922 260 968

-0.92176 296 -0,82431 427 +O. 03100'516 +0.85781 859 +0.89595 756

480 481 482 483 484

+0.61601 -0.33001 -0.97263 -0.72101 +O. 19350

671 928 707 682 297

-0.78773 -0.94397 -0.23232 +0.69291 +O. 98109

308 419 978 756 969

435 436 437 438 439

+0.99389 +0.62986 -0.31325 -0.96837 -0.73316

198 458 741 198 982

+0.11035 -0.77670 -0,94966 -0.24951 +0.68004

728 497 826 093 560

485 486 487 488 489

+0.93011 +0.81158 -0.05311 -0.86898 -0.88591

702 578 369 067 083

+0.36726 -0.58423 -0.99858 -0.49484 +0.46385

329 328 847 603 557

440 441 442 443 444

+0.17610 +0.92347 +0.82180 -0.03542 -0.86008

529 001 066 843 478

+0.98437 +0.38367 -0.56977 -0.99937 -0,51015

134 061 511 222 112

490 491 492 493 494

-0.08833 +0.79045 +O. 94250 +0.22802 -0.69610

866 167 438 291 177

+0.99609 +0.61252 -0.33419 -0.97365 -0.71794

050 441 379 577 312

445 446 447 448 449

-0.89398 -0.10595 +0.77948 +0.94827 +0.24522

316 754 495 257 276

+0.44810 +0.99437 +0.62642 -0.31745 -0.96946

056 066 095 729 676

495 496 497 498 499

-0.98023 -0i36314 +0.58781 +0.99834 +0.49099

370 328 939 363 533

+0.19784 +0:93173 +0.80899 -0.05753 -0.87116

312 331 219 262 220

450

-0.68328 373

-0,73015 296

500

-0.46777 181

-0.88384 927

I

181

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS 5

sin z

cos 2

sin z

z

500 501 502 503 504

-0.46777 -0.99647 -0.60902 +0.33836 +0.97465

181 170 011 176 539

-0.88384 -0.08392 +0.79315 +0.94101 tO.22371

927 940 478 611 157

550 551 552 553 554

505 506 507 508 5 09

+0.71485 -0.20217 -0.93333 -0.80638 +0.06195

535 940 135 275 042

-0.69927 -0.97934 -0.35901 +0.59139 +0.99807

236 850 615 399 923

555 556 55 7 558 559

510 511 512 513 514

+0.87332 +0.88177 +0.07951 -0.79584 -0,93950

667 040 849 235 941

+0.48713 -0.47167 -0.99683 -0,60550 +0.34252

502 887 339 389 310

515 516 517 518 519

-0,21939 +0.70242 +0.97844 +0.35488 -0,59495

585 924 413 199 701

+0.97563 +0.71175 -0.20651 -0.93491 -0.80375

520 521 522 523 524

-0.99779 -0.48326 +0.47557 +0.99717 +0.60197

528 517 670 555 580

525 526 527 528 529

-0.34667 -0.97659 -0.70863 +0.21084 +0.93647

530 531 532 533 534

-0.21948 -0.93954 -0.79578 +O. 07960 +0.88181

Table 4.8 CO8

z

408 038 759 864 305

-0.97561 -0.34243 +0.60557 +O. 99682 +0.47159

608 814 585 620 913

87328 261

-0.80643 623 -0.93329 888 -0.20209 084

-0.48723: -0.99808 -0.59132 +0.35910 +0.97936

400 483 107 055 678

560 561 562 563 564

+0.71491 +0.97463 +OS 33827 -0.60909 -0.99646

859 516 666 184 411

~0.69920 771 -0.22379 971 -0.94104 671 -0.79309 970 +O. 08401 951

593 358 172 110 753

565 566 567 568 5 69

-0.46769 +0.49107 +0.99834 +0.58774 -0,36322

187 411 883 623 754

+0.88389 +0.87111 +0.05744 -0.80904 -0.93170

157 780 234 534 046

+0.06636 +0.87547 +0.87967 +0.07510 -0.79851

701 403 426 603 433

570 571 572 573 5 74

-0.98025 -0.69603 +O. 22811 +0.94253 +O. 79039

158 684 096 460 628

-0.19775 +0.71800 +0.97363 +0.33410 -0.61259

448 607 514 856 589

773 735 787 000 255

-0.93798 -0.21507 +0.70557 +0.97752 +0.35074

430 583 237 059 088

575 576 577 578 579

-0,08842 -0.88595 -0.86893 -0.05302 +0.81163

874 278 592 338 861

-0.99608 -0.46377 +0.49492 +0.99859 +0.58415

251 546 461 327 989

+0.80111 655 -0,07078 230 -0.87760 424 -0.87756 088 -0.07069 210

-0.59850 -0.99749 -0.47938 +0.47946 +0.99749

837 179 586 522 818

580 581 582 583 584

+0.93008 +0.19341 -0.72107 -0.97261 -0.32993

380 424 948 606 391

-0.36734 -0.98111 -0.69285 +0.23241 +0.94400

740 719 235 774 403

+0.80117 068

592 557 965 828 415

585 586 588 589

+0.61608 +On99568 +0.45984 -0.49876 -0.99881

795 139 996 541 816

+0.78767 -0.09283 -0.88799 -0.86673 -0.04860

737 623 663 702 339

+o.

+On 06186 016

535 536 537 538 539

+0.21075 160 -0.70870 168 -0.97657 790

+0.59843 -0.35082 -0.97753 -0.70550 +0.21516

540 541 542 543 544

-0.34659 +0.60204 +0.99716 +0.47549 -0.48334

290 801 876 715 434

+0.93801 +0.79845 -0.07519 -0,87971 -0.87543

565 989 621 726 032

590 591 592 593 594

-0.58056 +0.37146 +0.98196 +OS 68965 -0.23671

210 008 357 428 997

+0.81421 +0.92844 +0.18907 -0.72413 -0.97157

597 893 022 878 792

545 546 547 548 549

-0.99780 -0.59488 +0.35496 +0.97846 +0.70236

128 432 654 280 487

-0.06627 +0.80381 +0.93487 +0.20642 -0.71181

678 133 901 324 710

595 596 597 598 599

-0.94545 -0.78494 +O. 09724 +0.89002 +0.86452

497 304 191 309 115

-0.32575 +0.61956 +O. 99526 +0.45591 -0.50259

281 794 078 545 644

550

-0.21948 408

-0.97561 608

600

+0.04418 245

+O. 93644 083

501

-0,99902 348

182

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.8 X

CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS sin x

cos

x

X

cos

sin x

x

600 601 602 603 604

+0.04418 -0.81677 -0.92679 -0.18472 +0.72718

245 739 586 249 389

-0.99902 -0.57695 +0.37556 +0.98279 +0.68644

348 294 547 072 271

650 651 652 653 654

+0.30475 -0.63678 -0.99286 -0.43611 +0.52160

320 449 546 050 244

-0.95243 -0.77104 +0.11923 +0.89989 +0.85318

136 183 999 312 866

605 606 607 608 609

+0.97052 +0.32156 -0.62303 -0.99482 -0.45197

075 532 579 067 201

-0.24101 -0.94688 -0.78219 +0.10164 +0.89203

756 740 333 568 212

655 656 657 658 659

+0.99975 +0.55873 -0.39598 -0.98663 -0.67018

651 905 051 742 443

+0.02206 -0.82934 -0,91825 -0.16293 +0.74219

648 352 891 130 460

610 611 612 613 614

+0.50641 +0.99920 +0.57333 -0.37966 -0.98359

763 923 248 351 862

+0.86228 +0.03976 -0.81932 -0.92512 -0.18037

834 064 281 465 115

660 661 662 663 664

+0.26243 +0.95377 +0.76821 -Oil2363 -0.90181

303 077 607 295 440

+0.96495 +0.30053 -0.64019 -0,99232 -0.43212

021 504 066 802 358

615 616 617 618 619

-0.68321 +0.24531 +0.94830 +0.77942 -0.10604

769 043 128 830 746

+0.73021 +0.96944 +0.31737 -0,62649 -0,99436

475 458 153 144 107

665 666 667 668 669

-0.85087 -0.01764 +0.83180 +0.91649 +0.15856

185 165 821 743 314

+0.52537 +0.99984 +0.55506 -0.40004 -0.98734

329 437 315 057 884

620 621 622 623 624

-0.89402 -0i86003 -0.03533 +0.82185 +0.92343

368 865 805 218 531

-0.44801 +0.51022 +0.99937 +0:56970 -0.38375

972 890 542 079 412

670 671 672 673 674

-0.74515 -0.96377 -0.29631 +0.64358 +0.99177

337 931 100 428 114

-0.66689 +0:26670 +0.95509 +0:76537 -0.12802

314 104 151 525 348

625 626 627 628 629

+0.17601 -0.73323 -0.96834 -0.31317 +0.62993

627 132 941 153 482

-0.98438 -0.67997 +0.24959 +0.94969 +0.77664

726 929 850 658 801

675 676 677 678 679

+0.42812 -0.52913 -0.99991 -0,55137 +0.40409

819 384 266 639 279

-0.90371 -0,84853 -0.01321 +0.83425 +0.91471

802 838 646 660 800

630 631 632 633 634

+0.99388 +0.44405 -0.51403 -0.99952 -0.56605

200 865 017 202 794

-0.11044 -0.89599 -0.85777 -0.03091 +0.82436

716 772 210 477 546

680 681 682 683 684

+0.98804 +0.66358 -0.27096 -0.95639 -0.76251

092 878 382 354 945

+0.15419 -0.74809 -0I96258 -0.29208 +0.64696

188 754 953 115 529

635 636 637 638 639

+0.38783 +0.98515 +0.67672 -0,25388 -0.95107

721 661 757 168 328

+0.92172 +0.17165 -0li3623 -0.96723 -0I30896

789 795 352 528 539

685 686 687 688 689

+0.13241 +0.90560 +0.84618 +0.00879 -0,83668

151 393 828 102 866

+0.99119 +0.42412 -0.53288 -0.99996 -0.54767

483 441 404 136 882

640 641 642 643 644

-0.77385 +0.11484 +0.89795 +0.85548 +0.02649

250 470 421 876 089

+0.63336 +0.99338 +OI44008 -0.51782 -0.99964

586 346 889 138 905

690 691 692 693 694

-0.91292 -0.14981 +0.75102 +0.96138 +0.28784

065 760 706 090 558

+0.40813 +0.98871 +0.66027 -0.27522 -0,95767

710 365 143 130 684

645 646 647 648 649

-0.82686 -0.92000 -0.16729 +o.i3922 +0.96610

259 241 626 130 221

-0.56240 +0.39191 +0.98590 +0.67346 -0.25815

400 270 667 260 988

695 696 697 698 699

-0.65033 -0.99059 -0.42011 +0.53662 +0.99999

364 911 233 379 047

-0.75964 +0.13679 +0.90747 +0.84382 +0.00436

871 694 211 161 541

650

+0.30475 320

-0.95243 136

700

+0.54397 052

-0,83910 433

183

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS 5

sin x

cos x

X

sin x

Table 4.8 cos 5

700 701 702 703 704

+0.54397 -0.41217 -0,98936 -0.65694 +0.27947

052 342 702 115 339

-0.83910 -0.91110 -0.14544 +0.75394 +0.96015

433 541 039 186 344

750 751 752 753 754

+0.74507 -0.15868 -0.91654 -0.83174 +0.01776

295 219 566 127 220

-0.66698 -0.98732 -0.39993 +0.55516 +0.99984

298 971 006 345 224

705 706 707 708 709

+0.95894 +0.75676 -0.14117 -0.90932 -0.84143

137 309 969 251 841

+0.28360 -0.65368 -0.98998 -0.41609 +0.54035

437 925 399 202 304

755 756 757 758 759

+0.85093 +0.90176 +0.12351 -0.76829 -0.95373

519 229 330 325 453

+0.52527 -0.43223 -0.99234 -0.64009 +0.30065

069 231 292 802 004

710 711 712 713 714

+O. 00006 +0.84150 +0.90927 +0,14106 -0,75684

029 356 234 032 190

+l. 00000 +0.54025 -0.41620 -0.99000 -0.65359

000 157 166 100 799

760 761 762 763 764

-0,26231 +0.67027 +0.98661 +0.39586 -0.55883

668 392 776 979 905

+0.96498 +0.74211 -0.16305 -0.91830 -0.82927

184 379 026 665 614

715 716 717 718 719

-0.95890 -0,27935 +0.65703 +0.98934 +0.41206

717 761 205 947 355

+0.28372 +0.96018 +0.75386 -0.14555 -0.91115

000 713 264 968 511

765 766 767 768 769

-0.99975 -0I52149 +0.43621 +0:99287 +0.63669

384 956 901 983 152

+0.02218 +0.85325 +0.89984 +0.11912 -0.77111

703 155 053 028 861

720 721 722 723 724

-0.54407 -0.99998 -0.53652 +0.42022 +0.99061

170 994 204 174 560

-0.83903 +0.00448 +0.84388 +0.90742 +0.13667

873 599 631 145 750

770 771 772 773 774

-0.30486 -0.96613 -0.73914 +0.16741 +0.92004

804 333 009 514 966

-0.95239 -0,25804 t0.67355 +0.98588 +0.39180

460 339 173 649 176

725 726 727 728 729

+0.65024 -0.28796 -0.96141 -0.75094 +0.14993

204 105 408 744 682

-0.75972 -0.95764 -0.27510 +0.66036 +0.98869

712 212 538 198 558

775 776 777 778 779

+0.82679 -0.02661 -0.85555 -0.89790 -0.11472

477 142 119 114 492

-0,56250 -0.99964 -0.51771 +0.44019 +0.99339

370 585 822 716 730

730 731 732 733 734

+0.91296 985 +0.83662 262 -0~00891160 -0.84625 253 -0.90555 279

+0.40802 -0.54777 -0.99996 -0.53278 +0.42423

702 970 029 200 360

780 781 782 783 784

+0.77392 +0.95103 +0.25376 -0,67681 -0,98513

886 602 505 634 591

007 +0.63327 -0.30908 255

735 736 737 738 739

-0.13229 +0.76259 +0.95635 +0.27084 -0.66367

199 745 831 775 898

+0.99121 +0.64687 -0,29219 -0.96262 -0,74801

079 335 647 220 752

785 786 787 788 789

-0,38772 +0.56615 +0.99951 +0.51392 -0.44416

606 733 829 674 668

+0.92177 +0.82429 -0.03103 -0,85783 -0.89594

465 720 529 408 417

740 741 742 743 744

-0.98802 -0.40398 +0.55147 +0.99991 +0.52903

232 250 697 106 153

+0.15431 +0.91476 +0.83419 -0,01333 -0,84860

102 672 011 703 217

790 791 792 793 794

-0.99389 -0.62984 +0.31328 +0.96837 +0.73314

531 117 604 950 932

-0.11032 +0.77672 +0.94965 +0.24948 -0.68006

732 396 881 174 770

745 746 747 748 749

-0.42823 -0.99178 -0.64349 +0.29642 +0.96381

715 657 199 616 146

-0,90366 -0.12790 +0.76545 +0.95505 +0.26658

639 390 285 577 483

795 796 797 798 799

-0.17613 -0.92348 -0.82178 +0.03545 +0.86010

497 158 349 855 016

-0.98436 -0.38364 +0.56979 +0.99937 +0.51012

603 277 988 115 519

750

+0.74507 295

-0,66698 298

800

+0.89396 965

-0.96726 589 -0i73615 192 +0.17177 673

-0.44812 751

184

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.8 5

800 801 802 803 804

CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS sin x

+0.89396 +0.10592 -0.77950 -0.94826 -0.24519

cos 5

965 756 384 300 354

sin

cos 2

2

-0.44812 -0.99437 -0.62639 +O. 31748 +0.96947

751 385 745 587 415

850 851 852 853 854

+0.98022 +0.36311 -0.58784 -0.99834 -0.49096

773 519 378 189 907

-0.19787 -0.93174 -0.80897 +0.05756 +0.87117

267 426 447 271 700

237 975 042 368 112

855 856 857 858 859

+0.46779 +0.99647 +0.60899 -0.33839 -0.97466

845 423 620 013 214

+0.88383 +0.08389 -0.79317 -0.94100 -0.22368

517 936 314 591 219

805 806 807 808 809

+0.37955 1 9 6 -0.57343 1 2 6 -0.99920 443

+0.73013 -0,18048 -0.92517 -0.81925 +0.03988

810 811 812 813 814

-0.50631 +0.45207 +0.99483 +0.62294 -0.32167

365 956 291 147 949

+0.86234 +0.89197 +0.10152 -0.78226 -0.94684

940 762 573 845 862

860 861 862 863 8 64

-0.71483 +0.20220 +0.93334 +0.80636 -0.06198

427 893 217 493 051

+0.69929 +0.97934 +0.35898 -0.59141 -0.99807

390 241 802 830 736

815 816 817 818 819

-0.97054 -0.72710 +0.18484 +0.92684 +0.81670

981 111 099 114 782

-0.24090 +0.68653 +0.98276 +0.37545 -0.57705

054 039 844 372 142

865 866 867 868 869

-0.87334 -0.88175 -0.07948 +0.79586 +0.93949

135 618 845 060 908

-0.48710 +0.47170 +0.99683 +0.60547 -0.34255

870 545 579 989 142

820 821 822 823 824

-0.04430 -0.86458 -0.88996 -0.09712 +0.78501

291 174 811 190 774

-0.99901 -0,50249 +0.45602 +0.99527 +0.61947

814 220 276 249 329

870 871 872 873 874

+O. 21936 644 -0.70245 -0.97843 -0.35485 +0.59498

070 790 381 124

-0.97564 -0.71173 +0.20654 +0.93492 +0.80373

254 241 122 180 959

825 826 827 828 829

+0.94541 +0.23660 -0.68974 -0.98194 -0.37134

569 282 159 076 812

-0.32586 -0.97160 -0.72405 +0.18918 +0.92849

680 646 561 862 371

875 876 877 878 879

+0.99779 +0.48323 -0.47560 -0.99717 -0.60195

328 878 322 782 173

-0. 06639 -0.87548 -0.87965 -0.07507 +0.79853

709 859 992 597 248

830 831 832 833 834

+0.58066 +0.99881 +0.49866 -0.45995 -0.99569

027 229 090 702 258

+0.81414 -0.04872 -0.86679 -0.88794 -0.09271

596 383 716 118 618

880 881 8 82 883 884

+0.34670 +O. 97660 +0.70861 -0.21086 -0.93648

601 383 660 947 312

+0.93797 +O. 21504 -0.70559 -0.97751 -0.35071

385 639 373 423 265

835 836 837 839

-0.61599 +0.33004 +0.97264 +0.72099 -0.19353

297 774 407 594 254

+0.78775 +0.94396 +0.23230 -0.69293 -0.98109

165 424 046 929 386

885 886 887 888 889

-0.80109 +0.07081 +0.87761 +0.87754 +0.07066

851 237 869 643 203

+0.59853 +0.99748 +0.47935 -0.47949 -0.99750

252 965 940 167 031

840 841 842 843 844

-0.93012 -0.81156 +0.05314 +0.86899 +0.88589

809 816 379 559 685

-0.36723 +0.58425 +0.99858 +0.49481 -0.46388

525 775 687 983 228

890 891 892 893 894

-0.80118 -0.93643 -0,21072 +0.70872 +0.97657

871 025 213 294 141

-0.59841 +0.35085 +0.97754 +0.70548 -0.21519

177 380 600 692 358

845 846 847 848 849

+0.08830 -0.79047 -0.94249 -0.22799 +0.69612

863 014 431 356 342

-0.99609 -0.61250 +0.33422 +0.97366 +0.71792

316 058 221 264 213

895 896 89 7 898 899

+0.34656 -0.60207 -0,99716 -0.47547 +0.48337

463 208 649 063 073

-0.93802 -0.79844 +0.07522 +0.87973 +0.87541

610 174 627 159 575

850

+0.98022 773

-0.19787

267

900

+0.99780 327

838

+0.68330 573

2

.to. 98357 687

+0.06624 670

185

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCNLLAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS sin x cos x X sin x

Table 4.8

900 901 902 903 904

+0.99780 +0.59486 -0.35499 -0.97846 -0.70234

327 009 472 902 341

+0.06624 -0.80382 -0.93486 -0.20639 +0.71183

670 926 831 374 827

950 951 952 953 954

+0.94546 +OS78492 -0.09727 -0.89003 -0.86450

479 436 191 684 600

cos x +0.32572 -0.61959 -0.99525 -0.45588 +0.50262

431 160 784 862 250

905 906 907 9 08 909

+0.21951 +O. 93955 +0.79576 -0.07963 -0.88182

349 070 933 869 727

+0.97560 +O. 34240 -0.60559 -0.99682 -0.47157

947 981 984 380 255

955 956 957 958 959

-0.04415 +0.81679 +0.92678 +0.18469 -0.72720

233 478 454 287 458

+0.99902 +0.57692 -0.37559 -0.98279 -0,68642

481 832 341 629 079

910 911 912 913 914

-0.87326 -0.06183 +0.80645 +0.93328 +0.20206

792 008 406 805 131

+0.48724 +0.99808 +0.59129 -0.35912 -0.97937

032 669 676 869 287

960 961 962 963 964

-0.97051 -0.32153 +0.62305 +0.99481 +0.45194

349 677 937 760 512

+0.24104 +0.94689 +0.78217 -0.10167 -0.89204

682 709 455 567 574

915 916 917 918 919

-0.71493 -0.97462 -0.33824 +0.60911 +0.99646

966 841 829 575 158

-0.69918 +0.22382 +0.94105 +0.79308 -0.08404

616 909 690 134 955

965 966 967 968 969

-0,50644 -0.99921 -0.57330 +0.37969 +0.98360

362 043 778 140 406

-0.86227 -0.03973 +0.81934 +0.92511 +0.18034

308 052 009 320 150

920 921 922 923 924

+0.46766 -0.49110 -0.99835 -0.58772 +O. 36325

523 037 056 184 562

-0.88390 -0.87110 -0.05741 +O.80906 +0.93168

567 299 224 306 952

970 971 972 973 974

+0.68319 -0.24533 -0.94831 -0.77940 +0.10607

568 966 084 942 744

-0.73023 -0.96943 -0.31734 +0.62651 +0.99435

535 718 294 493 787

925 926 927 928 929

+0.98025 +0.69601 -0.22814 -0,94254 -0.79037

754 520 031 467 781

+0.19772 -0.71802 -0.97362 -0.33408 +0.61261

493 705 827 015 972

975 976 977 978 979

+0.89403 +0.86002 +0.03530 -0.82186 -0.92342

718 327 793 936 374

+0.44799 -0.51025 -0.99937 -0.56967 +0.38378

277 482 648 601 195

930 931 932 933 934

+0.08845 +0.88596 +0,86892 +0.05299 -0.81165

877 676 100 328 622

+0.99607 +0.46374 -0.49495 -0.99859 -0.58413

984 875 080 487 542

980 981 982 983 984

-0.17598 +O. 73325 +0.96834 +O. 31314 -0.62995

660 181 189 290 823

+0.98439 +O. 67995 -0.24962 -0.94970 -0.77662

256 719 769 602 902

935 936 937 938 939

-0.93007 -0,19338 +0.72110 +0.97260 +0.32990

273 467 037 905 546

+0.36737 +0.98112 +0.69283 -0.23244 -0.94401

544 302 061 706 398

985 986 987 988 989

-0.99387 -0.44403 +0.51405 +0.99952 +0.56603

867 164 603 296 309

+0.11047 +0.89601 +0.85775 +0.03088 -0.82438

712 111 661 464 252

940 941 942 943 944

-0,61611 -0.99567 -0.45982 +0.49879 +0.99881

169 859 319 154 962

-0.78765 +0.09286 +0.88801 +0.86672 +0.04857

880 625 049 199 328

990 991 992 993 994

-0.38786 -0.98516 -0.67670 +0.25391 +0.95108

499 179 538 083 260

-0.92171 -0.17162 +0.73625 +0.96722 +0.30893

620 825 392 763 672

945 946 947 948 949

+0.58053 -0.37148 -0,98196 -0. 68963 +0.23674

755 806 927 246 926

-0,81423 -0.92843 -0.18904 +0.72415 +0.97157

347 773 062 957 078

995 996 997 998 999

+O. 77383 -0.11487 -0,89796 -0,85547 -0.02646

341 465 748 315 075

-0.63338 -0.99338 -0.44006 +0.51784 +0.99964

919 000 182 716 985

950

+0.94546 479

+0.32572 431

1000

X

For 2>1000 see Example 16.

+0.82687 954

+0.56237 908

I

186

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.9 CIRCULAR TANGENTS, COTANGENTS, SECANTS AND COSECANTS FOR RADIAN ARGUMENTS

tan x

cot x

csc x

66 32 27 52

sec x 1.00000 1.00005 1.00020 1.00045 1.00080

00 00 00 02 05

100.00166 50.00333 33.33833 25.00666

19.98333 16.64666 14.26237 12.47332 11.08109

06 19 33 19 49

1.00125 1.00180 1.00245 1.00320 1.00406

13 27 50 86 37

467 582 934 732 189

9.96664 9.05421 8.29329 7.64892 7.09612

44 28 49 55 94

1.00502 1.00608 1.00724 1.00850 1.00988

0.15113 0.16137 0.17165 Oil8196 0.19231

522 946 682 953 984

6.61659 6.19657 5.82557 5.49542 5.19967

15 54 68 56 16

0.20 0.21 0.22 0.23 0.24

0.20271 0.21314 0.22361 0.23414 0.24471

004 244 942 336 670

4.93315 4.69169 4.47188 4.27088 4.08635

0.25 0.26 0.27 0.28 0.29

0.25534 0.26602 0.27675 0.28755 0.29841

192 154 814 433 279

0.30 0.31 0.32 0.33 0.34

0.30933 0.32032 0.33138 0.34252 0.35373

0.35 0.36 0.37 0.38 0.39

2

0.00 0.01 0.02 0.03 0.04

0.00000 0.01000 0.02000 0.03000 0.04002

0000 0333 2667 9003 1347

99.99666 49.99333 33.32333 24.98666

0.05 0.06 0.07 0.08 0.09

0.05004 0.06007 0.07011 0.08017 0.09024

1708 2104 4558 1105 3790

D.10 0.11 0.12 0.13 0.14

0.10033 0.11044 0.12057 0.13073 0.14092

0.15 0.16 0.17 0.18 0.19

$-1-

cot x

csc x--2-1

67 35 39 79

0.00000 0.00333 0.00666 0.01000 0.01333

000 335 684 060 476

0.00000 0.00166 0.00333 0.00500 0.00666

000 668 349 053 791

20.00833 16.67667 14.29738 12.51334 11.12612

58 09 76 32 53

0.01666 0.02000 0.02334 0.02667 0.03001

944 480 096 805 621

0.00833 0.01000 0.01167 0.01334 0.01501

576 420 334 330 419

09 07 35 99 07

10.01668 9.10926 8.35336 7.71401 7.16624

61 83 70 72 39

0.03335 0.03669 0.04003 0.04338 0.04672

558 628 845 223 776

0.01668 0.01835 0.02003 0.02170 0.02338

614 925 365 946 680

1.01135 1.01293 1.01462 1.01642 1.01832

64 80 61 16 55

6.69173 6.27674 5.91078 5.58566 5.29495

24 65 21 93 84

0.05007 0.05342 0.05677 0.06013 0.06348

516 458 615 000 628

0.02506 0.02674 0.02842 0.03011 0.03180

578 653 915 379 054

49 81 35 77 78

1.02033 1.02246 1.02469 1.02704 1.02950

88 26 78 58 78

5.03348 4.79708 4.58232 4.38639 4.20693

95 57 93 73 71

0.06684 0.07020 0.07357 0.07693 0.08030

512 667 105 841 889

0.03348 0.03518 0.03687 0.03857 0.04027

955 092 477 124 044

3.91631 3.75909 3.61326 3.47760 3.35106

74 41 32 37 28

1.03208 1.03477 1.03759 1.04052 1.04357

50 89 10 27 57

4.04197 3.88983 3.74908 3.61852 3.49708

25 14 94 56 77

0.08368 0.08705 0.09044 0.09382 0.09721

264 978 046 483 302

0.04197 0.04367 0.04538 0.04709 0.04881

250 754 569 707 181

625 751 941 487 688

3.23272 3.12180 3.01759 2.91949 2.82696

81 50 80 61 00

1.04675 1.05005 1.05347 1.05703 1.06072

16 22 94 51 13

3.38386 3.27805 3.17897 3.08600 2.99861

34 83 74 99 68

0.10060 0.10400 0.10740 0.11080 0.11421

519 147 202 697 648

0.05053 0.05225 OI05397 0.05570 0.05744

003 186 744 689 034

0.36502 0.37640 0.38786 0.39941 0.41105

849 285 316 272 492

2.73951 2.65672 2.57822 2.50367 2.43276

22 80 89 59 50

1.06454 1.06849 1.07258 1.07681 1.08118

02 38 47 50 74

2.91632 2.83869 2.76536 2.69599 2.63027

08 75 87 57 48

0.11763 0.12104 0.12447 0.12790 0.13133

070 976 383 306 759

0.05917 0.06091 0.06266 0.06441 0.06617

792 976 601 678 222

0.40 0.41 0.42 0.43 0.44

0.42279 0.43463 0.44657 0.45862 0.47078

322 120 255 102 053

2.36522 2.30080 2.23927 2.18044 2.12413

24 12 78 95 20

1.08570 1.09036 1.09518 1.10015 1.10527

44 89 36 15 57

2.56793 2.50872 2.45242 2I39882 2.34775

25 20 03 48 15

0.13477 0.13822 0.14167 0.14513 0.14859

758 318 456 185 524

0.06793 0.06969 0.07146 0707324 0.07502

246 763 789 336 418

0.45 0.46 0.47 0.48 0.49

0.48305 0.49544 0.50796 0.52061 0.53338

507 877 590 084 815

2.07015 2.01837 1.96863 1.92082 1.87480

74 22 61 05 73

1.11055 1.11600 1.12161 1.12740 1.13335

94 60 91 22 91

2.29903 2.25251 2.20805 2.16553 2.12483

27 55 98 72 00

0.15206 0.15554 0.15902 0.16251 0.16600

486 089 348 280 901

0.07681 0.07860 0.08040 0.08220 0.08401

051 247 022 390 366

0.50

0.54630 249

"-:'I

OQ

1.83048 77

1.13949 39

"-:)21

m

2.08582 96

0.16951 228

0.08582 964

[(-:)7

[(-291 Compilation of tan z and cot z from National Bureau of Standards, Table of circular and hyperbolic tangents and cotangents for radian arguments, 2d printing. Columbia Univ. Press, New York, N.Y., 1947 (with permission).

,

187

ELEMENTARY TRANSCENDENTAL FUNCTIONS

X

CIRCULAR TANGENTS, COTANGENTS, SECANTS AND COSECANTS FOR RADIAN ARGUMENTS tan x cot x sec x

Table 4.9 csc 5

0.50 0.51 0.52 0.53 0.54

0.54630 0.55935 0.57256 0.58591 0.59942

249 872 183 701 962

1.83048 1.78776 1.74653 1.70672 1.66825

772 154 626 634 255

1.13949 1.14581 1.15231 1.15900 1.16589

39 07 38 77 70

2.08582 2.04843 2.01255 1.97810 1.94501

96 63 78 89 07

0.55 0.56 0;57 0.58 OI59

0.61310 0.62694 0.64096 0.65516 0.66955

521 954 855 845 565

1.63104 1.59502 1.56013 1.52632 1.49352

142 471 894 503 784

1.17298 1.18028 1.18778 1.19551 1.20345

68 21 81 06 53

1.91319 1.88257 1.85311 1.82473 1.79739

00 90 45 78 41

0.60 0.61 0.62 0.63 0.64

0.68413 0.69891 0.71390 0.72911 0.74454

681 886 901 473 382

1.46169 1.43078 1.40073 1.37152 1.34310

595 125 873 626 429

1.21162 1.22003 1.22868 1.23758 1.24673

83 59 47 16 39

1.77103 1.74560 1.72106 1.69737 1.67449

22 45 62 57 37

0.65 0.66 0.67 0.68 0.69

0.76020 0.77610 0.79225 0.80866 0.82533

440 491 417 138 611

1.31543 1.28848 1.26222 1.23661 1.21162

569 559 118 155 759

1.25614 1.26583 1.27580 1.28605 1.29660

92 52 04 34 31

1.65238 1.63101 1.61034 1.59034 1.57100

34 05 23 84 01

0.70 0.71 0.72 0.73 0.74

0.84228 0.85952 0.87706 0.89491 0.91308

838 867 790 753 953

1.18724 1.16342 1.14016 1.11742 1.09518

183 833 258 140 285

1.30745 1.31863 1.33013 1.34196 1.35415

93 17 09 77 38

1.55227 1.53413 1.51656 1.49954 1.48304

03 35 54 35 60

0.75 0.76 0.77 0.78 0.79

0.93159 0.95045 0.96966 0.98926 1.00924

646 146 833 154 629

1.07342 1.05213 1.03128 1.01085 0.99083

615 158 046 503 842

1.36670 1.37962 1.39293 1.40664 1.42076

11 24 10 08 67

1.46705 1.45154 1.43650 1.42190 1.40775

27 43 25 99 03

0.80 0.81 0.82 0.83 0.84

1.02963 1.05045 1.07171 1.09343 1.11563

857 514 372 292 235

0.97121 0.95196 0.93308 0.91455 0.89635

460 830 500 085 264

1.43532 1.45032 1.46580 1.48175 1.49821

42 96 02 42 08

1.39400 1.38066 1.36771 1.35513 1.34292

78 78 62 96 52

0.85 0.86 0.87 0.88 0.89

1.13833 1.16155 1.18532 1.20966 1.23459

271 586 486 412 946

0.87847 0.86091 0.84365 0.82667 0.80997

778 426 058 575 930

1.51519 1.53271 1.55080 1.56948 1.58878

02 39 46 63 44

1.33106 1.31953 1.30833 1.29745 1.28688

09 53 72 63 25

0.90 0.91 0.92 0.93 0.94

1.26015 1.28636 1.31326 1.34087 1.36923

822 938 370 383 448

0.79355 0.77738 0.76146 0.74578 0.73033

115 169 169 232 510

1.60872 1.62933 1.65065 1.67270 1.69552

58 92 49 52 44

1.27660 1.26661 1.25691 1.24747 1.23830

62 84 05 40 10

0.95 0.96 0.97 0.98 0.99

1.39838 1.42835 1.45920 1.49095 1.52367

259 749 113 827 674

0.71511 0.70010 0.68530 0.67070 0.65630

188 485 649 959 719

1.71914 1.74361 1.76897 1.79525 1.82252

92 84 37 95 32

1.22938 1.22071 1.21228 1.20409 1.19613

40 57 91 77 51

1.00

1.55740 772

*

*See page U.

[(-54)11

0.64209 262

1.85081 57 [(-:)I1

1.18839 51

188

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR TANGENTS, COTANGENTS, SECANTS AND COSECANTS FOR RADIAN ARGlUMENTS

Table 4.9

tan

I'

csc E.

sec x

cot .r

R'

1.00 1.01 1.02 1.03 1.04

1.55740 1.59220 1.62813 1.66524 1.70361

77 60 04 40 46

0.64209 0.62805 0.61420 0.60051 0.58698

262 942 141 260 722

1.85081 1.88019 1.91070 1.94243 1.97542

57 15 89 08 47

1.18839 1.18087 1.17356 1.16645 1.15954

51 20 01 42 90

1.05 1.06 1.07 1.08 1.09

1.74331 1.78442 1.82702 1.87121 1.91709

53 48 82 73 18

0.57361 0.56040 0.54733 0.53441 0.52162

970 467 693 147 342

2.00976 2.04552 2.08279 2.12166 2.16223

32 49 43 31 06

1.15283 1.14632 1.13999 1.13384 1.12787

98 17 02 11 01

1.10 1.11 1.12 1.13 1.14

1.96475 2.01433 2.06595 2.11975 2.17587

97 82 53 01 51

0.50896 0.49644 0.48403 0.47175 0.45958

811 096 759 371 520

2.20460 2.24890 2.29524 2.34378 2.39466

44 16 97 77 75

1.12207 1.11644 1.11098 1.10569 1.10055

33 69 71 05 37

1.15 1.16 1.17 1.18 1.19

2.23449 2.29579 2.35998 2.42726 2.49789

69 85 11 64 94

0.44752 0.43557 0.42373 0.41198 0.40033

802 829 221 610 638

2.44805 2.50413 2.56310 2.62518 2.69063

57 48 57 99 21

1.09557 1.09074 1.08607 1.08154 1.07715

35 67 04 17 79

1.20 1;21 1.22 1.23 1.24

2.57215 2.65032 2.73275 2.81981 2.91192

16 46 42 57 99

0.38877 0.37731 0.36593 0.35463 0.34341

957 227 119 310 486

2.75970 2.83270 2.90997 2.99188 3.07885

36 55 35 25 30

1.07291 1.06881 1.06485 1.06102 1.05732

64 46 01 06 39

1.25 1.26 1.27 1.28 1.29

3.00956 3.11326 3.22363 3.34135 3.46720

97 91 32 00 57

0.33227 0.32120 0.31020 0.29928 Oi28841

342 577 899 023 670

3.17135 3.26993 3.37517 3.48778 3.60853

77 04 57 15 36

1.05375 1.05032 1.04700 1.04382 1.04076

79 05 98 41 14

1.30 1.31 1.32 1.33 1.34

3.60210 3.74708 3.90334 4.07230 4.25561

24 10 78 98 79

0.27761 0.26687 0.25619 0.24556 0.23498

565 440 034 088 350

3.73833 3.87822 4.02940 4.19329 4.37153

41 33 74 31 10

1.03782 1.03499 1.03229 1.02970 1.02723

00 85 53 88 77

1.35 1.36 1.37 1.38 1.39

4.45522 4.67344 4.91305 5.17743 5.47068

18 12 81 74 86

0.22445 0.21397 0.20353 0.19314 0.18279

572 509 922 574 234

4.56607 4.77923 5.01379 5.27312 5.56133

06 14 49 60 39

1.02488 1.02263 1.02050 1.01848 1.01656

07 65 39 18 93

1.40 1.41 1.42 1.43 1.44

5.79788 6.16535 6.58111 7.05546 7.60182

37 61 95 38 61

0.17247 0.16219 0.15194 0.14173 0.13154

673 663 983 413 734

5.88349 6.24592 6.65666 7.12597 7.66731

01 80 08 85 76

1.01476 1.01306 1.01147 1.00999 1.00861

51 85 85 43 52

1.45 1.46 1.47 1.48 1.49

8.23809 8.98860 9.88737 10.98337 12.34985

28 76 49 93 64

0.12138 0.11125 0.10113 0.09104 0.08097

732 194 908 6660 2601

8.29856 9;04406 9.93781 11.02880 12.39027

45 58 87 66

1.00734 1.00616 1.00510 1.00413 1.00327

05 95 15 62 29

1.50 1.51 1.52 1.53 1.54

14.10141 16i42809 19.66952 24.49841 32.46113

99 17 78 04 89

0.06087 0.05084 0.04081 0.03080

1343 0061 8975 6066

14.13683 16.45849 19.69493 24.51881 32.47653

29 92 14 14 83

1.00251 1.00185 1.00129 1.00083 1.00047

13 09 15 27 44

1.55 1.56 1.57 1.58 1.59

48.07848 92.62049 +1255.76559 - 108.64920 - 52.06696

25 63 15 36 96

0.02079 0.01079 + 0.00079 - 0;00920 - 0.01920

9325 6746 6327 3933 6034

48.08888 92.62589 +1255.76598 - 108.65380 - 52.07657

10 45 97 55 18

1.00021 1.00005 1.00000 1.00004 1.00018

63 83 03 24 44

1.60

-

34.23253 27

- 0.02921

1978

-

For 2>1.6, use 4.3.44.

.

-

0.07091 - - . - . ~.-A .A. O

is

34.24713 56

1.00042 66

189

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES TO TENTHS OF A DEGREE e 0. oo

sin 0

Table 4.10 goo- 0

cos 0

0.1 0.2 0.3 0.4

0.00000 0.00174 0.00349 0.00523 0.00698

00000 53283 06514 59638 12602

00000 65898 15224 31420 97962

1.00000 0,99999 0.99999 0.99998 0.99997

00000 84769 39076 62922 56307

00000 13288 57790 47427 05395

90. Oo 89.9 89.8 89. 7 89.6

0. 5 0.6 0. 7 0. 8 0.9

0.00872 0.01047 0.01221 0.01396 0.01570

65354 17841 70008 21803 73173

98374 16246 35247 39145 11821

0.99996 0.99994 0,99992 0.99990 0.99987

19230 51693 53696 25240 66324

64171 65512 60452 09304 81661

89.5 89.4 89.3 89.2 89.1

1. 0 1.1 1.2 1.3 1.4

0.01745 0.01919 0.02094 0.02268 0.02443

24064 74423 24198 73335 21781

37284 99690 83357 72781 52653

0.99984 0.99981 0.99978 0.99974 0.99970

76951 57121 06834 26093 14897

56391 21644 74845 22698 81183

89.0 88.9 88. 8 88.7 88.6

1.5 1.6 1.7 1.8 1.9

0.02617 0.02792 0.02966 0.03141 0.03315

69483 16387 62440 07590 51783

07873 23569 85111 78128 88526

0.99965 0.99961 0.99955 0.99950 C.99945

73249 01150 98601 65603 02159

75557 40354 19384 65732 41757

88.5 88.4 88.3 88.2 88.1

2.0 2.1 2.2 2.3 2.4

0.03489 0.03664 0.03838 0.04013 0.04187

94967 37087 78090 17925 56537

02501 06556 87520 32560 29200

0.99939 0.99932 0.99926 0.99919 0.99912

08270 83937 29164 43951 28300

19096 78656 10621 14446 98858

88.0 87.9 87.8 87. 7 87.6

2.5 2.6 2.7 2.8 2.9

0.04361 0.04536 0.04710 0.04884 0.05059

93873 29881 64507 97697 29400

65336 29254 09643 95613 76713

0.99904 0.99897 0.99888 Oi99880 0.99871

82215 05697 98749 61373 93571

81858 90715 61970 41434 84186

87.5 87.4 87; 3 87.2 87.1

3.0 3.1 3.2 3.3 3.4

0.05233 0.05407 0.05582 0.05756 0.05930

59562 88129 15049 40269 63735

42944 84775 93164 59567 75962

0,99862 0.99853 0.99844 0;99834 0.99823

95347 66703 07641 18166 98279

54574 26212 81981 14028 23765

87.0 86.9 86. 8 86.7 86.6

3.5 3.6 3.7 3. 8 3.9

0,06104 0.06279 0.06453 0.06627 0.06801

85395 05195 23082 39004 52906

34857 29313 52958 00000 65248

0.99813 0;99802 0.99791 Oi99780 0.99768

47984 67284 56182 14682 42788

21867 28272 72179 92050 35605

86.5 86.4 86.3 86.2 86.1

4. 0 4.1 4.2 4.3 4. 4

0.06975 0.07149 0.07323 0.07497 0.07671

64737 74443 81971 87268 90281

44125 32686 27632 26328 26819

0.99756 40502 0.99744 07829 0.99731 44772 0799718 51335 0.99705 27522

59824 30944 24458 25116 26920

86.0 85.9 85.8 85. 7 85.6

4. 5 4.6 4.7 4.8 4.9

0.07845 0.08019 0.08193 0.08367 0.08541

90957 89243 85086 78433 69231

27845 28859 30041 32315 37367

0.99691 0.99677 0.99663 0.99649 0.99634

33128 56247 18037 49504 90906

85.5 85.4 85.3 85.2 85.1

5. 0

0.08715 57427 47658

0.99619 46980 91746

85.0

cos 6

90’ - R

*

[‘-?I

For conversion from radians to degrees see Example 14.

*See page

n.

73337 88784 73868 28592 52961

0

190

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.10

CIRCULAR SINES .4ND COSINES TO TENTHS OF A DEGREE sin e

e

90' - e

cos 0

5. '0 5.1 5.2 5.3 5.4

0.08715 0.08889 0.09063 0.09237 0.09410

57427 42968 25801 05874 83133

47658 66442 97780 46562 18514

0.99619 0.99604 0.99s588 0.99572 0.99556

46980 10654 43986 46981 19646

91746 10770 15970 84582 03080

85.0' 84.9 84.8 84.7 84.6

5.5 5. 6 5.7 5.8 5.9

0.09584 0.09758 0.09931 0.10105 0.10279

57525 28997 97497 62971 25367

20224 59149 43639 82946 87247

0.99539 0.99522 0.99505 0.99488 0.99470

61983 73999 55699 07088 28171

67179 81831 61226 28788 17174

84.5 84. 4 84.3 84.2 84.1

6.0 6.1 6.2 6.3 6.4

0.10452 0.10626 0.10799 0.10973 0.11146

84632 40713 93557 43110 89322

67653 36233 06023 91045 06325

0.99452 0.99433 0.99415 0.99396 0.99376

18953 79441 09639 09554 79191

68273 33205 72315 55180 60596

84.0 83. 9 83. 8 83. 7 83.6

6.5 6.6 6.7 6.8 6.9

0.11320 0.11493 0.11667 0.11840 0.12013

32137 71504 07370 39683 68388

67907 92867 99333 06501 34647

0.99357 0.99337 0.99317 0.99296 0.99275

18556 76587 27656 00396 06495 38486 55081 06537 ,73419 29446

83.5 83. 4 83. 3 83.2 83.1

7.0 7.1 7.2 7.3 7.4

0.12186 0.12360 0.12533 0.12706 0.12879

93434 14767 32335 46086 55965

05147 40493 64304 01350 77563

0.99254 0.99233 0.99211 0.99189 0.99167

61516 19378 47013 44425 11623

41322 85489 14478 90030 83090

83. 0 82.9 82.8 82.7 82.6

7.5 7.6 7.7 7.8 7.9

0.13052 0.13225 0.13398 0.13571 0.13744

61922 63902 61854 55724 45460

20052 57122 18292 34304 37147

0.99144 0.99121 0.99098 0.99074 0.99050

48613 55402 31997 78404 94632

73810 51542 14836 71444 38309

82.5 82.4 82. 3 82.2 82.1

8. 0 8.1 8.2 8.3 8.4

0.13917 0.14090 0.14262 0.14435 0,14608

31009 12319 89337 62010 30285

60065 37583 05512 00973 62412

0.99026 0.99002 0.98977 0.98952 0.98927

80687 36577 62309 57890 23329

41570 16558 07789 68969 62988

82. 0 81.9 81.8 81.7 81.6

8. 5 8.6 8.7 8.8 8.9

0.14780 Oil4953 0.15126 0.15298 0.15471

94111 53434 08202 58362 03862

29611 43710 47219 84038 99468

0.98901 0.98875 0.98849 0.98822 0.98795

58633 63810 38868 83814 98657

61917 47006 08684 46553 69389

81.5 81. 4 81.3 81. 2 81.1

9.0 9.1 9.2 9.3 9.4

0.15643 0.15815 0.15988 0.16160 0,16332

44650 80672 11876 38211 59622

40231 54484 91835 03361 41622

0.98768 0.98741 0.98713 0.98685 0.98657

83405 38067 62650 57164 21616

95138 50911 72988 06807 06969

81.0 80. 9 80. 8 80. 7 80. 6

9.5 9.6 9.7 9.8 9.9

0.16504 0.16676 0,16848 0.17020 0.17192

76058 87467 93795 94991 91002

60678 16102 65003 66033 79410

0.98628 0.98599 0.98570 0.98540 0.98510

56015 60370 34690 78984 93261

37231 70505 88854 83490 54774

80.5 80.4 80.3 80.2 80.1

C.98480 77530 12208 sin 0

80.0

10. 0

0.17364 81776 66930

goo- 0

cos 0

*

"-3

8

191

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES TO TENTHS OF A DEGREE cos e

sin e

e

Table 4.10

90'

-e

10. oo 10.1 10.2 10. 3 10. 4

0.17364 Oil7536 0.17708 Oil7880 0.18051

81776 67260 47403 22151 91452

66930 91987 19583 16350 50560

0.98480 0.98450 0.98419 0.98388 0.98357

77530 31799 56079 50379 14708

12208 74437 69242 33542 13386

80.0' 79.9 79.8 79.7 79. 6

10.5 10. 6 10.7 10. 8 10.9

0.18223 0.18395 0.18566 0.18738 0.18909

55254 13506 66153 13145 54429

92147 12720 85577 85725 89891

0.98325 0.98293 0.98261 0.98228 0.98195

49075 53491 27965 12507 87126

63955 49554 43615 28689 96444

79.5 79.4 79.3 79.2 79.1

11.0 11.1 11.2 11.3 11.4

0.19080 0.19252 0.19423 0.19594 0.19765

89953 19665 43512 61442 73403

76545 25907 19972 42518 79126

0.98162 0.98129 0.98095 0.98061 0.98027

71834 26639 51553 46585 11746

47664 92245 49192 46613 21722

79.0 78.9 78.8 78. 7 78. 6

11.5 11. 6 11.7 11.8 11.9

0.19936 0.20107 0.20278 0.20449 0.20620

79344 79211 72953 60518 41853

17197 45965 56512 41790 96630

0.97992 0.97957 0.97922 0.97886 0.97850

47046 52495 28106 73887 89851

20830 99344 21766 61685 01778

78.5 78.4 78.3 78.2 78.1

12. 0 12.1 12.2 12.3 12.4

0.20791 0.20961 0.21132 0.21303 0.21473

16908 85629 47964 03862 53271

17759 03822 55389 74977 67063

0.97814 0.97778 0.97741 0.97704 0.97667

76007 32367 58942 55744 22783

33806 58606 86096 35264 34168

78.0 77. 9 77.8 77.7 77.6

12.5 12.6 12.7 12.8 12.9

0.21643 0.21814 0.21984 0.22154 0.22325

96139 32413 62043 84976 01160

38103 96543 52838 19467 10951

0.97629 0.97591 0.97553 0.97514 0.97476

60071 67619 45439 93543 11941

19933 38747 45857 05563 91222

77.5 77.4 77.3 77.2 77.1

13. 0 13.1 13.2 13. 3 13. 4

0.22495 0.22665 0.22835 0.23004 0.23174

10543 13074 08701 97371 79034

43865 36855 10656 88104 94157

0.97437 0.97397 0.97357 0.97317 0.97277

00647 59672 89028 88727 58782

85235 79052 73160 77088 09397

77.0 76.9 76.8 76. 7 76.6

13. 5 13. 6 13. 7 13. 8 13.9

0.23344 0.23514 0.23683 0.23853 0.24022

53638 21131 81460 34575 80424

55905 02590 65619 78581 77264

0.97236 0.97196 0.97154 0.97113 0.97071

99203 10005 91199 42799 64815

97677 78546 97646 09636 78191

76. 5 76.4 76. 3 76.2

14. 0 14.1 14.2 14. 3 14. 4

0.24192 0.24361 0.24530 0.24699 0.24868

18955 50117 73858 90127 98871

99668 86023 78803 22743 64855

0.97029 0.96987 0.96944 0.96901 0.96858

57262 20152 53498 57314 31611

75996 84747 95139 06870 28631

76. 0 75.9 75. 8 75.7 75.6

14. 5 14. 6 14.7 14. 8 14.9

0.25038 0.25206 0.25375 0.25544 0.25713

00040 93582 79445 57579 27931

54441 43114 84806 35791 54696

0.96814 0.96770 0.96726 0.96682 0.96637

76403 91704 77527 33886 60793

78108 81971 75877 04459 21329

75.5 75.4 75. 3 75.2 75.1

15. 0 90' - e

0.25881 90451 02521 COS e

0.96592 58262 89068 sin 9

75.0

*See page n.

76,1

e

192

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.10

CIRCULAR SINES AND COSINES TO TENTHS OF A DEGREE

e

sin e

COS

e

90°

-e

15.0' 15.1 15.2 15. 3 15. 4

0.25881 0.26050 0.26218 0.26387 0.26555

90451 45086 91786 30499 61174

02521 42648 40865 65373 86809

0.96592 0.96547 0.96501 0.96455 0.96409

58262 26308 64944 74184 54042

89068 79225 72311 57798 34110

75. oo 74.9 74.8 74.7 74.6

15. 5 15. 6 15. 7 15. 8 15. 9

0.26723 0.26891 0.27060 0.27228 0.27395

83760 98206 04459 02470 92186

78257 15266 75864 49574 92432

0.96363 0.96316 0.96269 0.96221 0.96174

04532 25667 17464 79935 13095

08623 97658 26479 29285 49211

74.5 74.4 74; 3 74.2 747 1

16. 0 16. 1 16. 2 16. 3 16. 4

0.27563 0.27731 0.27899 0.28066 0.28234

73558 46533 11060 67089 14568

16999 02378 39229 20788 42876

0.96126 0.96077 0.96029 0.95980 0.95931

16959 91541 36856 52919 39745

38319 57594 76943 75187 40058

74.0 73.9 73.8 73.7 73.6

16. 5 16. 6 16. 7 16. 8 16.9

0.28401 0.28568 0.28736 0.28903 0.29070

53447 83674 05198 17969 21935

03923 04974 49712 44472 98252

0.95881 0.95832 0.95782 0.95731 0.95681

97348 25744 24948 94975 35840

68193 65133 45315 32067 57607

73.5 73.4 73.3 73.2 73.1

17. 0 17.1 17. 2 17. 3 17. 4

0.29237 0.29404 0.29570 0.29737 0.29904

17047 03252 80500 48740 07922

22737 32304 44047 77786 56087

0.95630 0.95579 0.95527 0.95476 0.95424

47559 30147 83621 07995 03285

63035 98330 22344 02797 16277

73. 0 72.9 72.8 72. 7 72. 6

17. 5 17. 6 17. 7 17. 8 17. 9

0.30070 0.30236 0.30403 0.30569 0.30735

57995 98907 30609 53049 66177

04273 50445 25490 63106 99807

0.95371 0.95319 0.95266 0.95212 0.95159

69507 06677 14812 93927 44038

48227 92947 53586 42139 79438

72.5 72.4 72.3 72.2 72.1

18. 0 18.1 18.2 18. 3 18. 4

0.30901 0.31067 0.31233 0.31399 0.31564

69943 64296 49185 24559 90369

74947 30732 12233 67405 47102

0.95105 0.95051 0.94997 0.94942 0.94887

65162 57316 20515 54776 60116

95154 27784 24653 41904 44497

72.0 71. 9 71.8 71. 7 71.6

18. 5 18. 6 18. 7 18.8 18.9

0.31730 0.31895 0.32061 0732226 0.32391

46564 93092 29905 56952 74181

05092 98070 85676 30511 98149

0.94832 0.94776 0.94721 0.94664 0.94608

36552 84100 02777 92601 53588

06199 09586 46029 15696 27545

71. 5 71.4 71. 3 71.2 71.1

19.0 19.1 19.2 19. 3 19. 4

0.32556 0.32721 0.32886 0.33051 0.33216

81544 78989 66467 43927 11318

57157 79104 38583 13223 83703

0.94551 0.94494 0.94437 0.94380 0.94322

85755 89121 63702 09515 26579

99317 57531 37481 83229 47601

71. 0 70. 9 70.8 70. 7 70. 6

19.5 19.6 19. 7 19. 8 19.9

0.33380 0.33545 0.33709 0.33873 0.34037

68592 15697 52584 79202 95502

33771 50255 23082 45291 13050

0.94264 0.94205 0.94147 0.94088 0.94028

14910 74527 05448 07689 81270

92178 87297 12038 54225 10419

70.5 70.4 70.3 70.2 70.1

20.0

0.34202 01433 25669

0.93969 26207 85908 sin e

70.0 e

90'

-e

[( -;)41

'See page n.

193

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES TO TENTHS OF A DEGREE sin

0

COB

Table 4.10

-

goo 0

0

20. oo 20.1 %O. 2 20.3 20.4

0.34202 0.34365 0.34529 0.34693 0.34857

01433 96945 81989 56515 20473

25669 85616 98535 73256 21815

0.93969 0.93909 0.93849 0.93788 0.93728

26207 42520 30227 89346 19894

85908 94709 59556 11898 91892

70.0' 69.9 69.8 69.7 69.6

20.5 20.6 20.7 20.8 20.9

0.35020 0.35184 0.35347 0.35510 0,35673

73812 16484 48437 69624 79993

59467 04702 79257 08137 19625

0.93667 0,93605 0.93544 0.93482 0.93420

21892 95357 40308 56763 44743

48398 38973 29867 96014 21030

69.5 69.4 69.3 69.2 69.1

21. 0 21.1 21.2 21.3 21.4

0.35836 0.35999 0.36162 0.36325 0.36487

79495 68081 45700 12304 67843

45300 20051 82092 72978 37620

0.93358 0.93295 0.93232 0.93169 0.93105

04264 35348 38012 12275 58158

97202 25489 15512 85549 62528

69.0 68.9 68. a 68.7 68.6

21.5 21.6 21.7 21.8 21.9

0.36650 0.36812 0.36974 0.37136 0.37298

12267 45526 67572 78355 77825

24297 84678 73829 50235 75809

0.93041 0.92977 OI92913 0.92848 0.92783

75679 64858 25715 58268 62538

82025 88251 34056 80914 98920

68.5 68.4 68.3 68.2 68.1

22.0 22.1 22.2 22. 3 22.4

0.37460 0.37622 0.37784 0.37945 0.38107

65934 42631 07868 61595 03763

15912 39366 18467 29005 50274

0.92718 0.92652 0.92587 0.92520 0.92454

38545 86308 05848 97183 60336

66787 71837 09995 85782 12313

68. 0 67.9 67.8 67.7 67.6

22.5 22.6 22.7 22.8 22.9

0.38268 0.38429 0.38590 0.38751 0.38912

34323 53226 60423 55864 39501

65090 59804 24319 52103 40206

0.92387 0.92321 0.92253 0.92186 0.92118

95325 02171 80894 31515 54055

11287 12981 56246 88501 65721

67.5 67.4 67. 3 67.2 67.1

23. 0 23.1 23.2 23. 3 23.4

0.39073 0.39233 0.39394 0.39554 0.39714

11284 71166 19095 55025 78906

89274 03561 90951 62965 34781

0.92050 0.91982 0.91913 0.91844 0.91775

48534 14973 53392 63813 46256

52440 21738 55234 43087 83981

67.0 66.9 66.8 66. 7 66.6

23.5 23. 6 23.7 23.8 23.9

0.39874 0.40034 0.40194 0.40354 0.40514

90689 90325 77766 52963 15867

25246 56895 55960 52390 79863

0.91706 0.91636 0.91566 0.91495 0.91425

00743 27295 25933 96678 39552

85124 62240 39561 49825 34264

66.5 66.4 66.3 66.2 66.1

24.0 24.1 24.2 24.3 24.4

0.40673 0.40833 0.40992 0.41151 0.41310

66430 04603 30338 43586 44298

75800 81385 41573 05109 24542

0.91354 0.91283 0.91212 0.91140 0.91068

54576 41772 01161 32766 36608

42601 33043 72273 35445 06177

66.0 65.9 65.8 65.7 65.6

24. 5 24. 6 24.7 24. 8 24.9

0.41469 0,41628 0.41786 0.41945 0.42103

32426 07922 70738 20824 58133

56239 60401 01077 46177 67491

0.90996 0.90923 0.90850 0.90777 0.90704

12708 61090 81775 74785 40142

76543 47069 26722 32909 91465

65.5 65.4 65.3 65.2 65.1

25. 0

0.42261 82617 40699

0.90630 77870 36650

65.0 e

90° - e

'See page n.

sin 8

194

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.10

CIRCULAR SINES AND COSINES TO TENTHS OF A DEGREE sin R

R

-

goo 6

cos R

25.0' 25. 1 25. 2 25. 3 25. 4

0.42261 0.42419 0.42577 0.42735 0.42893

82617 94227 92915 78633 51334

40699 45390 65073 87192 03146

0.90630 0.90556 0.90482 0.90408 0.90333

77870 87990 70524 25496 52928

36650 11140 66020 60778 63301

65.0' 64.9 64.8 64.7 64. 6

25.5 25. 6 25.7 25. 8 25. 9

0.43051 0.43208 0.43365 0.43523 0.43680

10968 57488 90845 10993 17883

08295 01982 87544 72328 67702

0.90258 0.90183 0.90107 0.90031 0.89955

52843 25264 70213 87714 77789

49861 05114 22092 02194 55180

64.5 64.4 64. 3 64.2 64.1

26. 0 26.1 26.2 26. 3 26.4

0.43837 0.43993 0.44150 0.44307 0.44463

11467 91698 58527 11908 51791

89077 55915 91745 24180 84927

0.89879 0.89802 0.89725 0.89648 0.89571

40462 75757 83696 64303 17602

99167 60616 74328 83441 39413

64. 0 63. 9 63.8 63.7 63.6

26. 5 26.6 26. 7 26. 8 26. 9

0.44619 0.44775 0.44931 0.45087 0.45243

78131 90878 89986 75406 47093

09809 38770 15897 89431 11783

0.89493 0.89415 0.89337 0.89258 0.89179

43616 42368 13883 58184 75296

02025 39368 27838 52125 05214

63. 5 63.4 63. 3 63.2 63.1

27. 0 27.1 27.2 27. 3 27.4

0.45399 0.45554 0.45709 0.45864 0.46019

04997 49072 79270 95544 97847

39547 33516 58694 84315 83852

0.89100 0.89021 0.88941 0.88861 0.88781

65241 28046 63732 72326 53851

88368 11127 91298 54949 36401

63. 0 62.9 62.8 62.7 62. 6

27.5 27. 6 27. 7 27. 8 27. 9

0.46174 0.46329 0.46484 0.46638 0.46792

86132 60351 20457 66403 98142

35034 19862 24620 39891 60573

0.88701 0.88620 0.88539 0.88458 0.88376

08331 35792 36257 09752 56300

78222 31215 54416 15084 88693

62.5 62.4 62.3 62.2 62.1

28. 0 28. 1 28. 2 28. 3 28. 4

0.46947 0.47101 0.47255 0.47408 0.47562

15627 18812 07648 82090 42090

85891 19410 69054 47116 70275

0.88294 0.88212 0.88130 0.88047 0.87964

75928 68660 34520 73535 85728

58927 17668 64992 09162 66617

62. 0 61. 9 61. 8 61.7 61.6

28. 5 28. 6 28. 7 28. 8 28. 9

0.47715 0.47869 0.48022 0.48175 0.48328

87602 18579 34974 36741 23832

59608 40607 43189 01715 55002

0.87881 0.87798 0.87714 0.87630 0.87546

71126 29754 61637 66800 45270

61965 27981 05589 43864 00018

61.5 61. 4 61. 3 61.2 61.1

29. 0 29.1 29.2 29. 3 29. 4

0.48480 0.48633 0.48785 0.48938 0.49090

96202 53804 96591 24517 37536

46337 23490 38733 48846 15141

0.87461 0.87377 0.87292 0.87206 0.87121

97071 22230 20772 92724 38111

39396 35465 69810 32121 20189

61.0 60. 9 60.8 60.7 60.6

29. 5 29.6 29. 7 29. 8 29. 9

0.49242 0.49394 0.49545 0.49697 0.49848

35601 18665 86684 39610 77397

03467 84231 32408 27555 53830

0.87035 0.86949 0.86863 0.86776 0.86689

56959 49295 15144 54533 67489

39900 05219 38191 68928 35603

60.5 60.4 60. 3 60.2 60.1

30. 0

0.500oo

ooooo oaooo

0.86602 54037 84439

60. 0

90'

e

-R

'See page

n.

195

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES TO TENTHS OF A DEGREE

sin e

e

Table 4.10 90'

cos B

-e

30. 0" 30.1 30; 2 30. 3 30.4

0.50000 0.50151 0.50301 0.50452 0.50603

00000 07371 99466 76238 37641

00000 59457 30235 15019 21164

0.86602 0.86515 0.86427 0.86339 0.86251

54037 14205 48019 55506 36692

84439 69704 53705 06772 07257

60.0' 59.9 59. 8 59. 7 59.6

30.5 30. 6 30. 7 30. 8 30. 9

0.50753 0.50904 0.51054 0.51204 0.51354

83629 14157 29179 28648 12520

60704 50371 11606 70572 58170

0.86162 0.86074 0.85985 0.85895 0.85806

91604 20270 22715 98969 49057

41526 03944 96873 30664 23645

59.5 59.4 59.3 59.2 59.1

31.0 31.1 31.2 31.3 31.4

0.51503 0,51653 0.51802 0.51951 0.52100

80749 33288 70093 91118 96318

10054 66642 73130 79509 40576

0.85716 0.85626 0.85536 0.85445 0.85355

73007 70846 42601 88301 07972

02112 00328 60507 32807 75327

59.0 58.9 58.8 58.7 58. 6

31.5 31.6 31.7 31.8 31.9

0.52249 0.52398 0.52547 0.52695 0.52843

85647 59059 16510 57954 83347

15949 70079 72268 96678 22347

0.85264 0.85172 0.85081 0.84989 0.84897

01643 69341 11094 26929 16876

54092 43048 24051 86864 29141

58.5 58.4 58.3 58.2 58.1

32.0 32.1 32.2 32.3 321 4

0.52991 0.53139 0.53287 0.53435 0.53582

92642 85795 62760 23493 67949

33205 18083 70730 89826 78997

0.84804 0.84712 0.84619 Oi84526 0.84432

80961 19213 31661 18332 79255

56426 82137 27564 21856 02015

58. 0 57.9 57. 8 57.7 57. 6

32.5 32.6 32.7 32.8 32. 9

0.53729 Oi53877 0.54024 0.54170 0.54317

96083 07850 03204 82102 44499

46824 06863 77655 82740 50671

0.84339 OI84245 0.84151 OI84056 0.83961

14458 23970 07819 66034 98645

12886 07148 45306 95684 34413

57.5 57.4 57.3 57.2 57.1

33. 0 33.1 33.2 33. 3 33.4

0.54463 Oi54610 0.54756 0.54902 0.55048

90350 19610 32234 28179 07400

15027 14429 92550 98132 84996

0.83867 OI83771 0.83676 0.83580 0.83484

05679 87166 43134 73613 78632

45424 20439 58962 68270 63407

57.0 56.9 56.8 56.7 56. 6

33. 5 33. 6 33.7 33.8 33.9

0.55193 0.55339 0.55484 0.55629 0.55774

69853 15492 44274 56155 51089

12058 43344 47999 00305 79690

0.83388 0.83292 0.83195 0.83098 0.83001

58220 12407 41221 44692 22850

67168 10099 30483 74328 95367

56.5 56. 4 56.3 56.2 56.1

34.0 34.1 34.2 34. 3 34.4

0.55919 0.56063 0.56208 0.56352 0.56496

29034 89945 33778 60489 70034

70747 63242 52131 37571 24938

0.82903 0.82806 0.82708 0.82609 0.82511

75725 03346 05742 82944 34982

55042 22494 74562 95764 78295

56. 0 55.9 55.8 55.7 55.6

34. 5 34.6 34.7 34.8 34.9

0.56640 0.56784 0.56927 0.57071 0.57214

62369 37450 95234 35676 58734

24833 53101 30844 84432 45516

0.82412 0.82313 0.82214 0.82114 0.82015

61886 63685 40410 92091 18758

22016 34442 30737 33704 73772

55.5 55.4 55.3 55.2 55.1

35. 0

0.57357 64363 51046 COS e

0.81915 20442 88992 sin e

55. 0

goo- e

*

0%

P W

"-67'21

B

196

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table4.10

CIRCULAR SJNES AND COSINES TO TENTHS OF A DEGREE

e

sin

e

COS

goo- e

e

35. O0 35.1 35.2 35.3 35.4

0.57357 0.57500 0,57643 0.57785 0.57928

64363 52520 23161 76243 11723

51046 43279 69793 83505 42679

0.81915 0.81814 0.81714 0.81613 0.81512

20442 97174 48983 75900 77957

88992 25023 35129 80160 28554

55. oo 54.9 54.8 54.7 54.6

35.5 35.6 35.7 35.8 35.9

0.58070 0.58212 0.58354 OI58495 0.58637

29557 29701 12113 76749 23567

10940 57289 56118 87215 35789

0.81411 0.81310 0.81208 0.81106 0.81004

55183 07610 35268 38189 16404

56319 47028 91806 89327 45796

54.5 54.4 54.3 54.2 54.1

36.0 36.1 36. 2 36. 3 36. 4

0.58778 0.58919 0.59060 0.59201 0.59341

52522 63573 56676 31787 88866

92473 53342 19925 99220 03701

0.80901 0.80798 0.80696 0.80592 0.80489

69943 98838 03121 82822 37973

74947 98031 43802 48516 55914

54. 0 53.9 53.8 53.7 53.6

36. 5 36. 6 36.7 36.8 36. 9

0.59482 0.59622 0.59762 0.59902 0.60042

27867 48749 51469 35985 02253

51341 65616 75521 15586 25884

0.80385 0.80281 0.80177 0.80073 0.79968

68606 74751 56442 13709 46584

17217 91115 43754 48733 87091

r3.5 53.4 53.3 53.2 53.1

37.0 37.1 37.2 37.3 37. 4

0.60181 0.60320 0.60459 0.60598 0.60737

50231 79077 91148 84002 58397

52048 45282 62375 65711 23287

0.79863 0.79758 0.79652 0.79547 0.79441

55100 39288 99180 34808 46205

47293 25229 24196 54896 35418

53.0 52.9 52.8 52.7 52.6

37.5 37.6 37.7 37.8 37.9

0.60876 0.61014 0.61152 0.61290 0.61428

14290 51639 70401 70536 52000

08721 01268 85831 52976 98943

0.79335 0.79228 0.79122 0.79015 0.78908

33402 96433 35329 50123 40848

91235 55i9i 67490 75690 34691

52.5 52.4 52.3 52.2 52. 1

38.0 38.1 38.2 38.3 38.4

0.61566 0.61703 0.61840 0.61977 0.62114

14753 58751 83953 90317 77802

25658 40749 57554 95140 78310

0.78801 0.78693 0.78585 0.78477 0.78369

07536 50219 68931 63705 34573

06722 61337 75402 33083 25840

52.0 51.9 51.8 51.7 51.6

38.5 38. 6 38. 7 38.8 38. 9

0.62251 0.62387 0.62524 0.62660 0.62796

46366 95967 26563 38113 30576

37620 09386 35705 64461 49338

0.78260 0.78152 0.78043 0.77933 0,77824

81568 04724 04073 79649 31485

52414 18819 38330 31474 26021

51.5 51.4 51.3 51.2 51.1

39. 0 39.1 39.2 39.3 39. 4

0.62932 0.63067 0.63202 0.63338 Oi63473

03910 58074 93026 08726 05132

49837 31286 64851 27550 02268

0.77714 0.77604 0.77494 0.77384 0.77273

59614 64070 44887 02097 35734

56971 66546 04180 26506 97351

51.0 50.9 50.8 50.7 50.6

39.5 39. 6 39.7 39.8 39.9

0.63607 0.63742 0.63876 0.64010 0.64144

82202 39897 78175 96994 96315

77764 48690 15598 84955 69158

0.77162 "0.77051 0.76939 0.76828 0.76716

45833 32427 95550 35235 51518

87720 75789 46895 93523 15300

50.5 50.4 50.3 50.2 50.1

40. 0

0.64278 76096 86539

0.76604 44431 18978

50.0 e

WO-e

*See page n.

COS

e

197

ELEMENTARY TRANSCENDENTAL FUNCTIONS

CIRCULAR SINES AND COSINES TO TENTHS OF A DEGREE sin e

e

COS

Table 4.10 goo- e

e

40.0' 40.1 40. 2 4.0.3 40. 4

0.64278 0. 64412 0.64545 0.64678 0.64811

76096 36297 76877 97795 99010

86539 61387 23951 10460 63131

0.76604 0.76492 0.76379 0.76266 0.76153

44431 14009 60286 83296 83075

18978 18432 34642 95688 36737

50.0' 49.9 49.8 49.7 49.6

40.5 40.6 40. 7 40.8 40.9

0.64944 0.65077 0.65209 0.65342 0.65474

80483 42172 84038 06039 08137

30184 65851 30392 90105 17340

0.76040 0.75927 0.75813 0.75699 0.75585

59656 13073 43361 50556 34691

00031 34881 97652 51756 67640

49.5 49.4 49. 3 49.2 49.1

41.0 41.1 41. 2 41.3 41.4

0.65605 0.65737 0.65868 0.66000 0.66131

90289 52457 94601 16679 18653

90507 94096 18680 60937 23652

0.75470 OI75356 0,75241 0.75126 0.75011

95802 33923 49088 41335 10696

22772 01638 95724 03511 30460

49.0 48.9 48.8 48.7 48.6

41.5 41.6 41. 7 41.8 41.9

0.66262 0.66392 0,66523 0.66653 0.66783

00482 62126 03546 24702 25554

15737 52242 54361 49452 71047

0.74895 0.74779 0.74663 0.74547 0.74431

57207 80904 81822 59996 15462

89002 98532 85391 82862 31154

48.5 48.4 48.3 48.2 48.1

42.0 42. 1 42.2 42. 3 42.4

0.66913 0.67042 0.67172 0.67301 0.67430

06063 66189 05893 25135 23875

58858 58799 22990 09773 83723

0.74314 Oi74197 0.74080 0.73963 0.73845

48254 58409 45962 10949 53406

77394 75616 86750 78610 25884

48.0 47.9 47.8 47.7 47.6

42.5 42.6 42. 7 42.8 42.9

0.67559 0.67687 0.67815 0.67944 0.68072

02076 59696 96698 13042 08689

15660 82661 68071 61517 58918

0.73727 Oi73609 0.73491 0.73372 0.73254

73368 70871 45951 98645 28987

10124 19734 49960 02876 87379

47.5 47.4 47.3 47.2 47.1

43.0 43.1 43.2 43.3 43.4

0.68199 0.68327 0.68454 0.68581 0.68708

83600 37736 71059 83529 75108

62499 80799 28689 27376 04423

0.73135 0.73016 0.72896 0.72777 0.72657

37016 22766 86274 27576 46709

19170 20752 21412 57210 70976

47. 0 46.9 46.8 46.7 46.6

43. 5 43.6 43.7 43. 8 43.9

0.68835 0.68961 0.69088 0,69214 0.69340

45756 95437 24110 31738 18282

93754 35670 76858 70407 75813

0.72537 0.72417 0.72296 0.72176 0.72055

43710 18614 71459 02280 11116

12288 37468 09568 98362 80330

46. 5 46.4 46.3 46.2 46.1

44.0 44.1 44.2 44. 3 44.4

0.69465 0.69591 0.69716 0.69841 0.69966

83704 27965 51028 52854 33405

58997 92314 54565 31006 13365

0.71933 0.71812 0.71691 0.71569 0.71447

98003 62977 06076 27337 26796

38651 63189 50483 03736 32803

46. 0 45.9 45.8 45.7 45.6

44.5 44. 6 44.7 44.8 44.9

0.70090 0.70215 0.70339 0.70463 0.70587

92642 30529 47028 42099 15706

99851 95162 10504 63595 78681

0.71325 0.71202 0.71079 0.70957 0.70833

04491 54182 60459 90996 94738 72992 07365 36521 98377 24529

45. 5 45.4 45. 3 45.2 45.1

45.0

0.70710 67811 86548

0.70710 67811 86548

45. 0

cos B

WO-8

*See page

n.

sin B

9

198

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.11

0.5 1.0 1.5 2.0

CIRCULAR TANGENTS, COTANGENTS, TO FIVE TENTHS OF tan 6' cot 8 0.00000 00000 00000 0 OIOO872 68677 90759 114.58865 01293 09608 57.28996 16307 59424 0.01745 50649 28217 38.18845 92970 25609 0.02618 59215 69187 28.63625 32829 15603 0.03492 07694 91747

2.5 3.0 3.5 4. 0 4.5

0.04366 0.05240 0.06116 0.06992 0.07870

09429 77792 26201 68119 17068

08512 83041 50484 43510 24618

22.90376 19.08113 16.34985 14.30066 12.70620

55484 66877 54760 62567 47361

31198 28211 99672 11928 74704

1.00095 1.00137 1.00186 1.00244 1.00309

269 235 869 190 220

22.92558 19.10732 16.38040 14.33558 12.74549

563 261 824 703 484

87.5 87. 0 86.5 86.0 85.5

5. 0 5.5 6. 0 6.5 7.0

0.08748 0.09628 0.10510 0.11393 0.12278

86635 90481 42352 56083 45609

25924 97538 65676 01645 02904

11.43005 10.38539 9.51436 8.77688 8.14434

23027 70801 44542 73568 64279

61343 38159 22585 69956 74594

1.00381 1.00462 1.00550 1.00646 1.00750

984 509 828 973 983

11.47371 10.43343 9.56677 8.83367 8.20550

325 052 223 147 905

85. 0 84.5 84. 0 83.5 83. 0

7.5 8. 0 8.5 9.0 9.5

0.13165 0.14054 0.14945 0.15838 0.16734

24975 08347 10013 44403 26090

87396 02391 49128 24536 81419

7.59575 7.11536 6.69115 6.31375 5.97576

41127 97223 62383 15146 43644

25150 84209 17409 75043 33065

1.00662 1.00982 1,01110 1.01246 1.01390

896 757 613 513 510

7.66129 7.18529 6.76546 6.39245 6.05885

758 653 908 322 796

82.5 82. 0 81.5 81.0 80. 5

10.0 10.5 11.0 11.5 12.0

0.17632 0.18533 0.19438 0.20345 0.21255

69807 90449 03091 22994 65616

08465 31534 37718 23699 70022

5.67128 5.39551 5.14455 4.91515 4.70463

18196 71743 40159 70310 01094

17709 19137 70310 71205 78454

1.01542 1.01703 1.01871 1.02048 1.02234

661 027 670 657 059

5.75877 5.48740 5.24084 5.01585 4.80973

049 427 307 174 435

80. 0 79.5 79. 0 78.5 78.0

12.5 13.0 13.5 14. 0 14.5

0.22169 0.23086 0.24007 0.24932 0.25861

46626 81911 87590 80028 75843

42940 25563 80116 43180 55890

4.51070 4.33147 4.16529 4.01078 3.86671

85036 58742 97700 09335 30948

62057 84155 90417 35844 98738

1.02427 1.02630 1.02641 1.03061 1.03290

951 411 519 363 031

4.62022 4.44541 4.28365 4.13356 3.99392

632 148 757 550 916

77.5 77. 0 76. 5 76. 0 75.5

15. 0 15.5 16. 0 16.5 17. 0

0.26794 0.27732 0.28674 0.29621 0.30573

91924 45440 53857 34949 06814

31122 59838 58808 62080 58660

3.73205 3.60588 3.48741 3.37594 3.27085

08075 35087 44438 34225 26184

68877 60874 40908 91246 84141

1.03527 1.03774 1.04029 1.04294 1.04569

618 221 944 891 176

3.86370 3.74197 3.62795 3.52093 3.42030

331 754 528 652 362

75. 0 74.5 74.0 73.5 73. 0

17. 5 18. 0 18.5 19.0 19.5

0.31529 0.32491 0.33459 0.34432 0.35411

87888 96962 53195 76132 85725

78983 32906 02073 89665 30698

3.17159 3.07768 2.98868 2.90421 2.82391

48023 35371 49627 08776 28856

63212 75253 42893 75823 00801

1.04852 1.05146 1.05449 1.05762 1.06084

913 222 231 068 870

3.32550 3.23606 3.15154 3.07155 2.99574

952 798 530 349 431

72.5 72.0 71. 5 71. 0 70. 5

20.0 20.5 21.0 21.5 22.0

0.36397 0.37388 0.38386 0.39391 0.40402

02342 46794 40350 04756 62258

66202 84804 35416 14942 35157

2.74747 2.67462 2.60508 2.53864 2.47508

74194 14939 90646 78956 68534

54622 26824 93801 64307 16296

1.06417 1.06760 1.07114 1.07478 1.07653

777 936 499 624 474

2.92380 2.85545 2.79042 2.72850 2.66946

440 095 811 383 716

70. 0 69.5 69.0 68.5 68. 0

22.5

0.41421 35623 73095

2.61312 593 sec 8

67.5

e 0. oo

goo- 8

2.41421 35623 73095 tan 0

SECANTS AND COSECANTS A DEGREE sec e csc e 1.00000 000 0 1.00003 808 114.59301 348 1.00015 233 57.29868 850 1.00034 279 38.20155 001 1.00060 954 28.65370 835

1.08239 220 csc 8

goo- e

90.0' 89.5 89.0 88.5 88. 0

8

ELEMENTARY TRANSCENDENTAL FUNCTIONS

199

CIRCULAR TANGENTS, COTANGENTS, SECANTS AND CO SECANTS TO FIVE TENTHS OF A DEGREE tan e cot e sec tl csc e

Table 4.11 90° - e

e 22.5O 23.0 23. 5 24. 0 24.5

0.41421 0.42447 0.43481 0.44522 0.45572

35623 48162 23749 86853 62555

73095 09604 60933 08536 32584

2.41421 2.35585 2.29984 2.24603 2.19429

35623 23658 25472 67739 97311

73095 23753 36257 04216 65038

1.08239 1.08636 1.09044 1.09463 1.09894

220 038 110 628 787

2.61312 2.55930 2.50784 2.45859 2.41142

593 467 285 334 102

67.5' 67.0 66.5 66.0 65.5

25.0 25.5 26.0 26.5 27. 0

0.46630 0.47697 0.48773 0.49858 0.50952

76581 55326 25885 16080 54494

54998 98160 65861 53431 94429

2.14450 2.09654 2.05030 2.00568 1.96261

69205 35990 38415 97082 05055

09558 88174 79296 59020 05150

1.10337 1.10792 1.11260 1.11740 1.12232

792 854 194 038 624

2.36620 2.32282 2.28117 2.24115 2.20268

158 050 203 845 926

65.0 64.5 64.0 63.5 63.0

27. 5 28.0 28.5 29. 0 29.5

0.52056 0.53170 0.54295 0.55430 0.56577

70505 51746 94316 61479 56996 38437 90514 52769 27781 87770

1.92098 1.88072 1.84177 1.80404 1.76749

21269 64655 08860 77552 40162

71166 46332 33458 71424 42891

1.12738 1.13257 1.13789 1.14335 1.14895

195 005 318 407 554

2.16568 2.13005 2.09573 2.06266 2.03077

057 447 853 534 204

62.5 62.0 61.5 61.0 60.5

30.0 30.5 31; 0 31.5 32.0

0.57735 0.58904 0.60086 0.61280 0.62486

02691 50164 06190 07881 93519

89626 20551 27560 39932 09327

1.73205 1.69766 1.66427 1.63185 1.60033

08075 31193 94823 16871 45290

68877 26089 50518 28789 41050

1.15470 1.16059 1.16663 1.17282 1.17917

054 210 340 770 840

2.00000 1.97029 1.94160 1.91388 1.88707

000 441 403 086 991

60.0 59.5 59.0 58.5 58.0

32.5 33.0 33.5 34.0 34.5

0.63707 0.64940 0.66188 0.67450 0.68728

02608 75931 55611 85168 09586

07493 97510 95691 42426 01613

1.56968 1.53986 1.51083 1.48256 1.45500

55771 49638 51936 09685 90286

17490 14583 14901 12740 72445

1.18568 1.19236 1.19920 1.20621 1.21340

905 329 494 795 641

1.86115 1.83607 1.81180 1.78829 1.76551

900 846 103 165 728

57.5 57.0 56.5 56.0 55.5

35.0 35.5 36. 0 36.5 37.0

0.70020 0.71329 0.72654 0.73996 0.75355

75382 30678 25280 10750 40501

09710 97005 05361 28487 02794

1.42814 1.40194 1.37638 1.35142 1.32704

80067 82944 19204 24379 48216

42114 76336 71173 45808 20410

1.22077 1.22832 1.23606 1.24400 1.25213

459 691 798 257 566

1.74344 1.72205 1.70130 1.68117 1.66164

680 082 162 299 014

55.0 54.5 54.0 53.5 53.0

37. 5 38. 0 38.5 39.0 39.5

0.76732 0.78128 0.79543 0.80978 0.82433

69879 56265 59166 40331 63858

78960 06717 67828 95007 17495

1.30322 1.27994 1.25717 1.23489 1.21309

53728 16321 22989 71565 70040

41206 93079 18954 35051 92932

1.26047 1.26901 1.27777 1.28675 1.29596

241 822 866 957 700

1.64267 1.62426 1.60638 1.58901 1.57213

963 925 793 573 369

52.5 52.0 51.5 51.0 50.5

40. 0 40.5 41. 0 41.5 42. 0

0.83909 0.85408 0.86928 0.88472 0.90040

96311 06854 67378 52645 40442

77280 63466 16226 55944 97840

1.19175 1,17084 1.15036 1.13029 1.11061

35925 95661 84072 43863 25148

94210 12539 21009 61753 29193

1.30540 1.31508 1.32501 1.33519 1.34563

729 700 299 242 273

1.55572 1.53976 1.52425 1.50916 1.49447

383 904 309 050 655

50.0 49.5 49.0 48.5 48.0

42.5 43. 0 43.5 44.0 44.5

0.91633 0.93251 0.94896 0.96568 0.98269

11740 50861 45667 87748 72631

17423 37661 14880 07074 15690

1.09130 1.07236 1.05378 1.03553 1.01760

85010 87100 01252 03137 73929

69271 24682 80962 90569 72125

1.35634 1.36732 1.37859 1.39016 1.40203

170 746 847 359 206

1.48018 1.46627 1.45273 1I43955 1.42671

723 919 967 654 819

47.5 47.0 46.5 46.0 45.5

45.0

1.00000 00000 00000

1.41421 356

45.0 e

goo- e

cat

e

1.00000 00000 00000 tan 0 ~

3

1

1.41421 356 csc e

sec e

i

200

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.12

CIRCULAR FUNCTIONS FOR THE ARGUMENT iz sin

X

r2 x

1-x

tan % x

0.00 0.01 0.02 0.03 0.04

0.00000 0.01570 0.03141 0.04710 0.06279

00000-00000 73173 11820 07590 78128 64507 09642 05195 29313

00000 67575 29384 66090 37607

1.00000 0.99987 0.99950 0.99888 0.99802

00000 66324 65603 98749 67284

00000 81660 65731 61969 28271

00000 59864 55700 97264 56195

0.00000 0.01570 0.03142 0.04715 0.06291

00000 92553 62660 88028 46672

00000 23664 43351 77480 53649

00000 91632 14782 47448 75722

1.00 0.99 0.98 0.97 0.96

0. 05 0. 06 0. 07 0. 08 0. 09

0.07845 OlO9410 0.10973 0.12533 0.14090

90957 83133 43110 32335 12319

27844 18514 91045 64304 37582

94503 31847 26802 24537 66116

0.99691 OI99556 0.99396 0.99211 0.99002

73337 19646 09554 47013 36577

33127 03080 55179 14477 16557

97620 01290 68775 83105 56725

0.07870 0.09452 0.11040 0.12632 0.14232

17068 78311 10278 93784 10757

24618 79282 15818 46108 02942

44806 04901 94497 17478 94229

0.95 0.94 0.93 0. 92 0.91

0.10 0.11 0.12 0.13 0. 14

0.15643 0.17192 0.18738 0.20278 0.21814

44650 91002 13145 72953 32413

40230 79409 85724 56512 96542

86901 54661 63054 48344 55202

0.98768 0.98510 0.98228 0.97922 0.97591

83405 93261 72507 28106 67619

95137 54773 28688 21765 38747

72619 91802 68108 78086 39896

0.15838 0.17452 0.19076 0.20709 0.22352

44403 79388 02022 00444 64828

24536 94365 18566 27938 97149

29384 08461 74856 70402 10184

0.90 0.89 0.88 0.87 0.86

0.15 0. 16 0. 17 0.18 0.19

0.23344 0.24868 0.26387 0.27899 0.29404

53638 98871 30499 11060 03252

55905 64854 65372 39229 32303

41177 78824 89696 25185 95777

0.97236 0.96858 0.96455 0.96029 0.95579

99203 31611 74184 36856 30147

97676 28631 57798 76943 98330

60183 11949 09366 07175 12664

0.24007 0.25675 0.27356 0.29052 0.30764

87590 63603 90430 68567 01696

80116 67726 82237 31916 59898

03926 78332 23655 45432 29067

0.85 0.84 0. 83 0. 82 0.81

0.20 0.21 0.22 0. 23 0. 24

0.30901 0.32391 0.33873 0.35347 OI36812

69943 74181 79202 48437 45526

74947 98149 45291 79257 84677

42410 41440 38122 12472 95915

0.95105 0.94608 0.94088 0.93544 0.92977

65162 53588 07689 40308 64858

95153 27545 54225 29867 88251

57211 31853 47232 32518 40366

0.32491 0.34237 0.36002 0.37786 0.39592

96962 65257 21530 85117 80087

32906 28683 95756 75820 97721

32615 05965 62634 93670 26049

0.80 0.79 0. 78 0.77 0. 76

0. 25 0.38268 34323 65089 77173 0.92387 0. 26 0.39714 78906 34780 61375 0.91775 0.27 0.41151 43586 05108 77405 0.91140 0.28 0.42577 92915 65072 64886 0.90482 0. 29 0.43993 91698 55915 14083 0.89802

95325 46256 32766 70524 75757

11286 83981 35445 66019 60615

75613 14114 24821 52771 63093

0.41421 0.43273 0.45151 Or47056 0.48989

35623 86422 73130 42812 49450

73095 47425 86983 12251 22477

04880 93197 28945 49308 05270

0.75 0.74 0.73 0.72 0. 71

0.30 0; 31 0. 32 0. 33 0. 34

0.45399 0.46792 0.48175 0.49545 0.50904

04997 98142 36741 86684 14157

39546 60573 01715 32407 50371

79156 37723 27498 53805 30028

0.89100 0.88376 0.87630 0.86863 0.86074

65241 56300 66800 15144 20270

88367 88693 43863 38191 03943

86236 42432 58731 24777 63716

0.50952 0.52947 0.54975 0.57038 0.59139

54494 27451 46521 99296 83513

94428 82014 92770 73294 99471

81051 63252 07429 88698 09817

0.70 0. 69 0.68 0. 67 0. 66

0.35 0. 36 0. 37 0.38 0. 39

0.52249 0.53582 0.54902 0.56208 0.57500

85647 67949 28179 33778 52520

15948 78996 98131 52130 43278

86499 61827 74352 60010 56590

0.85264 0.84432 0.83580 0.82708 0.81814

01643 79255 73613 05742 97174

54092 02015 68270 74561 25023

22152 07855 25847 82492 43213

0.61280 0.63461 0.65687 0.67959 0.70281

07881 92975 72224 92982 17712

39931 44148 01279 24526 40357

99664 10071 37691 52184 33761

0.65 0.64 0. 63 0.62 0. 61

0. 40 0. 41 0. 42 0.43 0. 44

0.58778 0.60042 Oi61290 0.62524 0.63742

52522 02253 70536 26563 39897

92473 25884 52976 35705 48689

12917 04976 49336 17290 71017

0.80901 0.79968 0.79015 0.78043 0.77051

69943 46584 50123 04073 32427

74947 87090 75690 38329 75789

42410 53868 36516 73585 23080

0.72654 0.75082 0.77567 0.80115 0.82727

25280 12380 95110 10705 19459

05360 38764 49613 58751 72475

88589 68575 10378 23382 63403

0. 60 0. 59 0.58 0.57 0. 56

0.45 0. 46 0.47 0. 48 0. 49

0.64944 0.66131 0.67301 0.68454 0.69591

80483 18653 25135 71059 27965

30183 23651 09773 28688 92314

65572 87657 33872 67373 32549

0.76040 0.75011 0.73963 0.72896 0.71812

59656 10696 10949 86274 62977

00030 30459 78609 21411 63188

93817 54151 69747 52314 83037

0.85408 0.88161 0.90992 OI93906 0.96906

06854 85923 99881 25056 74171

63466 63189 77737 17492 93793

63752 11465 46579 35255 27618

0.55 0. 54 0. 53 0. 52 0. 51

0. 50 0.70710 67811 86547 52440 0.70710 67811 86547 52440 1.00000 00000 00000 00000 0.50 cos i j x X sin T z c0t;x 1 -x 2 L

-1

[(

- 4111

I

201

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.12

CIRCULAR FUNCTIONS FOR THE ARGUMENT i x 7r

77

sec 2 x

csc - x

1-x

2

1.00000 1.00012 1.00049 1.00111 1.00197

00000 33827 36832 13587 71730

00000 39761 37144 85243 71142

00000 81169 42400 76109 10978

0.05 12.70620 47361 74704 64602 1.00309 0.06 10.57889 49934 05635 52417 1.00445 0.07 9.05788 66862 38928 19329 1.00607 0.08 7.91581 50883 05826 84427 1.00794 0.09 7.02636 62290 41380 19848 1.01007

21984 78193 57361 79708 68726

82825 57019 86291 09297 13784

50283 12.74549 48431 82374 28619 0.95 51480 10.62605 37962 83115 99865 0.94 90575 9.11292 00161 49841 72675 0.93 28943 7.97872 97555 59476 60149 0.92 19104 7.09717 00264 69225 38129 0. 91

0. 00 0.01 0.02 0.03 0.04

,

7r

c o t 2”

X

00

63.65674 31.82051 21.20494 15.89454

11628 59537 87896 48438

71580 73958 88751 65303

99500 03934 52283 44576

m

63.66459 31.83622 21.22851 15.92597

53060 52090 50958 11099

00564 97622 16816 08654

58546 95566 17580 59358

1.00 0.99 0. 98 0.97 0.96

0.10 0.11 0.12 0.13 0.14

6.31375 5.72974 5.24218 4.82881 4.47374

15146 16467 35811 73521 28292

75043 24314 13176 92759 11554

09898 86192 73758 97818 62415

1.01246 1.01511 1.01803 1.02121 1.02467

51257 57576 21481 80406 75534

88002 62501 91042 26567 55900

93136 87437 38259 47910 33566

6.39245 5.81635 5.33671 4.93127 4.58414

32214 10329 14122 53949 38570

99661 24944 92458 49859 27373

54704 03199 78659 96253 56913

0.90 0. 89 0.88 0.87 0.86

0.15 0.16 0.17 0.18 0.19

4.16529 3.89474 3.65538 3.44202 3.25055

97700 28549 43546 25766 08012

90417 29859 52259 69218 99836

20387 33474 73004 62809 37634

1.02841 1.03243 1.03674 1.04134 1.04625

51936 58714 49162 80947 16303

65208 17339 32016 70681 39647

54585 88710 53065 14007 78848

4.28365 4.02107 3.78970 3.58434 3.40089

75697 22333 11465 36523 40753

31185 75967 59780 72161 61802

03924 50952 81919 57038 31848

0. 85 0. 84 0.83 0.82 0.81

0.20 0.21 0.22 0.23 0.24

3.07768 2.92076 2.77760 2.64642 2.52571

35371 09892 68539 32102 16894

75253 98816 14974 86631 47304

40257 40048 88865 86514 99451

1.05146 1.05698 1.06283 1.06901 1.07552

22242 70790 39243 10439 73070

38267 93232 36113 98926 22247

21205 61183 96396 01199 78234

3.23606 3.08720 2.95213 2.82905 2.71647

79774 66268 47928 56388 18916

99789 08416 09339 91501 65871

69641 38088 97327 64260 74307

0. 80 0.79 0.78 0.77 0. 76

0.25 0.26 0.27 0.28 0..29

2.41421 2.31086 2.21475 2.12510 2.04125

35623 36538 44978 81731 39671

73095 82410 13361 57202 21703

04880 63708 51875 76115 26026

1.08239 1.08961 1.09720 1.10518 1.11355

22002 58646 91341 35787 15511

92393 48705 29537 56399 90413

96880 30888 26252 59380 37268

2.61312 2.51795 2.43004 2.34863 2.27304

59297 36983 88648 46560 15214

52753 10349 55296 54351 61957

05571 34110 52041 86300 72361

0.75 0.74 0.73 0. 72 0. 71

0.30 0.31 0.32 0.33 0.34

1.96261 1.88867 1.81899 1.75318 1.69090

05055 13416 32472 66324 76557

05150 31067 81066 72237 85011

58230 67620 27571 08332 24674

1.12232 1.13152 1.14115 1.15123 1.16178

62376 17133 30035 61494 82810

34360 97749 92241 81376 72765

80715 42882 17245 51287 98515

2.20268 2.13707 2107574 2.01833 1.96447

92645 26325 96076 18280 66988

85266 27611 48793 89559 67248

62156 85837 05903 43676 48330

0. 70 0. 69 0.68 0. 67 0. 66

0.35 0.36 0.37 0.38 0.39

1.63185 1.57574 1.52235 1.47145 1.42285

16871 78599 45068 53158 60774

28789 68651 96131 19969 31870

61767 08688 24085 04283 59031

1.17282 1.18437 1.19644 1.20907 1.22227

76966 39497 79450 20434 01770

14008 36918 89806 06541 86068

94955 17500 17366 15436 14117

1.91388 1.86627 1.82141 1.77909 1.73911

08554 47167 79214 54854 45497

30942 00567 74081 79867 30640

72280 54120 38479 33350 74960

0. 65 0.64 0. 63 0. 62 0.61

0.40 0.41 0.42 0.43 0.44

1.37638 1.33187 1.28919 1.24820 1.20879

19204 49515 22317 40363 23504

71173 02597 85066 53049 09609

53820 59439 67042 43751 13115

1.23606 1.25049 1.26557 1.28134 1.29783

79774 29154 44560 42308 62271

99789 09784 72090 20677 84727

69641 85573 15648 31999 12712

1.70130 1.66550 1.63156 1.59937 1.56881

16167 01910 87575 90408 45035

04079 65749 13749 68062 05365

86436 08074 73007 88301 75750

0.60 0.59 0.58 0.57 0.56

0.45 0.46 0.47 0.48 0.49

1.17084 1.13427 1.09898 1.06489 1.03191

95661 73492 56505 18403 99492

12539 55405 36301 24791 80495

22520 46422 56382 86700 57182

1.31508 1.33313 1.35202 1.37180 1.39251

69998 59054 53634 11480 27141

90784 50172 40027 64918 49012

80424 40410 12805 28453 49662

1.53976 1.51214 1.48585 1.46081 1.43696

90432 58610 64735 98491 16493

22366 31226 81717 22513 57094

30748 40092 76608 12750 20394

0.55 0.54 0.53 0.52 0. 51

0.50 1 -x

1.00000 00000 00000 00000 1.41421 35623H 73095 04880 tan;x csc 3 x

1.41421 35623 7r73095 04880 0.50 sec 2 x

X

I

I

202

ELEMENTARY TRANSCENDENTAL FUNCTIONS Table 4.13 r

2rr sin -

2ar

cos y

s=3

HARMONIC ANALYSIS . 2nr 27r sin cos s=4 1.00000 00000 to. 00000 00000 0.00000 00000 -1.00000 00000

sin 2w -

2TT

cos 7 9-5

0.95105 65163 t 0.30901 69944 0.58778 52523 -0.80901 69944

1 2

0.86602 54038 -0.50000 00000

s=6

S=7

1 2 3 4

0.86602 54038 +0.50000 00000 0.86602 54038 -0.50000 00000 0.00000 00000 -1.00000 00000

0.78183 14824 +O. 62348 98019 0.97492 79122 -0.22252 09340 0.43388 37391 -0.90096 88679

0.70710 1.00000 0.70710 0.00000

67812 0.70710 67812 00000 to. 00000 00000 67812 -0.70710 67812 00000 -1.00000 00000

1 2 3 4 5

0.64278 0.98480 0.86602 0,34202

0.58778 0.95105 0.95105 0.58778 0.00000

52523 0.80901 65163 to. 30901 65163 -0; 3090i 52523 -0.80901 00000 -1.00000

0.54064 0.90963 0.98982 0.75574 0.28173

08174 19953 14419 95743 25568

1 2 3 4 5 6 7

0.50000 0.86602 1.00000 0.86602 0.50000 0.00000

54038 00000 54038 00000 00000

0.46472 0.82298 0.99270 0.93501 0.66312 0.23931

31720 0.88545 60257 38659 0.56806 47468 88741 to. 12053 66803 62427 -0.35460 48871 26582 -0.74851 07482 56643 -0.97094 18174

0.43388 0.78183 0.97492 0.97492 0.78183 0.43388 0. ooooo

37391 14825 79122 79122 14825 37391

ooooo -i; ooooo ooooo

0.40673 0.74314 0.95105 0; 99452 0.86602 0.58778 0.20791

66431 48255 65163 18954 54038 52523 16908

0.36124 0.67369 0.89516 0.99573 0.96182 0.79801 0.52643 0.18374

16662 56436 32913 41763 56432 72273 21629 95178

0.93247 0.73900 0; 4i5i3 +O. 09226 -0.27366 -0.60263 -0.85021 -0.98297

22294 89172 83558 83595 29901 46364 71357 30997

0.34202 0.64218 0.86602 0.98480 0.98480 0.86602 0.64278 0.34202 0.00000

01433 76097 54038 77530 77530 54038 76097 01433 00000

0.30901 0.58778 0.80901 0.95105 1.00000 0.95105 0.80901 0.58778 0.30901 0.00000

9=20 69944 0.95105 52523 0.80901 69944 0.58778 65163 0.30901 00000 to. 00000 65163 -0.30901 69944 -0.58778 52523 -0.80901 69944 -0.95105 00000 -1.00000

65163 69944 52523 69944 00000 69944 52523 69944 65163 00000

0.29475 0; 56332 0.78183 0.93087 0.99720 0.97492 0.86602 0.68017 0.43388 0.14904

51744 00580 14825 37486 37972 79122 54038 27378 37391 22662

0.25881 0.50000 0.70710 0.86602 0.96592 1.00000 0.96592 0; 86601 0.70710 0.50000 o; 2588i 0.00000

90451 00000 61812 54038 58263 00000 58263 54038 67812 00000 90451 00000

s=9 76097 0.76604 77530 + 0.17364 54038 -0.50000 01433 -0.93969

44431 81777 00000

26208

9=12 00000

15

0.91354 0.66913 to. 30901 -0.10452 -0.50000 -0.80901 -0.97814

69944

69944 69944 00000

S

26208 44431 00000 81777 81777 00000 44431 26208 00000

0.32469 0.61421 0.83716 0.96940 0.99658 0.91577 0.73572 0.47594 0.16459

28058 87143 98019 10244 00936 09340 00000 18718 88679 08262

0.28173 25568 0.54064 08174 0.15574 95743 0; 90963 19953 0.98982 14419 0.98982 14419 0.90963 19953 0.75574 95743 0.54064 08174 0.28173 25568 0.00000 00000

=19

S

94692 27127 64782 02659 44930 33266 39107 73930 45903

0. 94581 72417 0. 78914 05094 0. 54694 81581

+O. 24548 54872

-0. 08257 93455 -0. 40169 54247 -0. 61728 15716 -0. 87947 37512 -0. 98636 13034

10

10 11

=21 0.95557 0.82623 0.62348 0.36534 +O. 07473 -0.22252 -0.50000 -0.73305 -0.90096 -0.98883

S

s=-240.

1 2 3

10 11 12

96592 58263 0. 86602 54038 0. 70710 67812 0. 50000 00000 0. 25881 90451 + 0. 00000 00000 -0. 25881 90451 -0. 50000 00000 70710 67812 0. 86602 54038 -0. 96592 58263 -1. 00000 00000

-

8

0.24868 0.48175 0.68454 0.84432 0.95105 0.99802 0.98228 0.90482 0.77051 0.58778 0.36812 0.12533

98812 36741 71059 79255 65163 67284 72507 70525 32428 52523 45527 32336

-22 0.95949 0.84125 0.65486 0; 41541 +O. 14231 -0.14231 -0.41541 -0.65486 -0.84125 -0.95949 -1.00000

0.84125 +O. 41541 -0.14231 -0.65486 -0.95949

35328 50130 48383 07340 29736

0.90096 0.62348 +O. 22252 -0.22252 -0.62348 -0.90096 ...-

88679 98019 09340 09340 98019 88679 -- - . .

s=17 -.

=16

0.38268 . . ~34324 ~ . ~ 0.,9238795325 0. 70710 67812 0. 70710 67812 0.92387 95325 0. 38268 34324 1.00000 00000 + 0. 00000 00000 0; 92387 95325 -0. 38268 34324 0.70710 67812 -0. 70710 67812 0.38268 34324 -0. 92387 95325 0; 00000 00000 -1. 00000 00000

~~

S

$=ll

~=14

54576 06064 69944 84633 00000 69944 76007

s= 18

0.93969 0.76604 0.50000 +O. 17364 -0.17364 -0.50000 -0.76604 -0.93969 -1.00000

69944

s=13

0.86602 54038 0.50000 00000 +o.ooooo 00000 -0,50000 00000 -0.86602 54038 -1.00000 00000

S=

8-8

29736 35328 07340 50130 48383 48383 50130 07340 35328 29736 00000

-26 0.96858 31611 0.87630 66801 0.72896 86274 0.53582 67950 0.30901 69944 to. 06279 05196 -0; ii738 13746 -0.42577 92916 -0.63742 39898 -0; 80901 69944 -0.92977 64859 -0.99211 47013

s=23

0.26979 67711 0.51958 39500 0.73083 59643 0.88788 52184 0.97908 40877 0.99766 87692 0; 94226 09221 0.81696 98930 0.63108 79443 0.39840 10898 0.13616 66491

0.96291 0.85441 0.68255 0.46006 t 0.20345 -0.06824 -0.33487 -0.57668 -0.77571 -0.91721 -0.99068

72874 94046 31432 50378

60131

24134 96122 03221 12907 13015 59460

203

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.14

INVERSE CIRCULAR SINES AND TANGENTS X

arcsin x

arctan x

2

arcsin x

arctan z

0.000 0.001 0.002 0. 003 0.004

0.00000 0,00100 0.00200 0.00300 0.00400

00000 00001 00013 00045 00106

00 67 33 00 67

0.00000 0.00099 0.00199 0.00299 0.00399

00000 99996 99973 99910 99786

00 67 33 00 67

0.050 0.051 0.052 0.053 0.054

0.05002 0.05102 0.05202 0.05302 0.05402

08568 21344 34632 48442 62784

06 17 28 51 97

0.04995 0.05095 0.05195 0.05295 0.05394

83957 58518 32065 04578 76036

22 77 61 05 42

0.005 0.006 0.007 0.008 0.009

0.00500 0.00600 0.00700 0.00800 0.00900

00208 00360 00571 00853 01215

34 01 68 36 04

0.00499 0.00599 0.00699 0.00799 0.00899

99583 99280 98856 98293 97570

34 02 70 40 12

0.055 0.056 0.057 0.058 0.059

0.05502 0.05602 0.05703 0.05803 0.05903

77669 93107 09107 25679 42835

81 15 14 92 64

0.05494 0.05594 0.05693 0.05793 0.05893

46421 15712 83890 50936 16829

07 34 60 23 64

0.010 0,011 0.012 0.013 0.014

0.01000 0,01100 0.01200 0.01300 0.01400

01666 02218 02880 03661 04573

74 45 19 95 74

0.00999 0.01099 0.01199 0.01299 0.01399

96666 95563 94240 92677 90854

87 66 50 41 41

0.060 0.061 0.062 0.063 0.064

0.06003 0.06103 0.06203 0.06304 0.06404

605'84 78936 97902 17491 37713

45 52 01 09 94

0.05992 0.06092 0.06192 0.06291 0.06391

81551 45081 07400 68489 28327

21 38 58 26 89

0.015 0.016 0.017 0.018 0.019

0.01500 0.01600 0.01700 0.01800 0.01900

05625 06827 08189 09721 11433

57 45 40 42 52

0.01499 0.01599 0.01699 0.01799 0.01899

88751 86348 83626 80563 77141

52 76 17 78 62

0.065 0.066 0.067 0.068 0.069

0.06504 0.06604 0.06705 0.06805 0.06905

58580 80101 02286 25146 48691

75 69 97 79 36

0.06490 0.06590 0.06690 0.06789 0.06889

86896 44176 00148 54791 08087

93 90 29 63 46

0.020 0.021 0.022 0.023 0.024

0.02000 0.02100 0.02200 0.02300 0.02400

13335 15438 17750 20283 23045

73 06 53 16 97

0.01999 0.02099 0.02199 0.02299 0.02399

73339 69138 64516 59456 53935

73 17 97 20 92

0.070 0.071 0.072 0.073 0.074

0.07005 0.07105 0.07206 0.07306 0.07406

72930 97875 23535 49921 77043

88 58 68 42 03

0.06988 0.07088 0.07187 0.07287 0.07386

60016 10558 59695 07407 53674

35 85 56 09 06

0.025 0.026 0.027 0.028 0.029

0.02500 0.02600 0.02700 0.02800 0.02900

26048 29302 32815 36599 40663

99 25 77 58 72

0.02499 0.02599 0.02699 0.02799 0.02899

47936 41437 34418 26861 18744

19 08 68 07 33

0.075 0.076 0.077 0.078 0.079

0.07507 0.07607 0.07707 0.07807 0.07908

04910 33534 62925 93093 24048

77 87 62 26 07

0.07485 0.07585 OiO7684 0.07784 0.07883

98477 41796 83614 23909 62663

11 89 08 37 48

0.030 0.031 0.032 0.033 0.034

0.03000 0.03100 0.03200 0.03300 0.03400

45018 49673 54638 59924 65540

23 15 51 37 77

0.02999 0.03099 0.03198 0.03298 0.03398

10048 00753 PO840 80288 69077

57 89 39 21 46

0.080 0.081 0.082 0.083 0.084

0.08008 0.08108 0.08209 0.08309 0.08409

55800 88360 21738 55944 90989

34 35 40 79 83

0.07982 0.08082 0.08181 0.08281 0.08380

99857 35471 69486 01882 32642

12 05 04 86 31

0,035 0.036 0.037 0.038 0.039

0,03500 0.03600 0,03700 0.03800 0.03900

71497 77805 84473 91512 98932

75 38 72 81 73

0.03498 0.03598 0.03698 0.03798 0.03898

57188 44600 31295 17251 02450

29 82 22 64 25

0.085 0.086 0.087 0.088 0.089

0.08510 0.08610 0.08711 0.08811 0.08911

26883 63637 01260 39763 79156

84 15 09 00 23

0.08479 0.08578 0.08678 0.08777 0.08876

61745 89172 14904 38923 61208

23 45 84 27 65

0.040 0.041 0.042 0.043 0.044

0.04001 0.04101 0.04201 0.04301 0.04401

06743 14955 23578 32622 42097

54 31 12 04 16

0.03997 0,04097 0.04197 0.04297 0.04397

86871 70494 53301 35270 16382

23 77 05 30 71

0.090 0.091 0.092 0.093 0.094

0.09012 0.09112 0.09213 0.09313 0.09413

19450 60655 02781 45839 89840

15 11 49 68 07

0.08975 0.09075 0.09174 OIO9273 0.09372

81741 00503 17475 32638 45972

90 96 79 38 74

0.045 0.046 0.047 0.048 0.049

0.04501 0.04601 0.04701 0.04801 0.04901

52013 62381 73210 84511 96293

56 33 57 37 83

0.04496 0.04596 0.04696 0.04796 0.04896

96618 75957 54381 31868 08400

52 97 30 77 65

0.095 0.096 0.097 0.098 0.099

0.09514 0.09614 0.09715 0.09815 0.09916

34793 80709 27598 75471 24339

06 05 48 75 32

0.09471 0.09570 0.09669 0.09768 0.09867

57459 67080 74816 80648 84557

88 87 76 65 66

0.050

0.05002 08568 06

0.04995 83957 22

0.100

0.10016 74211 62

[(-!)6]

[);-(

1'

[(-y]

0.09966 86524 91

For use and extension of the table see Examples 21-25. For other inverse func tionssee 4.4 and 4.3.45. :=1.57079 63267 95 2 Compilation of arcsin x from National Bureau of Standards, Table of arcsin x. Columbia Univ. Press, New York, N.Y ., 1945 (with permission),

204

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.14 X

INVERSE CIRCULAR SINES AND TANGENTS arcsin x

arctan x

arcsin x

X

arctan x

0.100 0.101 0.102 0.103 0.104

0.10016 0.10117 0.10217 0.10318 0.10418

74211 25099 77012 29961 83957

62 11 25 53 41

0.09966 0.10065 0.10164 0.10263 0.10362

86524 86531 84558 80587 74599

91 58 83 89 97

0.150 0.151 0.152 0.153 0.154

0.15056 0.15157 0.15259 0.15360 0.15461

82727 97940 14716 33066 53001

77 40 20 23 61

0.14888 0.14986 0.15084 0.15182 0.15279

99476 77989 53616 26338 96139

09 58 21 59 37

0.105 0.106 0.107 0.108 0.109

0.10519 0.10619 0,10720 0.10821 0.10921

39010 95131 52329 10617 70003

40 00 72 08 62

0.10461 0.10560 0.10659 0.10758 0.10857

66576 56498 44346 30103 13750

33 23 99 93 39

0.155 0.156 0.157 0.158 0.159

0.15562 0.15663 0.15765 0.15866 0.15967

74533 97672 22431 48819 76848

44 86 01 05 15

0.15377 0.15475 0.15572 0.15670 0.15768

63001 26906 87838 45780 00713

20 78 86 19 58

0.110 0.111 0.112 0.113 0.114

0.11022 0.11122 0.11223 0.11324 0.11424

30499 92116 54863 18752 83793

88 41 77 55 32

0.10955 0.11054 0.11153 0.11252 0.11350

95267 74637 51840 26859 99674

74 38 74 25 40

0.160 0.161 0.162 0.163 0.164

0,16069 0,16170 0.16271 0,16373 0.16474

06529 37874 70893 05599 42001

52 35 88 34 99

0.15865 0.15963 0.16060 0.16157 0.16255

52621 01487 47294 90024 29661

86 91 61 91 78

0.115 0.116 0.117 0.118 0.119

0,11525 0.11626 0.11726 0.11827 0.11928

49996 17373 85933 55688 26648

68 23 61 42 32

0.11449 0.11548 0.11647 0.11745 0.11844

70267 38620 04714 68531 30052

67 60 73 63 90

0.165 0.166 0.167 0.168 0.169

0.16575 0.16677 0.16778 0.16880 0.16981

80113 19943 61505 04810 49868

10 96 87 17 19

0.16352 0.16449 0.16547 0.16644 0.16741

66188 99587 29841 56935 80850

21 25 97 49 93

0.120 0.121 0.122 0.123 0.124

0.12028 0.12129 0.12230 0.12331 0.12431

98823 72225 46865 22751 99897

95 97 07 92 22

0.11942 0.12041 0.12140 0.12238 0.12337

89260 46135 00659 52814 02582

18 12 40 72 82

0.170 0.171 0.172 0.173 0.174

0.17082 0.17184 0.17285 0.17387 0.17489

96691 45290 95678 47864 01862

29 84 23 87 19

0.16839 0.16936 0.17033 0.17130 0.17227

01571 19080 33360 44396 52169

48 34 78 07 54

0.125 0.126 0.127 0.128 0.129

0.12532 0.12633 0.12734 0.12835 0.12936

78311 58006 38990 21277 04875

68 02 98 29 72

0.12435 0.12533 0.12632 0.12730 0.12829

49945 94884 37381 77418 14977

47 45 58 71 71

0.175 0.176 0.177 0.178 0.179

0.17590 0.17692 0.17793 0.17895 0.17996

57681 15334 74832 36187 99410

64 66 75 40 13

0.17324 0.17421 0.17518 0.17615 0.17712

56664 57864 55752 50312 41528

52 43 68 74 10

0.130 0.131 0.132 0.133 0.134

0.13036 0.13137 0.13238 0.13339 0.13440

89797 76052 63651 52606 42926

03 01 45 16 95

0.12927 0.13025 0.13124 0.13222 0.13320

50040 48 82588 96 12605 10 40070 89 64968 35

0.180 0.181 0.182 0.183 0.184

0,18098 0.18200 0.18302 0.18403 0.18505

64512 31505 00402 71212 43949

47 97 20 76 25

0.17809 0.17906 0.18002 0.18099 0.18196

29382 13858 94941 72613 46859

31 94 59 91 59

0,135 0.136 0.137 0.138 0.139

0.13541 0.13642 0.13743 0.13844 0.13945

34624 27710 22194 18087 15401

67 15 25 85 83

0.13418 0,13517 0.13615 0.13713 0.13811

87279 06986 24071 38516 50303

52 49 35 25 34

0.185 0.186 0.187 0.188 0.189

0.18607 0.18708 0.18810 0.18912 0.19014

18623 95246 73830 54387 36928

31 57 71 40 36

0.18293 0.18389 0.18486 0.18583 0.18679

17662 85005 48874 09250 66119

35 94 16 85 87

0.140 0.141 0.142 0.143 0.144

0.14046 0.14147 0.14248 0.14349 0.14450

14147 14334 15975 19079 23659

10 56 13 77 42

0.13909 0,14007 0.14105 0.14203 0.14301

59414 65832 69539 70518 68749

82 92 90 03 65

0.190 0.191 0.192 0.193 0.194

0.19116 Oil9218 0.19319 0.19421 0.19523

21465 31 08009 99 96574 17 87169 63 79808 18

0.18776 0.18872 0.18969 0.19065 0.19161

19465 69270 15520 58198 97288

14 59 22 05 15

0,145 0.146 0,147 0.148 0.149

0.14551 0.14652 0.14753 0.14854 0,14955

29725 04 37287 64 46358 19 56947 71 69067 22

0.14399 0,14497 0.14595 0.14693 0.14791

64217 09 56902 74 46789 00 33858 33 18093 19

0,195 0.196 0.197 0,198 0,199

0.19625 0.19727 0.19829 0.19931 0.20033

74501 71261 70100 71030 74061

0.19258 0.19354 0.19450 0.19547 0.19643

32774 64641 92873 17453 38367

60 55 18 71 38

0.150

0.15056 82727 77

0.14888 99476 09

0.200

0.20135 79207 90

r(-w

“-:”I

L 4 J

r[(-;)41 ( - 8)41 L 4 J

64 85 69 03 80

0.19739 55598 50 y p 5 1

2

= 1.57079

63267 95

-

ELEMENTARY TRANSCENDENTAL FUNCTIONS

INVERSE CIRCULAR SINES AND TANGENTS X

arctan x

arcsin x

Table 4.14

arcsin x

X

arctan x

0.200 0.201 0.202 0.203 0.204

0.20135 0.20237 0.20339 0.20442 0.20544

79207 86480 95890 07451 21175

90 31 97 90 10

0.19739 0.19835 0.19931 0.20027 0.20123

55598 69131 78950 85040 87384

50 40 44 06 69

0.250 0.251 0.252 0.253 0.254

0.25268 0.25371 0.25474 0.25577 0.25681

02551 31886 63988 98871 36548

42 28 49 33 08

0.24497 0.24591 0.24686 0.24780 0.24873

86631 96179 01284 01933 98113

27 19 51 77 53

0.205 0.206 0.207 0.208 0.209

0.20646 0.20748 0.20850 0.20952 0.21055

37072 55156 75438 97931 22647

61 48 81 68 22

0.20219 0.20315 0.20411 0.20507 0.20603

85968 80777 71793 59003 42391

83 01 81 83 73

0.255 0.256 0.257 0.258 0.259

0.25784 0.25888 0.25991 0.26095 0.26198

77032 20336 66475 15461 67307

07 66 22 18 97

0.24967 0.25061 0.25155 0.25249 0.25343

89810 77010 59702 37870 11502

38 99 05 29 51

0.210 0.211 0.212 0.213 0.214

0.21157 0.21259 0.21362 0.21464 0.21566

49597 78794 10251 43979 79990

58 93 46 39 96

0.20699 0.20794 0.20890 0.20986 0.21082

21942 97639 69469 37416 01465

20 97 83 57 06

0.260 0.261 0.262 0.263 0.264

0.26302 0.26405 0.26509 0.26613 0.26716

22029 79638 40148 03573 69927

08 02 31 53 28

0.25436 0.25530 0.25624 0.25717 0.25811

80585 45106 05051 60408 11163

53 23 53 40 83

0.215 0.216 0.217 0.218 0.219

0.21669 0.21771 0.21874 0.21976 0.22078

18298 58914 01850 47119 94733

42 06 19 15 28

0.21177 0.21273 0.21368 0.21464 0.21559

61600 17806 70070 18375 62706

20 92 19 04 53

0.265 0.266 0.267 0.268 0.269

0.26820 0.26924 0.27027 0.27131 0.27235

39223 11474 86696 64900 46102

20 95 22 75 31

0.25904 0.25997 0.26091 0.26184 0.26277

57304 98818 35692 67913 95468

89 68 33 04 05

0.220 0.221 0.222 0.223 0.224

0,22181 0.22283 0,22386 0.22489 0.22591

44704 97046 51770 08889 68415

97 62 66 55 75

0.21655 0.21750 0.21845 0.21941 0.22036

03049 39389 71712 00001 24243

76 87 05 53 57

0.270 0.271 0.272 0.273 0.274

0.27339 0.27443 0.27547 0.27651 0.27754

30314 17551 07827 01155 97549

67 69 21 13 38

0.26371 0.26464 0.26557 0.26650 0.26743

18344 36530 50011 58777 62813

62 10 84 27 84

0.225 0.226 0.227 0,228 0.229

0.22694 0.22796 0.22899 0,23002 0.23105

30361 94740 61563 30844 02595

79 17 45 22 07

0.22131 0.22226 0.22321 0.22416 0.22511

44423 60526 72538 80444 84229

48 61 37 19 53

0.275 0.276 0.277 0.278 0.279

0.27858 OI27962 0.28067 0.28171 0.28275

97023 99592 05269 14069 26005

92 75 90 43 45

0.26836 0.26929 0.27022 0.27115 0.27208

62109 56650 46425 31422 11628

06 49 71 39 19

0.230 0.231 0.232 0.233 0.234

0.23207 0.23310 0,23413 0.23516 0,23618

76828 53557 32794 14552 98843

63 56 53 26 48

0.22606 0.22701 0.22796 0.22891 0.22986

83879 79380 70718 57877 40844

94 96 22 34 03

0.280 0.281 0.282 0.283 0.284

0.28379 0.28483 0.28587 0.28692 0.28796

41092 59343 80773 05397 33228

08 51 93 58 75

0.27300 0.27393 0.27486 0.27578 0.27671

87030 57618 23377 84298 40366

87 19 99 14 55

0.235 0.236 0.237 0.238 0.239

0.23721 0.33824 0.23927 0.24030 0.24133

85680 75077 67045 61598 58749

94 44 78 80 37

0.23081 0.23175 0.23270 0.23365 0.23459

19604 94143 64447 30501 92293

03 10 07

80 19

0.285 0.286 0.287 0.288 0.289

0.28900 0.29004 0.29109 0.29213 0.29318

64281 98570 36110 76915 20999

74 89 61 30 43

0.27763 0.27856 0.27948 0.28041 0.28133

91571 37900 79341 15882 47512

20 08 26 83 95

0.240 0.241 0.242 0.243 0.244

0.24236 0.24339 0.24442 0.24545 0.24648

58510 60894 65915 73585 83917

39 77 47 45 73

0.23554 0.23649 0.23743 0.23837 0.23932

49807 03029 51947 96545 36809

21 83 10 10 95

0.290 0.291 0.292 0.293 0.294

0.29422 0.29527 0.29631 0.29736 0.29840

68377 19064 73073 30420 91120

49 01 57 76 25

0.28225 0.28317 0.28410 0.28502 0.28594

74219 95991 12816 24683 31580

81 65 76 46 14

0.245 0,246 0.247 0.248 0.249

0.24751 0.24855 0.24958 0.25061 0.25164

96925 12621 31018 52130 75970

34 33 81 88 69

0.24026 0.24121 0.24215 0.24309 0.24403

72727 04284 31467 54261 72654

81 90 47 82 29

0.295 0.296 0.297 0.298 0,299

0.29945 0.30050 0.30154 0.30259 0.30364

55186 22634 93479 67735 45417

70 85 45 30 24

0.28686 0.28778 0.28870 0.28962 0.29053

33495 30417 22334 09235 91109

23 18 53 83 69

0.250

0.25268 02551 42

0.24497 86631 27

0.300

0.30469 26540 15

2

= 1.57079 638267 95

0.29145 67944 78

206

ELEMENTARY TRANSCENDENTAL FUNCTIONS

INVERSE CIRCULAR SINES AND TANGENTS

Table 4.14 X

arctan x

arcsin x

arcsin x

X

arctan x

0.300 0.301 0.302 0: 303 0.304

0.30469 0.30574 0.30678 0.30783 0.30888

26540 11118 99168 90704 85740

15 95 60 09 46

0.29145 0.29237 0.29329 0.29420 0.29512

67944 39729 06453 68104 24672

78 79 47 62 09

0.350 0.351 0.352 0.353 0.354

0.35757 0.35863 0.35970 0.36077 0.36184

11036 88378 69995 55905 46125

46 55 85 70 51

0.33667 0.33756 0.33845 0.33934 0.34023

48193 54100 54442 49211 38398

87 58 85 81 61

0.305 0.306 0.307 0.308 0.309

0.30993 0.31098 0.31203 0.31309 0.31414

84292 86376 92005 01196 13964

78 19 83 91 68

0.29603 0.29695 0.29786 0.29877 0.29969

76144 22511 63761 99883 30866

75 55 46 52 80

0.355 0.356 0.357 0.358 0.359

0.36291 0.36398 0.36505 0.36612 0.36719

40672 39564 42819 50454 62486

71 82 39 05 46

0.34112 0.34200 0.34289 0.34378 0.34467

21994 99990 72378 39149 00294

49 70 56 42 69

0.310 0.311 0.312 0.313 0.314

0.31519 0.31624 0.31729 Or31835 0.31940

30324 50291 73881 01108 31990

41 43 12 88 18

0.30060 0.30151 0.30242 0130334 0.30425

56700 77373 92875 03195 08322

42 55 41 25 38

0.360 0.361 0.362 0: 363 0.364

0.36826 Oi36933 0.37041 0; 37148 0.37255

78934 99815 25147 54949 89237

37 54 84 16 46

0.34555 Or34644 0.34732 0.34820 0.34909

55805 05674 49891 88449 21339

82 30 68 54 52

0.315 0.316 0.317 0.318 0.319

0.32045 0.32151 0.32256 0.32361 0.32467

66540 04775 46710 92361 41743

50 38 42 24 51

0.30516 0.30607 0.30697 0.30788 0.30879

08246 02955 92441 76691 55696

16 99 31 62 46

0.365 0.366 0.367 0.368 0.369

0.37363 0.37470 0.37578 Or37685 0.37793

28030 71347 19204 71621 28616

75 12 71 69 34

0.34997 0.35085 0.35173 Or35261 0.35350

48553 70082 85919 96054 00481

30 60 21 93 64

0.320 0.321 0.322 0.323 0.324

0.32572 0.32678 0.32784 0.32889 0.32995

94872 51765 12436 76902 45178

95 31 42 11 29

0.30970 0.31060 0.31151 0.31242 0.31332

29445 97928 61134 19053 71675

42 14 29 60 84

0.370 0.371 0.372 0.373 0.374

0.37900 0.38008 0.38116 0.38224 0.38331

90206 56411 27249 02738 82897

96 93 69 73 61

0.35437 0.35525 0.35613 0.35701 0.35789

99191 92175 79426 60937 36698

23 68 98 18 38

0.325 0.326 0.327 0.328 0.329

0.33101 0.33206 0.33312 0.33418 0.33524

17280 93225 73029 56707 44276

89 91 38 38 04

0.31423 0.31513 0.31603 0.31694 0.31784

18990 60988 97658 28991 54976

84 47 63 30 47

0.375 0.376 0.377 0.378 0.379

0.38439 0.38547 0.38655 0.38763 0.38871

67744 57299 51579 50604 54393

96 45 83 92 57

0.35877 0.35964 0.36052 0.36139 0.36227

06702 70942 29409 82096 28995

71 35 56 58 76

0.330 0.331 0.332 0.333 0.334

0.33630 0.33736 0.33842 0.33948 0.34054

35751 31150 30487 33781 41047

54 09 98 50 05

0.31874 0.31964 0.32055 0.32145 0.32235

75604 90864 00747 05244 04343

21 60 81 03 49

0.380 0.381 0.382 0.383 0.384

0.38979 0.39087 0.39195 0.39304 0.39412

62964 76337 94530 17563 45455

74 42 68 64 51

0.36314 0.36402 0.36489 0.36576 0.36663

70099 05400 34890 58562 76408

46 09 12 04 40

0.335 0.336 0.337 0.338 0.339

0.34160 0.34266 0.34372 0.34479 0.34585

52301 67559 86840 10158 37531

02 88 15 39 21

0.32324 0.32414 0.32504 0.32594 0.32684

98036 86313 69164 46580 18551

48 34 46 25 19

0.385 0.386 0.387 0.388 0.389

0.39520 0.39629 0.39737 0.39846 0.39954

78225 15893 58477 05998 58474

54 06 48 24 89

0.36750 0.36837 0.36924 0.37011 0.37098

88421 94594 94920 89390 77999

81 90 36 92 35

0.340 0.341 0.342 0.343 0.344

0.34691 0.34798 0.34904 0.35010 0.35117

68975 04507 44144 87902 35798

27 29 03 30 98

0.32773 0.32863 0.32953 0.33042 0.33131

85067 46120 01700 51797 96403

81 66 37 60 04

0.390 0.391 0.392 0.393 0.394

0.40063 0.40171 0.40280 0.40389 0.40497

15927 78374 45836 18333 95884

01 28 44 27 67

0.37185 0.37272 0.37359 0.37445 0.37532

60738 37601 08580 73668 32860

49 18 36 96 01

0.345 0.346 0.347 0.348 0.349

0.35223 0.35330 0.35437 0.35543 0.35650

87850 44075 04488 69108 37952

97 25 84 81 29

0.33221 0.33310 0.33399 0.33489 0.33578

35507 69101 97176 19722 36731

47 67 49 83 63

0.395 0.396 0.397 0.398 0.399

0.40606 0.40715 0.40824 0.40933 0.41042

78510 66231 59066 57035 60160

57 00 02 81 60

0.37618 0.37705 0.37791 0.37878 0.37964

86146 33521 74978 10510 40109

53 62 43 12 93

0.350

0.35757 11036 46

0.33667 48193 87

0.400

0.41151 68460 67 ~ ( - 8 ~ 1

[)(:-

51

L'

2 = 1.57079 63267 95

4'

I

0.38050 63771 12

r ( -8 ~ 3 7 L 4 1

207

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.14

INVERSE CIRCULAR SINES AND TANGENTS X

arcsin x

arctan x

arctan x

arcsin x

X

0.400 0.401 0.402 0.403 0.404

0.41151 0.41260 0.41370 0.41479 0.41588

68460 81956 00668 24616 53822

67 42 29 80 54

0.38050 0.38136 0.38222 0.38308 0.38394

63771 81487 93250 99056 98896

12 02 97 39 72

0.450 0.451 0.452 0.453 0.454

0.46676 0.46788 0.46900 0.47012 0.47124

53390 54404 61761 75486 95604

47 09 03 20 59

0.42285 0.42368 0.42451 0.42534 0.42617

39261 52156 58823 59257 53454

33 87 89 92 56

0.405 0.406 0.407 0.408 0.409

0.41697 0.41807 0.41916 0.42026 0.42135

88306 28088 73190 23632 79435

20 50 29 45 96

0.38480 0.38566 0.38652 0.38738 0.38824

92765 80656 62562 38477 08395

46 14 34 69 85

0.455 0.456 0.457 0.458 0.459

0.47237 0.47349 0.47461 0.47574 0.47686

22141 55121 94570 40513 92976

29 50 53 79 80

0.42700 0.42783 0.42865 0.42948 0,43031

41409 23118 98576 67780 30725

43 21 60 36 28

0.410 0.411 0; 412 0.413 0.414

0.42245 0.42355 0.42464 0.42574 0.42684

40621 07211 79225 56685 39613

87 31 49 70 30

0.38909 0.38995 0.39080 0.39166 0.39251

72310 30215 82104 27971 67810

55 54 62 64 48

0.460 0.461 OI462 0.463 0.464

0.47799 0.47912 0.48024 0.48137 0.48250

51985 17564 89741 68540 53988

19 68 12 46 75

0.43113 0.43196 0.43278 0.43361 0.43443

87407 37821 81965 19833 51422

19 96 51 80 81

0.415 0.416 0.417 0.418 0.419

0.42794 0.42904 0.43014 0.43124 0.43234

28029 21956 21415 26427 37015

74 53 30 72 57

0.39337 0.39422 0.39507 0.39592 0.39677

01615 29379 51097 66763 76371

09 43 52 44 29

0.465 0.466 0.467 0.468 0.469

0.48363 0.48476 0.48589 0.48702 0.48815

46112 44937 50489 62796 81884

18 02 67 64 55

0.43525 0.43607 0.43690 0.43772 0.43854

76728 95747 08474 14907 15041

58 19 74 40 36

0.420 0.421 0.422 0.423 0.424

0.43344 0.43454 0.43565 0.43675 0.43785

53200 75005 02450 35559 74353

70 03 60 49 90

0.39762 0.39847 0.39932 0.40017 0.40102

79915 77389 68788 54105 33336

22 43 14 66 29

0.470 0.471 0.472 0.473 0.474

0.48929 0.49042 0.49155 0.49269 0.49382

07780 40510 80101 26582 79978

14 26 88 08 07

0.43936 0.44017 0.44099 0.44181 0.44263

08872 96398 77613 52515 21100

85 14 55 43 17

0.425 0.426 0.427 0.428 0.429

0.43896 0.44006 0.44117 0.44227 0.44338

18856 69088 25073 86833 54391

10 44 36 39 16

0.40187 0,40271 0.40356 0.40440 0.40525

06474 73514 34450 89278 37990

40 42 79 00 60

0.475 0.476 0.477 0.478 0.479

0.49496 0.49610 0.49723 0.49837 0.49951

40317 07626 81934 63268 51655

17 82 59 16 34

0.44344 0.44426 0.44507 0.44589 0.44670

83364 39303 88916 32196 69143

20 99 06 95 24

0.430 0.431 0.432 0.433 0.434

0.44449 0.44560 0.44670 0.44781 0.44892

27769 06990 92078 83054 79943

36 78 31 92 67

0.40609 0.40694 0.40778 0.40862 0.40946

80583 17050 47386 71587 89646

18 34 77 18 31

0.480 OI481 0.482 OI483 0.484

0.50065 0.50179 0.50293 0.50407 0.50522

47124 49702 59418 76300 00377

05 34 39 52 13

0.44751 0.44833 0.44914 0.44995 0.45076

99751 24018 41941 53515 58739

57 60 03 61 11

0.435 0.436 0.437 0.438 0.439

0.45003 0.45114 0.45226 0.45337 0.45448

82767 91550 06314 27084 53882

71 28 71 44 99

0.41031 0.41115 0.41199 0.41283 0.41366

01558 07319 06924 00366 87642

96 97 22 64 17

0.485 0.486 0.487 0.488 0.489

0.50636 0.50750 0.50865 0.50979 0.51094

31676 70228 16060 69201 29681

79 19 14 57 57

0.45157 0.45238 0.45319 0.45400 0.45480

57608 50120 36271 16059 89480

36 20 55 33 51

0.440 0.441 0.442 0.443 0.444

0.45559 0.45671 OI45782 0.45894 0.46005

86733 25661 70688 21838 79137

96 07 11 99 71

0.41450 0.41534 0.41618 0.41701 0.41785

68745 43672 12417 74976 31343

85 70 83 36 48

0.490 0.491 0.492 0.493 0.494

0.51208 0.51323 0.51438 0.51553 0.51668

97529 72774 55445 45573 43186

34 22 69 34 93

0.45561 0.45642 0.45722 0.45803 0.45883

56532 17211 71514 19440 60984

11 17 78 06 16

0.445 0.446 0.447 0.448 0.449

0.46117 0.46229 0.46340 0.46452 0.46564

42608 12275 88162 70294 58695

35 10 25 19 40

0.41868 0.41952 0.42035 0.42118 0.42202

81514 25484 63248 94802 20141

38 34 66 67 75

0.495 0.496 0.497 0.498 0.499

0.51783 0.51898 0.52013 0.52129 0.52244

48316 60991 81242 09100 44594

32 55 77 26 47

0.45963 0.46044 0,46124 0.46204 0.46284

96144 24917 47301 63293 72890

30 71 65 45 44

0.450

0.46676 53390 47

0.42285 39261 33

0.500

0.52359 87755 98

r L

(- 8 ~ 1 A A

r(- 7 ) 1 i L

J

2

= 1,57079 63267 95

4

J

0.46364 76090 01 r( - 8 ~ 7 L

4

J

208

ELEMENTARY TRANSCENDENTAL FUNCTIONS

INVERSE CIRCULAR SINES AND TANGENTS

Table 4.14 5

arcsin x

arctan x

arcsin x

2

arctan x

0.500 0.501 0.502 0.503 0.504

0.52359 0.52475 0.52590 0.52706 0.52822

87755 38615 97203 63552 37691

98 51 91 20 54

0.46364 0.46444 0.46524 0.46604 0.46684

76090 72889 63286 47278 24863

01 58 62 61 09

0.550 0.551 0.552 0.553 0.554

0.58236 0.58356 0.58476 0.58596 0.58716

42378 20792 08688 06104 13082

69 89 33 84 43

0.50284 0.50361 0.50437 0.50514 0.50590

32109 06410 74226 35557 90402

28 37 73 57 12

0.505 0.506 0.507 0.508 0. 509

0.52938 0.53054 0.53170 0.53286 0.53402

19653 09468 07169 12787 26354

22 69 56 56 61

0.46763 0.46843 0.46923 0.47002 0.47082

96037 60799 19147 71077 16589

63 83 34 82 00

0.555 0.556 0.557 0.558 0.559

0.58836 0.58956 0.59076 0.59197 0.59317

29661 55882 91785 37411 92803

37 10 32 92 04

0.50667 0.50743 0.50820 0.50896 0.50972

38759 80629 16011 44903 67306

68 53 02 52 43

0.510 0.511 0.512 0.513 0.514

0.53518 0.53634 0.53751 Oi53867 0.53984

47902 77464 15071 60756 14552

76 20 30 57 69

0.47161 0.47240 0.47320 0.47399 0.47478

55678 88344 14584 34396 47777

62 48 38 20 82

0.560 0.561 0.562 0.563 0.564

0.59438 0.59559 0.59680 0.59801 0.59922

58000 33044 17978 12842 17681

01 41 05 95 37

0.51048 0.51124 0.51200 0.51276 0.51352

83219 92641 95572 92011 81958

17 21 04 19 22

0.515 0.516 0.517 0.518 0.519

0.54100 0.54217 0.54334 0.54451 0.54568

76492 46608 24935 11504 06350

49 96 25 67 69

0.47557 0.47636 0.47715 0.47794 0.47873

54727 55242 49320 36961 18161

17 22 97 45 73

0.565 0.566 0.567 0.568 0.569

0.60043 0.60164 0.60285 0.60407 0.60528

32535 57448 92463 37623 92971

81 99 89 71 89

0.51428 0.51504 0.51580 0.51655 0.51731

65412 42374 12842 76817 34297

69 25 52 18 96

0.520 0.521 0.522 0: 523 0.524

0.54685 0.54802 0.54919 0.55036 0.55154

09506 21007 40885 69176 05913

96 28 61 11 07

0.47951 OI48030 0.48109 0.48187 0.48266

92919 61234 23102 78523 27495

93 17 64 54 12

0.570 0.571 0.572 OI 573 0.574

0.60650 0.60772 0.60894 0.61016 0.61138

58552 34408 20584 17125 24076

13 36 75 74 01

0.51806 0.51882 0.51957 0.52032 0.52108

85284 29776 67774 99277 24285

57 79 41 27 22

0.525 0.526 0.527 0.528 0.529

0.55271 0.55389 0.55506 0.55624 0.55742

51130 04864 67148 38017 17507

97 46 37 69 59

0.48344 0.48423 0.48501 0.48579 0.48657

70015 06083 35696 58854 75554

67 0 4 40 29

8

0.575 0.576 0.577 0.578 0.579

0.61260 0.61382 0.61505 0.61627 0.61750

41480 69384 07833 56872 16548

49 37 09 37 17

0.52183 0.52258 0.52333 0.52408 0.52483

42798 54815 60338 59366 51898

14 96 62 09 38

0.530 0,531 0.532 0.533 0.534

0.55860 0.55978 0.56096 0.56214 0.56332

05653 02490 08055 22382 45509

43 72 18 69 33

0.48735 0.48813 0.48891 0.48969 0.49047

85795 89575 86893 77747 62137

05 18 19 65 12

0.580 0.581 0.582 0.583 0.584

0.61872 0.61995 0.62118 0.62241 0.62364

86906 67994 59858 62545 76102

72 52 34 21 44

0.52558 0.52633 0.52707 0,52782 0.52857

37935 17477 90524 57076 17134

52 57 63 82 28

0.535 0.536 0.537 0.538 0.539

0.56450 0.56569 0.56687 0.56806 0.56924

77471 18305 68047 26734 94404

34 17 44 97 76

0.49125 0.49203 0.49280 0.49358 0.49435

40060 11515 76502 35018 87062

25 68 10 23 83

0.585 0.586 0.587 0.588 0.589

0.62488 0.62611 0.62734 0.62858 0.62982

00577 36018 82473 39990 08619

61 60 54 87 28

0.52931 0.53006 0.53080 0.53154 0.53229

70697 17765 58340 92420 20007

19. 76 23 86 93

0.540 0.541 0.542 0.543 0.544

0.57043 0,57162 0,57281 0.57400 0.57519

71094 56840 51680 55653 68796

00 08 58 28 15

0.49513 0.49590 0.49668 0.49745 0.49822

32634 71732 04355 30502 50171

68 62 48 17 59

0.590 0.591 0.592 0.593 0.594

0.63105 0.63229 0.63353 0.63477 0.63602

88407 79405 81662 95228 20152

78 66 50 17 84

0.53303 0.53377 0.53451 0.53525 0.53599

41101 55702 63811 65427 60552

77 73 18 53 20

0.545 0.546 0.547 0.548 0.549

0.57638 0.57758 0.57877 0.57997 0.58116

91147 22745 63628 13835 73406

36 29 51 79 12

0.49899 0.49976 0.50053 0.50130 0.50207

63362 70074 70305 64056 51324

71 50 98 22 28

0.595 0.596 0.597 0.598 0.599

0.63726 0.63851 0.63975 0.64100 0.64225

56487 04281 63587 34455 16938

00 42 17 66 57

0.53673 0.53747 0.53821 0.53894 0.53968

49185 31328 06980 76143 38817

66 39 90 74 48

0.550

0.58236 42378 69

0.50284 32109 28

0.600

0.64350 11087 93

ri-7m

L'

4'

I

"I"':-

57079 63267 95

0.54041 95002 71

209

ELEMENTARY TRANSCENDENTAL FUNCTIONS

INVERSE CIRCULAR SINES AND TANGENTS X

0.600 0.601 0.602 0.603 0.604

arcsin x 0.64350 0.64475 0.64600 0.64725 0.64851

11087 16956 34595 64059 05401

arctan x 93 07 63 60 26

0.54041 0.54115 0.54188 0.54262 0.54335

95002 44700 87910 24633 54871

0.650 0.651 0.652 0.653 0.654

arctan x

arcsin x

X

71 04 15 69 37

Table 4.14

0.70758 0.70890 0.71021 0.71153 0.71285

44367 10818 92154 88447 99773

25 82 53 93 14

0.57637 0.57707 0.57777 0.57848 0.57918

52205 78870 99113 12935 20337

91 95 37 07 94

I

I I

0.605 0.606 0.607 0.608 0.609

0.64976 0.65102 0.65228 0.65353 0.65479

58674 23932 01230 90622 92163

24 51 34 38 58

0.54408 0.54481 0.54555 0.54628 0.54701

78623 95892 06676 10978 08798

92 10 70 51 38

0.655 0.656 0.657 0.658 0.659

0.71418 0.71550 0.71683 0.71815 0.71948

26204 67817 24688 96892 84506

76 97 45 45 75

0.57988 0.58058 0.58128 0.58197 0.58267

21323 15895 04053 85800 61138

94 01 13 31 57

0. 610 0.611 0.612 0.613 0.614

0.65606 0.65732 0.65858 0.65985 0.66111

05909 31915 70237 20931 84055

25 05 00 44 09

0.54774 0.54846 0.54919 0.54992 0.55065

00137 84995 63375 35276 00699

16 75 05 01 59

0.660 0.661 0.662 0.663 0.664

0.72081 0.72215 0.72348 0.72481 0.72615

87608 06276 40587 90622 56459

70 21 76 40 74

0.58337 0.58406 0.58476 0.58545 0.58615

30069 92596 48720 98443 41768

94 49 31 49 17

0.615 0.616 0.617 0.618 0.619

0.66238 0.66365 0.66492 0.66619 0.66746

59665 47818 48573 61988 88121

02 67 84 69 78

0.55137 0.55210 0.55282 0.55354 0.55427

59646 12118 58116 97640 30692

79 61 10 33 38

0.665 0.666 0.667 0.668 0.669

0.72749 0.72883 0.73017 0.73151 0.73286

38180 35864 49593 79449 25515

01 02 16 44 49

0.58684 0.58754 0.58823 0.58892 0.58961

78696 09230 33372 51125 62489

50 63 77 11 89

0.620 0.621 0.622 0.623 0.624

0.66874 0.67001 0.67129 0.67257 0.67385

27032 78778 43421 21020 11636

02 71 53 54 20

0.55499 0.55571 0.55643 0.55715 0.55787

57273 77384 91026 98201 98910

39 48 82 62 07

0.670 0.671 0.672 0.673 0.674

0.73420 0.73555 0.73690 0.73825 0.73961

87874 66610 61807 73551 01927

53 44 69 41 39

0.59030 0.59099 0.59168 0.59237 0.59306

67469 66065 58281 44118 23579

35 77 44 66 77

0,625 0.626 0.627 0.628 0.629

0.67513 0.67641 0.67769 0.67898 0.68026

15329 32161 62193 05488 62108

37 29 62 41 12

0.55859 0.55931 0.56003 0.56075 0.56147

93153 80932 62249 37105 05501

44 97 97 74 63

0.675 0.676 0.677 0.678 0.679

0.74096 0.74232 0.74367 0.74503 0.74639

47022 08922 87716 83492 96338

03 43 32 13 96

0.59374 0.59443 0.59512 0.59580 0.59649

96667 63383 23729 77710 25326

11 05 99 32 49

0.630 0.631 0.632 0.633 0.634

0.68155 0.68284 0.68413 0.68542 0.68671

32115 15574 12547 23100 47295

63 24 66 04 93

0.56218 0.56290 0,56361 0.56433 0.56504

67439 22919 71943 14513 50631

00 24 75 97 37

0.680 0.681 0.682 0.683 0.684

0.74776 0.74912 0.75049 0.75186 0.75323

26346 73605 38206 20242 19805

60 52 91 68 42

0.59717 0.597E6 0.59854 0.59922 0.59990

66580 01476 30014 52198 68031

93 11 52 66 06

0.635 0.636 0.637 0.638 0. 639

0.68800 0.68930 0.69060 0,69189 0.69319

85200 36878 02396 81821 75218

35 74 97 37 73

0.56575 0.56647 0.56718 0.56789 0.56860

80297 03513 20281 30602 34478

42 63 53 67 63

0.685 0.686 0.687 0.688 0.689

0.75460 0.75597 0.75735 0.75872 0.76010

36988 71885 24592 95204 83816

49 95 63 10 68

0.60058 0.60126 0.60194 0.60262 0.60330

77514 80650 77443 67894 52006

26 81 31 35 54

0.640 0.641 0.642 0.643 0.644

0,69449 0,69580 0.69710 0.69840 0.69971

82656 04201 39923 89889 54169

27 68 13 23 09

0.56931 0.57002 0.57073 0.57143 0.57214

31911 22901 07451 85562 57237

01 42 52 98 47

0.690 0.691 0.692 0.693 0.694

0.76148 0.76287 0.76425 0.76564 0.76703

90527 15434 58636 20231 00322

48 36 00 84 15

0.60398 0.60466 0.60533 0.60601 0.60668

29782 01224 66336 25119 77577

53 96 52 88 76

0.645 0.646 0.647 0.648 0.649

0.70102 0.70233 0.70364 0.70495 0.70626

32832 25948 33589 55824 92726

27 84 34 80 76

0.57285 0.57355 0.57426 0.57496 0.57567

22476 81282 33656 79600 19116

73 48 48 51 38

0,695 0.696 0.697 0.698 0.699

0.76841 0.76981 0.77120 0.77260 0.77399

99008 16391 52574 07661 81757

00 29 75 95 30

0.60736 0.60803 0.60870 0.60938 0.61005

23712 63528 97025 24209 45081

89 01 88 28 01

0.650

0.70758

44367 25

0.57637

52205 9 1

0.700

0.77539 74966 11

E = 1.57079 63267 95 2

0.61072 59643 89

I

210

ELEMENTARY TRANSCENDENTAL FUNCTIONS

INVERSE CIRCULAR SINES AND TANGENTS

Table 4.14 X

arcsin x

arctan x

arcsin x

X

0.700 0.701 0.702 0.703 0.704

0.77539 0.77679 0.77820 0.77960 0.78101

74966 87394 19149 70339 41072

11 52 57 20 23

0.61072 0.61139 0.61206 0.61273 0.61340

59643 67900 69854 65507 54863

89 75 44 83 79

0.750 0.751 0.752 0.753 0.754

0.705 0.706 0.707 0.708 0.709

0.78242 0.78383 0.78524 0.78666 0.78807

31458 41608 71633 21647 91762

43 47 95 44 45

0.61407 0.61474 0.61540 0.61607 0.61674

37925 14695 85176 49371 07284

25 10 29 78 52

0.755 0.756 0.757 0.758 0.759

0.710 0.711 0.712 0.713 0.714

0.78949 0.79091 0.79234 0.79376 0.79519

82093 92755 23866 75542 47901

46 96 39 24 99

0.61740 0.61807 0.61873 0.61939 0.62006

58917 04273 43356 76168 02712

52 76 27 09 26

0.715 0.716 0.717 0.718 0.719

0.79662 0.79805 0.79948 0.80092 0.80236

41065 55152 90285 46586 24179

16 32 08 13 26

0.62072 0.62138 0.62204 0.62270 0.62336

22991 37009 44769 46274 41526

0.720 0.721 0.722 0.723 0.724

0.80380 0,80524 0.80668 0.80813 0.80958

23189 43742 85965 49986 35935

33 33 35 66 64

0.62402 0.62468 0.62533 0.62599 0.62665

0.725 0.726 0.727 0.728 0.729

0.81103 0.81248 0.81394 0.81540 0.81685

43942 74140 26660 01637 99207

88 11 28 58 37

0.730 0.731 0.732 0.733 0.734

0.81832 0.81978 0.82125 0.82272 0.82419

19506 62672 28844 18163 30770

0.735 0.736 0.737 0.738 0.739

0.82566 0.82714 0.82862 Oi83010 0.83158

0.740 0.741 0.742 0.743 0.744

arctan x

0.84806 0.84957 0.85109 0.85260 0.85413

20789 52355 10007 93916 04254

81 56 70 63 45

0.64350 0.64414 0.64477 0.64541 0.64605

11087 08016 98804 83456 61974

93 53 75 20 52

0.85565 0.85718 0.85870 0.86024 0.86177

41195 04914 95588 13398 58524

04 02 84 74 85

0.64669 0.64733 0.64796 0.64860 0.64923

34363 00626 60767 14789 62697

37 40 30 75 45

0.760 0.761 0.762 0.763 0.764

0.86331 0.86485 0.86639 0.86794 0.86949

31150 31459 59639 15879 00370

16 55 86 89 42

0.64987 0.65050 0.65113 0.65176 0.65240

04494 40183 69769 93255 10645

12 48 28 25 18

86 97 70 14 45

0.765 0.766 0.767 0.768 0.769

0.87104 0,87259 0.87415 0.87571 0.87727

13304 54876 25283 24724 53401

26 26 38 65 29

0.65303 0,65366 0.65429 0.65492 0.65555

21942 27151 26276 19320 06286

83 99 46 05 59

30529 13287 89802 60077 24116

77 26 10 48 63

0.770 0.771 0.772 0.773 0.774

0.87884 0.88040 0.88198 0.88355 0.88513

11516 99276 16888 64562 42511

69 42 33 55 51

0.65617 0.65680 0.65743 0.65805 0.65868

87179 62003 30762 93459 50098

91 87 31 11 15

0.62730 0.62796 0.62861 0.62927 0.62992

81922 33499 78848 17975 50882

76 11 95 54 17

0.775 0.776 0.777 0.778 0.779

0.88671 0.88829 0.88988 0.89147 0.89306

50950 90095 60166 61386 93979

00 19 70 58 43

0.65931 0.65993 0.66055 0.66118 0.66180

00683 45218 83707 16154 42563

33 55 72 79 67

32 31 52 44 85

0.63057 0.63122 0.63188 0.63253 0.63318

77572 98048 12315 20375 22232

15 79 41 38 04

0.780 0.781 0.782 0.783 0.784

0.89466 0.89626 0.89786 0.89947 0.90108

58172 54195 82279 42661 35578

34 03 83 72 41

0.66242 0.66304 0.66366 0.66428 0.66490

62938 77282 85600 87896 84174

33 73 83 62 09

66809 26424 09761 16968 48191

86 94 92 01 83

0.63383 0.63448 0.63512 0.63577 0.63642

17888 07348 90616 67693 38584

78 99 06 42 50

0.785 0.786 0.787 0.788 0.789

0.90269 0.90431 0.90593 0.90755 0.90917

61270 19980 11956 37444 96699

38 87 01 80 17

0.66552 0.66614 0,66676 0.66738 0.66799

74437 58690 36936 09181 75427

26 12 71 07 24

0.83307 0.83455 0.83604 0.83754 0.83903

03583 83294 87477 16286 69878

42 24 24 83 93

0.63707 OI63771 0.63836 0.63900 0.63965

03292 61821 14174 60355 00366

76 64 63 21 89

0.790 0.791 0.792 0.793 0.794

0.91080 0,91244 0.91407 0.91571 0.91736

89974 17527 79620 76517 08485

07 48 46 23 19

0,66861 0.66922 0.66984 0.67045 0.67107

35679 89941 38217 80511 16827

28 25 24 32 61

0.745 0.746 0.747 0.748 0.749

0.84053 0.84203 0.84353 0.84504 0.84655

48410 52041 80932 35244 15141

98 95 39 42 77

0.64029 0.64093 0.64157 Oi64221 0.64286

34213 61897 83423 98795 08015

19 63 76 14 33

0.795 0.796 0.797 0.798 0.799

0.91900 0.92065 0.92231 0.92396 0.92563

75795 78720 17539 92532 03983

02 67 49 24 15

0.67168 0.67229 0.67290 0.67352 0.67413

47170 71543 89950 02397 08886

20 22 79 05 15

0.750

0.84806 20789 81

0.64350 11087 93

0.800

0.92729 52180 02

’

r[(-;I51 (- 7)5i L

5 = 1.57079 63267 95 2

5

J

0.67474 09422 24

L 4 J

21 1

ELEMENTARY TRANSCENDENTAL FUNCTIONS

INVERSE CIRCULAR SINES AND TANGENTS arcsin x

arctan x

Table 4.14

arcsin x

X

arctan x

0.800 0.m 0.802 0.303 0.804

0.92729 0.92896 0.93063 0.93231 0.93399

52180 37414 59980 20178 18310

02 22 83 64 25

0.67474 0.67535 0.67595 0.67656 0.67717

09422 04009 92652 75354 52120

24 49 08 19 01

0.850 0.851 0.852 0.853 0.854

1.01598 1.01788 1.01979 1.02170 1.02362

52938 65272 36361 66824 57289

15 25 62 41 29

0.70449 0.70507 0.70565 0.70623 0.70681

40642 43293 40219 31425 16914

42 58 63 16 73

0.805 0.806 0.807 0.808 0.809

0.93567 0.93736 0.93905 0.94074 0.94244

54682 29604 43392 96363 88840

12 66 28 49 95

0.67778 0.67838 0.67899 0.67959 0.68020

22953 87859 46841 99904 47052

77 65 90 74 41

0.855 0.856 0.857 0.858 0.859

1.02555 1.02748 1.02941 1.03136 1.03331

08395 20794 95147 32127 32420

76 40 10 41 77

0.70738 0.70796 0.70854 0.70912 0.70969

96692 70764 39133 01805 58784

96 42 73 50 34

0.810 0.811 0.812 0.813 0.814

0.94415 0.94585 0.94757 0.94928 0.95100

21151 93626 06601 60416 55415

54 48 38 29 87

0.68080 0.68141 0.68201 0.68261 0.66321

88289 23619 53046 76576 94212

16 25 96 55 31

0.860 0.861 0.862 0.863 0.864

1.03526 1.03723 1.03920 1.04117 1.04316

96724 25749 20218 80867 08445

81 68 39 05 30

0.71027 0.71084 0.71141 0.71199 0.71256

10074 55681 95609 29862 58446

87 72 52 92 55

0.815 0.816 0; 817 0.818 0; 819

0,95272 0.95445 0.95618 0.95792 0.95966

91949 70370 91039 54318 60576

40 88 18 04 23

0.68382 0.68442 0.68502 0.68562 0.68621

05958 11819 11799 05903 94134

54 54 62 10 31

0.865 0.866 0.867 0.868 0.869

1.04515 1.04714 1.04915 1.05116 1.05317

03716 67458 00463 03538 77506

61 63 62 76 61

0.71313 0.71370 0.71428 0.71485 0.71542

81365 98623 10225 16176 16481

07 14 41 56 25

1.05520 1.05723 1.05927 1.0613! 1.0633

23205 41489 33231 99317 40653

49 91 01 03 78

0.71599 0.71656 0.71712 0.71769 0.71826

11144 00169 83563 61329 33471

16 99 41 12 82

0.820 0.821 0.822 0.823 0.824

0.96141 0.96316 0.96491 0.96667 0.96843

10187 03531 40991 22958 49827

64 36 79 76 60

0.68681 0.68741 0.68801 0.68860 0.68920

76497 52997 23637 88423 47358

59 28 73 31 39

0.870 0.871 0.872 0.873 0.874

0.825 0.826 0.827 0.828 0. 829

0.97020 0.97197 0.97375 0.97553 0.97731

21999 39880 03884 14428 71937

29 56 00 17 77

0.68980 0.69039 0.69098 0.69158 0.69217

00447 47694 89104 24681 54429

34 55 41 33 71

0.875 0.876 0.877 0.878 0.879

1.06543 1.06750 1.06958 1.07166 1.07376

58165 52793 25500 77266 09094

11 43 24 67 07

0.71882 0,71939 0.71996 0.72052 0.72109

99996 60907 16208 65906 10005

22 02 94 70 03

0.830 0.831 0.832 0.833 0. 834

0.97910 0.98090 0.98270 0.98450 0.98631

76843 29583 30600 80344 79274

68 19 05 64 13

0.69276 0.69335 0.69395 0.69454 0.69513

78353 96458 08747 15226 15898

97 54 85 33 44

0.880 0.881 0.882 0.883 0.884

1.07586 1.07797 1.08008 1.08221 1.08435

22004 17041 95270 57780 05681

54 59 75 22 59

0.72165 0.72221 0.72278 0.72334 0.72390

48508 81422 08750 30498 46670

65 30 71 64 83

0.835 0.836 0. 837 0. 838 0.839

0.98813 0.98995 0.99177 0.99360 0.99544

27852 26551 75847 76228 28187

56 06 95 94 22

0.69572 0.69630 0.69689 0.69748 0.69807

10768 99841 83121 60612 32319

63 36 11 34 55

0.885 0.886 0.887 0.888 0.889

1.08649 1.08864 1.09080 1.09297 1.09515

40110 62227 73218 74295 66698

49 36 22 43 56

0.72446 OI72502 0.72558 0.72614 0.72670

57272 62307 61780 55697 44062

04 01 53 34 23

0.840 0.841 0.842 0.843 0.844

0.99728 0.99912 1.00097 1.00283 1.00469

32223 88847 98574 61930 79448

72 18 39 35 46

0.69865 0.69924 0.69983 0.70041 0.70100

98247 58399 12781 61398 04252

21 85 94 02 59

0.890 0.891 0,892 0.893 0.894

1.09734 1.09954 1.10175 1.10396 1.10619

51695 30581 04685 75362 44002

23 99 30 43 56

0.72726 0.72782 0.72837 0.72893 0.72949

26879 04155 75893 42098 02775

97 34 12 11 09

0.845 0.846 0.847 0.848 0.849

1.00656 1.00843 1.01031 1.01220 1.01408

51670 79147 62439 02114 98751

67 75 41 56 50

0.70158 0.70216 0.70274 0.70333 0.70391

41350 72695 98292 18146 32261

19 35 60 49 58

0.895 0.896 0.897 0.898 0.899

1.10843 1.11067 1.11293 1.11520 1.11748

12027 80894 52092 27151 07636

75 12 94 85 13

0.73004 0.73060 0.73115 0.73170 0.73226

57928 07564 51686 90299 23408

87 24 02 00 01

0.850

1.01598 52938 15

0.70449 40642 42

0.900

1.11976 95149 99

2

= 1.57079 63267 95

[(-:"I

0.73281 51017 87 (- 8 m

r

1

I I

I

I

i

i

II I

212

ELEMENTARY TRANSCENDENTAL FUNCTIONS

INVERSE CIRCUI.AR SINES AND TA NGENTS

Table 4..14

arcsin .t

T

arctan .t

T

f (4

arctan 2

arcsin T

0.900 0.901 0.902 0.903 0.904

1.11976 1.12206 1.12437 1.12670 1.12903

95149 91337 97886 16524 49026

99 93 21 29 45

0.73281 0.73336 0.73391 0.73447 0.73502

51017 73133 89759 00900 06562

87 38 38 70 16

0.950 0.951 0.952 0.953 0.954

1.25323 1.25645 1.25970 1.26298 1.26630

58975 42223 47250 84259 64000

03 06 03 28 67

0.75976 0.76028 0.76081 0.76133 0.76186

27548 81166 29540 72674 10574

76 70 28 43 14

1.00421 1.00412 1.00404 1.00395 1.00387

42513 90197 38274 86742 35601

02 55 04 15 52

0.905 0.906 0.907 0.908 0.909

1.13137 1.13373 1.13610 1.13848 1.14087

97213 62953 48166 54823 84946

39 96 99 12 83

0.73557 0.73612 0.73666 0.73721 0.73776

06748 01464 90715 74506 52841

62 89 84 30 13

0,955 0.956 0.957 0.958 0.959

1.26965 1.27304 1.27647 1.27994 1.28345

97812 97667 76222 46878 23838

42 20 92 88 00

0.76238 0.76290 0.76342 0.76395 0.76447

43244 70690 92916 09927 21729

37 08 23 81 78

1. 00378 1.00370 1.00361 1.00353 1.00344

84851 34492 84523 34944 85754

78 58 57 39 69

0.910 0.911 0.912 0.913 0.914

1.14328 1.14570 1.14813 1.15057 1.15303

40618 23976 37219 82610 62474

50 58 91 10 12

0.73831 0.73885 0.73940 0.73995 0.74049

25725 93163 55160 11721 62850

17 30 36 22 76

0.960 0.961 0.962 0.963 0.964

1.28700 1.29059 1.29423 1.29792 1.30165

22175 57917 48124 10987 65939

87 69 14 43 20

0.76499 0.76551 0.76603 0.76655 0.76707

28327 29724 25927 16941 02769

11 78 75 02 55

1.00336 1.00327 1.00319 1.00310 1.00302

36954 88542 40518 92883 45635

10 28 88 53 89

0.915 0.916 0.917 0.918 0.919

1.15550 1.15799 1.16049 1.16300 1.16553

79206 35274 33215 75647 65266

90 19 50 25 04

0.74104 0.74158 0.74212 0.74267 0.74321

08553 48835 83700 13153 37199

83 32 10 04 05

0.965 0.966 0.967 0.968 0.969

1.30544 1.30928 1.31317 1.31713 1.32115

33771 36776 98896 45907 05615

97 35 52 19 54

0.76758 0. 76810 0.76862 0.76913 0.76965

83418 58892 29196 94335 54315

33 33 53 92 49

1.00293 1.00285 1.00277 1.00268 1.00260

98775 52302 06215 60515 15201

61 33 71 39 02

0.920 0.921 0.922 0.923 0.924

1.16808 1.17063 1.17321 1.17580 1.17841

04852 97273 45487 52550 21615

14 16 95 71 31

0.74375 0.74429 0.74483 0,74537 0.74591

55842 69089 76944 79411 76495

99 76 25 35 97

0.970 0.971 0.972 0.973 0.974

1.32523 1.32937 1.33359 1.33789 1.34226

08092 85940 74601 12711 42528

80 93 02 79 47

0.77017 0.77068 0.77120 0.77171 0.77222

09140 58815 03345 42735 76990

20 06 05 14 34

0.925 0.926 0.927 0.928 0.929

1.18103 1.18367 1.18633 1.18900 1.19170

55939 58892 33953 84725 14936

97 09 44 71 35

0.74645 0.74699 0.74753 0.74807 0.74860

68203 54537 35503 11107 81353

00 35 92 62 36

0.975 0.976 0.977 0.978 0.979

1.34672 1.35126 1.35590 1.36064 1.36549

10414 67425 69996 80777 69629

93 45 85 70 42

0.77274 0.77325 0.77376 0.77427 0.77478

06115 30116 48996 62761 71417

63 01 45 95 51

1.00209 1.00201 1.00192 1.00184 1.00175

51400 08777 66538 24682 83208

25 78 49 01 02

0.930 0.931 0.932 0.933 0.934

1.19441 1.19714 1.19989 1.20266 1.20544

28444 29249 21492 09472 97647

77 00 75 92 69

0.74914 0.74968 0.75021 0.75075 0.75128

46246 05790 59991 08855 52384

06 63 99 06 76

0.980 0.981 0.982 0.983 0.984

1.37046 1.37555 1.38077 1.38614 1.39167

14844 04644 39033 32129 15119

72 29 32 70 16

0.77529 0.77580 0.77631 0.77682 0.77733

74968 73418 66774 55040 38220

12 77 45 17 91

1.00167 1.00159 1.00150 1.00142 1.00133

42116 01406 61077 21129 81563

16 08 45 93 16

0.935 0.936 0.937 0.938 0.939

1.20825 1.21108 1.21394 1.21681 1.21971

90645 93272 10524 47598 09898

07 10 70 22 74

0.75181 0.75235 0.75288 0.75341 0.75394

90586 23463 51022 73268 90205

03 79 96 49 30

0.985 0.986 0.987 0.988 0.989

1.39737 1.40326 1.40937 1.41572 1.42233

40056 84832 59766 16538 60557

99 96 46 31 98

0.77784 0.77834 0.77885 0.77936 0.77986

16321 89347 57303 20194 78024

67 44 23 04 85

1.00125 1.00117 1.00108 1.00100 1.00091

42376 03570 65143 27096 89428

80 52 98 82 72

0.940 0.941 0.942 0.943 0.944

1.22263 1.22557 1.22854 1.23153 1.23455

03055 32932 05645 27575 05382

22 59 81 05 02

0.75448 0.75501 0.75554 0.75607 0.75659

01838 08172 09212 04964 95431

34 55 86 22 57

0.990 0.991 0.992 0.993 0.994

1.42925 1.43653 1.44422 1.45240 1.46119

68534 14207 07408 56012 69689

70 77 32 67 63

0.78037 0.78087 0.78138 0.78188 0.78238

30800 78526 21207 58848 91453

67 49 32 15 98

1.00083 1.00075 1.00066 1.00058 1.00050

52139 15228 78695 42540 06762

33 31 32 02 08

0.945 0.946 0.947 0.948 0.949

1.23759 1.24066 1.24376 1.24689 1.25004

46027 56791 45292 19509 87811

74 62 24 90 06

0.75712 0.75765 0.75818 0.75871 0.75923

80619 60534 35179 04559 68681

86 05 08 90 48

0.995 0.996 0.997 0.998 0,999

1.47075 1.48132 1.49331 1.50754 1.52607

46131 37665 72818 02279 12396

83 90 71 20 26

0.78289 0.78339 0.78389 0.78439 0.78489

19029 41580 59111 71627 79133

81 64 47 31 14

1.00041 1.00033 1.00025 1.00016 1.00008

71361 36336 01689 67417 33520

15 91 01 11 89

0.950

1.25323 58975 03

0.75976 27548 76

1.000

1.57079 63267 95

[(-;)41 For arctan

,c,

.1>1see Example 22.

0.78539 81633 97

[(-p71 5=1.57079 63267 95

1.00000 00000 00

I!-([

51

I I

213

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.15

HYPERBOLIC FUNCTIONS X

sinh

0.00 0. 01 0.02 0. 03 0. 04

0.00000 0000 0.01000 0167

2

cosh z

tanh x

coth x m

0.02000 1333 0.03000 4500 0. 04001 0668

1.00000 1.00005 1.00020 1.00045 1.00080

0000 0000 0007 0034 0107

0.00000 0.00999 0.01999 0.02999 0.03997

000 967 733 100 868

100.00333 50.00666 33.34333 25.01333

33 65 27 19

0. 05 0. 06 0. 07 0.08 0.09

0.05002 0.06003 0.07005 0. 08008 0.09012

0836 6006 7181 5361 1549

1.00125 1.00180 1.00245 1.00320 1.00405

0260 0540 1001 1707 2734

0.04995 0.05992 0.06988 0.07982 0.08975

838 810 589 977 779

20.01666 16.68666 14.30904 12.52665 11.14109

39 19 00 53 49

0.10 0.11 0.12 0. 13 0. 14

0.10016 0.11022 0.12028 0.13036 0.14045

6750 1968 8207 6476 7782

1.00500 1.00605 1.00720 1.00846 1.00981

4168 6103 8644 1907 6017

0.09966 0.10955 0.11942 0.12927 0.13909

800 847 730 258 245

10.03331 9.12754 8.37329 7.73559 7.18946

11 62 50 23 29

0.15 0. 16 0. 17 0.18 0.19

0.15056 0.16068 0.17082 0.18097 0.19114

3133 3541 0017 3576 5232

1.01127 1.01282 1.01448 1.01624 1. 01810

1110 7330 4834 3787 4366

0.14888 oIi5864 0.16838 Oil7808 0,18774

503 850 105 087 621

6.71659 6.30324 5.93891 ~. 5.61542 5.32633

18 25 07 64 93

0.20 0.21 0. 22 0. 23 0. 24

0.20133 0.21154 0.22177 0.23203 0.24231

6003 6907 8966 3204 0645

1.02006 1.02213 1.02429 1. 02656 1.02893

6756 1153 7764 6806 8506

0.19737 0.20696 0.21651 0,22602 0.23549

532 650 806 835 575

5.06648 4.83169 4.61855 4.42422 4.24636

96 98 23 37 11

0. 25 0.26 0.27 0.28 0.29

0.25261 0.26293 0.27329 0.28367 0.29408

2317 9250 2478 3035 1960

1.03141 1.03399 1,03667 1.03945 1.04234

3100 0836 1973 6777 5528

0.24491 0.25429 0.26362 0.27290 0.28213

866 553 484 508 481

4.08298 3.93243 3.79326 3.66427 3.54440

82 24 93 77 49

0.30 0.31 0. 32 0. 33 0.34

0,30452 0.31498 0.32548 0.33602 0.34658

0293 9079 9364 2198 8634

1.04533 1.04843 1.05163 1.05494 1.05835

8514 6035 8401 5931 8957

0.29131 0.30043 0.30950 0.31852 0.32747

261 710 692 078 740

3.43273 3.32848 3.23094 3.13951 3.05364

84 38 55 26 59

0.35 0.36 0.37 0. 38 0.39

0.35718 0.36782 0.37850 0. 38921 0,39996

9729 6544 0142 1590 1960

1.06187 1.06550 1.06923 1.07307 1.07701

7819 2870 4473 2999 8834

0,33637 0.34521 0.35399 0.36270 0.37136

554 403 171 747 023

2.97286 2.89675 2.82492 2.75704 2.69280

77 36 49 28 32

0. 40 0.41 0.42 0. 43 0. 44

0.41075 0.42158 0.43245 0.44337 0.45433

2326 3767 7368 4214 5399

1.08107 1.08523 1.08950 1.09388 1.09837

2372 4018 4188 3309 1820

0.37994 0.38847 0.39693 0.40532 0.41364

896 268 043 131 444

2.63193 2.57418 2.51933 2.46717 2.41753

24 36 32 85 52

0.45 0. 46 0.47 0.48 0.49

0.46534 0.47639 0.48749 0.49864 0.50984

2017 5170 5962 5505 4913

1.10297 1.10767 1.11249 1.11742 1.12247

0169 8815 8231 8897 1307

0.42189 0.43008 0.43819 0,44624 0.45421

901 421 932 361 643

2.37023 2.32512 2.28206 2.24092 2.20159

55 60 66 84 36

0.50

0.52109 5305

I“’,[

1.12762 5965

0.46211 716

2.16395 34

[(-;PI [(-291 For coth T , L I .1use 4.5.67. Compilation of tanh e and coth z from National Bureau of Standards, Table of circular and hyperbolic tangents and cotangents for radian arguments, 2d printing. Columbia Univ. Press, New York, N.Y., 1947 (with permission).

214

ELEMENTARY TRANSCENDENTAL FUNCTIONS

HYPERBOLIC FUNCTIONS

Table 4.15 X

sinh x

cosh x

tanh x

coth x

A

0.50 0. 51 0. 52 0.53 0. 54

0.52109 0.53239 0.54375 0.55516 0.56662

5305 7808 3551 3669 9305

1.12762 1.13289 1.13827 1.14376 1.14937

5965 3387 4099 8639 7557

0.46211 0.46994 0.47770 0.48538 0.49298

716 520 001 109 797

2.16395 2.12790 2.09336 2.06023 2.02844

34 77 40 68 71

0.55 0. 56 0.57 0.58 0.59

0.57815 0.58973 0.60137 0.61307 0.62483

1604 1718 0806 0032 0565

1.15510 1.16094 1.16689 1.17296 1.17915

1414 0782 6245 8399 7850

0.50052 0.50797 0.51535 0.52266 0.52989

021 743 928 543 561

1.99792 1.96859 1,94039 1.91326 1.88716

13 14 39 98 42

0.60 0. 61 0. 62 0. 63 0.64

0.63665 0.64854 0.66049 0.67250 0.68459

3582 0265 1802 9389 4228

1.18546 1.19189 1.19843 1.20510 1.21188

5218 1134 6240 1190 6652

0.53704 0.54412 0,55112 0.55805 0,56489

957 710 803 222 955

1.86202 1.83780 1.81446 1.79194 1.77022

55 59 04 70 62

0. 65 0.66 0. 67 0. 68 0.69

0.69674 Oi70897 0.72126 0.73363 0.74606

7526 0500 4371 0370 9732

1.21879 1.22582 1.23297 1.24024 1.24764

3303 1834 2949 7362 5801

0.57166 0.57836 0.58497 0.59151 0.59798

997 341 988 940 200

1.74926 1.72901 1.70946 1,69056 1.67229

10 67 05 16 11

0. 70 0.71 0.72 0.73 0.74

0.75858 0.77117 0.78384 0.79658 0.80941

3702 3531 0477 5809 0799

1.25516 1.26281 1.27059 1.27849 1.28652

9006 7728 2733 4799 4715

0.60436 0.61067 0.61690 0.62306 0.62914

778 683 930 535 516

1.65462 1.63752 1.62098 1.60496 1.58945

16 73 38 81 83

0. 75 0.76 0: 77 0. 78 0.79

0.82231 0.83530 0.84837 0.86153 0.87477

6732 4897 6593 3127 5815

1.29468 1.30297 1131138 1.31993 1.32862

3285 1324 9661 9138 0611

0.63514 0.64107 0.64692 0.65270 0.65840

895 696 945 671 904

1.57443 1.55987 1.54576 1.53208 1I51881

38 51 36 17 27

0. 80 0. 81 0. 82 0. 83 0. 84

0.88810 0.90152 0.91503 0.92863 0.94232

5982 4960 4092 4727 8227

1.33743 1.34638 1.35546 1.36468 1.37403

4946 3026 5746 4013 8750

0.66403 0.66959 0.67506 0.68047 0.68580

677 026 987 601 906

1.50594 1.49345 1.48132 1.46955 1.45813

07 06 81 95 18

0. 85 0. 86 0. 87 0. 88 0.89

0.95611 0.96999 0.98397 0.99805 1.01223

5960 9306 9652 8397 6949

1. 38353 1.39316 1.40293 1.41284 1.42289

0892 1388 1201 1309 2702

0.69106 0.69625 0.70137 0.70641 0.71139

947 767 413 932 373

1.44703 1.43624 1.42577 1.41558 1.40569

25 99 26 98 13

0. 90 0: 91 0. 92 0.93 0.94

1.02651 1.04089 1.05538 1.06997 1.08467

6726 9155 5674 7734 6791

1.43308 1I44342 1.45390 1.46453 1.47530

6385 3379 4716 1444 4627

0.71629 OI72113 0.72589 0.73059 0.73522

787 225 742 390 225

1.39606 il38670 1.37760 1.36874 1.36013

73 82 51 95 29

0.95 0.96 0.97 0.98 0.99

1.09948 1.11440 1.12943 1.14457 1.15982

4318 1794 0711 2572 8891

1.48622 1.49729 1.50851 1.51988 1.53140

5341 4680 3749 3670 5582

0.73978 0.74427 0.74870 0.75306 0.75736

305 687 429 591 232

1.35174 1.34358 1.33564 1,32790 1.32037

76 60 08 50 20

1.00

1.17520 1194

"!4- 1

1.54308 0635

w21

0.76159 416

1,31303 53

w21

215

ELEMENTARY TRANSCENDENTAL FUNCTIONS

HYPERBOLIC FUNCTIONS X

sinh x

cosh x

Table 4.15 tanh x

coth x

1.00 1. 01 1. 02 1.03 1.04

1.17520 1.19069 1.20629 1.22202 1.23788

1194 1018 9912 9437 1166

1.54308 1.55490 1.56689 1.57903 1.59133

0635 9997 4852 6398 5848

0.76159 0.76576 0.76986 0.77390 0.77788

4.i6 202 654 834 807

1.31303 1.30588 1.29892 1.29214 1.28553

53 87 64 27 20

1. 05 1. 06 1. 07 1.08 1. 09

1.25385 1.26995 1.28618 1.30254 1.31902

6684 7589 5491 2013 8789

1.60379 1.61641 1.62919 1.64213 1.65524

4434 3400 4009 7538 5283

0.78180 0.78566 0.78946 0.79319 0.79687

636 386 122 910 814

1.27908 1.27280 1.26668 il2607i 1.25489

91 90 67 75 70

1.10 1.11 1.12 1.13 1.14

1.33564 1.35239 1.36928 1.38631 1.40347

7470 9717 7204 1622 4672

1.66851 1.68195 1.69556 1.70934 1.72329

8554 8678 6999 4878 3694

0.80049 Oi80406 0.80756 0.81101 0.81441

902 239 892 926 409

1.24922 1.24368 1.23828 1.23301 1.22787

08 46 44 63 66

1.15 1.16 1.17 1.18 1.19

1.42077 1.43822 1.45581 1.47354 1.49142

8070 3548 2849 7732 9972

1.73741 1.75170 1.76617 1.78082 1.79565

4840 9728 9790 6471 1236

0.81775 0.82103 0.82427 0.82745 0.83057

408 988 217 161 887

1.22286 1.21796 1.21319 1.20852 1.20397

15 76 15 99 96

1.20 1.21 1.22 1. 2 3 1.24

1.50946 1.52764 1.54597 1.56446 1.58311

1355 3687 8783 8479 4623

1.81065 1.82584 1.84120 1.85676 1.87249

5567 0966 8950 1057 8841

0.83365 0.83667 0.83965 0.84257 0.84545

461 949 418 933 560

1.19953 1.19520 1.19096 1.18683 1.18279

75 08 65 19 42

1.25 1.26 1.27 1.28 1.29

1.60191 1.62088 1.64001 1.65930 1.67875

9080 3730 0470 1213 7886

1.88842 1.90453 1.92084 1.93733 1.95402

3877 7757 2092 8513 8669

0.84828 0.85106 0.85379 0.85648 0.85912

364 411 765 492 654

1.17885 1.17499 1.17123 1.16756 1.16397

10 96 77 29 29

1.30 1.31 1.32 1.33 1.34

1.69838 1.71817 1.73814 1.75828 1.77859

2437 6828 3038 3063 8918

1.97091 1.98799 2.00527 2.02276 2.04044

4230 6884 8340 0324 4587

0.86172 0.86427 0.86678 0.86924 0.87167

316 541 393 933 225

1.16046 1.15703 1.15369 1.15041 iIi4722

55 86 01 79 02

1.35 1.36 1.37 1.38 1.39

1.79909 1.81976 1.84062 1.86166 1.88288

2635 6262 1868 1537 7374

2.05833 2.07642 2.09472 2.11324 2.13196

2896 7039 8828 0090 2679

0.87405 0.87639 0.87869 0.88095 0.88317

329 307 219 127 089

1.14409 1.14104 1.13805 1.13513 1.13228

50 05 50 66 37

1.40 1. 41 1.42 1.43 1.44

1.90430 1.92590 1.94770 1.96969 1.99188

1501 6060 3212 5135 4029

2.15089 2.17004 2.18941 2.20900 2.22881

8465 9344 7229 4057 1788

0,88535 0.88749 0.88959 0.89166 0.89369

165 413 892 660 773

1.12949 1.12676 1.12410 1.12149 1.11894

47 80 21 54 66

1.45 1.46 1.47 1.48 1.49

2.01427 2.03686 2.05965 2.08265 2.10586

2114 1627 4828 3996 1432

2.24884 2.26909 2.28958 2.31029 2.33123

2402 7902 0313 1685 4087

0.89569 0.89765 0.89957 0.90146 0.90332

287 260 745 799 474

1.11645 1.11401 1.11163 1.10930 1.10702

41 67 30 17 16

1.50

2.12927 9455

2.35240 9615 [(-:I31

0.90514 825

['-y]

1.10479 1 4

[(-y]

216

ELEMENTARY TRANSCENDEN TAL FUNCTIONS

Table 4.15 X

HYPERBOLIC FUNCTIONS sinh x

cosh x

tanh x

coth x

1.50 1. 51 1.52 1. 53 1.54

2.12927 2.i5291 2.17675 2.20082 2.22510

9455 0408 6654 0577 4585

2.35240 2.37382 2.39546 2.41735 2.43948

9615 0386 8541 6245 5686

0.90514 0.90693 0.90869 0.91042 0.91212

825 905 766 459 037

1.10479 1.10260 1.10047 1.09838 1.09634

14 99 60 86 65

1. 55 1.56 1.57 1.58 1.59

2.24961 2.27434 2.29930 2.32449 2.34991

1104 2587 1506 0357 1658

2.46185 2.48447 2.50734 2.53046 2.55383

9078 8658 6688 5455 7270

0.91378 0.91542 0.91702 0.91860 0.92014

549 046 576 189 933

1.09434 1.09239 1.09048 1.08861 1.08678

87 42 19 09 01

1.60 1.61 1. 62 1. 63 1.64

2.37556 2.40146 2.42759 2.45397 2.48059

7953 1807 5809 2572 4735

2.57746 2.60134 2.62549 2.64990 2.67457

4471 9421 4508 2146 4777

0.92166 0.92316 0.92462 0.92606 0.92747

855 003 422 158 257

1.08498 1.08323 1.08152 1,07984 1.07819

87 58 04 18 90

1. 65 1.66 1. 67 1. 68 1. 69

2.50746 2.53458 2.56196 2.58959 2.61748

4959 5932 0366 0998 0591

2.69951 2.72472 2.75020 2.77596 2.80200

4868 4912 7431 4974 0115

0.92885 0.93021 0.93155 0.93286 0.93414

762 718 168 155 721

1.07659 1.07501 1,07347 1.07197 1.07049

13 78 77 04 51

1. 70 1. 71 1.72 1. 73 1.74

2.64563 2.67404 2.70273 2.73168 2.76091

1934 7843 1158 4749 1511

2.82831 2.85491 2.88179 2.90896 2.93643

5458 3635 7306 9159 1912

0.93540 0.93664 0.93786 0.93905 0.94022

907 754 303 593 664

1.06905 1.06763 1,06625 1.06489 1.06357

10 75 38 93 34

1.75 1. 76 1. 77 1. 78 1.79

2.79041 2.82019 2.85026 2.88060 2.91124

4366 6265 0186 9136 6148

2.96418 2.99224 3.02059 3. 04924 3.07820

8310 1129 3175 7283 6318

0.94137 0.94250 0.94360 0,94469 0.94576

554 301 942 516 057

1,06227 1,06100 1.05976 1.05854 1.05735

53 46 05 25 01

1. 80 1; 81 1.82 1. 83 1.84

2.94217 2.97339 3.00491 3. 03673 3.06886

4288 6648 6349 6545 0417

3.10747 3.13705 3.16694 3.19715 3.22767

3176 0785 2100 0113 7844

0.94680 0.94783 0.94883 0.94982 0.95079

601 185 842 608 514

1.05618 1.05503 1. 05392 1.05282 1.05175

26 95 02 43 13

1. 85 1. 86 1.87 1.88 1.89

3.10129 3.13403 3.16708 3.20045 3.23414

1178 2071 6369 7378 8436

3.25852 3.28970 3.32121 3.35304 3.38522

8344 4701 0031 7484 0245

0.95174 0.95267 0.95359 0.95449 0.95537

596 884 412 211 312

1. 05070 1.04967 1.04866 1.04767 1.04671

05 17 42 76 15

1.90 1; 91 1.92 1.93 1.94

3.26816 3.30250 3.33717 3.37218 3.40752

2912 4206 5754 1022 3510

3.41773 3.45058 3.48378 3.51732 3.55122

1531 4593 2716 9220 7460

0.95623 0.95708 0.95791 0.95873 0.95953

746 542 731 341 401

1.04576 1,04483 1.04393 1.04304 1.04217

53 88 14 28 25

1.95 1. 96 1.97 1.98 1.99

3.44320 3.47923 3.51560 3.55233 3.58941

6754 4322 9816 6874 9168

3.58548 3.62009 3.65506 3.69040 3.72611

0826 2743 6672 6111 4594

0.96031 0.96108 0.96184 0.96258 0.96331

939 983 561 698 422

1.04132 1.04048 1.03966 1.03886 1.03808

02 55 79 72 29

2. 00

3.62686 0408

3.76219 5691

0,96402 758

1.03731 47

,

21 7

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.15

HYPERBOLIC FUNCTIONS X

sinh x

tanhx .

cosh x

coth x

2.0 2.1 2.2 2.3 2.4

3.62686 4.02185 4.45710 4.93696 5.46622

0408 6742 5171 1806 9214

3.76219 4.14431 4.56790 5.03722 5.55694

5691 3170 8329 0649 7167

0.96402 0.97045 0.97574 0.98009 0.98367

75801 19366 31300 63963 48577

1.03731 1.03044 1.02485 1.02030 1.01659

47207 77350 98932 78022 60756

2.5 2.6 2.7 2.8 2.9

6.05020 6.69473 7.40626 8.19191 9.05956

4481 2228 3106 8354 1075

6.13228 6.76900 7.47346 8.25272 9.11458

9480 5807 8619 8417 4295

0.98661 0.98902 0.99100 0.99263 0.99396

42982 74022 74537 15202 31674

1.01356 1.01109 1.00907 1.00742 1.00607

73098 43314 41460 31773 34973

3.0 3.1 3.2 3.3 3.4

10.01787 11.07645 12.24588 13.53787 14.96536

4927 1040 3997 7877 3389

10.06766 11.12150 12.28664 13.57476 14.99873

1996 0242 6201 1044 6659

0.99505 0.99594 0.99668 0.99728 0.99777

47537 93592 23978 29601 49279

1.00496 1.00406 1.00332 1.00272 1.00223

98233 71152 86453 44423 00341

3.5 3.6 3.7 3.8 3.9

16.54262 18.28545 20.21129 22.33940 24.69110

7288 5361 0417 6861 3597

16.57282 18.31277 20.23601 22.36177 24.71134

4671 9083 3943 7633 5508

0.99817 0.99850 0.99877 0.99899 0.99918

78976 79423 82413 95978 08657

1.00182 1.00149 1.00122 1.00100 1.00081

54285 42872 32532 14040 98059

4.0 4.1 4.2 4.3 4.4

27.28991 30.16185 33.33566 36.84311 40.71929

7197 7461 7732 2570 5663

27.30823 30.17843 33.35066 36.85668 40.73157

2836 0136 3309 1129 3002

0.99932 0.99945 0.99955 0.99963 0.99969

92997 08437 03665 18562 85793

1.00067 1.00054 1.00044 1.00036 1.00030

11504 94581 98358 82794 15116

4.5 4.6 4.7 4.8 4.9

45.00301 49.73713 54.96903 60.75109 67.14116

1152 1903 8588 3886 6551

45.01412 49.74718 54.97813 60.75932 67.14861

0149 3739 3865 3633 3134

0.99975 0.99979 0.99983 0.99986 0.99988

32108 79416 45656 45517 91030

1.00024 1.00020 1.00016 1.00013 1.00011

68501 20992 54618 54666 09093

5. 0 5.1 5.2 5.3 5.4

74.20321 82.00790 90.63336 100.16590 110.70094

0578 5277 2655 9190 9812

74.20994 82.01400 90.63887 100.17090 110.70546

8525 2023 9220 0784 6393

0.99990 0.99992 0.99993 0.99995 0.99995

92043 56621 91369 01692 92018

1.00009 1.00007 1.00006 1. 00004 1.00004

08040 43434 08668 98333 07998

5.5 5.6 5.7 5.8 5.9

122.34392 135.21135 149.43202 165.14826 182.51736

2746 4781 7501 6177 4210

122.34800 135.21505 149.43537 165.15129 182.52010

9518 2645 3466 3732 3655

0.99996 0.99997 0.99997 0.99998 0.99998

65972 26520 76093 16680 49910

1.00003 1.00002 1.00002 1.00001 1.00001

34040 73488 23912 83323 50092

6.0

201.71315 7370

201.71563 6122

0.99998 77117

"-:I

1.00001 22885

"-:)21

218

ELEMENTARY TRANSCENDENTAL FUNCTIONS

6. 0 6.1 6.2 6.3 6.4

201.71315 222.92776 246.37350 272.28503 300.92168

7370. 3607 5831 6911 8157

HYPERBOLIC FUNCTIONS cosh x tanh x 0.99998 77117 201.71563 6122 222.93000 6475 0.99998 99391 0.99999 17629 246.37553 5262 0.99999 32560 272.28687 3215 300.92334 9715 0.99999 44785

6.5 6.6 6. 7 6. 8 6.9

332.57006 367.54691 406.20229 448.92308 496.13685

4803 4437 7128 8938 3910

332.57156 367.54827 406.20352 448.92420 496.13786

8242 4805 8040 2713 1695

0.99999 0.99999 0.99999 0.99999 0.99999

54794 62988 69697 75190 79687

1.00000 1.00000 1.00000 1.00000 1.00000

45207 37012 30303 24810 20313

7.0 7.1 7.2 7. 3 7.4

548.31612 605.98312 669.71500 740.14962 817.99190

3273 4694 8904 6023 9372

548.31703 605.98394 669.71575 740.15030 817.99252

5155 9799 5490 1562 0624

0.99999 0.99999 0.99999 0.99999 0.99999

83369 86384 88852 90873 92527

1.00000 1.00000 1.00000 1.00000 1.00000

16631 13616 11148 09127 07473

7.5 7. 6 7.7 7.8 7.9

904.02093 999.09769 1104.17376 1220.30078 1348.64097

0686 7326 9530 3945 8762

904.02148 999.09819 1104.17422 1220.30119 1348.64134

3770 7778 2357 3680 9506

0.99999 0.99999 0.99999 0.99999 0.99999

93882 94991 95899 96642 97251

1.00000 1.00000 1.00000 1.00000 1.00000

06118 05009 04101 03358 02749

8. 0 8.1 8.2 8.3 8.4

1490.47882 1647.23388 1820.47501 2011.93607 2223.53326

5790 5872 6339 2653 1416

1490.47916 1647.23418 1820.47529 2011.93632 2223.53348

1252 9411 0993 1170 6284

0.99999 0.99999 0.99999 0.99999 0.99999

97749 98157 98491 98765 98989

1.00000 1.00000 1.00000 1.00000 1.00000

02251 01843 01509 01235 01011

8.5 8. 6 8.7 8.8 8.9

2457.38431 2715.82970 3001.45602 3317.12192 3665.98670

8415 3629 5338 7772 1384

2457.38452 2715.82988 3001.45619 3317.12207 3665.98683

1884 7734 1923 8505 7772

0.99999 0.99999 0.99999 0.99999 0.99999

99172 99322 99445 99546 99628

1.00000 1.00000 1.00000 1.00000 1.00000

00828 00678 00555 00454 00372

9. 0 9.1 9.2 9. 3 9.4

4051.54190 4477.64629 4948.56447 5469.00955 6044.19032

2083 5908 8852 8370 3746

4051.54202 4477.64640 4948.56457 5469.00964 6044.19040

5493 7574 9892 9795 6471

0.99999 0.99999 0.99999 0.99999 0.99999

99695 99751 99796 99833 99863

1.00000 1.00000 1.00000 1. 00000 1.00000

00305 00249 00204 00167 00137

9.5 9.6 9.7 9.8 9.9

6679.86337 7382.39074 8158.80356 9016.87243 9965.18519

7405 8924 8366 6188 4028

6679.86345 7382.39081 8158.80362 9016.87249 9965.18524

2257 6653 9649 1640 4202

0.99999 0.99999 0.99999 0.99999 0.99999

99888 99908 99925 99939 99950

1.00000 1.00000 1.00000 1.00000 1.00000

00112 00092 00075 00061 00050

Table 4.15

sinh x

X

10.0

11013.23287 4703

11013.23292 0103

For z>>O, sinh x-cosh x -1 ~e x . For x>10, tanh *See page n.

0.99999 99959

2-1-28-22,

coth 2-1+2e-22

coth x

1.00001 1.00001 1.00000 1.00000 1.00000

22885 00610 82372 67441 55216

1.00000 00041 to 10D.

219

ELEMENTARY TRANSCENDENTAL FUNCTIONS

2

EXPONENTIAL AND HYPERBOLIC FUNCTIONS FOR THE ARGUMENT erZ e--rr sinh T X cosh T X

TX

Table 4.16 tanh T X

0.00 0.01 0.02 0.03 0. 04

1.00000 00000

1.00000 00000

0.00000

00000

1.03191 1.06484 1.09883 1.13390

46153 77733 19803 07803

0.96907 0.93910 0.91005 0.88191

24263 13674 72407 13783

0.03142 0.06287 0.09438 0.12599

10945 32029 73698 47010

1.00000 1.00049 1.00197 1.00444 1.00790

00000 35208 45704 46105 60793

0.00000 0.03140 0.06274 0.09396 0.12500

00000 55952 93000 97111 63906

0.05 0.06 0.07 0.08 0.09

1.17008 1.20743 1.24596 1.28573 1.32676

87875 17210 64399 09795 45892

0.85463 0.82820 0.80258 0.77776 0.75371

59992 41813 98355 76792 32120

0.15772 0.18961 0.22168 0.25398 0.28652

63942 37699 83022 16502 56886

1.01236 1.01781 1.02427 1.03174 1.04023

23933 79512 81377 93294 89006

0.15580 0.18629 0.21643 0.24616 0.27544

03292 43856 36952 60434 21974

0.10 0.11 0.12 0.13 0.14

1.36910 1.41280 1.45789 1.50441 1.55243

77706 23184 13610 94029 23694

0.73040 0.70781 0.68592 0.66470 0.64415

26910 31080 21659 82576 04440

0.31935 0.35249 0.38598 0.41985 0.45414

25398 46052 45975 55727 09627

1.04975 1.06030 1.07190 1.08456 1.09829

52308 77132 67634 38303 14067

0.30421 0.33244 0.36009 0.38711 0.41349

61929 55730 15776 92833 76928

0.15 0.16 0.17 0.18 0.19

1.60197 1.65310 1.70586 1.76030 1.81648

76513 41518 23348 42750 37088

0.62422 0.60492 0.58621 0.56808 0.55051

84336 25628 37756 36059 41583

0.48887 0.52409 0.55982 0.59611 0.63298

46088 07945 42796 03346 47753

1.11310 1.12901 1.14603 1.16419 1.18349

30425 33573 80552 39405 89335

0.43919 0.46420 0.48848 0.51203 0.53484

97777 24748 66406 69673 18637

0.20 0.21 0.22 0.23 0.24

1.87445 1.93427 1.99601 2.05971 2.12544

60876 86325 03910 22948 72203

0.53348 0.51698 0.50099 0.48550 0.47048

80911 85988 93958 47001 92177

0.67048 0.70864 0.74750 0.78710 0.82747

39982 50169 54976 37973 90013

1.20397 1.22563 1.24850 1.27260 1.29796

20893 36157 48934 84975 82190

0.55689 0.57818 0.59872 0.61849 0.63751

33069 66683 05188 64181 86920

0.25 0.26 0.27 OI28 0.29

2.19328 2.26327 2.33550 2.41004 2.48696

00507 77398 93782 62616 19609

0.45593 0.44183 0.42817 0.41492 0.40209

81278 70677 21192 97945 70227

0.86867 0.91072 0.95366 0: 99755 1.04243

09615 03361 86295 82336 24691

1.32460 1.35255 1.38184 1; 41248 1.44452

90893 74038 07487 80280 94918

0.65579 0.67333 0.69014 0; 70624 0.72164

42026 21140 36583 19035 15276

0.30 0.31 0.32 0.33 0.34

2.56633 2.64823 2.73275 2.81996 2.90996

23952 59064 33366 81081 63054

0.38966 0.37760 0.36593 0.35461 0.34364

11374 98638 13069 39395 65907

1.08833 1.13531 1.18341 1.23267 1.28315

56289 30213 10148 70843 98573

1.47799 1.51292 1.54934 1.58729 1.62680

67663 28851 23218 10238 64481

0.73635 0.75041 0.76381 0.77659 0.78876

85995 03695 50706 17313 00021

0.35 0.36 0.37 0.38 0.39

3.00283 3.09867 3.19756 3.29961 3.40491

67606 11407 40381 30643 89460

0.33301 0.32271 0.31273 0.30306 0.29369

84355 89833 80681 58385 27474

1.33490 1.38797 1.44241 1; 49827 1.55561

91626 60787 29850 36129 30993

1.66792 1.71069 1.75515 1.80133 1.84930

75980 50620 10531 94514 58467

0.80033 0.81135 0.82181 0.83175 0.84118

99933 21279 70068 52873 75743

0.40 0.41 0.42 0.43 0.44

3.51358 3.62572 3.74143 3.86084 3.98405

56243 03579 38283 02496 74810

0.28460 0.27580 0.26727 0.25901 0.25100

95433 72607 72113 09757 03946

1.61448 1.67495 1.73707 1.80091 1.86652

80405 65486 83085 46370 85432

1.89909 1.95076 2.00435 2. a5992 2.11752

75838 38093 55198 56127 89378

0.85013 0.85861 0.86665 0.87426 0.88146

43239 57589 17947 19762 54241

0.45 0.46 0.47 0.48 0.49

4.11120 4.24241 4.37780 4.51752 4.66170

71429 47373 97717 58864 09873

0.24323 0.23571 0.22842 0.22136 0.21451

75614 48138 47266 01040 39731

1.93398 2.00334 2.07469 2.14808 2.22359

47907 99617 25226 28912 35071

2.17722 2; 23906 2.30311 2.36944 2.43810

23522 47756 72491 29952 74802

0.88828 0: 89472 0.90081 0.90657 0.91201

07899 62194 93236 71557 61950

0.50

4.81047 73810

[(-8

0.20787 95764

2.30129 89023

2.50917 84787

0.91715 23357

[(-t)lI [(-231 [( -231 [(-:)91 Compiled from British Association for the Advancement of Science, Mathematical Tables, vol. I. Circular and hyperbolic functions, exponential, sine and cosine integrals, factorial function and allied functions, Hermitian probability functions, 3d ed. Cambridge Univ. Press, Cambridge, England, 1951 (with permission). Known errors have been corrected.

220

ELEMENTARY TRANSCENDENTAL FUNCTIONS

FOR THE ARGUMENT T X cosh TX tanh T X 2.50917 84787 0.91715 23357 2.58272 61407 0.92200 08803 2.65882 30610 0.92657 65378 2.73754 43503 0.93089 34251 2.81896 77098 0.93496 50714

0.50 0.51 0.52 0.53 0.54

EXPONENTIAL AND HYPERBOLIC FUNCTIONS e-rz erx sinh TX 4.81047 73810 0.20787 95764 2.30129 89023 4.96400 19160 0.20145 03654 2.38127 57753 5.12242 61276 0.19521 99944 2.46360 30666 5.28590 63869 0.18918 23136 2.54836 20366 5.45460 40558 0.18333 13637 2.63563 63461

0.55 0.56 0.57 0.58 0.59

5.62868 5.80832 5.99369 6.18497 6.38237

56460 29831 33767 97951 10460

0.17766 0.17216 0.16684 0.16168 0.15668

13694 67343 20350 20156 15832

2.72551 2.81807 2.91342 3.01164 3.11284

21383 81244 56709 88897 47314

2.90317 2.99024 3.08026 3.17333 3.26952

35077 48587 77058 09054 63146

0.93880 0.94242 0.94583 0.94904 0.95207

44259 38675 52160 97460 82009

0.60 0.61 0.62 0.63 0.64

6.58606 6.79625 7.01315 7.23697 7.46794

19627 35967 34158 55091 07985

0.15183 0.14713 0.14258 0.13817 0.13390

58020 98890 92093 92710 57214

3.21711 3.32455 3.43528 3.54939 3.66701

30804 68538 21032 81191 75386

3.36894 3.47169 3.57787 3.68757 3.80092

88823 67428 13125 73901 32600

0.95493 0.95761 0.96014 0.96252 0.96477

08086 72978 69151 84417 02118

0.65 0.66 0.67 0.68 0.69

7.70627 7.95222 8.20601 8.46790 8.73815

72563 01304 21768 38986 37941

0.12976 0.12575 0.12186 0.11809 0.11444

43423 10461 18713 29793 06500

3.78825 3.91323 4. 04207 4.17490 4.31185

64570 45422 51527 54597 65720

3.91802 4.03898 4.16393 4.29299 4.42629

07993 55883 70240 84390 72220

0.96688 0.96886 0.97073 0.97249 0.97414

01293 56859 39783 17255 52857

0.70 0.71 0.72 0.73 0.74

9.01702 9.30480 9.60176 9.90819 10.22441

86109 36103 28381 94054 57779

0.11090 0.10747 0.10414 10092 0.09780

12784 13709 75422 65114 50993

4.45306 4.59866 4.74880 4.90363 5.06330

36663 61197 76480 64470 53393

4.56396 4.70613 4.85295 5.00456 5.16111

49447 74906 51901 29584 04386

0.97570 0.97716 0.97853 0.97983 0.98104

06726 35718 93563 31019 96015

0.75 0.76 0.77 0.78 0.79

10.55072 10.88744 11.23491 11.59347 11.96347

40742 63743 50371 30285 42604

0.09478 0.09184 0.08900 0.08625 0.08358

02248 89025 82388 54299 77587

5.22797 5.39779 5.57295 5.75360 5.93994

19247 87359 33992 87993 32508

5.32275 5.48964 5.66196 5.83986 6.02353

21495 76384 16379 42292 10095

0.98219 0.98326 0.98427 0.98522 0.98612

33800 87071 96111 98912 31297

0.80 0.81 0.82 0.83 0.84

12.34528 12.73927 13.14584 13.56539 13.99832

39392 89270 81133 27988 70916

0.08100 0.07849 0.07606 0.07371 0.07143

25922 73785 96451 69955 71077

6.13214 6.33039 6.53488 6.74583 6.96344

06735 07743 92341 79017 49919

6.21314 6.40888 6.61095 6.81955 7.03488

32657 81528 88792 48972 20996

0.98696 0.98775 0.98849 0.98919 0.98984

27033 17946 34022 03509 53014

0.85 0.86 0.87 0.88 0.89

14.44507 14.90608 15.38180 15.87271 16.37928

83157 74333 94795 40119 55735

0.06922 0.06708 0.06501 0.06300 0.06105

77313 66855 18571 11981 27239

7.18792 7.41950 7.65839 7.90485 8.15911

52922 03739 88112 64069 64248

7.25715 7.48658 7.72341 7.96785 8.22016

30235 70594 06683 76050 91487

0.99046 0.99103 0.99158 0.99209 0.99257

07591 90830 24938 30818 28142

0.90 0.91 0.92 0.93 0.94

16.90202 17.44144 17.99808 18.57248 19.16521

41717 57711 28034 46925 83968

0.05916 0.05733 0.05556 0.05384 0.05217

45113 46965 14735 30919 78557

8.42142 8.69205 8.97126 9.25932 9.55652

98302 55373 06650 08003 02706

8.48059 8.74939 9.02682 9.31316 9.60869

43415 02338 21384 38922 81263

0.99302 0.99344 0.99384 0.99421 0.99456

35419 70066 48468 86036 97268

0.95 0.96 0.97 0.98 0.99

19.77686 20.40804 21.05935 21.73145 22.42500

89693 01345 48847 60946 71560

0.05056 0.04900 0.04748 0.04601 0.04459

41212 02956 48354 62446 30738

9.86315 10.17951 10.50593 10.84271 11.19020

24240 99195 50247 99250 70411

9.91371 10.22852 10.55341 10.88873 11.23480

65453 02151 98601 61696 01149

0.99489 0.99520 0.99550 0.99577 0.99603

95797 94443 05263 39591 08084

1.00

23.14069 26328

Table 4.16 2

~

3

OI

0.04321 39183 1

11.54873 93573

"-:"I

11.59195 32755

"-:"I

0.99627 20762 c55)41

221

ELEMENTARY TRANSCENDENTAL FUNCTIONS

X

arcsinh x

0.00

0.00000 0000

0.01 0.02 0.03 0.04

0.00999 0.01999 0.02999 0.03998

9833 8667 5502 9341

INVERSE HYPERBOLIC arctanh x X 0.00000 0000 0.50 0.01000 0333 0.51 0.02000 2667 0.52 0.03000 9004 0.53 0.04002 1353 0.54

FUNCTIONS arcsinh x 0.48121 1825 0.49013 8161 0.49902 8444 0.50788 2413 0.51669 9824

0.05 0.06 0. 07 0.08 0.09

0.04997 0.05996 0. 06994 0.07991 0.08987

9190 4058 2959 4912 8941

0.05004 0.06007 0.07011 0.08017 0.09024

1729 2156 4671 1325 4188

0.55 0.56 0.57 0.58 0.59

0.52548 0.53422 0.54293 0.55159 0.56023

0.10 0.11 0.12 0.13 0.14

0.09983 0.10977 0.11971 0.12963 0.13954

4079 9366 3851 6590 6654

0.10033 0.11044 0.12058 0.13073 0.14092

5347 6915 1028 9850 5576

0.60 0.61 0.62 0.63 0.64

0.15 0.16 0.17 0.18 0.19

0.14944 0.15932 0.16919 0.17904 0.18887

3120 5080 1636 1904 5015

0.15114 0.16138 0.17166 0,18198 0.19233

0436 6696 6663 2689 7169

0.20 0.21 0.22 0. 23 0. 24

0.19869 0.20848 0.21826 0.22801 0.23775

0110 6350 2908 8972 3749

0.20273 0.21317 0.22365 0.23418 0.24477

0.25 0.26 0. 27 0.28 0.29

0.24746 0.25715 0.26682 0.27646 0.28608

6462 6349 2667 4691 1715

0.30 0. 31 0. 32 0. 33 0.34

0.29567 0.30523 0.31477 0.32428 0.33376

0.35 0. 36 0. 37 0. 38 0. 39

Table 4.17

arctanh 2 0.54930 0.56272 0.57633 0.59014 0.60415

6144 9769 9754 5160 5603

0448 4074 0505 9562 1077

0.61838 0.63283 0.64752 0.66246 0.67766

1313 3186 2844 2707 6068

0.56882 0.57738 0.58589 0.59437 0.60282

4899 0892 8932 8911 0733

0.69314 0.70892 0.72500 0.74141 0.75817

7180 1359 5087 6144 3745

0.65 0.66 0. 67 0.68 0.69

0.61122 0.61958 0.62791 0,63620 0,64445

4314 9584 6485 4970 5005

0.77529 0.79281 0.81074 0.82911 0.84795

8706 3631 3125 4038 5755

2554 1346 6109 9466 4112

0.70 0.71 0.72 0. 73 0. 74

0.65266 0.66083 0,66897 0.67707 0.68512

6566 9641 4227 0332 7974

0.86730 0.88718 0.90764 0.92872 0.95047

0527 3863 4983 7364 9381

0.25541 0,26610 0.27686 0.28768 0.29856

2812 8407 3823 2072 6264

0.75 0.76 0.77 0; 78 0.79

0.69314 0.70112 0.70907 0.71697 0.72484

7181 7988 0441 4594 0509

0.97295 0.99621 1,02032 1.04537 1.07143

5074 5082 7758 0548 1684

3048 8020 5980 6295 8352

0.30951 0.32054 0.33164 0.34282 0.35409

9604 5409 7108 8254 2528

0.80 0.81 0.82 0.83 0.84

0.73266 0.74045 0.74820 0.75592 0.76359

8256 7912 9563 3300 9222

1.09861 1.12702 1.15681 1.18813 1.22117

2289 9026 7465 6404 3518

0.34322 0.35264 0.36203 0.37140 0.38073

1555' 5330 9121 2391 4624

0.36544 0.37688 0.38842 0.40005 0.41180

3754 5901 3100 9650 0034

0. 85 0. 86 0.87 0.88 0.89

0.77123 Or77883 0.78640 OI79392 0.80141

7433 8046 1177 6950 5491

1.25615 1.29334 1.33307 1.37576 1.42192

2811 4672 9629 7657 5871

0. 40 0. 41 0.42 0. 43 0. 44

0.39003 Oi39930 0.40854 0.41774 0.42691

5320 4001 0208 3500 3454

0.42364 0.43561 0.44769 0.45989 0.47223

8930 1223 2023 6681 0804

0.90 0. 91 0.92 0.93 0.94

0.80886 0.81628 0.82365 0.83100 0.83830

6936 1421 9091 0091 4575

1.47221 1.52752 1.58902 1.65839 1.73804

9490 4425 6915 0020 9345

0. 45 0. 46 0. 47 0. 48 0. 49

0.43604 Or44515 0.45421 0.46325 0.47224

9669 1759 9359 2120 9713

0.48470 0.49731 0.51007 0.52298 0.53606

0279 1288 0337 4278 0337

0.95 0. 96 0.97 0.98 0.99

0.84557 0.85280 0.86000 0.86716 0.87428

2697 4617 0498 0507 4812

1.83178 1.94591 2.09229 2.29755 2.64665

0823 0149 5720 9925 2412

0.50

0.48121 1825

0.54930 6144

1.00

0,88137 3587 );-( 51

[(-;)51

-

[(-,PI

For use of the table see Examples 26-28. &o(x) (Legendre Function-Second

[

E0

Kind) =arctanh z(Izll) Compiled from Harvard Computation Laboratory, Tables of inverse hyperbolic functions. Harvard Univ. Press, Cambridge, Mass., 1949 (with permission).

222

ELE%CG"ARY

Table 4.17

TRANSCENDENTAL FUNCTIONS

INVERSE HYPERBOLIC FUNCTIONS ~

arccosh x -__

arccosh x 1. 50 1.51 1.52 1.53 1.54

arcsinh x 1.19476 3217 1.20029 7449 1.20580 6263 1.21128 9840 1.21674 8362

0.86081 0.85849 0.85618 0.85389 0.85161

788 554 806 528 706

968 798 099 841 994

1.55 1.56 1.57 1.58 1.59

1.22218 1.22759 1.23297 1.23833 1.24367

2008 0958 5390 5478 1400

0.84935 0.84710 0.84486 0.84264 0.84043

324 368 823 676 913

0.96794 0.96487 0.96182 0.95880 0.95579

529 415 625 131 904

1.60 1.61 1.62 1.63 1.64

1.24898 1.25427 1.25953 1.26477 1.26999

3328 1436 5895 6877 4549

0.83824 0.83606 0.83389 0.83174 0.82960

520 483 788 424 376

6154 1729 5013 6208 5514

0.95281 0.94986 0.94692 0.94401 0.94111

918 146 561 139 853

1.65 1.66 1. 67 1. 68 1.69

1.27518 1.28036 1.28550 1.29063 1.29573

9081 0639 9389 5495 9120

0.82747 0,82536 0.82326 0.82117 0,81909

632 179 005 097 443

1.01597 1.02235 1. 02871 1.03503 1.04133

3134 9270 4123 7896 0792

0.93824 0.93539 0.93256 0.92975 Oi92696

678 589 563 576 604

1.70 1.71 1.72 1.73 1.74

1.30082 1.30587 1.31091 1.31593 1.32092

0427 9576 6727 2038 5666

0.81703 0.81497 0.81293 0.81091 0.80889

032 850 888 132 572

1.25 1. 26 1.27 1.28 1.29

1.04759 1.05382 1.06002 1.06619 1.07233

3013 4760 6237 7645 9185

0.92419 0.92144 0.91871 0,91600 0.91331

624 613 550 411 175

1.75 1.76 1. 77 1.78 1.79

1.32589 1.33084 1.33577 1.34068 1.34557

7767 8496 8006 6450 3978

0.80689 0.80489 0,80291 0.80095 0.79899

197 994 954 066 318

1. 30 1. 31 1. 32 1.33 1134

1.07845 1.08453 1.09058 1.09661 iIi0260

1059 3467 6610 0688 5899

0.91063 0.90798 0.90534 0.90272 0.90012

821 328 676 843 810

1.80 1. 81 1.82 1.83 1.84

1.35044 1.35528 1.36011 1.36491 1.36970

0740 6886 2562 7914 3089

0.79704 0.79511 0.79318 0.79127 0.78937

701 203 816 527 328

1.35 1. 36 1. 37 1.38 1. 39

1.10857 1.11451 1,12042 1.12630 1.13215

2442 0515 0317 2042 5887

0.89754 0.89498 0.89243 0.88990 0.88738

557 064 313 284 959

1.85 1.86 1.87 1.88 1.89

1.37446 1.37921 1.38393 1.38864 1.39333

8228 3477 8975 4863 1280

0.78748 0.78560 0.78373 0.78187 0.78002

209 160 170 231 334

1. 40 1.41 1. 42 1.43 1.44

1.13798 1.14378 1.14955 1.15529 1,16101

2046 0715 2086 6351 3703

0.88489 0.88241 0.87995 0.87750 0.87507

320 348 026 336 261

1.90 1.91 1.92 1.93 1.94

1.39799 1.40264 1.40727 1.41188 1.41647

8365 6254 5083 4987 6099

0.77818 0.77635 0.77453 0.77272 0.77093

468 625 796 971 142

1.45 1.46 1.47 1. 48 1. 49

1.16670 1.17236 1.17800 1.18361 1.18920

4331 8425 6174 7765 3384

0.87265 0.87025 0.86787 0.86550 0.86315

784 888 557 774 523

3.95 1.96 1.97 1.98 1.99

1.42104 1.42560 1.43013 1.43465 1.43915

8552 2476 8002 5259 4374

0.76914 0.76736 0.76559 0.76383 0.76208

300 437 544 612 633

1. 50

1.19476 3217 ri44-1 L 4 J

0.86081 788

2.00

1.44363 5475 [(-!)31

1. 00 1.01 1. 02 1. 03 1. 04

arcsinh x 0.88137 3587 0.88842 7007 0.89544 5249 0.90242 8496 0.90937 6928

1.00000 0.99667 0.99338 0.99011 0.98687

000 995 621 848 641

1. 05 1. 06 1. 07 1. 08 1; 09

0.91629 0.92317 0.93001 0.93682 0.94360

0732 0094 5204 6251 3429

0.98365 0.98046 0.97730 0.97415 0.97103

1.10 1.11 1.12 1. 1 3 1.14

0.95034 0.95705 0.96373 0.97037 0.97698

6930 6950 3684 7331 8088

1.15 1.16 1. 1 7 1. 18 1. 19

0.98356 0.99011 0.99662 1.00310 1.00955

1. 20 1.21 1. 22 1.23 1. 24

X

(22-

l)t

X

(XZ-

l)+

0.76034 600

223

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.17

IVVERSE HYPERBOLIC FUNCTIONS

z-I 0.50 0.49 0.48 0.47 0.46

arcsinh z-ln z 0.75048 82946 0.74839 16011 0.74632 48341 0.74428 85962 0.74228 34908

arccosh 2-ln z 0.62381 07164 0.62685 90940 0.62981 77884 0.63268 90778 0.63547 51194

0.45 0.44 0.43 0.42 0.41

0.74031 0.73836 0.73646 0.73458 0.73274

01215 90921 10057 64641 60676

0.63817 0.64079 0; 64334 0.64580 0.64819

79566 95268 16670 61207 45429

0.40 0; 39 0.38 0.37 0.36

0.73094 oi72917 0.72743 0.72573 0.72407

04145

57167 78524 70912

0.65050 0.65274 0.65491 0.65701 0.65904

0.35 0.34 0.33 0.32 0.31

0.72245 0.72066 0.71932 0.71781 0.71634

40117 91873 31846 65636 98766

0.30 0.29 0.28 0.27 0.26

0.71492 0.71353 0.71219 0.71089 0.70963

36678 84725 48165 32154 41742

0.25

0.70841 81861

oiooi

c2>

arcsinh 2-ln z 0.70841 81861 0.70724 57326 0.70611 72820 0.70503 32895 0.70399 41963

arccosh 2-ln z 0.67714 27078 0.67842 57947 0.67965 18411 0.68082 14660 0.68193 52541

2 2 2

_.

0.18 0.17 0.16

0.70300 0.70205 ol7oIi5 0.70029 0.69948

04288 23983 OSoOZ 51134 66000

0.68299 0.68399 0.68494 0.68584 0.68668

85051 95004 89477 81952 85249

3 3 3 3 3

0.15 3.14 0.13 0.12 0.11

0.69872 0.69801 0.69734 0.69672 0.69615

53043 15527 56533 78946 85462

0.68747 41175

0.66101 0.66290 0.66473 0.66650 0.66820

11555 72458 78974 41577 70226

3 3 3 3 3

0.10 _._ _

0.09 0.08 0.07 0.06

0.69563 0169516 0.69474 0.69436 0.69404

78573 60572 33542 99357 59680

0.69063 0.69111 0.69154 0.69191 0.69224

77531 60018 33269 99235 59631

10 11 13 14 17

0.66984 0.67142 0.67294 0.67440 0.67580

74382 63038 44732 27575 19258

3 3 4 4 4

0.05 0.04 0.03 0.02 0.01

0.69377 0.69354 0.69337 0.69324 0.69317

15954 69408 21047 71656 21796

0.69252 0.69274 0; 69292 0.69304 0.69312

15938 69403 21046 71656 21796

20 25 33 50

0.67714 27078

4

0.00

0.69314 71806



2-1

2 2 2 2 2

0.25 0.24 0.23 0.22 0.21

2

0.20

_

0 . 1-9.

2

37571 74947 69555 25981 48518

0.69314 71806

4 4 4 5 5 5 5

100 m

<.!.>=nearestinteger to .I,.

ROOTS x,, OF COL) x,, cosh X ~ = I n I 2 3 4 5

Xfi

4.73004 7165320 10.99560 14.13716 17.27875

07

46 78 55 96

ROOTS x,, OF COL) x,, n Xf1 1

2 3 4 5

'See page n.

1.87510 ~ 4.69409 7.85475 10.99554 14.13716

Cod1

41 ._.

11 74 07 84

~,,=-l

Table 4.111

224

ELEMENTARY TRANSCENDENTAL FUNCTIONS

Table 4.19 XI --x

HOOTS X riOF tan x t i AXt, :

22

Xti

27

xs

29

0.00 0.05 0.10 0.15 0.20

3.14159 2.99304 2.86277 2.75032 2.65366

6.28319 5.99209 5.76056 5.58578 5.45435

9.42478 9.00185 8.70831 8.51805 8.39135

12.56637 12.02503 11.70268 11.52018 11.40863

15.70796 15.06247 14.73347 14.56638 14.46987

18.84956 18.11361 17.79083 17.64009 17.55621

21.99115 21.17717 20.86724 20.73148 20.65782

25.13274 24.25156 23.95737 23.83468 23.76928

28.27433 27.33519 27.05755 26.94607 26.88740

0.25 0.30 0.35 0.40 0.45

2.57043 2.49840 2.43566 2.38064 2.33208

5.35403 5.27587 5.21370 5.16331 5.12176

8.30293 8.23845 8.18965 8.15156 8.12108

11.33482 11.28284 11.24440 11.21491 11.19159

14.40797 14.36517 14.33391 14.31012 14.29142

17.50343 17.46732 17.44113 17.42129 17.40574

20.61203 20.58092 20.55844 20.54146 20.52818

23.72894 23.70166 23.68201 23.66719 23.65561

26.85142 26.82716 26.80971 26.79656 26.78631

0.50 0.55 0.60 0.65 0.70

2.28893 2.25037 2.21571 2.18440 2.15598

5.08698 5.05750 5.03222 5.01031 4.99116

8.09616 8.07544 8.05794 8.04298 8.03004

11.17271 11.15712 11.14403 11.13289 11.12330

14.27635 14.26395 14.25357 14.24475 14.23717

17.39324 17.38298 17.37439 17.36711 17.36086

20.51752 20.50877 20.50147 20.49528 20.48996

23.64632 23.63871 23.63235 23.62697 23.62235

26.77809 26.77135 26.76572 26.76096 26.75688

0.75 0.80 0.85 0.90 0.95

2,13008 2.10638 2.08460 2.06453 2.04597

4.97428 4.95930 4.94592 4.93389 4.92303

8.01875 8.00881 7.99999 7.99212 7.98505

11.11496 11.10764 11.10116 11.09538 11.09021

14.23059 14.22482 14.21971 14.21517 14.21110

17.35543 17.35068 17.34648 17.34274 17.33939

20.48535 20.48131 20.47774 20.47457 20.47172

23.61834 23.61483 23.61173 23.60897 23.60651

26.75333 26.75023 26.74749 26.74506 26.74288

1.00

2.02876

4.91318

7.97867

11.08554

14.20744

17.33638

20.46917

23.60428

26.74092

-1.00 -0.95 -0.90 -0.85 -0.80

2.02876 2.01194 1.99465 1.97687 1.95857

4.91318 4.90375 4.89425 4.88468 4.87504

7.97867 7.97258 7.96648 7.96036 7.95422

11i08554 11.08110 11.07665 11.07219 11.06773

14.2bj44 14.20395 14.20046 14.19697 14.19347

17.33638 17.33351 17.33064 17.32777 17.32490

20.46917 20.46673 20.46430 20.46187 20.45943

23.60428 23.60217 23.60006 23.59795 23.59584

26,74092 26.73905 26.73718 26.73532 26.73345

-0.75 -0.70 -0.65 -0.60 -0.55

1.93974 1.92035 1.90036 1.87976 1.85852

4.86534 4.85557 4.84573 4.83583 4.82587

7.94807 7.94189 7.93571 7.92950 7.92329

11.06326 11.05879 11.05431 11.04982 11.04533

14.18997 14.18647 14.18296 14.17946 14.17594

17.32203 17.31915 17.31628 17.31340 17.31052

20.45700 20.45456 20.45212 20.44968 20.44724

23.59372 23.59161 23.58949 23.58738 23.58526

26,73159 26.72972 26.72785 26.72598 26.72411

-0.50 -0.45 -0.40 -0.35 -0.30

1.83660 1.81396 1.79058 1.76641 1.74140

4.81584 4.80575 4.79561 4.78540 4.77513

7.91705 7.91080 7.90454 7.89827 7.89198

11.04083 11.03633 11.03182 11.02730 11.02278

14.17243 14.16892 14.16540 14.16188 14.15835

17.30764 17.30476 17.30187 17.29899 17.29610

20.44480 20.44236 20.43992 20.43748 20.43503

23.58314 23.56102 23.57891 23.57679 23.57467

26.72225 26.72038 26.71851 26.71664 26.71477

-0.25 -0.20 -0.15 -0.05

1.71551 1.68868 1.66087 1.63199 1.60200

4.76481 4.75443 4.74400 4.73351 4.72298

7.88567 7.87936 7.87303 7.86669 7.86034

11.01826 11.01373 11.00920 11.00466 11.00012

14.15483 14.15130 14.14777 14.14424 14.14070

17.29321 17.29033 17.28744 17.28454 17.28165

20.43259 20.43014 20.42769 20.42525 20.42280

23.57255 23.57043 23.56831 23.56619 23.56407

26.71290 26.71102 26.70915 26.70728 26.70541

0.00 0.05 0.10 0.15 0.20

1.57080 1.53830 1.50442 1.46904 1.43203

4.71239 4.70176 4.69108 4.68035 4.66958

7.85398 7.84761 7.84123 7.83484 7.82844

10.99557 10.99102 10.98647 10.98192 10.97736

14.13717 14.13363 14.13009 14.12655 14.12301

17.27875 17.27586 17.27297 17.27007 17.26718

20.42035 20.41790 20.41545 20.41300 20.41055

23.56194 23.55982 23.55770 23.55558 23.55345

26.70354 26.70166 26.69979 26.69792 26.69604

20 10 7 5

0.25 0.30 0.35 0.40 0.45

1.39325 1.35252 1.30965 1.26440 1.21649

4.65878 4.64793 4.63705 4.62614 4.61519

7.82203 7.81562 7.80919 7.80276 7.79633

10.97279 10.96823 10.96366 10.95909 10.95452

14.11946 14.11592 14.11237 14.10882 14.10527

17.26428 17.26138 17.25848 17.25558 17.25268

20.40810 20.40565 20.40320 20.40075 20.39829

23.55133 23.54921 23.54708 23.54496 23.54283

26.69417 26.69230 26.69042 26.68855 26.68668

4 3 3 3 2

0.50 0.55 0.65 0.70

1.16556 1.11118 1.05279 0.98966 0.92079

4.60422 4.59321 4.58219 4.57114 4.56007

7.78988 7.78344 7.77698 7.77053 7.76407

10.94994 10.94537 10.94079 10.93621 10.93163

14.10172 14.09817 14.09462 14.09107 14.08752

17.24978 17.24688 17.24398 17.24108 17.23817

20.39584 20.39339 20.39094 20.38848 20.38603

23.54071 23.53858 23.53646 23.53433 23.53221

26.68480 26.68293 26.68105 26.67918 26.67730

2 2 2 2

0.75 0.80 0.85 0.90 0.95

0.84473 0.75931 0.66086 0.54228 0.38537

4.54899 4.53789 4.52678 4.51566 4.50454

7.75760 7.75114 7.74467 7.73820 7.73172

10.92704 10.92246 10.91788 10.91329 10.90871

14.08396 14.08041 14.07686 14.07330 14.06975

17.23527 17.23237 17.22946 17.22656 17.22366

20.38357 20.38112 20.37867 20.37621 20.37376

23.53008 23.52796 23.52583 23.52370 23.52158

26.67543 26.67355 26.67168 26.66980 26.66793

1 1

1.00

0.00000

4.49341

7.72525

10.90412

14.06619

17.22075

20.37130

23.51945

26.66605

1

-0.10

0.60

13

For A-0, see ji s of Table 10.6.

24

x5

<x>=nearest integer to

A.

-1 -

1 1 1 1

-

1 1 2 2 2

-

-

2

2

3 3 3

-

4 5 7 -10 -20 0

1

1 1

1

I

225

ELEMENTARY TRANSCENDENTAL FUNCTIONS

ROOTS x,, OF cot x,,

x

21

22

23

24

25

=Xzn

26

Table 4.20 $7

*8

2Y

0.00 0.05 0.10 0.15 0.20

1.57080 1.49613 1.42887 1.36835 1.31384

4.71239 4.49148 4.30580 4.15504 4.03357

7.85398 7.49541 7.22811 7.04126 6.90960

10.99557 10.51167 10.20026 10.01222 9.89275

14,13717 13.54198 13.21418 13.03901 12.93522

17.27876 16.58639 16.25936 16.10053 16.01066

20.42035 19.64394 19.32703 19.18401 19.10552

23.56194 22.71311 22.41085 22.28187 22.21256

26.70354 25.79232 25.50638 25.38952 25.32765

0.25 0.30 0.35 0.40 0.45

1.26459 1.21995 1.17933 1.14223 1.10820

3.93516 3.85460 3.78784 3.73184 3.68433

6.81401 6.74233 6.68698 6.64312 6.60761

9.81188 9,75407 9.71092 9.67758 9.65109

12.86775 12.82073 12.78621 12.75985 12.73907

15.95363 15.91443 15.88591 15.86426 15.84728

19.05645 19.02302 18.99882 18.98052 18.96619

22.16965 22.14058 22.11960 22.10377 22.09140

25.28961 25.26392 25.24544 25.23150 25.22062

0.50 0.55 0.60 0.65 0.70

1.07687 1.04794 1.02111 0.99617 0.97291

3.64360 3.60834 3.57756 3.55048 3.52649

6.57833 6.55380 6.53297 6.51508 6.49954

9.62956 9,61173 9.59673 9.58394 9,57292

12.72230 12.70847 12.69689 12.68704 12.67857

15.83361 15.82237 15.81297 15.80500 15.79814

18.95468 18.94523 18.937 34 18.93065 18.92490

22.08147 22.07333 22.06653 22.06077 22.05583

25.21190 25.20475 25.19878 25.19373 25.18939

0.75 0.80 0.85 0.90 0.95

0.95116 0.93076 0.91158 0.89352 0,87647

3.50509 3.48590 3.46859 3.45292 3.43865

6.48593 6.47392 6.46324 6.45368 6.44508

9.56331 9.55486 9.54738 9.54072 9.53473

12.67121 12.66475 12.65904 1?.65395 12.64939

15.79219 15.78698 15.78237 15.77827 15.77459

18.91991 18.91554 18.91168 18.90825 18.9051 8

22.05154 22.04778 22.04447 22.04151 22.03887

25.18563 25.18234 25,17943 25.17684 25.17453

1.00

0.86033

3.42562

6.43730

9.52933

12.64529

15.77128

18.90241

22.03650

25.17245

X5

26

27

50

29

A-1

21

22

23

24

*



1.00 0.95 0.90 0.85 0.80

0.86033 0.84426 0.82740 0.80968 0.79103

3.42562 3.41306 3.40034 3.38744 3.37438

6.43730 6.42987 6.42241 6.41492 6.40740

9.52933 9.52419 9.51904 9.51388 9.50871

12.64529 12.64138 12.63747 12.63355 12.62963

15.77128 15.76814 15.76499 15.76184 15.75868

18.90241 18.89978 18.89715 18.89451 18.89188

22.03650 22.03424 22.03197 22.02971 22.02745

25.17245 25.17047 25.16848 25.16650 25.16452

0.75 0.70 0.65 0.60 0.55

0.77136 0.75056 0.72851 0.70507 0.68006

3.36113 3.34772 3.33413 3.32037 3.30643

6.39984 6.39226 6.38464 6.37700 6.36932

9.50353 9.49834 9.49314 9.48793 9.48271

12.62570 12.62177 12.61784 12.61390 12.60996

15.75553 15.75237 15.74921 15.74605 15.74288

18.88924 18.88660 18.88396 18.88132 18.87868

22.02519 22.02292 22.02066 22.01839 22.01612

25.16254 25.16055 25.15857 25.15659 25.15460

2 2 2

0.50 0.45 0.40 0.35 0.30

0.65327 0.62444 0.59324 0.55922 0.52179

3.29231 3.27802 3.26355 3.24891 3.23409

6.36162 6.35389 6.34613 6.33835 6.33054

9.47749 9.47225 9.46700 9.46175 9.45649

12.60601 12.60206 12.59811 12.59415 12.59019

15.73972 15.73655 15.73338 15.73021 15.72704

18.87604 18.87339 18.87075 18.86810 18.86546

22.01386 22.01159 22.00932 22.00705 22.00478

25.15262 25.15063 25.14864 25.14666 25.14467

2 2 3 3 3

0.25 0.20 0.15 0.10 0.05

0.48009 0.43284 0.37788 0.31105 0.22176

3.21910 3.20393 3.18860 3.17310 3.15743

6.32270 6.31485 6.30696 6.29906 6.29113

9.45122 9.44595 9.44067 9.43538 9.43008

12.58623 12.58226 12.57829 12.57432 12.57035

15.72386 15.72068 15.71751 15.71433 15.71114

18.86281 18.86016 18.85751 18.85486 18.85221

22.00251 22.00024 21.99797 21.99569 21.99342

25.14268 25.14070 25.13871 25.13672 25.13473

4 5 7

10 20

0.00

0.00000

3.14159

6.28319

9.42478

12.56637

15.70796

18.84956

21.99115

25.13274

00

*



=nearest integer to A.

For A-' > .20, t h e maximum er r o r i n linear interpolation is (- 4)7; five-point interpolation gives 5D. For A - l S .20,

'See page

n.

1

1 1 1 1 1

1

5. Exponential Integral and Related Functions WALTERGAUTBCHIAND WILLIAM F. CAHILL

Contents Mathematical Properties . . . . . . . . . 5.1. Exponential Integral . . . . . . 5.2. Sine and Cosine Integrals . . . .

........... . . . . . . . . . . . . . . . . . . . . . . . . Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 5.3. Use and Extension of the Tables . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . Table 5.1. Sine, Cosine and Exponential Integrals ( 0 5 ~ 5 1 0 .) z-lSi(z), x-2[Ci(z)-ln 2-71 z-'[Ei(z)-ln 2-71, z-'[E1(z)+ln z+r], z=0(.01).5, 10s Si(z), Ci(z), 10D; Ei(z), El@), 9D; 2=.5(.01)2 Si(z), Ci(z), 10D; ze-z Ei(z), zez El@), 9D; 2=2(.1)10

. . .

Page

228 228 231 233 233 235 238

Table 5.2. Sine, Cosine and Exponential Integrals for Large Arguments ( 1 0 5 ~ 5 ~. ). . . . . . . . . . . . . . . . . . . . . . . . zf(z), 9 D ; s2g(z), 7D; ze-"Ei(z), 8 D ; zeZEl(z), 10D f(z)=-si(x) cos z+Ci(x) sin 2, g(z)=-si(z) sin z-Ci(x) cos z

243

K1= .1(- ,005)O

. . .

244

. . . . . . . . .

245

Table 5.5. Exponential Integrals E,($) for Large Arguments (2 5x5 C O ) . (z+n)eZEn(z),n=2, 3, 4, 10, 20, ~-1=.5(-.05).1(-.01)0, 5D

248

Table 5.6. Exponential Integral for Complex Arguments (I zI <29) zezEl(z), z=z+iy, z=-19(1)20, y=0(1)20, 6 D

.

.

249

(I ZI

<5)

.

251

Table 5.3. Sine and Cosine Integrals for Arguments Si(rx), Cin(rz), z=0(.1)10, 7D Table 5.4. Exponential Integrals EA(z)(0 5 s s 2 ) . E,(z)--a:In 2, En(z),n = 3 , 4 , 10, 20, r=0(.01).5 E&), n=2, 3, 4, 10, 20, 2=.5(.01)2, 7D

TX

(0 5 s 5 10)

Table 5.7. Exponential Integral for Small Complex Arguments ezEl(z), z=z+iy, z=-4(.5)-2, y=0(.2)1, 6 D El(z)+ln z, z=z+iy, z=-2(.5)2.5, y=0(.2)1, 6 D

The authors acknowledge the assistance of David S. Liepman in the preparation and checking of the tables, Robert L. Durrah for the computation of Table 5.2, and Alfred E. Beam for the computation of Table 5.6.

-

Guest worker, National Bureau of Standards, from the American University. Purdue University.) * National Bureau of Standards. (Presently NASA.)

(Presently

227

I

5. Exponential Integral and Related Functions Mathematical Properties

I

5.1. Exponential Integral Definitions

5.1.2

Ei(z)=-l

-

J-2

5.1.3

r2at

Explicit Expressions for a . ( z ) and Bn(z)

5.1.8

a,(z)=n!z-"-le-g ( l + z + az2 +

. . . +a) 2"

5.1.9 22

& ( ~ ) = n ! z - " - ~ {[l-z+--. e~

-1

!-a t - 1 t

li(z)=x fi=Ei(ln

2

J - m

2!

1

5 dt t

z)

. . + ( - l ) " a2"l

(z>o) I

(z>l)

Y

5.1.5

In 5.1.1 it is assumed that the path of integration excludes the origin and does not cross the negative real axis. Analytic continuation of the functions in 5.1.1, 5.1.2, and 5.1.4 for n>O yields multi-valued functions with branch points at z=O and z = - . ~ They are single-valued functions in the z-plane cut along the negative real axis.' The function li(z), the logarithmic integral, has an additional branch point a t z=1. 5.1.7

Intemlationr

El(- zf i0)=-Ei(z) 7 i?r, -Ei (z) =a[ El(- z+i0) +E1(- z8

(z>O)

Some authors [5.14], [5.16] use the entire function

E(l-e-l)dt/t

as the basic function and denote it by

Ein(z). We have Ein(z)=El(z)+ln 4

io)]

r

FIGURE 5.1. y=Ei(z) and y=El(z).

z+r.

Various authors define the integralm:J

( e l / t ) d t in the

z-plane cut along the positive real axis and denote italso by Ei(z). For z=z>O additional notations such as Ei(z). (e.g., in [5.10],[5.25]),E*@) (in [5.2]),Ei*(z) (in [5.6])are then used to designate the principal value of the integral. Correapondingly, E l @ ) is often denoted by -Ei(-z).

FIGURE 5.2. y=E,(z) n=o, 1 , 2 , 3, 5, 10

229

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

Symmetry Relation -

5.1.13

I

E,(Z)=E,(z) Recurrence Relations

5.1.14 1

(n=1,2,3,. . .)

E,+,(z)=n [e-'-zE,(z)] 5.1.15

za,(z)=e-z+m,-l(z)

(n=1,2,3,

. . .)

(n=1,2,3,

. . .)

5.1.16

FIGURE 5.3. y=an(x)

zfl,(z)=(-l)"e'-e-'+n/3,-,(z)

n=0(1)6

Inequalities [5.8], [5.4]

5.1.17

5.1.21

FIGURE 5.4. y=&,(x) n=O, 1, 2, 5 , 10, 15 Continued Fraction

Series Expansions

5.1.10

x" Ei(x)=r+lnx+c n-1 nn! OD

5.1.22 (x>O)

5.1.11 Special Values

5.1.12

5.1.23 5.1.24

r=.57721 56649

. . , is Euler's constant.

5.1.25

==

E"(0)

1

E&) a0(z)=-) e-' Z

(n> 1)

e-' =-

2

Bo(z)=;sinh 2 z

230

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

5.1.36

Derivatives

5.1.26

E,-,(z) (n=1,2,3,. . .)

A=")

dz

(a>O, b real)

5.1.27 5.1.37 (- l)"(n- 1)!

+

2"

(n=1,2,3,. . .)

+9E1(-u+ib)

(a>O,b real)

5.1.38

Definite and Indefinite Integrals

(For more extensive tables of integrals see i5.31, [5.61, [5.11], [5.12], [5.13]. For integrals involving iZ,(x) see [5.9].)

5.1.29

5.1.30

S,=tg2 e""Ldt=enbEl(ab)

(a>O, b>O)

5.1.31

+cons t.

sa&

5.1.43

b dt=h a

5.1.32 5.1.33

1

dx=-- a 9(efuEl(-x+ia))+const.

(a>O)

dz=-9(efaEl(-x+ia))+const.

(a>O)

5.1.44

S&

S,@Ei(t)dt=2 In 2

Relation to Incomplete Gamma Function (see 6.5)

1

5.1.45

E,(z)=z n - l r (1-12,

5.1.46

( Y n ( Z) =Z- n - lr ( n +

5.1.47

~ ~ (= 2 z)- n - I [ r

Z)

1, Z )

(n+ 1, -2) -r (n+ 1, z)]

Relation to Spherical Bessel Functions (see 10.2)

5.1.35

dt=n-arctan

b - +9E1(-a+ib) a

(a>& b>O)

231

EXPONENTIAL INTEGRAL AND RELATED F"CTI0NS

Number-Theoretic Significance of li

(2)

a0=-.57721 566 a,= .99999 193 %=--.24991 055

~

(Assuming Riemann's hypothesis that all nonreal zeros of l(z) have a real part of $) 5.1.50

li (z)-r(x)=O(@

In x)

5.1.54

(x+w>

aa= .05519 968 a(=-.00976 004 as= .00107 857

1
r(x) is the number of primes less than or equal to 2.

x 10-5

I4z) 1<5

a, =2.334733 %= ,250621 5.1.55

bl =3.330657 ba= 1.681534

lOIz< =

Ic (4I <10-' b, =5.03637 ba =4.19 160

a, =4.03640 %= 1.15 198

5.1.56

1Ix<=

I€(X)l<2

FIGUR.E 5.5. y=li(z) and y=r(z)

al= 8.57332 87401

bl= 9.57332 23454

18.05901 69730 aa= 8.63476 08925 a,= .26777 37343

ba=25.63295 61486 ba=21.09965 30827 b,= 3.95849 69228

Q=

Asymptotic Expandon

5.1.51

~

x 10-8

5.2. Sine and Cosine Integrals Definitions Representation of E&) for Large n

5.2.1

5.1.52 e-'

E.(z)=#+(rSla)e+

n

n(n-22) (itn)'

Polynomial and Rational Approximations

5.1.53

El(2)+In

Olxll

X=~O

+qz+w2+aaz+a d +a d +42) le(x)1<2X10-'

The approximation 5.1.53 is from E. E. Allen, Note 169, MTAC 8, 240 (1954); approximations 5.1.54 and 5.1.56 are from C. Hastings, Jr., Approximations for digital computers, Princeton Univ. Press, Princeton, N.J., 1955; approximation 5.1.55 is from C. Hastings, Jr., Note 143, MTAC 7, 68 (1953) (with permission). 6

6

Some authors (5.141,[5.16] use the entire function

ia(l--cos t)dt/t aa the basic function and denote it by Cin(z). We have Cin(z)= -Ci(z)+ln 7

The notations Sih(z)=

La

z+y.

sinh t dt/t,

(cosh t- 1)dt/thave also been proposed [5.14.1

232

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

si ( z ) =Si(z ) --

lr

5.2.5

2

Auxiliary Functions

5.2.6

f(z)=Ci(z) sin z-si(z) cos z

5.2.7

g(z)=-Ci(z) cos z-si(z) sin z

Sine and Cosine Integrals in Terms of Auxiliary Functions

5.2.8

lr

Si(z)=j-f(z)

cos z-g(z) sin z

Symmetry Relations

5.2.20

5.2.9

Ci(z)=f(z) sin z-g(z) cos z

Ci(- z ) =Ci ( z ) --ilr Ci (Z) =Ci ( z )

Integral Representations

5.2.10

-

Si(-z)= --Si(z), Si@)=Si(z)

5.2.19

si(z)=-

r

con

e-z

1

cos ( z sin t)d t

. O

Relation to Exponential Integral

1

e-z cost

Z
5.2.21

~i ( 2 ) =2i [ E(iz) ~ - E ~(- iz)I

I

5.2.11 Ci(z)+El(z)=Jr

(O<arg

sin ( z sin t)dt

+E

(larg z

I<E)

0

i

5.2.22 Si(iz)=s [Ei(z)+E1(z)l

(z>O)

Y 4

5.2.24 Ci(iz)=$1 [Ei(z)-E,(z)]+iE

(z>O)

Value at Infinity

lim Si(z)=2A

5.2.25

1-1 m

Integrals

(For more extensive tables of integrals see [5.31, [5.6], [5.11], [5.12], [5.131.)

FIGURE 5.6. y=Si(z) and y=Ci(x)

5.2.26

lm

5.2.27

lm

+s

OD

0

*See page 11.

Si(z)=r

(larg zl
(z)

--1 In (1+aZ) 242

5.2.29

lrn

5.2.30

lm

m

C JE+t n-0

(z)

dt=-Ci

5.2.28s e-alCi (t)dt=

Series Expansions

5.2.15

dt=-si

e-af si (t)dt=--

cos t Ci (t)dt=

1

a

arctan a

(9a>O)*

(ga>O) lr

sin t si (t)dt=-- 4

233

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

5.2.37

lSx<

5.2.38

1S x <

5.2.33

+ 9 E l ( a + i b ) (a real, b>O)

x8+a,xs+a2x4+a3x2+a4 x8+ blxe+ b2s4+ b3xZ+ b 4

Asymptotic Expansions

5.2.34

le(x)1<5X10-' 38.027264 bl= 40.021433

uI=

5.2.35

(h-265.187033

b2 =322.6249 11

~az335.677320

b3=570.236280

~

1

Rational Approximations *

5.2.36

1I x <

€

(XI

1 <2

5.2.39

=

38.102495 1

b4=157.105423

sx<

+ + +

1 x8+al xe azx4 a3x2 a4 g(x)=p (x8+b1x6+b2x4+b3xz+b4

Q)

lt(x) 1<3X 10-' al= 42.242855 bl= 48.196927 b2= 482.485984 a2=302.757865

1 x4+a1x2+a2 '(x)=~(x4+b1x2+b2

I

4

x 10-4

U1=7.24 1163

b1=9.068580

~4=2.463936

b2=7.157433

a3=352.018498 ~ 4 = 21.821899

b3=1114.978885 b4= 449.690326

Numerical Methods 5.3. Use and Extension of the Tables Example 1. Compute Ci (.25) to 5D. From Tables 5.1 and 4.2 we have

Ci (.25) = (.2q2(- .249350)

+ (-

1.38629) +.577216=-32466.

Example 3. Compute Si (20) to 5D. Since 1/20=.05 from Table 5.2 we find f(20) =.049757, g(20) =.O02464. From Table 4.8, sin 20= ,912945, cos 20= .408082. Using 5.2.8 ~i(20)=z-j(20) cos 20-g(20) sin 20 4 =1.570796- .022555=1.54824*

Example 4. Compute E,(x), n = l ( l ) N , to 5s for x= 1.275, N = 10. If is less than about five, the recurrence relation 5.1.14 can be used in increasing order of n without serious loss of accuracy. By quadratic interpolation in Table 5.1 we get 'See page 11. El (1.275) =.1408099, and from Table 4.4, e-l."' From C. Hastings, Jr., Approximations for digital =.2794310. The recurrence formula 5.1.14 then computers, Princeton Univ. Press, Princeton, N.J., 1955 (with permission). I yields

Example 2. Compute Ei (8) to 5s. From Table 5.1 we have xe-*Ei (5) = 1.18185 for x=8* From 4*49 e8=2*98096Xlo3* Thus Ei (8) =440.38.

234

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS n

1 2 3 4 5

E”(1.275) ,1408099 ,0998984 ,0760303 .0608307 ,0504679

En(1.275) 6 .0430168 7 .0374307 8 .0331009 9 .0296534 10 ,0268469

(1)

lWEn(lO) (2)

1 2 3 4 5 6 7 8 9 10

.41570 ,38300 .35505 .330% .310% .28% -27667 .25= -25084 .2=

.41570 .38302 .35488 .33041 .30898 .29005 ,27325 -25822 ,24472 .23253

-

106E,,(12.3) . 191038 . 183498 . 176516 . 170042 . 164015 . 158397 . 153144 . 148226 . 143608

. 191038

n 12 13 14 15 16 17 18 19 20

11 . 199213 10 ,208098 9 ,217793 8 ,228406 7 ,240073 6 ,252951 5 .267234 4 ,283155 3 .300998 2 .321117 1 .343953 From Tables 5.2 and 5.5 we find E1(12.3)=.343953 X 10+’, Eao(12.3) =,143609X 10-e as a check. Example 7. Compute a,(2) to 6s for n= 1(1)5. The recurrence formula 5.1.15 can be used for all x > O in increasing order of n without loss of

Interpolating directly in Table 5.4 for n=10 we get E10(1.275)=.0268470 as a check. Example 5. Compute E,,(z), n=1(1)& to 5 s for z=10, N=lO. If, as in this example, z is appreciably larger than five and N l z , then the recurrence relation 5.1.14 may be safely used in decreasing order of n([5.5]). From Table 5.5 for P = . 1 we get (z-k 10)e”Elo(z)=1.02436 so that Elo(lO) =2.32529 X10-6. Using this as the initial value we obtain column (2). lVE.(lO)

1OBEn (12.3)

n 12

accuracy.

1

From 5.1.25 we have a0(2)=3 e-a

= .0676676, so we get

arm

n

0 1 2 3 4 5

,0676676 .101501 ,169169 .32 1421 ,710510 1.84394

Independent calculation with 5.1.8 yields the same result for q ( 2 ) . The functions ao(z) and al(z) can be obtained from Table 10.8 using 5.1.48, 5.1.49. Example 8. Compute Pn(z), n=O(l)N to 6 s for z=1, N=5. Use the recurrence relation 5.1.16 in increasing order of n if x>.368N+ .184 In N + .821 and in decreasing order of n otherwise [5.5]. From 5.1.9 with n=5 we get &(1)=-.324297 correctly rounded to 6D. Using the recurrence formula 5.1.16 in decreasing order of n and carrying 9D we get the values in column (2).

From Table 5.2 we get xeZE1(z)=.915633 so that E1(10)=4.15697X10-6 as a check. Forward recurrence starting with E1(1O)=4.1570X 10-’ yields the values in column (1). The underlined figures are in error. Example 6. Compute E,(z), n = l ( l ) N , to 5s for z=12.3, N=20. If N is appreciably larger than z, and z appreciably larger than five, then the recurrence relation 5.1.14 should be used in the backward direction to generate E,,(z)for n<no, and in the forward direction to generate E,,(z) for n>nol where no=(x).

From 5.1.52, with no=12, z=12.3, we have

x3 (1f ,02032- .r)0043- .OOOOl) = 1.91038X lO-’.

e-iz.a

E,,,,(%)

0 2.35040 2 1 -.73575 9269 2 ,87888 3849 3 -.44950 9722 4 .55236 3499 5 -.32434 3774

-

Using the recurrence relation 5.1.14, as indicated, we get I

1

2.35040 2389 -.73575 8881 37888 4629 -.44950 7385 ,55237 2 8 z -.32429 7-

Using forward recurrence instead, starting with

EXPONENTIAL INTEGRAL

f10(l)=2 sinh 1=2.350402 and again carrying 9D, we obtain column (1). The underlined figures are in error. The above shows that three significant figures are lost in forward recurrence, whereas about three significant figures are gained in backward recurrence! An alternative procedure is to start with an arbitrary value for n sufficiently large (see also [5.1]). To illustrate, starting with the value zero a t n = l l we get n

6dl)

n

6dU

11 10 9 8 7 6

0. -280560 -.206984 .319908 -.253812 .40465

5 4 3 2 1 0

-.324297 -552373 -,449507 ,878885 --.735759 2.350402

-

k

f'k'

0 1 2 3

( z o ) / k!

.059898 . 008174 -. 001859 . 000088

I

-.

f.

+.

-.

1078952' 012795; 000155i 000212i

(Az)~f'L'(zo)/k! .059898 . 003460 -. 000094 000003

-.

-. 107895i +. 0024351' +. O O O l l O i -.

000004i I

f(~)=.063261 -.105354i e-'=.031510 -.0220753 E1 ( 2 ) = - .OOO332 - .0047 16i

Repeating the calculation with z0=3+6i and Az= .2578+ .8943i we get the same result. An alternative procedure is to perform bivariate interpolation in the real and imaginary parts of

The functions flo(z) and &(z) can be obtained from Table 10.8 using 5.1.48, 5.1.49.

zezEl(2).

Example 9. Compute El(z) for z=3.2578 f6.8943i. From Table 5.6 we have for z0=z0+iy0=3+7i

+ 12.7i.

Example 10. Compute El(z)for z=-4.2

Using the formula at the bottom of Table 5.6 .711093 ezE1(z) -3,784225 12.7i .010389 .278518 +-1.90572 12.7i 2.0900 12.7i = - .0184106- .0736698i E ~ ( zB--1.87133-4.705403. )

+

+

ezOEl(zo)=,059898-. 107895i. From Taylor's formula withf(z)=ezEl(z) we have

+

+

References [5.1] F. J. Corbat6, On the computation of auxiliary functions for two-center integrals by means of a high-speed computer, J. Chem. Phys. 24,452-453 (1956). [5.2] A. Erdblyi et al., Higher transcendental functions, vol. 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953).

[5.3]A. Erdblyi et al., Tables of integral transforms, vols. 1, 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1954). [5.4] W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys. 38, 77-81 (1959). [5.5] W. Gautschi, Recursive computation of certain integrals, J. Assoc. Comput. Mach. 8, 21-40 (1961).

l

[5.6] W. Grobner and N. Hofreiter, Integraltafel (Springer-Verlag, Wien and Innsbruck, Austria, 1949-50). [5.7] C. Hastings, Jr., Approximations for digital computers (Princeton Univ. Press, Princeton, N.J., 1955). [5.8] E. Hopf, Mathematical problems of radistive equilibrium, Cambridge Tracts in Mathematics and Mathematical Physics, No. 31 (Cambridge Univ. Press, Cambridge, England, 1934). [5.9] V. Kourganoff, Basic methods in transfer problems (Oxford Univ. Press, London, England, 1952). [5.10] F. Losch and F. Schoblik, Die Fakultat und verwandte Funktionen (B. G. Teubner, Leipzig, Germany, 1951). (5.111 N. Nielsen, Theorie des Integrallogarithmus (B.G. Teubner, Leipzig, Germany, 1906).

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

236

[5.12] F. Oberhettinger, Tabellen zur Fourier Transformation (Springer-Verlag, Berlin, Gottingen, Heidelberg, Germany, 1957). [5.13] 1. M. Ryshik and I. S. Gradstein, Tables of series, products and integrals (VEB Deutscher Verlag der Wissenschaften, Berlin, Germany, 1957). [ 5.141 S. A. Schelkunoff, Proposed symbols for the modified cosine and exponential integral, Quart. Appl. Math. 2, 90 (1944). [5.15] J. Todd, Evaluation of the exponential integral for large complex arguments, J. Research NBS 62, 313-317 (1954) R P 2508. [ 5.161 F. G. Tricomi, Funzioni ipergeometriche confluenti (Edizioni Cremonese, Rome, Italy, 1954).

Tables [5.17] British Association for the Advancement of Science, Mathematical Tables, vol. I. Circular and hyperbolic functions, exponential, sine and cosine integrals, etc., 3d ed. (Cambridge Univ. Press, Cambridge, England, 1951). Ei(z)--In 2, -El(z) --In 2, Ci(z)-ln 2, Si(z), z=0(.1)5, 11D; Ei(z), ~=5(.1)15, 13-14D; Z=5(.1)15, 10-11s;E~(z), Si(z), Ci (z), z = 5(. 1)20(.2)40, 10D. [5.18] L. Fox, Tables of Weber parabolic cylinder functions and other functions for large arguments, Mathematical Tables, vol. 4, National Physical Laboratory (Her Majesty's Stationery Office, London, England, 1960). e-zEi(z), eZE1(z), z-1= 0(.001).1, 10D;f(z), ~ ( z ) ~-1=0(.001).1, , 10D. [5.19] J. W. L. Glaisher, Tables of the numerical values of the sine-integral, cosine-integral and exponential integral, Philos. Trans. Roy. SOC. London 160, 367-388 (1870). Si(z), Ci(z), Ei(z), - E ~ ( z ) ~=0(.01)1, , 18D, ~=1(.1)5(1)15, 11D. [5.20] B. S. Gourary and M. E. Lynam, Tables of the auxiliary molecular integrals An(z) and the auxiliary functions C,(z), The Johns Hopkins Univ. Applied Physics Laboratory, CM Report 905, Baltimore, Md. (1957). a n ( z ) , n!en(z), z=.05(.05)15, n=0(1)18, 9s. [5.21] F. E. Harris, Tables of the exponential integral Ei(z), Math. Tables Aids Comp. 11, 9-16 (1957). El(z),eZEl(z), Ei(z), e-zEi(z), z= 1(1)4(.4)8(1) 50, 18-19s. [5.22] Harvard University, The Annals of the Computation Laboratory, vols. 18, 19; Tables of the generalized sine- and cosine-integral functions, parts I, I1 (Harvard Univ. Press, Cambridge, Mass.,

K

(12,

Es(a, z) =

&, Ec(a, 5) =

0 I z < 10. [5.241 A. V. Hershey, Computing programs for the complex exponential integral, U.S. Naval Proving Ground, Dahlgren, Va., NPG Report No. 1646 (1959). - E I ( - z ) , . z = z + ~ Y , z=- 20(1)20, y=0(1)20, 135. [5.251 E. Jahnke and F. Emde, Tables of functions, 4th ed. (Dover Publications, Inc., New York, N.Y., 1945). -El(z), Ei(z), ~=0(.01)1(.1)5(1)15, P 6s; Si (z) , C i h ), z = 0 (-01)1(. 1)5( 1)15(5) 100( 10) 200(100)10a(1)7, generally 4-55; maxima and minima of Ci(z) and si(z), O
[i@..(7)],z=i eh, q= .01(.01),2, rl

U-

Oo (2') 6Oo(l0)

90°, 5-6D. [5.29] G. F. Miller, Tables of generalized exponential integrals, Mathematical Tables, vol. 3, National Physical Laboratory (Her Majesty's Stationery Office, London, England, 1958). (z+n)e'h'.(z), 2- 0(.01)1, n= 1(1)8, Z= O(. 1)20, n= l(1) 24, ~-1=0(.001).05,n = 1(1)24; 8D. [5.30] J. Miller, J. M. Gerhauser, and F. A. Matsen, Quantum chemistry integrals and tables (Univ. of Texas Press, Austin, Tex., 1959). U&), ~ = . 1 2 5 (, 125)25, n = O( 1)16, 14s;B.(z), z=O(. 125)24.875, n=0(1)16, 12-145. [5.31] J. Miller and R. P. Hurst, Simplified calculation of the exponential integral, Math. Tables Aids Comp. 12, 187-193 (1958). e-ZEi(z), Ei(z), efEl(2) , El(2) , z = .2(.05)5(.1)10(.2)20(.5)50(1)80; 10s. [5.32] National Bureau of Standarde, Tables of sine, sinu sin z&, Sc(a, Z)= 6D; Ss(a, z ) = l cosine and exponential integrals, vol. I (U.S. Government Printing Office, Washington, D.C., cos zdz, 6D; Cs(a, 2) sin z&, 1940). Si(z), Ci(z), Ei(z), E~(z), z=0(.0001)2, ~=0(.1)10; 9D. C c ( a , z ) = c y (1-cos z)&, 6 D ; u = w , [5.33] National Bureau of Standards, Tables of sine, cosine and exponential integrals, vol. I1 (U.S. GovernO
1

=c7

237

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

($)

[5.34] National Bureau of Standards, Table of sine and El - ,1-- i ; e x p ( g ) El(%); AE=.2(.2)2, cosine integrals for arguments from 10 to 100, T=25(25)1000, T= 150(10)390, 3-4s; 5-1, Applied Math. Series 32 (US. Government Printing Office, Washington, D.C., 1954). Si(z), exp(-x-I), x exp (-z-l), EI(z-~),exp (-P)dt, Ci(z), z=10(.01)100, 10D. z-1exp (Z-~)EI(Z-I),1-2-1 exp (z-l)El(x-l); z=.01 [5.35] National Bureau of Standards, Tables of functions (.0001).1, 5-6s. and of zeros of functions, Collected short tables [5.38] V. I. Pagurova, Tables of the exponential integral of the Computation Laboratory, Applied Math. Series 37 (U.S. Government Printing Office, E&) e-zw-’du. Translated from the RusWashington, D.C., 1954). E,@), n=0(1)20, sian by D. G. Fry (Pergamon Press, New York, z=0(.01)2(.1)10, 4-95; E2(z)-z In 2, z=O(.Ol) N.Y.; Oxford, London, England; Paris, France, .5, 7s; E3(z)+3x2 In x, z=O(.Ol).l, 75. 1961). E,(z), n=0(1)20, z=0(.01)2(.1)10, 4[5.36] National Bureau of Standards, Tables of the ex9 s ; Ez(z)-z In z, z=0(.01)5, 7s; E&)+3z1 In z, ponential integral for complex arguments, Applied Math. Series 51 (U.S. Government Printing z=O(.Ol).l, 75; e*E,,(z), n=2(1)10, z=lO (.1)20, Office, Washington, D.C., 1958). El(z)+ln z, 7D; ezE,(z), u=O(.l) 1, z= .01(.01)7(.05) 12(.1)20, 7SorD. 6D, ~ = 0 ( . 0 2 ) 1 , y=0(.02)1, ~ = - 1 ( . 1 ) 0 , y= [5.39] Tablitsy integral’nogo sinusa i kosinusa (Iedat. O(.l)l; EI(z), 6D, ~ = 0 ( . 0 2 ) 4 ,1/=0(.02)3(.05)10, Akad. Nauk SSSR., Moscow, U.S.S.R., 1954). 2=0(1)20, 1/=0(1)20, x= -3.1(.1)0, y=0(.1)3.1, ~=-4.5(.5)0, y=0(.1)4(.5)10, Z= -10(.5) -4.5, Si(z), Ci(z), s=0(.0001)2(.001) 10(.005) 100, 7D; Ci(z)-ln z, s=O(.OOOl).Ol, 7D. y=0(.5)10, x= -20(1)0, y=0(1)20; ezEl(z), 6D, [5.40] Tablitsy integral’noi pokaeatel’nol funktsii (Izdat. 2=4(.1) 10, y=0(.5) 10. Akad. Nauk SSSR., Moscow. U.S.S.R.. 1954). [5.37] S. Oberlander, Tabellen von Exponentialfunktionen Ei(z), El(z),z=0(.0001)1.3(.00i)3(.0005)10(.1)15, und-integralen eur Anwendung rtuf Gebieten der “T\ Thermodynamik, Halbleitertheorie und Gaskinetik (Akademie-Verlag, Berlin, Germany, [5.41] D. K. Trubey, A table of three exponential intekT exp -AE kT -AE , grals, Oak Ridge National Laboratory Report 1959). AE -9 -, kT A E kT (F) 2750, Oak Ridge, Tenn. (June 1959). El@), E ~ ( z ) ,E ~ ( z ) , Z= 0 (.0005). 1(-001)2 (.01) 10(. 1)20, El(%), & J T e x p ( s E ) dT,gexp(%)x

1

~

=sr

(

)’ zexp

0s.

238

EXPONENTIAL I N T ~ G R A L AND RELATED FUNCTIONS

SINE, COSINE AND EXPONENTIAL INTEGRALS

Table 5.1

0.00 0. 01 0.02 0.03 0. 04

x- *Si(x) 1.00000 00000 0.99999 44444 0.99997 77781 0.99995 00014 0.99991 11154

0.05 0.06 0. 07 0. 08 0. 09

0.99986 0.99980 0.99972 0.99964 0.99955

0.10 0: 11 0.12 0.13 0. 14

2

z-Z[Ci(z) -In

2-71

z-l[Eihc) -In x-rl. .

-0.25000 -0.24999 -0.24999 -0.24999 -0.24998

00000 89583 58333 06250 33339

1.'00000 1,00250 1.00502 1.00755 1.01008

0000 5566 2306 0283 9560

11215 00216 78178 45127 01094

-0.24997 -0.24996 -0.24994 -0.24993 -0.24991

39598 25030 89639 33429 56402

1.01264 1.01520 1.01777 1,02036 1.02295

0.99944 0.99932 0.99920 0.99906 0.99891

46111 80218 03455 15870 17512

-0.24989 -0.24987 -0.24985 -0.24982 -0.24979

58564 39923 00480 40244 59223

0. 15 0.16 0.17 0.18 0.19

0.99875 0.99857 0.99839 0.99820 0.99799

08435 88696 58357 17486 66151

-0.24976 -0.24973 -0.24969 -0.24966 -0.24962

0.20 0.21 0.22 0.23 0.24

0.99778 0.99755 0.99731 0.99706 0.99680

04427 32390 50122 57709 55242

0.25 0. 26 0.27 0.28 0. 29

0.99653 0.99625 0.99595 0.99565 0.99533

0. 30 0.31 0.32 0.33 0.34

z-~[E1(z)+lnz+r] 1.00000 00000

0.99750 0.99502 0.99254 0.99008

55452 21392 97201 82265

0202 2272 5836 0958 7705

0.98763 0.98519 0.98276 0.98035 0.97794

75971 77714 86889 02898 25142

1.02556 1.02818 1.03081 1.03346 1.03611

6141 6335 8352 2259 8125

0.97554 0.97315 0: 97078 0.96841 0.96606

53033 85980 23399 64710 09336

57422 34850 91516 27429 42598

1.03878 1.04146 1.04415 1,04686 1.04957

6018 6006 8158 2544 9234

0.96371 0.96138 0.95905 0.95674 0.95443

56702 06240 57383 09569 62237

-0.24958 -0.24954 -0.24949 -0.24944 -0.24940

37035 10749 63752 96056 07674

1.05230 1.05504 1.05780 1.06057 1.06334

8298 9807 3833 0446 9719

0.95214 0.94985 0.94758 0.94531 0.94306

14833 66804 17603 66684 13506

42813 20519 88464 46750 95489

-0.24934 -0.24929 -0.24924 -0.24918 -0.24912

98618 68902 18540 47546 55938

1.06614 1.06894 1.07176 1.07459 1.07743

1726 6539 4232 4879 8555

0.94081 0.93857 0.93635 0.93413 0.93192

57528 98221 35046 67481 94997

0.99501 0; 99467 0.99432 0; 99396 0.99360

34793 64779 85570 97288 00064

-0.24906 -0.24900 -0.24893 -0.24886 -0.24879

43727 10933 57573 83662 89219

1.08029 1.08316 1.08604 1.08894 1.09185

5334 5293 8507 5053 5008

0.92973 0.92754 0.92536 0.92319 0.92103

17075 33196 42845 45510 40684

0. 35 0.36 0.37 0. 38 0. 39

0.99321 0.99282 0.99242 0.99201 0.99158

94028 79320 56078 24449 84579

-0.24872 -0.24865 -0.24857 -0.24850 -0.24842

74263 38813 82887 06507 09693

1.09477 1.09771 1.10066 1.10363 1.10660

8451 5458 6108 0481 8656

0.91888 0.91674 0.91460 0.91248 0.91036

27858 06533 76209 36388 86582

0. 40 0.41 0.42 0.43 0.44

0.99115 0.99070 0.99025 0.98978 0.98930

36619 80728 17063 45790 67074

-0.24833 -0.24825 -0.24816 -0.24808 -0.24799

92466 54849 96860 18528 19870

1.10960 1.11260 1.11562 1.11866 1.12170

0714 6735 6800 0991 9391

0.90826 0.90616 0.90407 0.90199 0.89992

26297 55048 72350 77725 70693

0.45 0. 46 0.47 0.48 0.49

0.98881 0.98831 0.98780 0.98728 0.98675

81089 88008 88010 81278 67998

-0.24790 -0.24780 -0.24771 -0.24761 -0.24751

00913 61685 02206 22500 22600

1.12477 1.12784 1.13094 1.13404 1.13716

2082 9147 0671 6738 7432

0.89786 0.89581 0.89376 0.89173 0.88970

50778 17511 70423 09048 32920

0. 50

0.98621 48361

See Examples 1-2

-0.24741 02526 y

a

=0.57721 56649

1.14030 2841

0.88768 41584

"-!"3

239

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

SINE, COS17VE AND EXPONEN'TIAL INTEGRALIS 5

Si(x)

Ci(x)

Table 5.1

Ei(x)

Eib)

0.50 0.51 0. 52 0.53 0. 54

0.49310 0.50268 0.51225 0.52179 0.53132

74180 77506 15212 84228 81492

-0.17778 -0.16045 -0.14355 -0.12707 -0.11099

40788 32390 37358 07938 04567

0.45421 0.48703 0.51953 0.55173 0.58364

9905 2167 0633 0445 5931

0.55977 0.54782 0.53621 0.52495 0.51400

3595 2352 9798 1510 3886

0.55 0. 56 0. 57 0.58 0.59

0.54084 0.55033 0.55981 0.56926 0.57870

03951 48563 12298 92137 85069

-0.09529 -0.07998 -0.06503 -0.05044 -0.03618

95274 55129 65744 14815 95707

0.61529 0.64667 0.67781 0.70872 0.73940

0657 7490 8642 5720 9764

0,50336 0.49301 0,48296 0.47317 0.46364

4081 9959 0034 3433 9849

0. 60 0. 61 0. 62 0.63 0. 64

0.58812 0.59752 0.60691 0.61627 0.62561

88096 98233 12503 27944 41603

-0.02227 -0.00867 +O. 00460 0. 01758 0.03026

07070 52486 59849 17424 03686

0.76988 1290 0.80015 0320 0.83022 6417 ol86Oii 8716 0,88983 5949

0.45437 0.44535 0.43656 0.42799 0.41965

9503 3112 1854 7338 1581

0. 65 0. 66 0. 67 0. 68 0.69

0.63493 0.64423 0.65351 0.66277 0.67200

50541 51831 42557 19817 80721

0.04264 0.05475 0.06659 0.07815 0.08946

98293 77343 13594 76659 33195

0.91938 0.94877 0.97801 1.00711 1.03607

6468 8277 9042 6121 6576

0.41151 OI40358 0.39585 0,38830 0.38095

6976 6275 2563 9243 0010

0.70 0. 71 0. 72 0.73 0.74

0.68122 0.69041 0.69958 0.70873 0.71785

22391 41965 36590 03430 39660

0.10051 0.11131 0.12187 0.13220 0.14229

47070 79525 89322 32879 64404

1.06490 1.09361 1.12220 1.15068 1.17905

7195 4501 4777 4069 8208

0.37376 0.36675 0.35991 0.35323 0.34671

8843 9981 7914 7364 3279

0.75 0.76 0.77 0.78 0.79

0.72695 0.73603 0.74508 0.75411 0.76311

42472 09067 36664 22494 63804

0.15216 0.16180 0.17123 0.18045 0.18946

36010 97827 98110 83335 98290

1.20733 1.23551 1.26360 1.29161 1.31954

2816 3319 4960 2805 1753

0.34034 0.33411 0.32803 0.32208 0.31627

0813 5321 2346 7610 7004

0.80 0.81 0.82 0.83 0.84

0.77209 0.78105 0.78997 0.79888 0.80776

57855 01921 93293 29277 07191

0.19827 0.20688 0.21530 0.22352 0.23156

86160 88610 45859 96752 78824

1.34739 1.37518 1.40290 1.43056 1.45816

6548 1783 1910 1245 3978

0.31059 0.30504 0.29961 0.29429 0.28910

6579 2539 1236 9155 2918

0.85 0.86 0.87 0. 88 0. 89

0.81661 0.82543 0.83423 0.84300 0.85175

24372 78170 65953 85102 33016

0.23942 0.24709 0.25459 0.26192 0.26907

28368 80486 69153 27264 86687

1.48571 1.51321 1.54067 1.56808 1.59546

4176 5791 2664 8534 7036

0.28401 0.27904 0.27417 0,26941 0.26474

9269 5070 7301 3046 9496

0.90 0.91 0.92 0.93 0.94

0.86047 0.86916 0.87782 0.88645 0.89506

07107 04808 23564 60839 14112

0.27606 0.28289 0.28955 0.29606 0.30241

78305 32065 77018 41358 52458

1.62281 1.65012 1.67741 1.70467 1.73191

1714 6019 3317 6891 9946

0.26018 0.25571 0.25133 0.24704 0.24285

3939 3758 6425 9501 0627

0.95 0.96 0.97 0.98 0.99

0.90363 0.91218 0.92070 0.92919 0.93765

80880 58656 44970 37370 33420

0.30861 0.31466 0.32056 0.32631 0.33193

36908 20547 28495 85183 14382

1.75914 1.78635 1.81355 1.84074 1.86793

5612 6947 6941 8519 4543

0.23873 0.23470 0.23075 0.22689 0.22309

7524 7988 9890 1167 9826

1. 00

0.94608 30704

0.33740 39229

1.89511 7816

0.21938 3934 ~

4

1

240

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

SINE, COSINE AND EXPON'ENTIAL INTEGRALS

Table 5.1 X

Si(x)

Ci(x)

1.00 1. 01 1.02 1. 03 1. 04

0.94608 0.95448 0.96285 0.97119 0.97949

30704 26820 19387 06039 84431

0.33740 0.34273 0.34793 0.35300 0.35793

39229 82254 65405 10067 37091

1.89511 1.92230 1.94948 1.97667 2.00387

7816 1085 7042 8325 7525

Ei(x) 0.21938 3934 0.21574 1624 0.21217 1083 0.20867 0559 0.20523 8352

1. 05 1. 06 1. 07 1.08 1. 09

0.98777 0.99602 1.00423 1.01241 1.02056

52233 07135 46846 69091 71617

0.36273 0.36741 0.37196 0.37638 0.38069

66810 19060 13201 68132 02312

2.03108 2.05830 2.08554 2.11280 2.14007

7184 9800 7825 3672 9712

0.20187 0.19857 0.19533 0.19216 0.18904

2813 2347 5403 0479 6118

1.10 1.11 1.12 1.13 1. 14

1.02868 1.03677 1.04482 1.05284 1.06083

52187 08583 38608 40082 10845

0.38487 0.38893 0.39288 0.39671 0.40043

33774 80142 58645 86134 79090

2.16737 2.19470 2.22205 2.24943 2.27684

8280 1672 2152 1949 3260

0.18599 0.16299 0.18005 0.17716 0.17433

0905 3465 2467 6615 4651

1.15 1.16 1. 17 1.18 1.19

1.06878 1.07670 1.08459 1.09244 1.10026

48757 51696 17561 44270 29760

0.40404 0.40754 0.41093 0.41421 0.41738

53647 25593 10390 23185 78816

2.30428 2.33176 2.35928 2.38684 2.41444

8252 9062 7800 6549 7367

0.17155 0.16882 0.16615 0.16352 0.16094

5354 7535 0040 1748 1567

1.20 1.21 1. 22 1. 23 1.24

1.10804 1.11579 1.12351 1.13119 1.13883

71990 68937 18599 18994 68160

0.42045 0.42342 0.42629 0.42906 0.43172

91829 76482 46760 16379 98802

2.44209 2.46978 2.49752 2.52531 2.55315

2285 3315 2442 1634 2836

0.15840 0.15592 0.15347 0.15108 0.14872

8437 1324 9226 1164 6188

1.25 1.26 1. 27 1.28 1.29

1.14644 1.15402 1.16155 1.16906 1.17652

64157 05063 88978 14023 78340

0.43430 0.43677 0.43915 0.44144 0.44363

07240 54665 53815 17205 57130

2.58104 2.60899 2.63700 2.66507 2.69320

7974 8956 7673 5997 5785

0.14641 0.14414 0.14191 0.13971 0.13756

3373 1815 0639 8989 6032

1.30 1.31 1.32 1.33 1.34

1.18395 1.19135 1.19870 1.20602 1.21331

80091 17459 88649 91886 25418

0.44573 0.44775 0.44967 0.45151 0.45326

85675 14723 55955 20863 20753

2.72139 2.74965 2.77798 2.80637 2.83484

8880 7110 2287 6214 0677

0.13545 0.13337 0.13133 0.12932 0.12735

0958 2975 1314 5224 3972

1.35 1.36 1.37 1. 38 1.39

1.22055 1.22776 1.23493 1.24207 1.24916

87513 76460 90571 28180 87640

0.45492 0.45650 0.45800 0.45941 0.46075

66752 69811 40711 90071 28349

2.86337 2.89198 2.92067 2.94943 2.97828

7453 8308 4997 9263 2844

0.12541 0.12351 0.12164 0.11980 0.11799

6844 3146 2198 3337 5919

1.40 1. 41 1.42 1.43 1.44

1.25622 1.26324 1.27022 1.27717 1.28407

67328 65642 81004 11854 56658

0.46200 0.46318 0.46427 0.46529 0.46624

65851 12730 78995 74513 09014

3.00720 3.03621 3.06530 3.09448 3.12375

7464 4843 6691 4712 0601

0.11621 0.11447 0.11275 0.11106 0.10940

9313 2903 6090 8287 8923

1.45 1.46 1.47 1.48 1.49

1.29094 1.29776 1.30455 1.31130 1.31801

13902 82094 59767 45473 37788

0.46710 0.46790 0.46862 0.46927 0.46984

92094 33219 41732 26848 97667

3,15310 3.18255 3.21209 3.24172 3,27145

6049 2741 2355 6566 7042

0.10777 0.10617 0,10459 0,10304 0,10151

7440 3291 5946 4882 9593

1. 50

1.32468 35312

ri- t35-1

L' 4 '

J

0.47035 63172

r( -

L'

5)21 5' J

Ei(x)

3.30128 5449

0.10001 9582

241

EXPONENTIAL INTEGRAL AND RELATED FUNC'I!IONS

SINE, COSINE AND EXPONENTIAL INTEGRALS X

Si(x)

Ci(x)

Ei(x)

Table 5.1 El (5)

1.50 1.51 1.52 1.53 1.54

1.32468 1.33131 1,33790 1.34445 1.35096

35312 36664 40489 45453 50245

0.47035 0.47079 0.47116 0.47146 0.47169

63172 32232 13608 15952 47815

3.30128 3.33121 3.36124 3.39137 3.42161

5449 3449 2701 4858 1576

0.10001 0.09854 0.09709 0.09566 0.09426

9582 4365 3466 6424 2786

1.55 1.56 1.57 1.58 1.59

1.35743 1.36386 1.37025 1.37660 1.38291

53577 54183 50823 42275 27345

0.47186 0.47196 0.47200 0.47197 0.47188

17642 33785 04495 37932 42164

3.45195 3.48240 3.51296 3.54363 3.57442

4503 5289 5580 7024 1266

0.09288 0.09152 0.09018 0.08687 0.08758

2108 3960 7917 3566 0504

1.60 1.61 1.62 1.63 1.64

1.38918 1.39540 1.40159 1.40773 1.41384

04859 73666 32640 80678 16698

0.47173 0.47151 0.47124 0.47091 0.47052

25169 94840 58984 25325 01507

3.60531 3.63633 3.66746 3.69871 3.73009

9949 4719 7221 9099 1999

0.08630 0.08505 0.08382 0.08261 0.08142

8334 6670 5133 3354 0970

1.65 1.66 1.67 1.68 1.69

1.41990 1.42592 1.43190 1.43784 1.44373

39644 48482 42202 19816 80361

0.47006 0.46956 0.46899 0.46837 0.46769

95096 13580 64372 54812 92169

3.76158 3.79320 3.82495 3.85682 3.88882

7569 7456 3310 6783 9528

0.08024 0.07909 0.07795 0.07683 0.07573

7627 2978 6684 8412 7839

1.70 1.71 1.72 1.73 1.74

1.44959 1.45540 1.46117 1.46690 1.47258

22897 46507 50299 33404 94974

0.46696 0.46618 0.46534 0.46445 0.46351

83642 36359 57385 53716 32286

3.92096 3.95322 3.98562 4.01816 4.05084

3201 9462 9972 6395 0400

0.07465 OIO7358 0.07253 0.07150 0.07048

4644 8518 9154 6255 9527

1.75 1.76 1.77 1.78 1.79

1.47823 1.48383 1.48939 1.49491 1.50038

34189 50249 42379 09830 51872

0.46251 0.46147 0.46038 0.45924 0.45804

99967 63568 29839 05471 97097

4.08365 4.11660 4.14970 4.18294 4.21633

3659 7847 4645 5736 2809

0,06948 0.06850 0.06753 0.06657 0.06563

8685 3447 3539 8691 8641

1.80 1.81 1.82 1.83 1.84

1.50581 1.51120 1.51655 1.52185 1.52711

67803 56942 18633 52243 57165

0.45681 0.45552 0.45419 0.45281 0.45139

11294 54585 33436 54262 23427

4.24986 4.28355 4.31738 4.35137 4.38551

7557 1681 6883 4872 7364

0.06471 0.06380 0,06290 0.06202 0.06115

3129 1903 4715 1320 1482

1.85 1. 86 1.87 1.88 1.89

1.53233 1.53750 1.54263 1.54772 1.55277

32813 78626 94066 78621 31800

0.44992 0.44841 0.44685 0.44526 0.44362

47241 31966 83813 08948 13486

4.41981 4.45427 4.48888 4.52366 4.55860

6080 2746 9097 6872 7817

0.06029 0.05945 0.05862 0.05780 0.05699

4967 1545 0994 3091 7623

1.90 1. 91 1.92 1.93 1.94

1.55777 1.56273 1.56764 1.57252 1.57735

53137 42192 98545 21801 11591

0.44194 0.44021 0.43845 0.43665 0.43481

03497 85005 63991 46388 38088

4.59371 4.62898 4.66442 4.70003 4.73582

3687 6242 7249 8485 1734

0.05620 0.05542 0.05465 0.05389 0.05314

4378 3149 3731 5927 9540

1.95 1.96 1.97 1. 98 1.99

1.58213 1.58687 1.59157 1.59623 1.60084

67567 89407 76810 29502 47231

0.43293 0.43101 0.42906 0.42707 0.42504

44941 72752 27288 14273 39391

4.77177 4.80791 4.84422 4.88071 4.91738

8785 1438 1501 0791 1131

0.05241 0.05169 0.05097 0.05027 0,04958

4380 0257 6988 4392 2291

2.00

1.60541 29768

0.42298 08288

4.95423 4356

~

0.04890 0511

I

242

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

SINE, COSINE AND EXPONENTI-iL INTEGRALS

Table 5.1 X

Si(x)

Ci(x)

xe-"Ei(x)

xezE ~ ( x )

2. 0 2.1 2.2 2.3 2.4

1.60541 1.64869 1.68762 1.72220 1.75248

29768 86362 48272 74818 55008

0.42298 0.40051 0.37507 0.34717 0.31729

08288 19878 45990 56175 16174

1.34096 1.37148 1.39742 1.41917 1.43711

5420 6802 1992 1534 8315

0.72265 0.73079 0.73843 0.74562 0.75240

7234 1502 1132 2149 4829

2. 5 2. 6 2. 7 2. 8 2. 9

1.77852 1.80039 1.81821 1.83209 1.84219

01734 44505 20765 65891 01946

0.28587 0.25333 0.22008 0.18648 0.15289

11964 66161 48786 83896 53242

1.45162 1.46303 1.47166 1.47780 1.48174

5159 3397 2153 8187 6162

0.75881 0.76488 0.77063 0.77610 0,78130

4592 2722 6987 2123 0252

3. 0 3.1 3.2 3. 3 3.4

1.84865 1.85165 1.85140 1.84808 1.84191

25280 93077 08970 07828 39833

0.11962 0.08699 0.05525 +O. 02467 -0.00451

97860 18312 74117 82846 80779

1.48372 1.48398 1.48274 1,48017 1.47646

9204 9691 0191 4491 8706

0.78625 0.79097 0.79548 0.79979 0.80391

1221 2900 1422 1408 6127

3. 5 3. 6 3. 7 3. 8 3.9

1.83312 1.82194 1.80862 1.79339 1.77650

53987 81156 16809 03548 13604

-0.03212 -0.05797 -0.08190 -0.10377 -0.12349

85485 43519 10013 81504 93492

1.47178 1.46625 1.46003 1.45321 1.44590

2389 9659 0313 0902 5765

0.80786 0.81165 0,81529 0.81878 0.82214

7661 7037 4342 8821 8967

4. 0 4.1 4.2 4.3 4.4

1.75820 1.73874 1.71836 1.69731 1.67583

31389 36265 85637 98507 39594

-0.14098 -0.15616 -0.16901 -0.17950 -0.18766

16979 53918 31568 95725 02868

1.43820 1.43020 1.42195 1.41354 1.40501

8032 0557 6813 1719 2424

0.82538 0.82849 0.83149 0.83439 0.83718

2600 6926 8602 3794 8207

4. 5 4. 6 4.7 4.8 4.9

1.65414 1.63246 1.61100 1.58997 1.56955

04144 03525 51718 52782 89381

-0.19349 -0.19704 -0.19839 -0.19760 -0.19477

11221 70797 12468 36133 98060

1.39641 1.38780 1.37920 1.37066 1.36219

9030 5263 9093 3313 6054

0.83988 0,84249 0,84501 0.84745 0,84982

7144 5539 7971 8721 1778

5. 0 5.1 5.2 5. 3 5.4

1.54993 1.53125 1.51367 1.49731 1.48230

12449 32047 09468 50636 00826

-0.19002 -0.18347 -0.17525 -0.16550 -0.15438

97497 62632 36023 59586 59262

1.35383 1.34558 1.33748 1.32953 1.32175

1278 9212 6755 7845 3788

0.85211 0.85432 0.85648 0.85856 0,86059

0880 9519 0958 8275 4348

5. 5 5.6 5.7 5. 8 5.9

1.46872 1.45666 1.44619 1.43735 1.43018

40727 83847 75285 91823 43341

-0.14205 -0.12867 -0.11441 -0.09944 -0.08393

29476 17494 07808 06647 26741

1.31414 1.30671 1.29947 1.29241 1.28555

3566 4107 0536 6395 3849

0.86256 0.86447 0.86633 0.86813 0.86989

1885 3436 1399 8040 5494

6. 0 6.1 6.2 6. 3 6.4

1.42468 1.42086 1.41870 1.41817 1.41922

75513 73734 68241 40348 29740

-0.06805 -0.05198 -0.03587 -0.01988 -0.00418

72439 25290 30193 82206 14110

1.27888 1.27240 1.26612 1,26002 1.25411

3860 6357 0373 4184 5417

0.87160 0.87327 0.87489 0.87647 0.87801

5775 0793 2347 2150 1816

6. 5 6.6 6. 7 6. 8 6. 9

1.42179 1.42581 1.43120 1.43786 1.44570

42744 61486 53853 84161 24427

+O. 01110 15195

1.24839 1.24284 1.23748 1.23228 1.22726

1155 8032 2309 9952 6684

0.87951 0.88097 0.88240 0.88379 0.88515

2881 6797 4955 8662 9176

7. 0

1.45459 66142

0.07669 52785

0.02582 0.03985 0.05308 0.06539

31381 54400 07167 23140

1.22240 8053

[(-;I6]

0.88648 7675

[(-€Y

243

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS Table -5.1

SINE, COSINE AND EXPONEN rIAL INTEGRALS

Ci(r)

Si(1)

I

qe-lEi(t)

(I)

CPI].:,

7.0 7.1 7.2 7.3 7.4

1.45459 1.46443 1.47508 1.48643 1.49834

66142 32441 90554 64451 47533

0.07669 0.08690 0.09595 0.10378 0.11035

52785 68881 70643 86664 76658

1.22240 1.21770 1.21316 1.20877 1.20452

8053 9472 6264 3699 7026

0.88648 0.88778 0.88905 0.89029 0.89150

7675 5294 3119 2173 3440

7.5 7.6 7.7 7.8 7.9

1.51068 1.52331 1.53610 1.54893 1.56167

15309 37914 92381 74581 10702

0.11563 0.11959 0.12224 0.12358 0.12363

32032 75293 58319 59542 80071

1.20042 1.19645 1.19261 1.18890 1.18531

1500 2401 5063 4881 7334

0.89268 0.89384 0.89497 0.89608 0.89717

7854 6312 9666 8737 4302

8.0 8.1 8.2 8.3 8.4

1.57418 1.58636 1.59809 1.60927 1.61980

68217 66225 85106 75419 65968

0.12243 0.12001 0.11644 0.11176 0.10607

38825 66733 00055 72931 09196

1.18184 1.17849 1.17524 1.17210 1.16906

7987 2509 6676 6376 7617

0.89823 0.89927 0.90029 0.90129 0.90227

7113 7888 7306 6033 4695

8.5 8.6 8.7 8.8 8.9

1.62959 1.63856 1.64665 1.65379 1.65993

70996 96454 45309 21861 35052

0.09943 0.09193 0.08367 0.07475 0.06528

13586 62396 93696 97196 03850

1.16612 1.16327 1.16052 1.15785 1.15526

6526 9354 2476 2390 5719

0.90323 0,90417 0.90509 0.90600 0.90688

3900 4228 6235 0459 7415

1.66504 00758

0.05534 0.04506 0.03455 0.02391 0.01325

75313 93325 49134 33045 24187

1.15275 1.15032 1.14797 1.14568 1.14347

9209 9724 4251 9889 3855

0.90775 0.90861 0.90944 0.91027 0.91107

7602 1483 9530 2177 9850

33423 69801 45697 96169 40566

+O. 00267 80588

1.14132 1.13923 1.13720 1.13524 1.13332

3476 6185 9523 1130 8746

0.91187 0.91265 0.91341 0.91416 0.91490

2958 1897 7043 8766 7418

1.65834 75942

-0.04545 64330

9.0 ~.

i; 66908 43056

9.1 9.2 9.3 9.4

1.67204 94480 1.67392 95283 1.67472 91725

9.5 9.6 9.7 9.8 9.9

1.67446 1.67315 1.67084 1.66756 1.66338

10.0

-0.00770 -0.01780 -0.02751 -0.03676

70361 40977 91811 39563

[ -46)

1.13147 0205

0.91563 3339 ( 41

Table 5.2 SINE, COSINE AND EXPONENTIAL INTEGRALS FOR LARGE ARGUMEY'IS

If09

A-]

tr-lEi( I )

l"0)

Ic~El(1)



0.100 0.095 0.090 0; 085 0.080

0.98191 0.98353 0.98509 Oi98660 0.98803

0351 4427 9171 1776 9405

0.94885 0.95323 0.95748 0.96160 0.96557

39 18 44 17 23

1.13147 1.12249 1.11389 1.10564 1.09773

021 671 377 739 775

0.91563 0.91925 0.92293 0.92665 0.93044

33394 68286 15844 90998 09399

10 11 11 12 13

0.075 0.070 0.065 0.060 0.055

0.98940 0.99070 0.99193 0.99308 0.99415

9188 8244 3695 2682 2385

0.96938 0.97302 0.97649 0,97976 0.98281

56 98 35 47 17

1.09014 1.08283 1.07578 1.06896 1.06236

087 054 038 548 365

0.93427 0.93817 0.94212 0.94614 0.95022

87466 42450 92486 56670 55126

13 14 15 17 18

0.050 0.045 0.040 0.035 0.030

0.99514 0.99604 0,99685 0.99758 0.99821

0052 3013 8722 4771 8937

0.98568 0.98830 0.99068 0.99282 0.99469

24 52 81 12 37

1.05595 1.04972 1.04366 1.03775 1.03198

591 640 194 135 503

0.95437 0.95858 0.96286 0.96722 0.97165

09099 41038 74711 35311 49596

20 22 25 29 33

0.025 0.020 0.015 0.010 0,005

0.99875 0.99920 0.99955 0,99980 0.99995

9204 3795 1207 0239 0015

0.99629 0.99761 0,99865 0.99940 0.99985

57 89 60 12 01

1.02635 1.02085 1.01547 1.01020 1.00505

451 228 157 625 077

0.97616 0.98075 0.98543 0.99019 0.99504

46031 54965 08813 42287 92646

40 50 67 100 200

0.000

1.00000 0000

[!-y]

1.00000 00

[(-y]

s i ( t ) = ; - j ( t )cos

l-Y(I)

5 =1.57079 63268 See Example 3.

sin

I

1.00000 000

[(-;)53

1.00000 00000

I":-

Ci(L)=f(l) sin l - y ( i ) cos <s>=nearest integer to

I.

L

0)

244

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

Table 5.3 X

SINE AND COSINE INTEGRALS FOR ARGUMENTS

Si (TX)

Cin(rx) 00

a

Si ( r x )

TZ

Cin(T.c)

28 67 75 10 77 63 93 42 68

5. 0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

1.63396 1.63088 1.62211 1.60871 1.59212 1.57408 1.55635 1.54064 1.52839 1.52065

48 98 92 21 99 24 75 82 53 96

3.32742 3.36670 3.40335 3.43582 3.46297 3.48419 3.49941 3.50911 3.51426 3.51619

23 50 81 68 82 47 45 89 89 81

1.64827 1.83737 2.00168 2.13821 2.24595 2.32581 2.38040 2.41370 2.43067 2.43680

75 48 51 22 41 82 96 98 75 30

6.0 6.1 6.2 6.3 6. 4 6.5 6. 6 6.7 6. 8 6. 9

1.51803 1.52060 1.52794 1.53921 1.55318 1.56843 1.58344 1.59679 1,60723 1.61383

39 20 77 04 17 12 97 62 30 85

3.51647 3.51674 3.51857 3.52330 3.53192 3.54500 3.56264 3.58447 3.60972 3.63727

44 38 25 06 30 55 55 72 10 15

16 13 38 03 40 10 60 16 62 05

2.43765 2.43844 2.44365 2.45676 2.48004 2.51446 2.55975 2.61452 2.67647 2.74269

34 23 73 95 47 40 53 59 93 41

7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 71 9

1.61608 1.61388 1.60756 1i59785 1;58578 1.57257 1.55954 1.54794 1.53885 1i53309

55 08 18 21 88 96 81 84 50

i3

3.66581 3.69395 3.72034 3i74385 3.76362 3.77914 3.79032 3.79749 3.80131 3.80274

26 05 97 98 13 01 64 22 21 91

1.67476 1.66968 1.65535 1.63369 1.60721 1.57870 1.55099 1.52667 1.50788 1.49612

18 11 02 82 88 92 62 49 19 20

2.80993 2.87498 2.93491 2.98737 3.03074 3.06427 3.08807 3.10310 3.11100 3.11393

76 49 77 63 73 25 51 38 53 95

8.0 8.1 8.2 8.3 8.4 8:_58.6 8. 7 8.8 8.9

1.53113 1,53306 1.53860 1.54713 1.55776 1;5694o - _1.58091 1.59117 1.59922 1.60433

13 26 67 99 52 54 06 06 11 29

3.80295 3.80315 3.80453 3.80812 3.81467 3;82466 3;83818 3.85496 3.87444 3.89576

56 83 88 16 97 68 15 61 05 52

4.0 4.1 4.2 4.3 4.4 4.5 4.6 A. 7 4.8 4.9

1.49216 1.49599 1.50687 1.52343 1.54382 1.56593 1.58755 1.60664 1.62147 1.63080

12 24 40 40 74 04 15 04 45 69

3.11435 3.11475 3.11746 3.12441 3.13699 3.15595 3.18134 3.21256 3.24843 3.28734

65 82 60 61 91 79 84 74 85 92

9. 0 9.1 9.2 9.3 9.4 9.5 9. 6 9.7 9.8 9.9

1,60607 1.60435 1.59942 1.59180 1.58232 1.57191 1.56161 1.55241 1.54519 1,54059

69 85 00 91 00 16 12 46 00 74

3.91792 3.93984 3.96047 3.97890 3.99443 4.00666 4.01551 4.02119 4.02422 4.02537

84 77 61 22 58 94 22 22 80 29

5.0

1.63396 48

3.32742 23

10.0

0.00000 0.31244 0.61470 0.89718 1.15147 1.37076 1.55023 1.68729 1.78166 1.83523

'00 18 01 92 74 22 35 94 12 65

0.00000 0.02457 0.09708 0.21400 0.36970 0.55679 0.76666 0.98995 1.21719 1.43932

1.2 1.3 1.4 1.5 1.6 1; 7 1.8 1.9

1.85193 1.83732 1.79815 1.74191 1.67621 1,60837 1.54487 1i49103 1.45072 1.42621

70 28 90 10 68 27 36 51 37 05

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2; 8 2.9

1.41815 1.42569 1.44667 1.47794 1.51568 1.55583 1.59441 1.62792 1.65355 1.66945

3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

0.0 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

[(-;)51

1.53902 91

4.02553 78

[(-;)71

[(-;)71

+

r+ln r=1.72194 55508 Ci (m) =Y In r +In z- Cin(a) Si(nr) are maximum values of Si(z) if n>O is odd, and minimum values if n>O is even. Ci

[()L+;) .],are

maximum values of Ci(z) if n>O is even, and minimum values if n>O is

245

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

Table 5.4

EXPONENTIAL INTEGRALS E,,(%) 2:

E ~ ( x- X) In z

(4

0.00 0.01 0.02 0.03 0.04

1.00000 0.99572 0.99134 0.98686 0.98229

00 22 50 87 39

0.50000 0.49027 0.48096 0.47199 0.46332

00 66 83 77 39

0.33333 0.32838 0.32352 0.31876 0.31408

33 24 64 19 55

Eio(2:) 0.11111 11 0.10986 82 0.10863 95 0.10742 46 0.10622 36

0.05263 0.05207 0.05153 0.05099 0.05045

16 90 21 11 58

0.05 0.06 0. 07 0. 08 0.09

0,97762 0.97285 0.96798 0.96301 0.95795

11 08 34 94 93

0.45491 0.44676 0.43883 0.43111 0.42360

88 09 27 97 96

0.30949 0.30498 0.30055 0.29620 0.29193

45 63 85 89 54

0.10503 0.10386 0.10270 0.10155 0.10042

63 24 18 44 00

0.04992 0.04940 0.04888 0,04837 0.04786

60 19 33 02 24

0.10 0.11 0.12 0; 13 0.14

0.95280 0.94755 0.94220 0.93676 0.93123

35 26 71 72 36

0.41629 0.40915 0.40219 0.39539 0.38876

15 57 37 77 07

0.28773 0.28360 0.27955 0.27556 0.27164

61 90 24 46 39

0.09929 0.09818 0,09709 0.09600 0.09493

84 96 34 95 80

0.04736 0.04686 0,04637 0.04588 0.04540

00 29 10 43 27

0.15 0. 16 0.17 0. 18 0; 19

0.92560 0,91988 0.91407 0.90817 0.90217

67 70 48 06 50

0.38227 0.37593 0.36974 0.36367 0.35774

61 80 08 95 91

0.26778 0.26399 0.26026 0.25660 0.25299

89 79 96 26 56

0.09387 0.09283 0.09179 0.09077 0.08975

86 12 56 18 95

0.04492 0.04445 0,04398 0.04352 0.04306

62 47 82 66 98

0.20 0.21 0.22 0.23 0.24

0,89608 0,88991 0.88364 0.87728 0.87083

82 09 33 60 93

0.35194 0.34626 0.34070 0.33525 0,32991

53 38 05 18 42

0.24944 0.24595 0.24252 0.23914 0.23581

72 63 16 19 62

0.08875 0.08776 0.08679 0.08582 0.08486

87 93 10 38 75

0.04261 0.04217 0.04172 0.04129 0.04085

79 07 82 03 71

0.25 0.26 0; 27 0. 28 0.29

0.86430 0.85767 OI85096 0.84416 0.83728

37 97 76 78 08

0.32468 0.31955 0.31453 0.30960 0.30477

41 85 43 86 87

0.23254 0.22932 0.22615 0.22303 0.21995

32 21 17 11 93

0.08392 0.08298 0.08206 0.08114 0.08024

20 72 30 92 57

0.04042 0.04000 0.03958 0.03916 0.03875

85 43 46 93 84

0.30 0.31 0.32 0; 33 0.34

0.83030 0.82324 0.81610 0.80886 0.80155

71 69 07 90 21

0.30004 0.29539 0.29083 0.28636 0.28197

18 56 74 52 65

0.21493 0.21395 0.21102 0.20814 0.20529

52 81 70 11 94

0.07935 0.07846 0.07759 0.07673 0.07587

24 93 60 27 90

0.03835 0.03794 0.03755 0.03715 0.03676

18 95 15 76 78

0.35 0. 36 0.37 0. 38 0.39

0.79415 0.78666 0.77909 0.77144 0.76370

04 44 43 07 39

0.27766 0.27344 0.26929 0.26521 0.26121

93 16 13 65 55

0.20250 0.19974 0.19703 0.19435 0.19172

13 58 22 97 76

0.07503 0,07420 0.07337 0,07255 0.07175

50 06 55 97 31

0,03638 0.03600 0.03562 0.03524 0.03487

22 06 31 95 98

0.40 0.41 0.42 0.43 0.44

0.75588 0.74798 0.73999 0.73193 0.72378

43 23 82 24 54

0.25728 0.25342 0.24963 0.24591 0.24225

64 76 73 41 63

0.18913 0.18658 0.18406 0.18158 0,17914

52 16 64 87 79

0.07095 0.07016 0.06938 0.06861 0.06785

57 71 75 67 45

0.03451 0.03415 0.03379 0.03343 0,03308

40 21 39 96 89

0.45 0. 46 0.47 0.48 0. 49

0,71555 0.70724 0.69886 0.69039 0.68184

75 91 05 21 43

0.23866 0.23513 0.23166 0.22825 0,22489

25 13 12 08 90

0.17674 0.17437 0.17204 0.16974 0.16747

33 44 05 10 53

0.06710 0.06635 0.06561 0.06489 0.06417

09 58 91 07 04

0.03274 0.03239 0.03205 0.03172 0.03139

20 87 90 29 03

0.50

0.67321 75

"-y]

See Examples 4-6.

E3

0.22160 44

[(- 5)5]

E4 (2)

0.16524 28

C'-y1

"-;'"I

0.06345 83

Ezo(2:)

0.03106 12

[)(: -

71

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

EXPONENTIAL INTEGRALS E,(%)

Table 5 .4 X

0.32664 0.32110 0.31568 0.31038 0.30518

39 62 63 07 62

E3(4 0.22160 44 0.21836 57 0.21518 18 0.21205 16 0.20897 39

0.16524 0.16304 0.16087 0.15873 0.15663

28 30 53 92 41

0.06345 0.06275 0.06205 0.06136 0.06068

83 42 80 96 89

Ezo(z) 0.03106 12 0.03073 56 0.03041 34 0.03009 46 0.02977 91

0.30009 0.29511 0.29023 0.28545 0.28077

96 79 82 78 39

0.20594 0.20297 0.20004 0.19716 0.19433

75 15 48 64 53

0.15455 0.15251 0.15050 0.14851 0.14655

96 50 00 39 65

0.06001 0.05935 0.05869 0.05804 0.05739

59 05 25 19 86

0.02946 0.02915 0.02885 0.02855 0.02825

70 81 25 01 08

0.60 0.61 0. 62 0. 63 0. 64

0.27618 0.27168 0.26727 0.26295 0.25871

39 55 61 35 54

0.19155 0.18881 0.18611 0.18346 0.18085

06 14 66 56 73

0.14462 0. 14272 0.14085 0.13900 0.13718

71 53 07 28 13

0.05676 0.05613 0.05551 0.05489 0.05428

26 36 18 69 89

0.02795 OiO2766 0.02737 0.02708 0,02680

48 18 19 50 12

0. 65 0. 66 0. 67 0. 68 0. 69

0.25455 0.25048 0.24648 0.24256 0.23872

97 44 74 67 06

0.17829 0.17576 0.17328 0.17083 0.16842

10 58 10 58 94

0.13538 0.13361 0.13187 0.13014 0.12845

55 53 01 95 33

0.05368 0.05309 0.05250 0.05192 0.05134

77 33 55 43 97

0.02652 0.02624 0.02596 0.02569 0.02542

04 25 75 54 62

0. 70 0. 71 0. 72 0. 73 0.74

0.23494 0.23124 0.22761 0.22404 0.22054

71 46 14 57 61

0.16606 0.16373 0.16143 0.15917 0.15695

12 03 60 78 49

0.12678 0.12513 0.12350 0.12190 0.12032

08 19 61 31 24

0.05078 0.05021 0.04966 0.04911 0.04857

15 96 40 47 15

0.02515 0.02489 0.02463 0.02437 0.02412

98 62 53 72 19

0.75 0.76 0.77 0. 78 0.79

0.21711 0.21373 0.21042 0,20717 0,20398

09 88 82 77 60

0.15476 0.15261 0.15049 0.14840 0.14634

67 25 17 37 79

0.11876 0.11722 0.11571 0.11421 0.11274

38 70 15 70 33

0.04803 0.04750 0. 04697 0.04645 0.04594

44 33 81 88 53

0. 02386 92

0.02361 0.02337 0.02312 0.02288

91 17 69 46

0.80 0.81 0. 82 0.83 0. 84

0,20085 0,19777 0.19475 0.19178 0.18886

17 36 04 10 41

0.14432 0.14233 0.14036 0.13843 0.13653

38 07 81 55 24

0.11129 0.10985 0.10844 0.10704 0.10567

00 67 33 93 44

0.04543 0.04493 0.04443 0.04394 0.04346

76 56 91 82 28

0.02264 0.02240 0.02217 0.02194 0.02171

49 78 31 08 11

0.85 0. 86 0. 87 0. 88 0.89

0.18599 0.18318 0.18041 0,17769 0.17502

86 33 73 94 87

0.13465 0.13281 0.13099 0.12920 0.12744

81 22 43 37 01

0.10431 0.10298 0.10166 0.10036 0.09907

85 12 22 12 80

0.04298 0.04250 0.04203 0.04157 0.04111

29 82 89 49 60

0.0214% 0.02125 0.02103 0.02081 0.02059

37 87 61 58 78

0. 90 0.91 0. 92 0.93 0. 94

0.17240 0.16982 0.16728 0.16479 0,16234

41 47 95 77 82

0.12570 0.12399 0.12230 0,12064 0,11901

30 19 63 59 02

0.09781 0.09656 0.09533 0.09411 0.09291

23 39 24 77 94

0.04066 0.04021 0.03976 0.03933 0.03889

22 35 98 11 73

0.02038 0.02016 0.01995 0.01974 0,01954

21 87 75 86 18

0.95 0.96 0. 97 0. 98 0.99

0.15994 0.15757 0.15524 0.15295 0.15070

04 32 59 78 79

0.11739 0.11581 0.11424 0.11270 0.11118

88 13 72 63 80

0.09173 0.09057 0.08942 0. 08828 0.08716

74 13 11 63 69

0.03846 0.03804 0.03762 0.03720 0.03679

83 41 46 98 96

0.01933 0.01913 0.01893 0.01873 0.01854

72 47 44 62 01

1.00

0.14849 55

E2@)

0. 50 0. 51 0. 52 0.53 0.54 0.55 0. 56 0. 57 0. 58 0.59

~

['-4?']

E4(.2)

0.10969 20

0,08606 25

(-617 3

(-614 3

[

3

I: 3

EioG)

0,03639 40

"-z?']

0,01834 60

c(

):-

41

247

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

EXPONENTIAL INTEGRALS E,(%)

E2 (.I )

Table 5.4.

1. 00 1. 0 1 1. 02 1. 03 1.04

0.14849 0.14631 0.14418 0.14207 0.14000

55 99 04 63 68

0.10969 0.10821 0.10676 0.10533 0.10392

20 79 54 42 38

E4 (.I,) 0.08606 25 0.08497 30 0.08389 81 0.08283 76 0.08179 1 3

0.03639 0.03599 0.03559 0.03520 0.03481

40 29 63 41 63

E24.1.) 0.01834 60 0.01815 39 0.01796 39 0. 01777 59 0.01758 98

1.05 1. 06 1. 07 1. 08 1.09

0.13797 0,13596 0.13399 0.13206 0.13015

13 91 96 22 62

0.10253 0.10116 0.09981 0.09848 0.09717

39 43 45 42 31

0.08075 0.07974 0.07873 0.07774 0.07676

90 06 57 42 59

0.03443 0.03405 0.03367 0.03330 0.03294

28 35 85 77 10

0.01740 0.01722 0.01704 0.01686 0.01668

57 35 33 49 84

1.10 1.11 1.12 1.13 1. 1 4

0.12828 0.12643 0.12462 0.12283 0.12107

11 62 10 50 75

0.09588 0.09460 0.09335 0.09211 0.09089

09 74 21 49 53

0.07580 0.07484 0.07390 0.07298 0.07206

07 83 85 12 61

0.03257 0.03221 0.03186 0.03151 0.03116

84 98 52 45 78

0.01651 0.01634 0.01616 0.01600 0.01583

37 09 99 07 33

1.15 1.16 1. 17 1. 18 1.19

0.11934 0.11764 0.11597 0.11432 0.11270

81 62 14 31 08

0.08969 0.08850 0.08734 0.08618 0.08505

32 83 02 88 37

0.07116 0.07027 0.06939 0.06852 0.06766

32 22 30 53 91

0.03082 0.03048 0.03015 0.02981 0.02949

49 58 05 89 10

0.01566 0.01550 0.01534 0.01518 0.01502

76 37 14 09 21

1.20 1.21 1.22 1.23 1.24

0.11110 0.10953 0.10798 0.10646 0.10496

41 25 55 27 37

0.08393 0.08283 0.08174 0.08067 0.07961

47 15 39 17 46

0.06682 0.06599 0.06516 0.06435 0.06355

42 04 75 55 40

0.02916 0.02884 0.02852 0.02821 0.02790

68 61 90 55 54

0.01486 0.01470 0.01455 0.01440 0. 01425

49 94 55 32 26

1.25 1. 26 1. 27 1.28 1.29

0.10348 . - - - . - 81 --

23-

47 16 26 76

0.06276 O;O6I98 0.06121 0.06045 0.05970

22 19 15

0.02759 OlO2729 0.02699 0.02669 0.02640

88

53 51 70 06

0.07857 . .. . -. 0.07754 0.07653 0.07553 0.07454

31

0.10203 0.10060 0.09919 0.09781

57 91 59

0.01410 oioi395 0.01381 0.01366 0.01352

00 55 26

1. 30 1. 31 1. 32 1. 33 1. 34

0.09644 0.09510 0.09377 0.09247 0.09119

55 15 80 47 13

0.07357 0.07261 0.07167 0.07074 0.06982

63 86 42 29 46

0.05896 0.05822 0.05750 0.05679 0.05609

09 99 85 64 36

0.02611 0.02582 0.02554 0.02526 0.02498

59 91 55 51 78

0.01338 0.01324 0.01310 0.01296 0.01283

11 12 27 57 01

1.35 1.36 1. 37 1. 38 1. 39

0.08992 0.08868 0.08745 0.08624 0.08506

75 29 71 99 10

0.06891 0.06802 0.06714 0.06627 0.06542

91 60 53 68 03

0.05539 0.05471 0.05403 0.05337 0.05271

98 51 93 22 37

0.02471 0.02444 0.02417 0.02390 0.02364

35 23 41 88 65

0.01269 0.01256 0.01243 0.01230 0.01217

59 31 17 17 31

1. 40 1.41 1. 42 1.43 1. 44

0.08388 0.08273 0.08160 0,08048 0.07937

99 65 04 13 89

0.06457 0.06374 0.06292 0.06211 0.06131

55 24 07 04 11

0.05206 0,05142 0,05078 0.05016 0.04954

37 22 89 37 66

0.02336 0.02313 0,02287 0.02262 0.02237

72 06 70 61 80

0.01204 0,01191 0.01179 0.01167 0.01154

58 98 52 19 99

1.45 1.46 1. 47 1. 48 1. 49

0.07829 0.07722 0.07616 0.07513 0,07410

30 33 94 13 85

0.06052 0,05974 0.05897 0.05822 0.05747

27 52 82 17 55

0,04893 0.04833 0.04774 0.04715 0.04657

74 61 25 65 80

0.02213 0.02189 0.02165 0.02141 0.02117

27 01 01 28 82

0.01142 0.01130 0.01119 0.01107 0.01095

91 96 14 44 86

1.50 1. 51 1.52 1.53 1. 54

0.07310 0.07210 0.07112 0.07016 0.06921

08 80 98 60 64

0.05673 0.05601 0.05529 0.05459 0.05389

95 35 73 08 39

0.04600 0.04544 0.04488 0.04433 0.04379

70 32 67 72 48

0.02094 0.02071 0.02048 0.02026 0.02004

61 67 97 53 33

0.01084 O;OiO73 0.01061 0.01050 0.01039

40 85 75 77

1.55 1.56 1. 57 1.58 1. 59

0.06826 0.06735 0.06645 0.06555 0.06467

07 87 02 49 26

0.05320 0.05252 0.05185 0,05119 0.05054

64 83 92 92 81

0.04325 0.04273 0.04220 0.04169 0.04118

93 07 87 35 47

0,01982 0.01960 0.01939 0.01917 0,01896

38 67 21 98 98

0.01026 0.01018 0.01007 0.00996 0.00966

90 15 50 97 56

1. 60

0.06380 32

1:

&(.r)

0.04990 57

25

0.04068 25

El 0 ( r )

55

0.01876 22

35

59

oi

0.00976 24

(-7% [(-37)31

248

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS Table 5.4

EXPONENTIAL INTEGRALS E,,(x)

EZ(L)

t

El(1)

E 4

ElO(1)

Ezo(~)

1.60 1.61 1.62 1.63 1.64

0.06380 0.06294 0.06210 0.06126 0.06044

32 64 20 98 97

0.04990 0.04927 0.04864 0.04802 0.04742

57 20 67 99 13

0.04068 0.04018 0.03969 0.03921 0.03873

25 66 70 36 64

0.01876 0.01855 0.01835 0.01815 0.01795

22 68 38 30 43

0.00976 0.00966 0.00955 0.00945 0.00936

24 04 95 96 07

1.65 1.66 1.67 1.68 1.69

0.05964 0.05884 0.05805 0.05728 0.05652

13 46 94 54 26

0.04682 0.04622 0.04564 0.04506 0.04449

09 84 39 72 82

0.03826 0.03779 0.03734 0.03688 0.03643

52 99 06 70 92

0.01775 0.01756 0.01737 0.01718 0.01699

79 37 16 16 37

0.00926 0.00916 0.00907 0.00897 0.00888

29 61 03 56 18

1.70 1.71 1.72 1.73 1.74

0.05577 0,05502 0.05429 0,05357 0.05286

06 94 88 86 86

0.04393 0.04338 0.04283 0.04229 0.04176

67 27 61 67 45

0.03599 0.03556 0.03512 0.03470 0.03428

70 04 93 37 34

0.01680 0.01662 0.01644 0.01626 0.01608

79 42 24 27 50

0.00878 0.00869 0.00860 0.00851 0.00842

90 72 63 64 74

1.75 1.76 1.77 1.78 1.79

0.05216 0.05147 0.05079 0.05012 0.04946

87 88 86 81 70

0.04123 0.04072 0.04020 0.03970 0.03920

93 11 97 51 71

0.03386 0.03345 0.03305 0.03265 0.03225

84 86 39 44 98

0.01590 0.01573 0.01556 0.01539 0.01522

92 54 34 34 53

0.00833 0.00825 0.00816 0.00808 0.00799

94 22 60 07 63

1.80 1.81 1.82 1.83 1.84

0.04881 0.04817 0.04753 0.04691 0.04629

53 27 92 46 87

0.03871 0.03823 0.03775 0.03728 0.03681

57 08 22 00 39

0.03187 0.03148 0.03110 0.03073 0.03035

02 55 56 04 99

0.01505 0.01489 0.01473 0.01457 0.01441

90 45 18 10 19

0.00791 0.00783 0.00774 0.00766 0.00758

28 02 84 74 74

1.85 1.86 1.87 1.88 1.89

0.04569 0.04509 0.04450 0.04392 0.04334

15 28 24 03 63

0.03635 0.03590 0.03545 0.03501 0.03457

40 01 21 00 37

0.02999 0.02963 0.02927 0.02892 0.02857

41 28 61 38 59

0.01425 0.01409 0.01394 0.01379 0.01364

46 90 51 29 24

0.00750 0.00742 0.00735 0.00727 0.00719

81 97 21 53 93

1.90 1.91 1.92 1.93 1.94

0.04278 0.04222 0.04167 0.04112 0.04059

03 22 18 91 38

0.03414 0.03371 0.03329 0.03288 0.03247

30

80 86 46 59

0.02823 0.02789 0.02755 0.02722 0.02690

23 30 79 70 02

0.01349 0.01394 0.01320 0.01305 0.01291

35 63 07 67 43

0.00712 0.00704 0.00697 0.00690 0.00683

42 98 62 33 12

1.95 1.96 1.97 1.98 1.99

0.04006 0.03954 0.03903 0.03852 0.03802

60 55 22 59 67

0.03207 0.03167 0.03128 0.03089 0.03051

27 46 17 39 12

0.02657 0.02625 0.02594 0.02563 0.02532

75 87 40 31 61

0.01277 0.01263 0.01249 0.01236 0.01222

34 41 64 01 54

0.00675 0.00668 0.00661 0.00655 0.00648

99 93 95 04 20

2.00

0.03753 43

Table 5.5 7-1

0. 50 0.45 0; 40 0.35 0.30 0. 25 0.20 0.15 0.10

0.09 0.08 0. 07 0.06 0.05 0.04 0.03 0.02 0.01 0. 00

0.03013 34

0.02502 28

0.01209 21

0.00641 43

EXPONENTIAL INTEGRALS E,(%) FOR LARGE ARGUMENTS

(i+2)erE20) (z+3)eZEl(s) 1.10937 1.11329 1.09750 1.10285 1.08533 1.09185 1.07292 1.08026 1.06034 1.06808 1.04770 1.05536 1.03522 1.04222 1.02325 1.02895 1.01240 1.01617 1. 01045 1. 00861 ..___ 1. 00688 1. 00528 1. 00384 1. 00258 1. 00152 1. 00071 1. 00019 00000

r"

-4111 4'

J

1.01377 1.01147 1.00927 1.00721 1.00531 1.00361 1.00217 1.00103 1.00027 1.00000

[(-y]

( ~ + 4 ) e ' E ~ ( r )(z+lO)eZElo(c) (z+20)eZE20(~) <x> 1.10937 1.07219 1.04270 1.10071 1.06926 1.04179 1.09136 1.06586 1.04067 1.08125 1.06187 1.03932 1.07031 1.05712 1.03762 1.05850 1.05138 1.03543 1.04584 1.04432 1.03249 5 1.03247 1.03550 1.02837 7 1.01889 1.02436 1.02222 10

1.01624 1.01366 1.01116 1.00878 1.00654 1.00451 1.00275 1.00133 1.00036 1.00000

1.02182 1.01917 1.01642 1.01360 1.01074 1.00790 1.00516 1.00271 1.00081

<.r>=nearest integer to .r.

1.02060 1.01883 1.01688 1.01472 1.01234 1.00973 1.00692 1.00401 1.00137

11 13 14 17

20 25 33 50 100 00

249

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

EXPONENTIAL INTEGRAL FOR COMPLEX ARGUMENTS

Talile 5.6

zeZEI(z)

.f

92 Y\X

0 1

-19

R

9

-18

.f

9 -17

w

J

9

d

-15

-16

1.059305 1.059090 1.058456 1.057431 1.056058

0.000000 0.003539 0.007000 0.010310 0.013410

1.063087 1.062827 1.062061 1.060829 1.059190

0.000091 0.004010 0.007918 0.011633 0.015079

1.067394 1.067073 1.066135 1.064636 1.062657

0.000002 0.004584 0.009032 0.013226 0.017075

1.072345 1.071942 1.070774 1.068925 1.066508

0.000006 0.005296 0.010403 0.015172 0.019486

1.078103 1.077584 1.076102 1.073783 1.070793

0.000014 0.006195 0.012118 0.017579 0.022432

9

1.054391 1.052490 1.050413 1.048217 1.045956

0.016252 0.018806 0.021055 0.022996 0.024637

1.057715 1.054981 1.052565 1.050037 1.047458

0.018202 0.020969 0.023364 0.025391 0.027066

1.060297 1.057655 1.054829 1.051905 1.048958

0.020512 0.023505 0.026044 0.028141 0.029824

1.063659 1.060510 1.057187 1.053795 1.050421

0.023272 0.026499 0.029167 0.031306 0.032960

1.067318 1.063538 1.059610 1.055664 1.051797

0.026598 0;030055 0.032823 0.034957 0,036527

10 11 12 1_ _3 14

1.043672 1.041402 1.039177 1.037018 -.~. ~ 1.034942

0.025993 0.027086 0.027940 0.028581 _ ~ ._ . 0.029034

1.046045 1.043212 1.040490 1.037901 --~. 1.035456

0.031130 0.032102 0.032781 0.033211

1.047129 1.043967 1.040965 1.038140

0.034183 0.035034 0.035567 0.035836

1.048081 1.044559 1.041259 1.038192

0.037609 0.038282 0.038616 0.038677

15 16 17 18 19

1.032959 1.031076 1.029296 1.027620 1.026046

0.029326 0.029477 0.029511 0.029445 0.029296

1.033123 1.031110 1.029222 1.027456 1.025809

0.031330 0.031368 0.031288 0.031110 0.030854

1.033162 1.031019 1.029025 1.027174 1.025459

0.033476 0.033377 0.033162 0.032855 0.032474

1.033049 1.030780 1.028685 1.026756 1.024981

0.035765 0.035502 0.035129 0.034672 0.034150

1.032754 1.030365 1.028180 1.026183 1.024360

0.038193 0.037735 0.037179 0.036552 0.035873

20

1.024570

0.029080

1.024275

0.030534

1.023872

0.032037

1.023349

0.033582

1.022695

0.035160

2

3 4 5 6 7 8

Y\X

- 14

1.044880 0.028412 1.042345 0.029461 1.039882 0.030245 0.030796 _1.037515 ~ . ~ ~ CO35259 O;ojii48

- 13

o;o3341I

i;o%Soi

O;o35888

-11

-12

i;oSS%S

O;O38Szo

-10

0 1 2 3 4

1.084892 1.084200 1.082276 1.079313 1.075560

0.000037 0.007359 0.014306 0.020604 0.026075

1.093027 1.092067 1.089498 1.085635 1.080853

0.000092 0.008913 0.017161 0.024471 0.030637

1.102975 1.101566 1.098025 1.092873 1.086686

0.000232 0.011063 0.020981 0.029507 0.036422

1.115431 1.113230 1.108170 1.101137 1.093013

0.000577 0.014169 0.026241 0.036189 0.043843

1.131470 1.127796 1.120286 1.110462 1.099666

0.001426 0.018879 0.033700 0.045218 0.053451

5 6 7 8 9

1.071279 1.066708 1.062046 1.057448 1.053021

0.030642 0.034303 0.037117 0.039174 0.040580

1.075522 1.069960 1.064412 1.059054 1.053997

0.035599 0.039405 0.042169 0.044041 0.045176

1.079985 1.073185 1.066578 1.060352 1.054606

0.041724 0.045552 0.048115 0.049644 0.050359

1.084526 1.076197 1.068350 1.061159 1.054687

0.049336 0.052967 0,055093 0.056057 0.056158

1.088877 1.078701 1,069450 1.061235 1.054046

0.058817 0.061886 0.063225 0.063322 0.062566

10 11 12 13 14

1.048834 1.044928 1.041320 1.038010 1.034989

0.041444 0.041867 0.041938 0.041734 0.041321

1.049303 1.044997 1.041080 1.037537 1.034344

0.045719 0.045801 0.045531 0.044999 0.044277

1.049380 1.044674 1.040464 1.036713 1.033378

0.050452 0.050084 0.049384 0.048452 0.047365

1.048933 1.043853 1.039389 1.035473 1.032040

0.055640 0.054695 0.053465 0.052056 0.050547

1.047807 1.042417 1.037766 1.033752 1.030282

0.061249 0.059584 0.057719 0.055758 0.053773

15 16 17 18 19

1.032241 1.029747 1.027486 1.025437 1.023580

0.040751 0.040066 0.039301 0.038481 0.037629

1.031474 1.028895 1.026579 1.024499 1.022628

0.043422 0.042477 0.041475 0.040444 0.039401

1.030414 1.027781 1.025438 1.023352 1.021489

0.046180 0.044941 0.043679 0.042417 0.041170

1.029026 1.026377 1.024043 1.021981 1.020155

0.048991 0.047428 0.045883 0.044374 0.042912

1.027274 1.024658 1.022375 1,020375 1.018617

0.051808 0.049894 0.048049 0.046282 0.044599

20

1.021896

0.036759

1.020942

0.038361

1,019824

0.039950

1.018533

0.041505

1.017066

0.043001

V\X

I I /

-9

-8

-7

-6

-5

0 1 2 3 4

1.152759 1.146232 1.134679 1.120694 1.106249

0.003489 0.026376 0.044579 0.057595 0.065948

1.181848 1,169677 1.151385 1.131255 1,111968

0.008431 0.038841 0.060814 0.074701 0.082156

1.222408 1.199049 1.169639 1.140733 1.115404

0.020053 0.060219 0.085335 0.098259 0.102861

1.278884 1.233798 1.186778 1.146266 1.114273

0.046723 0.097331 0.122162 0.130005 0.128440

1.353831 1.268723 1.196351 1.142853 1.105376

0.105839 0.160826 0.175646 0.170672 0.158134

5 6 7 8 9

1.092564 1.080246 1.069494 1.060276 1.052450

0.070592 0.072520 0.072580 0.071425 0.069523

1.094818 1.080188 1.067987 1.057920 1.049645

0.085055 0.084987 0.083120 0,080250 0.076885

1.094475 1.077672 1.064339 1.053778 1.045382

0.102411 0.099188 0.094618 0.089537 0.084405

1,089952 1.071684 1.057935 1.047493 1.039464

0.122397 0.114638 0.106568 0.098840 0.091717

1.079407 1.061236 1.048279 1.038838 1.031806

0.143879 0.130280 0.118116 0.107508 0.098337

10 11 12 13 14

1.045832 1.040241 1.035508 1.031490 1.028065

0.067197 0.064664 0.062063 0.059482 0.056975

1.042834 1.037210 1.032539 1.028638 1.025359

0.073340 0.069803 0.066381 0.063128 0.060070

1.038659 1.033231 1.028808 1.025171 1.022152

0.079462 0.074821 0.070524 0.066576 0.062962

1.033205 1.028260 1.024300 1.021090 1.018458

0.085271 0.079488 0.074315 0.069688 0.065542

1.026459 1.022317 1.019052 1.016439 1.014319

0.090413 0.083544 0.077561 0.072320 0,067702

15 16 17 18 19

1.025132 1.022608 1.020426 1.018530 1.016874

0.054573 0.052291 0.050135 0.048106 0.046201

1.022583 1.020219 1.018192 1.016444 1.014929

0.057215 0.054559 0.052094 0.049806 0.047684

1.019626 1.017494 1.015681 1.014129 1.012790

0.059658 0.056638 0.053874 0.051341 0.049015

1.016277 1.014452 1.012912 1.011600 1.010476

0.061817 0.058460 0.055424 0.052670 0.050161

1.012577 1.011130 1.009915 1.008887 1.008009

0.063610 0.059962 0.056694 0.053752 0.051092

20

1.015422

0.044413

1.013607

0.045714

1.011629

0.046875

1.009505

0.047870

1.007254

0.048675

For IzI> 4, linear interpolation will yield about four decimals, eight-point interpolation will yield about six decimals.

-

See Examples 9 10.

I

250

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

EXPONENTIAL INTEGRAL FOR COMPLEX ARGUMENTS zeZE1 (z) f -@ 9 a -..P w 9 -3 -2 -1

Table 5.6

I

54

-4

Y\X

0

4

1.438208 1.287244 1.185758 1.123282 1.085153

0.230161 0.263705 0.247356 0.217835 0.189003

1.483729 1.251069 1.136171 1.080316 1.051401

0.469232 0.410413 0.328439 0.262814 0.215118

1.340965 1.098808 1.032990 1.013205 1.006122

0.850337 0.561916 0.388428 0.289366 0.228399

0.697175 0.813486 0.896419 0.936283 0.957446

1.155727 0.578697 0.378838 0.280906 0.222612

0.577216 0.621450 0.798042 0.875873 0.916770

0.000000 0.343378 0.289091 0.237665 0.198713

5 6 7 8 9

1.061263 1.045719 1.035205 1.027834 1.022501

0.164466 0.144391 0.128073 0.114732 0.103711

1.035185 1.025396 1.019109 1.014861 1.011869

0.180487 0.154746 0.135079 0.119660 0.107294

1.003172 1.001788 1.001077 1.000684 1.000454

0.187857 0.159189 0.137939 0.121599 0.108665

0.969809 0.977582 0.982756 0.986356 0.988955

0.183963 0.156511 0.136042 0.120218 0.107634

0.940714 0.955833 0.965937 0.972994 0.978103

0.169481 0.147129 0.129646 0.115678 0.104303

10 11 12 13 14

1.018534 1.015513 1.013163 1.011303 1.009806

0.094502 0.086718 0.080069 0.074333 0.069340

1.009688 1.008052 1.006795 1.005809 1.005022

0.097181 0.088770 0.081673 0.075609 0.070371

1.000312 1.000221 1.000161 1.000119 1.000090

0.098184 0.089525 0.082255 0.076067 0.070738

0.990887 0.992361 0.993508 0.994418 0.995151

0.097396 0.088911 0.081769 0.075676 0.070419

0.981910 0.984819 0.987088 0.988891 0.990345

0.094885 0.086975 0.080245 0.074457 0.069429

15 16 17 18 19

1.008585 1.007577 1.006735 1.006025 1.005420

0.064959 0.061086 0.057640 0.054555 0.051779

1.004384 1.003859 1.003423 1.003057 1.002747

0.065803 0.061786 0.058227 0.055052 0.052202

1.000070 1.000055 1.000043 1.000035 1.000028

0.066102 0.062032 0.058432 0.055224 0.052349

0.995751 0.996246 0.996661 0.997011 0.997309

0.065838 0.061812 0.058246 0.055066 0.052214

0.991534 0.992518 0.993342 0.994038 0.994631

0.065024 0.061135 0.057677 0.054583 0.051801

20

1.004902

0.049267

1.002481

0.049631

1.000023

0.049757

0.997565

0.049640

0.995140

0 1 2

3

1

Y\”

2

3

0.049284

5

4

4

0.596347 0.673321 0.777514 0.847468 0.891460

0,000000 0.147864 0.1865.70 0.181226 0,165207

0.722657 0.747012 0.796965 0.844361 0.881036

0.000000 0.075661 0.118228 0.132252 0.131686

0.786251 0 797036 0:82305$ 0.853176 0.880584

0.000000 0 045686 01078753 0.096659 0.103403

0.825383 0.831126 0.846097 0.865521 0.885308

0.000000 0.030619 0.055494 0.072180 0.081408

0.852111 0.855544 0.864880 0.877860 0.892143

0.000000 0.021985 0.040999 0.055341 0.064825

5 6 7 8 9

0.919826 0.938827 0.952032 0.961512 0.968512

0.148271 0.132986 0.119807 0.108589 0.099045

0.907873 0.927384 0.941722 0.952435 0.960582

0.125136 0.116656 0.107990 0.099830 0.092408

0.903152 0.921.006 0.934958 0.945868 0.954457

0.103577 0.100357 0.095598 0.090303 0.084986

0.903231 0.918527 0.931209 0.941594 0.950072

0.085187 0.085460 0.083666 0.080755 0.077313

0.906058 0.918708 0.929765 0.939221 0.947219

0.070209 0.072544 0.072792 0.071700 0.069799

10 11 12 13 14

0.973810 0.977904 0.981127 0.983706 0.985799

0.090888 0.083871 0.077790 0.072484 0.067822

0.966885 0.971842 0.975799 0.979000 0.981621

0.085758 0.079836 0.074567 0.069873 0.065679

0.961283 0.966766 0.971216 0.974865 0.977888

0.079898 0.075147 0.070769 0.066762 0.063104

0.957007 0.962708 0.967423 0.971351 0.974646

0.073688 0.070080 0.066599 0.063300 0.060206

0.953955 0.959626 0.964412 0.968464 0.971911

0.067447 0.064878 0.062242 0.059630 0.057096

15 16 17 18 19

0.987519 0.988949 0.990149 0.991167 0.992036

0.063698 0.060029 0.056745 0.053792 0.051122

0.983791 0.985606 0.987138 0.988442 0.989561

0.061921 0.058539 0.055485 0.052717 0.050199

0.980414 0.982544 0.984353 0.985902 0.987237

0.059767 0.056723 0.053941 0.051394 0.049057

0.977430 0.979799 0.981827 0.983574 0.985089

0.057322 0.054644 0.052162 0.049861 0.047728

0.974858 0.977391 0.979579 0.981478 0.983135

0.054671 0.052371 0.050200 0.048160 0.046245

20

0.992784

0.048699

0.990527

0.047900

0.988395

0.046909

0.986410

0.045749

0.984587

0 1 2

3

Y\X 0

7

6

9

8

0.044449

10

4

0.871606 0.873827 0.880023 0.889029 0.899484

0.000000 0.016570 0.031454 0.043517 0.052380

0.886488 0.888009 0.892327 0.898793 0.906591

0.000000 0.012947 0.024866 0.034995 0.042967

0.898237 0.899327 0.902453 0.907236 0.913167

0,000000 0.010401 0.020140 0.028693 0.035755

0.907758 0.908565 0.910901 0.914531 0.919127

0.000000 0.008543 0.016639 0.023921 0.030145

0.915633 0.916249 0.918040 0.920856 0.924479

0.000000 0.007143 0.013975 0.020230 0.025717

5 6 7 8 9

0.910242 0.920534 0.929945 0.938313 0.945629

0.058259 0.061676 0.063220 0.063425 0.062714

0.914952 0.923283 0.931193 0.938469 0.945023

0.048780 0.052667 0.054971 0.056047 0.056211

0.919729 0.926481 0.933096 0.939359 0.945154

0.041242 0.045242 0.047942 0.049570 0.050349

0.924336 0.929836 0.935365 0.940731 0.945812

0.035208 0.039123 0.041986 0.043936 0.045128

0.928664 0.933175 0.937807 0.942398 0.946833

0.030334 0.034063 0.036944 0.039060 0.040514

10 11 12 13 14

0.951965 0.957427 0.962128 0,966178 0.969673

0.061408 0.059735 0,057855 0.055877 0.053874

0.950850 0.955987 0.960495 0.964444 0.967903

0.055725 0.054790 0.053560 0.052146 0.050627

0.950427 0.955176 0.959421 0.963201 0.966559

0.050481 0.050135 0.049444 0.048514 0.047425

0.950535 0.954870 0.958814 0.962379 0.965591

0.045711 0.045818 0.045563 0.045038 0.044319

0.951035 0.954959 0.958586 0.961913 0.964949

0.041413 0.041861 0.041948 0.041755 0.041347

15 16 17 18 19

0.972699 0.975326 0.977617 0.979622 0.981384

0.051894 0.049966 0.048109 0.046332 0.044641

0.97093. 0.9735‘11 0.975940 0.978009 0.979839

0.049062 0.047489 0.045935 0.044419 0.042951

0.969539 0.972185 0.974538 0.976632 0.978500

0.046236 0.044992 0.043724 0.042456 0.041205

0.968477 0.971067 0.973393 0.975481 0.977357

0.043463 0.042516 0.041512 0.040477 0.039431

0.967710 0.970214 0.972484 0.974540 0.976402

0.040780 0.040095 0.039329 0.038508 0.037653

20

0.982938

0.043036

0.981465

0.041538

0.980169

0.039980

0.979047

0.038388

0.978090

0.036781

1 2

3

,

9

d

* If~ > 1 or 0 y>10

then (see [5.15]) 0.711093 0.278518 +0.010389+,, lCj<3x l ~ - e . eZEl(z)= z-tO.415775 z+2.29428 z+6.2900 +

El(iy)=-Ci(y)+i si(y) (y real) *See page

11.

251

EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

Table 5.6

EXPONENTIAL INTEGRAL FOR COMPLEX ARGUMENTS tezEl(z) 4 # .f 9 9H 4

w

12

11

Y\"

13

4

d

14

15

0 1 2 3 4

0.922260 0.922740 0.924143 0.926370 0.929270

0.000000 0.006063 0.011902 0.017321 0.022171

0.927914 0.928295 0.929416 0.931205 0.933560

0.000000

0.005212 0.010258 0.014991 0.019295

0.932796 0.933105 0.934013 0.935473 0.937408

0.000000 0.004528 0.008932 0.013098 0.016934

0.937055 0.937308 0.938055 0.939261 0.940870

0.000000 0.003972 0.007847 0.011540 0.014974

0.940804 0.000000 0.941014 0.003512 0.941636 0.006949 0.942643 0.010242 0.943994 0.013331

5 6 7 8 9

0.932672 0.936400 0.940297 0.944229 0.948093

0.026361 0.029857 0.032670 0.034847 0,036453

0.936356 0.939462 0.942757 0.946132 0.949500

0.023091 0.026339 0.029036 0.031205 0.032887

0.939729 0.942338 0.945140 0.948047 0.950985

0.020373 0.023378 0.025934 0.028052 0.029756

0.942816 0.945024 0.947419 0.949933 0.952502

0.018095 0.020867 0.023273 0.025315 0.027004

0.945640 0.947522 0.949582 0.951765 0.954018

10 11 12 13 14

0.951816 0.955347 0.958659 0.961739 0.964583

0.037566 0.038261 0.038612 0.038684 0.038534

0.952792 0.034134 0.955958 0.035004 0.958968 0.035552 0.961800 0.035833 0.964447 0.035893

0.953895 0.956729 0.959454 0.962049 0.964499

0.031081 0.032068 0.032761 0.033201 0.033428

0.955075 0.957610 0.960073 0.962443 0.964702

0.028365 0.029426 0.030221 0.030781 0.031140

0.956296 0.025949 0.958563 0.027052 0.960787 0.027915 0.962947 0.028564 0.965026 0.029024

15 16 17 18 19

0.967199 0.969597 0.971789 0.973792 0.975621

0.038211 0.037756 0.037200 0.036572 0.035893

0.966907 0.969184 0.971285 0.973220 0.974999

0.035775 0.035515 0.035144 0.034687 0.034166

0.966799 0.968947 0.970946 0.972802 0.974521

0.033479 0.033384 0.033172 0,032865 0.032485

0.966843 0.968860 0.970752 0.972521 0.974172

0.031327 0.031370 0.031293 0.031117 0.030862

0.967011 0.968897 0.970680 0.972359 0.973936

0.029320 0.029476 0.029512 0.029448 0.029301

20

0.977290

0.035179

0.976634 0.033597

0.976112

0.032049

0.975709

0.030542

0.975414

0.029086

16

Y'X

17

0.000000 0.002290 0.004549 0.006745 0.008853

0.954371 0.954467 0.954752 0.955219 0.955856

0.000000 0.002085 0.004144 0.006151 0.008084

0.950600 0.013121 0.951995 0.015296 0.953545 0.017265 0.955212 0.019019 0.956960 0.020555

0.952782 0.011904 0.953995 0.013916 0.955349 0.015753 0.956815 0.017409 0.958363 0.018878

0.954793 0.955853 0.957043 0.958337 0.959712

0.010847 0.012709 0.014425 0.015986 0.017387

0.956650 0.957581 0.958631 0.959779 0.961004

0.009922 0.011649 0.013253 0.014723 0.016056

0.023797 0.024917 0.025823 0.026534 0.027070

0.958758 0.021878 0.960576 0.022998 0.962391 0.023927 0.964181 0.024679 0.965931 0.025271

0.959966 0.961598 0.963238 0.964868 0.966472

0.020163 0.021270 0.022207 0.022984 0.023616

0.961144 0.962612 0.964097 0.965582 0.967052

0.018628 0.019712 0.020645 0.021436 0.022094

0.962288 0.017250 0.963611 0.018305 0.964956 0.019227 0.966310 0.020021 0.967658 0.020694

0.967280 0.969038 0.970712 0.972300 0.973800

0.027453 0.027700 0.027831 0.027862 0.027809

0.967628 0.969264 0.970832 0.972328 0.973751

0.025720 0.026041 0.026249 0.126361 0.026388

0.968039 0.024114 0.969558 0.024493 0.971023 0.024765 0.972430 0.024943 0.973775 0.025038

0.968496 0.969906 0.971273 0.972594 0.973863

0.022629 0.023052 0.023375 0.023607 0.023760

0.968990 0.970297 0.971571 0.972808 0.974004

0.021255 0.021712 0.022075 0.022352 0.022552

0.975215

0.027685

0.975099

0.026343

0.975057

0.975079

0.023842

0.975155

0.022684

0.000000 0.003128 0.006196 0.009150 0.011940

0.947100 0.947250 0.947693 0.948416 0.949395

5 6 7 9

0.948226 0.949842 0.951624 0.953527 0.955509

0.014529 0.016886 0.018994 0.020847 0.022445

10 11 12 13 14

0.957530 0.959559 0,961568 0.963534 0.965443

15 16 17 18 19 20

0.000000 0.002804 0.005560 0.008223 0.010754

0.949769 0.949897 0.950277 0.950898 0.951741

0.025062

EXPONENTIAL INTEGRAL FOR SMALL eZE,(z) 4 Y Y B H f

Y"," 0.0 0.2 0.4 0.6 0.8 1.0 Y\X

0.0 0.2 0.4 0.6 0.8 1.0

20

0.952181 0.952291 0.952619 0.953156 0.953887

0.944130 0.944306 0.944829 0.945678 0.946824

Y\X 0.0 0.2 0.4 0.6 0.8 1.0

19

0.000000 0.002527 0.005016 0.007430 0.009735

0 1 2 3 4

8

18

0.016169 0.018725 0.020980 0.022931 0.024582

-4.0 -0.359552 -0,347179 -0.333373 -0.918556 -0.303109 -0.287369

-0.057540 -0.078283 -0.096648 -0.112633 -0.126301 -0.137768

-2.0 -4.261087 -4.219228 -4.094686 -3.890531 -3.611783 -3.265262

0.000000 0.636779 1.260867 1.859922 2.422284 2.937296

-3.5 -0.420509 -0.400596 -0.379278 -0.357202 -0.334923 -0.312894

-3 ..5 -2.895820 -2.867070 -2.781497 -2.641121 -2.449241 -2.210344

0.5 -0.133374 -0.126168 -0.104687 -0.069328 -0,020743 +0.040177

0.000000 0.157081 0.312331 0.463961 0.610264 0.749655

-0.094868 -0.119927 -0.141221 -0.158890 -0.173169 -0.184355

0.000000 0.462804 0.917127 1.354712 1.767748 2.149077

-3.0 -0.494576 -0.462493 -0.429554 -0.396730 -0.364785 -0.334280

0.000000 0.126210 0.251143 0.373547 0.492229 0.606074

-0.156411 -0.185573 -0.208800 -0.226575 -0.239500 -0.248231

El(t)+ln t -1.0 -1.895118 -1.875155 -1.815717 -1.718135 -1.584591 -1.418052

1.0 0.219384 0.224661 0.240402 0.266336 0.302022 0.346856

cc3MPLEX ARGUMENTS

0.000000 0.342700 0.679691 1.005410 1.314586 1.602372

1.5 0.505485 0.509410 0.521123 0.540441 0.567061 0.600568

0.000000 0.103432 0.205962 0.306707 0.404823 0.499516

9

.I

-2.5 -0.580650 -0.528987 -0.478303 -0.429978 -0.384941 -0.343719

-0.257878 -0.289009 -0.310884 -0.324774 -0.332047 -0.334043

Table 5.7 4

.d

-2.0 -0.670483 -0.587558 -0.510543 -0.441128 -0.380013 -0.327140

0

-0.5 -1.147367 -1.133341 -1.091560 -1.022911 -0.928842 -0.811327

0.000000

0.258840 0.513806 0.761122 0.997200 1.218731

2.0 0,742048 0.000000 0.745014 0.086359 0.753871 0.172075 0.768490 0.256515 0.788664 0.339075 0.814107 0.419185

-0.425168 -0.451225 -0.463193 -0.464163 -0.457088 -0.444528

-0.577216 -0.567232 -0.537482 -0.488555 -0.421423 -0.337404

U.000000

0.199556 0.396461 0.588128 0.772095 0.946083

2.5 0.941206 0.943484 0.950289 0.961532 0.977068 0.996699

0.00~000

0.073355 0.146246 0.218215 0.288822 0.357653

6. Gamma Function and Related Functions PHILIPJ. DAVIB'

Contents

. . . , . 6.1. Gamma Function. , . , . . . 6.2. Beta Function . . . . . . . . 6.3. Psi (Digamma) Function. . . . 6.4. Polygamma Functions. . . . . 6.5. Incomplete Gamma Function. . 6.6. Incomplete Beta Function. , . Numerical Methods . , . , . . . . . Mathematical Properties.

,

. . . . . . . .

. , . . , . . . , . , . , . . . . .

. . . . .. . , . . . . . . . . . . , . . . , . .... . . . . . . . . . .

. . .. . . . . . . .. . . . .

. . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . .

6.7. Use and Extension of the Tables. 6.8. Summation of Rational Series by Means of PoIygamma Functions.. . . , . . . . , . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . Gamma, Digamma and Trigamma Functions ( 1 5 2 5 2 ) . . .

References. . Table 6.1.

,

.

. .

,

r(s),In r(z), $(s),

,

$I@),

. . .

,

x=1(.005)2,

Page

255 255 258 258 260 260 263 263 263 264 2 65 267

10D

. . .

27 1

$"(z), $J3)(z),2=1(.01)2, IOD Table 6.3. Gamma and Digamma Functions for Integer and HalfInteger Values (15 n l 101) , . . . . . . . . . . . . . . . . .

272

Table 6.2. Tetragamma and Pentagamma Functions ( 1 5 2 5 2 )

11s

l/P(n), 9s r(n+i), 8s

$(n), 10D

n!/[(27r)hn+4]e-", 8D

lnn-$(n),

8D

n==l(1) 101

Table 6.4. Logarithms of the Gamma Function (1 Sn5 101).

log10 r(n), 8s log10 r(n+i), 8s

,

... .

274

log10 r(n+$), 8s In r (n)- (n-

4) In n+n,

8D

loglo r(n++), 8s It= l(1) 101

1

National Bureau of Standards. 253

254

GAMMA FUNCTION AND RELATED FUNCTIONS

Page

Table 6.5. Auxiliary Functions for Gamma and Digamma Functions ( 6 6 5 ~ 5 .~ .) . . . . . . . . . . . . . . . . . . . . . 2!/[(2*)

k++e-z],

2-'=.015 (-.001)0,

In

276

r (2)- (2- 3) In z+x, In z-+(x)

8D

. . . .

276

Table 6.7. Gamma Function for Complex Arguments. In r(z+iy), 2=1(.1)2, y=0(.1)10, 12D

. . . . . . .

277

Table 6.8. Digamma Function for Complex Arguments

.

Table 6.6. Factorials for Large Arguments (1001n51000)

,

n!, n= 100(100) 1000, 20s

+ ( ~ + i y ) ,2=1(.1)2, y=O(.1)10,

.

. . . .

,

288

5D

9 + ( 1 + i 9 l 1 10D g + ( l + i y ) - l n y, y - * = . l l (-.Ol)O,

8D

The author acknowledges the assistance of Mary Orr in the preparation and checking of the tables; and the assistance of Patricia Farrant in checking the formulas.

,

6. Gamma Function and Related Functions Mathematical Properties 6.1. Gamma (Factorial) Fuwtion Euler's Integral

r ( z ) = lt'-'e-'dt

6.1.1

= p i mt'-le-k'

Y

I

4

(32>0)

(3z>o,gk>0)

dt

Euler's Formula

6.1.2

r (z)=

iim 11-9-

n!nz

z(z+l)

. . (z+n)

(z#O,-l,-2,.

, .)

*

Euler's Infinite Product

6.1.3

1 -=ze"

6

1 1 1 2 3 4 = ,57721 56649. .

Y=

(lzl<~))

11-1 [(l+i)e-z/n]

r (2)

lim I+-+-+-+.

m+-[

.

. . +--1n m. l

l

m

Y is known as Euler's constant and is given to 25 decimal places in chapter 1. r(z) is single valued and analytic over the entire complex plane, save 1, 2, . . . ) where it for the points z=-n(n=O, possesses simple poles with residue (- l)"/n!. Its reciprocal i/r(z) is an entire function possessing 1, 2, . . .). simple zeros a t the points z= -n(lt=O,

Hankel'a Contour Integral

FIQURE 6.1. Gamma function. ,~ = r ( z ) , - -

6.1.10

+

The path of integration C starts at w on the real axis, circles the origin in the counterclockwise direction and returns to the starting point.

r(n+))=

*

- - , V=l/r(z)

. . (4n-3) r ( t )

1.5.9.13.

4"

..

r ($1

. . (273-1)

r (3)

1*4.7*10. (3n-2) 3"

6.1.11 r(n+$>-

Factorial and II Notations

n(z)=z!=r(z+1)

6.1.5

r(n+)) =

6.1.13

r (n+ p ) =2-54-11.3". . (3n-1) r ($1

6.1.14

r(n+t) =

Integer Values

6.1.6

r(n+1)=1.2.3

6.1.7 lim -=o=1 L-)o

r(-z)

. . . (n-l)n=n!

1 (-n-l)!

(n=O, 1, 2,

1.3.5.7.

6.1.12

2"

. . .)

Fractional Valuer

6.1.8 ~ ~ f ) - 2 S0- c - " d t = r t = 1 . 7 7 2 4 538509, , .=(-))!

3.7.11.15.

. . (4n-1) r ( t )

4"

256

GAMMA FUNCTION AND RELATED FUNCTIONS

Recurrence Formulas

6.1.15

r(z+i)=Zr(Z)=Z!=Z(Z-i)!

6.1.16

. . . (i+z)r(i+z)

r(n+~)=(n-i++)(n-2++) = (n- 1+z)!

. . . (l+z)z!

=(n-l+z)(n-2++)

Reflection Formula

r (2)r (1- 2 ) = -w(-

6.1.17

Z)

r (2) =

csc T Z

Power Series

6.1.33

+

In r (1 z) =-ln(l+ z ) + z (1-7) Duplication Formula

r ( 2 2 ) = ( 2 ~ ) - +2 2 4 r(z)r(z+i)

6.1.18

Triplication Formula

6.1.19 r(3z)=(2+13ae-+

f(n)is the Riemann Zeta Function (see chapter 23). Series Expansion 2 for 1 /r( 2 )

r(z)r(z++)r(z++)

Gauss' Multiplication Formula

6.1.34

k 1 Binomial Coefficient

(?E)=- =

6.1.21

Z!

w!(z-w)!

r(z+l)

r (w+i) r (~--w+i)

Pochhammer's Symbol

6.1.22 (z)o= 1,

Gamma Function in the Complex Plane

6.1.23

r(Z)=r(2);In r(Z)=ln rfz)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Ck

1.00000 0.57721 -0.65587 -0.04200 0. 16653 -0.04219 -0.00962 0.00721 -0.00116 -0.00021 0.00012 -0.00002 -0.00000 0.00000 -0.00000 0.00000 0.00000 -0.00000 0.00000 0.00000 -0.00000 0.00000 -0.00000 -0.00000 0.00000 0.00000

00000 56649 80715 26350 86113 77345 19715 89482 51675 52416 80502 01348 12504 11330 02056 00061 00050 00011 00001 00000 00000 00000 00000 00000 00000 00000

000000 015329 202538 340952 822915 555443 278770 466630 918591 741149 823882 547807 934821 272320 ' 338417 160950 020075 812746 043427 077823 036968 005100 000206 000054 000014 000001

* The coefficients c k are from H. T. Davis, Tables of higher mathematical functions, 2 vols., Principia Press, Bloomington, Ind., 1933, 1935 (with permission) ; with corrections due t o H. E. Sslzer.

257

GAMMA FUNCTION AND RELATED FUNCTIONS

Polynomial Approximations 8

6.1.35

Error Term for Asymptotic Expansion

6.1.42

O_<Xll

If

R,(z)= In U]=-. U2= ~3=-.

57486 46 .95123 63 69985 88

6.1.36

..

b2=

b3=--. b4=

In (2r)

+b8z8+e(z)

where

le(z)I ~ 3 ~ 1 0 - 7

1652 5891 6937 6857

In z+z-+

then

I ' ( ~ + l ) = ~ ! = l + h ~ z + b 2 ~ 2 +.

57719 . 98820 89705 .91820

(z)-(z-+)

-42455 49 10106 78

~ 4 =

U5=-.

O l X S l

b1=-.

r

b5=-.

bs= b7=-. b8=

75670 .48219 19352 .03586

K ( z )=upper boundlz2/(u2+z2)I

4078 9394 7818 8343

U20

For z real and positive, R, is less in absolute value than the first term neglected and has the same sign.

Stirling's Formula

6.1.43

6.1.37

Wln r(iy)=&?ln

r(--iv)

-& In (2r) -+y-+ln 6.1.38

1

6.1.44

I

6.1.47

y,

(y-+

=I

1

. .

Asymptotic Formulas

6.1.39

For B, see chapter 23 6.1.41

In

r(+(z-+) 1

1 122 360.9

In z-z++ln (%)+--- 1 1

+126oz'-m+*

* *

(2-00

in larg z l < r )

From C. Hastings, Jr., Approximations for digital computers, Princeton Univ. Press, Princeton, N.J., 1955 (with permission).

+;pib)

(a(a+b-1)2-a+b-1)~+

as z+along any curve joining z=O and z = = , providingz# --a, --a-l, . . , ; z # -b, -b-1,

....

258

GAMMA FUNCTION AND RELATED FUNCTIONS

Continued Fraction

6.1.48 In

In z-+In

r(z)+Z-(Z-+)

(2a)

1 az=-' 53 a3=-'195 q=-112 al=Fo' ~ 210 37 1

a(=-

22999 29944523 109535241009 22737' "= 19733142' "= 48264275462 Wallis' Formula 4

6.1.49

. .

1 . 3 . 5 . . (2n-1) 2 .4 .6 . . (2n)

lr

FIGURE 6.2. Psi function. y=$(z)=dln I'(z)/dz Some Definite Integrals

6.1.50

Integer Values

=(z-+)

In z-z+*

I

In 2 r

6.2. Beta Function

Fractional Values

#(+)=-~-2

6.2.1

In 2=--1.96351

00260 21423

6.3.4

Recurrence Formulas

6.3. Psi (Digamma) Function 6.3.1

+(z)=d[ln

'

r (z)]/dz=r'(z)/r (2)

* Some authors employ the special double factorial notation as follows: ( 2 4 ! ! =2.4.6 . . . (2n)=2nn i (2n-1) ! ! = 1 . 3 . 5 . . . ( 2 n - i ) = ~ t 2" r(n+t) d Some authors write $(z) =- In r ( z + 1)and similarly for dz the polygamma functions.

6.3.5

+(Z+l)=#(Z)+;

1

6.3.6 1 #(n+z)= (n-l)+z

+(n-2)+2 1

+...

...

259

G A M M A FUNCTION AND RELATED FUNCTIONS

I

Reflection Formula

+r cot

+( 1- 2 ) = + ( z )

6.3.7

6.3.19

r2

Duplication Formula

=In Y+2+120y'+2Tj2ya+ 1 1 1

$(22)=++(2)+++(2++) +In 2

6.3.8

* *

12Y

Psi Function in the Complex Plane

=m

$4;)

6.3.9

6.3.10 $?t+(iy)= 9 + ( - i y ) = 9 + ( 1+iy) = 9 + ( 1-iy)

6.3.12

+ 1.462

Y+(++iy) =+r tanh r y

-0.504 -1.573 -2.611 -3.635 -4.653 -5.667 -6.678

Series Expansions

6.3.14 $(1+ z ) =

-rfg(-1)"{(n)zn-'

(121<1)

n=2

~0=1.46163 21449 68362

6.3.15

+

$(1

2 ) =+2 -1-

&r

cot

Irz

r(zo)= ,88560 31944 10889

- (1-22) -'+1--Y

-n5 k(2n +1)-11zsn =l

$0.886 -3.545 $2.302 -0.888 $0.245 -0.053 $0.009 -0.001

6.3.20

(I 2 I <2)

x*= -n+ On n)-l+o[(ln n Y 2 ] Definite Integrals

6.3.21

6.3.16 w+2)=--Y+n2

"

. . .)

2 n(n+z) (~#-1,-2,-3,

6.3.17

+c

(-l)"+'[{(2n+

1)-lIy2"

zl<

n=l

);

(lY1<2)

6.3.22

m

n-'(n2+yZ)-l

=--Y+y2 n=l

(-

W
Asymptotic Formula8

6.3.18

~

6 From W. Sibagaki, Theory and applications of the gamma function, Iwanami Syoten, Tokyo, Japan, 1952 (with permission).

260

GAMMA FUNCTION AND RELATED FUNCTION8

1 6.4.9

6.4. Polygamma Functions' 6.4.1

(n=1,2,3,

. . .]

(n+ l)! (n+ 2) ! -__ l! l(n+2)z+-2r{(n+3)z2-

...)

(Wz>O)

*

Series Expansions

-

6.4.10

$(")(z),(n=O,l, . . .), is a single valued analytic function ovcr the crit,irc complex plane save at. the points z=-m(m=0,1,2, . , . ) where it possesses poles of order (n+ 1).

Asymptotic Formulas

6.4.11 Integer Values

6.4.2 J.'"'(l)=(-l)"+'n!l(n+l)

6.4.3 $(m)(n+l)= (-l)mm!

(n=1',2,3,

r++

6.4.12 1)+

L

+'(z)--+ 1 2

1

1 * * '

+2"+1+

-+---+--1 1 1 1 2z2 6 9 30z5 422'

1 302'

+...

+;I"]

Fractional Values

6.4.4

$(")(+I = (-

1)n+1n!(2"+1-l){(n+l) (n=1,2,

6.4.5

.)

,

n

$'(n+&) =+9-4

..

0

)

6.4.14

(2k- 1)- 2

k-1

(z+w

Recurrence Formulo

6.4.6

$(n)

(z+ 1)= p( aj

+ (-

l)"n!z-"-l

6.5. Incom lete Gamma Function gee also 26.4)

Reflection Formula

6.5.1

6.4.7 #")( 1-2)

+(-

l)"+'J.'"'(Z)

in I arg z I
a"

=(- 1)%-

d2"

cot r z 6.5.2

6.4.8

Multiplication Formula

6.5.3 b=1, 8=0,

(w~>o)

y ( a , r ) = ~ ( az>r(a) , =S;-'ta-ldt 0

r (a, r ) = r (a)-?(a,

n=O n>O

1 m

r )=

z

e-gta-ldt

6.5.4 r -8

nre the tetra-, pentn-, and hesngamma functions rrspectircly. Some authors write $(L) =d[In r ( s + l ) l / d t , and simi1:irly for the polygamma functions. 7

$f is known aa the trignmma function.

*See posp 11.

$f', $(a),

y * ( a , r ) =r-aP(a, r ) =-y(a,

r (U)

$(a)

y*

r)

is a single valued analytic function of a and

z possessing no finite singularities.

GAMMA FUNCTION AND RELATED FUNCTIONS

261

a

t S

4

3

2

I

a

*

-I

-3

-4

-S

From F. G. Tricomi, Sulla funrione gamma incompleta, Annali di Matematica, IV, 33, 1950 (with permission).

G A M M A FUNCTION AND RELATED FUNCTIONS

262 6.5.5

6.5.16

y

(3, z2)=2

erf z

e-12 dt=&

Probability Integral of the x2-Distribution

6.5.17

r(+,z2)=21me-tadt=fierfc z

6.5.6 (Pearaon's Form of the Incomplete Gamma Function)

Recurrence Formulas

6.5.7

C ( z , ~ ) = ~ ~cos t "t -dt~

(%
6.5.8

S ( z , ~ ) = l ~ tsin " - t~dt

(9?'a
+ 1,z) =ay (a,z)-rue-'

6.5.22

y (U

6.5.23

y*(a-l,z)=zy*(a,z)

+-re-'(a>

Derivatives and Differential Equations

6.5.24

6.5.11 Incomplete Gamma Function as a Confluent Hypergeometric Function (see chapter 13)

6.5.12

6.5.26 an [ ~ - ~ r ( a , z )(-l)nz-"-nr(a+n,z) ]= azn

y(a,z)=u-lzae-zM(l, l+a,x) =a-'z"

M(a, l+a,-2)

Special Values

6.5.13

(n=O, 1,2, . . .)

6.5.27 an

- [&fly* (~,z)]=e~f-~y*(a-n, z) axn

6.5.28 Series Developments

For relation to the Poisson distribution, see 26.4. 6.5.14

.

(n=O, 1 , 2 , . .)

6.5.29 y*(-n,

z)=z"

263

GAMMA FUNCTION AND RELATED FUNCTIONS

Definite Integrals

6.5.36

*

Continued Fraction

6.5.31

6.6. Incomplete Beta Function

I

Asymptotic Expansions

6.5.32

Iz(a,b)=Bz(a,b)/B(a,b)

For statistical applications, see 26.5.

Suppose R,(u,z)=u,+,(u,z)+ . . . is the rcmniiitlcr nftc.r R tcrms in this scrics. Then if u,z

111i d

sign I?, (a,2 ) =sign

I,(U,b)= 1-11-,(b,u)

6.6.3

Relation to Binomial Expansion

For binomial distribution, see 26.1.

U ,1+1 ( U , z ) .

-x---

Rccurrence Formulas

= (-l)nZa+n

6.5.33

6.5.34

Y(U,Z)

n=O

(u+n)n!

{ (4-

limes)= ii+m

6.5.35

l?(z+l,z.)-e-*zz

(a-++ =)

6.6.5

I z ( ~ , b ) = d z (1,b) ~ -f( l - ~ ) I z ( U , b - 1)

6.6.7

( ~ + b ) I , ( u , b=) d , ( ~1,b) ++ b I z ( ~ , b +1)

0 for a>1 3 for a=l 1 for O I a < l

-+. . 24

x-z++,+-2 J G 1

2

3

24

(z+m

ill I nrg zl

Relation to Hypergeom etric Function

6.608

B z ( ~ , b ) = ~ - ' r " F ( ~ , l - a- b+;l ;

2)

6.7. Use and Extension of the Tables

Example 1. Compl1tc r(6.38) to 8s. Usiw tlic l ' i ~ ( ' t l ~ l ' ~ ' l i Crclntiori c 6.1.16 a l l d Table 6.1 wc

Thc error of linear intcrpolation in the table of tlic fuiictioiifz is smaller than 10-7 in this region. HellCC, fz(56.38) =.92041 67 atid In r ( 5 6 . 3 8 ) ~

h Y C )

169.85497 42.

r(6.38)=[(5.38)(4.38)(3.38)(2.38)(1.38)]F(1.38) = 232.43671.

~ Example 2. C o m ~ l l t cIn r(56.38), l l s i Table 6.4 atid liiicnr intrrpolntion i n .f2. \Yc have In r(56.38) = (56.38-+) In (56.38)- (56.38)

Direct interpolation i n Table 6.4 of log,, r(n) climitiatcs the ncccssity of employing logarithms. Ho~wver,tlic error of linear interpolation is .002 so that loglo r(n) is obtained with R rclativc error of 10-6. *See page 11.

I

264

GAMMA FUNCTION AND RELATED FUNCTIONS

Example 3. Compute q(6.38) to 8s. Using the recurrence relation 6.3.6 and Table 6.1. 1 1 1 1 1 q(6.38) =-+-+-+-+-+~(1.38) 5.38 4.38 3.38 2.38 1.38 =1.77275 59.

Example 4. Compute q(56.38). Using Table 6.3 we have q(56.38) =In 56.38-f3(56.38). Ths error of linear interpolation in the table of the function f3 is smaller than BXlO-' in this region. Hence,f3(56.38)= .00889 53 andq(56.38) = 4.023219. Example 5. Compute In r(1-i). From the reflection principle 6.1.23 and Table 6.7, In r (1-i) =In r (1 i) = - .6509 .30 16i.

+

6.8. Summation of Rational Series by Means of Polygamma Functions

An infinite series whose general term is a rational function of the index may always be reduced to a finite series of psi and polygamma functions. The method will be illustrated by writing the explicit formula when the denominator contains a triple root. Let the general term of an infinite series have the form

where

+

Example 6. Compute In I'(+++i). Taking the logarithm of the recurrence relatiw 6.1.15 we have,

In

r(++&i) =In r (g++i) -1n (&++i) --.23419+ -

.03467i -(+ In ++i arctan 1) = .11239- .750733

+ +

where p (n) is a polynomial of degree m 2r 3s -2 a t most and where the constants a$,b1, and y c are distinct. Expand un in partial fractions as follows

The logarithms of complex numbers are found from 4.1.2. Example 7. Compute In r(3+7i) using the duplication formula 6.1.18. Taking the logarithm of 6.1.18, we have

-+In 2 ~ = - .91894 (#+7i) In 2= 1.73287+ 4.852033 In r(g+$i)=-3.31598+ 2.32553i In r(2+$3)=-2.66047+ 2.938693 In

r (3+7i) =-5.16252+10.116253

m

Then, we may express

c

n-1

U,,

in terms of the

constants appearing in this partial fraction expansion as follows

Example 8. Compute In r(3+7i) to 5D using the asymptotic formula 6.1.41. We have

In (3+7i)=2.03022 15-tl.16590 4%. Then, (2.5+ 7i) In (3+ 7i) = -3. 0857779-t 17.12631 19i - (3 7i) = -3.0000000- 7. oooooooi 6 In (2n)= . 9189385 . 0043103- . 0100575i [12(3+7i)]-'= -[360(3+7i)3]-'= .0000059. 0000022i

+

In r(3+7i)=-5.

16252

+ l O . 11625i

Higher order repetitions in the denominator are handled similarly. If the denominator contains

265

GAMMA FUNCTION AND RELATED F m C T I O N S

Therefore

only simple or double roota, omit the corresponding lines.

S=

16+(1)

-16+(1$) ++'( 1)++'(lQ)

=.018499.

Example 9. Find Example 11.

'=z OJ

Since

1 (n2+ 1) (n2+4)

(see also 6.3.13).

we have aI=l, a2=+,aa=t, al=+,az=-l,

aa=Q.

i --i --i Hence, al=-, a*=- 7 a3==7 6

Thus,

i

a 4 = ~ 7

I

~=-$$(2)+$(13)-8+(1t)=.047198.

and therefore

Example 10.

s=-

--i 6

+i) -+(l

[+( 1

i -i)]+E [+( 1+2i) -+( 1-2i)l.

By 6.3.9, this reduces to 1 3

s=- Y + ( l + i ) - s we have, &=O,

82=9, bll=-l6,

I From Table 6.8,

blz=l6, bZ1=1, bz2=l.

1

9+(1+2i).

s=.13876.

References Texts

[6.1] E. Artin, Einfiihrung in die Theorie der Gammafunktion (Leipzig, Germany, 1931). [6.2] P. E. Btihmer, Differenzengleichungen und bestimmte Integrale, chs. 3, 4, 5 (K. F. Koehler, Leipzig, Germany, 1939). [6.3] G. Doetsch, Handbuch der Laplace-Transformation, vol. 11, pp. 52-61 (Birkhauser, Basel, Switzerland, 1955). (6.41 A. Erd6lyi e t al., Higher transcendental functions, vol. 1, oh. 1, oh. 2, sec. 5; vol. 2, oh. 9 (McGrawHill Book Co., Inc., New York, N.Y., 1953). [6.5] C. Hastings, Jr., Approximations for digital computers (Princeton Univ. Press, Princeton, N.J., 1955). [6.6]'F. Losch and F. Schoblik, Die Fakultat und verwandte Funktionen (B. G. Teubner, Leipzig, Germany, 1951). [6.7] W. Sibagaki, Theory and applications of the gamma function (Iwanami Syoten, Tokyo, Japan, 1952).

[6.8]E. T. Whittaker and G. N. Watson, A course of modern analysis, ch. 12, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952).

I

Tables [6.9] A. Abramov, Tables of In r ( z ) for complex argument. Translated from the Russian by D. G. Fry (Pergamon Press, New York, N.Y., 1960). In r($+iy),~ = 0 ( . 0 1 ) 1 0 ,y=0(.01)4, 6D. [6.10] Ballistic Research Laboratory, A table of the factorial numbers and their reciprocals from l! through lOOO! to 20 significant digits. Technical Note NO. 381, Aberdeen Proving Ground, Md., 1951. [6.1I ] British Association for the Advancement of Science, Mathematical tables, vol. 1, 3d ed., pp. 40-59 (Cambridge Univ. Press, Cambridge, England, 1951). The gamma and polygamma functions.

Also 1+lslog10 ( t ) ! d t , z=O(.O1)1, 10D. [6.12] H. T. Davis, Tables of the higher mathematical functions, 2 vols. (Principia Press, Bloomington, Ind., 1933, 1935). Extensive, many place tables of the gamma and polygamma functions up t o $(4) (2) and of their logarithms. [6.13] F. J. Duarte, Nouvelles tables de loglo nl h 33 d6cimales depuis n = l jusqu'h n=3000 (Kundig, Geneva, Switzerland; Index Generalis, Paris, France, 1927).

266

QAMMA FUNCTION AND RELATED FUNCTIONS

[6.14] National Bureau of Standards, Tables of nl and r(n++) for the first thousand values of n, Applied Math. Series 16 (U.S. Government Printing Office, Waahington,D.C., 1951). nl,lSS;r(n++), 8s.

[6.15] National Bureau of Standards, Table of Coulomb wave functions, vol. I, pp. 114-135, Applied Math. Series 17 (U.S. Government Printing Office, Washington, D.C., 1952).

~ [ r J ( i + i 9 ) / r ( i + i 9 ~ , 9 = o ( . o (.0i)6 o 5 ) 2 (.02)10(.1) 20 (.2)60(.5)110,10D;arg r(l+i7),9=O(.Ol)l(.O2) 3 (.05) 10(.2)20 (.4)30(.5)85, 8D. [0.16] National Bureau of Standards, Table of the gamma function for complex arguments, Applied Math. Series 34 (US. Government Printing Office, Washington, D.C., 1954). In r(z+iy), z=0(.1)10, y=O(.l)lO,

12D.

[6.18] E. S. Pearson, Table of the logarithms of the complete r-function, arguments 2 to 1200, Tracts for Computers No. VIII (Cambridge Univ. Press, Cambridge, England, 1922). Logl0 r(p),p=2(.1) 5(.2)70(1) 1200, 10D. [6.19] J. Peters, Ten-place pendix, pp. 58-68 New York, N.Y., ( n ! ) ~ 'n=1(1)43, ,

logarithm tables, vol. I, Ap(Frederick Ungar Publ. Co., 1957). n!, n=1(1)60, exact; 54D; Loglo(nl), n=1(1)1200,

18D. K6.201 J. P. Stanley and M. V. Wilkes, Table of the reciprocal of the gamma function for complex argument (Univ. of Toronto Press, Toronto, Canada, 1950). Z= -.5(.01).5, y=O(.O1)1, 6D.

[6.21] M. Zyczkowski, Tablice funkcyj eulera i pokrewnych Contains an extensive bibliography. (Panstwowe Wydawnictwo Naukowe, Warsaw, (6.171 National Physical Laboratory, Tables of Weber Poland, 1954). Extensive tables of integrals parabolic cylinder functions, pp. 226-233 (Her involving gamma and beta functions. Majesty's Stationery Office, London, England, 1955). Real and imaginary parts of In r(*k+*ia), k=0(1)3, For references to tabular material on the incomplete u=O(. 1)5(.2)20, 8D ; (Ir(4 tia)/r(t+tia) [)-]Ia gamma and incomplete beta functions, see the references in chapter 26. 8D. U= 0(.02) 1(. 1) q . 2 )20,

+

267

GAMMA FUNCTION AND RELATED FUNCTIONS

Table 6.1

GAMMA, DIGAMMA A N D TRIGAMMA FUNCTIONS

r (2.1

L

In r ( ~ , )

+Cl,)

!b'(X)

1.000 1.005 1.010 1.015 1.020

1.00000 0.99713 0.99432 0.99156 0.98884

00000 85354 58512 12888 42033

0,00000 -0.00286 -0.00569 -0.00847 -0.01121

00000 55666 03079 45187 84893

-0.57721 -0.56902 -0.56088 -0.55280 -0.54478

56649 09113 54579 85156 93105

1.64493 1.63299 1.62121 1.60958 1.59811

40668 41567 35283 91824 81919

0,000 0.005 0.010 0.015 0.020

1.025 1.030 1.035 1.040 1.045

0.98617 0.98354 0.98097 0.97843 0.97594

39633 99506 15606 82009 92919

-0.01392 -0.01658 -0.01921 -0.02179 -0.02434

25067 68539 18101 76511 46490

-0.53682 -0.52892 -0.52107 -0.51327 -0.50553

70828 10873 05921 48789 32428

1.58679 1.57562 1.56459 1.55371 1.54296

76993 49154 71163 16426 58968

0.025 0.030 0.035 0.040 0.045

1.050 1.055 1.060 1.065 1.070

0.97350 0.97110 0.96874 0.96642 0.96415

42656 25663 36495 69823 20425

-0.02685 -0.02932 -0.03175 -0.03414 -0.03650

30725 31868 52537 95318 62763

-0.49784 -0.49020 -0.48262 -0.47509 -0.46761

49913 94448 59358 38088 24199

1.53235 1.52188 1.51154 1.50133 1.49124

73421 35001 19500 03259 63164

0.050 0.055 0.060 0.065 0.070

1.075 1.080 1.085 1.090 1.095

0.96191 0.95972 oi95757 0.95545 0.95338

83189 53107 25273 94882 57227

-0.03882 -0.04110 -07045% -0.04556 -0.04773

57395 81702

11367

76622 21556

29148 57114

64i% 17635 47988

1.48128 1.47145 iA6173 1.45214 1.44266

0.075

38141

-0.46018 -0.45279 -oh4546 -0.43818 -0.43094

763i7

19988 31755

0.085 0.090 0.095

1,100 1.105 1.110 1.115 1.120

0.95135 0.94935 0.94739 0.94547 0.94359

07699 41778 55040 43149 01856

-0.04987 -0.05197 -0.05403 -0.05606 -0.05806

24413 33384 86341 85568 33325

-0.42375 -0.41661 -0.40951 -0.40246 -0.39545

49404 16193 42761 23611 53339

1.43329 1.42404 1.41490 1.40587 1.39695

91508 79514 76482 63535 22213

0.100 0.105 0.110 0.115 0.120

1.125 1.130 1.135 1.140 1.145

0.94174 0.93993 0.93815 0.93641 0.93471

26997 14497 60356 60657 11562

-0.06002 -0.06194 -0.06383 -0i06569 -0.06751

31841 83322 89946 53867 77212

-0.38849 -0.38157 -0.37469 -0.36786 -0.36107

26633 38268 83110 - _- _ . 56106 52291

1,38813 1.37941 1.37080 iI36229 1.35387

34449 82573 49288 17670 71152

0.125 0.130 0.135 .___. 0.140 0.145

1.150 1.155 1.160 1.165 1.170

0.93304 0.93140 0.92980 0.92823 0.92669

09311 50217 30666 47120 96106

-0.06930 -0.07106 -0.07278 -0.07447 -0.07612

62087 10569 24716 06558 58106

-0.35432 -0.34761 -0.34095 -0.33432 -0.32774

66780 94768 31528 72413 12847

1.34555 1.33733 1.52920 1.32117 1.31322

93520 68900 81752 16859 59322

0.150 0.155 0.160 0.165 0.170

1.175 1.180 1.185 1.190 1.195

0.92519 0.92372 0.92229 0.92088 0.91951

74225 78143 04591 50371 12341

-0.07774 -0.07933 -0.08089 -0.08242 -0.08391

81345 78240 50733 00745 30174

-0.32119 -0.31468 -0.30821 -0.30178 -0.29539

48332 74438 86809 81156 53259

1.30536 1.29760 1.28991 1.28232 1.27480

94548 08248 86421 15358 81622

0.175 0.180 0.185 0.190 0.195

1.200 1.205 1.210 1.215 1.220

0.91816 0.91685 0.91557 0.91432 0.91310

87424 72606 64930 61500 59475

-0.08537 -0.08680 -0.08820 -0.08956 -0.09090

40900 34780 13651 79331 33619

-0.28903 -0.28272 -0.27643 -0.27019 -0.26398

98966 14187 94897 37135 37000

1.26737 1.26002 1.25275 1.24556 1.23845

72054 73755 74090 60671 21360

0.200 0.205 0.210 0.215 0.220

1.225 1.230 1.235 1.240 1.245

0.91191 0.91075 0.90962 0.90852 0.90744

56071 48564 34274 10583 74922

-0.09220 -0.09348 -0.09472 -0.09593 -0.09711

78291 15108 45811 72122 95744

-0.25780 -0.25166 -0.24556 -0.23949 -0.23345

90652 94307 44243 36791 68341

1.23141 1.22445 1.21756 1.21074 1.20400

44258 17702 30254 70707 28063

0.225 0.230 0.235 0.240 0.245

1.250

0.90640 24771

-0.09827

18364

-0.22745

35334

1.19732 91545

0.250

U!

For 2>2 see Examples 1-4.

In y !

93380

o. m o_ - . _

Y

log,, e=0.43429 44819

Compiled from H. T. Davis, Tables of the higher mathematical functions, 2 vols. (Principia Press, Bloomington, Ind., 1933, 1935) (with permission). Known error has been corrected.

268

GAMMA FUNCTION AND

Table 6.1

RELATED FUNCTIONS

GAMMA, DIGAMMA AND TRIGAMMA FUNCTIONS

5

1.250 1.255 1.260 1.265 1.270

0.90640 0.90538 0.90439 0.90343 0.90250

1.275 1.280 1.285 1.290 1.295

r (2)

*(435334

In r(x)

24771 57663 71178 62946 30645

-0.09827 -0.09939 -0.10048 -0.10154 -0.10258

18364 41651 67254 96809 31932

-0.22745 -0.22148 -0.21554 -0.20964 -0.20376

0.90159 0.90071 0.89986 0.89904 0.89824

71994 84765 66769 15863 29947

-0.10358 -0.10456 -0.10550 -0.10642 -0.10731

74224 25269 86634 59872 46519

1.300 1.305 1.310 1.315 1.320

0.89747 0.89672 0.89600 0.89530 0.89464

06963 44895 41767 95644 04630

-0.10817 -0.10900 -0.10981 -0.11058 -0.11133

1.325 1.330 1.335 1.340 1.345

0.89399 0.89337 0.89278 0.89221 0.89167

66866 80535 43850 55072 12485

1.350 1.355 1.360 1.365 1.370

0.89115 0.89065 0.89018 0.88973 0.88931

1.375 1.380 1.385 1.390 1.395

*' (4

34266 61686 14193 88437

1.19732 1.19072 1.18418 1.17772 1.17131

91545 50579 94799 14030 98301

0.250 0.255 0.260 0.265 0.270

-0.19792 -0.19211 -0.18634 -0.18059 -0.17487

81118 88983 08828 37494 71870

1.16498 1.15871 1.15250 1.14635 1.14027

37821 22990 44385 92764 59053

0.275 0.280 0.285 0.290 0.295

48095 66107 02045 57384 33587

-0.16919 -0.16353 -0.15790 -0.15231 -0.14674

08889 45526 78803 05782 23568

1.13425 1.12829 1.12238 1.11654 1.11075

34350 09915 77175 27706 53246

0.300 0.305 0.310 0.315 0.320

-0.11205 -0.11274 -0.11341 -0.11404 -0.11465

32100 54356 01772 75756 77697

-0.14120 -0.13569 -0.13020 -0.12475 -0.11932

29305 20180 93416 46279 76069

1.10502 1.09934 1.09372 1.08816 1.08265

45678 97037 99497 45379 27136

0.325 0.330 0.335 0.340 0.345

14420 59235 45324 71116 35074

-0.11524 -0.11579 -0.11632 -0.11682 -0.11730

08974 70951 64980 92401 54539

-0.11392 -0.10855 -0.10321 -0.09789 -0.09259

80127 55827 00582 11840 87082

1.07719 1.07178 1.06643 1.06112 1.05587

37361 68773 14226 66696 19286

0.350 0.355 0.360 0.365 0.370

0.88891 0.88853 0.88818 0.88785 0.88754

35692 71494 41041 42918 75748

-0.11775 -0.11817 -0.11857 -0.11894 -0.11929

52707 88209 62331 76353 31538

-0.08733 -0.08209 -0.07687 -0.07168 -0.06652

23825 19619 72046 78723 37297

1.05066 1.04550 1.04040 1.03533 1.03032

65216 97829 10578 97036 50881

0.375 0.380 0.385 0.390 0.395

1.400 1.405 1.410 1.415 1.420

0.88726 0.88700 0.88676 0.88654 0.88635

38175 28884 46576 89993 57896

-0.11961 '-0.11990 -0.12017 -0.12041 -0.12063

29142 70405 56559 88823 68406

-0.06138 -0.05627 -0.05118 -0.04611 -0.04107

45446 00879 01337 44589 28433

1.02535 i102043 1.01555 1.01072 1.00593

65905 36002 55173 17518 17241

0.400 0.405 0.410 0.415 0.420

1.425 1.430 1.435 1.440 1.445

0.88618 0.88603 0.88590 0.88580 0.88572

49081 62361 96587 50635 23397

-0.12082 -0.12099 -0.12114 -0.12125 -0.12135

96505 74307 02987 83713 17638

-0.03605 -0.03106 -0.02609 -0.02114 -0.01621

50697 09237 01935 26703 81479

1.00118 0.99648 0.99181 0.98719 0.98261

48640 06113 84147 77326 80318

0.425 0.430 0.435 0.440 0.445

1.450 1.455 1.460 1.465 1.470

0.88566 0.88562 0.88560 0.88560 0.88563

13803 20800 43364 80495 31217

-0.12142 -0.12146 -0.12148 -0.12148 -0.12145

05907 49657 50010 08083 24980

-0.01131 -0.00643 -0.00158 +0.00325 0.00806

64226 72934 05620 39677 64890

0.97807 0.97357 0.96911 0.96469 0.96031

87886 94874 96215 86921 62091

0.450 0.455 0.460 0.465 0.470

1.475 1.480 1.485 1.490 1.495

0.88567 0.88574 0.88583 0.88594 0.88607

94575 69646 55520 51316 56174

-0.12140 -0.12132 -0.12122 -0.12110 -0.12095

01797 39621 39528 02585 29852

0.01285 0101762 0.02237 0.02710 0.03180

71930 62684 3901.3 02758 55736

0.95597 0.95166 0.94739 0.94316 0.93896

16896 46592 46509 12052 38700

0.475 0.480 0.485 0.490 0.495

1.500

0.88622 69255

0.93480 22005

0.500

y!

-0.12078 22376

In y!

0.03648 99740

*

d

lny!

log,, e=0.43429 44819 *See page XI.

!I

G A M M A FUNCTION AND RELATED FUNCTIONS

269

GAMMA, DIGAMMA AND TRIGAMMA FUNCTIONS z

* (x)

In r(.r)

Table 6.1 *' (x)

1,500 1.505 1.510 1.515 1.520

0.88622 0.88639 0.88659 0.88680 0.88703

(x) 69255 89744 16850 49797 87833

-0.12078 -0.12058 -0.12037 -0.12013 -0.11986

22376 81200 07353 01860 65735

0.03648 0.04115 0.04579 0.05041 0.05502

99740 36543 67896 95527 21146

0.93480 0.93067 0.92658 0.92252 0.91850

22005 57588 41142 68425 35265

0.500 0.505 0.510 0.515 0.520

1.525 1.530 1.535 1.540 1.545

0.88729 0.88756 0.88786 0.88817 0.88851

30231 76278 25287 76586 29527

-0.11957 -0.11927 -0.11893 -0.11858 -0.11820

99983 05601 83580 34900 60534

0.05960 0.06416 0.06871 0.07323 0.07773

46439 73074 02697 36936 77400

0.91451 0.91055 0.90663 0.90274 0.89888

37552 71245 32361 16984 21253

0.525 0.530 0.535 0.540 0.545

1.550 1.555 1.560 1.565 1.570

0.88886 0.88924 0.88963 0.89005 0.89048

83478 37830 91990 45387 97463

-0.11780 -0.11738 -0.11693 -0.11647 -0.11598

61446 38595 92928 25388 36908

0.08222 0.08668 0.09113 0.09556 0.09997

25675 83334 51925 32984 28024

0.89505 0.89125 0.88749 0.88375 0.88005

41371 73596 14249 59699 06378

0.550 0.555 0.560 0;565 0.570

1.575 1.580 1.585 1.590 11595

0.89094 OI89141 0.89191 0.89242 0189296

47686 95537 40515 82141 19949

-0.11547 -0.11494 -0.11438 -0.11380 -0.11321

28415 00828 55058 92009 12579

0.10436 0.10873 0.11309 0.11742 oI12174

38544 66023 11923 77690 64754

0.87637 0.87272 0.86911 0.86552 0186196

50766 89402 18871 35815 36921

0.575 0.580 0.585 0.590 01595

1.600 1.605 1.610 1.615 1.620

0.89351 0.89408 0.89468 0.89529 0.89592

53493 82342 06085 24327 36685

-0.11259 -0.11195 -0.11128 -0.11060 -0.10990

17657 08127 84864 48737 00610

0.12604 0.13033 0.13459 0.13884 0.14307

74528 08407 67772 53988 68404

0.85843 0.85492 0.85145 0.84800 0.84457

18931 78630 12856 18488 92455

0.600 0.605 0.610 0.615 0.620

1.625 1.630 1.635 1.640 1.645

0.89657 0.89724 0.89793 0.89864 0.89936

42800 42326 34930 20302 98138

-0.10917 -0.10842 -0.10765 -0.10687 -0.10606

41338 71769 92746 05105 09676

0.14729 0.15148 0.15566 0.15983 0.16398

12354 87158 94120 34529 09660

0.84118 0.83781 0.83446 0.83115 0.82785

31730 33330 94315 11790 82897

0.625 0.630 0.635 0.640 0.645

1.650 1.655 1.660 1.665 1.670

0.90011 0.90088 0.90166 0.90247 0.90329

68163 30104 83712 28748 64995

-0.10523 -0.10437 -0.10350 -0.10261 -0.10170

07282 98739 84860 66447 44301

0.16811 0.17222 0.17632 0.18040 0.18447

20776 69122 55933 82427 49813

0.82459 0.82134 0.81812 0.81493 0.81176

04826 74802 90092 48001 45875

0.650 0.655 0.660 0.665 0.670

1.675 1.680 1.685 1.690 1.695

0.90413 0.90500 0.90588 0.90678 0.90770

92243 10302 18996 18160 07650

-0.10077 -0.09981 -0.09884 -0.09785 -0.09684

19212 91969 63351 34135 05088

0.18852 0.19256 0.19658 0.20058 0.20457

59282 12015 09180 51931 41410

0.80861 0.80549 0.80239 0.79931 0.79626

81094 51079 53282 85198 44350

0.675 0.680 0.685 0.690 0.695

1.700 1.705 1.710 1.715 1.720

0.90863 0.90959 0.91057 0.91156 0.91258

87329 57079 16796 66390 05779

-0.09580 -0.09475 -0.09368 -0.09259 -0.09147

76974 50552 26573 05785 88929

0.20854 0.21250 0.21645 0.22037 0.22429

78749 65064 01462 89037 28871

0.79323 OI79022 0.78723 0.78427 Oi78132

28302 34645 61012 05060 64486

0.700 0.705 0.710 0.715 0.720

1.725 1.730 1.735 1I740 1.745

0.91361 0.91466 0.91573 0.91682 0.91793

34904 53712 62171 60252 47950

-0.09034 -0.08919 -0.08802 -0.08683 -0.08562

76741 69951 69286 75466 89203

0.22819 0.23207 0.23594 0.23980 0.24364

22037 69593 72589 32061 49038

0.77840 0.77550 0.77262 0.76976 0.76692

37011 20396 12424 10915 13714

0.725 0.730 0.735 0.740

1.750

0.91906 25268

0.76410 18699

0.750

Y!

-0.08440 11210

In y!

0.24747 24535 d In y!

c'-231 log,,, e=0.43429 44819

01745 .Y

270

GAMMA FUNCTION AND RELATED FUNCTIONS

Table 6.1

GAMMA, DIGAMMA AND TRIGAMMA FUNCTIONS

r (4

(4

25268 92224 48846 95178 31277

-0.08440 -0.08315 -0.08188 -0.08060 -0.07929

11210 42192 82847 33871 95955

!+) 0.24747 24535 0.25128 59559 0.25508 55103 0.25887 12154 0.26264 31686

0.76410 0.76130 0.75852 0.75576 0.75302

18699 23773 26870 25950 19003

0.750 0.755 0.760 0.765 0.770

0.92498 0.92622 0.92748 0.92876 0.93006

57211 73062 78926 74904 61123

-0.07797 -0.07663 -0.07527 -0.07389 -0.07249

69782 56034 55386 68509 96070

0.26640 0.27014 0.27387 0.27759 0.28129

14664 62043 74769 53776 99992

0.75030 0.74759 0.74491 0.74224 0.73960

04040 79107 42268 91617 25271

0.775 0.780 0.785 0.790 0.795

1.800 1.805 1.810 1,815 1.820

0.93138 0.93272 0.93407 0.93545 0.93684

37710 04811 62585 11198 50832

-0.07108 -0.06964 -0.06819 -0.06672 -0.06523

38729 97145 71969 63850 73431

0.28499 0.28866 0.29233 0.29598 0.29962

14333 97707 51012 75138 70966

0.73697 0.73436 0.73177 0.72919 0.72663

41375 38093 13620 66166 93972

0.800 0.805 0.810 0.815 0.820

1.825 1.830 1.835 1.840 1;845

0.93825 0.93969 0.94114 0.94261 0.94410

81682 03951 17859 23634 21519

-0.06373 -0.06220 -0.06066 -0.05910 -0.05752

01353 48248 14750 01483 09071

0.30325 0.30686 0.31046 0.31405 0.31763

39367 81205 97335 88602 55846

0.72409 0.72157 0.71907 0.71658 0.71411

95297 68426 11662 23333 01788

0.825 0.830 0.835 0.840 0.845

1.850 1.855 1i860 1.865 1.870

0.94561 0.94713 0.94868 0.95025 0.95184

11764 94637 70417 39389 01855

-0.05592 -0.05430 -0.05267 -0.05102 -0.04935

38130 89276 63117 60260 81307

0.32119 0.32475 0.32829 0.33182 0.33533

99895 21572 21691 01056 60467

0.71165 0.70921 0.70679 0.70438 0.70199

45396 52546 21650 51138 39461

0.850 0.855 0.860 0.865 0.870

1.875 1.880 1.885 1.890 1.895

0.95344 0.95507 0.95671 0.95837 0.96006

58127 08530 53398 93077 27927

-0.04767 -0.04596 -0.04424 -0.04251 -0.04075

26854 97497 93824 16423 65875

0.33884 0.34233 0.34581 0.34928 0.35273

00713 22577 26835 14255 85596

0.69961 0.69725 0.69491 0.69258 0.69027

85089 86512 42236 50790 10717

0.875 0.880 0.885 0.890 0.895

1.900 1.905 1.910 1.915 1.920

0.96176 0.96348 0.96523 0.96699 0.96877

58319 84632 07261 26608 43090

-0.03898 -0.03719 -0.03538 -0.03356 -0.03172

42759 47650 81118 43732 36054

0.35618 0.35961 0.36304 0.36645 0.36985

41612 83049 10646 25136 27244

0.68797 0.68568 0.68341 0.68116 0.67892

20582 78965 84465 35696 31293

0.900 0.905 0.910 0.915 0.920

1.925 1.930 1.935 1.940 1.945

0.97057 0.97239 0.97423 0.97609 0.97797

57134 69178 79672 89075 97861

-0.02986 -0.02799 -0.02609 -0.02419 -0.02226

58646 12062 96858 13581 62778

0.37324 0.37661 0.37998 0.38334 0.38668

17688 97179 66424 26119 76959

0.67669 0.67448 0.67228 0.67010 0.66793

69903 50194 70846 30559 28044

0.925 0.930 0.935 0.940 0.945

1.950 1.955 1.960 1.965 1.970

0.97988 0.98180 0.98374 0.98570 0.98768

06513 15524 25404 36664 49838

-0.02032 -0.01836 -0.01639 -0.01439 -0.01239

44991 60761 10621 95106 14744

0.39002 0.39334 0.39665 0.39996 0.40325

19627 54805 83163 05371 22088

0.66577 0.66363 0.66150 0.65938 0.65728

62034 31270 34514 70538 38134

0,950 0.955 0.960 0.965 0.970

1.975 1.980 1.985 1.990 1.995

0.98968 0.99170 0.99375 0.99581 0.99789

65462 84087 06274 32598 63643

-0.01036 -0.00832 -0.00626 -0.00419 -0.00210

70060 61578 89816 55291 58516

0.40653 0.40980 0.41306 0.41631 0.41955

33970 41664 45816 47060 46030

0.65519 0.65311 0.65105 0.64900 0.64696

36104 63266 18450 00505 08286

0.975 0.980 0.985 0.990 0.995

2.000

1.00000 00000

0.64493 40668

1,000

z

1.750 1.755 1.760 1.765 1.770

0.91906 0.92020 0.92137 0.92255 0.92376

1.775 1.780 1;785 1.790 1.795

Y!

In r(a.)

0.00000 00000

In y!

0.42278 43351 d ;t;; I n y !

“-y]

log,, e=0.43429 44819

$J‘

Y

271

GAMMA FUNCTION AND RELATED FUNCTIONS

TETRAGAMMA A N D PENTAGAMMA FUNCTIONS

1.00 1.01 1.02 1.03 1.04

-2.40411 -2.34039 -2.27905 -2.21996 -2.16303

38063 86771 42052 85963 63855

6.49393 6.25106 6.01969 5.79918 5.58891

94023 18729 49890 38573 68399

0.00 0.01 0.02 0.03 0.04

1.50 1.51 1.52 1.53 1.54

-0.82879 -0.81487 -0.80129 -0.78803 -0.77509

66442 76121 51399 87419 83287

1.40909 1.37489 1.34177 1.30967 1.27856

(,v) 10340 70527 21104 56244 88154

0.50 0.51 0.52 0.53 0.54

1.05 1.06 1.07 1.08 1.09

-2.10815 -2.05523 -2.00419 -1.95493 -1.90737

80219 94833 19194 13213 82154

5.38832 5.19686 5.01404 4.83939 4.67247

23132 56970 67303 69702 74947

0. 05 0.06 0.07 0.08 0.09

1.55 1.56 1.57 ~.~ 1.58 1.59

-0.76246 -0.75012 -0.73807 -0i72630 -U.71480

41904 69793 76946 76669 85441

1.24841 1.21917 1.19082 1.16332 1.13663

46160 75841 38216 08979 77770

0.55 0.56 0;57 0.58 0.59

1.10

73783 75731 13035 45852 67342

4.51287 4.36020 4.21411 4.07424 3.94028

67903 88083 11755 35447 60737

0.10 0.11 0.12 0.13 0.14

1.60 1.61 1.62 1.63 1.64

-0.70357 22779

1.12 1.13 1.14

-1.86145 -1.81709 -1.77423 -1.73279 -1.69272

1.11074 1.08561 1.06121 iI03isZ 1.01452

47490 33658 63792 76835 22608

0.60 0.61

1.15 1.16 1.17 1.18 1.19

-1.65397 -1.61647 -1.58017 -1.54503 -1,51100

01677 02206 49731 50903 36723

3.81193 3.68891 3.57095 3.45780 3.34922

80220 64540 50416 29554 38402

0.15 0.16 0.17 0.18 0.19

1.65 1.66 1.67 1.68 1.69

-0.65107 -0.64125 -0.63166 -0.62226 -0.61308

17793 90881 04061 96973 11332

0.99217 0.97046 0.94937 0.92886 0.90893

61290 62927 06973 81843 84502

0. 65 0. 66 0.67 0. 68 0.69

1.20 1.21 1.22 1;23 1.24

-1.47803 -1.44608 -1.41512 -1,38510 -1.35598

61144 99765 48602 22950 56308

3.24499 3.14490 3.04875 2.95636 2.86754

48647 58422 84139 52925 95589

0.20 0.21 0.22 0.23 0.24

1.70 1.71 1.72 1.73 1.74

-0.60408 -0.59528 -0.58667 -0.57823 -0.56997

90841 81112 29593 85490 99702

0.88956 0.87072 0.85239 0.83456 0.81722

20066 01433 48922 89940 58660

0.70 0.71 0. 72 0.73 0.74

1.25 1.26 1.27 1.28 1.29

-1.32773 -1,30033 -1.27372 -1.24790 -1.22282

99375 19112 97857 32496 33691

2.78214 2.69999 2.62093 2.54484 2.47158

40092 05478 96227 97000 67746

0.25 0.26 0.27 0.28 0.29

1.75 1.76 1.77 1.78 1.79

-0.56189 -0.55397 -0.54621 -0.53861 -0.53116

24756 14738 25238 13291 37320

0.80034 O;78392 0.76793 0.75237 0.73721

95719 47929 68005 14300 50564

0. 75 0.76 0.77 0.78 0.79

1.30 1.31 1.32 1.33 1.34

-1.19846 -1.17479 -1.15179 -1.12943 -1.10769

25147 42923 34794 59642 86881

2.40102 2.33304 2.26752 2.20436 2.14345

39143 08348 35032 37678 90132

0.30 0.31 0.32 0.33 0.34

1.80 1.81 1.82 1.83 1.84

-0.52386 -0.51671 -0.50970 -0.50283 -0.49609

57084 33630 29242 07396 32712

0.72245 0.70807 0.69407 0.68042 0.66712

45705 73565 12710 46226 61527

0. 80 0.81 0.82 0.83 0. 84

1.35 1.36 1.37 1.38 1.39

-1,08655 -1.06599 -1.04599 -1.02652 -1.00757

95925 75682 24073 47586 60850

2.08471 2.02802 1.97332 1.92051 1.86951

18367 97472 48830 37473 69616

0.35 0.36 0.37 0.38 0.39

1.85 1.86 1.87 1.88 1.89

-0.48948 70921 -0.48300 88813 -0.47665 54207 -0.47042 35909 -0.46431 03677

0.65416 0.64153 OI6292i 0.61720 0.60549

50169 07680 33389 30270 04793

0. 85 0.86 0. 87 0.88 0. 89

1.40 1.41 1.42 1.43 1.44

-0.98912 -0.97116 -0.95366 -0.93662 -0.92002

86236 53479 99322 67177 06808

1.82025 1.77266 1.72667 1.68221 1.63423

90339 81419 59295 73161 03178

0.40 0.41 0.42 0.43 0.44

1.90 1.91 1.92 1.93 1.94

-0.45831 -0.45242 -0.44665 -0.44098 -0.43542

28188 81007 34549 62055 37563

0.59406 0.58292 0.57205 0.56144 0.55108

66772 29238 08299 23020 95304

0.90 0.91 0.92 0.93 0.94

1.45 1.46 1.47 1.48 1.49

-0.90383 -0.88806 -0.87268 -0.85768 -0.84306

74031 30426 43070 84281 31376

1.59765 1.55743 1.51852 1.48085 1.44439

58792 77157 21649 80478 65370

0.45 0.46 0.47 0.48 0.49

1.95 1.96 1.97 1.98 1.99

-0.42996 -0.42460 -0.41934 -0.41417 -0.40909

35876 32537 03805 26631 78630

0.54098 0.53112 0.52149 0.51208 0.50290

49774 13668 16733 91127 71324

0.95 0.96 0.97 0.98 0.99

1.50

-0.82879

66442

1.40909 10340 d' In y! du'

0.50

2.00

-0.40411 38063

0.49393 94023 d' In y!

1.00

iiii

$b(:j) (.r)

!b"(.l,)

I'

*

T a l k 6.2

-

I'

Y

!b"(R.)

CP In y! dva

&i4

-.

0.62 --

0.63 0.64

Y

[(-65)41

Compiled from H. T. Davis, Tables of the higher mathematical functions, 2 vols. (Principia Press, Bloomington, Ind., 1933, 1935) (with permission). *Bee page

11.

*

272

GAMMA FUNCTION AND RELATED FUNCTIONS

‘rai)ic6.:3

V.tI,C E X

GAMMA A 1’ (//)

11

1poo

3 4 5

( 0)6IOOOOO 00000

( 1)2.40000 00000

I’(//

+ 4)

$00

0)1.00000 000

(-1) 8.86226 93

( - 2)4.16666 667

04 10 28 1)5.23427 78

(

( 2)2.87885 28

9 10 11 12 13 14 15

1I

6 3.62880 713.99168 8 4.79001 9 6.22702 10 8.71782

00000 00000 60000 08000 91200

31

:; 41 42 43 44 45

566 285 795 669 024

1.01397 1.01196 1.01046 1.00929 1.00836

285 0.08564 180 776 0.07312 581 565 0.06380 006 843 0.05658 310 536 0.05083 250

( 7)1.18994 f 8)1.36843 9’ 1.71054 1 0 2.30923 11)3.34838

23 37 21 18 61

2.35175 2.44266 2.52599 2.60291 2.67434

25891 16800 50133 80902 66617

1.00760 1.00696 1.00642 1.00596 1.00557

243 700 958 911 019

0.04614 0.04224 0.03895 0.03613 0.03370

268 497 434 924 354

(1215.18999 8I56349 1.49861 2.77243 5.40624

85 74 21 23 30

2.74101 2I80351 2.86233 2.91789 2.97052

33283 33283 68577 24133 39922

1.00522 1.00491 1.00463 1.00439 1.00417

124 343 988 519 501

0.03157 0.02970 0.02803 0.02654 0.02520

539 002 490 657 828

(-19) 4.11031 762

(19)1.10827 2.38280 5.36130 1.25990 3.08677

98 16 36 63 05

3.02052 3.06814 3.11359 3.15707 3.19874

39922 30399 75853 58462 25129

1.00397 1.00379 1.00362 1.00347 1.00333

584 480 953 806 872

0.02399 0.02289 0.02189 0.02097 0.02013

845 941 663 798 331

( - 2 6 6.44695 029

(25)7.87126 49 27 2.08588 52 28 5.73618 43 25 69

3.23874 3.27720 3.31424 3I34995 3.38443

25129 40513 10884 53741 81327

1.00321 1.00309 1.00298 1100287 1.00278

011 0.01935 403 105 0.01863 281 050 0.01796 342 758 0.01734 046 154 0.01675 925

1.47092 4.63340 1.50585 5.04462 39)1.74039

26 61 70 09 42

3.41777 3.45002 3.48127 3.51158 3.54099

14660 95305 95305 25608 43255

1.00269 1.00260 1.00252 1.00245 1.00238

170 0.01621 574 748 0.01570 637 837 0.01522 803 392 0.01477 796 372 0.01435 374

6.17839 2.25511 8.45668 3.25582 1.28605

94 58 42 34 02

3.56956 3.59734 3.62437 3.65068 3.67632

57541 35319 05589 63484 73740

1.00231 1.00225 1.00219 1.00213 1.00208

744 474 534 899 546

0.01395 0.01357 0.01321 0.01287 0.01255

318 438 560 530 208

5.20850 2.16152 9.18649 3.99612 1.77827

35 90 81 67 64

3.70132 3.72571 3.74952 3.77278 3.79551

73740 76179 71417 29557 02284

1.00203 1.00198 1.00193 1.00189 1.00185

455 606 983 570 354

0.01224 0.01195 0.01167 0.01140 0.01115

469 200 297 668 226

8.09115 3.76238 1.78713 8.66760 4,29046

74 82 44 18 29

3.81773 3.83947 3.86074 3.88158 3.90198

24506 15811 81768 15102 96734

1.00181 1.00177 1.00173 1.00170 1.00166

321 460 759 210 803

0.01090 0.01067 0.01045 0.01023 0.01003

895 602 283 879 333

( - 6)2.75573 192 ( - 7 2.75573 I - 8 2.50521 912.08767 -10 1.60590 (-11) 1.14707

(-

( 2 5 ) 1.55112 26 4.03291 2811.08888 29 3.04888 3018.84176

10043 46113 69450 34461 19937

(32) 2.65252 8.22283 2.63130 8.68331 2.95232

85981 80542 83693 76188 79904

40 1.03331 1411 3.71993 43j1.37637 44 5.23022 4612.03978

47966 32679 53091 61747 82081

(-44 (-45 (-47

8.15915 3.34525 1.40500 6.04152 2.65827

28325 26613 61178 63063 15748

-48)1,22561 1-50 2.98931 [-5217,11740 -53 1.65521 ( -551 3.76184

( 6 4 ) 3.04140 93202

,/’,!,( 1 1 )

0.57721 0.27036 0.17582 0.13017 0.10332

76684 43351 14780 14780 25891

192 084 570 438 456

( -271 2.47959 626

I

(-29 9.18368 986 1-30 3.27988 924 ‘-31 1.13099 629 ( -33) 3.76998 763

(-34) 1.21612 504

744 083 673 087 288

46 47 48 49 50 51

755 712 452 830 399

1.70611 1.87278 2.01564 2.14064 2.25175

(17)1.21645 10041

26

1.08443 1104220 1.02806 1.02100 1.01678

43 07 46 84

-15) 2.81145 725 20

0))

56649 43351 43351 76684 76684

3)1.87125 4)1.40344 5 1.19292 611.13327

4 4.03200 00000

( 513.62880 00000

,fI

-0.57721 +0142278 0.92278 1.25611 1.50611

(-65) 3.28794 942

I

I’

I 1

( 6 5 ) 2.16668 38

3.92198 96734

1.00163 530

0.00983 596

from H. T. Davis, Tables of the higher mathematical functions, 2 vols. (Principia Press, Bloomington, Ind., 1933, 1935) (with permission).

$(H) compiled

*See page II.

273

QAMMA FUNCTION AND RELATED FUNCTIONS

6513.28794 942 ( 65)2.16668 38 3.92198 3.94159 3.96082 - 7oj2.33924 515 3.97969 - 72)4.33193 547 72)1,70809 63 3.99821

51 f 64)3.04140 93202 52 ( 66)1155111 87533 53 f 6718.06581 75171 69)4.27488 32841 54 71)2,30843 69734 55

(-

I

56 57 58 59 60

530 383 355 438 628

0.00983 0.00964 0.00946 0100928 0.00911

596 620 363 784 846

( 73)9.47993 44 4.01639 65470 1.00148 919 0.00895 514

80)1.38683 11855

4.03425 4I05179 79)1.8Gl67 93 4.06903 ( 81)1.07199 92 4.08598

36899 75495 89288 80814

1.00146 1I00143 1.00141 1.00138

304 780 341 984

0.00879 0100864 0.00849 0.00835

758 546 852 648

51 4.10265 4.11904 06 4.13517 19 4.15105 98 4.16667

47481 81907 72229 02388 52388

1.00136 1.00134 1.00132 1.00130 1.00128

704 498 362 292 286

0.00821 0.00808 0.00795 0.00783 0.00771

912 619 750 284 203

4.18205 4.19721 4.21213 97 2.05634 36 4.22684 ( 9911.42915 88 4.24133

98542 13693 67425 26248 53785

1.00126 1.00124 1.00122 1.00120 1.00119

341 455 623 845 118

0.00759 0.00748 0.00737 0.00726 0.00715

489 125 096 388 986

4.25562 4.26970 4.28359 4129729 4.31080

10927 55998 44887 31188 66323

1.00117 1.00115 1.00114 1.00112 1.00111

439 807 220 675 172

0.00705 0.00696 0.00686 0.00677 0.00668

878 052 495 197 148

4.32413 4.33729 4.35028 4.36310 4.37576

99657 78604 48734 53862 36140

1.00109 1.00108 1.00106 1.00105 1.00104

709 283 894 540 220

0.00659 0.00650 0.00642 0.00634 0.00626

337 756 395 247 302

4.38826 4.40060 123)4.32425 79 4.41280 4142485 4.43675

36140 92931 44150 26078 73697

1.00102 1.00101 1.00100 1.00099 1.00098

933 677 452 255 087

0.00618 0.00610 0.00603 0.00596 0.00589

554 995 619 419 389

4.44852 4.46014 4.47164 4.48300 4.49424

20756 99825 42354 78718 38268

1.00096 1.00095 1.00094 1.00093 1.00092

946 831 741 676 635

0.00582 0.00575 0.00569 0.00562 0.00556

522 814 258 850 584

4.50535 4.51634 4.52721 4153796 4.54860

49379 39489 35142 62023 45002

1.00091 1.00090 1.00089 1.00088 1.00087

617 620 646 691 757

0.00550 457 0,00544 463 0.00538 598 0.00532 858 0.00527 239

4.55913 4.56954 4.57985 154 9.36780 21 4.59006 156)9.32096 31 4.60016

08160 74827 67610 08426 18527

1.00086 1.00085 1.00085 1.00084 1.00083

843 947 070 210 368

0.00521 0.00516 0.00511 0.00505 0.00500

1

8216.48559 84 3.98864 86 2.49290 ( 88j1.58299 ( 90)1.02102

10

( 91)6.68774 50

96 2.48003 55424 9811.71122 45243

71 72 73 74 (10514i47011 54615 ( -10612I23707 868 75 (107)3.30788 54415 ( -108)3.02307 930 76 77 78 79 80

1.00163 1.00160 1.00157 1.00154 1.00151

73)1.26964 03354

81)8.32098 71127 61 62 63 64 87)1198260 83154 65 ( 89)1.26886 93219 66 67 68 69 70

96734 75166 82858 62103 47288

-110 4.03077 240 -11215,30364 789 -114 6.88785 441 -11618.83058 257 -117)1.11779 526

109)2,48091 40811

116)8.94618 21308

(110)2.15925 (112 1.65183 (11411,28016 (116 1.00493 ( 1171 7.98921

-129 3.54974 456 ‘129)2.60868 -13114.12760 995 131 2.25650 1331 1.97444 135 1.74738 1371 1,56390

64 12 92 28 57

05 86 50 38 85

1

91 138 1.48571 59645 92 140 1.35200 15277 93 11421 1.24384 14055 94 144 1.15677 25071 95 146 1.08736 61567

738 350 072 901 833

101 (157)9.33262 15444 ( -158)1.07151 029 (158)9.36756 79 4.61016 18527 1.00082 542 0.00495 866 d ln(/i-l)! * -dn 1/ ( / I - 1) ! (//-$)! (//-l)! /) ~~

‘See page U.

r ( / / ) =( Z T ) + / / ’ +

’-”.fl

(tt)

+()/)=I

r)’=2.50662 82746 31001

274

GAMMA FUNCTION AND RELATED FUNCTIONS

LOGARITHMS OF THE GAMMA FUNCTION

Table 6.4

(%++I

log,, r

I?

-0.04915 t0.07578 0.44375 0.96663 1.60345

851 023 702 576 79

-0.05245 t0.12363 0.52157 1.06564 1.71885

506 620 621 43 68

2.07918 2.85733 3.70243 4.60552 5.55976

12 25 05 05 30

2.33045 3.13208 3.99739 4.91820 5.88824

66 89 04 91 59

2.45921 3.27213 4.14719 5.07661 6.05433

95 28 41 30 66

2.58998 3.41389 4.29850 5.23635 6.22163

11 12 13 14 15

6.55976 7.60115 8.68033 9.79428 10.94040

30 57 70 03 8

6.90248 7.95684 9.04792 10.17286 11.32921

63 40 45 3 0

7.07552 8.13622 9.23313 10.36346 11.52483

59 37 38 8 6

7.24966 8.31660 9.41927 10.55493 11.72126

16 17 18 19 20

12.116500 13.320620 14.551069 15.806341 17.085095

12.514847 13.727922 14.966804 16.230045 17.516352

12.715167 13.932651 15.175689 16.442861 17.732896

21 22 23 24 25

18.386125 19.708344 21.050767 22.412494 23.792706

18.824561 20.153619 21.502573 22.870550 24.256751

26 27 28 25 30

25.190646 26.605619 28.036983 29.484141 30.946539

31 32 33 34 35

71

log,, r

(7%)

1 2 3 4 5

0.00000 0.00000 0.30103 0.77815 1.38021

000 000 000 125

6 7 8 9 10

log,,

I-

(,2+$)

log,, r (n+5) -0.04443 477 +0.17741 398 0.60338 271 1.16765 41 1.83666 09

f 2 (7~)

1.00000 0.96027 0.94661 0.93972 0.93558

000 923 646 921 323

86 73 39 60 27

0.93281 0.93083 0.92934 0.92819 0.92726

466 524 980 400 910

15 83 06 3 5

0.92651 0.92588 0.92534 0.92488 0.92449

221 137 753 990 327

12.916241 14.138090 15.385245 16.65631 1 17.950042

0.92414 0.92383 0.92356 0.92332 0.92310

619 993 769 409 485

19.044649 20.377088 21.729270 23.100338 24.4895 04

19.265313 20.601105 21.956492 23.330629 24.722740

0.92290 0.92272 0.92256 0.92241 0.92227

649 615 149 055 169

25.660444 27.080949 28.517642 29.969940 31.437301

25.896045 27.319290 28.758623 30.213468 31.683290

26.132 109 27.558078 29.000035 30.457412 31.929681

0.92214 0.92202 0.92191 0.92181 0.92171

350 481 460 198 621

32.423660 33.915022 35.420172 36.938686 38.470165

32.919221 34.415228 35.924878 37.447757 38.983473

33.167590 34.665900 36.177784 37.702829 39.240648

33.416347 34.916950 36.431055 37.958255 39.498167

0.92162 0.92154 0.92146 0.92138 0.92131

661 262 371 944 942

36 37 38 39 40

40.014233 41.570535 43.138 737 44.718520 46.309585

40.531658 42.091963 43.664060 45.247636 46.842 397

40.790876 42.353169 43.927200 45.512661 47.109258

41.050429 42.614701 44.190658 45.777995 47.376420

0.92125 0.92119 0.92113 0.92107 0.92102

329 073 146 524 182

41 42 43 44 45

47.911645 49.524429 51.147678 52.781147 54.424599

48.448061 50.064362 51.691044 53.327866 54.974597

48.716713 50.334761 51.963150 53.601639 55.249999

48.985659 50.605448 52.235536 53.875686 55.525670

0.92097 0.92092 0.92087 0.92083 0.92079

101 262 648 244 035

46 47 48 49 50

56.077812 57.740570 59.412668 61.093909 62.784105

56.631014 58.296908 59.972075 61.656322 63.349462

56.908011 58.575464 60.252157 61.937899 63.632504

57.185269 58.854276 60.5 32491 62.219723 63.9 15788

0.92075 0.92071 0.92067 0.92063 0.92060

010 156 462 919 518

51

64.483075

65.051318

65.335796

65.620510

0.92057 250

log,, (n-l)! log,, (n-;) ! log,, (.-$) ! log,, (n-i) ! In r(n)=ln (n-1) != (n-+) In n-n+f2(?2) In 10=2.30258 509299 log,, r(n) compiled from E. S. Pearson, Table of the logarithms of the complete r-function, arguments 2 to 1200. Tracts for Computers No. VIII (Cambridge Univ. Press, Cambridge, England, 1922) (with permission).

275

GAMMA FUNCTION AND RELATED FUNCTIONS

LOGARITHMS OF THE GAMMA FUNCTION

Table 6.4

n 51 52 53 54 55

64.483075 66.190645 67.906648 69.630924 71.363318

65.051318 66.761 717 68.480496 70.207494 71.942561

65.335796 67.047603 68.767762 70.496116 72.232512

65.620510 67.333720 69.055256 70.784961 72.522683

.f2 ( 0.92057 0.92054 0.92051 0.92048 0.92045

250 108 084 173 367

56 57 58 59 60

73.103681 74.851869 76.607744 78.371172 80.142024

73.685548 75.436313 77.194720 70.960637 80.733936

73.976805 75.728854 77.488522 79.255677 81.030194

74.268279 76.021606 77.782531 79.550922 81.326654

0.92042 0.92040 0.92037 0.92035 0.92032

661 051 530 095 741

61 62 63 64 65

81.920175 83.705505 85.497896 87.297237 89.103417

82.5 14493 84.302190 86.096910 87.898542 89.706978

82.811950 84.600825 86.396705 88.199479 90.009038

83.109604 84.899655 86.696691 88.500604 90.311284

0.92030 0.92028 0.92026 0.92024 0.92022

464 261 127 061 057

66 67 68 69 70

90.916330 92.735874 94.561949 96.394458 98.233307

91.522113 93.343845 95.172075 97.006708 98.847650

91.825280 93.648101 95.477405 97.313096 99.155080

92.128629 93.952538 95.782913 97.619659 99.462684

0.92020 0.92018 0.92016 0.92014 0.92012

115 231 401 625 900

log,, r

(71)

log,, r

(n++)

log,, r (n+f)

log,, r

(n+-5)

71 72 73 74 75

100.07841 101.92966 103.78700 105I65032 107.51955

100.69481 102.54810 104.40744 106.2 7274 108.14393

101.00327 102.85758 104.71791 106.58420 108.45636

101.31190 103.16722 105.02855 106.89582 108.76895

0.92011 0.92009 0.92008 0.92006 0.92004

223 593 008 465 964

76 77 78 79 80

109.39461 111.27543 113.16192 115.05401 116i95164

110.02091 111.90363 113.79200 115.68594 117.58540

110.33430 112.21797 114.10727 116.00214 117.90250

110.64785 112.53246 114.42269 116.31848 118.2 1976

0.92003 0.92002 0.92000 0.91999 0.91998

502 078 690 338 019

81 82 83 84 85

118.85473 120.76321 122.67703 124.59610 126.52038

119.49029 121.40056 123.31614 125.23696 127.16296

119.80830 121.71946 123.63591 125.55760 127.48445

120.12646 122.0 3850 123.95583 125.87838 127.80610

0.91996 0.91995 0.91994 0.91993 0.91991

733 479 254 059 892

86 87 88 89 90

128.44980 130.38430 132h82 134.2 6830 136.21769

129.09407 131.03025 132797143 134.91756 136.86857

129.41642 131.35344 133.29545 135.24239 137.19421

129.73891 131.67676 133.61959 135.56735 137.51999

0.91990 0.91989 0.91988 0.91987 0.91986

752 638 550 486 446

91 92 93 94 95

138.17194 140.13098 142.09477 144.06325 146.03638

138.82442 140.78505 142.75041 144.72 044 146.6951 1

139.15086 141.11228 143.07842 145.04923 147.02467

139.47743 141.43964 143.40657 145.37815 147.35435

0.91985 0.91984 0.91983 0.91982 0.91981

428 433 459 505 572

96 97 98 99 100

148.01410 149.99637 151.98314 153.97437 155.97000

148.67435 150.65813 152.64639 154.63909 156.63619

149.00467 150.98920 152.97820 154.971 64 156.96946

149.33511 151.32039 153.31013 155.30430 157.30285

0.91980 0.91979 0.91978 0.91978 0.91977

659 764 887 028 186

159.30574

0.91976 361

101

157.97000

158.63763

log,, (n-l)!

log,() (n+) !

158.97163

log,,

(n-t) !

In r(n)=In (n-l)!=(n-+) In n-n+.fdn)

log,, (n-+) ! In 10=2.30258 509299

276 Table 6.5

GAMMA FUNCTION AND RELATED FUNCTIONS

AUXILIARY FUNCTIONS FOR GAMMA AND DIGAMMA FUNCTIONS

fl(4 1.00125 1.00116 1.00108 1,00100 1.00091

077 735 391 050 708

0.92018 0.92010 0.92002 0.91993 0.91985

852 519 186 853 520

f 3 (x) 0.00751 875 0.00701 633 0.00651 408 0.00601 200 0.00551 008

<x>

0.015 0.014 0.013 0.012 0.011 0.010 0.009 0.008 0.007 0.006

1.00083 1.00075 1.00066 1.00058 1.00050

368 028 689 350 012

0.91977 0.91968 0.91960 0.91952 0.91943

186 853 520 187 853

0.00500 0.00450 0.00400 0.00350 0.00300

833 675 533 408 300

100 111 125 143 167

0.005 0.004 0.003 0.002 0.001

1.00041 1.00033 1.00025 1.00016 1.00008

675 339 003 668 334

0.91935 0.91927 0.91918 0.91910 0.91902

52C 187 853 520 187

0.00250 0.00200 0.00150 0.00100 0.00050

208 133 075 033 008

200 250 333 500 1000

0.000

1.00000 000

2-1

f 2 (XI

0.91893 853

"-:"I

67 71 77 83 91

0.00000 000

00

I"'!-

[( -? 1 x != (2T)$a;:+te -"f 1(x)

r (5) = ( 2 ~*a;:) te-zj'l

(z)

(2r))=2.50662 82746 31001

=nearest integer to x.

Table 6.6

FACTORIALS FOR LARGE ARGUMENTS

n

n!

9.3326 7.8865 3.0605 6.4034 1.2201

21544 78673 75122 52284 36825 r(n+l>

n

39441 64790 16440 66238 99111

52682 50355 63604 95262 00687

n!

1.2655 2.4220 7.7105 6.7526 4.0238

72316 40124 30113 80220 72600

22543 75027 35386 96458 77093

07425 21799 00414 41584 77354

W+l)

Compiled from Ballistic Research Laboratory, A table of the factorial numbers and their reciprocals from l! to lOOO! to 20 significant digits, Technical Note No. 381, Aberdeen Proving Ground, Md.(1951) (with permission).

277

GAMMA FUNCTION AND RELATED FUNCTIONS

Table 6.7

GAMMA FUNCTION FOR COMPLEX ARGUMENTS

2=1.0

9 1 n r(z)

?I

0.0 0.1 0.2 0.3 0.4

0.5 0.6 0.7 0; 8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

5.0

9 1 n r(z) 00000 00 29404 17 22226 44 06721 68 58263 16

5.0 5.1 5.2 5.3 5.4

-

-

0.00000 0.05732 0.11230 0.16282 0.20715

6.13032 6.27750 6.42487 6.57242 6.72016

41445 24635 30533 85885 21547

53 84 35 29 03

3.81589 3.97816 4.14237 4.30850 4.47650

85746 15 38691 88 74050 86 21885 83 25956 68

-

0.24405 0.27274 0.29282 0.30422 0.30707

82989 05 38104 9 1 63511 87 56029 76 43756 42

5.5 5.6 5.7 5.8 5.9

-

6.86806 7.01613 7.16436 7.31275 7.46128

72180 48 75979 76 74421 06 12034 30 36194 29

4.64634 4.81799 4.99142 5.16659 5.34347

42978 41933 03424 19085 91013

-

0.30164 0.28826 0.26733 0.23921 0.20430

03204 68 66142 39 05805 8 1 67844 65 07241 49

6.0 6.1 6.2 6.3 6.4

7.60995 7.75877 7.90772 8.05680 8.20600

96929 51 46746 55 40468 98 35089 04 89631 00

5.52205 5.70228 5.88415 6.06762 6.25267

31255 15 61315 35 11702 39 21500 13 37967 05

+

0.16293 0.11546 0.06219 0.00341 0.06061

97694 87935 86983 66314 28742

80 89 29 77 95

6.5 6.6 6.7 6.8 6.9

8.35533 8.50478 8.65434 8.80401 8.95379

65025 11 23991 25 30931 23 51829 10 54158 79

6.43928 6.62742 6.81707 7.00820 7.20081

16159 76 18579 12 14837 44 81345 02 01014 93

87864 3 1 41504 7 1 42092 96 81922 04 81411 84

0.12964 0.20345 0.28184 0.36461 0.45158

63163 94738 56584 40489 81524

10 33 26 50 41

7.0 7.1 7.2 7.3 7.4

9.10368 9.25366 9.40375 9.55393 9.70421

06798 79950 45067 74783 42849

32 15 08 21 72

7.39485 7.59032 7.78719 7.98545 8.18508

62984 36 62351 84 99928 77 82004 68 20125 03

2.54990 2.68737 2.82558 2.96448 3.10401

68424 95 61537 50 56411 9 1 14617 89 54399 01

0.54260 0.63751 0.73616 0.83843 0.94420

44058 09190 63516 89130 54730

52 46 79 96 39

7.5 7.6 7.7 7.8 7.9

9.85458 -10.00503 -10.15558 -10.30621 -10.45692

24074 86 94267 90 30186 86 09489 48 10687 39

8.38605 8.58835 8.79196 8.99687 9.20305

30880 35709 60705 36442 97799

3.24414 3.38482 3.52603 3.66772 3.80988

42995 90 90223 77 43067 09 81104 88 12618 23

1.05335 1.16576 1.28135 1.40001 1.52165

07710 67132 17459 02965 22746

69 86 32 76 73

8.0 8.1 8.2 8.3 8.4

-10.60771 -10.75857 -10.90952 -11.06054 -11.21163

13103 96829 42693 32217 47589

15 95 78 92 48

9.41050 9.61920 9.82913 10.04027 10.25261

83803 12 37472 42 05671 62 38971 80 91518 09

3.95246 4.09546 4.23884 4.38258 4.52667

71261 89 13204 5 1 14660 7 1 69752 28 88647 16

1.64619 1.77355 1.90365 2.03642 2.17179

26242 69 09225 9 1 10190 19 07096 93 14436 05

8.5 8.6 8.7 8.8 8.9

-11.36279 -11.51402 -11.66532 -11.81669 -11.96812

71628 87756 79970 32818 31369

04 02 81 48 01

10.46615 10.68085 10.89672 11.11373 11.33188

20903 24 88047 12 57081 77 95241 57 72758 53

4.67109 4.81583 4.96086 5.10617 5.25176

95934 29197 37766 81606 30342

09 96 87 63 30

2.30969 2.45007 2.59287 2.73802 2.88548

80565 85299 37713 74148 56389

73 47 19 20 27

9.0 9.1 9.2 9.3 9.4

-12.11961 -12.27117 -12.42278 -12.57446 -12.72619

61192 08338 59312 01059 20940

81 67 81 08 29

11.55115 11.77153 11.99300 12.21556 12.43920

62762 41183 86662 80464 06390

5.39760 5.54369 5.69002 5.83657 5.98334

62389 64183 29483 58764 58655

84 04 73 54 32

3.03519 3.18711 3.34118 3.49736 3.65561

69999 22793 43443 80186 99647

22 89 27 15 12

9.5 9.6 9.7 9.8 9.9

-12.87798 -13.02982 -13.18172 -13.33367 -13.48567

06720 46547 28939 42765 77234

44 89 51 47 95

12.66389 12.88964 13.11642 13.34423 13.57307

50701 28 02037 08 51346 66 91814 77 18794 55

3.81589 85746 15

10.0

00000 77805 62923 62509 93748

0.19094 0.26729 0.35276 0;44597 0.54570

54991 87 00682 14 86908 60 87835 49 51286 05

0.65092 0.76078 0.87459 0.99177 1.11186

31993 39588 04638 27669 45664

02 41 95 59 26

1.23448 1.35931 1.48608 1.61459 1.74464

30515 47 22484 65 96127 57 53960 00 42761 74

-

1.87607 2.00876 2.14258 2.27743 2.41323

-

-

-

-

9 1 n r(z)

-

0.00000 0.00819 0.03247 0.07194 0.12528

--

9 l n r(z)

!/

00

65 18 00 21

6.13032 41445 53

-

-

-

-

-

--

-

-13.63773 21882 47

70 05 89 37 53

89 62 87 29 25

02 09 85 79 90

13.80291 29742 30

Linear interpolation will yield about three figures; eight-point interpolation will yield about eight figures. For z outside the range of the table, see Examples 5-8.

W In r (z) =In 1 r(z) I

9In r ( 2 ) =arg r (z)

278

GAMMA FUNCTION AND RELATED FUNCTIONS

GAMMA FUNCTION FOR COMPLEX ARGUMENTS

Table 6.7

.,.=1.1

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 276 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0

4 In r(z)

0.00000 0.04206 0.08230 0.11905 0.15086

00000 00 65443 76 97383 98 06275 18 79240 09

Y 5.0 5.1 5;2 5.3 5.4

0.17666 0.19566 0.20740 0.21167 - 0.20843

11398 43 16788 64 35526 60 10325 55 91333 00

5.5 5.6 5.7 5.8 5.9

78153 48 80620 74 85646 26 73364 97 26519 67

-

78257 67 55175 74 33222 12 93047 38 08957 00

6.0 6.1 6.2 6.3 6.4

- 1.18885 84815 22 - 1.30787 15575 95

-

0.04217 34907 11

%'lnr(z)

?I

0.0 0.1 0.2 0.3 0.4

-

0.04987 0.05702 0.07824 0.11291 0.16008

24412 60 02290 38 35801 68 43470 17 21257 99

0.21858 0.28718 0.36464 0.44978 0.54157

96764 09 99839 43 38731 53 83131 87 54093 11

0.63908 0.74153 0.84825 0.95868 - 1.07235

--

0.19781 0.18000 0.15525 0.12383 - 0.08605

--

5.96893 6.11415 6.25959 6.40526 6.55114

91493 52 43840 05 93585 61 53566 40 41480 20

9 In r(z) 3.96198 63258 60 4.12446 68364 90 4.28888 73284 80 4.45521 12743 47 4.62340 34819 04

6.69722 6.84350 6.98998 7.13663 - 7.28347

79531 89 94110 69 15495 70 77586 96 17659 19

4.79343 4.96525 5.13885 5.31419 5.49124

00232 04 81683 67 63238 91 39750 77 16322 40

--

7.43047 7.57764 7.72498 7.87247 8.02011

76136 25 96383 95 24519 72 09237 38 01645 61

5.66997 5.85035 6.03236 6.21597 6.40116

07803 94 38321 46 40835 50 56726 90 35407 92

8.16789 8.31582 8.46388 8.61208 8.76041

55118 88 25159 69 69271 17 46838 95 19021 72

6.58790 6.77617 6.96594 7.15720 7.34992

33956 67 16773 32 55256 30 27497 24 17993 20

8.90886 9.05744 9.20613 9.35494 9.50386

48649 60 00129 63 39357 92 33637 73 51603 25

7.54408 7.73966 7.93664 8.13500 8.33473

17375 09 22151 13 34464 25 61862 70 17082 71

9 In r(z)

1.42911 03402 04 1.55233 58336 11 1.67734 40572 49

0.06275 56777 30 0.12329 53847 15 0.18890 25358 69

6.5 6.6 6.7 6.8 6.9

1.80395 1.93203 2.06142 2.19203 2.32375

99248 63 22878 13 99239 46 82866 29 68617 01

0.25935 0.33446 0.41402 0.49786 0.58582

93780 23 29085 79 40321 50 66085 82 64745 04

7.0 7.1 7.2 7.3 7.4

2.45649 2.59018 2.72473 2.86010 - 2.99622

70097 26 01959 43 65306 67 35591 81 52529 98

0.67775 0.77349 0.87292 0.97592 1.08236

04868 09 56148 91 80949 66 26515 07 17859 08

7.5 7: 6 7.7 7.8 7.9

9.65289 63148 29 9;80203 39359 83 9.95127 52455 81 -10.10061 75726 94 -10.25005 83482 21

8.53580 8.73819 8.94190 9.14690 9.35317

17842 76 86648 33 50606 84 41251 84 94376 01

11644 50 -- 3.13305 3.27053 57144 30 3.40863 75892 32

1.19213 1.30513 1.42127 1.54045 1.66258

51297 05 88581 77 51595 43 17547 76 14631 94

8.0 8.1 8.2 8.3 8.4

-10.39959 50947 80 -ioI54922 54469 17 -10.69894 70966 06 -10.84875 78390 24 -10.99865 55435 72

9.56071 9.76949 9.97950 10.19072 10.40315

49872 49 51583 85 47158 43 87913 49 28704 84

77368 25 45962 20 05882 64 14278 81 47017 40

1.78758 1.91537 2.04588 2.17904 2.31478

18092 68 46664 26 59340 24 52440 32 56943 26

8.5 8.6 8.7 8.8 8.9

-11.14863 -11.29870 -11.44885 -11.59907 -11.74937

81551 38 36905 72 02353 71 59405 42 90196 53

10.61676 10.83154 11.04748 11.26457 11.48278

27802 52 46772 22 50362 14 06394 86 85664 18

4.53177 4.67407 4.81672 4.95971 5.10303

96812 84 71584 70 93009 83 95242 44 23779 21

2.45304 2.59375 2.73687 2.88232 3.03007

36058 25 83010 13 19016 54 91437 48 72080 09

9.0 9.1 9.2 9.3 9.4

-11.89975 -12.05021 -12.20073 -12.35133 -12.50199

77460 43 04501 83 55171 88 13844 58 65394 43

11.70212 11.92257 12.14411 12.36673 12.59043

61836 32 11355 62 13354 15 49565 33 04241 06

5.24665 5.39056 5.53476 5.67923 5.82396

34450 28 92519 72 71881 64 54339 89 28961 29

3.18006 3.33224 3.48657 3.64299 3.80148

55643 29 58288 43 16324 07 84993 84 37357 79

9.5 9.6 -~ 9.7 9.8 9.9

-12.65272 -12.80352 -12.95439 -13.10532 -13.25631

95175 33 89000 52 33123 60 14220 44 19372 14

12.81518 13.04099 13.26783 13.49570 13.72459

64072 43 18113 65 57709 12 76423 49 69974 44

3.96198 63258 60

10.0

-

--

-

-

-

3.54731 92273 03

- 3.68654 63804 17

--

3.82628 3.96651 - 4.10720 - 4.24832 - 4.38985

--

-

5.96893 91493 52

+ 0.00751 65191 79

-

-13.40736 36048 74

13.95449 36168 27

279

GAMMA FUNCTION AND RELATED FUNCTIONS

Table 6.7

GAMMA FUNCTIOY FOR COMPLEX ARGUMENTS

r=1.2 J In r(z)

9 In r(z)

Y

9In r ( z )

Y

4 In

r(z)

-

0.08537 0.09169 0.11050 0.14135 0.18352

40900 03 75124 13 89067 86 09532 62 07443 57

-

0.00000 0.02865 0.05586 0.08025 - 0.10066

00000 00 84973 21 39903 67 91592 09 05658 03

5.0 5.1 5.2 5.3 5.4

-

5.80731 5.95057 6.09410 6.23788 6.38192

52672 85 66519 39 47211 91 94064 81 11972 10

4.10609 4.26883 4.43349 4.60005 4.76847

64053 70 00575 53 40204 01 23089 91 02339 50

0.23614 0.29824 0.36884 0.44697 - 0.53174

32688 51 98509 35 83560 49 73864 90 22756 96

0.11610 0.12588 0.12948 0.12663 0.11720

77219 87 00935 13 68069 28 80564 16 77278 71

5.5 5.6 5.7 5.8 5.9

11003 82 06038 24 16425 98 65682 97 81209 15

4.93871 5.11075 5.28455 5.46008 5.63732

43339 56 23127 64 29803 68 61980 02 28266 55

-

0.62233 0.71803 0.81823 0.92237 1.03001

46814 87 95313 44 34133 20 79303 78 06294 86

0.10119 0.07868 0.04983 - 0.01483 + 0.02611

48344 90 85726 52 92764 14 57562 65 15201 47

6.0 6.1 6.2 6.3 6.4

7.25082 7.39636 7.54208 7.68798 7.83406

94030 54 38562 29 52390 70 76072 47 52949 57

5.81623 5.99679 6.17897 6.36275 6.54810

46788 41 44733 73 57929 16 30441 11 14200 83

1.5 1.6 1.7 1.8 1.9

-

--

6.52619 6.67069 6.81541 6.96034 7.10548

1.0 1.1 1.2 1.3 1.4

--

7.98031 8.12672 8.27329 8.42002 8.56690

28978 26 52570 99 74450 10 47512 17 26702 20

6.73499 6.92341 7.11333 7.30473 7.49759

68651 55 60416 24 62984 34 56416 32 27064 69

8.71392 8.86109 9.00839 9.15583 9.30340

68896 74 32795 24 78818 89 69016 37 66975 98

7.69188 7.88759 8.08470 8.28319 8.48303

67310 43 75313 86 54778 77 14729 22 69297 94

0.0 0.1 0.2 0.3 0.4

0.5 0.6 0.7 0.8 0.9

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 219 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0

-

- 1.14073 52341 62

--

-

1.25421 1.37015 1.48829 1.60844

22047 39 01536 37 83245 09 01578 57

0.07278 0.12495 0.18241 0.24494 0.31234

23932 61 51937 38 21090 01 25273 48 49712 35

6.5 6.6 6.7 6.8 6.9

1.73038 1.85397 1.97906 2.10553 2.23325

78680 93 79144 87 72374 32 01371 17 56848 33

0.38442 0.46101 0.54192 0.62700 0.71610

80719 73 09100 87 29484 31 37140 16 23338 39

7.0 7.1 7.2 7.3 7.4

2.36214 2.49211 2.62307 2.75497 2.88773

55727 43 23232 46 77928 95 19177 39 16568 77

0.80907 0.90579 1.00612 1.10996 1.21718

69945 69 43715 71 90561 43 29987 33 49784 62

7.5 7.6 7.7 7.8 7.9

-

9.45110 9.59892 9.74686 9.89492 -10.04309

37743 60 47746 01 64719 23 57641 38 96669 84

8.68422 8.88673 9.09055 9.29565 9.50203

37525 82 43171 55 14530 96 84265 39 89238 50

3.02130 3.15562 3.29066 3.42636 3.56269

00992 07 57049 65 16590 00 53170 56 77297 54

1.32769 1.44137 1.55815 1.67794 1.80064

01044 18 93510 29 91278 68 08829 56 07379 67

8.0 8.1 8.2 8.3 8.4

-10.19138 -10.33977 -10.48828 -10.63688 - 10.78559

53082 31 99221 46 08443 04 55067 01 14331 66

9.70967 9.91855 10.12866 10.33998 10.55250

70361 08 72443 36 44054 34 37387 77 08134 40

32317 85 90860 24 3.97512 51741 07 4.11364 37264 61 4.25263 90859 57

1.92617 2.05448 2.18547 2.31908 2.45526

91533 49 06211 84 33836 08 91746 67 29835 70

8.5 8.6 8.7 8.8 8.9

-10.93439 -11.08329 -11.23229 -11.38138 -11.53055

62350 38 76070 93 33237 11 12352 53 92646 46

10.76620 10.98107 11.19709 11.41426 11.63257

15360 05 21389 38 91694 76 94790 19 02129 90

4.39208 4.53196 4.67225 4.81293 4.95399

75003 42 69393 70 69332 23 84293 30 36651 50

2.59393 2.73503 2.87852 3.02434 3.17242

28374 55 96019 03 67976 01 04316 86 88424 26

9.0 9.1 9.2 9.3 9.4

-11.67982 -11.82917 -11.97861 -12.12813 -12.27773

54041 57 77123 44 43111 70 33832 78 31694 04

11.85198 12.07251 12.29413 12.51683 12.74059

88011 32 29482 35 06252 48 00607 77 97329 36

-

5.09540 5.23716 5.37924 5.52163 5.66433

60548 36 00880 20 12391 93 58863 97 12381 00

3.32274 3.47523 3.62985 3.78657 3.94533

25560 43 41545 72 81537 79 08902 31 04167 32

9.5 9.6 9.7 9.8 9.9

-12.42741 -12.57716 -12.72700 -12.87690 -13.02688

19659 29 81225 64 00401 42 61685 35 50046 68

12.96542 13.19130 13.41821 13.64615 13.87511

83615 35 49005 92 85311 47 86543 64 48849 16

-

5.80731 52672 85

4.10609 64053 70

10.0

-- 3.69962 3.83710

-

-

--

-

-13.17693 50906 38

14.10507 70446 23

280

GAMMA FUNCTION AND RELATED FUNCTIONS

GAMMA FUNCTION FOR COMPLEX ARGUMENTS

Table 6.7

,I-=

Y

r(2)

0.03225 84033 35 0.04549 95427 81 0.05544 82296 06

- 0.30082 34434 02 - 0.36553 39002 19 - 0.43754 53407 27 0.51609 74046 40

-

0.06126 0.06229 0.05805 0.04820 0.03257

78750 55 79103 48 28252 04 73993 35 37450 94

5.5 5.6 5.7 5.8 5.9

- 0.60048 45154 05

-

0.01107 52190 48

0.04941 70710 23 0.08822 25250 96 0.13255 01649 50

6.0 6.1 6.2 6.3 6.4

0.18223 0.23711 0.29699 0.36171 0.43110

70479 17 09920 47 65855 44 93463 93 85022 51

6.5 6.6 6.7 6.8 6.9

0.50499 0.58323 0.66565 0.75212 0.84250

87656 67 13926 09 47394 67 44759 30 35670 42

7.0 7.1 7.2 7.3 7.4

0.93666 1.03447 1.13583 1.24061 1.34872

21049 03 70464 53 18965 15 63628 56 60013 87

7.5 7.6 7.7 71 8 7.9

1.46006 1.57453 1.69204 1.81251 1.93586

18633 96 01525 07 18960 57 26335 69 21235 97

8.0 8.1 8.2 8.3 8.4

2.06201 2.19089 2.32243 2.45657 2.59324

40693 37 58627 45 83465 44 55932 86 47004 59

8.5 8.6 8.7 8.8 8.9

-10,72005 77580 15 -10.86779 71917 09 -11.01564 42292 16 -11.16359 64236 64 -11.31165 14105 63

10.91447 11.12944 11.34557 11.56283 11.78123

04638 39 32237 30 00727 24 79415 00 40512 20

2.73238 2.87394 3.01785 3.16407 3.31255

56006 34 08855 80 56433 48 73073 22 55163 23

9.0 9.1 9.2 9.3 9.4

-11.45980 -11.60806 -11.75641 -11.90485 -12.05339

69041 59 06939 74 06415 49 46773 52 07978 49

12.00074 12.22136 12.44306 12.66585 12.88971

59040 23 12739 31 81981 38 49686 64 01243 51

3.46324 3.61609 3.77105 3.92809 4.08717

19848 78 03828 59 62237 32 67607 19 08902 55

9.5 9.6 9.7 9.8 9.9

-12.20201 -12.35073 -12.49953 -12.64841 -12.79738

70627 34 15923 02 25649 49 82148 10 68295 12

13.11462 13.34058 13.56757 13.79559 14.02462

24431 99 09350 03 48342 95 35935 62 68767 33

4.24823 90621 27

10.0

- 0.10817 48095 08 61080 85 20636 90 10081 49 12771 78

0.5 0.6 0.7 0.8 0.9

-

0.24420 93680 45

0.11383 0.13070 0.15843 - 0.19649

-

1.0 1.1 1.2 1.3 1.4

-

1.5 1.6 1.7 1.8 1.9

- 1.08986 76158 16

-

2.0 2.1 2.2 2.3 2.4

- 1.65517 68709 10 - 1.77442 71431 91 - 1.89533 34239 28 - 2.01776 14331 34 - 2.14159 19646 87 - 2.26671 88222 04 - 2.39304 70725 18 - 2.52049 15659 37 - 2.64897 56799 18 - 2.77843 02497 03 - 2.90879 26554 06 - 3.04000 60402 26 - 3.17201 86387 60 - 3.30478 31979 94 - 3.43825 64765 05 - 3.57239 88099 07 - 3.70717 37325 19 - 3.84254 76469 59 - 3.97848 95346 95 - 4.11497 07016 98 - 4.25196 45543 38 - 4.38944 64012 12 - 4.52739 32778 30 - 4.66578 37904 84 - 4.80459 79774 65

3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0

9 In r(z)

-

In

-

3.0 3.1 3.2 3.3 3.4

.%In r(z)

Y 5.0 5.1 5.2 5.3 5.4

.@

0.0 0.1 0.2 0.3 0.4

2.5 2.6 2.7 2.8 2.9

In r(z)

1.3

0.69006 0.78427 - 0.88259 - 0.98458

1.19809 1.30898 1.42227 - 1.53773

62005 12 03001 02 13601 03 61322 90 86148 04 54162 82 19237 14 44011 63

- 4.94381 71850 33

-

5.08342 5.22340 5.36373 5.50441

39564 42 19323 94 57615 52 10199 31

5.64541 41381 33

J

0.00000 00000 00

- 0.01671 99199 34

+ 0.01627 90894 04

- 5.64541 41381 33

4.24823 4.41126 4.57620 4.74303 4.91171

90621 27 31957 95 66023 67 39118 17 10050 12

5.08220 5.25448 5.42851 5.60427 5.78172

49501 77 39434 72 72533 50 51684 12 89485 09

5.96085 6.14161 6.32399 6.50795 6.69349

07788 45 37268 52 17016 49 94158 99 23498 81

6.88056 7.06915 7.25924 7.45081 7.64382

67176 38 94350 45 80896 76 09123 38 67501 64

7.83827 8.03413 8.23138 8.43001 8.63000

50411 67 57901 50 95458 91 73795 19 08640 04

8.83132 9.03396 9.23790 9.44313 9.64964

20546 97 34708 43 80780 23 92714 58 08601 22

81051 96 08670 60 19248 88 84612 54

9.85739 10.06639 10.27661 10.48804 10.70066

70516 25 24378 12 19810 47 10011 24 51627 91

-

23355 37 35606 66 64370 51 02140 03

-

6.35492 47217 66

5.78673 5.92835 - 6.07026 - 6.21246

- 6.49765 03305 97

6.64062 79133 72 6.78384 88113 55 - 6.92730 48028 21

- 7.07098 80742 52

- 7.21489 11938 62 - 7.35900 70872 13 - 7.50332 90147 58 - 7.64785 05510 98 - 7.79256 55658 27 - 7.93746 82058 02 - 8.08255 28787 24 - 8.22781 42379 13 - 8.37324 71681 76 - 8.51884 67726 68 - 8.66460 83606 78 - 8.81052 74362 48 - 8.95659 96875 66 - 9.10282 09770 73 - 9.24918 73322 19 - 9.39569 49368 29 - 9.54234 01230 14 - 9.68911 93636 11 - 9.83602 92650 88 - 9.98306 65608 89 -10.13022 -10.27751 -10.42491 -10.57242

-12.94643 67480 34

14.25466 45529 28

281

GAMMA FUNCTION AND RELATED FUNCTIONS

Table 6.7

GAMMA FUNCTION FOR COMPLEX ARGCMENTS

r=1.4 Y

&ln

r(z)

P In r(z)

alnr(z)

Y

f In r(z)

-

0.12473 0.14000 0.16515 0.19978

21357 76 01552 88 59551 89 93616 12

-

00000 00 40017 43 08056 66 93840 03 47989 49

5.0 5.1 5.2 5.3 5.4

-

5.48319 5.62258 5.76231 - 5.90236 - 6.04272

80511 50 51037 75 08530 59 26637 68 85898 90

4.38842 4.55177 4.71703 4.88416 5.05313

59888 87 72808 10 54898 14 59286 80 51119 86

0.24337 0.29530 0.35492 0.42158 - 0.49462

34438 09 16779 62 46161 10 20669 55 85345 46

- 0.01124 72025 18 - 0.00401 77865 38 0.00775 78473 84 0.02441 65124 32 0.04618 11610 42

5.5 5.6 5.7 5.8 5.9

-

6.18339 6.32435 6.46560 6.60711 6.74889

73257 62 81614 11 09417 01 60288 99 42683 24

5.22391 5.39646 5.57075 5.74676 5.92446

06968 84 14275 35 70829 41 84279 33 71670 92

1.0 1.1 1.2 1.3 1.4

-

0.57345 0.65748 0.74620 0.83914 0.93588

12921 03 16506 41 06322 98 04638 04 32199 21

0.07317 0.10545 0.14300 0.18575 0.23362

82199 73 58409 92 11986 37 57618 52 80933 40

6.0 6.1 6.2 6.3 6.4

-

6.89092 7.03320 7.17572 7.31847 7.46144

69567 80 58135 18 29534 78 08625 98 23750 25

6.10382 6.28481 6.46741 6.65160 6.83734

59013 94 80874 01 79988 09 06901 96 19628 28

1.5 1.6 1.7 1.8 1.9

-

1.03605 1.13933 1.24542 1.35407 1.46505

77156 27 54742 88 63479 49 41615 64 26007 14

0.28650 0.34425 0.40674 0.47383 0.54537

41540 26 53337 92 45404 a7 07041 21 20299 26

6.5 6.6 6.7 6.8 6.9

- 7.60463 06520 25

1.57816 1.69322 1.81008 1.92859 2.04863

14562 85 32702 19 03838 54 23663 09 37884 08

0.62122 0.70126 0.78534 0.87333 0.96512

82885 81 23803 49 13608 50 70735 61 64991 00

7.0 7.1 7.2 7.3 7.4

- 2.17009 23032 73

0.0 0.1 0.2 0.3 0.4

0.5 0.6 0.7 0.8 0.9

2.0 2.1 2.2 2.3 2.4

- 0.11961 29141 72

+

0.00000 0.00597 0.01097 0.01405 0.01439

-

-

7.74802 7.89163 8.03543 8.17942

91624 64 16647 23 21899 02 50262 34

7.02461 7.21340 7.40368 7.59543 7.78862

83323 73 69984 03 58155 67 32663 20 84351 12

8.32360 8.46796 8.61250 8.75721 8.90209

47045 82 59849 44 38438 82 34627 90 02169 54

7.98325 8.17928 8.37669 8.57548 8.77562

09839 40 11291 83 96196 29 77156 28 71692 98

3.04712 96653 17

-

9.48318 23233 58 9.62883 14889 78

8.97710 9.17988 9.38397 9.58934 9.79598

02057 23 95050 80 81856 34 97875 68 82575 76

- 9.77462 35841 76

2.29286 2.41686 2.54201 2.66822

69947 17 69570 58 00734 84 19640 86

1.06059 1.15962 1.26210 1.36794 1.47704

19035 92 08468 95 60952 18 54704 02 16591 47

7.5 7.6 7.7 7.8 7.9

2.79543 2.92358 3.05262 3.18249 3.31314

50784 95 79116 75 43245 92 29542 71 67001 61

1.58930 1.70463 1.82296 1.94420 2.06828

19987 43 82510 60 63729 35 62885 89 16678 1 0

8.0 8.1 8.2 8.3 8.4

-

9.92055 -10.06662 -10.21282 -10.35915

50889 05 26112 05 28810 76 27447 20

10.00387 10.2130il 10.42335 10.63490 10.84764

79341 91 35337 97 01372 94 31773 72 84263 58

3.44454 3.57663 - 3.70940 - 3.84279 - 3.97678

22757 38 98160 21 25331 00 64130 02 99482 49

2.19511 2.32465 2.45680 2.59153 2.72875

97123 13 09517 70 90502 77 06235 98 50671 88

8.5 8.6 8.7 8.8 8.9

-10.50560 -10.65218 -10.79888 -10.94570 -11.09264

91591 10 91868 81 99915 05 88327 39 30623 27

11.06157 11.27666 11.49290 11.71027 11.92878

19846 19 02694 74 00045 92 82098 57 21916 70

4.0 4.1 4.2 4.3 4.4

-

39012 79 10946 69 62246 51 56949 47 74683 75

2.86842 3.01048 3.15487 3.30155 3.45047

43947 56 30870 18 79501 77 79836 24 42563 18

9.0 9.1 9.2 9.3 9.4

-11.23969 -11.38684 -11.53411 -11.68148 -11.82895

01199 39 75293 27 28946 97 38972 65 82920 01

12.14839 12.36911 12.59092 12.81381 13.03776

95336 59 80877 89 59658 40 15312 39 33912 29

4.5 4.6 4.7 4.8 4.9

- 4.79184 09340 18

3.60157 3.75482 3.91017 4.06758 4.22701

97913 33 94580 13 98712 52 92973 81 75662 27

9.5 9.6 9.7 9.8 9.9

-11.97653 -12.12420 -12.27198 -12.41984 -12.56780

39045 38 86282 47 04214 52 73048 02 73587 55

13.26277 13.48882 13.71590 13.94401 14.17313

03893 53 15982 45 63127 03 40430 46 45087 16

4.38842 59888 87

10.0

2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

5.0

-

-

4.11135 4.24646 4.38208 4.51820 4.65479 4.92931 5.06720 5.20549 5.34416

67880 70 69267 30 43497 23 30732 30

5.48319 80511 50

- 9.19232 75409 21 - 9.33767 97419 53

-12.71585 87212 03

14.40325 76321 42

282

GAMMA FUNCTION AND RELATED FUNCTIONS

GAMMA FUNCTION FOR COMPLEX ARGUMENTS

Table 6.7

r=1.5 d In r ( z )

Y

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1: 6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

--

4 In r(z)

00229 09 92990 12 21987 69 48816 77 42552 05

4.52667 4.69038 4.85599 5.02345 5.19275

02683 19 46594 51 23475 89 93914 30 29984 42

6.01157 6.15078 6.29030 6.43012 6.57022

79223 61 41337 33 17435 55 01693 96 93551 39

5.36384 5.53669 5.71128 5.88757 6.06554

14702 24 41510 65 13794 95 44426 18 55330 63

6.71061 6.85128 6.99220 7.13338 7.27481

97369 14 22117 36 81085 67 91616 09 74856 07

6.24516 6.42641 6.60926 6.79368 6.97965

77083 65 48526 40 16403 83 35022 65 65928 01

7.41648 7.55838 7.70051 7.84285 7.98541

55529 97 61727 29 24706 26 78711 49 60804 40

7.16715 7.35616 7.54665 7.73860 7.93200

77597 60 45152 22 50081 65 79984 87 28323 86

7.0 7.1 7.2 7.3 7.4

--

8.12818 8.27114 8.41430 8.55766 8.70119

10705 51 70647 52 85238 40 01333 52 67916 34

8.12681 8.32303 8.52064 8.71960 8.91991

94190 02 82082 45 01697 48 67728 67 99676 60

90329 08 91860 05 69784 97 85397 98 39035 70

7.5 7.6 7.7 7.8 7.9

- 8.84491 35986 81 - 8.98880 58456 98 - 9.13286 90053 22 - 9.27709 87224 65 -

9.12156 9.32451 9.52876 9.73429 9.94108

21668 12 62284 17 54395 97 35008 92 45113 82

0.00000 0.00378 0.00839 0.01460 0.02315

00000 00 68415 10 39012 17 80536 11 34211 15

5.0 5.1 5.2 5.3 5.4

0.23418 0.28208 0.33728 0.39923 0.46739

63474 70 36136 63 34790 33 54301 20 08704 08

0.03466 0.04969 0.06866 0.09191 0.11969

89612 75 46638 36 64150 66 83319 43 06415 60

5.5 5.6 5.7 5.8 5.9

0.54121 0.62021 0.70391 0.79189 0.88375

88685 47 70896 71 84698 97 44573 28 56946 74

0.15214 0.18935 0.23137 0.27816 0.32969

09934 52 73091 01 07067 73 75270 32 99180 52

6.0 6.1 6.2 6.3 6.4

0.97915 1.07776 1.17931 1.28355 - 1.39024

09391 81 48736 47 53061 81 01134 19 41643 92

0.38589 0.44666 0.51189 0.58148 0.65532

47712 67 10201 49 54441 75 71805 09 11610 93

6.5 6.6 6.7 6.8 6.9

-

1.49919 1.61022 1.72317 1.83789 1.95426

63725 85 69592 23 49667 28 60327 96 04180 71

0.73328 0.81524 0.90111 0.99075 1.08407

06816 91 92850 60 21116 92 68430 94 43370 92

2.07215 12706 83

-

2.43397 68277 27 2.55701 34593 17

1.18095 1.28130 1.38502 1.49201 1.60219

- 2.19146 31061 38 - 2.31210 04795 77

9 In r(z)

5.32063 5.45809 5.59594 5.73414 5.87269

22376 35 03928 11 53175 79 37050 76 35254 45

-

r(z)

-

0.12078 0.12545 0.13938 0.16238 0.19412

--

9 In

Y

-

9.42149 08057 13

3.0 3.1 3.2 3.3 3.4

- 2.68113 86746 74 2.80628 2.93239 3.05941 3.18729

69972 89 85022 62 82284 63 56630 57

1.71546 1.83175 1.95097 2.07306 2.19793

69204 67 51411 18 96800 61 50684 28 91011 06

8.0 8.1 8.2 8.3 8.4

-

9.56604 9.71074 9.85560 -10.00060 -10.14575

12192 67 60753 60 16271 36 42619 46 04950 41

10.14912 10.35839 10.56888 10.78057 10.99345

29545 01 36845 06 19135 53 31993 69 34334 60

3.5 3.6 3.7 3.8 3.9

3.31598 3.44544 3.57562 3.70650 - 3.83803

42885 64 11840 65 66733 10 40135 44 91197 27

2.32553 2.45577 2.58861 2.72398 2.86182

26824 38 96733 92 67421 82 32197 35 09608 36

8.5 8.6 8.7 8.8 8.9

-10.29103 -10.43646 -10.58201 -10.72770 -10.87352

69636 22 04212 40 77325 09 58681 09 19000 77

11.20750 11.42272 11.63909 11.85659 12.07521

88298 51 59143 12 15140 53 27478 60 70166 56

- 3.97020 03195 93

4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0

--

-

4.10295 4.23628 4.37015 4.50454

81356 26 50905 75 55336 09 54845 89

3.00207 3.14468 3.28961 3.43680 3.58620

42115 08 94828 47 54314 23 27461 51 40415 07

9.0 9.1 -~ 9.2 9.3 9.4

-11.01946 -11.16552 -11.31170 -11.45800 -11.60442

29973 44 64215 28 95229 33 97367 84 45796 38

12.29495 12.51578 12.73770 12.96070 13.18476

19944 46 56196 58 60868 20 18385 99 15581 47

4.63943 4.77479 4.91061 5.04687 5.18354

24943 00 55187 51 48059 11 17934 63 90163 32

3.73777 3.89146 4.04724 4.20506 4.36488

37568 62 80616 79 47663 05 32380 55 43223 09

9.5 9.6 9.7 9.8 9.9

-11.75095 -11.89758 -12.04433 -12.19118 -12.33813

16459 94 86050 76 31977 78 32337 59 65886 95

13.40987 13.63602 13.86321 14.09142 14.32063

41617 61 87918 31 48100 75 17910 27 95157 82

4.52667 02683 19

10.0

5.32063 00229 09

-12.48519 12016 51

14.55085 79659 84

283

GAMMA FUNCTION AND RELATED FUNCTIONS

G)AMMA FUNCTION FOR COMPLEX ARGUMENTS

Table 6.7

.~=1.6 Ye In r(z)

Y 0.0 0.1 0.2 0.3 0.4

-

0.11259 0.11687 0.12968 0.15085 0.18012

-

0.37068 83847 40 0.43448 55339 80

9 In r(z)

In r(z)

Y

-

17656 97 93076 07 70233 13 38452 14 29875 82

0.00000 0.01272 0.02614 0.04092 0.05771

00000 00 17953 11 08547 67 98346 69 47266 93

5.0 5.1 5.2 5.3 5.4

--

38696 89 00046 70 38858 50 05247 67 57347 10

4.66298 4.82709 4.99309 5.16092 5.33057

63139 40 89421 23 00410 26 64732 77 61938 29

0.21715 76591 72

0.07705 0.09944 0.12527 0.15488 0.18851

74009 90 39491 75 90746 90 59553 99 04588 87

5.5 5.6 5.7 5.8 5.9

- 5.83943 60752 49

5.50200 5.67519 5.85009 6.02670 6.20497

82001 33 24850 30 99922 08 25740 71 29518 79

21960 58 58588 66 82809 44 95833 61 01661 20

0.22632 0.26845 0.31495 0.36583 0.42110

83631 44 42738 89 11405 00 95580 78 63293 75

6.0 6.1 6.2 6.3 6.4

0.91903 1.01326 1.11052 1.21057 1.31318

10002 05 27864 52 43845 66 08228 70 11150 50

0.48071 0.54459 0.61268 0.68490 0.76114

20031 31 72874 22 83586 73 11588 51 48080 60

6.5 6.6 6.7 6.8 6.9

1.41815 1.52531 1.63449 1.74555 1.85836

60399 85 59861 47 89215 98 85219 99 24696 22

0.84132 0.92534 1.01310 1.10450 1.19945

42695 09 23984 61 14934 56 44515 88 56127 07

7.0 7.1 7.2 7.3 7.4

7.93255 8.07413 - 8.21592 - 8.35792 - 8.50012

1.97279 2.08873 2.20609 2.32478 2.44471

09238 15 51557 24 63358 10 44606 95 74052 94

1.29786 1.39963 1.50467 1.61290 1.72424

13618 36 05453 39 47448 81 84436 93 91120 48

7.5 7.6 7.7 7.8 7.9

1.83861 1.95593 2.07613 2.19913 2.32487

72327 21 62824 65 26817 55 57221 55 74784 17

8.0 8.1 8.2 8.3 8.4

- 8.78511 32665 62 - 8.92789 17384 38 - 9.07085 27813 87 - 9.21399 18168 02 - 9.35730 44352 92 - 9.50078 63884 89 - 9.64443 35813 39 - 9.78824 20648 48 - 9.93220 80292 58

3.56955 42495 22 3.69866 25626 62

2.45329 2.58431 2.71789 2.85396 2.99247

27106 82 87608 00 54457 96 49506 80 17222 46

8-5 8.6 8.7 8.8 8.9

-10.07632 -10.22059 -10.36501 -10.50957 -10.65427

3.82845 3.95888 4.08994 4.22158 4.35379

04545 47 63415 67 07464 23 61190 90 66759 32

3.13336 3.27658 3.42209 3.56983 3.71976

23649 89 55399 89 18672 73 38320 36 56948 92

9.0 9.1 9.2 9.3 9.4

4.5 4.6 4.7 4.8 4.9

-- 4.48654 82548 65 4.61981 81847 38 - 4.75358 51673 33 91705 81 -- 4.88782 5.02253 13317 74

3.87184 4.02602 4.18226 4.34053 4.50078

34062 62 45248 92 81404 46 48000 81 64388 72

9.5 9.6 9.7 9.8 9.9

5.0

-

4.66298 63139 40

10.0

0.5 0.6 0.7 0.8 0.9

- 0.26155 99560 50 - 0.31289 07142 69

1.0 1.1 1.2 1.3 1.4

0.50382 0.57825 - 0.65736 - 0.74076 - 0.82810

1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4

-

-

--

-

--

--

- 2.56582 00865 46 37258 40 -- 2.68802 2.81126 51983 53 - 2.93548 64586 59 - 3.06063 40331 69 -- 3.18665 85710 48 3.31351 44463 00 - 3.44115 94046 31 -

--

-

5.15767 38696 89

-

5.15767 5.29324 5.42921 5.56558 5.70232

9 In r(z)

- 5.97689 88014 04 - 6.11470 18170 24 - 6.25283 36319 59 - 6.39128 33226 66 - 6.53004 04959 33 - 6.66909 52554 28 - 6.80843 81708 20 - 6.94806 02492 33 - 7.08795 29088 41 - 7.22810 79544 00 - 7.36851 75545 64 - 7.50917 42208 19 - 7.65007 07879 17 - 7.79120 03956 68

6.38488 46780 37 6.56641 21003 90 6.74953 03284 11 6.93421 52011 79 7.12044 32570 25 7.30819 7.49743 7.68816 7.88034 8.07395

17047 52 83963 44 18010 64 09808 67 55670 43

64719 90 27171 08 30888 20 17887 32 32493 99

8.26898 8.46541 8.66321 8.86237 9.06288

57380 27 21983 05 61583 45 93155 10 38358 78

- 8.64252 21222 97

9.26471 9.46784 9.67227 9.87797 10.08493

23369 30 78710 61 39098 48 43290 61 33943 44

10.29313 10.50256 10.71321 10.92505 11.13808

57475 61 63937 51 06886 60 43268 31 33302 08

77975 98 78196 20 46660 67 50232 55 56879 66

11.35228 11.56764 11.78414 12.00178 12.22054

40372 42 30924 55 74364 58 42963 80 11767 06

-10.79911 -10.94408 -11.08918 -11.23442 -11.37977

35626 11 56506 53 90522 76 09602 86 86562 21

12.44040 12.66136 12.88341 13.10652 13.33070

58504 89 63509 22 09632 56 82170 40 68786 75

-11.52525 -11.67086 -11.81658 -11.96241 -12.10836

95066 64 09597 45 05418 21 58543 24 45707 60

13.55593 13.78220 14.00950 14.23781 14.46714

59442 57 46327 06 23791 60 88286 23 38298 57

--

-12.25442 44338 60

14.69746 74295 03

284

GAMMA FUNCTION AND RELATED FUNCTIONS

Table 6.7

GAMMA FUNCTION FOR COMPLEX ARGUMENTS

.~=1.7 d l r l r(z)

In r(z)

0.0 0.1 0.2 0.3 0.4

-

0.09580 76974 07 0.09977 01624 55 - 0.11161 35203 43 0.13120 82417 20 - 0,15834 67099 43

0.00000 0.02095 0.04250 0.06524 0.08970

00000 00 53101 47 99781 99 48506 20 54480 34

Y 5.0 5.1 5.2 5.3 5.4

0.5 0.6 0.7 0.8 0.9

-

Y

-

J

0.19275 0.23410 0.28203 0.33614 0.39604

44989 43 41754 11 01468 30 32007 35 36829 33

0.11638 0.14573 0.17810 0.21382 0.25312

82473 83 09476 06 70108 82 42284 85 66649 29

5.5 5.6 5.7 5.8 5.9

0.46133 0.53162 0.60653 0.68572 0.76884

26441 19 06562 78 43029 30 05552 37 93610 19

0.29619 0.34317 0.39413 0.44912 0.50817

91243 57 32455 42 44205 39 88915 80 05624 82

6.0 6.1 6.2 6.3 6.4

-

0.85561 0.94573 1.03895 1.13503 1.23375

48134 32 52538 42 26210 76 13039 83 66975 90

0.57124 0.63832 0.70935 0.78428 0.86303

72307 84 60866 03 84280 02 36123 89 23052 04

6.5 6.6 6.7 6.8 6.9

2.0 2.1 2.2 2.3 2.4

--

1.3?493 1.43838 1.54394 1.65147 1.76084

36116 09 46369 05 85411 53 87389 10 18623 15

0.94552 1.03169 1.12144 1.21470 1.31138

91079 51 46541 37 72591 94 42030 73 27144 41

7.0 7.1 7.2 7.3 7.4

2.5 2.6 2.7 2.8 2.9

-

1.87191 1.98459 2.09876 2.21434 2.33125

64452 44 17246 80 65571 99 84448 82 26629 53

1.41140 07152 26 1,51467 73744 45 1.62113 35114 76 1.73069 18813 34 1.84327 73680 71

7.5 7.6 7.7 7.8 7.9

2.44940 2.56872 2.68915 2.81062 2.93309

14805 61 34658 89 28670 03 90603 59 60594 79

1.95881 2.07724 2.19847 2.32246 2.44914

71071 34 05531 98 95064 74 81077 41 28100 87

8.0 8.1 8.2 8.3 8.4

3.05650 3.18079 3.30594 3.43189 3.55860

20770 24 91341 33 27115 93 14379 84 68105 24

2.57844 2.71030 2.84468 2.98150 3.12073

23336 16 76079 67 17064 22 97744 80 89551 42

3.68605 3.81419 3.94300 4.07245 4.20250

29448 47 63503 82 57284 13 17902 59 70933 22

3.26231 3.40619 3.55233 3.70067 3.85118

4.33314 4.46434 4.59607 4.72832 4.86107

58930 01 40087 52 87027 47 85697 79 34372 26

4.00380 4.15851 4.31526 4.47402 4.63474

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0

-

-

--

4.99429 42740 24

%In r(z)

-

I In r(z)

4.99429 5.12797 5.26209 5.39663 5.53159

42740 24 31077 01 29486 79 77210 79 21994 12

4.79738 4.96193 5.12834 5.29658 5.46661

98064 85 49448 28 25830 88 04404 97 72692 91

5.66694 5.80267 5.93877 6.07522 - 6.21202

19505 53 32805 14 31855 28 93070 61 98903 76

5.63842 5.81196 5.98722 6.16416 6.34275

28098 55 77481 03 36749 88 30480 45 91548 66

6.90082 60067 27

6.52298 6.70481 6.88823 7.07320 7.25970

60784 05 86640 24 24881 89 38287 20 96365 25

-

-

--- 6.48662 6.34916 37463 25 02160 75 -- 6.62438 91385 04 6.76246 08200 42 -

- 7.03947 58582 98 7.17840 7.31759 7.45705 7.59675

19241 47 61209 77 07120 18 82876 82

7.44772 7.63723 7.82821 8.02063 8.21449

75087 22 56630 84 29137 39 86480 35 28045 37

7.73671 7.87690 8,01732 8.15798 8.29885

17475 34 42834 81 93640 69 07198 22 23295 23

8.40975 8.60640 8.80443 9.00381 9.20452

58520 62 87697 25 30279 13 05701 63 37958 73

- 8.43993 84073 80

-

8.58123 8.72273 8.86442 9.00631

33910 02 19301 22 88760 30 92716 38

9.40655 9.60988 9.81450 10.02039 10.22753

55438 14 90763 93 80646 38 65738 46 90498 84

9.14839 9.29066 9.43310 9.57572 9.71851

83421 51 14862 98 42680 75 24089 73 17806 54

10.43592 10.64552 10.85634 11.06835 11.28154

03060 85 55107 28 01750 59 01418 23 15743 00

8.5 8.6 8.7 8.8 8.9

9.86146 83980 47 -10,00458 84128 32 -10.14786 81072 85 -10.29130 38884 74 -10.43489 22827 58

11.49590 11.71141 11.92807 12.14585 12.36475

09457 89 50295 52 08891 58 58692 46 75866 47

83125 99 87555 93 29614 33 53013 46 17677 02

9.0 9.1 9.2 9.3 9.4

-10,57862 99305 96 -10.72251 35816 27 -10.86654 00900 14 -11.01070 64100 32 -11.15500 95918 83

12.58476 12.80586 13.02804 13.25129 13.47560

39218 81 30109 93 32377 08 32259 06 18323 86

99034 45 87339 90 87017 23 16031 94 05290 18

9.5 9.6 9.7 9.8 9.9

-11.29944 -11.44401 -11.58871 -11.73353 -11.87848

13.70095 13.92735 14.15477 14.38320 14.61264

81399 16 14505 47 12791 90 73474 23 95775 51

4.79738 98064 85

10.0

--

67777 28 51979 25 21674 47 50824 91 14J72 43

-12.02354 &208 09

14.84308 80868 68

285

GAMMA FUNCTION AND RELATED FUNCTIONS

GAMMA FUNCTION FOR COMPLEX .4RGUMENTS

Table 6.7

~=1.8 2!31r ( z )

Y

X In r ( r )

0.0 0.1 0.2 0.3 0.4

-

38729 14 57386 86 55297 09 76857 32 22486 30

0.00000 0.02858 0.05769 0.08782 0.11946

00000 00 63331 36 29209 31 58538 91 40495 57

5.0 5.1 5.2 5.3 5.4

0.5 0.6 0.7 0.8 0.9

0.16140 31015 52 -- 0.20006 82029 53 0.24498 08149 51

83729 82 35429 70 31014 17 73612 58 39650 50

5.5 5.6 5.7 5.8 5.9

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0

--

0.07108 0.07476 0.08577 0.10400 0.12929

--

0.29581 07721 71 0.35221 50054 25

0.15304 0.18897 0.22758 0.26916 0.31396

0.41384 0.48036 0.55143 0.62673 0.70596

67690 74 32669 52 15880 74 30272 43 59713 03

0.36216 0.41389 0.46927 0.52836 0.59119

05120 09 86472 00 90315 88 66950 54 63857 23

6.0 6.1 6.2 6.3 6.4

-

0.78884 0.87511 0.96452 1.05684 1.15188

75850 80 45440 57 30468 26 83111 80 37223 02

0.65777 0.72809 0.80213 0.87984 0.96117

76436 65 94297 11 42229 48 15616 08 10434 30

6.5 6.6 6.7 6.8 6.9

1.24943 1.34934 1.45143 1.55556 1.66161

97659 29 28469 99 40669 35 80105 11 15761 22

1.04606 1.13445 1.22628 1.32148 1.41997

48267 65 96865 98 86841 72 25078 65 05387 49

7.0 7.1 7.2 7.3 7.4

1.76944 1.87895 1.99003 2.10259 2.21654

28703 84 01786 38 10163 61 12619 95 43688 12

1.52168 1.62654 1.73449 1.84544 1.95935

16884 90 50508 69 04020 35 85788 28 17594 45

7.5 7.6 7.7 7.8 7.9

2.07613 2.19572 2.31807 2.44311 2.57078

36663 29 97074 49 70690 52 47704 17 36890 62

8.0 8.1 8.2 8.3 8.4

-

--

-

--- 2.33181 06516 27 2.44831 66432 13 2.56599 45147 78 --- 2.68478 15548 41 2.80461 97009 53

2 In r(z)

! I

-

-

4.83045 4.96226 5.09454 5.22728 5.36046

68451 13 53555 54 72216 70 53433 89 35143 73

4.92989 5.09490 5.26176 5.43043 5.60088

76263 84 86275 80 50781 04 56009 62 97905 12

5.77309 5.94703 6.12266 6.29996 6.47891

81726 78 21669 16 40498 86 69207 68 46681 58

6.71341 45046 23

6.65948 6.84164 7.02537 7.21065 7.39746

19384 99 41059 65 72437 42 80966 53 40550 43

6.85056 6.98801 7.12575 7.26377 7.40207

76090 92 82204 65 76814 17 77029 87 03441 98

7.58577 7.77556 7.96681 8.15950 8.35362

31298 85 39290 39 56346 11 79813 46 12360 30

7.54062 7.67944 7.81850 7.95781 8.09736

79930 63 33488 49 94055 06 94361 78 69787 03

8.54913 8.74603 8.94429 9.14390 9.34484

61778 15 40794 54 66893 74 62145 64 53042 25

8.23714 8.37714 8.51737 8.65781 8.79845

58220 35 99935 16 37469 39 15513 42 80804 75

9.54709 9.75064 9.95547 10.16156 10.36890

70341 42 48917 54 27618 74 49130 30 59844 02

10.57748 10.78727 10.99827 11.21046 11.42383

09733 12 52232 56 44124 32 45427 62 19293 59

- 5.49406 63619 68 - 5.62807 92920 13 -

5.76248 84380 56 5.89728 06145 63 6.03244 32737 64

44658 02 - 6.16796 - 6.30383 28019 05 - 6.44003 74202 92 - 6.57656 79546 04

-

9 In r(z)

8,93930 82029 08 --- 9.22159 9.08035 69727 14 96207 08 -- 9.36303 15461 81 9.50464 83091 20

-

2.92545 3.04723 3.16992 3.29346 3.41782

51190 19 78253 42 13469 31 24159 89 06949 39

2.70102 2.83378 2.96901 3.10665 3.24665

65631 50 79764 90 43304 05 38058 79 63186 51

8.5 8.6 8.7 8.8 8.9

9.64644 9.78841 9.93056 -10.07288 -10.21535

56228 63 93471 63 54816 43 01596 06 96421 85

11.63836 11.85404 12.07086 12.28881 12.50786

31904 38 52376 37 52667 34 07487 37 94213 31

-

3.54295 3.66884 3.79543 3.92270 4.05063

85286 89 07212 13 43338 26 85028 21 42744 24

3.38897 3.53355 3.68036 3.82935 3.98047

34693 93 84906 21 61916 47 29025 75 64181 31

9.0 9.1 9.2 9.3 9.4

-10.35800 03128 01 -10.50079 86719 24 -10.64375 13321 05 -10,78685 50132 67 -10.93010 65382 43

12.72802 12.94927 13.17160 13.39499 13.61944

92806 69 85734 79 57894 90 96541 43 91215 87

4.17918 4.30833 4.43805 4.56833 4.69913

44552 05 34763 48 72703 06 31585 96 97495 61

4.13369 4.28897 4.44626 4.60554 4.76676

59419 14 20315 17 65448 66 25879 92 44644 38

9.5 9.6 9.7 9.8 9.9

-11.07350 -11.21704 -11.36071 -11.50453 -11.64847

13.84494 14.07147 14.29902 14.52758 14.75715

33679 42 17848 17 39730 75 97368 21 90776 29

4.92989 76263 84

10.0

---

-

4.83045 68451 13

-

28285 39 09003 12 78605 47 09034 33 73069 06

-11.79255 44293 69

14.98772 21889 61

286

GAMMA FUNCTION AND RELATED FUNCTIONS

GAMMA FUNCTIOX FOR COMPLEX ARGUMENTS

Table 6.7

2=1.9 %lnr(z)

Y

9 In r(z)

0.03898 0.04242 0.05270 0.06974 0.09340

42759 23 16648 18 43596 13 53071 16 38158 25

0.00000 0.03569 0.07184 0.10889 0.14726

00000 00 47077 36 49288 73 51730 33 87453 39

Y 5.0 5.1 5.2 5.3 5.4

0.12349 0.15978 0.20201 0.24990 0.30315

16727 26 08372 30 20244 82 35004 09 95035 34

0.18735 0.22952 0.27407 0.32128 0.37139

90383 60 28050 02 56544 06 97690 64 36389 55

5.5 5.6 5.7 5.8 5.9

0.36147 0.42455 0.49209 0.56381 0.63943

78527 10 64621 11 86372 39 71504 20 71834 98

0.42457 0.48097 0.54071 0.60385 0.67046

34706 81 58618 37 13247 70 82827 52 72268 81

6.0 6.1 6.2 6.3 6.4

1.5 1.6 1.7 1.8 1.9

0.71869 0.80135 - 0.88717 - 0.97595 - 1.06749

82795 42 54698 30 97447 03 80247 42 27687 53

0.74056 0.81415 0.89123 0.97177 1.05574

47971 47 76239 52 58296 55 61401 47 45936 43

6.5 6.6 6.7 6.8 6.9

2.0 2.1 2.2 2.3 2.4

-

- 1.16160 13318 68 1.25811 51641 83 1.35687 89195 14 -- 1.45774 95259 72 1.56059 52554 63

1.14309 1.23379 1.32776 1.42496 1.52533

88592 34 01934 57 50714 39 65323 75 52787 28

2.5 2.6 2.7 2.8 2.9

--

48176 11 64947 51 73280 00 23595 80 39332 16

1.62881 1.73533 1.84483 1.95726 2.07254

10520 42 -- 2.21298 2.32673 87919 77 2.44172 77675 72 - 2.55788 36468 15

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0

-

--

-

9 In r ( z )

--

4.66612 4.79608 4.92654 5.05749 5.18891

81728 77 44074 24 53878 64 30552 47 02823 51

5.06052 5.22603 5.39337 5.56250 5.73340

77830 38 70297 75 36626 27 72499 47 82679 93

5.32078 5.45308 5.58582 5.71896 5.85249

08121 05 92008 98 07663 21 15389 41 82177 50

5.90604 6.08039 6.25643 6.43412 6.61345

80662 49 88340 38 35684 02 60432 49 07797 49

5.98641 6.12070 6.25535 6.39035 6.52569

81289 78 91879 56 98637 85 91465 66 65169 71

6.79438 6.97689 7.16097 7.34658 7.53371

30179 35 86894 96 43917 16 73625 14 54565 59

6.66136 6.79734 6.93363 7.07023 7.20711

19179 75 57285 54 87392 01 21291 12 74449 04

7.72233 7.91243 8.10397 8.29695 8.49134

71224 13 13806 57 78029 64 64920 80 80626 65

7.0 7.1 7.2 7.3 7.4

---

7.34428 7.48173 7.61944 7.75742 7.89565

65807 56 17598 49 55170 18 06825 11 03667 87

8.68713 8.88429 9.08281 9.28267 9.48385

36229 72 47573 07 35092 45 23655 74 42409 11

05662 06 09179 80 46926 69 05315 67 77068 08

7.5 7.6 71 7 7.8 71 9

-

8.03412 8.17284 8.31180 8.45098 - 8.59039

79462 62 70499 43 15468 79 55343 75 33269 14

9.68634 9.89012 10.09517 10.30148 10.50903

24629 88 07585 45 32398 33 43916 76 90590 64

2.67514 67111 48

2.19063 2.31146 2.43498 2.56113 2.68985

63887 13 78475 36 46022 00 05263 98 09205 60

8.0 8.1 8.2 8.3 8.4

10.71782 10.92782 11.13901 11.35140 11.56495

24352 78 00504 91 77608 39 17379 39 84588 29

2.79346 2.91277 3.03304 3.15421 3.27625

14569 24 62346 38 29224 14 66305 10 54337 96

2.82109 2.95480 3.09093 3.22943 3.37025

25566 19 37012 40 41220 91 50808 91 93162 16

8.5 8.6 8.7 8.8 8.9

- 9.43124 64604 23 9.57207 9.71309 9.85429 9.99566

78935 85 43338 13 16464 97 58330 75

11.77967 11.99553 12.21253 12.43065 12.64988

46963 13 75096 87 42358 42 24807 06 01110 27

3.39912 3.52277 3.64718 3.77231 3.89813

01294 42 40173 08 27007 49 39057 84 73167 71

3.51336 3.65869 3.80622 3.95589 4.10767

10185 24 57993 21 06560 50 39339 63 52859 66

9.0 9.1 9.2 9.3 9.4

-10.13721 -10.27892 -10.42081 -10.56285 -10.70505

30251 72 94790 52 15703 58 57891 26 87350 54

12.87020 13.09161 13.31410 13.53765 13.76225

52464 75 62520 42 17307 41 05165 78 16677 85

4.02462 4.15174 4.27948 4.40780 4.53669

44269 53 84023 59 39577 56 72434 44 57418 38

4.26152 4.41740 4.57528 4.73511 4.89687

56312 41 71132 72 30577 67 79308 60 72979 01

9.5 9.6 9.7 9.8 9.9

-10.84741 -10.98992 -11.13258 -11.27539 -11.41834

71130 08 77287 64 74849 48 33771 93 24904 66

13.98789 14.21456 14.44226 14.67096 14.90067

44603 16 83815 73 31243 75 85811 36 48382 65

5.06052 77830 38

10.0

--

-

9 In r(z)

1.66529 1.77173 1.87981 1.98944 2.10052

4.66612 81728 77

-

-- 8.73001 94457 32 8.86985 86090 10 -

9.00990 57226 31 9.15015 58714 69 9.29060 43111 75

-11.56143 19955 88

15.13137 21707 60

287

GAMMA FUNCTION A N D RELATED FUNCTIONS

GAMMA FUNCTIOX FOR COMPLEX ARGUMENTS

Table 6.7

.,.=2.0 91nr(z)

Y 0.0 0.1 0.2 0.3 0.4

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0

-

-

0.00000 0.00322 0.01286 0.02885 0.05107

00000 00 26151 39 59357 41 74027 79 93722 62

0.00000 0.04234 0.08509 0.12863 0.17335

00000 00 57120 74 33372 06 61223 10 05507 97

Y 5.0 5.1 5.2 5.3 5.4

0.07937 0.11354 0.15338 0.19863 0.24904

37235 30 77183 40 06308 81 06626 31 17059 66

0.21958 0.26767 0.31789 0.37051 0.42574

93100 95 56897 80 96132 02 53392 47 07261 44

5.5 5.6 5.7 5.8 5.9

0.30434 96090 22

0.48375 0.54471 0.60872 0.67588 0.74624

78429 30 46524 35 74700 17 39160 88 61166 63

6.0 6.1 6.2 6.3 6.4

0.81985 0.89672 0.97687 1.06028 1.14693

39537 67 82178 63 35612 07 11909 26 12720 53

6.5 6.6 6.7 6.8 6.9

1.23679 1.32983 1.42601 1.52528 1.62759

50341 04 65907 26 44920 94 30352 04 33595 36

7.0 7.1 7.2 7.3 7.4

1.73289 1.84113 1.95225 2.06621 2.18294

43555 35 34120 22 70264 63 12994 71 23322 91

7.5 7.6 7.7 7.8 7.9

- 0.36428 77010 76 - 0.42859 14442 42 - 0.49700 21701 52 - 0.56926 99322 58 - 0.64515 55533 76 - 0.72443 19760 33 - 0.80688 50339 42 - 0.89231 37613 78 - 0.98053 03476 69 - 1.07135 98302 14 - 1.16463 96040 42 - 1.26021 88108 76 - 1.35795 76568 48 - 1.45772 66961 57 - 1.55940 61080 61 - 1.66288 49866 52 - 1.76806 06566 17 - 1.87483 80234 65

- 1.98312 89631 02

9 In P(z)

ileInr(z)

Y In r(z)

-

4.50127 4.62939 4.75805 4.88723 5.01690

58755 42 88796 82 70222 52 13522 76 38831 33

5.18929 5.35533 5.52318 5.69281 5.86418

93415 60 82031 27 54439 62 16137 11 81052 00

-

5.14705 75299 57 51 27767 60518 81 5.40874 39987 03 5.54024 66615 82 5.67217 00274 24

6.03728 6.21208 6.38854 6.56665 6.74637

71248 73 16640 30 54709 43 30238 56 95048 97

5.80450 5.93722 6.07033 6.20381 6.33765

07366 29 60439 25 37820 31 23278 98 05713 36

6.92770 7.11059 7.29503 7.48100 7.66847

07748 95 33491 13 43738 76 16040 81 33815 76

6.47183 6.60636 6.74121 6.87639 7.01188

78858 22 41013 16 94789 19 46872 45 07803 50

7.85742 8.04784 8.23970 8.43299 8.62768

86143 76 67567 00 77898 07 22035 86 09788 99

7.14766 7.28375 7.42022 7.55676 7.69368

91771 18 16419 82 02668 81 74543 62 59017 46

8.82375 55706 27 9.02119 78914 05 9,21999 02960 14 9.42011 55664 09 9.62155 68973 45

7.83086 7.96830 8.10599 8.24393 8.38211

85862 69 87511 38 98924 36 57468 08 02798 83

-

9.82429 10.02832 10.23361 10.44016 10.64794

78825 87 25016 83 51072 54 04128 09 34810 35

9.07636 66296 28

10.85694 11.06716 11.27857 11.49116 11.70492

97125 60 48351 59 48933 86 62386 10 55194 45

9.21585 9.35555 9.49544 9.63552 9.77579

80388 55 14566 37 23361 92 62811 84 90392 11

11.91983 12.13589 12.35308 12.57138 12.79079

96725 52 59137 86 17297 01 48693 62 33364 76

-- 8.52051 76753 67 8.65915 23247 82 -- 8.79800 88177 87 8.93708 19330 47

-

2.09285 2.20393 2.31629 2.42987 2.54462

17530 93 05460 64 48844 77 92551 37 26813 03

2.30239 2.42452 2.54926 2.67657 2.80639

65434 67 09185 18 32043 52 20582 60 71597 50

8.0 8.1 8.2 8.3 8.4

2.66046 2.77736 2.89525 3.01410 - 3.13386

83499 73 32717 84 79709 78 62029 30 46968 42

2.93868 3.07340 3.21048 3.34989 3.49158

92920 59 03990 47 36221 88 33215 16 50837 57

8.5 8.6 8.7 8.8 8.9

-

3.25449 3.37595 3.49820 3.62122 3.74497

29213 81 28711 45 88720 59 74039 03 69383 89

3.63551 3.78164 3.92992 4.08032 4.23280

57202 41 32567 78 69172 45 71023 23 53645 81

9.0 9.1 9.2 9.3 9.4

9.91625 -10.05689 -10.19770 -10.33869 -10.47985

64956 49 46678 12 96994 20 78553 49 55166 49

13.01129 13.23287 13.45553 13.67924 13.90401

53818 23 94959 63 44022 19 90499 21 26078 95

-

3.86942 3.99455 4.12032 4.24671 4.37370

77912 99 19873 65 31366 90 63216 20 79930 87

4.38732 4.54384 4.70234 4.86276 5.02509

43808 43 19226 20 08252 48 89562 20 91831 32

9.5 9.6 9.7 9.8 9.9

-10.62117 -10.76266 -10.90431 -11.04611 -11.18806

91758 12 54322 81 09881 75 26442 29 72959 27

14.12981 14.35664 14.58449 14.81334 15.04319

44581 93 41900 46 15940 42 66565 09 95540 92

-

4.50127 58755 42

5.18929 93415 60

10.0

-

-11.33017 19298 27

15.27404 06485 34

288

GAMMA FUNCTION AND RELATED FUNCTIONS

DIGAMMA FUNCTION FOR COMPLEX ARGUMEN‘I‘S

Talde 6.8

x=l.O Y

Y* ( 2 )

#* ( 2 )

0.16342 0.32064 0.46653 0.59770

Y 5.0 5.1 5.2 5.3 5.4

1.61278 1.63245 1.65175 1.67068 1.68926

48446 69889 20861 42228 67162

.f*(z) 1.47080 1.47276 1.47464 1.47646 1.47820

63572 65809 61258 34022 20154

0.71269 0.81160 0.89563 0.96655 1.02628

5.5 5.6 5.7 5.8 5.9

1.70751 1.72543 1.74303 1.76034 1.77735

21687 25175 90807 25988 32733

1.47989 1.48151 1.48308 1.48459 1.48605

0.09465 0.17219 0.24588 0.31576 0.38196

03206 05426 65515 20906 28134

1.07667 1.11938 1.15580 1.187 07 1.21413

6.0 6.1 6.2 6.3 6.4

1.79408 1.81053 1.82672 1.84265 1.85833

08018 44105 28842 45939 75219

1.48746 1.48883 1.49015 1.49143 1.49267

1.5 1.6 1.7 1.8 1.9

0.44469 0.50420 0.56072 0.61448 0.66570

79402 34618 00645 06554 39172

1.23772 1.25843 1.27675 1.29306 1.30766

6.5 6.6 6.7 6.8 6.9

1.87377 1.88898 1.90396 1.91872 1.93327

92858 71602 80964 87422 54582

1.49387 1.49504 1.49617 1.49727 1.49833

2.0 2.1 2.2 2.3 2.4

0.71459 0.76132 0.80607 0.84899 0.89021

15154 74328 84807 54079 42662

1.32081 1.33271 1.34353 1.35341 1.36246

7.0 7.1 7.2 7.3 7.4

1.94761 1.96175 1.97569 1.98944 2.00300

43346 12062 16663 10799 45959

1.49937 1.50037 1.50135 1.50230 1.50323

2.5 2.6 2.7 2.8 2.9

0.92985 0.96803 1.00485 1.04039 1.07474

78387 70243 21252 40175 51976

1.37080 1.37849 1.38561 1.39222 1.39838

7.5 7.6 7.7 7.8 7.9

2.01638 2.02959 2.04262 2.05549 2.06820

71585 35177 82397 57159 01717

1.50413 1.50501 1.50586 1.50669 1.50751

3.0 3.1 3.2 3.3 3.4

1.10798 1.14016 1.17137 1.20164 1.23104

07107 89703 24783 84581 94107

1.40413 1.40951 1.41455 1.4 1928 1.42374

8.0 8.1 8.2 8.3 8.4

2.08074 2.09313 2.10537 2.11746 2.12941

56749 61434 53524 69410 44191

1.50830 1.50907 1.50982 1.51056 1.51127

3.5 3.6 3.7 3.8 3.9

1.25962 1.28741 1.31446 1.34081 1.36649

36033 54995 61381 34679 26435

1.42794 1.4 3191 1.43566 1.43922 1.44259

8.5 8.6 8.7 8.8 8.9

2.14122 2.15289 2.16442 2.17582 2.18710

11731 04718 54716 92217 46687

1.51197 1.51266 1.51332 1.51398 1.51462

4.0 4.1 4.2 4.3 4.4

1.39153 1.41597 1.43983 1.46314 1.48593

62879 47255 61892 70060 17620

1.44580 1.44885 1.45175 1.45452 1.45716

9.0 9.1 9.2 99‘3 .4

2.19825 2.20928 2.22018 2.23097 2.24165

46616 19555 92160 90229 38740

1.51524 1.51585 1.51645 1.51703 1.51760

4.5 4.6 4.7 4.8 4.9

1.50821 1.53001 1.55135 1.57224 1.59272

34505 36052 24197 88550 07370

1.45969 1.46210 1.46441 1.46663 1.46876

9.5 9.6 9.7 9.8 9.9

2.25221 2.26266 2.27301 2.28325 2.29338

61882 83093 25085 09877 58823

1.51816 1.51871 1.51925 1.51978 1.52029

5.0

1.61278 48446

2.30341 92637

1.52080

$*c(Z)

0.0 0.1 0.2 0.3 0.4

-0.57721 ‘56649 -0.56529 77902 -0,53073 04055 -0,47675 48934 -0.40786 79442

0.5 0.6 0.7 0.8 0.9

-0.32888 -0.24419 -0.15733 -0.07088 t0.01345

1.0 1.1 1.2 1.3 1.4

0.00000

10.0

(-5) 5

=:

1 t h ry- 2Y # ( z ) to 5D, computed by M. Goldstein, Los Alamos Scientific Laboratory.

j+ (1+iy)

T CO

AUXILIARY FUKCTION FOR .%#(l+iy)

Y-’

0.11 0.10 0.09 0.08 0.07 0.06

f4(Y)

0.00100 0.00083 0;00067 0.00053 0.00040 0.00030

956 417 555 368 853 011


9 10 11 13 14 17

Y-’

0.05 0.04 0.03 0; 02 0.01

0.00

9’+(l+iy)=lny+ f4(y) =nearestinteger to y.

NY) 0.00020 0.00013 0.00007 0.00003 0.00000 0.00000

839 335 501 333 833 000

20 25 33 50 100 m

I

289

GAMMA FUNCTION AND RELATED FUNCTIONS

DIGAMMA FUNCTION FOR COMPLEX ARGUMENTS x=l.l

Table 6.8

x=1.2

!/

m(z)

0.0 0.1 0.2 0.3 0.4

-0 42375 -0.41451 -0.38753 -0.34490 -0.28961

0.00000 0.14258 0.28082 0.41099 0.53042

5.0 5.1 5.2 5.3 5.4

1.61498 1.63457 1.65378 1.67264 1.69115

Y$(z) 1.45097 1.45332 1.45557 1.45774 1,45983

0.5 0.6 0.7 0.8 0.9

-0.22498 -0.15426 -0.08023 -0.00509 +0.06954

0.63764 0.73229 0.81484 0.88630 0.94792

5.5 5.6 5.7 5.8 5.9

1.70933 1.72718 1.74473 1.76197 1.77893

1.0 1.1 1.2 1.3 1.4

0.14255 0.21327 0.28131 0.34649 0.40880

1.00102 1.04687 1.08660 1.12119 1.15146

6.0 6.1 6.2 6.3 6.4

1.5 1.6 1.7 1.8 1.9

0.46829 0.52507 0.57930 0.63111 0.68067

1.17810 1.20169 1.22269 1.24148 1.25839

2.0 2.1 2.2 2.3 2.4

0.72813 0.77363 0.81730 0.85928 0.89967

2.5 2.6 2.7 2.8 2.9

0.93858 0.97610 1.01234 1.04736 1.08124

a+)

Aqz)

v

0.00000

0.1 0.2 0.3 0.4

-0.28904 -0.28169 -0.26014 -0.22578 -0.18064

0.12620 0.24926 0.36640 0.47552

5.0 5.1 5.2 5.3 5.4

1.61756 1.63705 1.65617 1.67494 1.69336

9$(z) 1.43125 1.43396 1.43658 1.43910 1.44152

1.46184 1.46378 1,46565 1.46746 1.46921

0.5 0.6 0.7 0.8 0.9

-0.12710 -0.06753 -0.00412 +0.06130 0.12730

0.57530 0.66517 0.74519 0.81589 0.87806

5.5 5.6 5.7 5.8 5.9

1.71146 1.72924 1.74672 1.76390 1.78079

1.44386 1.44612 1.44829 1.45039 1.45243

1.79561 1.81201 1.82815 1.84404 1.85968

1.47090 1.47253 1.47411 1.47565 1,47713

1.0 1.1 1.2 1.3 1.4

0.19280 0.25707 0.31960 0.38012 0.43846

0.93260 0.98046 1.02252 1.05960 1.09240

6.0 6.1 6.2 6.3 6.4

1.79740 1.81375 1.82983 1.84567 1.86126

1.45439 1.45629 1.45813 1.45991 1.46164

6.5 6.6 6.7 6.8 6.9.

1.87508 1.89025 1.90519 1.91992 1.93443

1,47857 1.47996 1.48132 1.48263 1.48391

1.5 1.6 1.7 1.8 1.9

0.49459 0.54851 0.60028 0.64999 0.69774

1.12153 1.14752 1.17082 1.19179 1.21074

6.5 6.6 6.7 6.8 6.9

1.87661 1.89173 1.90663 1.92132 1.93579

1.46331 1.46493 1.46651 1.46803 1.46952

1.27368 1.28755 1.30021 1.31179 1.32243

7.0 7.1 7.2 7.3 7.4

1.94874 1.96284 1.97675 1.99047 2.00401

1.48515 1.48635 1.48752 1.48866 1.48977

2.0 2.1 2.2 2.3 2.4

0.74362 0.78775 0.83022 0.87114 0.91060

1.22794 1.24362 1.25796 1.27112 1.28323

7.0 7.1 7.2 7.3 7.4

1.95006 1.96413 1.97800 1.99169 2.00519

1.47096 1.47236 1.47372 1.47505 1.47634

1.33224 1.34131 1.34972 1.35753 1.36482

7.5 7.6 7.7 7.8 7.9

2.01736 2.03054 2.04356 2.05640 2.06908

1.49085 1.49190 1.49292 1,49392 1.49489

2.5 2.6 2.7 2.8 2.9

0.94868 0.98546 1.02103 1.05546 1.08881

1.29442 1.30478 1.31441 1.32337 1.33173

7.5 7.b 7.7 7.8 7.9

2.01852 2.03167 .~ 2.04465 2.05746 2.07012

IAOOI

3.0 3.1 3.2 3.3 3.4

1.11405 1.37162 1.14586 1.37800 -. - .- - . 1.17671 1.38398 1.20667 1.38960 1.23578 1.39489

8.0 8.1 8.2 8.3 8.4

2.08160 2.09397 2.10619 2.11826 2.13019

1,49584 1.49676 1,49855 1.49940

1 d9-1~7 -.,,,-.

3.0 3.1 3.2 3.3 3.4

1.12113 1.15250 1.18295 1.21254 1.24132

1.33955 1.34688 1.35377 1.36024 1.36635

8.0 8.1 8.2 8.3 8.4

2.08262 2.09496 2.10716 2.11921 2.13111

1.48341 1.48448 1.48553 1.48656 1.48756

3.5 3.6 3.7 3.8 3.9

1.26409 1.29164 1.31847 1.34461 1.37010

1.39989 1.40461 1.40907 1.41331 1.41732

8.5 8.6 8.7 8.8 8.9

2.14198 2.15363 2.16515 2.17654 2.18780

1.50024 1,50106 1.50186 1.50265 1.50341

3.5 3.6 3.7 3.8 3.9

1.26932 1.29659 1.32315 1.34905 1.37432

1.37211 1.37756 1.38272 1.38761 1.39226

8.5 8.6 8.7 8.8 8.9

2.14288 2.15451 2360i 2.17738 2.18862

1.48853 1.48949 1.49042 1.49133 1.49222

4.0 4.1 4.2 4.3 4.4

-.-

1.41924 1.44294 1.46611 1.48876

1.42114 1.42478 1.42824 1.43154 1.43469

9.0 9.1 9.2 9.3 9.4

2.19893 2.20995 2.22084 2.23161 2.24228

1.50416 1.50489 1.50561 1.50631 1.50699

4.0 4.1 4.2 4.3 4.4

1.39898 1.42306 1.44659 1.46959 1.49209

1.39667 1.40088 1.40489 1.40871 1.41236

9.0 9.1 9.2 9.3 9.4

2.19973 2.21073 2.22160 2.23236 2.24301

1.49310 1.49395 1.49478 1.49560 1.49640

4.5 4.6 4.7 4.8 4.9

1.51092 1.53261 1.55384 1.57463 1.59501

1.43771 1.44059 1.44335 1.44600 1.44854

9.5 9.6 9.7 9.8 9.9

2.25283 2.26326 2.27360 2.28382 2.29395

1.50766 1.50832 1.50896 1.50960 1.51021

4.5 4.6 4.7 4.8 4.9

1.51410 1.53565 1.55676 1.57743 1.59769

1.41586 1.41920 1.42240 1.42547 1.42842

9.5 9.6 9.7 9.8 9.9

2.25354 2.26397 2.27429 2.28450 2.29461

1.49718 1.49794 1.49869 1.49943 1.50015

5.0

1.61498

1.45097

10.0

2.30397

1.51082

5.0

1.61756

1.43125

10.0

2.30462

1.50085

-

... 1.39496

!/

0.0

PR$(Z)

1.47760 1.47882

1.48117 1.4823G

GAMMA FUNCTION AND RELATED FUNCTIONS

Table 6.8

DIGAMMA FUNCTION FOR COMPLEX ARGUMENTS x=1.4

x=1.3

Y$(z)

Y$(z)

y

W$(Z)

Y

W ( Z )

~ O 1.41163 ~ 1.62052 1.63990 1.41472 1.65891 1.41769 1.67758 1.42055 1.69591 1.42331

~ 0.0

0.11303 0.22372 0.32997 0.43011

5.0~ 5.1 5.2 5.3 5.4

-0.06138 -0.05646 -0.04192 -0.01844 +0.01295

0.00000 0,10223 0.20269 0.29974 0.39204

5.0 5.1 5.2 5.3 5.4

1.62386 1.64311 1.66200 1.68055 1.69878

1.39559 1.39891 1.40211 1.40519

-0.03520 +0.01541 0.07003 0.12718 0.18561

0.52298 0.60796 0.68491 0.75404 0.81582

5.5 5.6 5.7 5.8 5.9

1.71392 1.73161 1.74900 1.76611 1.78292

1.42597 1.42853 1.43101 1.43340 1.43571

0.5 0.6 0.7 0.9

0.05100 0.09436 0.14171 0.19183 0.24367

0.47862 0.55886 0.63250 0,69957 0.76033

5.5 5.6 5.7 5.8 5.9

1.71668 1.73428 1.75158 1.76860 1.78533

1.40817 1.41103 1.41380 1.41648 1.41907

1.0 1.1 1.2 1.3 1.4

0.24434 0.30262 0.35994 0.41593 0.47035

0.87085 0.91983 0.96341 1.00227 1.03698

6.0 6.1 6.2 6.3 6.4

1.79947 1.81575 1.83177 1.84754 1.86308

1.43794 1.44011 1.44220 1.44423 1.44619

1.0 1.1 1.2 1.3 1.4

0.29635 0.34918 0.40163 0.45331 0.50395

0,81517 0.86457 0.90903 0.94907 0.98517

6.0 6.1 6.2 6.3 6.4

1.80180 1.81800 1.83395 1.84966 1.86513

1.42157 1.42399 1.42634 1.42861 1.43081

1.5 1.6 1.7 1.8 1.9

0.52310 0.57409 0.62333 0.67084 0.71667

1.06809 1.09605 1.12126 1.14409 1.16483

6.5 6.6 6.7 6.8 6.9

1.87837 1.89344 1.90829 1.92293 1.93735

1.44810 1.44995 1.45174 1.45348 1.45517

1.5 1.6 1.7 1.8 1.9

0.55336 0.60144 0.64811 0.69337 0.73722

1.01778 1.04730 1.07409 1.09849 1.12075

6.5 6.6 6.7 6.8 6.9

1.88036 1.89537 1.91017 1.92475 1.93912

1.43294 1.43502 1.43702 1.43898 1.44087

2.0 2.1 2.2 2.3 2.4

0.76087 0.80353 0.84470 0.88447 0.92290

1.18373 1.20102 1.21688 1.23148 1.24495

7.0 7.1 7.2 7.3 7.4

1.95158 1.96560 1.97944 1.99309 2.00655

1.45681 1.45841 1.45996 1.46147 1.46294

2.0 2.1 2.2 2.3 2.4

0.77968 0.82078 0.86058 0.89913 0.93647

1.14113 1.15984 1.17707 1.19296 1.20768

7.0 7.1 7.2 7.3 7.4

1.95330 1.96727 1.98106 1.99467 2.00809

1.44271 1.44450 1.44625 1.44794 1.4495 9

2.5 2.6 2.7 2.8 2.9

0.96007 0.99604 1.03088 1.06464 1.09739

1.25743 1.26900 1.27976 1.28980 1.29918

7.5 7.6 7.7 7.8 7.9

2.01984 2.03296 2.04591 2.05869 2.07131

1.46438 1.46577 1.46713 1.46845 1.46974

2.5 2.6 2.7 2.8 2.9

0.97265 1.00775 1.04179 1.07484 1.10693

1.22133 1.23402 1.24585 1.25689 1.26723

7.5 7.6 7.7 7.8 7.9

2.02134 2.03442 2.04733 2.06008 2.07267

1.45119 1.45276 1.45428 1.45576 1.45721

3.0 3.1 3.2 3.3 3.4

1.12917 1.16004 1.19005 1.21923 1.24763

1.30797 1.31621 1.32396 1.33126 1.33814

8.0 8.1 8.2 8.3 8.4

2.08378 2.09610 2.10827 2.12029 2.13217

1.47100 1.47223 1.47342 1.47459 1.47573

3.0 3.1 3.2 3.3 3.4

1.13813 1.16846 1.19797 1.22670 1.25469

1.27693 1.28604 1.29461 1.30269 1.31032

8.0 8.1 8.2 8.3 8.4

2.08510 2.09739 2.10952 2.12151 2.13337

1.45862 1.46000 1.46134 1.46266 1.46394

3.5 3.6 3.7 3.8 3.9

1.27529 1.30223 1.32851 1.35413 1.37915

1.34464 1.35080 1.35663 1.36216 1.36742

8.5 8.6 3.7 8.8 8.9

2.14391 2,15552 2.16700 2.17834 2.18956

1.47685 1.47794 1.47900 1.48004 1.48106

3.5 3.6 3.7 3.8 3.9

1.28196 1.30855 1.33450 1.35983 1.38456

1.31753 1.32436 1.33084 1.33699 1.34283

8.5 8.6 8.7 8.8 8.9

2.14508 2.15666 2.16811 2.17943 2.19063

1.46519 1.46641 1.46760 1.4 6 8 7 7 1.46991

4.0 4.1 4.2 4.3 4.4

1.40357 1.42744 1.45077 1.47358 1.49590

1.37242 1.37718 1.38172 1.38606 1.39020

9.0 9.1 9.2 9.3 9.4

2.20066 2.21163 2.22249 2.23323 2.24386

1.48205 1.48302 1.48397 1.48490 1.48582

4.0 4.1 4.2 4.3 4.4

1.40873 1.43235 1.45546 1.47806 1.50019

1.34840 1.35370 1.35876 1.36359 1.36821

9.0 9.1 9.2 9.3 9.4

2.20170 2.21265 2.22349 2.23421 2.24481

1.47103 1.47212 1.47319 1.47423 1.47525

4.5 4.6 4.7 4.8 4.9

1.51775 1.53914 1.56010 1.58064 1.60078

1.39416 1.39795 1.40158 1.40507 1.40841

9.5 9.6 9.7 9.8 9.9

2.25437 2.26478 2.27508 2.28528 2.29537

1.48671 1.48758 1.48844 1.48927 1.49010

4.5 4.6 4.7 4.8 4.9

1.52185 1.54307 1.56387 1.58425 1.60425

1.37263 1.37686 1.38092 1.38481 1.38854

9.5 9.6 9.7 9.8 9.9

2.25531 2.26570 2.27598 2.28616 2.29623

1.47626 1.47724 1.47820 1.47914 1.48006

5.0

1.62052

1.41163

10.0

2.30537

1.49090

5.0

1.62386

1.39213

10.0

2.30621

1.48096

Y+(z)

W$(Z)

Y 0.0 0.1 0.2 0.3 0.4

-0.16919 -0.16323 -0.14567 -0.11748 -0.08009

0.5 0.6 0.7 0.8 0.9

~

.

Y

W$(Z)

0.1 0.2 0.3 0.4

0.8

291

GAMMA FUNCTION AND RELATED FUNCTIONS

DIGAMMA FUNCTION FOR COMPLEX ARGUMENTS x=1.5

%$(4

Table 6.8

x=1.6

-@$(%I

%$(z)

9$(z)

y

.%$(Z)

0.09325 0.18511 0.27432 0.35978

Y 5.0 5.1 5.2 5.3 5.4

1.62756 1.64667 1.66543 1.68386 1.70196

1.37278 1.37658 1.38025 1.38378 1.38719

0.0 0.1 0.2 0.3 0.4

0.12605 0.12955 0.13995 0,15687 0.17976

J$(z) 0,00000 0.08566 0.17023 0.25268 0.33214

Y 5.0 5.1 5.2 5.3 5.4

1.63162 1.65057 1.66919 1.68748 1.70546

1.35357 1.35773 1.36173 1.36558 1.36930

0.13189 0.16935 0.21064 0.25479 0.30091

0.44066 0.51640 0.58668 0.65144 0.71078

5.5 5.6 5.7 5.8 5.9

1.71976 1.73725 1.75445 1.77137 1.78801

1.39047 1.39364 1.39670 1.39965 1.40251

0.5 0.6 0.7 0.8 0.9

0.20790 0.24050 0.27674 0.31581 0.35697

0.40789 0.47942 0.54642 0.60875 0.66642

5.5 5.6 5.7 5.8 5.9

1.72313 1.74051 1.75760 1.77441 1.79095

1.37289 1.37635 1.37969 1.38293 1.38605

1.0 1.1 112 1.3 1.4

0.34824 0.39614 0.44411 0.49175 0.53878

0.76494 0.81424 0.85907 0.89980 0.93684

6.0 6.1 6.2 6.3 6.4

1.80439 1.82051 1.83638 1.85201 1.86741

1.40528 1.40796 1.41055 1.41306 1.41549

1.0 1.1 1.2 1.3 1.4

0.39957 0.44305 0.48692 0.53082 0.57445

0.71957 0.76840 0.81319 0.85423 0.89183

6.0 6.1 6.2 6.3 6.4

1.80724 1.82327 1.83906 1.85460 1.86992

1.38908 1.39200 1.39484 1.39759 1.40025

1.5 1.6 1.7 1.8 1.9

0.58497 0.63018 0.67432 0.71732 0.75916

0.97054 1.00127 1.02932 1.05500 1.07855

6.5 6.6 6.7 6.8 6.9

1.88258 1.89752 1.91225 1.92677 1.94109

1.41786 1.42015 1.42237 1.42453 1.42663

1.5 1.6 1.7 1.8 1.9

0.61757 0,66001 0.70167 0.74244 0.78228

0.92629 0.95790 0.98693 1.01363 1.03824

6.5 6.6 6.7 6.8 6.9

1.88501 1.89989 1.91455 1.92900 1.94326

1.40284 1.40534 1.40778 1.41014 1.41244

2.0 2.1 2.2 2.3 2.4

0.79983 0.83935 0.87772 0.91 499 0.95118

1.10020 1.12015 1.13857 1.15563 1.17146

7.0 7.1 7.2 7.3 7.4

1.95521 1.96914 1.98287 1.99643 2.00981

1.42866 1.43065 1.43257 1.43445 1.43628

2.0 2.1 2.2 2.3 2.4

0.82115 0.85905 0.89597 0,93193 0.96694

1.06096 1.08197 1.10144 1.11953 1.13635

7.0 7.1 7.2 7.3 7.4

1.95731 1.97118 1.98487 1.99837 2.01169

1.41467 1.41684 1.41895 1.42101 1.42301

2.5 2.6 2.7 2.8 219

0.98634 1.02050 1.05370 1.08598 1.11738

1.18618 1.19990 1.21271 1.22469 1.23592

7.5 7.6 7.7 7.8 7.9

2.02301 2.03604 2.04891 2.06162 2.07417

1.43805 1.43978 1.44147 1.44312 1.44472

2.5 2.6 2.7 2.8 2.9

1.00102 1.03421 1.06653 1.09801 1.12867

1.15204 1.16668 1.18039 1.19324 1.20530

7.5 7.6 7.7 7.8 7.9

2.02485 2.03784 2.05066 2.06332 2.07583

1.42496 1.42686 1.42871 1.43051 1.43227

3.0 3.1 3.2 3.3 3.4

1.14794 1.17769 1.20667 1.23491 1.26245

1.24647 1.25639 1.26574 1.27457 1.28290

8.0 8.1 8.2 8.3 8.4

2.08657 2.09882 2.1 1092 2.12288 2.13470

1.44628 1.44781 1.44930 1.45075 1.45217

3.0 3.1 3.2 3.3 3.4

1.15856 1.18770 1.21611 1.24383 1.27089

1.21664 1.22733 1.23741 1.24693 1.25594

8.0 8.1 8.2 8.3 8.4

2.08819 2I10040 2.11246 2.12439 2.13617

1.43398 1.43565 1.43728 1i43888 1.44043

3.5 3.6 3.7 3.8 3.9

1.28931 1.31552 1.34112 1.36612 1.39055

1.29080 1.29828 1.30537 1.31212 1.31853

8.5 8.6 8.7 8.8 8.9

2.14638 2.15794 2.16936 2.18065 2.19182

1.45355 1.45 491 1.45623 1.45753 1.45879

3.5 3.6 3.7 3.8 3.9

1.29731 1.32311 1.34833 1.37297 1.39707

1.26448 1.27257 1.28026 1.28757 1.29454

8.5 8.6 8.7 8.8 8.9

2.14782 2.15934 2.17073 2.18199 2.19313

1.44195 1.44344 1.44489 1.44631 1.44770

4.0 4.1 4.2 4.3 4.4

1.41443 1.43779 1.46065 1.48302 1.50493

1.32464 1.33047 1.33603 1.34134 1.34642

9.0 9.1 9.2 9.3 9.4

2.20286 2.21379 2.22460 2.23530 2.24588

1.46003 1.46124 1.46242 1.46358 1.46471

4.0 4.1 4.2 4.3 4.4

1.42065 1.44373 1.46632 1.48844 1.51012

1.30117 1.30750 1.31354 1.31932 1.32485

9.0 9.1 9.2 9.3 9.4

2.20415 2.21504 2.22583 2.23650 2.24706

1.44905 1.45038 1.45168 1.45295 1.45420

4.5 4.6 4.7 4.8 4.9

1.52639 1.54742 1.56804 1.58826 1.60810

1.35128 1.35594 1.36041 1.36470 1.36882

9.5 9.6 9.7 9.8 9.9

2.25635 2.26672 2.27698 2.28714 2.29720

1.46582 1.46691 1.46798 1.46902 1.47004

4.5 4.6 417 4.8 4.9

1.53136 1.55219 1757262 1.59265 1.61232

1.33014 1.33522 1.34009 1.34476 1.34925

9.5 9.6 9.7 9.8 9.9

2.25751 2.26785 2.27809 2.28822 2.29826

1.45542 1.45661 1.45778 1.45892 1.46005

5.0

1.62756

1.37278 10.0

2.30716

1.47105

5.0

1.63162

1.35357 10.0

[(-:I91

[(-:I61

2.30820 [(-;)41

[(-;PI

0.0 0.1 0.2 0.3 0.4

0.03649 0.04062 0.05284 0.07266 0.09932

0.5 0.6 0.7 0.8 0.9

-@$(Z)

O*OOOOO

[(-:)“I

[(-{)2]

.f J. (1.5+iY)

=+H

tanh *U---

4Y 4y2+ 1

%$(z)

1.46115

292

GAMMA FUNCTION AND RELATED FUNCTIONS

Table 6.8

DIGAMMA FUNCTION FOR COMPLEX ARGUMENTS x=1.8

x=1.7 &(z)

91L(z)

?/

Y$(z)

Y

W ( Z )

0.20855 0.21156 0.22050 0.23511 0.25494

&(z) 0,00000 0.07918 0.15747 0.23407 0.30824

y

0.0 0.1 0.2 0.3 0.4

5.0 5.1 5.2 5.3 5.4

1.63603 1.65482 1.67328 1.69142 1.70926

1.33453 1.33902 1.34335 1.34752 1.35154

0.0 0.1 0.2 0.3 0.4

0.28499 0.28760 0.29537 0.30809 0.32541

0.00000 0.07358 0.14644 0,21792 0.28740

5.0 5.1 5.2 5.3 5.4

1.64078 1.65939 1.67769 1.69567 1.71336

1.31566 1.32048 1.32513 1.32961 1.33393

0.5 0.6 0.7 0.8 0.9

0.27945 0.30803 0.34001 0.37474 0.41161

0.37937 0.44701 0.51086 0.57074 0.62661

5.5 5.6 5.7 5.8 5.9

1.72680 1.74405 1.76102 1.77772 1.79416

1.35543 1.35918 1.36280 1.36630 1.36969

0.5 0.6 0.7 0.8 0.9

0.34693 0.37215 0.40053 0.43155 0.46469

0.35437 0.41842 0.47928 0.53675 0.59076

5.5 5.6 5.7 5.8 5.9

1.73076 1.74787 1.76472 1.78130 1.79762

1.33810 1.342 13 1.34603 1.34979 1.35344

1.0 1.1 1.2 1.3 1.4

0.45005 0.48957 0.52973 0.57018 0.61063

0.67852 0.72661 0.77107 0.81211 0.84996

6.0 6.1 6.2 6.3 6.4

1,81034 1.37297 1.82627 1.37614 1.84196 1.37922 1.85742 1.38220 1.87266 1.38509

1.0 1.1 1.2 1.3 1.4

0.49947 0.53546 0.57226 0.60955 0.64706

0.64131 0.68847 0.73237 0.77316 0.81103

6.0 6.1 6.2 6.3 6.4

1.81369 1.82952 1.84511 1.86047 1.87561

1.35697 1.36038 1.36369 1.36690 1.37001

1.5 1.6 1.7 1.8 1.9

0.65085 0.69065 0.72990 0.76849 0.80636

0.88488 0.91710 0.94685 0.97436 0.99982

6.5 6.6 6.7 6.8 6.9

1.88767 1.90246 1.91705 1.93143 1.9456’

1.38789 1.39061 1.39326 1.39582 1.39832

1.5 1.6 1.7 1.8 1.9

0.68455 0.72184 0.75879 0.79528 0.83122

0.84617 0.87877 0.90903 0.93713 0.96326

6.5 6.6 6.7 6.8 6.9

1.89053 1.90525 1.91975 1.93406 1.94817

1.37303 1.37596 1.37881 1.38158 1.38426

2.0 2.1 2.2 2.3 2.4

0.84345 0.87973 0.91519 0.94981 0198362

1.02342 1.04533 1.06570 1.08468 1.10238

7.0 7.1 7.2 7.3 7.4

1.95961 1.97342 1.98704 2.00048 2.01375

1.40074 1.40310 1.40539 1.40762 1.40980

2.0 2.1 2.2 2.3 2.4

0.86655 0.90123 0.93523 0.96853 1.00111

0.98757 1.01022 1.03136 1.05110 1.06957

7.0 7.1 7.2 7.3 7.4

1.96210 1.97583 1.98939 2.00277 2.01598

1.38688 1.38942 1.39189 1.39430 1.39664

2.5 2.6 2.7 2.8 2.9

1.01661 1.04879 1.08020 1.11084 1.14075

1.11893 1.13441 1.14893 1.16257 1.17539

7.5 7.6 7.7 7.8 7.9

2.02685 2.03979 2.05256 2.06518 2.07764

1.41191 1.41398 1.41599 1.41794 1.41986

2.5 2.6 2.7 2.8 2.9

1.03299 1.06416 1.09463 1.12442 1.15353

1.08687 1.10310 1.11836 1.13270 1.14622

7.5 7.6 7.7 7.8 7.9

2.02903 2.04191 2.05463 2.06719 2.07960

1.39 892 1.40115 1.40332 1.40543 1.4 0749

3.0 3.1 3.2 3.3 3.4

1.16993 1.19842 1.22625 1.25342 1.27997

1.18747 1.19886 1.20962 1.21981 1.22945

8.0 8.1 8.2 8.3 8.4

2.08996 2.10212 2.11415 2.12603 2113778

1.42172 1.42354 1.42531 1.42704 1.42874

3.0 3.1 3.2 3.3 3.4

1.18200 1.20982 1.23703 1.26363 1.28965

1.15898 1.17103 1.18243 1.19322 1.20345

8.0 8.1 8.2 873 8.4

2.09187 2.10399 2.11597 2.12781 2.13952

1.40950 1.41146 1.41338 1.41525 1.41708

3.5 3.6 3.7 3I8 3.9

1.30592 1.33129 1.35610 1.38037 i;40413

1.23859 1.24727 1.26338 1.27087

8.5 8.6 8.7 8.8 8.9

2.14939 2.16087 2.17222 2.18345 2.19456

1.43039 1.43200 1.43358 1.43513 1.43664

3.5 3.6 3.7 3.8 3.9

1.31511 1.34003 1.36441 1.38829 1.41168

1.21317 1.22241 1.23119 1.23956 1.24754

8.5 8.6 8.7 8.8 8.9

2.15109 2.16253 2.17385 2.18504 2.19611

1.41886 1.42061 1.42231 1.42398 1.42561

4.0 4.1 4.2 4.3 4.4

1.42738 1.27800 1.45015 1.28481 1747246 1.29132 1.49432 1.29755 1.51574 1.30351

9.0 9.1 9.2 9.3 9.4

2.20555 2.21642 2.22717 2.23781 2.24834

1.43811 1.43956 1.44097 1144235 1.44371

4.0 4.1 4.2 4.3 4.4

1.43459 1.45704 1.47904 1.50062 1.52178

1.25516 1.26243 1126939 1.27605 1.28242

9.0 9.1 9.2 9.3 9.4

2.20707 2.21790 2.22862 2.23923 2.24974

1.42720 1.42876 1.43029 1.43178 1.43324

4.6 4.6 4.7 4.8 4.9

1.53675 1.55736 1.57758 1.59742 1.61690

1.30922 1731470 1.31996 1.32501 1.32 986

9.5 9.6 9.7 9.8 9.9

2.25877 2.26908 2.27930 2.28941 2.29942

1.44503 1.44633 1.44760 1.44885 1.45 007

4.5 4.6 4.7 4.8 4.9

1.54254 1.56292 1.58291 1.60255 1.62183

1.28854 1.29440 1.30004 1.30545 1.31065

9.5 9.6 9.7 9.8 9.9

2.26013 2127042 2.28061 2.29069 2.30068

1.43468 1.43608 1.43745 1.43880 1.44012

Y

*See page

11.

m ( Z )

293

GAMMA FUNCTION AND RELATED FUNCTIONS

DIGAMMA FLXCTION FOR COMPLEX iRGUMENTS

Table 6.8

y

M$(z)

24$(%)

Y$(Z)

0.0 0..~ 1 0.2 0.3 0.4

0.35618 0.35847 0.36528 0.37644 0.39169

0.00000 0.06870 0.13681 0.20377 0.26908

5.0 5.1 5.2 5.3 5.4

1.64585 1.66428 1.68240 1.70022 1.71775

1.29698 1.30212 1.30707 1.31185 1.31647

0.0 0.1 0.2 0.3 0.4

0.42278 0.42480 0.43081 0.44068 0.45420

0.00000 0.06441 0.12833 0.19130 0.25288

x = 2 .0 Y 5.0 5.1 5.2 5.3 5.4

0.5 0.6 0.7 0.8 0.9

0.41071 0.43309 0.45842 0.48625 0.51614

0.33229 0.39306 0.45110 0.50624 0.55838

5.5 5.6 5.7 5.8 5.9

1.73500 1.75197 1.76868 1.78513 1.80133

1.32092 1.32522 1.32938 1.33341 1.33730

0.5 0.6 0.7 0.8 0.9

0.47111 0.49110 0.51380 0.53887 0.56594

0.31269 0.37042 0.42583 0.47874 0.52904

5.5 5.6 5.7 5.8 5.9

1.73951 1.75633 1.77290 1.78921 1.80528

1.30389 1.30846 1.31288 1.31715 1.32129

1.0 1.1 1.2 1.3 1.4

0.54770 Oi58053 0.61431 0.64872 0.68351

0.60749 0.65359 0.69677 0.73714 0.77483

6.0 6.1 6.2 6.3 6.4

1.81728 1.83300 1.84848 1.86374 1.87878

1.34107 1.34473 1.34827 1.35170 1.35503

1.0 1.1 1.2 1.3 1.4

0.59465 0.62468 0.65572 0.68751 0.71980

0.57667 0.62165 0.66400 0.70380 0.74116

6.0 6.1 6.2 6.3 6.4

1.82111 1.83671 1.85208 1.86723 1.88217

1.32536 1.32918 1.33295 1.33660 1.34015

1.5 1.6 1.7 1.8 1.9

0.71846 0.75338 0.78814 0.82261 0.85669

0.80999 0.84278 0.87335 0.90188 0.92851

6.5 6.6 6.7 6.8 6.9

1.89361 1.90824 1.92266 1.93688 1.95092

1.35826 1.36140 1.36445 1.36741 1.37029

1.5 1.6 1.7 1.8 1.9

0.75239 0.78510 0.81779 0.85033 0.88262

0.77618 0.80899 0.83973 0.86853 0.89551

6.5 6.6 6.7 6.8 6.9

1.89690 1.91143 1.92576 1.93990 1.95385

1.34358 1.34692 1.35017 1.35332 1.35639

2.0 2.1 2.2 2.3 2.4

0.89031 0.92342 0.95598 0.98795 1.01932

0.95338 0.97664 0.99840 1.01879 1.03792

7.0 7.1 7.2 7.3 7.4

1.96476 1.97843 1.99192 2.00523 2.01838

1.37308 1.37581 1.37846 1.38104 1.38355

2.0 2.1 2.2 2.3 2.4

0.91459 0.94617 0.97731 1.00798 1.03814

0.92081 0.94454 0.96681 0.98775 1.00743

7.0 7.1 7.2 7.3 7.4

1.96761 1.98120 1.99462 2.00786 2.02094

1.35937 1.36227 1.36509 1.36784 1.37052

2.5 2.6 2.7 2.8 2.9

1.05008 1.08022 1.10975 1.13867 1.16698

1.05588 1.07278 1.08868 1.10367 1.11782

7.5 7.6 7.7 7.8 7.9

2.03136 2.04418 2.05684 2.06935 2.08171

1.38599 1.38838 1.39070 1.39297 1.39518

2.5 2.6 2.7 2.8 2.9

1.06779 1.09690 1.12548 1.15352 1.18102

1.02597 1.04344 1.05992 1.07548 1.09020

7.5 7.6 7.7 7.8 7.9

2.03385 2.04661 2.05921 2.07167 2.08397

1.37313 1.37567 1.37815 1.38056 1.38292

3.0 3.1 3.2 3.3 3.4

1.19470 iI22184 1.24841 1.27442 1.29990

1.13119 1.14384 1.15583 1.16719 1.17798

8.0 8.1 8.2 8.3 8.4

2.09393 2.10600 2.11793 2.12973 2.14139

1.39734 1.39944 1.40149 1.40350 1.40546

3.0 3.1 3.2 3.3 3.4

1.20798 1.23442 1.26034 1.28575 1.31067

1.10413 1.11733 1.12985 1.14174 1.15304

8.0 8.1 8.2 8.3 8.4

2.09613 2;108i5 2.12003 2.13178 2.14339

__. ~-

3.5 3.6 3.7 3.8 3.9

1.32485 1.34929 1.37324 1.39670 1.41970

1.18823 1.19798 1.20727 1.21613 1.22458

8.5 8.6 8.7 8.8 8.9

2.15292 2.16432 2.17560 2.18675 2.19778

1.40738 1.40925 1.41108 1.41286 1.41461

3.5 3.6 3.7 3.8 3.9

1.33510 1.35905 1.38254 1.40558 1.42818

1.16379 1.17403 1.18379 1.19310 1.20200

8.5 8.6 8.7 8.8 8.9

2.15487 2.16623 2.17746 2.18858 2.19957

1.39593 1.39793 1.39988 1.40179 1.40366

4.0 4.1 4.2 4.3 4.4

1.44226 1.46437 1.48606 1.50734 1.52822

1.23265 1.24037 1.24775 1.25482 1.26160

9.0 9.1 9.2 9.3 9.4

2.20870 2.21950 2.23019 2.24077 2.25124

1.41632 1.41800 1.41964 1.42124 1.42281

4.0 4.1 4.2 4.3 4.4

1.45036 1.47212 1.49348 1.51446 1.53505

1.21050 1.21864 1.22643 1.23389 1.24105

9.0 9.1 9.2 9.3 9.4

2.21045 2.22121 2;22121 2.23187 2.24241 2.25284

1.40548 1.40727 iT40527 1.40902 1.41074 1.41241

4.5 4.6 4.7 4.8 4.9

1.54872 1.56885 1.58861 1.60803 1.62710

1.26810 1.27434 Lzeo33 1.28610 1.29164

9.5 9.6 9.7 9.8 9.9

2.26160 2.27186 2.28202 2.29207 2.30203

1.42435 1.42586 1.42733 1.42878 1.43020

4.5 4.6 4.7 4.8 4.9

1.55527 1.57514 1.59466 1.61385 1.63270

1.24792 1.25452 1.26086 1.26696 1.27283

9.5 9.6 9.7 9.8 9.9

2.26318 2.27340 2.28353 2.29356 2.30349

1.41406 1.41566 1.41724 1.41879 1.42030

1.29698

10.0

2.31190

1.43159

5.0

1.65125

1.27849

10.0

2.31332

1.42179

(-5)4 C 3 I

(-5)2 [ 3 I

x=1.9 ?/

5.0

9$(Z)

[(-;)41

J$(t)

y

1.65125 1.66948 1.68742 1.70506 1.72242

1.27849 1.28394 1.28919 1.29426 1.29916

1.38522 1.38746 1.38966 1.39180 1.39389 ~~

\

7. Error Function and Fresnel Integrals WALTERGAUTSCHI Contents Mathematical Properties . . . . . . . . . . . 7.1. Error Function . . . . . . . . . . . . . 7.2.. Repeated Integrals of the Error Function 7.3. Fresnel Integrals . . . . . . . . . . . . 7.4. Definite and Indefinite Integrals . . . . Numerical Methods . . . . . . . . . . . . . 7.5. Use and Extension of the Tables . . . . References Table 7.1.

......... ......... . . . . . . . . . . ......... . . . . . . . . . . .........

. . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . Error Function and its Derivative ( 0 5 2 1 2 ) . . . . . .

(2/11;;)e-z2,erf 2=(2/~;;) Jz

e-t2dt, 2=0(.01)2,

Page

297 297 299 300 302 304 304 308 310

IOD

0

Table 7.2. Derivative of the Error Function ( 2 5 2 5 1 0 )

. . . . . .

312

......

316

(2/J;;)e-z2, 2=2(.01)10, 8s

Table 7.3. Complementary Error Function ( 2 5 2 5 a).

l-

zeZ2erfc 2=(2/J;;)ze~~ e-"dt, f2=.25(-.005)0, erfc&,

n=1(1)10,

7D

15D

. . .

317

. . . . . . . . . . . .

319

Table 7.4. Repeated Integrals of the Error Function ( 0 2 ~ 5 5 )

~=0(.1)5,n=1(1)6,10, 11, 6 s

Table7.5. Dawson's Integral ( 0 5 z S m ) . e-z21e12dt, 2=0(.02) 2, IOD ze-z21etPdt,z - ~ =.25(- .005)0, 9D

* Guest worker, National Bureau of Standards, from The American University. (Presently Purdue University.)

ERROR FU.WCTION AND FRESNEL INTEGRALS

296

Page -

Table 7.6. (3/I'(1/3))JZe-''dt

( 0 1 ~ 1 2 . 3 ).

. . . . . . . . . . . .

320

. . . . . . . . . . . . . .

321

0

Z= 0 (.02) 1.7(.04)2.3,

Table 7.7.

7D

Fresnel Integrals ( 0 2 x 1 5 ) .

C ( Z ) = cos ~

(it')d t , S ( z ) = l s i n ( i

Table 7.8. Auxiliary Functions

z=O(

t z ) d t , ~ = 0 ( . 0 2 ) 5 , 7D

(02x1a) . . . . . . . . . . . . .

323

.02) 1, x-'=1(- .02)O, 15D

Table 7.9. Error Function for Complex Arguments ( 0 < ~ < 3 . 9 , 0 1 ~ 1 .3 ) 325 W ( Z ) = ~ -erfc ~ ~

( - i z ) , z=s+iy, ~=0(.1)3.9,y=0(.1)3, 6D

Table 7.10. Complex Zeros of the Error Function ( l I n 1 1 0 )

z,,erf z,=O,n=l(l)lO,

329

.

329

8D

Table 7.11. Complex Zeros of Fresnel Integrals (OSn15) z,, z:, C(z,).=O, S(~:)=O,n=0(1)5,

Table 7.12.

. . . .

.

, ,

,

4D

Maxima and Minima of Fresnel Integrals (0
C ( J W ) , C(y'GJZ), S ( J w ) , S(J=),

n=0(1)5,

. . .

329

6D

The author acknowledges the assistance of Alfred E. Beam, Ruth E. Capuano, Lois K. Cherwinski, Elizabeth F. Godefroy, David S. Liepman, Mary Orr, Bertha H. Walter, and Ruth Zucker in the preparation and checking of the tables.

7. Error Function and Fresnel Integrals I I

,

Mathematical Properties

Series Expansions

7.1. Error Function Definitions

7.1.1 7.1.2

erfc z=--,

2

f " e-12dt=l-erf

47r J z

(

l:

7.1.3 w(z)=e-z2 1+-

)

z

eC2dt =e-"erfc

(-iz)

In 7.1.2 the path of integration is subject to the For In-+(x),see chapter 10.

restriction arg t+a with lal<-" 4 as t + w along the path. (a=? is permissible if 9't2 remains bounded 4

Symmetry Relations

7.1.10

erf (-z)=-erf z - erf z=erf z

7.1.11

w(-z) -2e-~2-w(z)

7.1.9

to the left.) Integral Representation

7.1.4

-

-

w(2 ) =w (-

7.1.12

2)

Y 7 6

5 4

3 2 I

-x 0

2

4

6

.R

10

I2

1.4

16

18

20

22

FIGURE 7.1. y = e z u l " e-"dt. P=2(1)6

297

24

298

ERROR FUNCTION AND FRESNEL INTEGRALS

I

7.1.15

Y .2

.6

1.0

1.4

1.E

2.2

2.6

3.0

3.0

xp)and HP) are the zeros and weight factors of the Hermite polynomials. For numerical values see chapter 25. Value at Infinity

erf z+l

7.1.16

z+a

in larg zl
Maximum and Inflection Points for Dawson's Integral [7.31]

F(x) =e-$

etaat

F(.92413 88730 . . . )=.54104 42246

.. . F(1.50197 52682 , . . )=.42768 66160 . . .

7.1.17 7.1.18

Derivatives

7.1.19

7.1.20 W("+2) ( 2 ) +22w("f') (2)

+2 (n+ 1)w'"'

( 2 ) =0

(n=O, 1,2, FIaoRE

w( 2 ) ,

7.3. Altitude Chart of w ( z ) .

w(0)( 2 ) =

w'(2) =-22 w( 2 )

. . .)

+-62i

(For the Hermite polynomials Hn(z)see chapter 22.) Inequalities [7.11], [7.17]

Relation to Confluent Hypergeometric Function (see chapter 13)

7.1.13

7.1.21

The Normal Distribution Function With Mean m and Standard Deviation U (see chapter 26)

(For other inequalities see t7.21.)

7.1.22

-

2

e

--(t-m), ~a

dt=f(l+erf(F))

S-m

Continued Fractions

7.1.14

1

I

Asymptotic Expansion - m

m-1

1* 3 .

. . (2m-I) (222)"

299

ERROR FUNCTION AND FRESNEL INTEGRALS

Idinite Series Approximation for Complex Error Function [7.18]

If E&) is the remainder after n terms then

7.1.29

7.1.24

1 . 3 . . . (2n-1) R, (2)= (- 1)" (222)n 6,

e-22

erf (x+iy)=erf x + G [(l-cos 2xy)+i sin 2xy]

where For z real, &(x) is leas in absolute value than the first neglected term and of the same sign. Rational Approximations* ( 0
fn(X, Y ) = ~ x - ~ cosh x ny COS 2xyfn sinh ny sin 2xy gn(X, Y ) = ~ X cosh ny sin 2xy+n sinh ny cos 2xy le(x, y)l = l O - l E ( erf (x+iy)l 7.2. Repeated Integrals of the Error Function

m

)

I

I

Definition

7.1.25

erf x=1-(alt+a2t2+aat3)

e-z2+t(2),

t=-

I e (2)I 5 2.5 x 10-6 p= .47047

al= .34802 42

7.1.26

1

7.2.1

in-' erfc t dt

1+ P X i-1

a2= - .09587 98 a3=.74785 56

(n=O, 1,2, . .

I

.

I

erfc z=- 2 e-,', io erfc z=erfc z

6 Differential Equation

erf x= 1- (alt +a2t2+a,t3+a4t4+a~t6) e-21+ e(x) ,

t=- 1

7.2.2

1+PZ

1t (z) I I1.5x 10-7 p=.32759 11 a1=.25482 9592 az=-.28449 6736 a3=1.42141 3741 a,=--1.45315 2027 a6=1.06140 5429

y=Ain erfc z+Bin erfc

(-2)

( A and B are constants.) Expression as a Single Integral

7.1.27 Power Series 8

I e (z) 1 5 5 x 10-4 al= .278393 as= .000972

a2=.230389 a(= .078108

Recurrence Relations

7.2.5

7.1.28 1

erf x=1- [1+alx+azx2+ It (z)

. . +aexE] lE+e(x)

in erfc z=--

Z

n

in-l

erfc z+-

1 in-2erfc z 2n

(n=l, 2 , 3 , . . . )

153x 10-7

al=.07052 30784 a3=,00927 05272 a&=,00027 65672

a2=.04228 20123 a4=.00015 20143 a6=.00004 30638

4 Approximations 7.1.254.128 are from C. Hastings, Jr., Approximations for digital computers. Princeton Univ. Press, Princeton, N. J., 1955 (with permission).

7.2.6

2(n+l) (n+2)i"+2 erfc z 1 =(2n+1+2z2)in erfc z - ~

erfc z (n=1,2,3,.

. .)

~~

* The

terms in this series corresponding to b=n+2, , are understood to be zero.

n+4, n+6,

..

ERROR FUNCTION AND FRESNEL INTEGRALS

300

Relation to the Confluent Hypergeometric Function (see chapter 13)

Value at Zero

7.2.7

c+ 1

in erfc O= 2~

(n=-l,O,

1,2, * . .)

7.2.12

1)

i Relation to Parabolic Cylinder Functions (see chapter 19)

e-i@

7.2.13

in erfc z=- (2"47r)

* D-7&-1(28)

Asymptotic Expansion

7.2.14

in erfc z--

2

e-z2

fi (2z)"f'

2 (-l)fy2m+n)!

__.

md)

n!m!(22)2"

7.3. Fresnel Integrals Definition

7.3.1

7.3;2 FIQURE7.4. Repeated Integrals of the Error Function. y-2'17 (%+1)

in

The following functions are also in use:

erfc z

7.3.3

n-0, 1, 2, 4, 8, 14, 22

Derivatives

d 7.2.8 -in erfc z=-i*-l dz

erfc z

(n=O,1,2, , , .)

7.3.4

sin t2dt,

SZ(x)=k$' 7 27

7.2.9 Auxiliary Functions

d" - (ell erfc ~ ) = ( - 1 ) " 2 ~ n ! e ~ ' erfc i~ z dz"

7.3.5

(n=O,1,2, . . .)

Relation to Hh.(z) (see 19.14)

7.2.10

in erfc z=-

1 (2"-'7r)+

Hhn (422) Interrelations

Relation to Hermite Polynomials (see chapter 22) i-n

1-2-11 (-1)V erfc z+i" erfc (-z)==H,(iz)

7.3.7

0

dt

301

ERROR FUNCTION AND FRESNEL INTEGRAL8

7.3.14

I

, ~

~

7.3.9

1

C ( z ) = 2 f f ( z )sin

G

zZ)-g(z)

6

1

7-3-10 S ( Z ) = ~ - ~COS ( Z ) z2)-g(z)

cos

G

sin

22)

(% 22)

Series Expansions Symmetry Relations

7.3.12

C(Z)=cos

(; 51 3(.-. 1). (4n+1)

z4n+l

5 1 3(- . 1). . (4n+3)

Z4n,+3

22)

n7r2n

n-0

+sin

(z

zz)

C(-Z) = - C ( Z ) , S(-Z) =-S(

7.3.18

C(iz)= ~ C ( Z ) ,S(i2)=-iS(z)

7.3.19

C(Z,=CqT, S ( z ) = S ( z )

-~

Value at Infinity

'

7.3.20

S(z)=z =

7'3*13

(-1)n(T/2y+1 (2n+1)!(4n+3)

1

1

(x-+ 1

SCx)C>ti

Q ( 4 3 ,

Derivatives

24n+3

7.3.21

V

.a

Z)

*

n p + l

n-0

7-3-17

t

df(z)dx ---azg(z) ,

d g o --7rzf(x) - 1

dx

Relation to Error Function (see 7.1.1, 7.1.3)

7.3.22 C(Z)+iS(z)=-l + 2 i erf

[$ (1-i)z-j

="{ 2

l-eGz~W

[$ (l+i)z]}

[$ (l+i)z]}

7.3.23

g(z)=1{

w

7.3.24

j(x)=Y{

w[$

(l+i)z]}

Relation to Coduent Hypergeometric Function (see chapter 13)

7.3.25

0

.6

1.2

1.8

2.4

3.0

FIQURE 7.5. Fresnel Integrals. y=C(r), u = S W

3.6

Relation to Spherical Bessel Functions (see chapter 10)

ERROR FUNCTION AND FRESNEL INTEGRALS

302

7.4.2

Asymptotic Expansions

7.3.27

- +5 1

azf(2)

m=l

(- 1)"

1* 3

. . . (4m-1) (T22)2m

larg

(2-,

7.3.28 nzgi2)-

2 (-1lm 1

*

4<

a)

7.4.3

3 . . . (4mfl) ( d ) 2 m+ l

m =O

7.4.4

If R h f ) ( z )RAg)(z) , are theremainders aftern terms in 7.3.27, 7.3.28, respectively, then 7.3.29 Rj&"(z)=(-1)"

=-r(n+')

2an+i

(ga>O;n=O, I, 2,

. , .)

7.4.5

.

1 * 3 . . (4n-1) (TZ2)zn

so-

e(f),

.

tZn+le-otzdt =- n!

(9a>O;n=O, 1 , 2 , . .)

2an+l

7.4.6

7.4.7

p(f'l
7.3.31

7.4.8

For x real, R i f ) ( z )and Rj&E)(z) are less in absolute value than the first neglected term and of the same sign. Rational Approximations ( O s x 5 m ) 7.3.32

+

1 .926x 1.7922+3.1O4$

'(')=2+

+

c (2)

7.4.9 . .

1 t ( 2 )1 I2 x 10-3

7.4.10

7.3.33 1 g(x)=2+4. 142x4- 3 . 4 9 2 4~ ~6.6702

+ It (x) I 5 2 x 10-3

(For more accurate approximations see (7.11.) 7.4. Definite and Indefinite Integrals

For a more extensive list of integrals see [7.5], 17 81, [7.151. 7.4.1

1

7.4.12 Jo

4 Approximations 7.3.32,7.3.33 are based on those given in C. Hastings, Jr., Approximations for calculating Fresnel integrals, Approximation Newsletter, April 1956, Note 10. [See also MTAC 10, 173, 1956.1

1

7.4.B

*See page

11.

e-al'dt t +1

r 4

ea[l-(erf y=-

.\lii)2]

(a>O)

303

ERROR FUNCTION AND FRESNEL INTEGRALS

7.4.24

7.4.14

7.415

+T2 "-8 2

(;$)T

(Wa>O)

7.4.20 e-2@J

(ga>O, Wb>O)

7.4.21

7.4.22

7.4.31 7.4.23

l"{

[;-C(t)]2+[;-S(t)]2}dt=;

1

ERROR FUNCTION AND FRESNEL INTEGRALS

304

7.4.38

7.4.33

s c o s (a2+2bx+c)dx

=df

7.4.34

{cos ( P ) C [ d z

+sin

I 1

b

+w

(---+ iax)]+const. 1

Serf xdx=x erf x+- 6e-Z2+const.

7.4.35

"

s e a erf bxdx=- em erf a +conSt. s e a erf

44

dx=;{e"

erf

-sin

1

d'z +const.

(ax+ b)]}+const.

s s i n (ax2+2bx+c)dx

(a#O)

7.437

S

7.4.39

=&

7.4.36

(v) [dg

(ax+b)]

{cos

4;

(y) S[

(ax Sb)]

(p)

C [ g (az+b)]}+const.

7.4.40 rC(x)dx=xC(x)

-1 sin (i x2)+const. 7i

(a#O)

Numerical Methods erf .745=.70467 80779+ (.5) (.00652 58247) X [l-(.005)(.74)+(.00000 83333)(.0952)] Example 1. Compute erf ,745 and e-(.74a)2 =.70792 8920 using Taylor's series. With the aid of Taylor's theorem and 7.1.19 e-(.746)2=? 4-(.65258 24665) [1--.0074 it can be shown that 2 (.000025)(.0952) (.OOOOO 00833) (.74) (1.904811 erf (xo+ph)=erf xo =.57405 7910. 2 e-gph [1-phxo+~p2h2(2G-1)]+~

7.5. Use and Extension of the Tables

+

+-

6

+

As a check the computation was repeated with q=.75, p=-.5. Example 2. Compute erfc x to 5 s for x=4.8. We have 1/2=,0434028. With Table 7.2 and linear interpolation in Table 7.3, we obtain

+

1-2phxo p2h2 (2g -1)

-2 P"3%(25-3)]+7 3 It[< 1.2X 10-lo, Iql< 3.2 x 10-lo if h= 1W, (PIS a. With zo=.74, p=.5 and using Table 7.1

where

I

5 ;; erfc 4 . 8 = 1 (1.11253)(10-10)(.552669)J 4.8 =(1.1352)10-11.

ERROR FUNCTION AND FRESNEL INTEGRAL8

Example 3.

Compute e-"~ee"ddtto 5s for

305 2

From Table 7.1 we have - e-(1.72)2=.058565.

fi

Thus,

~=6.5.

With I/."= .0236686 and linear interpolation in Table 7.5

i erfc 1.72 = (.058565)(6.0064X 1011)/1.0087X1013 =3.4873X10-3

i2erfc 1.72 = (.058565)(1.2920X 1011)/1.0087X1013 =7.5013X10-4

e-(6.5J2['5e"dt= (506143) /(6.5) = .077868.

i3 erfc 1.72 = (.058565)(2.6031 X 1010)/1.0087X1013 Example 4. Compute i2 erfc 1.72 using the recurrence relation and Table 7.1. By 7.2.1, using Table 7.1,

=1.5114X 10-4.

Example 6. Compute C(8.65) using Table 7.8. With x=8.65, 1/z=.115607 we have from Table 7.8 by linear interpolation f(8.65) = .036797, g(8.65)=.000159.

i-lerfc 1.72 = .05856 50.

From Table 4.6

Using the recurrence relation 7.2.5 and Table 7.1 i erfc 1.72=-(1.72)(.01499

72)+(.5)(.05856 50)

sin

(5 x2)=-.961382,

cos

(x 9-x

--.275218.

Using 7.3.9

= ,0034873

i2erfc 1.72=- (.86)(.0034873)+(.25)(.01499 72) = .0007502.

C(8.65) = .5+ (.036797) (- .961382) - (.000159) (-.275218)=.46467.

Example 7. Compute Sl(l.l)to 10D. Using 7.3.8 and 7.3.10 we obtain by 6-pt interpolation in Table 7.8

Note the loss of two significant digits. Example 5. Compute ik erfc 1.72 for k=1, 2, 3 by backward recurrence. Let the sequence w?(x)(p=m, m-1, . . ., 1, 0, -1) be generated by backward use of the recurrence relation 7.2.5 starting with W E + ~ = O , wE+]=l. Then, for any fixed k, (see [7.7]),

S , ( l . l ) = S (1.1

=5(.87767 30169)=.31865 57172.

Example 8. Compute &(5.24) to 6D. Enter Table 7.7 in the column headed by Using Aitken's scheme of interpolation

5.20310 5.31808 5.08938 5.43432 4.97881

With x=1.72, m=15 we obtain

d:)

53 80 01 70 11

,43280 .41573 ,45093 ,38898 .46980

06 .03689 42 07808 80 ,42732 63 97 88 ,16061 99 691 63 ,42718 63 44 --.19432 70 758 60 6 52 ,42717 71 61 94 .26308 89 674 70 9 39

U.

-.

,42717 67

S2(5.24)=.427177

17 16 15 14 13

0 1 3.44 (1) 4.3834 (2) 2.5389

12 11 10

Q

8

(3) (4) (4) (5) (6)

2.1011 1.3831 9.8W5 6.4143 4.1668

7 6 5 4

3

(7) 2.5879 2 (8) 1.5569 1 (8) 8.9787 0 (9) 4.9570 -1 (10) 2.6031

:

(111 1.2820

:1

12) 2.5530

Example 9. Compute s2(5.24) using Taylor's series and Table 7.8. Using- 7.3.21 we can write Taylor's series forf, (U) =j

(,/?) and g2

(U) =g

(,/:)

in the form

ERROR FUNCTION AND FRESNEL INTEGRALS

306

By numerical integration, using Table 9.1, Jlo

+"C21 (u-u

)2 - u-uo)a+ 0

+;(

where

(k=O, 1, 2,

Using the fact that the remainder terms of the asymptotic expansion are less in absolute value than the first neglected terms, we can estimate

1

...

9

Finally,

J: PO at

( t ) -=-14659 t 6720

fa ( ~ g = ) ,168270, g2 ( ~ g = ) -014483.

Hence, applying the series above,

Jz[1-

c 2

1

at =7.33 x 10-7.

(10)-8 2 (10) I

5953819 cos 10-sin 10 --2688000 c & --23107 cos 10+sin 10,,.02298 2150400

fZ(5.24) =.170436, g2(5.24) =.015030.

~ g ,

&G

using Tables 7.8 and 4.8. Hence

Using the 4th formula a t the bottom of Table 7.8

m

&( 5.24) =.5 - (.170436) (-503471) - (.015030)(-364012) =.42718.

Yo(t)?=.41826

00-,02298 78=.39527 22.

The answer correct to 8D is .39527 290 (Table 11.2).

Example 10. Compute &(2) using 7.3.16. Generating the values of Jn+i(2) as described in chapter 10 we find

Example 12. Compute w(.44+.67i) using bivariate linear interpolation. By linear interpolation in Table 7.9 along the z-direction at y=.6 and y=.7

+J7/2(2) +Jll/2(2) +Jl6/2(2) f * * * =.49129+.06852+ .00297+.00006= .56284.

&(2) 'J8/2(2)

[

92 t-1512

+3'

. . .).

Consulting Table 7.8 we chose U = ,, 1/.185638 =5.386819, thus having U -uo= 5.24 -5.386819 = - .146819. From Table 7.8

m

Example 11. Compute

Yo(t)at ~ = . 4 1 8 2 6 00.

dt by numerical

4.44

+.6i) = .6(.522246+ .167880i) +.4(.498591

+.202666i)= .512784+.181794i is the integration using Tables 9.1 and 7.8. [Yo(t) Bessel function of thesecond kind definedin9.1.16.1 ' w(.44+.7i) =.6(.487556+.1479753)+.4(.467521 We decompose the integral into three parts +.179123i)=.479542+.1604343. ~

By linear interpolation along the y-direction at x= .44

+$' f + O ( t ) p 1 w(.44+ .67i) =.3(3127844- ,181794i) +.7 (.479542 10

where sin ( t

-:)

?oCt)=(1-&)

represents the first two terms of the asymptotic expansion 9.2.2.

+.160434i)=.489515+.166842im

+

The correct answer is .489557 .1668893. Example 13. Compute g w ( z ) for z=.44+.61i. Bivariate linear interpolation, as described in Example 12, is most accurate if z lies near the center or along a diagonal of one of the squares of the tabular grid [7.6]. It is not as accurate for z near the midpoint of a side of a square, as in this example. However, we may introduce an a d -

307

ERROR FUNCTION AND FRESNEL INTEGRALS

Example 16. Compute w(7+2i). Using the second formula at the end of Table 7.9

iary square (see diagram) which contains z close to its center. Bivariate linear interpolation can then be applied within this auxiliary square. The values of w(z) needed at z=C1, and 2=C2 are easily approximated by the average of the four neighboring tabular values. Furthermore the parts to be used are given by

-lzo--lI I 20- 51I -P1+ P21

*05176536 ‘42.27525 f 2 8 i

lzo--XzI

Example 17. Compute erf (2+i). From 7.1.3, 7.1.12 we have

m = P l - P2

r

)=.021853+.07501Oi.

erf z= 1-e-%o(iz) = 1-e+z2(cos

2 q

-i sin 2q)w(y+ix)

(z=x+iy).

Using Tables 7.9, 4.4, 4.6 erf (2+i)=l-e-a

(cos 4-i sin 4)w(lf2i) =1.003606-.0112590~.

where z= zo+ .1(pl+ip2). Thus, with zo= .4+ .Si, t1=.45+.65i, 5;=.45+.55i, pl=..4, p2=.l, we get from Table 7.9

W W ( =+(.522246 ~~) +.498591+.487556+.467521) = ,493979

+

Example 18. Compute Sl((a+i)8). From 7.3.22, 7.3.8, 7.3.18 we have

[( ~ + i ) -

+

W W ( ( ~ +(.522246 ) ,498591 ,561252 f.533157) B

I;

1 1-i S,(z)=s-- 4 efz2w

= ,528812

+

W W ( Z ) [l- (.4 .1)]{[l- (.4 - 41.522246 B

{ [l-(.4-

+ (.4 -.1).528812 } + (.4+ . l ) X

.1)] .493979+(.4--.1).498591}

=.509789.

The correct answer is .509756. Straightforward bivariate interpolation gives .509460. Example 14. Compute Yw(.39+.61i) to 6 D using Taylor’s series. From 7.1.20, Let z=.39+.61i1 z0=.4+.6i. and using Table 7.9, we have ~ ( 2 0= ) .522246+

8and making use of 7.1.11,

Setting 2 =(i+i) 7.1.12, and Table 7.9 S1

((i+i)

@)= ---2 4

e-2

3 1 3 i) (cos z-i sin 2) w (2+2

+T l+i e2 (cos -+i 23 sin 22) w (:+: i)

.16788Oi

w’(zo)=--.21634+ ,367383, Z-Zo= (- 1+i) 10-’ -.215 -. 185i, (2- z0)2= -2ix 1 0 - ~

~ W ” ( G )=

~~(~)=.167880-.0021634-.0036738 +.0000430= .162086.

Example 15. Compute w(.4 - 1.34. From 7.1.11, 7.1.12

2

=-.990734-.681619i.

Example 19. Compute

som

e-(1/4)12-3c cos (2t)dt

using Table 7.9. Setting b=y+ix, c=O in 7.4.2 and using 7.1.3, 7.1.12 we find

(a>(),z, y real). Using Tables 7.9, 4.4 and 4.6 ~ ( . 4 - 1.3i) =4.33342 +8.042013.

Hence from Table 7.9

308

ERROR FUNCTION AND FRESNEL INTEGRALS

References Texta

[7.1] J. Boersma, Computation of Fresnel integrals, Math. Comp. 14, 380 (1960). [7.2] A. V. Boyd, Inequalities for Mills’ ratio, Rep. Statist. Appl. Res. Un. Jap. Sci. Engrs. 6, 44-46 (1959). [7.3] 0. Emersleben, Numerische Werte des Fehlerintegrals fiir &, Z. Angew. Math. Mech. 31, 393-394 (1951). [7.4] A. ErdBlyi et al., Higher transcendental functions, vol. 2 (McGraw-Hill Book Co., Inc., New York, N.Y., Toronto, Canada, London, England, 1953). [7.5] A. Erdblyi et al., Tables of integral transforms, vol. 1 (McGraw-Hill Book Co., Inc., New York, N.Y., Toronto, Canada, London, England, 1954). [7.6] W. Gautschi, Note on bivariate linear interpolation for analytic functions, Math. Tables Aids Comp. 13, 91-96 (1959). [7.7] W. Gautschi, Recursive computation of the repeated integrals of t,he error function, Math. Comp. 15, 227-232 (1961). [7.8] W. Grobner and N. Hofreiter, Integraltafel (Springer-Verlag, Wien and Innsbruck, Austria, 1949-50). [7.9] D. R. Hartree, Some properties and applications of the repeated integrals of the error function, Mem. Proc. Manchester Lit. Philos. Soc. 80, 85-102 (1936). [7.10] C. Hastings, Jr., Approximations for digital computers (Princeton Univ. Press, Princeton, N.J., 1955). [7.11] Y. Komatu, Elementary inequalities for Mills’ ratio, Rep. Statist. Appl. Res. Un. Jap. Sci. Engrs. 4, 69-70 (1955-57). [7.12] E. Kreysaig, On the zeros of the Fresnel integrals, Canad. J. Math. 9, 118-131 (1957). [7.13] Th. Laible, Hohenkarte des Fehlerintegrals, Z. Angew. Math. Phys. 2, 484-486 (1951). (7.141 F. Losch and F. Schoblik, Die Fakultat (B. G. Teubner, Leipzig, Germany, 1951). [7.15] F. Oberhettinger, Tabellen zur Fourier Transformation (Springer-Verlag, Berlin, Gottingen, Heidelberg, Germany, 1957). [7.16] J. R. Philip, The function inv erfc 8 , Austral. J. Phys. 13, 13-20 (1960). [7.17] H. 0. Pollak, A remark on “Elementary inequalities for Mills’ ratio” by Ydsaku Komatu, Rep. Statist. Appl. Res. Un. Jap. Sci. Engrs. 4, 110 (1955-57). e-z’dz and [7.181 J. B.Rosser, Theory and application of

so’

s,’

e-.’GdyLY

e-z2dz (Mapleton House, Brooklyn,

N.Y.,1948). [7.19] H. E. Salzer, Formulas for calculating the error function of a complex variable, Math. Tables Aids Comp. 5, 67-70 (1951). [7.20] H. E. Salzer, Complex zeros of the error function, J. Franklin Inst. 160, 209-211 (1955). [7.211 F. G. Tricomi, Funaioni ipergeometriche confluenti ( W o n i Cremonese, Rome, Italy, 1954).

17.22) G. N. Watson, A treatise on the theory of Bessel functions, 2d ed. (Cambridge Univ. Press, London, England, 1958). Tables [7.23] M. Abramowita, Table of the integral

s,’

e-u*

du, J.

Math. Phys. 30, 162-163 (1951). ~=0(.01)2.5, 8D. [7.241 P. C. Clemmow and Cara M. Munford, A table of d ( i r ) e + i r p z J m e-hi*h2dX for complex values of p, Philos. Trans. Roy. Soc. London { A } , 245, 189211 (1952). IpI =0(.01).8, arg p=0°(10)450, 4D. [7.25] R. B. Dingle, Doreen Arndt and S. K. Roy, The integrals ~ , ( z )= ( p l ) - ~ S 6~(c2+z~)-1e-~dc ,~

and ~ , ( z )= ( p t ) - ~ S ,en(c2+z2)-*e-#dc ~

and their tabulation, Appl. Sci. Res. B 6, 155-164 (1956). C(z), S(z), 2==0(1)20, 12D. [7.26] V. N. Faddeeva and N. M. Terent’ev, Tables of &~‘e~dt) values of the f u n c t i o n w ( z ) = e - ~ 2 ( l +2i for complex argument. Translated from the Russian by D. G. Fry (Pergamon Press, New York, N.Y., 1961). w(z),z=z+iy; z,y=0(.02)3; z=3 (. 1)5, y =O(. 1) 3 ; z=0 (. 1) 5, 1/ =3 (.I)5; 6D. [7.27] B. D. Fried and S. D. Conte, The plasma dispersion function (Academic Press, New York, N.Y. and London, England, 1961). i f i w ( z ) , i&&’(z), z=z+iy; ~ = 0 ( . 1 ) 9 . 9 , y= -9.1(.1)10; z=var. (.1)9.9, y=-lO(.1)-9.2; 6s. [7.28] K. A. Karpov, Tablitsy funktsii w ( z ) =e-s’h‘ s3dz v kompleksnoi oblasti (Izdat. Akad. Nauk SSSR., MOSCOW, U.S.S.R., 1954). Z=Z; ~=0(.001)2(.01)10; 6D; 8=2.5°(2.50)300(1.250)350(,6250)400; (.OOl)p; (.Ol)p;’ (.0002)5, O l p e < p i < p i ’ 5 5 , 5D; z= i y ; y =O( .001)3(.0002)5, 5s. z=pe”; p=pb

=l

[7.29] K. A. Karpov, Tablitsy funktsii F ( z ) e%!z v kompleksnoi oblasti (Iadat. Akad. Nauk SSSR., Moscow, U.S.S.R., 1958). z=peib; 0 =45°(.31250)48.750(.6250)550( 1.25’) 65O(2.5’)9Oo, ~ = ~ e ~ . ~OIP~
flection points.

309

ERROR FUNCTION AND FRESNEL INTEGRALS [7.32] W. Lash Miller and A. R. Gordon, Numerical evaluation of infinite series and integrals which arise in certain problems of linear heat flow, electrochemical diffusion, etc., J. Phys. Chem. 35, 2785-2884 (1931). F(x) = e - * ’ r

et2&;

x=O(.O1)1.99,

6D;

2=2(.01)4(.05)7.5(.1) 10(.2)12, 8s. [7.33] National Bureau of Standards, Tables of the error function and its derivative, Applied Math. Series 41, 2d ed. (U.S. Government Printing Office, Washington, D.C., 1954). (2/4r)e-z2, erf x, x=O(.OOOl) 1(.001)5.6, 15D; (2/4a)e-z2, erfc x, x=4(.01)10,

85.

[7.34] T. Pearcey, Table of the Fresnel integral (Cambridge Univ. Press, London, England, 1956). C

(E),(E), S

z=0(.01)50,

6-7D.

[7.35] Tablitsy integralov Frenelya (Iadat. Akad. Nauk SSSR.,MOSCOW, U.S.S.R., 1953). C(z), S(x), x = 0(.001)25, 7D; S(Z), s=O(.OOl) .58, 75; C(x), 2=0(.001) .101, 7s. [7.36] A. van Wijngaarden and W. L. Scheen, Table of Fresnel integrals, Verh. Nederl. Akad. Wetensch., Afd. Natuurk. Sec. I, 19, No. 4, 1-26 (1949). C(z), S(z), x=0(.01)20, 5D. (Also contains numerical values of the coefficients in Taylor and asymptotic expansions.)

310

ERROR FUNCTION AND FRESNEL INTEGRALS

ERROR FUNCTION AND ITS DERIVATIVE

Table 7.1

2

- p-2

erf D'

erf .c

0. 01 0.02 0.03 0. 04

\r 1.12837 1.12826 1.12792 1.12736 1.12657

91671 63348 79057 40827 52040

0.00000 0. 01128 0.02256 0.03384 0.04511

00000 34156 45747 12223 11061

0. 50 0. 51 0. 52 0. 53 0. 54

0.87878 0.86995 0.86103 0.85204 0.84297

25789 15467 70343 34444 51813

0.52049 0.52924 0.53789 0.54646 0.55493

98778 36198 86305 40969 92505

0. 05 0. 06 0. 07 0. oa 0. 09

1.12556 1.12432 1.12286 1.12118 1.11927

17424 43052 36333 06004 62126

0.05637 0. 06762 0.07885 0.09007 0.10128

19778 15944 77198 81258 05939

0. 55 0. 56 0. 57 0. 58 0. 59

0.83383 0.82463 0.81536 0.80604 0.79666

66473 22395 63461 33431 75911

0.56332 0.57161 0.57981 0.58792 0.59593

33663 57638 58062 29004 64972

0.10 0. 11 0.12 0. 13 0. 14

1.11715 1.11480 1.11224 1.10946 1.10647

16068 80500 69379 97934 82654

0.11246 0.12362 0.13475 0.14586 0.15694

29160 28962 83518 71148 70331

0. 60 0. 61 0. 62 0. 63 0. 64

0.78724 0.77777 0.76826 0.75872 0.74914

34317 51846 71442 35764 87161

0.60385 0.61168 0.61941 0.62704 0.63458

60908 12189 14619 64433 58291

0. 15 0.16 0. 17

41267 92726 57192 56008 11683

0.16799 0.17901 0.18999 0.20093 0.21183

59714 18132 24612 58390 98922

0. 65

0. 19

1.10327 1.09985 1.09623 1.09240 1.08837

0.73954 0.72992 0.72027 0.71061 0.70095

67634 18814 81930 97784 06721

0.64202 0.64937 0.65662 0.66378 0.67084

93274 66880 77023 22027 00622.

0.20 0.21 0.22 0. 23 0. 24

1.08413 1.07969 1.07506 1.07023 1.06522

47871 89342 61963 92672 09449

0.22270 0.23352 0.24429 0.25502 0.26570

25892 19230 59116 25996 00590

0. 71 0. 72 0. 73 0.74

0.69127 0.68159 0.67191 0.66224 0. 65258

48604 62792 88112 62838 24665

0.25 0. 26 0. 27 0.28 0.29

1.06001 1.05462 1.04904 1.04329 1.03736

41294 18194 71098 31885 33334

0.27632 0.28689 0.29741 0.30788 0.31828

63902 97232 82185 00680 34959

0.75 0. 76 0. 77 0.78 0. 79

0.64293 0.63329 0.62368 0.61408 0.60452

10692 57399 00626 75556 16696

0.71115 0.71753 0.72382 0.73001 0.73610

56337 67528 16140 04313 34538

0. 30 0. 31 0. 32 0. 33 0. 34

1.03126 1.02498 1.01855 1.01195 1.00519

09096 93657 22310 31119 56887

0.32862 0.33890 0.34912 0.35927 0.36936

67595 81503 59948 86550 45293

0. 80 0. 81 0.82 0.83 0. 84

0.59498 0.58548 0.57601 0.56659 3.55720

57863 32161 71973 08944 73967

0.74210 0.74800 0.75381 0.75952 0.76514

09647 32806 07509 37569 27115

0. 35 0. 36 0. 37 0. 38 0. 39

0.99828 0.99122 0.98401 0.97665 0.96916

37121 10001 14337 89542 75592

0.37938 0.38932 0.39920 0.40900 0.41873

20536 97011 59840 94534 87001

0. 85 0.86 0.87 0. 88 0. 89

0.54786 0.53858 0. 52934 0.52016 0.51103

97173 07918 34773 05514 47116

0.77066 0.77610 0.78143 0.78668 0.79184

80576 02683 98455 73192 32468

0.40 0. 41 0. 42 0.43 0.44

0.96154 0.95378 0.94590 0.93789 0.92977

12988 42727 06256 45443 02537

0.42839 0.43796 0.44746 0.45688 0.46622

23550 90902 76184 66945 51153

0. 90 0. 91 0. 92 0. 93 0. 94

0.50196 0.49296 0.48402 0; 475i5 0.46635

85742 46742 54639 33I32 05090

0. 80676 77215

0. 45 0.46 0.47 0.48 0.49

0.92153 0.91318 0.90473 0.89617 0.88752

20130 41122 08685 66223 57337

0.47548 0.48465 0.49374 0.50274 0.51166

17198 53900 50509 96707 82612

0. 95 0. 96 0. 97 0. 98 0. 99

0.45761 0.44896 0.44037 0.43187 0.42345

92546 16700 97913 55710 08779

0.82089 0.82542 0.82987 0.83423 0. 83850

0. 50

0.87878 25789

0.52049 98778

1. 00

0.41510 74974

0. 00

0. 18

0. 66

0. 67 0. 68 0. 69 0. 70

[(-55)1]

SE

Example 1.

z - 0.88622 69255

@-

0.79690 82124 0.80188 28258 0.81156 35586 0.81627 10190

[(-55)

08073 36496 02930 15043 80696

'1

0.84270 07929

311

ERROR FUNCTION AND FRESNEL INTEGRALS

ERROR FUNCTION AND ITS DERIVATIVE X

2

-

J

e-x2

2

erf x

p-22 , I II

Table 7.1

erf x

1. 00 1. 01 1.02 1: 03 1. 04

0.41510 0.40684 0.39867 0.39058 0.38257

74974 71315 13992 18368 98986

0.84270 0.84681 0.85083 0.85478 0.85864

07929 04962 80177 42115 99465

1. 50 1. 51 1. 52 1. 53 1. 54

0.11893 0; 11540 0.11195 0.10859 0; io53i

02892 38270 95356 63195 30683

0.96610 0.96727 0.96841 0.96951 0.97058

51465 67481 34969 62091 56899

1. 05 1. 06 1. 07 1. 08 1. 09

0.37466 0.36684 0.35911 0.35147 0.34392

69570 43034 31488 46245 97827

0.86243 0.86614 0.86977 0.87332 0.87680

61061 35866 32972 61584 31019

1.55 1. 56 1. 57 1. 58 1.59

0.10210 0.09898 0.09593 0.09295 0.09005

86576 19506 17995 70461 65239

0.97162 0.97262 0.97360 0.97454 0.97546

27333 81220 26275 70093 20158

1.10 1.11 1.12 1. 13 1.14

0.33647 0.32912 0.32186 0.31470 0.30764

95978 49667 67103 55742 22299

0.88020 0.88353 0.88678 0.88997 0.89308

50696 30124 78902 06704 23276

1. 60 1. 61 1. 62 1. 63 1. 64

0.08722 0.08447 0.08178 0.07917 0.07662

90586 34697 85711 31730 60821

0.97634 0.97720 0.97803 0.97884 0.97962

83833 68366 80884 28397 17795

1. 15 1.16 1. 17 1. 18 1.19

0.30067 0.29381 0.28704 0.28037 0.27381

72759 12389 45748 76702 08437

0.89612 0.89909 0.90200 0.90483 0.90760

38429 62029 03990 74269 82860

1. 65 1. 66 1. 67 1. 68 1. 69

0.07414 0.07173 0.06938 0.06709 0.06487

61034 20405 26972 68781 33895

0.98037 0.98110 0.98181 0.98249 0.98315

55850 49213 04416 27870 25869

1.20 1. 21 1.22 1. 23 1. 24

0.26734 0.26097 0.25471 0.24854 0.24248

43470 83664 30243 83805 44335

0.91031 0.91295 0.91553 0.91805 0.92050

39782 55080 38810 01041 51843

1. 70 1. 71 1.72 1.73 1. 74

0.06271 0.06060 0.05856 0.05657 0.05464

10405 86436 50157 89788 93607

0.98379 0.98440 0.98500 0.98557 0.98613

04586 70075 28274 84998 45950

1. 25 1. 26 1.27 1. 28 1. 29

0.23652 0.23065 0.22489 0; 21923 0.21367

11224 83281 58748 35317 10145

0.92290 0.92523 0.92751 0; 92973 0.93189

01283 59418 36293 41930 86327

1. 75 1. 76 1.77 1. 78 1.79

0.05277 0.05095 0.04918 0.04747 0.04580

49959 47262 74012 18791 70274

0.98667 0.98719 0.98769 0.98817 0.98864

16712 02752 09422 41959 05487

1. 30 1. 31 1. 32 1. 33 1.34

0.20820 0.20284 0.19757 0.19241 0.18734

79868 40621 88048 17326 23172

0.93400 0.93606 0.93806 0.94001 0.94191

79449 31228 51551 50262 37153

1. 80 1. 81 1. 82 1. 83 1. 84

0.04419 0.04262 0.04110 0.03963 0.03820

17233 48543 53185 20255 38966

0.98909 0.98952 0.98994 0.99034 0.99073

05016 45446 31565 68051 59476

1. 35 1. 36 1: 37 1. 38 1. 39

0.18236 0.17749 0.17271 0.16802 0.16343

99865 41262 40811 91568 86216

0.94376 0.94556 0.94731 0.94901 0.95067

21961 14366 23980 60353 32958

1. 85 1. 86 1. 87 1. 88 1. 89

0.03681 0.03547 0.03417 0.03292 0.03170

98653 88774 98920 18811 38307

0.99111 0.99147 0.99182 0.99215 0.99247

10301 24883 07476 62228 93184

1.40 1. 41 1. 42 1.43 1. 44

0.15894 0.15453 0.15022 0.14600 0.14187

17077 76130 55027 45107 37413

0.95228 0.95385 0.95537 0.95685 0.95829

51198 24394 61786 72531 65696

1. 90 1.91 1. 92 1.93 1. 94

0.03052 0.02938 0.02827 0.02721 0.02617

47404 36241 95101 14412 84752

0.99279 0.99308 0.99337 0.99365 0.99392

04292 99398 82251 56502 25709

1.45 1. 46 1. 47 1. 48 1.49

0.13783 0.13387 0.13001 0.12623 0.12254

22708 91486 33993 40239 00011

0.95969 0.96105 0.96237 0.96365 0.96489

50256 35095 28999 40654 78648

1.95 1. 96 1.97 1. 98 1: 99

0.02517 0; 02421 0.02328 0.02237 0: 02150

96849 41583 09986 93244 82701

0.99417 0.99442 0.99466 0.99489 0.99511

93336 62755 37246 20004 14132

1. 50

0.11893 02892

0.96610 51465

2. 00

0.02066 69854

[(-55111

U* =0.88622

2

69255

0.99532 22650

312

ERROR FUNCTION AND FRESNEL INTEGRALS

Table 7.2

DERIVATIVE OF THE ERROR FUNCTION ~

1

2

2

z

(?-I=

\= 2.00 I - 212.0666 985

~

3.00 4 1.3925 3.01 (1 411.3113 4)1.2345 3.02 3.03 - 4 1.1620 3.04 ( - 411.0936

:g:

i

2.70 2.71 2.72 2.73 2.74

2.25 I - 317.1423 190 2726 - 3 6.8273 562 316.5249 776 3 6.2347 440 2.29 3 5.9562 287

2.75

2-22 2.23

%:;

I-

1:

3 7.8118 164

3.50 3.51 3.52 3.53 3.54

3.05 I - 4)1.0290 362

3.55 3.56 3.57 3.58 3.59

1:;

I- I

558 778 949 992 247

3.15 3.16 3.17

I:

- 4 7.6992 476

3.20 3.21 3.22 3.23 3.24

4.0297 {I- 5513.7795

1:

11

1-

3 1.0060 4 9.5402 4 9.0450 4 8.5738 d8.1256 417.2937 416.9082 4 6.5418 416.1936

850 932 671 378

1: 1

- 4 5.8627 725

4 5.5484 4 5.2499 2.78 4 4.9665 2779 ( - 4j4.6974

911'3 2.44

1:

-

313.5556 3 3.3886 3 3.2288 J3.0760 3)2.9298

487 700 871 230 098

2.50 ( - 3)2.1782 842

e-t'

722 713 360 632

5 5.5356 515.1971 5 4.8783 5 4.5782 5 4.2956

429 360 532 082 707

3.65 3.66 3.67 3.68 3.69

[I

636 604 513.5441 831 5 3.3227 997 513.1146 217

3.70 3.71 3.72 3I73 3.74

3.25 3.26 3.27 3.28 ,3.29 ( -

025 351 500 5 2.3996 135 5j2.2470 263

z:;: 1: 1

I

2.9189 [1-- 555 2.5620 2.7349

3.75 ( - 718.8143 219 3.76 3.77 3.78 3.79 ( - 7j6.5193 709 3.80 3.81 3.82 3.83 3.84

I- I

7 4.1221 624 7 3.8162 867

3.3488 3.1630 2.9868 2.8199 2.6618

688 053 598 597 533

3.35 3.36 3.37 3.38 3.39

2.90 2.91 2.92 2.93 2.94 ( -

4 2.5121 412.3703 4 2.2360 4 2.1090 4j1.9887

089 144 761 184 824

3.40

:;3

921 235 243 910 894

3.90 3.91 3.92 3.93 3.94

2.95 2796 2.97 2.98 2.99

411.8750 4 1.7674 411.6656 4 1.5694 4 1.4784

262 231 619 459 919

3.45 I - 617.6430 199 3.46 ( - 6 7.1327 211 616.6551 620 3.47 6 6.2083 353 3.48 6 5.7903 503 3.49

3.95 3.96 3.97 3.98 3.99

3.00 ( - 4)1.3925 305

3.50 ( - 6)5.3994 268

4.00 ( - 7)1.2698 235

2.85 2.86 2.87 2.88 2.89 (-

2

3.60 3.61 3.62 3.63 3.64

2.80 2.81 2.82 2.83 2.84

2.40 2141

~

\=

2.60 2.61 2.62 2.63 2.64 2.65 2.66 2.67

8.9221 551 1--- 333 8.1672 8.5372 378 930

X

305 047 698 929 521

I-

2.55 I - 311.6922 136 2.56 2.57 2.58 2759 ( - 3j1.3777 304

2.20 2.21

p-.?

\=

2.04 ( - 2j1.7583 088

2.12 ( - 2 1.2605 554 2.13 2 1.2081 043 2.14 (1 211.1576 041

__2

.r

p--2

1:

1:

(-

-

$

=

1: 1 -

5 1.0764 5 1.0056 6 9.3923 3.43 6 8.7704 3.44 ( - 6j8.1881

3.85 3.86 3.87 3.88 3.89

1;

0.88622 692%

-

7)2.0447 548

313

ERROR FUNCTION AND FRESNEL INTEGRALS

Table 7.2

DERIVATIVE OF THE ERROR FUNCTION z

.% p - - P

.r

7 1.'2698 235 776 7 1.0816 394 8 9.9797 993 8 9.2060 694

[I 7 1.1720 (-

1-

4.59 4.10 ( 4.11 4.12 4.13 4.14 ( -

[I

I

815.6489 8'5.2036 8 4.7925 8 4.4130 8j4.0627

4.50

8 1.6143 811.4826 8 1.3614 8 1.2498 8 1.1472

1: 1

lie113 1.6552 1.5123 1.3814 1.2616

059 434 248 699 849

5.00 5.01 5.02 5.03 5.04

1.1520 1.0517 9.5997 8.7603 ( -10) 7.9927

559 423 127 264 363

5.05 5.06 5.07 5.08 5.09 (-12)6.3198 998

4.65

994 974 673 993 445

( - 9)1.8113 059

5.00

2

2

5.60 5.61 5.62 5.63 5.64

121 639 517 364 618

4.15 ( - 8)3.7395 414

4.25 4.26 4.27 4.28 4.29

e-?

e--zl

VT

x7r

4.00 4.01 4.02 4.03 4.04

2 -i=

-14) 1.7287 067

621 187 962 727 772

5.15 (-12) 3.4188 470 5.16 5.17 5.18 5.19

5.65 (-14)1.5441 499 5.66 5.67 5.68 5i69 ( -15)9.8105 529

2.8766 2.6183 2.3826 2.1678 1.9719

694 207 973 441 702

5.20 5.21 5.22 5.23 5.24

5.70 5.71

1.7934 1.6307 1.4825 1.3474 1.2245

357 388 049 759 007

5.25 5.26 5.27 5.28 5.29

5.75 5.76 5.77 5.78 5.79

1.1125 1.0105 9.1780 8.3337 7.5656

261 888 821 894 500

5.30 5.31 5.32 5.33 5.34

5.80 5.81 5.82 5.83 5.84

6.8669 6.2315 5.6537 5.1285 4.6511

377 074 456 259 675

5.35 5.36 5.37 5.38 5.39

5.85 5.86 5.87 5.88 5.89

4.2173 3.8233 3.4653 3.1402 2.8451

976 166 660 998 570

5.40 5.41 5.42 5.43 5.44

2.5772 2.3340 2.1134 1.9132 1.7317

379 811 428 785 254

5.45 5.46 5.47 5.48 5.49

-10) 4.5911 471830 3.8103 3.4702 3.1598

( -11) 1.5670 866

fi2

=

5.50 (-14)8.2233 160

0.88622 69255

6.00

I- 1

15 8.7544 193 -15 7.8104 192

(-16)2.6173 012

ERROR FUNCTION AND FRESNEL INTEGRALS

DERIVATIVE OF THE ERROR FUNCTION 2 2 -= e - x 2 2 _-2- e-X' \' = \=

Table 7.2 2 p--2

z

I-

X

7.51 7*50 7.52 7.53 7.54

6.54

i1

-25 4.2013 654 -25 3.6157 871 -25 3.1112 033

2.6310 2.3078 2.0238 1.7744 1.5555

921 100 447 651 031

7.05 7.06 7.07 7.08 7.09

7.55 7.56 7.57 7.58 7.59

1.3632 1.1945 1.0465 9.1667 8.0275

874 852 500 618 879

7.10 7.11 7.12 7.13 7.14

7.60 7.61 7.62 7.63 7.64

6.15 6.16 6.17 6.18 6.19

6.65 I-20) 7.0285 671526 5.3848 4.7118 4.1221

758 575 212 664 880

7.15 (-23)7.0832 963 7.16 7.17 7.18 7.19

7.65 7.66 7.67 7.68 7.69

6.20 6.21 6.22 6.23 6.24

6.70 6.71 6.72 6.73 6.74

3.6055 3.1530 2.7568 2.4098 2.1061

852 937 372 972 973

7.20 7.21 7.22 7.23 7.24

7.70 7.71 7.72 7.73 7.74

6.25 6.26 6.27 6.28 6.29

6.75 6.76 6.77 6.78 6.79

1.8404 1.6078 1.4043 1.2264 1.0707

021 278 634 013 765

7.25 7.26 7.27 7.28 7.29

7.75 f -27) 9.2820 251 7.76 7.77 7.78 7.79

6.30 6.31 6.32 6.33 6.34

6.80 6.81 6.82 6.83 6.84

9.3471 8.1577 7.1183 6.2100 5.4166

286 565 018 515 048

7.30 7.31 7.32 7.33 7.34

7.80 7.81 7.82 7.83 7.84

6.35 f -18)3.4722 886 6.36 6.37 6.38 6.39

6.85 6.86 6.87 6.88 6.89

4.7235 4.1184 3.5900 3.1288 2.7263

904 183 610 615 649

7.35 7.36 7.37 7.38 7.39

7.85 7786 7.87 7.88 7789

6.40 6.41 6.42 6.43 6.44

6.90 6.91 6.92 6.93 6.94

2.3751 2.0688 1.8bl.5 1.5685 1.3654

704 010 892 776 297

7.40 7.41 7.42 7.43 7.44

7.90 7.91

574 924 503 224 461

6.95 6.96 6.97 6.98 6.99

1.1883 1.0340 8.9957 7.8244 6.8042

540 356 684 565 967

7.45 7.46 7.47 7.48 7.49

7.95 7.96 7.97 7.98 7.99

( -19)5.0525 800

7.00

( -22)5.9159 - 630

7.50

6.10 -17)7.8047 211 6.11 6.12 6I13 6.14 ( -17)4.7832 911

6.45 (-19)9.6542 6.46 -19 8.4849 -19 7.4558 6.47 6.48 -19 6.5502 6.49 -19 5.7534

ii

6.50

( -25) 4.2013 654

c= 0.88622 69255 2

(-25)1.9796 292

(-25j1.0803 765

f -26) 2.0097 185

(-26)1I0837 147

-28) 4.0175 202 -28) 3.4265 874

8.00 (-28)1.8097 068

315

ERROR FUNCTION AND FRESNEL INTEGRALS

DERIVATIVE OF THE ERROR FUNCTION

2

-=

X

e-x2

I=

I 1

8.01 8.02 8.03 8I04

8.10 8.11 8.12 8.13 8.14

I- 1

2.0157 780 1.6987 713

-29 8.1112 334 29 6.9043 382

2 % ,

2

-32 4.7280 139 8 .51 -32 3.9884 601 8*50 3.3639 141 2.8365 973 2.3914 628

8.00

8.05 8.06 8.07 8.08 8.09

z

Table 7.2 -p

f?

9.00 9.01 9.02 9.03 9.04

9.50 9.51 9.52 9.53 9.54

9.05 -36 3.0384 441 9.06 1-3612.5351 317 9.07 1-3612I1147 690 9.08 -36 1.7637 559 9.09 -36)1.4707 105

9.55 9.56 9.57 9.58 9.59 (-40)1.2918 638

9.10 (-36)1.2261 088 9.11 -36 1.0219 837 9.12 37 8.5167 148 9.13 (-37 7.0959 960 9.14 (-37 5.9110 925

9.60 9.61 9.62 9.63 9.64

I1

-29)1.8891 933

8.60 8.61 8.62 8.63 8.64

8.15 8.16 8.17 8.18 (-30 9.8348 778 8.19 ( -3018.3497 786

8.65 8.66 8.67 8.68 8.69

9I18 9.19

8.20 8.21 8.22 8.23 8.24

8.70 8.71 8.72 8.73 8.74

9.20 (-37)1.9668 449 9.21 9.22 9.23 9.24 ( -38)9.4066 395

9.70 9.71 9.72 9.73 9.74

8.75 (' -34)6.3355 422 8.76 8.77 8.78 8.79 (-34)3.1411 074

9.25 9.26 9.27 9.28 9.29

9.75 9.76 9.77 9.78 9.79 ( -42)2.6784 979

8.30 8.31 8.32 8.33 8.34

8.80 8.81 8.82 8.83 8.84

9.30 (-38)3.0926 2.5674 9.31 2.1310 9.32 1.7684 9.33 9.34 (-38)1.4672

112 566 520 718 880

9.80 9.81 9.82 9.83 9.84

8.35 8.36 8.37 8.38 8.39

8.85 8.86 8.87 8.88 8.89

9.35 9.36 9.37 9.38 9.39

1.2171 1.0094 8.3703 6.9392 5.7517

545 602 932 997 311

9.85 I-43)8.2436 338 9.86 9.87 9.88 9.89 (-43j3I7428 271

8.40 8.41 8.42 8.43 8.44

8.90 8I91 8.92 8.93 8;94

9.40 9.41 9.42 9.43 9.44

4.7664 3.9491 3.2713 2,7093 2.2434

456 520 439 286 186

9.90 9.91 9.92 9.93 9.94

8.45 8.46 8.47 8.48 8.49

8.95 8.96 8.97 8.98 8.99

9.45 9.46 9.47 9.48 9.49

-39 1.8572 574 39 1.5372 589 1,2721 404 1.0525 343 8.7066 400

9.95 9.96 9I97 9.98 9.99

8.50 ( -32)4.7280 139

9.00 (-36)7.4920 734

9.50 ( -40)7.2007 555

8.25 (-30)3.1137 8.26 2.6398 8.27 2.2376 8.28 1.8963 8.29 (-30)1.6067

725 841 697 577 846

I- 1

I

*

4.9230 4.0993 3.4127 2.8406 -37)2.3639

619 592 918 437 423

I- 1

2- 0.88622 69255

-40 1.0662 -41 8.7992 -41 7.2599 -41)5.9886 -41)4.9390

907 901 363 802 403

9.65 9.66 9.67 9.68 9.69 (-41)1.8788 710 -41)1.5477 017

-43)3.0708 096

ji

-43 1.1381 -44 9.3271 -44 7.6417 -44 6.2596 -4415.1265

922 204 477 629 162

10.00 ( -44)4.1976 562

316

ERROR FUNCTION AND FRESNEL INTEGRALS

COMPLEMENTARY ERROR FUNCTION

Table 7.3

xez2erfc x 0.53406 72 0.53511 47 0.53617 29 0.53724 20 0.53832 23

0.250 0.245 0.240 0.235 0.230

x e z 2erfc x 0.51079 1 4 0.51163 07 0.51247 67 0.51332 94 0.51418 9 0

0.225 0.220 0.215 0.210 0,205

0.51505 0.51592 0.51681 0.51769 0,51859

55 92 01 83 40

2 2 2 2 2

0.095 0.090 0.085 0.080

0.53941 0.54051 0.54163 0.54276 0,54390

41 76 32 11 16

0.200 0.195 0.190 0.185 0.180

0.51949 0.52040 0.52132 0.52225 0.52318

74 85 75 45 98

2 2 2 2 2

0.075 0.070 0.065 0.060 0.055

0.54505 0.54622 0.54740 0.54859 0.54980

51 19 24 69 58

0.175 0.170 0.165 0.160 0.155

0,52413 0.52508 0.52604 0,52701 0.52799

33 55 63 59 46

2 2 2 3 3

0.050 0.045 0.040 0.035 0.030

0.55102 0.55226 0.55352 0.55479 0.55608

95 85 32 41 17

0.150 0.145 0.140 0.135 0.130

0.52898 0.52997 0.53098 0.53200 0.53303

25 98 67 35 02

3 3 3 3 3

0.025 0.020 0.015 0.005

0.55738 0.55870 0.56005 0,56140 0.56278

65 90 00 99 96

0.125

0.53406 72

3

0.000

0.56418 96

X -2

See Example 2.

"-? 1

<x>

2-2

2 2 2 2 2

0.125 0.120 0.115 0.110 0.105

0,100

0,010

<x> 3 3

3 3 3 3 3 3 3 4

6 7 8 10 14 W

<x> =nearest integer to x.

erfc 4%

n

1 2 3 4 5

0.01218 0.00039 0.00001 0,00000 0.00000

88821 27505 41444 05351 00208

n 6 7 8 9

84803 88282 02689 64662 26552

erfcx=-= 4.

10

erfc JG 0.00000 0.00000 0.00000 0.00000

00008 00000 00000 00000

25422 33136 01343 00055

0.00000 00000 00002

6"

e-t2dt=1-erf x

erfc ,/G compiled from 0. Emersleben, Numerische Werte des Fehlerintegrals fiir &G, Z. Angew. Math. Mech. 31,393-394, 1951 (with permission).

317

ERROR FUNCTION AND FRESNEL INTEGRALS

REPEAI'ED INTEGRALS OF THE ERROR FUNCTION

X

n=l

1.00000

0.0 0.1 0.2 0.3 0.4 0. 5 0. 6 01 7 0.8 0.9

1)4.46884

I- 1 I-

1 3.53855 1 2.76388 lj2.12869 1 1.61601 1 1.20884

I- I

-

VL =3

n -2 1.00000 1 7.93573 116.22654 1 4.82842 113.69906

1.00000

Table 7.4

n =4 1.00000 ( - 1) 7.36220

I- I

1 1.12021 2 8.03288

1.0 1.1 1.2 1.3 1. 4

-

(I 1

1.5 1.6 1.7 1.8 1. 9

3 8.02626 3 5.18140 3 3.29192

2.0

2. 1 2.2 2. 3 2.4 2. 5 2.6 2. 7 2.8 2.9

(-

411.27195

I - 51 4.81417

(-

-

1.96029 I:- 677)14.80916 9.79725

3.0 3.1 3.2 3.3 3.4

ir 1

7 2.31835 7 1,09748

512.05339 5 1.09167 6 5.70591 6 2.93172 6 1,48058

I- I -

7.34867 t- 77713.58429 1.71780

ir i

8 8.08871 8 3.74180

- 8)1.40460

3.5 3.6 3. 7 3. 8 3. 9 4. 0 4.1 4. 2 4. 3 4. 4

I-

9)3.22966

(-121 8.68930

4. 5 4.6 4.7 4.8 4.9 5. 0

(-121 2.35705

(-14)1.13080 (-13)2.62561

( -1) 5.64189 58355 see Exainples 4 and 5 .

(-14)5.61169

(-14)1.38998

(-15)3.83592

(-1)2.50000 00000

(-2)9.40315 97258

(-2)3.12500

ERROR FUNCTION AND FRESNEL INTEGRALS

REPEATED INTEGRALS OF THE ERROR FUNCTION

Table 7.4

1.00000

0. 5 0. 6 0. 7 0. 8 0. 9 1. 0 1.1 1.2 1. 3 1.4

n =6

n =5

X

0. 0 0.1 0.2 0. 3 0.4

1.00000

I - 1)1.43588

I - 2)8.84744

I

-

2) 9.41309 - 2) 6.09742 - 2 3.90166 - 21 2.46567

-

3 4.95614

- 3) 3. 01008

1.5 1.6 1.7 1. 8 1.9

- 3)1.80252

2. 0 21 1 2. 2 2. 3 2.4

I - 4)1.11492

n=ll

n =10

1.00000

1.00000

2) 7.95749 2)4.64127 2)2.67626 2)1.52533 ( - 3)8.59126

I - 3)5.90062

-

3)4.78106

-

4)2.11641 4 1.09146 51 5.55435 511.38116 5 2.78871

- 3 3.45489

-

312,04411

( - 5)2.04244

I - 6)6.74666

I - 5)6.51088 6)9.51547

( - 6)4186044

2.5 2. 6 21 7 2.8 2. 9

2.44418 I: 661 1.20988

3. 0

I- I

8 6.18684 8 2.82454

3.2

I: 99I 4.28380 1.81176

3. 4

3.5 3.6 3. 7 3.8 3.9 4.0 4.1 4.2 4. 3 4.4

-14 1.69612 -151 5.80461 -15 1.95316 -16) 6.46126

4.5 4. 6 4.7 4.8 4. 9

-14 1.01604 -15) 3.45215

5.0

(-15) 1.15173

(-16)3.70336

[12" r ('+ (-3)9.40315 97258

(-3)2.60416 66667

(-18)6.51829

I- 1

16 8.42124 -16 2. 7t.119

(-18)2.61062

1)I-l

(-6)8.13802 08333

(-6)1.69609 66316

319

ERROR FUNCTION AND FRESNEL INTEGRALS

DAWSON’S INTEGRAL X

e -X 2 h r e t 2 dt

5

e - ~ h rd d t

Table 7.5 X--2

xe-zsJox

el*c/t

<X>

00 02 04 06 08

0.00000 0.01999 0.03995 0.05985 0.07965

00000 46675 73606 62071 95389

1.00 1.02 1.04 1.06 1.08

0,53807 0.53637 0.53431 0.53192 0.52921

95069 44359 71471 50787 57454

0.250 0,245 0.240 0.235 0.230

0.60268 0.60046 0.59819 0.59588 0.59351

0777 6027 8606 1008 6018

2 2 2 2 2

0.10 0.12 0; 14 0.16 0.18

0.09933 0,11885 0.13818 0.15729 0.17616

59924 46083 49287 70920 19254

1.10 1.12 1.14 1.16 1.18

0.52620 0.52291 0.51935 0.51555 0.51152

66800 53777 92435 55409 13448

0.225 0.220 0.215 0.210 0.205

0.59110 0.58865 0.58616 0,58364 0.58109

6724 6517 9107 8516 9080

2 2 2 2 2

0. 20 0: 22 0. 24 0. 26 0.28

0.19475 0.21303 0.23099 0.24859 0.26581

10334 68833 28865 34747 41727

1.20 1.22 1.24 1.26 1.28

0.50727 0.50282 0.49820 0,49341 0.48847

34964 85611 27897 20827 19572

0.200 0.195 0.190 0.185 0.180

0.57852 0.57593 0.57332 0.57071 0.56809

5444 2550 5618 0126 1778

2 2 2 2 2

0.30 0.32 0.34 0.36 0.38

0.28263 0.29902 0.31496 0.33045 0.34544

16650 38575 99336 04051 71562

1.30 1.32 1.34 1.36 1.38

0.48339 0.47820 0.47290 0,46751 0.46204

75174 34278 38898 26208 28368

0.175 0.170 0.165 0.160 0.155

0.56547 0.56287 0,56027 0.55770 0.55516

6462 0205 9114 9305 6829

2 2 2 3 3

0.40 0.42 0.44 0.46 0.48

0.35994 0.37392 0.38737 0.40028 0,41264

34819 41210 52812 46599 14572

1.40 1.42 1.44 1.46 1.48

0,45650 0.45091 0.44528 0.43962 0.43394

72375 79943 67410 45670 20135

0.150 0.145 0.140 0.135 0.130

0,55265 0.55018 0.54776 0.54538 0.54305

7582 7208 0994 3766 9774

3 3 3 3 3

0.50 OI52 0.54 0.56 0.58

0.42443 0143566 0.44631 0.45637 0.46586

63835 16609 10184 96813 43551

1.50 1752 1.54 1.56 1.58

0.42824 OI42255 0.41686 0.41119 0.40555

90711 51804 92347 95842 40424

0.125 Oil20 0.115 0.110 0.105

0.54079 0153858 0.53643 0.53435 0.53233

2591 5013 8983 5529 4747

3 3 3 3

0.60 0.62 0.64 0.66 0.68

0.47476 0.48307 0.49080 0.49794 0.50451

32037 58219 32040 77064 30066

1.60 1.62 1.64 1.66 1.68

0.39593 0.39436 0.38883 0.38335 0.37792

98943 39058 23346 09429 50103

0.100 0.095 0.090 0,085 0.080

0.53037 0.52847 0.52663 0,52484 0.52311

5810 7031 5967 9575 4393

3 3 3 3 4

0.70 0.72 0.74 0.76 0.78

0.51050 0.51592 0.52078 0.52509 0.52886

40576 70382 93010 93152 66089

1.70 1.72 1.74 1.76 1.78

0.37255 0.36725 0.36202 0.35686 0.35178

93490 83182 58410 54206 01580

0.075 0.070 0.065 0.060 0. 055

0.52142 0.51978 0.51817 0.51661 0.51508

6749 2972 9571 3369 1573

4 4 4 4 4

0.80 0.82 0.84 0186 0.88

0.53210 0.53481 0.53702 0153873 0.53996

17071 60684 20202 26921 19480

1.80 1.82 1.84 1786 1.88

0.34677 0,34184 0.33700 0733223 0.32756

27691 56029 06597 96091 38080

0.050 0.045 0.040 01035 0.030

0.51358 0.51211 0.51067 0.50925 0.50786

1788 1971 0372 5466 5903

4 5 5 5 6

0.90 0.92 0.94 0.96 0.98

0,54072 0.54103 0.54090 0.54036 0.53941

43187 49328 94485 39857 50580

1.90 1.92 1.94 1.96 1.98

0.32297 0.31847 0.31405 0.30973 0.30549

43193 19293 71655 03141 14372

0.025 0.020 0.015 0.010 0.005

0.50650 0.50515 0150383 0,50253 0.50125

0473 8078 7717 8471 9494

6 7 8 10 14

1.00

0.53807 95069

2.00

0.30134 03889

0,000

0.50000 0000

0. 0. 0. 0. 0.

[( -271

See Example 3.

~

4

1

=nearest integer to x .

Compiled from J. B. Rosser, Theory and application of

soz

p-Z2dz

and

W I soz

e-P2@dy :f

3

m

P-~’~.v,

Mapleton House, Brooklyn, N.Y., 1948; and B. Lohmander and S. Rittsten, Table of the function p e - z P S0Z Pdt,Kungl. Fysiogr. Sallsk. i Lund Forh. 28,4552, 1958 (with permission).

I

,

320

ERROR FUNCTION AND FRESNEL INTEGRALS

rqrJs," 3

Table 7.6

e-t3dt

5

2

X

0. 00 0. 02 0. 04 0. 06 0. 08

0:00000 0.02239 0.04479 0.06718 0.08957

00 69 31 72 63

0.70 0.72 0. 74 0. 76 0. 78

0.'72276 0.73842 0,75360 0,76829 0.78247

69 49 34 12 88

1.40 1.42 1.44 1.46 1.48

0.'98973 0.99109 0.99229 0.99335 0.99429

54 36 70 97 49

0.10 0.12 0.14 0. 16 0.18

0.11195 0.13432 0,15667 0.17899 0.20127

67 36 11 22 90

0.80 0. 82 0. 84 0. 86 0. 88

0.79615 0.80932 0.82196 0.83408 0.84567

78 16 48 41 73

1.50 1.52 1.54 1.56 1.58

0.99511 0.99583 0.99645 0.99699 0.99746

49 14 52 62 38

0.20 0.22 0. 24 0.26 0.28

0.22352 0.24571 0.26783 0.28988 0.31184

24 24 80 71 70

0.90 '0.92 0.94 0.96 0.98

0,85674 0.86728 0.87730 0.88680 0,89580

42 62 62 89 05

1. 60 1.62 1.64 1. 66 1.68

0,99786 0,99821 0.99850 0.99875 0.99897

63 16 65 75 03

0. 30 0. 32 0. 34 0. 36 0. 38

0.33370 0.35544 0.37704 0.39850 0.41979

37 26 82 45 45

1.00 1.02 1. 04 1.06 1. 08

0.90428 0.91228 0.91979 0.92683 0.93341

86 25 27 11 06

1.70

0.99914 99

0.40 0. 42 0.44 0. 46 0.48

0.44090 0.46180 0.48248 0.50293 0.52312

07 52 96 51 25

1.10 1.12 1.14 1; 16 1.18

0.93954 0.94525 0.95054 0.95543 0.95995

56 09 27 76 30

1.70 1.74 1.78 1.82 1.86

0.99914 0.99942 0.99962 0.99975 0.99984

99 75 05 26 14

0.50 0. 52 0.54 0. 56 0. 58

0.54303 0.56264 0.58194 0.60090 0.61951

28 66 46 80 78

1.20 1.22 1.24 1.26 1.28

0,96410 0,96791 0.97140 0.97457 0.97746

64 62 05 79 66

1. 90 1.94 1.98 2. 02 2.06

0.99990 0.99993 0.99996 0.99997 0.99998

01 82 24 76 69

0. 60 0. 62 0. 64 0. 66 0. 68

0.63775 0,65560 0.67304 0.69006 0.70664

57 39 52 30 18

1.30 1.32 1.34 1. 36 1.38

0.98008 0.98245 0.98458 0.98649 0.98820

48 07 18 52 77

2.10 2.14 21 18 2.22 2.26

0.99999 0,99999 0.99999 0.99999 0.99999

25 57 77 87 93

0.70

0.72276 69

1.40

0.98973 54

2. 30

0.99999 97

Compiled from M. Abramowitz, Table of the integral (with permission).

S; e-u3du, J. Math. Phys. 30,162-163,1951

,

321

ERROR FUNCTION AND FRESNEL INTEGRALS

FRESNEL INTEGRALS

U=*2 z2

C(r)=C2(U)

S(r)=S2(U)

0. 00 0. 02 0. 04 0. 06 0. 08

0.00000 0.00062 0.00251 0.00565 0.01005

00 83 33 49 31

0.00000 0.02000 0.04000 0.05999 0.07999

00 00 00 98 92

0.00000 0.00000 0.00003 0.00011 0.00026

0.10 0. 12 0. 14 0.16 0. 18

0.01570 80 0; O M 95 0.03078 76 0.04021 24 0.05089 38

0.09999 0; 11999 0.13998 0.15997 0.17995

75 39 67 41 34

0. 20 0.22 0. 24 0. 26 0.28

0.06283 0.07602 0.09047 0.10618 0.12315

19 65 79 58 04

0.19992 0.21987 0.23980 0.25970 0.27957

0. 30 0. 32 0. 34 0. 36 0. 38

0.14137 0.16084 0.18158 0.20357 0.22682

17 95 41 52 30

0.40 0.42 0.44 0.46 0.48

0.25132 0.27708 0.30410 0.33238 0.36191

0.50 0.52 0.54 0.56 0.58

2

r

Table 7.7

U=; .E2

1.00 1.02 1.04 1.06 1.08

1.57079 1.63425 1.69897 1.76494 1.83217

0.00052 36

0; ooOS0 45

1.10

0.00143 67 0.00214 44 0.00305 31

1.12 1.14 1.16 1.18

1.90066 36 iI97040 69 2.04140 69 2.11366 35 2.18717 68

11 29 36 70 56

0.00418 0.00557 0.00723 0.00919 0.01148

76 30 40 54 16

1.20 1.22 1.24 1.26 1.28

2.26194 2. 33797 2.41525 2.49379 2.57359

0.29940 0.31917 0.33888 0.35851 0.37804

10 31 06 09 96

0.01411 0.01712 0.02053 0.02435 0.02862

70 56 11 68 55

1.30 1.32 1.34 1.36 1.38

74 85 62 05 15

0.39748 0.41678 0.43594 0.45494 0.47375

08 68 82 40 10

0.03335 0.03858 0.04430 0.05056 0.05736

94 02 85 42 63

0.39269 0.42474 0.45804 0.49260 0.52841

91 33 42 17 59

0.49234 0.51069 0.52878 0.54656 0.56401

42 69 01 30 31

0,06473 0.07267 0.08122 0.09037 0.10014

0.60 0.62 0.64 0.66 0.68

0.56548 0.60381 0.64339 0.68423 0.72633

67 41 82 89 62

0.58109 0.59777 0.61400 0.62976 0.64499

54 37 94 25 12

0.70 0.72 0.74 0.76 0.78

0.76969 0.81430 0.86016 0.90729 0.95567

02 08 81 20 25

0.65965 0.67370 0.68709 0.69977 0.71171

0.80 0.82 0.84 0.86 0.88

1.00530 1.05620 1.10835 1.16176 1.21642

96 35 39 10 47

0.90 0.92 0.94 0.96 0.98

1.27234 1.32952 1. 38795 1.44764 1.50859

50 20 56 59 28

-

see Example 8.

00 42 35 31 81

63 65 33 68 68

c( r )

=

cz (11)

0.77989 0.77926 0.77735 0,77414 0.76963

s ( . I , ) =s z ( I ) )

34 11 01 34 03

0.43825 0,45824 0.47815 0.49788 0.51736

91 58 08 84 86

0i71667 60 0.74824 94 0.73854 68 0.72759 68

0.76380 67

0.53649 oI55517 0.57331 0.59079 0.60752

79

92 28 66 74

67 33 64 62 27

0.71543 0.70211 0.68769 0.67223 0.65582

77 76 47 78 63

0.62340 0.63831 0.65216 0.66484 0.67626

09 34 19 56 72

2.65464 2.73695 2.82052 2.90534 2.99142

58 55 19 49 45

0.63855 0.62051 0.60181 0.58259 0.56297

05 11 95 73 59

0.68633 0.69495 0.70205 0.70755 0.71139

33 62 50 67 77

1.40 1.42 1.44 1.46 1.48

3.07876 3.16735 3.25720 3.34830 3.44067

08 37 33 95 23

0.54309 0.52310 0.50316 0.48342 0.46407

58 58 23 80 05

0.71352 0.71389 0.71248 0.70928 0.70428

51 77 78 16 12

24 89 06 08 09

1.50 1.52 1.54 1.56 1.58

3.53429 3.62916 3.72530 3.82268 3.92133

17 78 06 99 60

0.44526 0,42717 0.40997 0.39385 0.37895

12 32 99 29 96

0.69750 0.68898 0.67878 0.66697 0.65363

50 88 67 13 46

0.11054 0.12157 0.13325 0.14557 0.15853

02 59 28 29 54

1.60 1.62 1.64 1.66 1.68

4.02123 4.12239 4.22481 4.32848 4.43341

86 79 38 64 56

0.36546 0,35351 0.34325 0.33481 0.32830

17 20 29 32 61

0.63888 0.62286 0.60570 0.58758 0.56867

77 07 26 04 83

24 12 20 79 13

(1.17213 0.18636 0.20122 0.21668 0.23272

65 89 21 16 88

1.70 1.72 1.74 1.76 1.78

4.53960 4.64704 4.75574 4.86569 4.97691

14 39 30 87 11

0.32382 0,32145 0.32122 0.32318 0.32733

69 02 83 a7 25

0.54919 0.52934 0.50935 0.48946 0.46990

60 73 84 49 94

0.72284 0.73312 0.74251 0.75095 0.75840

42 83 54 79 90

0.24934 0.26649 0.28414 0.30227 0.32083

14 22 98 80 55

1.80 1.82 1.84 1.86 1.88

5.08938 5.20310 5.31808 5.43432 5.55182

01 58 80 70 25

0.33363 0.34203 0.35244 0.36476 0.37882

29 39 96 35 93

0.45093 0.43280 0.41573 0.39999 0.38579

88 06 97 44 25

0.76482 0.77015 0.77436 0.77741 0.77926

30 63 72 68 95

0.33977 0.35904 0.37859 0.39836 0.41827

63 93 81 12 21

1.90 1.92 1.94 1.96 1.98

5.67057 5.79058 5.91184 6.03437 6.15814

47 36 91 12 99

0.39447 0.41148 0.42963 0.44866 0.46830

05 24 33 69 56

0.37334 0.36285 0.35448 0.34838 0.34466

73 37 37 30 65

322

ERROR FUNCTION AND FRESNEL INTEGRALS

Table 7.7

s( J )

2.00 2.02 2.04 2.06 2.08

C(4 0.48825 0.50820 0.52782 0.54681 0.56482

34 04 73 06 79

0.34341 0.34467 0.34844 0.35470 0.36334

57 48 87 04 98

3.00 3.02 3.04 3.06 3.08

0.60572 0.60383 0.59823 0.58910 0.57674

08 73 78 11 01

0.49631 0.51619 0.53536 0.55311 0.56880

30 42 29 95 28

4.00 4.02 4.04 4.06 4.08

0.49842 0.51821 0.53675 0.55284 0.56543

60 54 05 04 47

0.42051 0.42301 0.43039 0.44217 0.45764

58 99 00 81 45

2.10 2.12 2.14 2.16 2.18

0.58156 0.59671 0.61000 0.62117 0.62999

41 75 60 32 53

0.37427 0.38730 0.40223 0.41880 0.43673

34 37 09 45 63

3.10 3.12 3.14 3.16 3.18

0.56159 0.54421 0.52525 0.50543 0.48552

39 58 53 56 76

0.58181 0.59165 0.59791 0.60033 0.59880

59 11 29 66 34

4.10 4.12 4.14 4.16 4.18

0,57369 56 0.57705 88 0.57527 76 0.56844 74 0.55700 75

0.47579 0.49545 0.51532 0.53405 0.55039

83 71 14 87 41

2.20 2.22 2.24 2.26 2.28

0.63628 0,63990 0.64075 0.63879 0.63403

60 31 25 28 83

0.45570 0.47535 0.49532 0.51521 0.53462

46 85 41 11 03

3.20 3.22 3.24 3.26 3.28

0.46632 0.44858 0.43306 0.42040 0.41113

03 96 55 05 97

0.59334 0.58416 0,57161 0.55618 0.53849

95 97 47 06 35

4.20 4.22 4.24 4.26 4.28

0.54171 0.52362 0.50396 0.48411 0.46549

92 06 08 63 61

0.56319 0.57157 0.57491 0.57295 0.56582

89 23 03 47 05

2.30 2.32 2.34 2. 36 2.38

0.62656 0.61649 0.60402 0.58940 0.57293

17 45 69 65 44

0.55315 0.57041 0.58602 0.59964 0.61095

16 28 84 89 96

3.30 3.32 3.34 3.36 3.38

0.40569 0.40431 0.40709 0.41393 0.42455

44 99 96 66 18

0.51928 0.49936 0.47960 0.46084 0.44393

61 95 04 46 82

4.30 4.32 4.34 4.36 4.38

0.44944 0.43712 0.42946 0.42704 0.43006

12 50 40 39 79

0.55399 0.53831 0.51990 0.50011 0.48041

59 55 77 73 08

2. 40 2.42 2. 44 2. 46 2. 48

0.55496 0.53588 0.51612 0.49614 0.47641

14 11 29 28 35

0.61969 0.62562 0.62859 0.62851 0.62535

00 11 38 43 98

3.40 3.42 3.44 3.46 3.48

0.43849 0.45514 0.47375 0.49348 0.51340

17 37 96 70 62

0.42964 0.41864 0.41143 0.40839 0.40967

95 11 69 28 54

4.40 4.42 4.44 4.46 4.48

0.43833 0.45123 0.46781 0.48679 0.50671

29 59 05 41 95

0.46226 0.44707 0.43599 0.42990 0.42931

80 06 33 86 16

2. 50 2.52 2.54 2.56 2. 58

0.45741 0.43961 0.42346 0.40939 0.39777

30 32 72 65 91

0.61918 0.61010 0,59834 0.58415 0.56790

18 76 06 75 42

3.50 3.52 3.54 3.56 3.58

0.53257 0.55006 0.56501 0.57668 0.58446

24 11 32 02 43

0.41524 0.42486 0.43808 0.45428 0.47265

80 72 83 17 92

4.50 4.52 4.54 4.56 4.58

0.52602 0.54318 0.55680 0.56578 0.56936

59 11 46 27 57

0.43427 0.44442 0.45897 0.47676 0.49637

30 34 36 89 56

2.60 2.62 2. 64 2. 66 2. 68

0.38893 0.38312 0.38052 0.38123 0.38525

75 73 80 50 32

0.54998 0.53087 0.51106 0.49110 0.47153

93 53 79 35 52

3.60 3.62 3.64 3.66 3.68

0.58795 0,58694 0.58147 0.57178 0.55838

33 64 10 75 18

0.49230 0.51224 0.53143 0.54888 0.56366

95 12 21 15 38

4.60 4.62 4.64 4.66 4.68

0.56723 0.55954 0.54691 0.53039 0.51135

67 81 86 13 38

0.51619 0,53457 0.54999 0.56113 0.56702

23 97 67 28 44

2. 70 2. 72 2.74 2. 76 2. 78

0.39249 0.40277 0.41581 0.43125 0.44865

40 39 68 85 46

0.45291 0.43578 0.42066 0.40798 0.39817

75 98 03 90 24

3.70 3.72 3.74 3.76 3.78

0.54194 0.52334 0.50357 0.48371 0.46487

57 49 70 94 19

0.57498 0.58220 0.58492 0.58296 0.57641

04 56 61 92 91

4.70 4.72 4.74 4.76 4.78

0.49142 0.47232 0.45572 0.44308 0.43554

65 71 30 30 28

0.56714 0.56146 0.55044 0.53504 0.51659

55 19 52 16 82

2.80 2. 82 2.84 ~- - . 2.86 2. 88

0.46749 0:48720 0.50717 0.52677 0.54538

17 04 21 66 21

0.39152 Oi38828 0,38856 0.39238 0.39964

84 41 43 50 80

3.80 3.82 3.84 3.86 3.88

0.44809 0.43434 0.42443 0.41894 0.41822

49 86 43 43 16

0.56561 0.55115 0.53384 0.51466 0.49472

87 74 32 22 45

4.80 4.82 4.84 4.86 4.88

0.43379 0.43802 0.44786 0.46244 0.48042

66 47 69 40 90

0.49675 0.47728 0.45995 0.44637 0.43780

02 00 75 74 82

2. 90 2.92 2.94 2. 96 2.98

0.56237 0.57718 0.58930 0.59830 0.60384

64 78 60 19 56

0.41014 0.42353 0.43941 0.45724 0.47643

06 87 39 45 06

3.90 3.92 3.94 3.96 3.98

0.42233 0.43105 0.44389 0.46007 0.47863

27 68 17 70 51

0.47520 0.45726 0.44198 0.43032 0.42301

24 13 92 79 17

4.90 4.92 4.94 4.96 4.98

0.50016 0.51979 0.53747 0.55150 0.56051

10 51 34 25 94

0.43506 0.43843 0.44761 0.46175 0.47951

74 48 56 67 78

3. 00

0.60572 08

0.49631 30

4.00

0.49842 60

0.42051 58

5.00

0.56363 12

z

'(') = 0.5+.3183099 For .I ,>5 S(3.)

0.49919 14

323

ERROR FUNCTION AND FRESNEL INTEGRALS

AUXILIARY FUNCTIONS 2 .$

f (x)= f 2 ( U )

Table 7.8

0. 00 0. 02 0.04 0. 06 0. 08

0.00000 0.00062 0.00251 0.00565 0.01005

00000 83185 32741 48667 30964

00000 30718 22872 76462 91487

0.50000 0.49969 0.49880 0.49739 0.49548

00000 41196 88057 07811 44294

00000 39303 20520 66949 00553

9 (4= Q2 ( U ) 0.50000 00000 00000 0.48031 40626 54163 0.46125 51239 79101 0.44281 99356 00196 0.42500 33536 38036

0.10 0.12 0.14 0.16 0.18

0.01570 0.02261 0.03078 0.04021 0.05089

79632 94671 76080 23859 38009

67949 05847 05180 65949 88155

0.49313 0.49037 0.48724 0.48378 0.48002

18256 27777 48761 35493 21268

06624 82254 11561 31728 70713

0.40779 0.39119 0.37518 0.35976 0.34491

85545 72364 98069 55566 28197

29930 96391 99885 09573 39391

0.20 0.22 0.24 0. 26 0.28

0.06283 0.07602 0,09047 0.10618 0.12315

18530 65422 78684 58316 04320

71796 16873 23386 91335 20720

0.47599 0.47172 0.46724 0.46257 0.45774

19056 22205 05176 24293 18508

49140 45221 22164 12303 40978

0.33061 0.31687 0.30365 0.29095 0.27876

91227 13200 57186 81914 42811

69034 89318 36191 92531 44593

0. 30 0. 32 0.34 0. 36 0. 38

0.14137 0.16084 0.18158 0.20357 0.22682

16694 95438 40553 52039 29895

11541 63797 77490 52619 89183

0.45277 0.44768 0.44248 0.43721 0.43187

10172 05805 96860 60487 60273

56087 06203 81319 95888 53913

0.26705 0.25582 0.24505 0.23472 0.22483

92929 83796 66166 90703 08578

81728 24420 57772 35799 07150

0. 40 0. 42 0.44 0.46 0. 48

0.25132 0.27708 0.30410 0.33238 0.36191

74122 84720 61688 05027 14736

87183 46620 67492 49800 93544

0.42648 0.42105 0.41560 0.41013 0.40466

46973 59227 24246 58491 68313

90789 36507 90070 35691 67950

0.21534 0.20626 0.19756 0.18923 0.18126

72003 34704 52322 82774 86555

95520 48744 49727 60398 47172

0. 50 0. 52 0. 54 0. 56 0.58

0.39269 0.42474 0.45804 0.49260 0.52841

90816 33267 42088 17280 58843

98724 65340 93392 82880 33803

0.39920 0. 39375 0.38833 0.36294 0.37759

50585 93295 76127 71004 42617

25702 63563 15400 26771 52882

0.17364 0.16634 0.15936 0.15269 0.14631

26996 70480 86623 48414 32329

13238 39628 13733 00876 91905

0. 60 0. 62 0. 64 0. 66 0. 68

0.56548 0.60381 0.64339 0.68423 0.72633

66776 41080 81754 a8799 62215

46163 19958 55190 51857 09960

0.37228 0.36702 0.36181 0.35666 0.35157

48922 41612 66571 64292 70288

35620 87842 25476 98472 80259

0.14021 0.13437 0.12880 0.12347 0.11838

18419 90361 35503 44874 13187

37684 59907 06985 03863 25611

0. 70 0. 72 0.74 0.76 0. 78

0.76969 0.81430 0.86016 0.90729 0.95567

02001 08158 80685 19583 24852

29499 10474 52885 56732 22015

0.34655 0.34159 0.33670 0.33188 0.32713

15463 26474 26065 33382 64271

82434 67053 33192 57734 72503

0.11351 0.10886 0.10441 0.10016 Oi09611

38821 23788 73696 97688 08389

06517 79214 22082 77848 91866

0. 80 0. 82 0. 84 0. 86 0.88

1.00530 1.05620 1.10835 1.16176 1.21642

96491 34501 38881 09632 46754

48734 36888 86479 97506 69968

0.32246 0.31786 0.31334 0.30889 0.30452

31553 45283 12993 39917 29200

61284 60796 49704 09068 36579

0.09223 0.08852 0.08498 0.08159 0.07836

21832 57381 37656 88446 38619

05037 23702 77045 61614 62362

0. 90 0.92 0.94 0. 96 0. 98

1.27234 1.32952 1.38795 1.44764 1.50859

50247 20109 56343 58947 27922

03866 99200 55971 74177 53819

0.30022 0.29600 0.29186 0.28780 0.28380

82096 98149 75359 10340 98467

95385 76518 51781 91658 20271

0.07527 0.07231 0.06949 0.06679 OiO6420

20035 67451 18433 13255 94813

30280 87932 26312 49021 13093

1. 00

1.57079 63267 94897

U

X

2

See Examples 6,7 , and 9.

0.27989 34003 76823

0.06174 08526 09645

324

ERROR FUNCTION AND FRESNEL INTEGRALS

AUXILIARY FUNCTIONS

Table 7 . 8 ,U- 1

!,-I=

2

x.?

1.00

0.98 0.96 0.94 0.92

0.63661 0.61140 0.58670 0.56251 0.53883

97723 96293 87822 72308 49753

67581 81825 13963 63995 31921

.f ).( 0.27989 0.27597 0.27197 0.26788 0.26371

34003 33733 11505 56989 60682

76823 36442 76851 47656 37287

9(4= 0.06174 08526 0.05933 31378 0.05693 89827 0.05456 06112 0.05220 03510

09645 64174 01255 91100 52931

0.90 0.88 0.86 0.84 0.82

0.51566 0.49299 0.47084 0.44919 0.42806

20156 83517 39836 89113 31349

17741 21455 43063 82565 39962

0.25946 0.25512 0.25069 0.24618 0.24157

14023 09512 40835 02994 92449

65674 80091 25766 44393 31459

0.04986 0.04754 0.04525 0.04299 0.04075

06317 39838 30354 05078 92107

93636 94725 03048 69390 68723

0.80 0.78 0.76 0.74 0. 72

0.40743 0.38731 0.36771 0.34861 0.33002

66543 94695 15805 29873 36899

15252 08436 19515 48488 95354

0.23689 0.23211 0.22725 0.22230 0.21726

07256 47216 14019 11393 45250

57089 24632 06110 53995 44609

0.03856 0.03640 0.03428 0.03220 0.03017

20343 19405 19524 51407 46086

27312 75704 44132 19129 88637

0. 70 0.68 0.66 0. 64 0. 62

0.31194 0.29437 0.27731 0.26075 0.24471

36884 29827 15728 94587 66404

60115 42770 43318 61761 98098

0.21214 0.20693 0.20164 0.19627 0.19082

23821 57789 60404 47584 37987

60229 65521 80635 00004 55563

0.02819 0.02626 0.02439 0.02257 0.02082

34743 48498 18186 74093 45674

19381 36510 13588 32978 44482

0.60 0.58 0. 56 0.54 0. 52

0.22918 0.21415 0.19964 0.18563 0.17214

31180 88914 39606 83256 19864

52329 24454 14474 22387 48194

0.18529 0.17969 0.17401 0.16827 0.16245

53067 17083 57076 02799 86594

79209 86674 89207 47273 19322

0.01913 0.01751 0.01596 0.01448 0.01307

61240 47623 29821 30628 70253

35536 30357 58470 73722 60097

0. 50 0. 48

0. 46 0.44 0. 42

0.15915 0.14667 0.13470 0.12324 0.11229

49430 71955 87438 95879 97278

91895 53491 32980 30364 45641

0.15658 0.15065 0.14466 0.13862 0.13253

43216 09597 24548 28400 62592

36302 56320 29603 34552 29647

0.01174 0.01049 0.00931 0.00822 0.00720

65939 31590 77420 09631 30137

24659 42015 66589 52815 00215

0. 40 0.38 0. 36 0. 34 0. 32

0.10185 0.09192 0.08250 0.07359 0.06518

91635 78951 59224 32456 98646

78813 29879 98839 85692 90440

0.12640 0,12023 0.11403 0.10780 0.10154

69204 90456 68174 43252 55126

94864 93806 47880 41741 32988

0.00626 0.00540 0.00461 0.00390 0.00327

36346 21018 72197 73235 02912

49122 72942 27002 12822 03254

3

0. 30 0.28 0. 26 0.24 0.22

0.05729 0.04991 0.04303 0.03666 0.03081

57795 09901 54966 92988 23969

13082 53618 12048 88373 82591

0.09526 0.08896 0.08264 0.07631 0.06997

41276 36786 73969 82087 87161

74844 39974 33180 00913 16730

0.00270 0.00220 0.00176 0.00139 0.00107

35642 41768 87922 37442 50825

68526 84885 53708 77909 02743

3 4 4 4 5

17 20 23 27 32

0. 20 0. 18

0.16 0.14 0.12

0.02546 0.02062 0.01629 0.01247 0.00916

47908 64806 74661 77475 73247

94703 24710 72610 38405 22093

0.06363 0.05727 0.05091 0.04455 0.03819

11887 75644 94597 81874 47805

04012 30652 59575 32960 44642

0.00080 0.00058 0.00041 0.00027 0.00017

86180 99686 45999 78633 50279

82883 10701 18234 97799 00844

5 6 6 7 8

39 48 61 80 109

0.10 0.08 0.06 0.04 0.02

0.00636 0.00407 0.00229 0.00101 O.QO025

61977 43665 18311 85916 46479

23676 43153 80523 35788 08947

0.03183 0.02546 0.01909 0.01273 0.00636

00214 44738 85179 23855 61974

15118 95252 38105 39770 14061

0.00010 0.00005 0.00002 0.00000 0.00000

13057 18732 18849 64845 08105

94484 17470 44630 30524 69272

10 13 17 25 50

157 245 436 982 3927

4)

m

= f 2 (U)

<.c>

1 1 1 1 1

3 3

10 11

12 14 15

325

ERROR FUNCTION AND FRESNEL INTEGRALS

Table 7.9

ERROR FUNCTION FOR COMPLEX ARGUMENTS w(z)=e-z2 erfc (-iz)

9w(z) Yw(2)

Y

WW(Z)

4w(zj

x=o.1

x=o

zQw(z)

z=z+iy

YUW(2)

x=0.2

9w(z) .fw(z)

Ww(2)

x=0.3

BzL'(2)

x=0.4

0.0 0.1 0.2 0.3 0.4

1.000000 0.896457 0.809020 0.734599 0.670788

0.000000 0.000000 0.000000 0.000000 0.000000

0.990050 0.888479 0.802567 0.729337 0.666463

0.112089 0.094332 0.080029 0.068410 0.058897

0.960789 0.864983 0.783538 0.713801 0.653680

0.219753 0.185252 0.157403 0.134739 0.116147

0.913931 0.827246 0.752895 0.688720 0.632996

0.318916 0.269600 0.229653 0.197037 0.170203

0.852144 0.777267 0.712146 0.655244 0.605295

0.406153 0.344688 0.294653 0.253613 0.219706

0.5 0.6 0.7 0.8 0.9

0.615690 0.567805 0.525930 0.489101 0.456532

0.000000 0.000000 0.000000 0.000000 0.000000

0.612109 0.564818 0.523423 0.486982 0.454731

0.051048 0.044524 0.039064 0.034465 0.030566

0.601513 0.555974 0.515991 0.480697 0.449383

0.100782 0.087993 0.077275 0.068235 0.060563

0.584333 0.541605 0.503896 0.470452 0.440655

0.147965 0.129408 0.113821 0.100647 0.089444

0.561252 0.522246 0.487556 0.456579 0.428808

0.191500 0.167880 0.147975 0.131101 0.116714

1.0 1.1 1.2 1.3 1.4

0.427584 0.401730 0.378537 0.357643 0.338744

0,000000 0.000000 0.000000 0.000000 0.000000

0.426044 0.400406 0.377393 0.356649 0.337876

0.027242 0.024392 0.021934 0.019805 0.017951

0.421468 0.396470 0.373989 0.353691 0.335294

0.054014 0.048393 0.043542 0.039336 0.035671

0.413989 0.390028 0.368412 0.348839 0.331054

0.079864 0.071628 0.064510 0.058329 0.052936

0.403818 0.381250 0.360799 0.342206 0.325248

0.104380 0.093752 0.084547 0.076538 0.069538

1.5 1.6 1.7 1.8 1.9

0.321585 0.305953 0.291663 0.278560 0.266509

0.000000 0.000000 0.000000 0.000000 0.000000

0.320825 0.305284 0.291072 0.278035 0.266042

0.016329 0.014905 0.013648 0.012536 0.011547

0.318561 0.303290 0.289309 0.276470 0.264648

0.032463 0.029643 0.027154 0.024948 0.022987

0.314839 0.300009 0.286406 0.273892 0.262350

0.048210 0.044051 0.040377 0.037118 0.034217

0.309736 0.295506 0.282417 0.270346 0.259186

0.063393 0.057978 0.053186 0.048931 0.045139

2.0 2.1 2.2 2.3 2.4

0.255396 0.245119 0.235593 0.226742 0.218499

0.000000 0.000000 0.000000 0,000000 0.000000

0.254978 0.244745 0.235256 0.226438 0.218224

0.010664 0.009874 0.009165 0.008526 0.007949

0.253732 0.243628 0.234251 0.225531 0.217404

0.021236 0.019669 0.018260 0.016991 0.015845

0.251677 0.241783 0.232592 0.224033 0.216047

0.031626 0.029304 0.027217 0.025335 0.023633

0.248844 0.239239 0.230300 0.221963 0.214172

0.041748 0.038706 0.035968 0.033498 0.031263

2.5 2.6 2.7 2.8 2.9

0.210806 0.203613 0.196874 0.190549 0.184602

0.000000 0.000000

0.210557 0.203387 0.196668 0.190360 0.184429

0.007427 0.006952 0.006520 0.006125 0.005764

0.209813 0.202710 0.196050 0.189796 0.183912

0.014806 0.013862 0.013002 0.012216 0.011498

0.208582 0.201589 0.195028 0.188861 0.183056

0.022090 0.020687 0.019409 0.018241 0.017172

0.206879 0.200039 0.193613 0.187566 0.181868

0.029234 0.027389 0.025706 0.024168 0.022759

3.0

0.179001 0,000000

Y

i

0.000000 0.000000 0.000000

0.178842 0.005433

~=0.6

x=n.Fi

0.178368 0.010839

~=0.7

0.177581 0.016192

x=0.8

0.176491 0.021466

x=o.9

0.0 0.1 0.2 0.3 0.4

0.778801 0.717588 0.663223 0.614852 0.571717

0.478925 0.408474 0.350751 0.303124 0.263563

0.697676 0.651076 0.608322 0.569238 0.533581

0.535713 0.459665 0.396852 0.344645 0.300989

0.612626 0.580698 0.549739 0.520192 0.492289

0.576042 0.497744 0.432442 0.377688 0.331535

0.527292 0.509299 0.489710 0.469480 0.449244

0.600412 0.522932 0.457569 0.402194 0.355082

0.444858 0.439421 0.430271 0.418736 0.405763

0.610142 0.536087 0.472773 0.418491 0.371813

0.5 0.6 0.7 0.8 0.9

0.533157 0.498591 0.467521 0.439512 0.414191

0.230488 0.202666 0.179123 0.159087 0.141945

0.501079 0.471453 0.444434 0.419766 0.397216

0.264268 0.233206 0.206787 0.184200 0.164793

0.466127 0.441712 0.418998 0.397906 0.378341

0.292432 0.259136 0.230646 0.206155 0.185005

0.429418 0.410264 0.391936 0.374518 0.358043

0.314828 0.280290 0.250532 0.224789 0.202429

0.392021 0.377977 0.363957 0.350182 0.336799

0.331544 0.296692 0.266427 0.240057 0.217004

1.0 1.1 1.2 1.3 1.4

0.391234 0.370363 0.351335 0.333942 0.318001

0.127202 0.114460 0.103395 0.093744 0.085288

0.376571 0.357637 0.340241 0.324229 0.309463

0.148036 0.133501 0.120838 0.109759 0.100026

0.360200 0.343375 0.327766 0.313273 0.299804

0.166660 0.150681 0.136706 0.124435 0.113620

0.342511 0.327900 0.314176 0.301294 0.289208

0.182932 0.165868 0.150877 0.137661 0.125971

0.323899 0.311537 0.299741 0.288519 0.277865

0.196783 0.178990 0.163281 0.149370 0.137012

1.5 1.6 1.7 1.8 1.9

0.303355 0.289866 0.277412 0.265890 0.255205

0.077851 0.071283 0.065461 0.060283 0.055661

0.295820 0.283192 0.271479 0.260598 0.250469

0.091443 0.083845 0.077096 0.071081 0.065701

0.287274 0.275602 0.264718 0.254554 0.245050

0.104054 0.095563 0.088001 0.081245 0.075190

0.277869 0.267228 0.257237 0.247851 0.239027

0.115594 0.106355 0.098103 0.090710 0.084068

0.267766 0.258203 0.249151 0.240586 0.232482

0.126002 0.116164 0.107348 0.099427 0.092291

2.0 2.1 2.2 2.3 2.4

0.245276 0.236031 0.227407 0.219347 0.211800

0,051521 0.047804 0.044454 0.041428 0.038686

0.241025 0.232204 0.223952 0.216219 0.208961

0.060876 0.056534 0.052617 0.049073 0.045859

0.236152 0.227810 0.219978 0.212616 0.205686

0.069748 0.064842 0.060409 0.056391 0.052741

0.230724 0.222905 0.215535 0.208581 0.202013

0.078085 0.072680 0.067785 0.063342 0.059298

0.224813 0.217552 0.210676 0.204160 0.197982

0.085845 0.080009 0.074712 0.069894 0.065500

2.5 2.6 2.7 2.8 2.9

0.204723 0.198074 0.191818 0.185924 0.180361

0.036196 0.033929 0.031859 0.029966 0.028231

0.202139 0.195717 0.189664 0.183950 0.178549

0.042936 0.040271 0.037836 0.035607 0.033561

0.199155 0.192992 0.187170 0.181662 0.176441

0.049417 0.046384 0.043608 0.041064 0.038728

0.195804 0.189928 0.184362 0.179084 0.174074

0.055610 0.052238 0.049156 0.046315 0.043708

0.192120 0.186554 0.181265 0.176237 0.171452

0.061486 0.057811 0.054439 0.051339 0.048485

3.0

0.175105 0.026636

0.173437 0.031680

see Examples 12-19.

0.171502 0.036577

w(x)=e-22 + 2 i-e - 2 2 ir

soet2 dt

0.169315 0.041306

0.166895 0.045851

2

~ ( - ~ + i y ) = w ( z + i y ) w ( z - i y ) = 2 e ~ ~ (cos - 2 ~ 2xy+i sin 2zy)-w(x+iy) w(iy)=ey2erfc y

326

ERROR FUNCTION AND FRESNEL INTEGRALS

ERROR FUNCTION FOR COMPLEX ARGUMENTS

Table 7.9

w(z)=e-z2erfc (-iz) 9w(z) Y

9 w ( z ) AN(.)

JW(2)

x=1.0

%W(Z)

Yw(2)

x=1.2

x=l.l

z=x+iy Ww(2) . U W ( Z )

x=1.3

%w(z) ,"W(t)

~=1.4

0.1 0.2 0.3 0.4

0.367879 0.373170 0.373153 0.369386 0.363020

U.607158 0.538555 0.478991 0.427225 0.382166

0.298197 0.312136 0.319717 0.322586 0.321993

0.593761 0.532009 0.477439 0.429275 0.386777

0.236928 0.257374 0.270928 0.279199 0.283443

0.572397 0.518283 0.469488 0.425667 0.386412

0.184520 0.209431 0.227362 0.239793 0.247908

0.545456 0.499216 0.456555 0.417491 0.381908

0.140858 0.168407 0.189247 0.204662 0.215711

0.515113 0.476535 0.440005 0.405823 0.374110

0.5 0.6 0.7 0.8 0.9

0.354900 0.345649 0.335721 0.325446 0.315064

0.342872 0.308530 0.278445 0.252024 0.228759

0.318884 0.313978 0.307816 0.300807 0.293259

0.349266 0.316128 0.286815 0.260847 0.237800

0.284638 0.283540 0.280740 0.276693 0,271752

0.351299 0.319910 0.291851 0.266757 0.244295

0.252654 0.254784 0.254895 0.253461 0.250858

0.349611 0.320368 0.293927 0.270040 0.248462

0.223262 0.228026 0.230578 0.231385 0.230826

0.344868 0.318022 0.293453 0.271015 0.250549

1.0 1.1 1.2 1.3 1.4

0.304744 0.294606 0.284731 0.275174 0.265967

0.208219 0.190036 0.173896 0.159531 0.146712

0.285402 0.277407 0.269401 0.261476 0.253697

0.217306 0.199046 0.182742 0,168151 0.155066

0.266189 0.260213 0.253985 0.247628 0.241233

0.224168 0.206108 0.189878 0.175271 0.162100

0.247381 0.243266 0.238695 0.233813 0.228733

0.228967 0.211343 0.195398 0.180957 0.167863

0.229205 0.226767 0.223710 0.220192 0.216340

0.231897 0.214902 0.199416 0.185299 0.172423

1.5 1.6 1.7 1.8 1.9

0.257128 0.248665 0.240578 0.232861 0.225503

0.135242 0.124954 0.115702 0.107361 0.099824

0.246112 0.238752 0.231635 0.224775 0.218176

0.143305 0.132711 0.123147 0.114495 0.106650

0.234870 0.228592 0.222436 0.216428 0.210587

0.150205 0.139441 0.129684 0.120822 0.112760

0.223542 0.218309 0.213086 0.207912 0.202818

0.155975 0.145167 0.135326 0.126353 0.118158

0.212253 0.208014 0.203684 0.199315 0.194947

0.160668 0.149927 0.140103 0.131106 0.122858

2.0 2.1 2.2 2.3 2.4

0.218493 0.211816 0.205457 0.199402 0.193634

0.092998 0.086801 0.081162 0.076021 0.071324

0.211839 0.205760 0.199935 0.194356 0.189014

0.099523 0.093035 0.087116 0.081706 0.076753

0.204926 0.199452 0.194166 0.189072 0.184165

0.105411 0.098700 0.092562 0.086936 0.081773

0.197827 0.192953 0.188208 0.183599 0.179131

0.110662 0.103795 0.097495 0.091706 0.086378

0.190608 0.186324 0.182112 0.177985 0.173954

0.115286 0.108325 0.101919 0.096015 0.090567

2.5 2.6 2.7 2.8 2.9

0.188139 0.182903 0.177910 0.173147 0.168602

0.067024 0.063080 0.059456 0.056118 0.053041

0.183901 0.179008 0.174324 0.169840 0.165546

0.072208 0.068031 0.064186 0.060639 0.057363

0.179444 0.174903 0.170538 0.166342 0.162310

0.077024 0.072651 0.068617 0.064890 0.061440

0.174805 0.170623 0.166582 0.162681 0.158916

0.081467 0.076933 0.072742 0.068863 0.065266

0.170024 0.166201 0.162487 0.158883 0.155389

0.085532 0.080873 0.076557 0.072553 0.068834

3.0

0.164261 0.050197

0.0

Y

~=1.5

0.161434 0.054331

x=1.6

0.158435 0.058243

0.155285 0.061926

~=1.7

0.152005 0.065375

~=1.8

x=1.9

0.0 0.1 0.2 0.3 0.4

0.105399 0.134049 0.156521 0.173865 0.186984

0.483227 0.451763 0.421076 0.391665 0.363828

0.077305 0.105843 0.128895 0.147272 0.161702

0.451284 0.426168 0.400837 0.375911 0.351803

0.055576 0.083112 0.105929 0.124612 0.139717

0.420388 0,400743 0.380161 0.359313 0.338676

0.039164 0.065099 0.087090 0.105522 0.120793

0.391291 0.376214 0.359721 0.342479 0.324985

0.027052 0.051038 0.071811 0.089592 0.104641

0.364437 0.353066 0.340004 0.325873 0.311161

0.5 0.6 0.7 0.8 0.9

0.196636 0.203461 0.207990 0.210664 0.211846

0.337720 0.313397 0.290847 0.270016 0.250823

0.172820 0.181177 0.187245 0.191423 0.194049

0.328777 0.306990 0.286517 0.267378 0.249556

0.151751 0.161171 0.168379 0.173725 0.177513

0.318584 0.299261 0.280846 0.263418 0,247012

0.133288 0.143369 0.151366 0.157578 0.162268

0.307609 0.290613 0.274180 0.258431 0.243439

0.117233 0.127644 0.136134 0.142949 0.148310

0,296240 0.281392 0.266823 0.252681 0.239067

1.0 1.1 1.2 1.3 1.4

0.211837 0.210881 0.209182 0.206902 0.204177

0.233171 0.216954 0.202067 0.188403 0.175862

0.195407 0.195734 0.195228 0.194053 0.192347

0.233009 0.217678 0.203494 0.190384 0.178275

0.180002 0.181414 0.181938 0.181733 0.180933

0.231630 0.217253 0,203847 0.191366 0.179762

0.165667 0.167977 0.169373 0.170003 0.169997

0.229244 0.215857 0.203272 0.191471 0.180425

0.152418 0.155452 0.157569 0.158906 0.159585

0.226046 0.213656 0.201914 0.190821 0.180367

1.5 1.6 1.7 1.8 1.9

0.201115 0.197806 0.194320 0.190717 0.187043

0.164349 0.153773 0.144054 0.135113 0.126883

0.190222 0.187772 0.185073 0.182189 0.179172

0.167092 0.156765 0.147226 0.138412 0.130262

0.179651 0.177983 0.176008 0,173792 0.171390

0.168980 0.158969 0.149674 0.141045 0.133033

0.169465 0.168500 0.167183 0.165579 0.163746

0.170099 0.160457 0.151458 0.143063 0.135234

0.159709 0.159369 0.158641 0.157593 0.156282

0.170534 0.161300 0.152637 0.144516 0.136908

2.0 2.1 2.2 2.3 2.4

0.183335 0.179623 0.175930 0.172276 0.168674

0.119298 0.112302 0.105842 0.099870 0.094343

0.176064 0.172901 0.169710 0.166513 0.163330

0.122723 0.115744 0.109277 0.103280 0.097713

0.168849 0.166206 0.163493 0,160737 0.157958

0.125590 0.118674 0.112243 0.106260 0.100689

0.161733 0.159580 0.157320 0.154982 0.152591

0.127931 0.121118 0.114761 0.108827 0.103285

0.154757 0.153059 0.151224 0.149281 0.147256

0.129781 0.123108 0.116858 0.111003 0.105519

2.5 2.6 2.7 2.8 2.9

0.165136 0.161669 0.158281 0.154975 0.151753

0.089222 0.084472 0.080061 0.075960 0.072142

0.160175 0.157060 0.153993 0.150981 0.148030

0.092541 0.087732 0.083254 0.079082 0.075191

0.155175 0.152402 0.149649 0.146927 0.144243

0.095499 0.090660 0.086143 0.081925 0.077982

0.150165 0.147722 0.145274 0.142834 0.140411

0.098107 0.093265 0.088735 0.084493 0.080519

0.145172 0.143045 0.140892 0.138725 0.136555

0.100378 0.095558 0.091037 0.086794 0.082809

3.0

0.148618 0.068585

0.145144 0.071558

See Examples 12-19.

w(-x+iy)=-w(x+iy)

0.141602 0.074293

24

0.138012 0.076794

0.134391 0.079065

w(x)=e-z2+4" e-22 Soe 2 12 dt w(x-iy)=2e~2-22(cos Bxy-ti sin Bxy)-w(x+iy)

327

ERROR FUNCTION AND FRESNEL INTEGRALS

Table 7.9

ERR OR FUNCTION FOR COMPLEX ARGUMENTS Au(z) Y

x=2.0

%u(z)

w(z)=e-Zz erfc (-iz) z=z+iy Ww(2) h ( z ) d w ( z ) .Yw(z) fw(z)

~=2.4 0.003151 0.017397 0.030792 0.043211 0.054585

0.265522 0.263201 0.259435 0.254478 0.248566

0.251953 0.243617 0.234952 0.226111 0.217219

0.064890 0.074132 0.082345 0.089576 0.095884

0.241914 0.234714 0.227129 0.219302 0.211349

0.109709 0.114251 0.118019 0.121092 0.123548

0.208376 0.199660 0.191133 0.182840 0.174814

0.105999 0.101336 0.203368 0.195438

0.168569 0.160680 0.153161 0.146009 0.139217

0.125454 0.126877 0.127873 0.128495 0.128792

0.167078 0.159646 0.152526 0.145721 0.139229

0.118109 0.119812 0.121096 0,122010 0.122597

0.165281 0.158299 0.151576 0.145120 0.138933

0.134959 0.134414 0.133669 0.132755 0.131699

0.132773 0.126667 0.120885 0.115413 0.110236

0.128805 0.128574 0.128130 0.127506 '0.126726

0.133045 0.127161 0.121569 0.116258 0.111218

0.122897 0.122945 0.122773 0.122411 0.121884

0.133015 0.127363 0.121972 0.116834 0.111942

0.130524 0.129252 0.127900 0.126483 0.125016

0.105339 0.100709 0.096330 0.092189 0.088273

0.125814 0.124792 0.123676 0.122484 0.121229

0.106436 0.101901 0.097601 0.093523 0.089658

0.121215 0.120424 0.119530 0.118548 0.117492

0.107286 0.102858 0.098648 0.094646 0.090842

0.0 0.1 0.2 0.3 0.4

0.018316 0.040201 0.059531 0.076396 0.090944

0.340026 0.331583 0.321332 0.309831 0.297529

0.012155 0.031936 0.049726 0.065521 0.079385

0.318073 0.311886 0.303894 0.294574 0.284327

0.5 0.6 0.7 0.8 0.9

0.103359 0.113836 0.122574 0.129768 0.135600

0.284786 0.271881 0.259031 0.246396 0.234096

0.091422 0.101765 0.110558 0.117948 0.124081

0.273482 0.262308 0.251016 0.239772 0.228703

0.081182 0.091245 0.099943 0.107383 0.113679

0.262499 0.252844 0.242947 0.232968 0.223037

0.072408 0.082092 0.090585 0.097963 0.104309

1.0 1.1 1.2 1.3 1.4

0.140240 0.143840 0.146541 0.148466 0.149725

0.222213 0.210805 0.199904 0.189529 0.179687

0.129097 0.133125 0.136286 0.138689 0.140432

0.217904 0.207442 0.197366 0.187705 0.178478

0.118941 0.123277 0.126788 0.129570 0.131709

0.213253 0.203692 0.194410 0.185446 0.176827

1.5 1.6 1.7 1.8 1.9

0.150415 0.150622 0.150418 0.149870 0.149032

0.170371 0.161572 0.153274 0.145457 0.138100

0.141604 0.142283 0.142540 0.142434 0.142021

0.169691 0.161343 0.153429 0.145938 0.138855

0.133284 0.134367 0.135021 0.135305 0.135269

2~. .0 2.1 2.2 2.3 2.4

0.147953 0.146675 0.145234 0.143660 0.141982

0.131180 0.124674 0.118558 0.112810 0.107408

0.141347 0.140453 0.139375 0.138145 0.136789

0.132164 0.125849 0.119891 0.114272 0.108973

2.5 2.6 2.7 2.8 2.9

0.140220 0.138395 0.136523 0.134619 0.132693

0.102329 0.097554 0.093062 0.088837 0.084859

0.135331 0.133791 0.132187 0.130533 0.128842

0.103977 0.099265 0.094822 0.090631 0,086677

3.0

0.130757 0.081113

Y

x=2.5

0.127125 0.082944

X-2.6

Yw(z)

%U().

x=2.1

0.123510 0.084568

0.119922 0.085992

~=2.7

0.109942 0.187620 0.113232 0.179965 0.115935 0.172510

0.116375 0.087227

~=2.8

X-2.9

0.1 0.2 0.3 0.4

0.001930 0.014698 0.026841 0.038226 0.048773

0.251723 0.250050 0.247092 0.243042 0.238092

0,001159 0.239403 (1.012635 0.238187 0.023653 0.235838 0.034087 0.232504 0.043849 0.228337

0.000682 0.011037 0.021057 0.030626 0.039656

0.228355 0.227458 0.225569 0.222800 0.219268

0.000394 0.009778 0.018918 0.027707 0.036064

0.218399 0.217722 0.216181 0.213858 0.210843

0.000223 0.008769 0.017134 0.025225 0.032967

0.209377 0.208854 0.207577 0.205607 0.203014

0.5 0.6 0.7 0.8 0.9

0.058437 0.067205 0.075088 0.082112 0.088317

0.232420 0.226190 0.219546 0.212614 0.205504

0.052885 0.061167 0.068691 0.075467 0.081521

0.223482 0.218077 0.212247 0.206103 0.199744

0.048090 0.055890 0.063043 0.069548 0.075416

0.215093 0.210387 0.205258 0.199804 0.194111

0.043930 0.051264 0.058046 0.064266 0.069927

0.207232 0.203119 0.198594 0.193741 0.188638

0.040304 0.047194 0.053611 0.059543 0.064986

0.199873 0.196262 0.192256 0.187927 0.183344

1.0 1.1 1.2 1.3 1.4

0.093751 0.098466 0.102518 0.105960 0.108848

0.198307 0.191099 0.183943 0.176889 0.169977

0.086885 0.091598 0.095702 0.099243 0.102264

0.193255 0.186707 0.180163 0.173670 0.167270

0.080670 0.085338 0.089451 0.093044 0.096155

0.188258 0.182311 0.176328 0.170357 0.164438

0.075043 0.079632 0.083718 0.087328 0.090492

0.183354 0.177950 0.172480 0.166990 0.161519

0.069944 0.074431 0.078462 0.082059 0.085245

0.178568 0.173654 0.168651 0.163603 0.158547

1.5 1.6 1.7 1.8 1.9

0.111233 0.113165 0.114690 0.115851 0.116689

0.163237 0.156692 0.150359 0.144249 0.138368

0.104811 0.106925 0.108647 0.110016 0.111067

0.160996 0.154872 0.148918 0.143147 0.137569

0.098820 0.101076 0.102957 0.104498 0.105730

0.158604 0.152882 0.147292 0.141851 0.136571

0.093239 0.095601 0.097608 0.099288 0.100671

0.156099 0.150758 0.145518 0.140395 0.135403

0.088044 0.090482 0.092584 0.094376 0.095882

0.153515 0.148534 0.143625 0.138807 0.134094

2.0 2.1 2.2 2.3 2.4

0.117239 0.117534 0.117606 0.117481 0.117184

0.132720 0.127305 0.122121 0.117164 0.112428

0.111834 0.112347 0.112635 0.112723 0.112633

0.132191 0.127015 0.122042 0.117271 0.112699

0.106683 0.107386 0.107864 0.10tr140 0.108238

0.131459 0.126522 0.121762 0.117180 0.112775

0.101783 0.102649 0,103293 0.103737 0.104002

0.130553 0.125851 0.121303 0.116911 0.112676

0.097127 0.098133 0.098922 0.099513 0.099925

0.129498 0.125027 0.120688 0.116484 0.112419

2.5 2.6 2.7 2.8 2.9

0.116737 0.116160 0.115471 0.114685 0.113816

0.107909 0.103597 0.099487 0.095570 0.091838

0.112389 0.112008 0.111508 0.110904 0.110210

0.108322 0.104136 0.100133 0.096309 0.092657

0.108177 0.107975 0.107648 0.107213 0.106682

0.108546 0.104489 0.100601 0.096876 0.093310

0.104105 0.104066 0.103898 0.103617 0.103236

0.108597 0.104674 0.100905 0.097284 0.093810

0.100177 0.100284 0.100261 0.100122 0.099879

0.108493 0.104707 0.101058 0.097546 0.094168

3.0

0.112878 0.088283

0o:

0.109439 0.089170

See Examples 12-19.

w(-x+iy)-w(x+iy)

0.106067 0.089898

0.102767 0.090479

0.099544 0.090921

=e-z2_2ie-z2Soe z 12 dt 4%)

4%

w(x- iy)=ze+-22 (cos 2xy+i

sin Bxy)-w(xtiy)

328

ERROR FUNCTION AND F R E S N E L INTEGRALS

Table 7.9

ERROR FUNCTION FOR COMPLEX ARGUMENTS w(z)=e - -i erfc (--iz)

v

2 w ( z ) AN(.) x -3.0

&I(.).aw(z)

#w(z) Jw(2)

z=x+iy .iRw(z) ./w(z)

X-3.2

x 3.1

x -3.3

2w(z)

'flW(2)

x=3.4

0.000123 0.007943 0.015627 0.023095 0.030279

0.201157 0.200742 0.199669 0.197980 0.195732

0.000067 0.007254 0.014338 0.021250 0.027929

0.193630 0.193292 0.192376 0.190915 0.188951

0.000036 0.006670 0.013225 0.019639 0.025862

0.186704 0.186421 0.185630 0.184354 0.182626

0.000019 0.006167 0.012252 0.018222 0.024032

0.180302 0.180061 0.179369 0.178245 0.176715

0.000010 0.005728 0.011394 0i016966 0.022403

0.174362 0.174152 0.173542 Oil72545 0.171181

0.037126 0.043598 0.049665 0.055311 0.060529

0.192984 0.189798 0.186239 0.182368 0.178243

0.034328 0.040407 0.046141 0.051509 0.056501

0.186532 0.183709 0.180534 0.177061 0.173340

0.031849 0.037565 0.042983 0.048083 0.052854

0.180484 0.177970 0.175128 0.172003 0.168637

0.029643 0.035022 0.040144 0.044989 0.049544

0.174808 0.172560 0.170006 0.167184 0.164132

0.027670 0.032738 0.037582 0.042185 0.046532

0.169475 0.167455 0.165151 0.162596 0.159821

1.4

0.065318 0.069685 0.073641 0.077202 0.080385

0.173918 0.169445 0.164866 0.160223 0.155551

0.061114 0.065350 0.069216 0.072722 0.075883

0.169418 0.165339 0.161145 0.156872 0.152553

0.050615 0.054428 0.057971 0.061246 0.064258

0.156858 0.153738 0.150490 0.147141 0.143717

1.5 1.6 1.7 1.8 1.9

0.083210 0.085697 0.087870 0.089749 0.091355

0.150880 0.146236 0.141640 0.137113 0.132667

0.078712 0.081229 0.083450 0.085394 0.087080

0.148217 0.143888 0.139588 0.135335 0.131146

0,074529 0.077055 0.079306 0.081297 0.083044

0.145545 0.141510 0.137488 0.133495 0.129548

0.070636 0.073158 0.075423 0.077445 0.079236

0.142882 0.139120 0.135357 0.131609 0.127892

2.0 2.1 2.2 2.3 2.4

0.092711 0.093835 0.094748 0.095467 0.096010

'0.128317 0.124071 0.119936 0.115919 0.112023

0.088525 0.127031

0.084562 0.085867 0.086974 0.087900 0.088657

0.125660 0.121840 0.118099 0.114442 0.110875

0.080811 0.082182 0.083364 0.084370 0.085213

0.124219 0.120600 0.117045 0.113560 0.110153

0.077263 oio78687 0.079930 0.081004 0;081921

0.122723 oiii929k 0.115919 0.112602 0.109349

2.5 2.6 2.7 2.8 2.9

0.096393 0.096632 0.096739 0.096729 0.096613

0.108249 0.104600 0.101076 0.097674 0.094395

0.089259 0.089719 0.090050 0.090263 0.090368

0.107403 0.104027 0.100751 0.097575 0.094499

0.085905 0.106827

0.082690 0.083324 0.083832 0.084225 0.084511

0.106166 0.103057 0.100026 0.097073 0.094202

3.0

0.096402 0.091236

0.090375 0.091523

0.087493 0.091513

0.084700 0.091413

0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.8 0.9 1.0

1.1 1.2 1.3

Y

x=3.5

0.093345 0.091434

~=3.6

x-3.9

xs3.8

x=3.7

0.000001 0.000001 ~~. .. .- 0.158821 ~ __ . 0;004406 0.004685 0.15867: 0.008786 0.009339 0.158235 0.013115 0.013935 0.157513 0.01737 0 0i018446 Oil56516

0.154273 0.154140 0.153743 0.153088 0.152 183

0.000000 0.004153 0.008282 0.012368 0.016389

0.155260 0.153760 0.152034 0;150102 0.147985

0.021529 0.025574 0.029486 0.033253 0.036861

0.151040 0.149672 0.148094 0.146324 0,144380

0.020326 0.147044 o;o24ibi oii45792 0.027880 0.144346 0.031469 0.142721 Oi034916 0.140931

0.042565 0.045962 0,049161 0.052159 0.054958

0.145703 0.143277 0.140727 0.138074 0.135336

0.040301 ... _ _ 0.043567 0.046653 0.049558 Oi052279

0.142279 __ 0.140039 0.137680 0.135218 0.132671

0.038212 0.041352 0.044328 0.047139 0.049783

0.138993 0.136922 0.134735 0.132448 0.130076

0.057557 0.059962 0.062177 0.064206 0.066058

0.132530 0.129674 0.126782 0.123869 0.120947

0.054819 0.057179 0.059362 0.061374 0.063219

0.130054 0.127384 0.124673 0.121935 0.119182 0.062222 0.063759 0.065156 0.066420 0.067556

0.114817 0,112212 0.109614 0,107031 0.104469

0.068572 0.069474 0.070267 0.070959 0.071555

0.101935 0.099433 0.096968 0.094543 0.092162

0.2 0.3 0.4

0.000005 0.005340 0;010633 0.015846 0.020944

0.168830 0.168645 0.168102 0.167212 0.165990

0.5 0.6 0.7 0.8 0.9

0.025897 Oi030677 0.035263 0.039637 0.043785

0.164456 0.162633 0.160548 0.158227 0.155698

0.024297 0.028812 0.033158 0.037316 0.041274

0.159725 0.158075 0.156181 0.154066 0.151755

0.022847 0.027118 0.031239 Oi035195 0.038974

1.0 1.1 1.2 1.3 1.4

0.047698 0.051370 0.054798 0.057984 0.060928

0.152988 0.150124 0.147132 0.144038 0.140862

0.045023 0.048556 0.051869 0.054962 0.057835

0.149271 0.146637 0.143878 0.141014 0.138067

1.5 1.6 1.7 1.8 1.9

0.063637 0.066116 0.068374 0.070419 0.072260

0.137628 0.134354 0.131058 0.127755 0.124460

0.060491 0.062936 0.065176 0.067217 0.069068

0,135056 0.131999 0.128913 0.125812 0.122709

2.0 2.1 2.2 2.3 2.4

0.073908 0.075373 0.076666 0.077796 0.078774

0.121185 0.117940 0.114735 0.111578 0,108474

0.070736 0.072232 0.073563 0.074739 0.075770

0.119617 0.116545 0.113503 0.110500 0.107540

2.5 2.6 2.7 2.8 2.9

0.079611 0.080316 0.080898 0.081366 0.081730

0.105431 0.102451 0.099538 0.096696 0.093927

0.076664 0.077430 0.078076 0.078612 0.079044

0.104631 0.101777 0.098981 0.096247 0.093577

0.073845 -.- . - . . 0.103777 - - .0.074663 0.101044 0.075366 0.098362 0.075961 0.095734 0.076455 0.093162

0.071149 0;0720!3 0.072764 0.073411 0.073959

3.0

0.081996 0.091230

0.079381 0.090973

0.076855 0.090649

0.074415 0.090265

0.0 0.1

~

~~~

~~

0.102875 0;100260 0.097688 0.095163 0.092688

0.149992 0.149871 0.149510 0.148913 0.148088

0.072061 0.089826

0.4613135 + 0.09999216 0.002883894)+e(z) If x>3.9 or y > 3 w ( z ) = i z (z2-0.1901 c(z)~.:2xlO-~ 635 2-1.7844927 i-5.5253437 0.5124242 +0.05176536 ,ll(z) <10-6 * 'Bee page 11. If x > 6 o r y > 6 w(z)=iz (2-0.2752551 9-2.724745 +

329

ERROR FUNCTION AND FRESNEL INTEGRALS

COMPLEX ZEROS OF THE ERROR FUNCTION

erf zn=O n

Xn

1 2 3 4 5

1.45061 2.24465 2.83974 3.33546 3.76900

zn=xn+ign

6 300 7 514 8 810 9 438 10 723 erf zn =erf ( -zn) =erf ?n =erf

1.88094 2.61657 3.17562 3.64617 4,06069

616 928 105 074 557

R'n

71

?/n

Table 7.10

4.15899 4.51631 4.84797 5.15876 5.45219 (-in)

YlL

840 940 031 791 220

4.43557 4.78044 5.10158 5.40333 5.68883

144 764 804 264 744

=O

From H. E. Salzer, Complex zeros of the error function, J. Franklin Inst. 260,209-211, 1955 (with permission).

COMPLEX ZEROS OF FRESNEL INTEGRALS

.rn

11

0

C(zn)=O

zn=zn+i?/n

S ( z 3 =O

zA=xi+fgE

?A

.rn

!In

0.0000

0.0000 0.2886 0.2443 0.2185 0.2008

1.7437 2.6515 3.3208 3.8759 5

4.3611

0.1909

4.4742

0.1877

MAXIMA AND MINIMA OF FRESNEL INTEGRALS Mn= C(

Table 7.11

Jm) mn = C( 3J-)

n

-111;= S( ,/E+)

Table 7 . 1 2

mi = A(' 4 4 c 4 )

4

0

0.779893 0.640807 0.605721 0.588128 0.577121

0.321056 0.380389 0.404260 0.417922 0.427036

M; 0.713972 0.628940 0.600361 0.584942 0.574957

0.343415 0.387969 0.408301 0.420516 0.428877

5

0.569413

0.433666

0.567822

0.435059

From G. N. Watson, A treatise on the theory of Bessel functions, 2d ed. Cambridge Univ. Press, Cam bridge, England, 1958 (with permission).

.

8 Legendre Functions IRENE A. S T E G U N ~ Contents Mathematical Properties . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Differential Equation . . . . . . . . . . . . . . . . . 8.2. Relations Between Legendre Functions . . . . . . . . . . 8.3. Values on the Cut . . . . . . . . . . . . . . . . . . . 8.4. Explicit Expressions . . . . . . . . . . . . . . . . . . 8.5. Recurrence Relations . . . . . . . . . . . . . . . . . 8.6. Special Values . . . . . . . . . . . . . . . . . . . . 8.7. Trigonometric Expansions . . . . . . . . . . . . . . . 8.8. Integral Representations . . . . . . . . . . . . . . . . 8.9. Summation Formulas . . . . . . . . . . . . . . . . . 8.10. Asymptotic Expansions . . . . . . . . . . . . . . . . 8.11. Toroidal Functions . . . . . . . . . . . . . . . . . . 8.12. Conical Functions . . . . . . . . . . . . . . . . . . . 8.13. Relation to Elliptic Integrals . . . . . . . . . . . . . . 8.14. Integrals . . . . . . . . . . . . . . . . . . . . . . . Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 8.15. Use and Extension of the Tables . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 8.1. Legendre Function-First Kind P,(z) ( ~ 1 1 ). . . . . . . ~=0(.01)1, n=0(1)3. 9. 10. 5-8D Table 8.2. Derivative of the Legendre Function-First Kind Pk(z) (211) . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~=0(.01)1, n=1(1)4. 9. 10. 5-7D Table 8.3. Legendre Function-Second Kind Qn(z) (211) . . . . . . ~=0(.01)1, n=0(1)3, 9, 10, 8D Table 8.4. Derivative of the Legendre Function-Second Kind Q:(z) ($21). . . . . . . . . . . . . . . . . . . . . . . . . . . . ~=0(.01)1, n=0(1)3. 9. 10. 6-8D Table 8.5. Legendre Function-First Kind PA(z) ( ~ 2 1 ). . . . . . . 2 ~ 1 ( ( . 2 ) 1 0n=0(1)5, , 9. 10. exact or 6 s Table 8.6. Derivative of the Legendre Function-First Kind P:(z) (521) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2=1(.2)10, n=1(1)5. 9. 10, 6 s Table 8.7. Legendre Function-Second Kind Qn(z) (221) . . . . . . ~=1(.2)10, n=0(1)3. 9. 10. 6 s Table 8.8. Derivative of the Legendre Function-Second Kind &:(z) (221) . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~=1(.2)10, n=0(1)3. 9, 10, 6 s

Page

332 332 332 333 333 333 333 334 335 335 335 335 336 337 337 337 339 339 340 342

344 346

348 350

351 352

353

The author acknowledges the assistance of Ruth E . Capuano. Elizabeth F . Godefroy. David S. Liepman. and Bertha €I. Walter in the preparation and checking of the tables and examples. 1 National Bureau of Standards.

33 1

8. Legendre Functions Mathematical Properties Notation

The conventions used are z=r+iy, z, 1~ real, and in particular, z always means a real number in the interval - 1 < z < + 1 with cos e=x where e is likewise a real number; n and m are positive integers or zero; v and p me unrestricted except where otherwise indicated. Other notations are :

8.1. Differential Equation 8.1.1 (1-22) d2w --22 dw -+[v(v+1)--1w=0 P 2 dz2 dz 1-22 Solutions

(Degree v and order p with singularities Lt as ordinary branch points-p, v arbiz=&l, trary complex constants.) P:(z ) ,

e(

z ) -Associated Legendre Functions (Spherical Harmonics) of the First and Second Kinds *

larg ( ~l)l
l)tr=

(For P:(z),p = O , chapter 22.)

(2-

larg zl
Legendre polynomials, see

13.1.2

Various other definitions of the functions occur as well EMmixing of definitions.

(For F(a, b; c; z ) see chapter 15.)

Alternate Forms

(Additional forms may be obtained by means of the trmsformation formulas of the hypergeometric function, see [8.1].)

The functions Y:(e, v ) = z E z ~ ( C 8) Mcalled surface harmonics of the first kind, tesseral for m
332

333

LEGENDRE FUNCTIONS

Wronskian

8.1.8

(Upper and lower signs according as 9~20.) 8.3.2 P:(z)=e*iifirP:(sfiO) 8.3.3

8.1.9

Negative Degree

8.2.1

P'-y-1(2)=P:(z)

-etffirQ:(z-iO)] 8.3.4

(2-

iO)]

(Formulas for P:(z) and @(z) are obtained with the replacement of z- 1 by (1-z)e*{*, (9-1) by (l-$)e*'", z + l by z + l for z=sfiO.) 8.4. Explicit Expressions

sin [7r(v+P)l}/sin [4v--Cc)l

8.3. Values on the Cut

(Z=COS

(Both P: and relations.) 8.5.1

*See page U.

+

@(z) =3 e -irr[e-+'fir@(5 io) +eiifirQ:

{ -7re*fir cos v?rP:(z) +&:(z>

*

=i*-le,-i~"[e-t'fir61:(s+iO)

W{Pn(z),Q n ( Z ) } =-(z2-1)-'

8.2. Relations Between Legendre Functions

8.2.2 Q!.-,(z)=

*

e)

&: satisfy the same recurrence Varying Order

*

LEGENDRE FUNCTIONS

334

8.6.9 { [ z+ (22- 1) Il2]v++

+

- [ 2 (22- 1)1/21 - v - + } 8.6.10

Q?(Z ) =i (3).

'I2( 2'-

1) -'I4[ Z+

(2'-

(sin e)-f

COS

1)1/2]-v-t

8.6.11

8.6.12

Pt(cos e)=($+

[(v+4>0]

8.6.13 &!(cos

e)=-($r)f

(sin e)-+ sir1 [(v++)e]

8.6.14

P;f (COS e) =( 3 ~-f) (Y +3) - (sin e) -4 sin [ ( Y

+3>8]

8.6.15 &;+(cos e)=(2T) +(2v+l)-l(sin

e>-+ cos [ ( Y + i > e*~

p=-v

p=O,

p=m=l, 2 , 3 ) . . .

(Rodriguer' Formula)

8.6.6 ~ ( z ) = ( 2 2 - l ) f r ndmPv(z), dz"

pc" (2) = (- 1)"( 1-2')

3'"

dmPv(x) dx"

8.6.7

Qy (2 ) =( z2-1)fm dmQv(Z) d 2"

8.6.18

1

where

I

W-l(Z)=O

) -

@(2) =( -1)"( 1-x2) 4" d"Q"(4 -

dx"

+ [ 2 + ( 2 ~ - 1 ) " ~ ] - ~ -1~ *See page 11.

v=n

P,(Z) =-

1 d"(22-1)" dz"

2"n!

,

335

LEGENDRE FUNCTIONS

v=o, 1

8.6.22 8.7. Trigonometric Expansions (o<e
8.8. Integral Representations ( z not on the real axis between -1 and

8.9.2

(i-z)

5 (2m +1)Pm(z)

m =O

Q m

- a)

(€)=I - ( n i l )[Pn+l(z>Q n ( € > - P n ( e ) & e + l ( € )

1

8.10. Asymptotic Expansions

For fixed z and Y and 9 ' p 3 w , 8.10.1-8.10.3 are asymptotic expansions if z is not on the real --CO iind -1 and + w and + l . (Upper or lower signs according as YzZO.)

risis between

'See page 11.

336

LEGENDRE FUNCTIONS

With p replaced by - p , 8.1.2 is an asymptotic expansion for P;”(z) for fixed z and v and %‘ p+w if z is not on the real axis between --OD and -1. For fixed z and p and 9 v - m , 8.10.4 and 8.10.6 are asymptotic expansions if z is not on the real axis between - and - 1 and + a and 1 ; 8.10.5 if z is not on the real axis between - and 1.

+

+

+ietvr cos p ~ [ z - ( z ~ - l ) t ] ~ - t F ( 3 + p ,+ - p ; ++v;

2 (22 -1)t

The related asymptotic expansion for Pfy(z) may be derived from 8.10.4 together with 8.2.1.

For other asymptotic expansions, see [8.7] and [8.9]. 8.11. Toroidal Functions (or Ring Functions)

(Only special properties are given : other properties and representations follow from the earlier sections.)

8.11.4 @-+(cosh v)= ‘See page XI.

cosh mt d t (cosh v+cosh t sinh n)”++

*

337

LEGESDRE FUNCTIONS

8.12. Conical Functions P!~+,A(cos

8.13.2

01, ~-t+,~(cos 0))

-

(Only specinl properties are given ns other properties tind representtitions follow from earlier (A, n r e d punmeter) iind sections with U = -3+& Z=COS

e.)

8.13.3

K

Q-+(z)=J&

(4s)

Q-i(cosh 7)=2e-'J2K(e-9

8.13.4

1

8.12.1

P-i(cosh v)=

8.13.5

P,(z) A ( z + & q + E (J-) z + (22-

1)

8.13.7

8.12.4

,

+Jo

r"

8.13.8

cosh Atdt d2(cosh t-cos

-I

8.12.5

P - ~ + ~ ~ ( - c oe>s cosh AA =-

*

A

e)

R

8.13.9

P-,(cos

Q-,(z>=K

8.13.10 [ Q - ~ + ~ A C O S8)

+~-)-tA(cos e ) i

8.13. Relation to Elliptic Integrals (see chapter 17) (%>O)

8.13.11

3

e) =-R K (sin -

(@)

-

(J?)]

P+(z) =:A [2E (d?)-K Ql(z)=K

8.13.12

*

(4q)-2.(J15) *

8.14. Integrals 8.14.1

J'

pl(z)Q,(z)dz=[(p-v)(p+v+1)]-'

1

[Ql(z)I2dz=( 2 ~ +l)-'+'(v+

8.14.3

Jm

8.14.4

J:lP.(r)Pp(z)dz=2 [ ( p - v ) ( p + v + ~ ) ] - '

8.14.5

1 J-l

2

[P,(z)I2dz=

r2-2((sin m)2+'(u+l) *'(Y-t

8-14-6

1)

{2 s i n m s i n ~ p [ + ( v + 1 ) - + ( p +

( p + v+

*

J:, Q l ( z ) Q P ( z ) d z = [ ( p - ~ ) ( p + ~ +l)]-l{[+(v+

I)-+(p+

1)][1 +cos

S_11[Q.(z)]2dz=(2~+1)-'{~ . ? - + ' ( v + l ) [ l f ( c o ~ u ~ ) ~ ] }

'Sec pope 11.

1 z0)

4) p~

(p+u+l#O;

8.14.7

l)]+rsin (rp-nv)}

cos v7rIY,

4.

sin (YR- p ~ ) *}

pz-1, - 2 ,

(v#-1.

-3,

-2, - 3 ,

. *

. .) . .)

338

LEGENDRE FUNCTIONS

8.14.8

J:T(z)

Qp(z)dz= [ ( Y -

p) (p+v+l)I-'

1-cos

2 -sin TV cos m[+(v+l) +(p+

(p7r-v7r)

7r

(9v>O,

J

1- (-1)'f" (n+ m)!

(I-n)(Z+n+l)(n-m)!

( I Zn)

8.14.11 J:ll?(z)fl(z)dz=O

l)]

9P>O,

PZY)

-1

8014.14

y1

I

(1-2%)-

'[e(2)]2dZ=

(n+m)!/m(n-m)!

I

8.14.16

I

8.14.17

P;"l(z)= (22- 1)-8" 8.14.18

Q;"(z) =(- I)"(z'-

1)-4"

s{* *

l m Q v ( z )W"

For other integrals, see [8.2], [8.4] and chapter 22.

1.5

-

I

FIQURE8.1.

P,,(COS

e).

n=0(1)3.

F~QURE 8.2.

P:(z). n=l(l)3, 251.

339

LEGENDRE FUNCTIONS

t

/

10-2

\

\\

tI-=

\

PO

FIQIJRE 8.3. Pn(z). n=0(1)3, 2 2 1 .

IO-~ I

\

I

I

I

I

I

2

3

4

5

6

\.

Q,

1 \I

7

0

I 9

I0

X

Numerical Methods 8.15. Use and Extension of the Tables Computation of P,(z)

FIQWBE 8.4. &,,(z). n=0(1)3, z
For all values of z there is very little loss of significant figures (except at zeros) in using the recurrence relation 8.5.3 for increasing values of n. Example 1. ComputeP,(s) fors= .31415 92654 and z=2.6 for n=2(1)8.

340 n

LEQENDRE FUNCTIONS

Pn(.31415 92654)

Example 3.

Pn(2.6)

1 .31415 92654 2. 6 2 --.35195 59340 9. 64 40.04 -. 39372 32064 3 4 .04750 63122 174.952 5 .34184 27517 786.74336 . 15729 86975 3604.350016 6 7 --.20123 39354 16729.51005 8 -.25617 29328 78402.55522 Computing P&) using Table 22.9 carrying ten significant figures, P8(.31415 92654)= - .25617 2933 and P8(2.6)=78402.55526. 0 1

1

Ten terms in the series for F

n

-;

v+-;

2

22

n

Q" (2.6)

0 1 2 3 4 5

.40546 51081 -05420 928 .00868 364 -00148 95 .00026 49 .00004 81

where Qo and Q1 are obtained using 8.4.2 and 8.4.4. Computation of P*)(z), &(z)

For all values of x, Pi+($)and Q++(x) are most easily computed by means of 8.13. Example 4. Compute Q-+(x) for x=2.6. Using 8.13.3 and interpolating in Table 17.1 for K(.5),we find

&,(.31415 92654)

,32515 34813 89785 00212 -. 58567 85953 2 3 .29190 60854 4 .59974 26989 Using the results of Example L together with 8.6.19. we find Q4(x)=$P4(x)1n 1+x 0 1

ri2 vF -'>

of 8.1.3 are necessary to obtain nine significant figures giving Q5(2.6)=4.8182 4468 X 10-6. Using 8.5.3 with increasing values of n carrying ten significant figures we obtain

Computation of Qn(z)

For x<1, use of 8.5.3 for increasing values of n leads to very little loss of significant figures. However, for x>1, the recurrence relation 8.5.3 should be used only for decreasing values of n , after having first obtained Qn using the formulas in terms of hypergeometric functions. Example 2. Compute Qn(x) for x= .31415 92654 and n=0(1)4. With the aid of 8.4.2 and 8.4.4 we obtai?,

Compute Q6(x) for s=2.6.

- I

(=--W3(x)

7 1 where W8=i P3+- PI,giving Q4(.31115 92654)= 3 ,59974 26989.

= (.74535 59925) (1.90424 1417)

=1.41933 7751.

On the other hand, at least nine terms in the necessary to obtain comparable accuracy.

References Texts [8.1] A. Erdblyi et al., Higher transcendental functions, vol. 1, ch. 3 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [8.2] E. W. Hobson, The theory of spherical and ellipsoidal harmonics (Chelsea Publishing Co., New York, N.Y., 1955). [8.3] J. Lense, Kugelfunktionen (Akademische Verlagsgesellschaft, Leipzig, Germany, 1950). [8.4] T. M. MacRobert, Spherical harmonics, 2d rev. ed. (Dover Publications, Inc., New York, N.Y., 1948). (8.51 W. Magnus and F. Oberhettinger, Formulas and theorems for the special functions of mathematical physics (Chelsea Publishing Co., New York, N.Y., 1949). [8.6] G. Prasad, A treatise on spherical harmonics and the functions of Bcssel and Lamb, Part I1 (Advanced) (Mahamandal Press, Benares City, India, 1932).

[8.7] L. Robin, Fonctions sph6riques de Legendre e fonctions sph6roidales. Tome I, 11, I11 (GauthierVillars, Paris, France, 1957). [8.8] C. Snow, Hypergeometric and Legendre functions with applications to integral equations of potential theory, NBS Applied Math. Series 19 (U.S. Government Printing Office, Washington, D.C., 1952). [8.9] R. C. Thorne, The asymptotic expansion of Legendre functions of large degree and order, Philos. Trans. Roy. Soc. London 249, 597-620 (1957).

Tables [8.10] H. Bateman, Some problems in potential theory, Mess. of Math., 617,52,73-75 (1922). Pn(cosh U), &,(cosh U), PL(qosh U), &:(cosh U ) ; cosh u = l . l , n=0(1)20, 10D; cosh u=1.2, 2, 3; n=O(l)lO, exact or 10D.

341

LEGENDRE FUNCTIONS

[8.11] Centre National d'fitfides des TBlBcommunications, Tables des fonctions de Legendre associees. Fonction associQ de premiere espece P."(cos 0) (aditions de La Revue d'optique, Paris, France, 1952). n=-+(.1)10, m=o(1)5, e=oo(lo)goo (variable number of figures). [8.12] Centre National d'fitudes des T61Bcommunications, Tables numerique des fonctions associQs de Legendre. Fonctions associees de premiere espece COS e) (gditions de La Revue d'Optiquo, Paris, France, 1959). n = -i(.l)lO, m=0(1)2, 8=Oo (1') 180' (variable number of figures). [8.13] G. C. Clark and S.W.Churchill, Table of Legendre polynomials P,(cos 0) for n=0(1)80 and O=O' (1') B O O , Engineering Research Institute Publications (Univ. of Michigan Press, Ann Arbor, Mich., 1957). [8.14] R. 0. Gumprecht and G. M. Sliepcevich, Tables of functions of the first and second partial derivatives of Legendre polynomials (Univ. of Michigan Press, Ann Arbor, Mich., 1951). Values of [zn, -(l-x2)nA]* 104 and n,lO4 for y=0'(10')170' (1')180', n=1(1)420, 5s. [8.15] M. E. Lynam, Table of Legendre functions for complex arguments TG-323, The Johns Hopkins Univ. Applied Physics Laboratory, Baltimore, Md. (1958). [8.16] National Bureau of Standards, Tables of associated Legendre functions (Columbia Univ. Press, New d York, N.Y., 1945). Pr(cos e), - P."(cos e), de n=1(1)10, m ( s n ) = 0 ( 1 ) 4 , t9=0"(1')90", 6s;

(-lp+l<

dx

Qr(z), n=0(1)10, m(
6 s or exact; i-nP:(iz),

=0(1)4,

i-n

d - P:(iz), n=1(1)10,

dz

d ~ = 0 ( . 1 ) 1 0 , 6 s ; P:++(z), L P,"-~(z),

(-l)mQ:-h(z),

d (-l)m+ldz

&:++,

n=

- 1(1)4,

m=0(1)4, z=1(.1)10, 4 6 s . [8.17] G. Prevost. Tables des fonctions spheriques et de leurs intAgrales (Gauthier-Villars, Bordeaux and 1 0 n = l(1)lO; Paris, France, 1933). Pn(x),[*Pn(t)dt,

P!(z),

so'

P'.(t)dt, n=0(1)8, j = O ( l ) n , r=0(.01)1,

5s. [8.18] H. Tallqvist, Sechsstellige Tafeln der 32 ersten Kugelfunktionen P,(cos e), Acta Soc. Sci. Fenn., Nova Series A, 11, 11 (1938). P,(COS e), n=1(1)32, 0'(10')90'; 6D. [8.19] H. Tallqvist, Act8 Soc. Sci. Fenn., Nova Series A, 11, 4(1937). P,(z), n = l ( l ) l 6 , x=O(.OO1)1, 6D. [8.20] H. Tallqvist, Tafeln der Kugelfunktionen P ~ ~ ( 0C) O S bis P ~ ~ ( c oO),s Soc. Sci. Fenn. Comment. Phys.Math., VI, lO(1932). P,(cos e), n=25(1)32, e=oo(lo)900, 5D. [8.21] H. Tallqvist, Tafeln der 24 ersten Kugelfunktionen P,(cos O), Soc. Sci. Fenn. Comment. Phys.-Math., VI, 3(1932). P.(COS e), n=1(1)24, e=oo(lo)90", 5D.

342

LEGENDRE FUNCTIONS

LEGENDRE FUNCTION-FIRST

Table 8.1

PO@)= 1

KIND

Pn(.%)

P1(2)=2

PlO(2)

0.00 0.01 0.02 0.03 0.04

90.00000 89.42703 88.85400 88.28086 87.70755

00 26 80 87 72

-0.50000 -0.49985 -0.49940 -0.49865 -0.49760

0.00000 -0.01499 -0.02998 -0.04493 -0.05964

00 75 00 25 00

0.00000 0.02457 0.04893 0.07285 0.09614

000 330 045 701 188

-0.24609 -0.24474 -0.24069 -0.23400 -0.22473

37 14 84 69 64

0.05 0; 06 0.07 0.08 0.09

87.13401 867 56018 85.98601 85.41143 84.83639

60 72 28 43 29

-0.49625 -0; 49460 -0.49265 -0.49040 -0.48785

-0.07468 -0; 08946 -0.10414 -0.11872 -0.13317

75 00 25 00 75

0.11857 Oil3996 0.16012 0.17885 0.19599

899 890 040 206 366

-0.21298 -0.19887 -0.18254 -0.16418 -0.14397

35 11 68 20 02

0.10 0.11 0.12 0.13 0.14

84.26082 83.68468 83.10789 82.53040 81.95215

95 44 74 77 37

-0.48500 -0.48185 -0.47640 -0.47465 -0.47060

-0.14750 -0.16167 -0.17568 -0,18950 -0.20314

00 25 00 75 00

0.21138 0.22489 0.23637 0.24572 0.25285

764 042 363 526 070

-0.12212 -0.09887 -0,07447 -0,04918 -0.02328

50 86 93 90 12

0.15 0.16 -- 0.17 0.18 0.19

81.37307 80.79310 80I212I8 79.63024 79.04721

34 38

io

02 58

-0.46625 -0.46160 -0I45665 -0.45140 -0.44585

-0.21656 -0.22976 -0.24271 -0.25542 -0.26785

25 00 75 00 25

0.25767 0.26013 0.26020 0.25785 0.25309

367 706 358 632 918

+0.00296 0.02925 0.05529 0,08080 0.10549

18 20 81 85 42

0.20 0.21 0.22 0.23 0.24

78.46304 77.87764 77.29096 76.70292 76.11345

10 77 70 82 96

-0.44000 -0.43385 -0.42740 -0.42065 -0.41360

-0.28000 -0.29184 -0.30338 -0.31458 -0.32544

00 75 00 25 00

0.24595 0.23647 0.22472 0.21078 0.19477

712 631 407 870 914

0.12907 0.15126 0.17181 0.19047 0.20700

20 74 75 36 49

0.25 0.26 0.27 0.28 0.29

75.52248 74.92993 74.33573 73.73979 73.14204

78 79 31 53 40

-0.40625 -0.39860 -0.39065 -0.38240 -0.37385

-0.33593 -0.34606 -0.35579 -0.36512 -0.37402

75 00 25 00 75

0.17682 0.15707 0.13569 0.11286 0.08879

442 305 215 642 707

0.22120 0.23287 0.24185 0.24801 0.25124

02 14 52 62 81

0.30 0.31 0.32 0.33 0.34

72.54239 71.94076 71.33707 70.73122 70.12312

69 95 51 45 59

-0.36500 -0.35585 -0.34640 -0.33665 -0.32660

-0.36250 -0.39052 -0.39808 -0.40515 -0.41174

00 25 00 75 00

0.06370 0.03760 +0.01135 -0.01539 -0.04219

038 634 691 566 085

0.25147 0.24665 0,24278 0.23369 0.22203

63 91 89 37 73

0.35 0.36 0.37 0.38 0.39

69.51268 68I89980 68.28438 67.66631 67.04550

49 39 27 73 06

-0.31625 -0.30560 ~ _ ._ - - . -0.29465 -0.28340 -0.27185

-0.41761 -0I42336 -0.42636 -0.43262 -0.43670

25 00 75 00 25

-0.06676 -0.09483 -0.12014 -0.14441 -0.16737

185 780 608 472 489

0,20732 0.18967 0.16986 0.14754 0.12310

00 83 48 72 73

0.40 0.41 0.42 0. 43 0.44

66.42182 65.79516 65.16541 64.53243 63.89611

15 52 25 99 88

-0.26000 -0.24785 -0.23540 -0.22265 -0.20960

-0.44000 -0.44269 -0.44478 -0.44623 -0.44704

00 75 00 25 00

-0.18676 -0.20832 -0.22581 -0.24101 -0.25369

356 609 900 269 426

0.09663 0.06904 0.04006 +0.01024 -0.02002

91 71 39 69 45

0.45 0.46 0.47 0.48 0.49

63.25631 62.61289 61.96570 61.31459 60.65941

61 25 35 80 84

-0.19625 -0.18260 -0.16865 -0.15440 -0.13985

-0.44716 75 -0.44666 ... 00 .. -0i44544 25 -0.44352 00 -0.44067 75

-0.26367 -0.27076 .-. . -0.27484 -0.27577 -0.27346

022 932 521 908 225

-0.05035 -0.06032 -0.10952 -0.13752 -8.16389

30 72 64 51 87

0.50

60.00000 00

-0.12500

-0.43750 00 [(-2)91

-0.26789

856

-0.18822 86 [(-;)41

[(-;PI[(-;)41

+

Pz(z) = 3( - 1 3z2) Pg(z)

P3(2)=

+

2 ( - 3 h2)

=&(1260- 18480Z2+72072~4- 10296036+48620Z8)

P ~ o ( z=) -( - 252 +13860Z2- 12012024+360360~6-43758W +1847562'0) 1024 (n+ 1)Pn+I(Z)= (2n+l)zPn(z) -nPn- I(Z) For coefficients of other polynomials, see chapter 22. 1

343

LEGENDRE FUNCTIONS LEGENDRE FUNCTION-FIRST

KIND Pn(r)

Table 8.1

Pl(X) = x

Po(x) = 1

0.50 0.51 0.52 0.53 0.54

arccos x 60.00000 59.33617 58.66774 57.99454 57.31636

00 03 85 51 11

-0.12500 -0.10985 -0.09440 -0.07665 -0.06260

P3(d -0.43750 -0.43337 -0.42848 -0.42260 -0.41634

00 25 00 75 00

-0.26789 -0.25900 -0.24682 -0.23139 -0.21283

856 667 215 939 321

PlO(X) -0.16822 86 -0.21010 83 -0.22914 92 -0,24498 73 -0.25728 92

0. 55 0. 56 0.57 0.58 0. 59

56.63298 55.94420 55.24977 54.54945 53.84299

70 22 42 74 18

-0.04625 -0.02960 -0.01265 +0.00460 0.02215

-0.40906 25 -0.40096 00 -0.39201 75 -0.36222 00 -0.37155 25

-0.19126 -0.16686 -0.13985 -0.11051 -0.07914

025 000 552 366 497

-0,26575 -0.27014 -0.27023 -0.26590 -0.25704

85 28 97 30 92

0.60 0.61 0.62 0.63 0.64

53.13010 52.41049 51.68386 50.94987 50.20818

24 70 55 75 05

0.04000 0.05815 0.07660 0.09535 0.11440

-0.36000 -0.34754 -0.33418 -0.31988 -0.30464

00 75 00 25 00

-0.04610 -0.01178 +0.02337 0.05890 0.09430

304 332 862 951 141

-0.24366 -0.22580 -0.20360 -0,17728 -0.14714

27 16 19 16 41

0.65 0.66 0.67 0.68 0.69

49.45839 48.70012 47.93293 47.15635 46.36989

81 72 52 69 11

0.13375 0.15340 0.17335 0.19360 0.21415

-0.28843 -0.27126 -0.25309 -0.23392 -0.21372

75 00 25 00 75

0.12901 0.16248 0.19412 0.22334 0.24952

554 693 981 410 270

-0.11358 -0.07707 -0.03818 +0.00243 0.04403

05 01 08 30 37

U. 70

0.71 0.72 0.73 0.74

45.57299 44.76508 43.94551 43.11360 42.26858

60 47 96 59 44

0.23500 0.25615 0.27760 0.29935 0.32140

-0.19250 -0.17022 -0.14688 -0.12245 -0.09694

00 25 00 75 00

0.27205 0.29036 0.30385 0.31199 0.31430

993 111 323 698 004

0.08580 0.12686 0.16625 0.20299 0.23605

58 31 89 76 08

0.75 0.76 0.77 0.78 0.79

41.40962 40.53580 39.64611 38.73942 37.81448

21 21 11 46 85

0.34375 0.36640 0.38935 0.41260 0.43615

-0.07031 -0.04256 -0.01366 +0.01638 0.04759

25 00 75 00 75

0.31033 0.29973 0.28226 0.25777 0.22625

185 981 712 224 012

0.26437 45 0.28693 1 9 0.~30271 79 0.31078 93 0.31029 79

0.80 0.81 0.82 0. 83 0.84

36.86989 35.90406 34.91520 33.90126 32.85988

76 86 62 20 04

0.46000 0.48415 0.50860 0.53335 0.55840

0.08000 0.11360 0.14842 0.18446 0.22176

00 25 00 75 00

0.18785 0.14292 0.09201 +0.03591 -0.02431

528 678 529 226 874

0.30052 0.28094 0.25124 0.21139 0.16170

98 87 52 19 50

0. 85 0.86 0; 87 0.88 0. 89

31.78833 30.68341 29.54136 28.35763 27.12675

06 71 05 66 31

0.58375 0.60940 0.63535 0.66160 0.68815

0.26031 0.30014 0.34125 0.38368 0.42742

25 00 75 00 25

-0.08730 -0.15134 -0.21433 -0.27376 -0.32665

820 456 544 627 610

0.10291 +0.03622 -0.03655 -0.11300 -0.18989

23 91 86 29 29

0. 90 0.91 0.92 0.93 0.94

25.84193 24.49464 23.07391 21.56518 19.94844

28 85 81 50 36

0.71500 0.74215 0.76960 0.79735 0.82540

0.47250 0.51892 0.56672 0.61589 0.66646

00 75 00 25 00

-0.36951 049 -0.39827 146 -0.40626 421 -0.39414 060 -0.34981 919

-0.26314 -0.32768 -0.37731 -0.40457 -0,40058

56 58 58 43 29

0.95 0.96 0.97 0.98 0.99

18.19487 16.26020 14.06986 11.47834 8.10961

23 47 77 09 44

0.85375 0.88240 0.91135 0.94060 0,97015

0.71843 0.77184 0.82668 0.88296 0.94074

75 00 25 00 75

-0.26842 -0.14220 +0.03750 0.26039 0.59724

-0.35488 -0.25524 -0.06749 +0.16470 0.52006

03 34 40 81 90

0.00000 00

1.00000 [(-[)4]

1.00000 00 [(-:)"I

X

1.00

Pz(z)

P9(d

182 642 397 609 553

1.00000 000

['-;"]

1.00000 00 [(-;P]

~

Pz(x)= 3( - 1+ 3 ~ 2 )

PQ(x)= PlO(X)

+

P ~ ( x=) ( - 3 5 2 2 )

& (1260- 18480$+7207&4-

10296W+4862019)

1 =m (-252+1386@~2- 12012O~'+36036W-43758~+18475Q10)

(n+ l)Pn+ 1(z) = (2n+ l)ZPn(x) - nPn- I (x) For c(w?fficientsof other polynomials, see chapter 22.

344

LEGENDRE FUNCTIONS Table 8.2

DERIVATIVE OF THE LEGENDRE FUNCTION-FIRST

P:(x) = 1 X

Pi (x)

KIND PL(r)

P6(x) = 3s

Pi(x)

P6(x)

Pio(x)

0. 00 0.01 0. 02 0. 03 0. 04

-1.50000 -1.49925 -1.49700 -1.49325 -1.48800

0.00000 -0.07498 -0.14986 -0.22452 -0.29888

00 25 00 75 00

2.46093 2.45011 2.41773 2.36405 2.28948

75 64 75 34 35

0.27023 0.53765 0.79949 1.05299

00 41 93 17 82

0. 05 0; 06 0. 07 0. 08 0. 09

-1.48125 -1.47300 -1.46325 -1.45200 -1.43925

-0.37281 -0,44622 -0.51899 -0.59104 -0,66224

25 00 75 00 25

2.19461 2.08018 1,94709 1.79639 1.62929

13 11 32 87 31

1.29552 1.52449 1.73750 1.93223 2.10657

05 98 05 25 29

0.10 0.11 0.12 0.13 0.14

-1.42500 -1.40925 -1.39200 -1.37325 -1.35300

-0.73250 -0.80170 -0.86976 -0.93655 -1.00198

00 75

25 00

1.44710 1.25130 1.04346 0.82528 0.59853

87 64 68 00 47

2.25858 2.38654 2.48895 2.56453 2.61230

73 80 24 90 18

0.15 0. 1 6 0.17 0.18 0.19

-1.33125 -1.30800 -1.28325 -1; 25700 -1.22925

-1.06593 -1.12832 -1.18902 -1.24794 -1.30496

75 00 25 00 75

0.36510 +0.12694 -0.11392 -0.35546 -0.59555

73 88 76 01 27

2.63150 2.62168 2.58266 2.51458 2.41783

28 25 81 04 68

0. 20 0.21 0.22 0.23 0.24

-1.20000 -1.16925 -1.13700 -1.10325 -1.06800

-1.36000 -1.41293 -1.46366 -1.51207 -1.55808

00 25 00 75 00

-0.83208 -1.06295 -1.28602 -1.49923 -1.70052

96 03 54 18 94

2.29315 2.14154 1.96431 1.76306 1.53966

33 35 51 37 43

0. 25 0.26 0.27 0.28 0.29

-1.03125 -0.99300 -0.95325 -0.91200 -0.86925

-1.60156 -1.64242 -1.68054 -1.71584 -1.74819

25 00 75 00 25

-1.88793 -2.05954 -2.21355 -2.34823 -2.46202

72 92 15 78 63

1.29625 1.03524 0.75926 0.47115 +0.17398

99 77 26 77 30

0.30 0.31 0. 32 0.33 0.34

-0.82500 -0.77925 -0.73200 -0.68325 -0.63300

-1.77750 -1,80365 -1.82656 -1.84610 -1.86218

00 75 00 25 00

-2.55347 -2.62129 -2.66437 -2.68178 =2.67279

51 80 95 96 74

-0.12903 -0,43453 -0.73903 -1.03894 -1.33065

87 90 23 72 96

0.35 0.36 0.37 0.38 0.39

-0.58125 -0.52800 -0.47325 -0.41700 -0.35925

-1.87468 -1.88352 -1.88857 -1.88974 -1.88691

75 00 25 00 75

-2.63688 -2.57375 -2.48336 -2.36588 -2.22176

47 82 07 14 52

-1.61052 -1.87493 -2.12030 -2.34318 -2.54023

81 10 43 21 74

0.40 0.41 0.42 0.43 0.44

-0.30000 -0.23925 -0.17700 -0.11325 -0.04800

-1.88000 -1.86888 -1.85346 -1.83362 -1.80928

00 25 00 75 00

-2.05172 -1.85672 -1.63802 -1.39715 -1.13592

01 35 69 86 50

-2.70832 -2.84451 -2.94616 -3.01090 -3.03674

36 75 13 51 96

0.45 0.46 0.47 0.48 0.49

+O. 01875

-1,78031 -1.74662 -1.70809 -1.66464 -1.61614

25 00 75 00 25

-0,85640 -0.56096 -0.25222 +0.06693 0.39337

91 76 53 30 29

-3.02208 -2.96573 -2.86699 -2.72566 -2.54206

63 83 80 30 98

0.50

[(-;'"I

-1.56250

00

0.72372 44 [( 7 ' 3 1

-2.31712

34

0.08700 0.15675 0.22800 0.30075 0.37500

[(-;I61 Ph(x)=i(-3+15x2)

Pk(x) = P'lo(x) =

00

Pi@)=; (-60+14h2)

& (1260-5544h2+36036h4-72072~6+43758W)

g4 (27720-48048h2+216216h4-350064he+184756W)

0.00000

345

LEGENDRE FUNCTIONS DERIVATIVE OF THE LEGENDRE FUNCTION-FIRST Pi(x)=l L

Pi (2)

0.50 0. 51 0. 52 0. 53 0.54

0.37500 0.45075 0.52800 0.60675 0.68700

0. 5 5 0; 56 0.57 0.58 0.59

o.xa75 0; 85200 0.93675 1.02300 1.11075

0.60 0.61 0.62 0.63 0.64

1.20000 1.29075 1.38300 1.47675 1.57200

0.65 0.66 0.67 0.68 0.69

1.66875 1.76700 1.86675 1.96800 2.07075

0.70 0. 71 0. 72 0.73 0. 74

-

P&(4 1,56250 i.50360 1.43936 1.36965 1.29438

P2X) 00 75 00 25 00

- 1.21343 75 - iIiZ672 00 - 1.03412 25 - 0.93554 00 - 0.83086 75 - 0.72000 00 - 0.60283 25 - 0.47926 00 - 0.34917 75 - 0.21248 00 +

Table 8.2

KIND P;(x)

P6(2)= 32

PiO(2)

- 2.31712 - 2.05232 -- 1.74978 1.41226 - 1.04315

0.72372 1.05439 1.38160 1,70137 2.00958

44 75 24 21 86

34 40 82 67 43

2.30201 E57431 2.82213 3.04106 3.22677

29 a7 05 49 77

3.37501 3.48166 3.54283 3.55487 3.51451

44 60 00 57 63

1.56397 2.00598 2.43034 2.82866 3.19230

82 31 08 68 45 07 74 13 16 84

-- 0.64649 0.22698

+

54 16 0.21005 92 0.65868 1 0 1.11234 92

0.06906 0.08118 0.23835 0.40256 0.57390

25 00 25 00 75

3.41888 3.26561 3.05294 2.77978 2.44582

50 51 03 82

3.51243 3.78017 3.98677 4.12369 4.18284

2.17500 2.28075 2.38800 2.49675 2.60700

0.75250 0,93844 1.13184 1.33279 1.54142

00 25 00 75 00

2.05168 1.59897 1,09043 0.53008 0.07667

93 66 73 28 36

4.15678 4.03888 3.82364 3.50693 3.08626

18 45 72 03 20

0. 75 0.76 0.77 0. 78 0.79

2.71875 2.83200 2.94675 3.06300 3.18075

1.75781 1,98208 2.21432 2.45466 2.70318

25 00 75 00 25

- 1.39984 - 2.09708 - 2.80201 - 3.49987

0.72287 14 93 32 52 45

2.56116 1.93351 1,20791 0,39215 0.50239

49 26 71 05 96

0.80 0.81 0. 82 0. 83 0.84

3.30000 3.42075 3.54300 3.66675 3.79200

2.96000 3.22521 3.49894 3.78127 4.07232

00 75 00 25 00

- 4.80308 - 5.36607 - 5.83686

-- 2.46122 3.48002 - 4.48547 - 5.43990

77 91 97 21 91

0. 85 0. 86 0.87 0. 88 0.89

3.91875 4.04700 4.17675 4.30800 4.44075

4.37218 4.68098 4.99880 5.32576 5.66195

75 00 25 00 75

03 07 93 51 90

0.90 0.91 0. 92 0. 93 0.94

4.57500 4.71075 4.84800 4.98675 5.12700

6.00750 6.36249 6.72704 - . - . .. 7.10124 7.48522

00 25

- 6.29851 - 7.00851 - 7.50840 - 7.72711 - 7.58303 - 6.98312 - 5.82184 - 3.98006 - 1.32394

0.95 0.96 0.97 0.98 0. 99

5.26875 5.41200 5.55675 5; 70300 5.85075

7.87906 8.28288 8.69677 9.12086 9.55523

25 00 75 00 25

7.04763 13.11571 20.70612 30.04600 41.38561

58 11 01 25 43

1.00

6.00000

w21

00 ..

75 00

10.00000 00

+

-

-

4.17348 81 26 64 10 6.18657 35

-- 6.38961 6.38285

68 06 6.16672 97 5.66983 23 4.84997 54

- 3.65335 - 2.02101 +

89 73 0.11150 20 2.81447 1 8 6.16433 35

10.24405 15.14351 20.95987 27.79800 35.77086

70 59 66 16 77

45.00000 00

[‘-y]

+

84

X

w21

+

- 1.46023

+

55.00000 00

P ~ ( x=)t (- 3 152’) Pi@)= g ( - 60+140~2) 1 Pi@) (1260- 554d0~*+360360~4-720720~~ +437580~8)

=m

P’lo(~) =

k4(27720-48048o;C2+216216O4-

79 03 04 73 2.29628 14

350064O-c6+184756O)

~

346

LEQENDRE FUNCTIONS Table 8.3 X

0.00 0. 01 0.02 0. 03 0. 04

LEGENDRE FUNCTION-SECOND

Qo ( 2 ) Ql(4 0.00000 000 -1.00000 000 0.01000 033 -0.99990 000 0.02000 267 -0.99959 995 0.03000 900 -0.99909 973 0.04002 135 -0.99839 915

Qz(4

0.00000 -0.01999 -0.03998 -0.05996 -0.07991

KIND

Qn(Z)

000 867 933 399 463

0.66666 0.66626 0.66506 0.66306 0.66027

667 669 699 829 179

Qd4 -0.40634 921 -0.40452 191 -0.39905 538 -0.38999 553 -0.37741 852

QIO(Z) 0.00000 000 -0.04056 181 -0.08068 584 -0.11993 860 -0.15789 513

Q3

(4

0. 05 0.05004 0. 06 0.06007 0. 07 0.07011 0.08 0.08017 0.09 0.09024

173 216 467 133 419

-0.99749 791 -0.09983 321 -0.99639 567 -0.11971 169 -0.99509 197 -0.13954 199 -0.99358 629 -0.15931 602 -0.99187 802 -0.17902 563

0.65667 0.65229 0.64711 0.64114 0.63439

917 261 475 873 817

-0.36143 -0,34216 -0.31978 -0.29448 -0.26647

026 562 750 565 538

-0.19414 -0.22828 -0.25995 -0.28879 -0.31447

321 745 321 038 701

0.10 0.11 0.12 0.13 0.14

0.10033 0.11044 0.12058 0.13073 0.14092

535 692 103 985 558

-0.98996 -0.98785 -0.98553 -0.98300 -0.98027

0.62686 0,61856 0.60948 0.59964 0,58903

720 044 299 048 905

-0.23599 -0.20330 -0.16869 -0.13245 -0.09491

595 891 616 792 050

-0.33672 - .. - . -0.35527 -0.36990 -0.38044 -0.38675

259 -- . 122 435 330 142

0.15 0.16 0.17 0.18 0.19

0.15114 0.16138 0.17166 0.18198 0.19233

044 670 666 269 717

-0.97732 893 -0.97417 813 -0.97081 667 -0.96724 312 -0.96345 594

-0.29546 923 -0.31449 610 -0.33339 158 -0.35214 699 -0.37075 353

-0.05638 -0.01721 +0.02223 0.06161 0.10057

395 959 260 670 361

-0.38873 -0.38634 -0.37958 -0.36850 -0.35318

587 905 962 308 198

0.20 0.20273 0; 21 0.21317 0.22 0.22365 0.23 0.23418 0.24 0.24477

255 135 611 947 411

-0.95945 -0.95523 -0.95079 -0,94613 -0.94125

349 402 566 642 421

-0.38920 232 -0.40748 439 -0.42559 062 -0.44351 180 -0.46123 857

0,50990 0.49420 0,47781 0.46074 0.44300

155 314 388 476 738

0,13874 0.17577 0.21130 0.24499 0.27652

395 093 336 861 557

-0.33376 -0.31043 -0.28343 -0.25302 -0.21951

565 947 378 221 969

0.25 0.26 0.27 0.28 0.29

0.25541 0.26610 0.27686 0.28768 0,29856

281 841 382 207 626

-0.93614 -0.93081 -0,92524 -0.91944 -0.91341

680 181 677 902 578

-0,47876 -0.49607 -0,51315 -0.53000 -0.54661

145 081 685 962 900

0.42461 0.40557 0.38591 0.36562 0.34474

393 719 059 819 467

0.30556 765 571 0.35502 089 0.37489 746 0.39122 551

-0.18327 -0.14469 -0.10417 . - .. - . -0.06219 -0,01920

994 251

0.30 0: 31 0.32 0.33 0.34

0.30951 0.32054 0.33164 0.34282 0.35409

960 -0.90714 412 541 -0.90063 092 711 -0.89387 293 825 -0.88686 668 253 -0.87960 a54

-0.56297 -0,57906 -0.59488 -0.61041 -0,62564

466 608 256 313 662

0.32327 0.30123 0.27864 0.25551 0,23187

542 647 459 723 261

0.40380 0.41246 0,41705 0.41749 0.41371

351 +0.02428 610 080 0.06776 975 981 0.11072 534 0.15262 723 822 0.19295 076 084

0.35 0.36 0.37 0.38 0.39

0.36544 0.37688 0.38842 0,40005 0,41180

375 590 310 965 003

-0.87209 -0.86432 -0.85628 -0.84797 -0.83939

469 108 345 733 799

-0.64057 -0.65517 -0.66944 -0.68337 -0.69694

159 633 887 690 784

0.2(2772 0,18310 0,15802 0.13251 0.10658

970 825 883 285 256

0.40567 0.39339 0.37693 0.35638 0.33189

128 336 227 546 317

0.23117 0.26680 0.29934 0.32833 0.35334

811 432 337 437 774

0.40 0.41 0.42 0.43 0.44

0,42364 0,43561 0.44769 0.45989 0.47223

893 122 202 668 080

-0.83054 -0.82139 -0.81196 -0.80224 -0.79221

043 -0.71014 940 -0.72296 935 -0.73538 443 -0.74739 845 -0.75897

0.08026 114 872 624 0.05357 267 670 +0.02654 221 600 -0.00080 418 958 -0.02843 939

0.30363 0,27184 0.23679 0,19877 0.15815

867 811 006 461 208

0.37399 0,38991 0.40082 0.40646 0.40665

123 596 218 477 845

0.45 0.46 0.47 0.48 0.49

0.48470 0.49731 0.51007 0.52298 0.53606

028 129 034 428 034

-0.05633 524 -0.06446 239 - 0 i i i 2 ~ 9 034 -0.14128 732 -0,16992 027

0.11531 136 0.07067 773 +oIoiiSi 030 -0.02210 100 -0.06923 897

0.40128 0.39028 0.37368 0.35158 0.32415

259 551 827 779 933

0.50

0.54930 614

-0.119865 477 [(-;)11

-0.11616 303 [1'-;)51

0.29165 814 [(-;)71

[(-;PI

647 084 028 382 042

-0.19866 -0.21821 -0.23768 -0.25705 -0,27631

-0.77012 . . . .- -0.78080 -0.79102 -0.80074 -0,80996 -0.72534 693 [(-;)41

264 885 596 567 958

243. 904 336 877 804

p-;q

-0.81866

327

0:ijiSZ

(n+l)Qn+l(~)= ( 2 n + l ) ~ Q n (-nQn-l(Z) ~)

Qo(z)=arctanh z (Table 4.17) is included here for complekneas.

949 . ..

173 468

347

LEGENDRE FUNCTIONS

LEGENDRE FUNCTION-SECOND Qo (4

2

Qi

(4

Q2

(4

KIND Q3

(4

I

Table 8.3

Qn(%)

QQ

(4

I

Qio(4

0.50 0.51 0.52 0;53 0.54

0.54930 0.56272 0.57633 Oi59014 0,60415

614 977 975 516 560

-0.72534 -0.71300 -0.70030 -0.68722 -0.67375

693 782 333 307 597

-0.81866 -0.82681 -0.83440 -0.84141 -0.84782

327 587 647 492 014

-0,19865 -0,22745 -0.25628 -0.28510 -0.31386

477 494 339 113 748

-0.11616 -0.16231 -0.20711 -0.24999 -0,29035

303 +0.29165 814 372 0,25442 027 759 0.21286 243 263 0.16748 087 406 0.11884 913

0.55 0.56 0.57 0.58 0.59

0.61838 0.63283 0.64752 0.66246 0.67766

131 319 284 271 607

-0.65989 -0.64561 -0,63091 -0.61577 -0.60017

028 342 198 163 702

-0.85360 -0.85873 -0.86319 -0.86695 -0.86998

014 186 116 267 970

-0.34253 -0.37107 -0.39942 -0.42753 -0,45537

994 413 362 983 186

-0.32762 -0.36122 -0.39060 -0.41523 -0.43463

069 172 386 901 218

0.06761 +0.01449 -0.03973 -0.09422 -0.14810

470 441 144 630 594

0.60 0.61 0.62 0.63 0.64

0.69314 0.70892 0.72500 0.74141 0. 75817

718 136 509 614 374

-0.58411 -0.56755 -0.55049 -0.53290 -0.51476

169 797 685 783 880

-0.87227 -0.87377 -0.87446 -0.87430 -0.87326

411 622 461 597 492

-0.48286 -0,50996 -0.53661 -0.56274 -0.58830

632 718 553 938 338

-0.44832 -0.45592 -0.45708 -0.45151 -0.43903

986 864 410 989 693

0.65 0.66 0.67 0.68 0.69

0,77529 0.79281 0.81074 0.82911 0.84795

871 363 313 404 576

-0.49605 -0.47674 -0.45680 -0.43620 -0.41491

584 300 211 245 053

-0.87130 -0.86838 -0.86445 -0.85948 -0.85341

380 239 768 352 027

-0.61320 -0.63739 -0.66077 -0.68327 -0.70481

855 196 634 969 480

-0.41952 -0.39296 -0.35943 -0.31915 -0.27244

271 048 834 810 363

-0.20044 -0.25030 -0,29671 -0,33872 -0,37537 -0,40576 -0.42904 -0.44442 -0.45121 -0.44884

847 577 648 031 391 815 673 606 636 377

0.70 0.71 0.72 0.73 0.74

0.86730 0.88718 0.90764 0.92872 0.95047

053 386 498 736 938

-0,39288 -0.37009 -0.34649 -0.32202 -0.29664

963 946 561 902 526

-0.84618 -0.83774 -0.82803 -0.81698 -0.80451

438 785 775 546 593

-0.72528 -0.74460 -0.76264 -0.77931 -0.79447

868 199 823 296 280

-0.21974 -0.16166 -0.09892 -0.03241 +O. 03684

878 443 467 178 038

-0,43687 -0.41503 -0,38323 -0.34160 -0.29049

329 236 471 431 884

0.75 0.76 0.77 0.78 0.79

0,97295 0,99621 1,02032 1. 04537 1.07143

507 508 776 055 168

-0.27028 -0.24287 -0.21434 -0.18461 -0.15356

369 654 763 097 897

-0.79054 -0.77498 -0.75773 -0.73868 -0.71769

669 679 539 011 507

-0.80799 -0.81973 -0; 82952 -0.83721 -0,84258

424 225 866 016 586

0.10764 0.17866 0; 24840 0.31523 0,37739

474 149 151 275 063

-0.23053 -0.16259 -0; 08787 -0.00787 +0.07559

218 543 565 146 560

0.80 0. 81 0. 82 0. 83 0. 84

1.09861 1.12702 1.15681 1.18813 1.22117

229 903 746 640 352

-0.12111 -0.08710 -0:05140 -0.01384 +0.02578

017 649 968 678 575

-0.69463 -0.66934 -0i64164 -0,61130 -0.57809

835 890 264 745 671

-0.84544 -0.84555 -0.84263 -0,83641 -0.82652

435 002 849 078 589

0.43299 0.48006 0.51654 0.54037 0.54946

312 146 781 123 418

0,16037 0.24398 0.32364 0,39624 0.45844

522 961 357 661 913

0.85 0. 86 0.87 0.88 0.89

1.25615 1.29334 1i33307 1.37576 1.42192

281 467 963 766 587

0.06772 0.11227 Oil5977 0.21067 0.26551

989 642 928 554 403

-0.54172 -0.50183 -0i45802 -0.40979 -0.35650

080 576 786 212 171

-0.81259 -0,79414 -0.77065 -0.74147 -0.70582

105 886 991 880 022

0.54183 0.51562 0.46925 0.40147 0.31159

191 828 273 508 776

0,50669 0.53731 0.54659 0.53099 0.48727

726 190 757 253 156

0.90 0.91 0; 92 0.93 0.94

1.47221 1.52752 1.58902 1.65839 1.73804

949 443 692 002 934

0.32499 0.39004 0.46190 0.54230 0,63376

754 723 476 272 638

-0.29736 -0.23134 -0.15708 -0.07268 +0.02458

306 775 489 272 593

-0.66270 -0.61090 -0.54880 -0.47419 -0.38399

962 890 000 336 297

0.19967 +0.06677 -0,08454 -0.24975 -0.42137

0.95 0.96 0.97 0.98 0.99

1.83178 1.94591 2.09229 2129755 2.64665

082 015 572 993 241

0.74019 0.86807 1.02952 1i25160 1.62018

178 374 685 873 589

,

I

I I ~

~

I

I

i

I

I

I ~

1.00

m

037 0.41282 291 934 0.30602 901 828 +0.16680 029 925 -0.00265 428 701 -0.19666 273

0.13888 288 -0.27356 330 -0.58752 240 -0.40421 0.27707 112 -0.13540 204 -0.72921 201 -0.60564 0.45181 370 +0.04408 092 -0.81464 729 -0.76587 0169108 487 0.29436 613 -0.78406 452 -0,81720 1.08264 984 0,70624 831 -0.48875 677 -0,59305

m

(iN;) Q ~ ( x=) 3x2-1 4 In -.

m

m

--

Q3(x)=-

m

X

4

(522-3)

2 In l+x - -6.2 ( 1 4 ) 2 +8

m

502 435 179 735 105

i

348

LECIENDRE FUNCTIONS

Table 8.4

DERIVATIVE OF THE LEGENDRE FUNCTION-SECOND

Qb(4

Qi(4

Qb,

QA

KIND Q;(z)

Qi (4

0.00 0.01 0.02 0.03 0.04

1.00000 1.00010 1.00040 1.00090 1.00160

000 001 016 081 256

0.00000 0.02000 0.04001 0,06003 0.08008

000 133 067 603 546

-2.00000 -1.99959 -1.99839 -1.99639 -1.99359

000 998 968 838 487

0.00000 -0.07999 -0.15993 -0.23978 -0.31948

000 200 599 392 767

0.00000 0.36520 0.72733 1.08336 1.43027

00 25 83 24 23

9;o (2) -4.06349 21 -4.04156 71 -3,97600 70 -3.86745 44 -3.71697 43

0.05 0.06 0.07 0.08 0.09

1.00250 1.00361 1.00492 1,00644 1.00816

627 301 413 122 615

0.10016 0.12028 0.14045 0.16068 0.18097

704 894 936 662 914

-1.98998 -1.98557 -1.98035 -1.97431 -1.96746

747 401 179 766 792

-0.39899 -0.47826 -0.55725 -0,63589 -0,71414

900 951 060 347 899

1.76512 2.08508 2.38737 2.66939 2.92866

98 14 90 94 44

-3.52604 -3.29655 -3.03075 -2.73130 -2.40117

61 13 84 45 40

0.10 0.11 0.12 0.13 0.14

1.01010 1.01224 1.01461 1.01719 1.01999

101 820 039 052 184

0.20134 0.22179 0.24233 0.26297 0.28372

545 422 428 462 443

-1.95979 -1,95130 -1,94198 -1.93182 -1.92081

839 431 044 094 942

-0.79196 -0.86930 -0.94609 -1.02230 -1.09787

777 001 554 373 345

3.16285 3.36984 3.54769 3.69467 3.80930

86 76 49 78 18

-2.04367 -1.66240 -1.26123 -0.84427 -0.41580

37 59 82 11 27

0.15 0.16 0.17 0.18 0.19

1.02301 1.02627 1.02976 1.03348 1.03745

790 258 007 491 202

0.30459 0.32559 0.34672 0.36800 0.38945

312 031 587 997 305

-1.90896 -1i89626 -1.88268 -1I86824 -1.85291

890 181 994 444 580

-1.17275 -1I24689 -1.32023 -1.39272 -1.46431

302 019 203 496 458

3.89031 3i93671 3.94778 3.92304 3.86234

48 92 25 76 02

+O. 01970 77

0.20 0. 21 0.22 0. 23 0.24

1.04166 1.04613 1.05086 1.05585 1.06112

667 453 171 471 054

0.41106 0.43285 0.45484 0.47703 0.49944

589 960 568 605 304

-1.83669 -1.81956 -1.80152 -1.78255 -1.76264

380 752 526 455 210

-1,53494 -1.60456 -1,67310 -1.74052 -1.80674

573 234 742 294 982

3.76577 3.63376 3.46700 3.26651 3.03359

0. 25 0.26 0.27 0.28 0. 29

1.06666 1.07250 1.07863 1.08506 1.09182

667 107 229 944 225

0.52207 0.54495 0.56809 0.59150 0.61519

948 869 454 152 472

-1.74177 -1.71993 -1.69710 -1.67327 -1.64842

372 437 801 761 510

-1.87172 -1.93539 -1.99768 -2.05854 -2.11790

780 537 972 661 027

0.30 0.31 0.32 0.33 0.34

1.09890 1,10631 1.11408 1.12220 1.13071

110 707 200 851 009

0.63918 0,66350 0.68815 0.71315 0.73853

993 370 335 706 396

-1.62253 -1.59557 -1.56753 -1.53839 -1.50811

126 570 678 152 553

-2,17568 -2.23182 -2.28625 -2.33890 -2.38969

0.35 0.36 0.37 0: 38 0.39

1.13960 1.14889 1.15861 1.16877 1.17938

114 706 430 045 436

0.76430 0.79048 0.81711 0.84419 0.87175

415 884 039 242 994

-1.47668 -1.44406 -1.41023 -1.37516 -1.33880

292 617 606 155 960

0.40 0.41 0.42 0.43 0.44

1.19047 1.20206 1.21418 1.22684 1.24007

619 756 164 333 937

0.89983 0.92845 0.95764 0.98743 1.01786

941 892 831 931 572

-1.30114 -1.26213 -1.22172 -1.17988 -1.13657

0.45 0.46 0.47 0.48 0.49

1.25391 1.26839 1.28353 1.29937 1.31596

850 168 228 630 263

1.04896 1.08077 1.11333 1.14668 1.18088

360 146 051 490 202

-1.09173 -1.04531 -0,99726 -0.94752 -0.89602

0. 50

1.33333 333

z

1.21597 281

(5)

Oi45767 0.89344 1.32231 1.73958

92 90 56 08

54 26 84 77 33

2.14059 2.52079 2.87577 3.20128 3.49331

45 94 54 51 81

2.76983 2.47712 2.15764 1.81383 1.44841

31 56 35 48 22

3.74813 3.96230 4.13277 4.25684 4.33229

48 97 26 84 46

334 672 944 860 914

1.06434 0.66482 +0.25327 -0.16667 -0.59123

02 02 32 95 78

4.35733 4.33070 4.25164 4.11997 3.93608

72 22 55 79 76

-2.43855 -2.48539 -2.53013 -2.57269 -2.61297

378 281 394 210 926

-1.01644 -1.43822 -1.85237 -2.25465 -2.64075

6? 04 43 05 25

3.70095 3.41617 3.08394 2.70708 2.28903

66 42 42 74 82

509 064 641 995 597

-2.65090 -2.68637 -2.71928 -2.74954 -2.77703

420 229 520 067 216

-3.00637 -3.34725 -3.65918 -3.93806 -4.17995

81 61 35 51 45

1.83383 1.34610 0.83104 +0.29437 -0.25765

54 61 35 81 92

613 874 854 634 868

-2.80164 -2.82327 -2.84178 -2.85705 -2.86895

855 375 630 896 817

-4.38109 -4.53797 -4.64734 -4.70629 -4.71228

69 26 21 25 35

-0.81838 -1.38069 -1.93714 -2.48003 -3.00140

00 01 78 04 86

-0.84270 745

-2.87734 353

-4.66319 54

-3.49322 79

,

349

LEGENDRE FUNCTION8

DERIVATIVE OF THE LEGENDRE FUNCTION-SECOND

Qh (XI

X

Q;(4

Qb) - 0.84270 74 - 0.78748 95 -- 0.73029 59 0.67104 20 - 0.60963 61 - 0.54597 91 - 0.47996 38 - 0.41147 39 - 0.34038 30 - 0.26655 35 - 0.18983 51 - 0.11006 36 - 0.02705 91

0. 50 0. 51 0. 52 0.53 0.54

1.33333 1.35153 1.37061 1.39062 1.41163

333 399 403 717 185

1.21597 1.25201 1.28905 1.32717 1.36643

281 210 905 756 680

0.55 0. 56 0. 57 0. 58 0.59

1.43369 1.45687 1.48126 1.50693 1.53397

176 646 204 189 760

1.40691 1.44868 1.49184 1.53648 1.58271

178 400 220 320 285

0. 60 0. 61 0.62 0.63 0.64

1.56250 1.59261 1.62443 1.65809 1.69376

000 029 145 982 694

1.63064 1.68041 1.73215 1.78601 1.84218

718 364 259 903 458

0. 65 0; 66 0. 67 0; 68 0. 69

1.73160 1.77179 1.81455 1.86011 1.90876

173 305 271 905 121

1,90083 1.96219 2.02649 2.09399 2.16500

983 705 344 499 099

0.24343 0.34156 0.44414 0.55151 0.66402

42 40 64 17 96

0. 70 0. 73 0. 74

1.96078 2.01653 2.07641 2.14086 2.21043

431 559 196 919 324

2.23984 , 2.31892 2.40266 2.49156 2.58619

955 413 159 187 998

0.78211 0.90623 1.03692 1.17478 1.32049

54 72 51 21 75

0.75 0.76 0.77 0. 78 0.79

2.28571 2.36742 2.45639 2.55362 2.66028

429 424 892 615 199

2.68724 2.79545 2.91175 3.03719 3.17305

079 751 493 894 446

1.47486 1.63879 1.81335 1.99979 2.19957

32 46 60 32 51

0. 80 0. 81 0. 82 0.83 0. 84

2.77777 2.90782 3.05250 3.21440 3.39673

778 204 305 051 913

3.32083 3.48236 3.65986 3.85608 4.07443

451 488 997 883 439

2.41444 2.64650 2.89827 3.17286 3.47409

73 26 40 02 64

0. 85 0. 86 0.87 0. 88 0.89

3.60360 3.84024 4.11353 4.43262 4.81000

360 578 352 411 481

4.31921 4.59595 4.91185 5.27647 5.70283

588 604 380 688 015

3.80679 4.17707 4.59287 5.06465 5.60654

0.90 0.91 0.92 0.93 0. 94

5.26315 5.81733 6.51041 7.40192 8.59106

789 566 667 450 529

6.20906 6.82129 7.57861 8.54217 9.81365

159 988 025 980 072

0.95 0. 96 0.97 0. 98 0.99

10.25641 12.75510 16.92047 25.25252 50.25125

026 204 377 525 628

11.57537 14.19080 18.50515 27.04503 52,39539

057 811 528 467 613

0: 71 0. 72

1.00

m

W

+ 0.05937 63

0.14946 05

Q4 (x)

-- 2.87734 35 2.88206 72 -- 2.88297 33 2,87989 70 - 2.87266 39 - 2.86108 89 -- 2.84497 53 2.82411 36 -- 2.76723 2.79828 02 56 -- 2.68846 2.73072 34 75 -- 2.58551 2.64017 05 - 2.52414 0800 92 --- 2.45567 2.37971 49 2.29579 49 -- 2.20342 26 2.10205 04

KIND QA(z)

Table 8.4

Q-b ( ~, X)

- 4.66319 - 4.55737 - 4.39368 - 4.17156 - 3.89102 - 3.55277 - 3.15819 - 2.70941 - 2.20934 - 1.66171 - 1.07108 - 0.44291 +

54 62 94 11 65 54 61 73 79 26

51 60 0.21644 47 0.89973 10 1.59875 12

Ql"(X',, - - - \

- 3.493228 - 3.947399 - 4.355894 - 4.710854 - 5.004695

-- 5.230233 5.380807 -- 5.433812 5.450406 - 5.326732 -- 4.829465 5.125950

- 4.436645 - 3.948368 - 3.367169 - 2.697375 - 1.945245 - 1.119087 - 0.229371

2.30438 3.00660 3.69447 4.35619 4.97914

77 55 22 14 99

23 51 72 54 96

5.54998 6.05466 6.47859 6.80675 7.02388

34 05 09 90 88

1.687501 2.682165 3.675339 4.644816 5.566082

27 65 20 - 0.65620 16 - 0.41542 09

7.11464 7.06387 6.85691 6.47990 5.92027

51 68 02 33 14

6.412431 7.155161 7.763836 8.206652 8.450921

-

5.16720 4.21227 3.05023 1.67989 0.10532

18 67 28 36 57

8.463693 8.212559 7.666669 6.798024 5.583115

-

1.99107 1.86981 1.73752 1.59336 1.43637

- 1.26549 -- 1.07947 0.87692

+

0.711177

0.15235 0.13562 0,45165 0.79955 1.18395

72 04 68 16 08

33 50 14 07 69

1,61061 2.08677 2,62171 3.22751 3.92032

19 72 45 63 16

-

6.23815 6.98747 7.89613 9.02883 10.49236

05 73 09 27 44

4.72224 5.66456 6.79318 8.17876 9.93658

63 11 58 62 04

-12.26944 -14.26758 -15.92348 -16.99643 -17.13329

98 89 54 22 84

-12.315713 -15.495090 -18.304274 -20.319071

12.4'7698 15.35932 20.00905 29.00735 55.11181

56 33 43 14 39

12.26978 15.57616 20.76422 30.50045 57.80864

50 37 38 90 53

-15.78782 -12.04072 4.11777 +12.32933 54.86521

62 38 87 89 05

-20.873659 -18.851215 -12.140718 + 4.242107 49.428990

m

+

m

+

1.66270 85 91 02 81 18

--- 3.60489 5.69098 7.87652 -10.09858

-

m

4.005017

+ 2.056070

- 0.258625 - 2.916594 - 5.871760 - 9.045801

Q)

LEGENDRE FUNCTIONS

Table 8.5

LEGENDRE FUNCTION-FIRST

Po(z)=l 5

(4

P2

p3(4 1.00 2.52 4.76 7.84 11.88

(4

P4

KIND P,(z)

P1(x)=z

(4

P5

p9 $1 1.00000 6.02754 5.03668 2.45973 8.97882

11

PlO(X)

1.0 1.2 1.4 1.6 1.8

1.00 1.66 2.44 3.34 4.36

2.0 2.2 2.4 2.6 2.8

5. 50 6.76 8.14 9.64 11.26

17.00 23.32 30.96 40.04 50.68

3. 0

63.00 77.12

2 3.21000 2 4.20727

3 1.68300 3 2.36169

3.462

13.00 14.86 18.94 16.84

3.8

21.16

111.24 93.16 131.48

1215.41672 2 6.86667 2 8.58472

1313.24050 3 4.36022 3 5.76676

4.0 4.2 4.4 4.6 4.8

23.50 25.96 28.54 31.24 34.06

154.00 178.92 206.36 236.44 269.28

5. 0 5.2 5.4 5.6 5.8

37.00 40.06 43.24 46.54 49.96

6. 0

531.00 586.52

66.2 .4 6.6 6.8

53.50 57.16 60.94 64.84 68.86

3 5.53538 3 6.32087

4 5.93572 4 7.00717

645.76 708.84 775.88

1317.18681 3 8.13847 3 9.18133

1418.22754 4 9.61180 5 1.11759

7.0 7.2 7.4 7.6 7.8

73.00 77.26 81.64 86.14 90.76

847.00 922.32 1001.96 1086.04 1174.68

8. 0 8.2 8.4 8.6 8.8

95.50 100.36 105.34 110.44 115.66

1268. 00 1366.12 1469.16 1577.24 1690.48

(llj4.87282

9.0 9.2 9.4 9.6 9.8

121.00 126.46 132.04 137.74 143.56

1809.00 1932.92 2062.36 2197.44 2338.28

(12)1;43817

10.0

1.00000 4.04700

1.00000 6.72552 1 2.09686

(:/!::4:::

1 2 3 3

1.00000

11

6 1.46256 6 2.69625

6 8.15181 4.77208 7 1.34978

11

( 9j7.23884

8 9.01781 9 1.21596

9 1.62372 2,14858 9 2.81890

(10)3162216

149.50 2485.00 (4)4.33754 (5)7.78769 (10)9.29640 ( 12)l. 76188 From National Bureau of Standards, Tables of associated Legendre functions. Columbia Univ. Press, New York, N.Y., 1945 (with permission).

351

LEGENDRE FUNCTIONS

DERIVATIVE OF THE LEGENDRE FUNCTION-FIRST

P;(z)= 1 2

Pi (4

1.0 1.2

6.000 9.300

KIND PA(x)

Table 8.6

Pi@) = 32

P;(2)

1.8

Pio(2) 1 5.50000 3 1.53586 4 1.13477 4 5.24824 5)1.85808

1

2.0 2.2 2.4 2.6 2.8

4.0 4.6 4.8

5.6 5.8

1

9 1.37071 9 1.78712 912.31006 9 2.96206 9 3.76947

11

10 1.72421 10 2.32397 10 4.10354 3.10217 10 5.38214

8.8

9.8 10.0 (2)7.485 (4)l. 74250 (5)3.91127 (10)8.40642 (12)l. 77028 k o m National Bureau of Standards, Tables of associated Legendre functions. Columbia Univ. prw, New York, N.Y., 1945 (with permission).

LEGENDRE FUNCTIONS

Table 8.7 5

LEGENDRE FUNCTION-SECOND 620

(4

Qio(x)

Q9 (2)

6218)

1. 0 W iI2 1.4 (-1 8.95880 -11 1.6 1.8 -1 6.26381

[

KIND Q ~ ( z )

:::t

W

f

- 912.07945

( -3) 1; 25217

10.0 (-1)l.00335 (-3)3.35348 (-4)l. 34486 (-6)5.77839 (-14)5.69010 (-15)2.71639 From National Bureau of Standards, Tables of associated Legendre functions. Columbia Univ. Press, New York, N.Y., 1945 (with permission).

DERIVATIVE OF THE LEGENDRE FUNCTION-SECOND 5

1. 0 1.2

- Qh (5)

- Q6 (4

KIND QA(z)

- Q6 (5) m

m

I

353

LEGENDRE FUNCTION8

Table 8.8

-Q;o(~) m

2.27273 1.04167

I-5) 7.82792

5. 0 5.2 5.4 5.6 5. 8

I - 1 O ) l . 28985

6. 0 6.2 6.4 6.6 6. 8 7.0 7.2 7.4 7.6 7; 8

-3 1.99230 -3 1.82834 -3 1.68195 -3 1.55083 (-3)lI 43304

-4 1.71573 -4 1.53040 -4 1.36949 1 -4 - j l .1.22923 10651

8.0 8.2 8.4 8. 6 8.8 9. 0 9.2 9.4 9.6 9.8 I

10.0 (-2)l. 01010 (-4)6. 74753 (-5)4.05782 (-6)2.32430 (-14)5.72014 (-15)3.00374 fiom National Bureau of Standards, Tables of associated Legendre functions. Columbia Univ. Press, New York, N.Y., 1945 (with permission).

9. Bessel Functions of Integer Order F. W. J. OLVER

Contents Page

Mathematical Properties . . . , , , . . , , . . . . . . . , . , Notation. . . . . . . . . ........ . ........ Bessel Functions J and Y . . . . . . . . . . . . . , . . . . . 9.1. Definitions and Elementary Properties . . . . . . , . . 9.2. Asymptotic Expansions for Large Arguments , , . . , . 9.3. Asymptotic Expansions for Large Orders . . , . . . . 9.4. Polynomial Approximations. . , . . . . . , . . , . . 9.5. Zeros. . . . , . . . . . . . . . . . . . . . . , . Modified Bessel Functions I and K . . . . . , . . . . . , , . , 9.6. Definitions and Properties . . , . . , . . . . . , . . 9.7. Asymptotic Expansions. . . . , . . . . . . . , . . 9.8. Polynomial Approximations. . , . . . . . . . . . . Kelvin Functions. . . . . . , . . . ....... .. ... 9.9. Definitions and Properties . , . . , , . . . . . . . 9.10. Asymptotic Expansions . . . . . . . . . , . . . 9.11. Polynomial Approximations , . . . . . , . . , , . Numerical Methods . . . . . . . . , . . . . . . . . . . . 9.12. Use and Extension of the Tables. .. . . ... References. . , . . . . . , , , . .., , ..,,.. . Table 9.1. Bessel Functions-Orders 0, 1, and 2 (05~517.5) . .. Jo(z), 15D, Ji@), J z P ) , Yo(z), Yi@), 10D Yz(z),8D z =0 (.1) 17.5 Bessel Functions-Modulus and Phase of Orders 0, 1, 2

. . .

358 358 358 358 364 365 369 370 374 374 377 378 379 379 381 384 385 385 388 390

.., . .. .

396

.

.

.

.

.

. . . .

.... ..

(~O
Bessel Functions-Auxiliary (05z52).

z=0(.1)2,

, ,

.

.

.

.

. .

.

,

.

.

I

Table for Small .Arguments

...... ... . ..... . *

.

397

.. ... ..

398

9

*

8D

Table 9.2. Bessel Functions-Orders 3-9 (0 5 ~ 5 2 0 .) n= 3(1) 9 Jn (5) 9 Yn (5) , 5D or 55 z= 0(.2)20,

* National Bureau of Standards, on leave from the National Physical Laboratory. 355

356

BESSEL FUNCTIONS O F INTEGER ORDER

Table 9.3. Bessel Functions-Orders 10, 11,20, and 21 ( 0 5 3 5 2 0 ) z-'oJ'o(z), z-"rJ11(z), 2'OY,,(Z) s=0(.1)10, 8s or 9s JlO(4,

JIl(4,

. .

Page

402

YlO(3)

~=10(.1)2O, 8D z-'OJ20(2), 2-21Jz1(2),z20Y20(z)

z=0(.1)20, 6 s or 7s Bessel Functions-Modulus and Phase of Orders 10,11,20, and21 ( 2 0 1 2 2 = ) . . . . . . . . . . . . . . . . . zfMn(x),On(z)--z n=10, 11, 8D n=20, 21, 6 D z-'= .05( - .002)0

Table 9.4. Bessel Functions-Various Orders (0 ,
. . . . .

406

407

10s

Table 9.5. Zeros and Associated Values of Bessel Functions and Their Derivatives ( O s n 1 8 , 1 5 9 5 2 0 ) . . . . . . . . jn,,, J X j n , , ) ;jb,,, Jn(jb,J, 5 D (10D for n=O) Yn,,, Yk(Yn,J; Y k . 8 , Yn(YbJ, 5D (8D for n=o> S = 1(1)20, n=0(1)8

Table 9.6. Bessel Functions Jo(jo,j), s=1(1)5.

. . .

409

. . . . . . . . . .

413

.....

414

z=O(.O2)1, 5D

Table 9.7. Bessel Functions-Miscellaneous 9th zero of z Jl(z)-xJo(z) A, A-'=O(.O2) , I , .2(.2)1,

Zeros (s= 1(1)5) 4D

zero of Jl(z)-Wo(z) 1(- .2).2, .1(- .02)0, 4 D 8th zero of Jo(s) Yo(kc) - Yo(z)Jo(kz) 5 D (8D for s=1) A-'=.8(--.2) .2, .1(-.02)0, 9th zero of J1(z) Yl(Az) - Yl(z)Jl(Xz) 5 D (8D for s=1) X-'=.8(-.2) .2, .1(-.02)0, 9th zero of J1(z) Yo(Az)- Yl(z)Jo(kz) 5D (8D for s = l ) .2, .l(-.02)0, X-'=.8(-.2) 9th

X= .5(.1) 1, A-'=

Table 9.8. Modified Bessel Functions of Orders 0, 1, and 2 (0 5 2 1 2 0 ) e-.I0(z), ezKo(z),e-"II(z), e"Kl(z) z=0(.1)10 (.2)20, 10D or 10s 2-2&

.

416

(422Ka (2) 7

~ = 0 ( . 1 ) 5 , IOD, 9 D

e-"12(2) , ezKz(2)

~ = 5 ( . 1 ) 1 0(.2)20, 9D, 8 D Modified Bessel Functions-Auxiliary Table for Large Arguments(20lzlco). . . . . . . . . . . . . . . de-"In(z),a-'de"K,(z), n=O, 1, 2 ~-'=.05(--.002)0, 8-9D Modified Bessel Functions-Auxiliary Table for Small hguments ( 0 5 2 5 2 ) . . . . . . . . . . . . . . . . Ko(z) +Io(z) lnz, z{ Kl(z)-Il(z) lnz} 2=0(.1)2, 8D

422

422

357

BESSEL FUNCTIONS O F INTEGER ORDER

Table 9.9.

Modified Bessel Functions-Orders e-”I,(z), e”K,(z), n= 3 (1)9 z=0(.2)10(.5)20, 5s

3-9 ( 0 5 z < 2 0 )

. . .

Table 9.10. Modified Bessel Functions-Orders 10, 11, 20 and 21 (05z<20) . . . . . . . . . . . . . . . . . . . . z-10110(2), 2-”111(z), zl°K1,(z) z=0(.2)10, 8s or 9 s e-“IlO(z),e-”Ill(z>,ezKlo(z) z=10(.2)20, IOD, 10D, 7D z-’O120(%)j z-”121

Page

423

425

(z) z20Kzo(z) J

5s to 7 s Modified Bessel Functions-Auxiliary Tabie for Large Arguments (2O}, 1n{n-lztezKlo(z)} In{z+e-z120(z)}, In{zfe-”Izl(z)}, ln{a-lz+e”K&)} ~-‘=.05(-.001)0, SD, 6D z=0(.2)20,

Table 9.11. Modified Bessel Functions-Various Orders (0 I n 5 100) In(%),K,(z), n=0(1)20(10)50, 100 s = l , 2, 5, 10, 50, 100, 9s or 10s

.

Table 9.12. Kelvin Functions-Orders 0 and 1 (O
427

428

430

430

432

MO(d, @0(4, Ml(4, e l ( d No(z),40(d, Nl (4, 41(2) z=0(.2)7,

6D

Kelvin Functions-Modulus ments ( 6 . 6 S z 5 a). . .

and Phase for Large Argu-

. . . . . . . . . . . . . .

432

z*e-z/~zMo(z)J e0(+ ( z / J z ) ,z+e-“/dzM1(z), e1(+ (zlJz) z+ez’JzNo(z>, 4dz) ( z / J z ), z+ez/dzNl(z),41(4 (s/Jz) ~-‘=.15(--.01)0, 5D

+

+

The author acknowledges the assistance of Alfred E. Beam, Ruth E. Capuano, Lois K. Cherwinski, Elimbeth F. Godefroy, David S. Liepman, Mary Orr, Bertha H. Walter, and Ruth Zucker of the National Bureau of Standards, and N. F. Bird, C. W. Clenshaw, a nd Joan M. Felton of the National Physical Laboratory in the preparation and checking of the tables and graphs.

I

9. Bessel Functions of Integer Order Mathematical Properties

I

Notation

The tables in this chapter are for Bessel functions of integer order; the text treats general orders. The conventions used are: z=z+iy; z, y real. n is a positive integer or zero. V, p are unrestricted except where otherwise indicated; v is supposed real in the sections devoted to Kelvin functions 9.9, 9.10, and 9.11. The notation used for the Bessel functions is that of Watson [9.15] and the British Association and Royal Society Mathematical Tables. The function Y,(z) is often denoted N y ( z )by physicists and European workers. Other notations are those of: Aldis, Airey:

G,(z)for -~TY,(Z),K,(z) for (-),K,(z). Clifford:

C,(z) for s-tnJ,(2&). Gray, Mathews and MacRobert [9.9]: Yn(z) for irYn(z)+ (In 2 - - ~ ) J , ( z ) ,

-Y,(z) for ?reyrf sec(vr)Y,(z),

Bessel Functions Jand Y

9.1. Definitions and Elementary Properties Differential Equation 22

9.1.1

d2w -+z dz2

dw --+(22-vy2)w=O dz

Solutions are the Bessel functions of the first kind Yy(z)(also called Weber’sfunction) and of the third kindH!”(z), HLa)(z) (also called the Hankel functions). Each is a regular (holomorphic) function of z throughout the z-plane cut along the negative real axis, and for fixed z ( # O ) each is an entire (integral) function of v. When v= An, J y ( z )has no branch point and is an entire (integral) function of z. Important features of the various solutions are as follows: Jv(z)(9i?v>O) is bounded as z+O in any bounded range of arg z. J y ( z )and J-,(z) are linearly independent except when v is an integer. J,(z)and Y,(z) are linearly independent for all values of v. Hi1)(z)tends to zero as I z / + w in the sector O<arg Z
J * y ( z ) , of the second kind

Gv(z)for +?rzHi’)(z). Relations Between Solutions

Jahnke, Emde and Losch t9.321: A&) for r(v+i)(+z)-YJv(z).

9.1.2

J ” ( 2 )cos

Y,(z)=

(rn)-J-r(Z)

sin (VT)

The right of this equation is replaced by its limiting value if v is an integer or zero.

Kh,(z) for (2/r)K,(~). 9.1.3 Heine:

HLl)(Z) =J,(z)+iY,(z)

K,( z ) for- +nY,(2 ) . Neumann: Yn(z) for +rY,(z)+(ln 2--r)Jn(z). Whittaker and Watson [9.18]:

9.1.5

J - (Z ) = ( -) “J,(Z )

Y-n( 2 ) = (-

Y,(2)

BESSEL FUNCTIONS OF INTEGER ORDER.

359

1,\,M,oIKJ \, \,

- 6

II

I

‘~

I

I

::I i L

FIGURE 9.3. J,(lO) and Yy(lO).

Y

c

FIGURE 9.4. Contour lines of the modulus and phase of the Hankel Function Hi”(x+iy)=M~e{So. From E. Jahnke, F. Emde, and F. Losch, Tables of higher functions, McGraw-Hill Book Co.,Inc., New York, N.Y.,1960 (with permission).

360

BESSEL FUNCTIONS OF INTEGER ORDER

Integral Representations

Limiting Forms for Small Arguments

9.1.18

When v is fixed and z+O

J~( z )

9.1.7

J , ( z ) - ( ~ ~ ) y / r ( ~ +(VZ-1, i) 9.1.8 9.1.9

-2, -3,

. . .)

Yo(z)--~iH~1)(z)-iH~2)(~)-(2/7r)ln z

-

=;

cos ( z sin e)ds=-

‘so’

lr

cos ( z cos e)ds

9.1.19

S,

4 y0(z)=p

f’

cos ( z cos e) {?+In (22 sin2 e)

1

9.1.20

-

Y,(Z ) -iH;l) (Z ) -iH;2) (Z ) - ( I / ~r)(v) (32) -” (9V>O)

Ascending Series

9.1.11

Y&)= -.

(t2”)” 7r

k=O

where g(n) is given by 6.3.2.

9.1.13

lm

Y o ( z ) = -?r ~

cos(s cosh t)dt (s>O)

9.1.24

9.1.26

In the last integral the path of integration must lie to the left of the points t = O , 1, 2, . . . .

361

O F INTEGER ORDER

1

and 4

9.1.34

Analytic Continuation

In 9.1.35 to 9.1.38, m is an integer.

1

9.1.35

J ,(Zem* i) =em,*

J&)

9.1.36

Yv(z emri)

=e-m”* “,(

z)

+2i sin (mvx) cot

(VT)

J,( z )

9.1.37

I sin(va)H!’)

(zem*t)

II

= -sin { (m- 1)v r 1~ j 1(z> )

1

sin(mv?r)H!’) ( z )

--e-v*t

9.1.38 sin(vT)H!’) (zemTt)=sin { (m+ 1)VT} H:) ( z ) +evsi sin(mv?r)H!’)(z) 9.1.39

Hi1)(Ze*i) = -e-v*tHjW

0

H?)(ze-*:) = -eV*tH!l’(z)

Formulas for Derivatives

9.1.30

9.1.31 1 Y!*)(2)= p { W v - k ( Z )

(k=O, 1,2, . . .)

-(;)

+(i)

9.1.41

eM-l/t)=

5

* * *

+(-)b%‘”+k(Z)l

(titO) m

(k=O, 1,2, . . .)

9.1.42 cos ( z sin 0) =Jo(z)4-2

Recurrence Relations for Cross-Products

k=l

J z k ( z cos ) (2kO)

m

9.1.43 sin ( z sin 8)=2

9.1.32

k-0

J P k f l ( zsin ) f(2k+l)e}

9.1.44

pv=Jv(a)Yv(b) -J”(b)YY(Q) pv y:(b) - J : (b)Yv(a)

m

=JY(a)

then 9.1.33

tV&)

k=-m

5fv-k+&)-

If

Generating Function and Associated Seriee ~Py-k+Z(Z)

r,=J:(a)Y,(b)-J,(b)Y:(a) s, =J: (a)Y:(a) -J : ( b )Yfl(a)

COS ( Z COS

e) = J ~ ( z+2)

c (-)V~~(Z)

COS

k=l

(2ke)

9.1.45 m

sin ( z cos e)=2 2v

2v pvc1-pV-l=-- a p v - ~r, v vfl

P v -7

Q’+l+T’=;

v TY+l+pv=J

vfl

pv+l

p v - 7 p,+1

I

9.1.46

k=O

(-)kJzk+I(z) cos { (2k+i)e}

I

~ = J O ( Z > + ~ J Z ( Z ) + ~ J ~ ( Z .) + . .~ J ~ ~ ( Z )

9.1.47 cos z=Jo(z)-2J2(2)f2J,(z)-2Jtj(2)+

..

a

BESSEL FUNCTIONS OF INTEGER ORDER

362

Other Differential Equations

w''

9.1.49

+(.---$)

v2-

L

w = z W v (Xz)

w=O,

Derivatives With Respect to Order

9.1.64

9.1.51 w' ' +h22p-2~=0,

W = Z ~W11,(2Xz*P/p)

9.1.52 W'

--2v-1

f

Z

9.1.65

+

w ' X'w =0 ,

w= zvVv(Xz,

9.1.53 AY (1- 2 p ) ~ w ' + ( ~ ? q ~ + p 2v ~- P ~ ) w = o , w =Z P V v (XZO) 9.1.54

+

+

w' ' (X2ezr- v2)w=0,

-csc (v.)

a

by J-v(z)-7rJv ( 2 ) ( v f O , f l , f2, *

. .)

9.1.66

w =Vv(Xec>

9.1.55 V')W'

2'( 2'-

+{

'+Z ( 2'-

9.1.67

3 V 2 ) W'

(z"v2)}w=O,

(z'-v2)*-

w=WR:(z)

9.1.56

where

CY

9.1.68

W=ZfnVn(2Xazf)

(-)nX2nZ-nW,

is any of the 2n roots of unity.

Differential Equations for Products

Vv(z), 9 , , ( z ) are any

In thefollowing6=2d;and d cylinder functions of orders

v, p

Expressions in Terms of Hypergeometric Functionr

9.1.69

respectively.

9.1.57 { 6'- 2 (v2+ p2) 6 2 +

(v2-

w

+4Za(6+1)(6+2)~=0,

w=V~(Z)~.,(Z)

9.1.70

9.1.58 6(6'

-4 v2)w +4 Za (6+1)w =0,

w =V,(Z )

(Z )

9.1.59 zawff'+2(49+1-4s)w'+(4v2-l)w=o,

w =zVv(2 ) .5PV( 2 )

as X, p+= through real or complex values; z, v being fixed. is the generalized hypergeometric function. For M(u, b, z ) and P(u, b; c; z ) see chapters 13 and

15.)

Upper Bounds

9.1.60

IJV

Is 1 ( v > O ) , I Jv (5)I Ill@

(5)

Connection With Legendre Functions ( v 2 1)

If p and 5 are fixed and positive values

U+=

through real

9.1.71 lim

{ v f i ~ (cos ; ~ :)}=~,(z)

(z>o)

363

BESSEL FUNCTIONS OF INTEGER ORDER

In 9.1.79 and 9.1.80, w=J(u2+v2-2uv cos ff), U-v cos a = w cos X , v sina=w sin x

9.1.72 lim

{Y’Q;’

(cos

:)I = - i r ~ , , ( x )

(Z>O)

the branches being chosen so that w+u and x+O as v+O. C(:)(cos a) is Gegenbauer’s polynomial (see chapter 22).

For P;’ and Q;”, see chapter 8. Continued Fractions

9.1.73

I

Multiplication Theorem

9.1.74 %?u(Xz)=X*u

Gegenbauer’s addition theorem.

( F1 (A2 -1)“(34n yv*n (z )

OD

k=O

k!

If U, v are real and positive and 0 < a ,< T , then w, x are real and non-negative, and the geometrical relationship of the variables is shown in the diagram. The restrictions Ive*‘”l< JuI are unnecessary in 9.1.79 when %?= J and Y is an integer or zero, and in 9.1.80 when V= J .

(I -1I<1)

If %?= J and the upper signs are taken, the restriction on X is unnecessary. This theorem will furnish expansions of V,(re*e) in terms of %?v&). Addition Theorems

Degenerate Form

Neumann’s

(U=

0):

9.1.81

The restriction Ivl<[u[ is unnecessary when %?= J and Y is an integer or zero. Special cases are

9.1.76

5Ji(z)

1= Ji(z)+2 k = 1

I

Neumann’s Expansion of an Arbitrary Function in a Series of Bessel Functions

c m

9-1-82f(z>=aoJ0(2)+2k = 1 anJn(z)

(Izl
where c is the distance of the nearest singularity off(z) from z=O,

9.1.83

an=,&

s,.,-.,

f(t)Odt)dt

(O
and O,(t) is Neumann’s polynomial. The latter is defined by the generating function Graf’s

9.1.84

9.1.79

O,(t) isapolynomialof degreen+l in l/t; Oo(t)=l/t, Gegenbauer’s

9.1.80

9.1.85 (n=1,2,. k=O

. .)

BESSEL FUNCTIONS OF INTEGER ORDER

364

9.2.6 Y,(z)=42/(*z) {P(v,2 ) sin x+Q(v,

also called a Neumann expansion, is investigated in [9.7] and [9.15] together with further generalizations. Examples of Neumann expansions are 9.1.41 to 9.1.48 and the Addition Theorems. Other examples are

9.2.7

9.1.87

H!"(z) =4-) . .

( v f O , -1,-2,.

. .)

9.1.88

2)

cos x} zl<.>

{ P ( Yz, ) + i Q ( v , z ) } etx (- r<arg 2<21)

9.2.8 H!2)(~)=J2/(nz){P ( Y ,z)-i&(v, z)}e-*x

n!(+z)-"n-1

Y n( 2 ) =-~-a

(+Z)kJJlc(Z)

&i (n-k)k!

where X = Z - ( + Y + ~ ) T

(-2?r<arg Z < T ) and, with4v2denoted by p ,

9.2.9

where +(n) is given by 6.3.2.

9.1.89

=p-1 -

82

9.2. Asymptotic Expansions for Large Arguments

*..

If v is real and non-negative and z is positive, the remainder after k terms in the expansion of P(v, z ) does not exceed the (k+l)th term in absolute value and is of the same sign, provided that k>+v-$. The same is true of &(u,z) provided that k>+v-i.

Principal Asymptotic Forms

When v is fixed and

(P-l)(P-9)(P-25)+ 3!(8~)~

(zI+o~

Asymptotic Expansions of Derivatives

With the conditions and notation of the preceding subsection

9.2.11 ~:(z)=J2/(-az){ -R(v, z ) sinx-S(v, z ) cos X}

9.2.12

Y:(Z)=J=){R(~,

(larg Zl
(larg zl<-a>

9.2.13 Hankel's Asymptotic Expansions

~ ! " ' ( z ) = J 2 / ( - a z ) { i ~ ( v , z ) - ~ ( v , z)}eix

When v is fixed and I z ( + o ~

(--a<arg

9.2.14

9.2.5

JV(z)=J2/(9rz){P(v,

2) COSX-Q(Y,

Z) s ~ X }

(larg zl<-a>

z<2~)

1

365

BESSEL FUNCTIONS OF INTEGER ORDER

I

9.2.15

,

9*2*29 P-1 (+v+:)r+2 (4x1

ev-x-

(p-

+

9.2.16

+ --p+ 3

h- 1) (cc-

9) b+35) +

3!(8~)~

82

(p-

+

1) (p-25) 6(4~)~

(p-

1) (p'- 114p+ 1073) 5 (42) 5

1) (5p3- 1535p2+54703p-375733) 14(42)

'

+.. .

9.2.30

...

Modulus and Phase

For real v and positive 9.2.17

The general term in the last expansion is given by

2

*

em) +

x (p-

9.2.18

1)(p-9).

. . { p-(2k-

3)' } { p- (2k+l) (2k- 1)2} * (2x)'k

N,= I H ~ ' " ( ~ ) ~ = . \ / { J : ' ( ~ ) + Y ~ ' ( S ) } 9.2.31 p,=arg H$l)'(z)=arctan { Y;(x)/J;(x)} 9.2.19

Jv(x)=My COS e,,

Yv(z)=Mysin 0,)

9.2.20

J:(z)=N, cos p,,

Yi(x)-N, sin p,.

+,-z-(+v-~)

9.2.2 1 M:e: =2/ (r ~ )

Wp; =2 ( 2-v2)/ (rz3)

W=M:2+AlZlelz=M:2+4/(?r~My)2 (~-V').A~,ML+~N~NL+XN~=O

+

(p, -0,)

=M,e:/M:

M,N, sin 9.2.25

x'M:'$zM:+

(py- 0,)

pZ+46p-63 6(4~)~

9.3. Asymptotic Expansions for Large Orders Principal Asymptotic Forms

=2/(rsM,Mi)

I n the following equations it is supposed that co through real positive values, the other variables being fixed.

=2/(rz)

v+

(x2- 2 )Mv-4/(7?M;) =O

9.2.26 $ ~ " ' + ~ ( 4 ~ + 1 - 4 ~ ' ) ~ ' + ( 4 ~ ' - 1 ) ~ = 0 ,w = x M ~

9.3.1

Asymptotic Expansions of Modulus and Phase

When v is fixed, xis large and positive, and p=4v2

9.3.2

9.2.28

1* 3 5 (p-1) 2' .4.6

+.. .

185~'- 2053p+ 1899 5 (42)

If v 2 0, the remainder after k terms in 9.2.28 does not exceed the (k+l)th term in absolute value and is of the same sign, provided that k > v - $ .

9.2.24

tan

p+3 2(42)

rf-+

p3f

I n the following relations, primes denote differentiations with respect to x.

9.2.23

I

I

-1 2 1. 4 3. 6. ,... .(2k-3) (2k)

M,= IHP ( 2 )1 =.\/I YXX)1 e,=arg H$')(z)=arctan { Y,(x)/J,(z))

9.2.22

I

(p-9)

(p-25)

GWe

+.

*

.I

--

Y ,(v sech a) *See page

11.

J + r v tanh a

I

366

BESSEL FUNCTIONS OF INTEGER ORDER

9.3.3

I

J,(v sec 0)= &(nv

tan 8) {cos (vtanP-up-b)+O

Y,(v sec S ) =

(v-l)}

(O<S<

14.

J2/(nv tan p) {sin (v tan ji?-~fi-+n)+O(v-~)} (O
h)

9.3.9 uo(t)=l (t = (3t -5t3?/24 (t) = (8lt2-462t4+ 385t')ll 152 UQ(t)= (30375t3-3 69603t5+7 65765t' -4 25425t8)/4 14720 Ud(t) = (44 65125t4-941 21676te$3499 22430t' -4461 85740t1°+ 1859 10725t'2)/398 13120 UZ

For zc5(t) and u,(t) see [9.4] or [9.21].

9.3.4

J,(v+ zv%)=2%-fS Ai( -2%) Yv(v+zvf6)= -2JSv-fs Bi(-2

+O(v-l) )6z >+O(v-l)

9.3.10 U*+I

(t) =itz(1 -tZ)t&(t)

(k=O, 1,. . .)

9.3.5

Also 9.3.11

-

Ji(v sech a)

9.3.6

9.3.12 Y:(vsech a)

where 9.3.13

In the last two equations b is given by 9.3.38 and 9.3.39 below .

vo(t)=l q ( t \ = ( -9t+7t3)/24 vz(t) = (- 135t2+594t4-455t')/l 152 ~ 3 ( t ) = (-42525t3+4 51737t5-8 83575t' $4 75475tQ)/4 14720

9.3.14

D d t ) =Uk(t) Debye's Asymptotic Expansions

(i) If CII is fixed and positive and v is large and positive

+w- 1) {

4%-I

( t )+tu:.-l (t) 1

(ii) If p is fixed, O < ~ < $ T positive

(k=1, 2,

..

.)

and v is large and

9.3.7

9.3.8

Y,(vsech a)-

where

=I- 81 cot28$462 cot4 Pf385 cot' S+ 11529.

...

367

BESSEL FUNCTIONS OF INTEGER ORDER

-3 cot /3+5

24v

cota 8-

...

g1(z)=--

Also

9.3.19 J;(V sec 0) = J(sin

17 1 a 70 ‘70

g2(z)=-afi)/(?rv){

-N(v, 8) sin -O(v, a) cos e }

9.3.20 ~ : ( v sec p) =J(sin 2 p ) / ( w ){ N(v, p ) cos e -O(V, p ) sin * } where 9.3.21

=1+ 135 cot2 /3+594 cot‘ 44-455 cot’ 8 1152~~

-

9 611 z7+24 -, 1000 3150

549

37 z 3150

110767 79 z5+z2 zs-693000 12375

The corresponding expansions for Hi1)(Y+ zv1I3) and H ! a ) ( ~ + ~ ~are 1 1 3obtained ) by use of 9.1.3 and 9.1.4; they are valid for -4?r<arg v<&r and -#a<arg v<+n, respectively.

I 9.3.27

j

I

* .

9.3.22 OD

varcl(i cot 8)-9 cot /3+7 cot8 I3- * . 24v

o(v,8)miC k-0 3k+I

Asymptotic Expansions in the Transition Regions

When z is fixed,

IvI

is large and larg v1<3.

where

9.3.23 211a

J”(v+zvlla)‘Vylla Ai (-21iaz) I1+

OD

fk(z>

}

ywla

9.3.29

h 2 ( z ) =- - 9

57 6 100 ‘70

9.3.24

2

23 699 2617 2’- -23+3500 3150 3150 27 46631 3889

ha(z)=-

h4(z)=m z

where

fl(2) =

l

o

-

1159

z7+m ~ ~ 24-11550~

9.3.30

9.3.25 1 -52

j*(z)=--9

3

-22 looz6 +35

--27 fi(z)-20000

21 -0,

23573 5903 947 z7+z4+147000 138600 346500

I

369 999443 31727 947 z’+za+z a ( 2 ) = ~”-0 173250 346500

BESSEL FUNCTIONS OF INTEGER ORDER

368

9.3.37

When v++ , these expansions hold uniformly with respect to z in the sector larg zIIR--E, where E is an arbitrary positive number. The corresponding expansion for H ; * ) ( v z ) is obtained by changing the sign of i in 9.3.37. Here

where 2113 a=-=.44730 73184, 321317(g) 2213 b=,41085 01939, 3113r(;)-

,

ao= 1

a1=

~ ~ = . 0 0 0 63735 9

3*a= ,77475 90021 3*b=.71161 34101

--225---

. . .,

1 --=.01428 "-70

equivalently,

.004,

a3=-.OOO35 57143.

38

..

. ., ... 15873 . . .,

y2=-.00093 60=5)1

23 ~1=-=.00730 3150 7300

. . .,

9.3.39

. .,

1213 =-.00118 48596. 23750 p3=-.OOO38 p2=.00043 78 . . ., 'l=-10

yo=1,

9.3.38

y3=.OOO44 40

the branches being chosen so that { is real when z is positive. The coefficients are given by 9.3.40

..,

947 =- 3 46500 -00273 30447 . . ., ~

6 ~ =.00060 47

. . .,

63=

- .OOO38

.,.

Uniform Asymptotic Expansions

These are more powerful than the previous expansions of this section, save for 9.3.31 and 9.3.32, but their coefficients are more complicated. They reduce to 9.3.31 and 9.3.32 when the argument equals the order.

where uk is given by 9.3.9 and 9.3.10, &=p0=1 and 9.3.41 A,=

.

(2~4-1)( 2 ~ + 3 ) . . (6s-1) s! (144)*

)

ps=--

6s+l 6s- 1 ' 8

9.3.42

=-- 5

1 5 1 I 4 8 ( 2 + ~ { 2 4 ( 2 2 1)3/2+8(22-1)b -

Tables of the early coefficients are given below. For more extensive tables of the coefficients and for bounds on the remainder terms in 9.3.35 and 9.3.36 see [9.38].

369

BESSEL FUNCTIONS O F INTEGER ORDER

Uniform Expansions of the Derivatives

With the conditions of the preceding subsection 9.3.43

For {<-10

use

9.3.44

Bi' ( v 2 t 3 { )

+

&({)

k -0

v2/3

CO(

7)

{) 'V

5 -{-'-1.33(12

{) -'I2,

d1({) = .OOO.

Maximum vaIues of higher coefficients:

I bi({) I =,003,

I

({) I =.0008,

/cl(T)/=.008 ({
C1({)--.003{*

1dz ({) 1 as

-00 1

{++

03.

9.4. Polynomial Approximations

where 9.4.1

9.3.46

J~(z) = 1-2.24999 9 7 ( ~ / 3 ) ~1.26562 + 08(~/3)~

2k+l ck({)=-{*

~1,{-38/2vZk-8+l a-0

(1-z2)-*}

- .31638 66(~/3)'+.04444 7 9 ( ~ / 3 ) ~

-*

2k

-.00394 44(~/3)'~+.0002100(~/3)"+e

dk( {) =Eh1,{-3s/2u2k-s{ (1 -z2) } 8-0

I

and v k is given by 9.3.13 and 9.3.14. For bounds on the remainder terms in 9.3.43 and 9.3.44 see [ 9.381.

0 1 2 3 4 5 6 7 8 9 10

1

1

0. 0180 -0. 004 0. 1587 . 0278 -. 004 . 1785 . 0351 -. 001 . 1862 . 0366 002 . 1927 I . 0352 I . 003 I . 2031 I . 0331 . 004 . 2155 . 0311 . 004 . 2284 . 0294 . 004 . 2413 . 0278 . 004 . 2539 . 0265 . 004 . 2662 , 0253 . 004 . 2781 ~

~

+.

-32x13

0. 007 .009 . 007 . 005 , 004 003 . 003 , 003 . 003 . 003 . 003

.

(e/<5X10-8

9.4.2

O<x<3

Yo(x) = ( 2 / ~ ln(&c)Jo(x) ) $.36746 691

+,60559 3 6 6 ( ~ / 3 ) ~.74350 384(~/3)~ +.25300 117( ~ / 3'-) ,04261 2 14( ~ / 3 ) ~ +.00427 916(~/3) .00024 846 ( ~ / 3"+ ) e l0-

lel<

9.4.3

1.4X 10-'

31x<

J,(z)=z-tf0 cos e,

Q)

Y,(z) = ~ - t f , sin eo

foz.79788 456- ,00000 077(3/~)-.00552 740(3/~)' - ,00009 512(3/~)~+.00137 237(3/~)~ -.00072 805(3/~)'+.00014 4 7 6 ( 3 / ~ ) ~ + e

0 1 2

8 9 10

0.0180 -0. 004 0. 1587 1323 . 0109 003 . 1087 0067 -. 002 .. 0044 -. 001 0903 . . ~ . .~ . ~. . . . 0031 0764 001 0658 .0022 -. 000 . 0017 000 . 0576 0511 . 0013 -. 000 .OOll 000 . 0459 0415 * 0009 -. 000 . 0007 . 0379 000

.

-.

.

-. -. -. -.

. . . .

0. 007 . 004 * 002 , 0~ 01 ~ * 001 . 000 ~~

*

000

. 000 000 . 000 *

. 000

It[<

1.6X 10-'

2 Equations 9.4.1 to 9.4.6 and 9.8.1 to 9.8.8 are taken from E. E. Allen, Analytical approximations, hlath. Tables Aids Comp. 8, 240-241 (1954), and Polynomial approximations to some modified Bessel functions, Math. Tables Aids Comp. 10, 162-164 (1956) (with permission). They were checked at the National Physical Laboratory by systematic tabulation; new bounds for the errurs, 6 , given here were obtained as a result.

I

370

BESSEL FUNCTIONS OF INTEGER ORDER

e0=x-.78539

*

et

J , , ~ ,j y ,s, y v . sand respectively, except that z=O 816- .04166 397(3/~) is counted as the f i s t zero of Jh(z). Since - .00003 954(3/~)’+.00262 573(3/~):)” Jo(z)=-Jl(z),it follows that --.00054 1 2 5 ( 3 / ~ ) ~.00029 333(3/~)~ .I * 9.5.1 j;,i=O, jo,s=j1,+1 ( ~ = 2 , 3, .) +.00013 558(3/x)‘+e

.

The zeros interlace according to the inequalities

le1<7X 10-8 9.4.4 2-l

9.5.2

-35x13

J~(s)=*-.56249 985(~/3)~+.21093 573(~/3)~ -.03954 289(~/3)’+ ,00443 319(~/3)* - .00031 7 6 1 ( ~ / 3 ) ~ ~ + . 0 0 0 10 01 9 ( ~ / 3 ) ” + ~

YV,

u s j : ,I < Y Y ,

I
i
O<x<3

xYl(z)=(2/n)xln(~x)J1(x)-.63661 98 .22120 91 ( ~ / 3 ) ~ + 2 . 1 6 8 209(x/3)‘ 7 - 1.31648 27(~/3)’+ .31239 51( ~ / 3 ) ’ -.04009 76(x/3)lo+ .00278 73(~/3)”+e

+

lel
9.4.6 COS

el,

a<

. .

l < L 2 a

The positive zeros of any two real distinct cylinder functions of the same order are interlaced, as are the positive zeros of any real cylinder function %‘,(z), defined as in 9.1.27, and the contiguous function %‘R,+l(z). If pu is a zero of the cylinder function 9.5.3

%‘,(z) = J v ( z )cos(d)

9.5.4

Yl(x)=x-+jl sin

+ Y,(z)sin(&)

e1

If

U,

+ .00017 105(3/~)~-.002495 1 1 ( 3 / ~ ) ~

+.00113 653(3/~)~-.00020033(3/~)’+t I€1<4X 10-8 4=2-2.35619 449+ .12499 612(3/~) .00005 650(3/~)~-.006378 7 9 ( 3 / ~ ) ~

+ +.00074 348(3/x)‘+ .00079 8 2 4 ( 3 / ~ ) ~ 166(3/~)’+a

l€1<9X 10-8

9.5.5

is a zero of % ( z ) then b.

W,(u,)=y

v”-l(c”)=:

W*+i(Ur)

The parameter t may be regarded as a continuous variable and pv, U, as functions p,(t), uv(t) of t. If these functions are fixed by 9.5.6

py(O)=O,

u”(o)=j:,

1

then 9.5.7 jw,*=pds),

YI,I=PY(Q-3)

*

’

J

9.5.8

For expansions of J&), Yo(s), J1(x), and Yl(x) in series of Chebyshev polynomials for the ranges j;,s= U”@ O < z 1 8 and O<S/x11, see [9.37].

l),

y;,s=uv(8-

3)

9.5. Zeros Real Zeros

When U is real, the functions J,(z), Jl(z),Yu(z) and Y:(z)each have an infinite number of real zeros, all of which are simple with the possible For non-negative U the 6th exception of z=O. positive zeros of these functions are denoted by

Infinite Produoto

I I

I

I

i

iI

~~(P,)=~”-l(P,)=--”+r(P*)

f i =,79788 456+.00000 1 5 6 ( 3 / ~ ) +,01659 667(3/~)’

-.00029

--

where t is a parameter, then

3 Is<

Jl(x)=x-+jl

a<

< y v , z < d , 2<jv, 2 < j L

le1<1.3X 10-8 9.4.5

..

j v , i<j,+i,i<jv,2<jv+il2<jv, a<

(s= 1,2,

. . .)

BESSEL FUNCTIONS OF INTEGER ORDER

371

McMahon's Expansions for Large Zeros

When v is fixed, S > > V

and p=4v2

9.5.12 ~l-1 4b-1) (7p-31) 3 w 3 8B

j ,s l Y., -P---

--32b-1)

(83p2-982p+3779) 15(sal5 64(p-1) (6949p3-1 53855p2+15 85743~-62 77237) 105CSP)'I

where P=(s++v-t)?r forjv.s,P = ( s + b - $ ) * asymptotic expansion of p Y ( t ) for large t. 9.5.13 .I JY,

for yY,*. With B=(t+$v-t>?r,

-

-64(6949p4+2 96492p3-12105(88')' 48002p2f74 where P ' = ( 5 + 3 v - + ) ? r for jL,slP ' = ( s + ~ v - ~ ) T for Y:,~, 9.5.12 and 9.5.13 see [9.4] or [9.40]. Asymptotic Expansions of Zeros and Associated Values for Large Orders

-+-

I

.

9.5.15 Y,~1-v+.93157 68~'/~+.26035 lv-'I3 .01198~-'- .0060~-~"- .OO 1v-'/'+ ..

.

+

9.5.16 j:,l-v+.80861

9.5.18

P'=(t+&+t>?r

14380~-58 53627) - *

I

..

1

I

for u Y ( t ) . For higher te,rms in

Uniform Asymptotic Expansions of Zeros and Associated Values for Large Orders

I

+

1.85575 7 1v1I3 1.033 15 Ov-'13 .00397~-'- .0908~-~"+. 0 4 3 ~ - ~ / ~, $.

9.5.17 y;,I-V+

I

the right of 9.5.12 is the

~ 4 - 34 (7p2+82p- 9) 32 (83pa+2075p2- 3039~4-3537) 3(8P'I3 15(SP') 8P'

I

s, Y v . s-B'---

9.5.14 j, v

-..

with {=v-2/aar

65dta+.07249 Ov-lt3 -.05097~-'+.0094v'~'~+

...

1.82109 80v1la+.940O0 7v-'la - .05808~-'- . 0 5 4 0 ~ ' ~ / ~ + ,

-+

. .

Ji(jY, 1) - 1.11310 2 8 ~ - ~ / ~ / (1.48460 1+ 6v-2/3 .43294v-4/3-- .1943v-2+ .019V-s/3+

. .) *

9.5.19

Y;(yy,1) -.95554 86~-~"/(1+.74526 1 ~ - ~ / ~ +.10910v-'~~-.O185v-~-..003v-~/3+ * . . ) 9.5.20

Jy(jL,J --.ti7488 51~-~/~(1--.16172 3v-2/a

+ . 0 2 9 1 8 ~ - ~ / ~ - . 0 0 6 8 ~ -.~.+. ) I

9.5.21

Y&:,1) -37319 4Ov-lt3(1- .36422 OV-~/'

+. 0 9 0 7 7 ~ ' ~ ~. 0+2 3 7 ~ - ~ +. . . )

Corresponding expansions for s =2, 3 are given in [9.40]. These expansions become progressively weaker as s increases; those which follow do not suffer from this defect.

where b0(f), co(f) appear in 9.;3.42 and 9.3.46. Tables of the leading coefficients+llow. More extensive tables are gven in [9.40]. The expansions of yy.a, Y:(yy.J, y:, and YY(y:,J corresponding to 9.5.22 to 9.5.25 are obtained by changing the symbols j, J, Ai, Ai', a , and a: to y, Y,-Bi, -Bi', b, and b: respectively.

I

372

BESSEL FUNCTIONS OF INTEGER ORDER

0.0 0.2 0.4 0.6 0.8 1. 0

1.000000 1. 166284 1.347557 1. 543615 1. 754187 1. 978963

1. 25992 1. 22076 1. 18337 1. 14780 1. 11409 1.08220

0.0143 .0142 .0139 . 0135 . 0131 0.0126

-0.007 -. 005 -. 004 -. 003 -. 003 -0.002

-0. 1260 -. 1335 -. 1399 -. 1453 -. 1498 -0. 1533

-0.010 -. 010 -. 009 -. 009 -. 008 -0. 008

0.000 .002 .004 . 005 . 006 0. 006

1. 0 1. 2 1. 4 1. 6 1. 8

1. 978963 2. 217607 2.469770 2.735103 3. 013256

1.08220 1.05208 1. 02367 0. 99687 . 97159

0.0126 .0121 .0115 . 0110 . 0105

-0. 002 -. 002 -. 001 -, 001 -. 001

-0. 1533 -. 1301 -. 1130 -. 0998 -. 0893

-0.008 -. 004 -. 002 -. 001 -. 001

0.006 . 004 .003 .002 .002

2. 0 2. 2 2. 4 2. 6 2. 8

-0. 001 -0. 001 .0086 ,0082

-0. 0807 -. 0734 -. 0673 -. 0619 -. 0573

-0.001

4.247441 4.584833

0.94775 , 92524 .90397 . 88387 . 86484

0.001 . 001 . 001 . 001 0.001

3. 0 3. 2 3. 4 3. 6 3. 8

4. 933192 5. 292257 5. 661780 6.041525 6.431269

0.84681 . 82972 .a1348 .79806 . 78338

. 0065

-0.0533 -. 0497 -. 0464 -. 0436 -. 0410

0.0062 . 0060 .0057 . 0055 . 0052

4.0 4. 2 4. 4 4. 6 4.8

8. 086150 8. 522912

5. 0 5. 2 5 .4 -. 5. 6 5. 8

8.968548 9.422900 9.885820 10.357162 10.836791

. 67758 . 66811

. 0043

-0. 0296 0282 -. 0270 0258 -. 0246

6.0 6. 2 S. 4 6. 6 6.8

11.324575 11. 820388 12.324111 12.835627 13.354826

0.65901 .65024 , 64180 .63366 . 62580

0.0042 ,0040 .0039 . 0037 . 0036

-0.0236 -. 0227 0218 -. 0209 0201

7. 0

13. 881601

0.61821

0.0035

-0.0194

~

-. -.

-. -.

-. -.

I 0.40 .35 .30 . 25 . 20

1. 528915 1. 541532 1. 551741 1. 559490 1. 564907

1. 62026 1. 65351 1. 68067 1. 70146 1. 71607

0.0040 -0. 0224 . 0029 -. 0158 . 0020 -. 0104 . 0012 -. 0062 .0006 -. 0033

0. 15 . 10 . 05 00

1. 568285 1.570048 1. 570703 1. 570796

1. 72523 1. 73002 1. 73180 1. 73205

0.0003 -0.0014 ,0001 -. 0004 . 0000 -. 0001 0000 . 0000

9

0328 0311

Complex Zeros of J,(5)

When U 2 - 1 the zeros of Jy(z)are all real. If and U is not an integer the number of complex zeros of Jy(z) is twice the integer part of (-Y); if the integer part of ( - Y ) is odd two of these zeros lie on the imaginary axis. If ~ 2 0 all , zeros of &(z) are real. U<-1

-.

Complex Zeros of YAs)

When U is real the pattern of the complex zeros of

Y.(z)and YL(z)depends on the non-integer part of U. Attention is confined here to the case u=n, a positive integer or zero.

373

BESSEL FUNCTIONS OF INTEGER ORDER

and Yk(z) FIGURE 9.5. Zeros of Y*(z) larg zI ST.

are given by the right of 9.5.22 with v=n and t=n-2/3b8or n-2i3FaJ where b,, 3, are the complex zeros of Bi(z) (see 10.4). Figure 9.5 is also applicable to the zeros of Yk(z).There are again an infinite number near the infinite curves, and n near each of the finite curves. Asymptotic expansions of the latter for large n are given by the right of 9.5.24 with v=n and f=n-2t3bi or n-2t33:;where and 'pi are the complex zeros of Bi'(z). Numerical values of the three smallest complex zeros of Yo(z),Yl(z) and Y:(z)in the region O < arg z
...

Figure 9.5 shows the approximate distribution of the complex zeros of Y,(z)in the region (arg zl
Y~=&$h3=&.54931

Complex Zeros of the Hankel Functions

The approximate distribution of the zeros of Hil)(z) and its derivative in the region larg zlS?r is indicated in a similar manner on Figure 9.6.

...

There are an infinite number of zeros near each of these curves. The two curves extending from z=-n to z=n and bounding an eye-shaped domain intersect the imaginary axis at the points f i ( n a + b ) , where a=m=.66274 2=.19146

n

10

-----, \

...

b=$d-In

-n

CUT

FIGURE 9.6. Zeros of & ( z )

...

and Hi1)'@).

..

I arg zI
and t,,=1.19968 . , . is the positive root of coth t =t. There are n zeros near each of these curves. Asymptotic expansions of these zeros for large n

..

Zeros of Yo(z) and Values of Yl(z)at the Zerosa Yl

Zero Real

-2.40301 -5.51987 -8.65367

Imag.

6632 6702 2403

+. 53988 2313 $.54718 0011 $. 54841 2067

f.

*

Real

-.

10074 7689 -.02924 6418 01490 8063

Imag.

88196 7710 +.58716 9503 -. 46945 8752

+.

Zeros of Yl(z) and Values of Yo(z) at the Zeros Yo

Zero Real

Imag.

-0.50274 3273 -3.83353 5193 -7.01590 3683

Real

+. 78624 3714 -. 45952 7684 +. 56235 6538 $. 04830 1909 $.55339 3046

-.02012

6949

Imag.

+1.31710 1937 -0.69251 2884 $0.51864 2833

Zeros of Y:(z) and Values of Yl(z)at the Zeros Yl

Zero Real

+O. 57678 5129 -1.94047 7342 -5.33347 8617

Imag.

Real

+.56721 9637

-.03179 4008

Imag.

+.90398 4792 -. 76349 7088 +. 58924 4865 +. 72118 5919 +. 16206 4006 -. 95202 7886 $.59685 3673

8 From National Bureau of Standards, Tables of the Bessel functions Y o ( i ) and Y,(z)for complex arguments, Columbia Univ. Press, New York, N.Y., 1950 (with permission).

BESSEL FUNCTIONS OF INTEGER ORDER

374

Modified Bessel Functions I and K

There are n zeros of each function near the finite curve extending from z=-n to z=n; the asymptotic expansions of these zeros for large n are given by the right side of 9.5.22 or 9.5.24 and +e-hi1%-2/3a8 or +e-2si1&n-*/3a’ 8’ with

is real and X is positive, the zeros of the function v

J,(z)Yu(Xz)-J&)

9.5.21

are real and simple. If X > l , expansion of the sth zero is

Differential Equation

9.6.1

Zeror of CroM-Producte

If

9.6. Definitions and Properties

Ydz) the asymptotic

where with 4va denoted by p , 9.5.29

22

#w -+z dz2

dw

--(22+v2)w=O

dz

Solutions are I*u(z)and KU(z).Each is a regular function of z throughout the z-plane cut along the negative real axis, and for fixed z( #O) each is an entire function of U. When v=&n, Iu(z) is an cntire function of z. Iu(z) (WY 20) is bounded as z+O in any bounded range of arg z. I J z ) and I-”(z)are linearly independent except when v is an integer. Kv(z)tends to zero as IzI-+= in the sector larg z 1 < 3 ~ , and for all values of U, Iu(z)and Ku(z)arelinearly independent. I”(@, Kv(z) are real and positive when v>-1 and z>O.

b=s$r/(X-l)

r=

(p-1)(p2-114~+1073)(X6-1) 5 (4X) (A -1)

The asymptotic expansion of the large positive zeros (not necessarily the sth) of the function

J:(z)y:(Xz) -J:(Xz) Y:(z)

9.5.30

0>1)

is given by 9.5.28 with the same value of B, but instead of 9.5.29 we have 9.5.31 r+3

(p2+46p-63)(X3-1) *= 6(4X)3(X-1) l85pa-2053p+ 1899) (A6- 1) 5(4X)’(X-l)

’=8x’ r=

(p3+

The asymptotic expansion of the large positive zeros of the function

J:( 2 )Y”(XZ)-Y: ( Z V ,(XZ)

9.5.32

is given by 9.5.28 with 9.5.33

B= (s-3>./(X-1>

P‘

5 (4X)6(X-

(p2+46p-63)X3-(p-l)(p-25) 6(4X)’(X- 1) l)r= (p3+ 185pa-2053p+ 1899)X’ -(p-1) (p2-114p+1073)

FIGURE 9.7. Io(z),KO(*), Il(z)and KI(z).

375

BESSEL FUNCTIONS OF INTEGER ORDER

9.6.5 Y,(ze+*')=et"+ l)*fI,(z) - (2/?r)e-+"{KK,(z) (-.<arg 9.6.6

z13d

I - , (Z ) = I , (Z ) ,K - ,(Z ) =K ,(Z )

Most of the properties of modified Bessel functions can be deduced immediately from those of ordinary Bessel functions by application of these relations.

Amending Seriee

9.6.11

-5

FIGURE 9.9. Iv(5)and K,(5).

1

where $(n) is given by 6.3.2.

Relation0 Between Solutione

I

9.6.13

L

9.6.2

m z > = 3 * I-,(Z)-Iv(Z) (*)

The right of this equation is replaced by its limiting value if U is an integer or zero.

t z2

Ko(z>=- Ih (3z)+YlIo(z)+(l!)? +(1+3>

(tzaIa (tz")" (21)9+(1+*+*) (3!)a+'

* *

376

BESSEL FUNCTION8 OF INTEGER ORDER

2, denotes I y ,eV*‘Ky or any linear combination of these functions, the coefficients in which are independent of z and v .

Integral Representations

9.6.16

I;(Z ) =I1 (Z ) ,

9.6.27

KL (Z ) = -K1(

Z)

Formulas for Derivatives

9.6.28

K o ( x ) = l mcos (5 sinh t)dt=

cos (zt)

Analytic Continuation

(x>O)

9.6.22

l

=csc (im)

(m an integer)

9.6.31

m

K.(z)=sec (tm)

Iy(zem*i) =emyriIy(z)

9.6.30

cos (z sinh t ) cosh (vt)dt

lm sin

(5 sinh

Ky( zemri) =e-mu*iKv( z ) -~isin ( m m )csc ( Y T ) I”(z )

t ) sinh (vt)dt

( l 9 v l < L r>O)

9.6.32

-

Iy(Z)=Iy(z),

( m an integer)

-

K,(H)=K,(z)

(v

real)

9.6.23 Generating Function and Associated Series

9.6.24

(9v>-3, 18% Z l < M Ku(z)=Jme-zco8ht cosh ( v t ) d t (larg Z I
9.6.25

9.6.34

5

ez c0ae=10(~)+2 I,(z) cos(k0) k=l

9.6.35 OD

e z sin

e=I o (2) +2 PI0 C (-)kI~k+l(z) sin{ (2k+l)e) +2

9.6.36 9.6.37 9.6.38

2 ( - ) v Z k ( z ) cos(2ke) k-1

... ez=Io(z)+211(2)+212(2)+213(2)+ . . . e- ’=Io(2 ) -211(2)+2 I2(2)-213(2 ) + . . . l=Io(z)-212(2)+214(2)-21e(z)+

9.6.39

cosh ~ = I ~ ( z ) + 2 1 ~ ( 2 ) + 2 I , (+21e(2)+ z) 9.6.40 *See page !I.

sinh 2=211(2)+21a(z)+215(2)+

...

...

377

BESSEL FUNCTIONS OF INTEGER ORDER

Other Differential Equations

9.6.50

The quantity X2 in equations 9.1.49 to 9.1.54 and 9.1.56 can be replaced by -A2 if at the same time the symbol V in the given solutions is replaced by 424

lim { v-pe-c*$ &: (cosh

For the definition of

:)}=K,(z)

P;' and &:, see chapter

8.

Multiplication Theorems

9.6.51

9.6.41

w=ei2Yu(z)

22~"+2(1f2z)w'+(fz-~)w=O,

Differential equations for products may be obtained from 9.1.57 to 9.1.59 by replacing 2 by iZ.

If %"=I and the upper signs are taken, the restriction on h is unnecessary. 9.6.52

Derivatives With Respect to Order

9.6.42

9.6.43

a

a

a

-I-u(z)-~Iv(z~} av Kv(z)=$7rcsc(v7r) { av -7rcot(wr)K,(z)

(v#O,f1,&2,

* *

.>

9.6.44 (-)"

;[

I.(z)] v-n =

9.6.54

Ko(z)=- {ln ( $ 2 ) + 7 } I o ( z ) + 2 k-1 C 12k(z> k OD

Zeros

9.6.45

9.6.46

Expressions in Terms of Hypergeometric Functions

9.6.47

Properties of the zeros of I v ( z ) and Ku(z)may be deduced from those of Jv(z)and H Y ( z ) respectively, by application of the transformations 9.6.3 and 9.6.4. For example, if v is real the zeros of Iu(z)are all for some posicomplex unless - 2 k < v < - ( 2 k - l ) tive integer k, in which event Iu(z)has two real zeros. The approximate distribution of the zeros of Kn(z)in the region -#r<arg z < $ n is obtainedon rotating Figure 9.6 through an angle -3. so that the cut lies along the positive imaginary axis. The zeros in the region -37r 4arg z 1#7rare their conjugates. K,(z)has no zeros in the region larg 21 this result remains true when n is replaced by any real number v.

sir;

9.7. Asymptotic Expansions

(di; is the generalized hypergeometric function. For M(a, b, zj, MO,&) and Wo,v(z) see chapter 13.) Connection With Legendre Functions

If p and z are fixed, Wz>O, and v+m through real positive values 9.6.49

lim{t+P;p

Asymptotic Expansions for Large Arguments

When v is fixed, Iz( is large and p=4v2 9.7.1 ~ , ( z ) - -{I-e* J G Z

p--1+(~-1)(~--9) €42 2!(82)'

- (p-')(p-g)b-'')+ 3!(8~)~

. . .I

(iargzl<+rr>

,

378

BESSEL FUNCTIONB OF INTEGER ORDER

9.7.2 9.7.10

9.7.3

When v-++ w , these expansions hold uniformly with respect to z in the sector larg zl<]a?~-e, where e is an arbitrary positive number. Here

9.7.4 and uk(t), vk(t) are given by 9.3.9, 9.3.10, 9.3.13 and 9.3.14. See [9.38] for tables of 7, uk(t), vk(t), and also for bounds on the remainder terms in 9.7.7 to 9.7.10. The general terms in the last two expansions can be written down by inspection of 9.2.15 and 9.2.16. If U is real and non-negative and z is positive the remainder after k. terms in the expansion 9.7.2 does not exceed the (k+l)th term in absolute value and is of the same sign, provided that klv-3.

9.7.5

9.8. Polynomial Approximations In equations 9.8.1 to 9.8.4, t=2/3.75. -3.75 1 2 1 3 . 7 5

9.8.1

Io(~)=1$3.51562 29t2+3.08994 24t4+ 1.20674 92t' -26597 32t8+ ,03607 68t'O-t ,00458 13t12+e

+

(e1<1.6X 10-7

9.8.2 3.75 I x < = x~e-zIo(x)=,39894 2284- .01328 5921-'

+.00225 3 1 w - .(I0157 565t-a f.00916 281t-'-.02057 706t-6 +.02635 5372-'--.01647 633t-' +.00392 3 7 7 P - k e I€[<

--1- 1 (P- 1)(P-45) . . . } 2.4 ( 2 ~ ) ~

9.8.3

+

(lmg 4 The general terms can be written down by inspection of 9.2.28 and 9.2.30.

1.9x10-7

-3.75 1 2 1 3 . 7 5 x-'I~(x)=~ ,87890 + 594Pf.51498 8692' +.15084 934t" t.02658 733P +.00301 532t'0+.00032 411t'2+e

le1<8X 10-e

9.8.4

x'e-"Il(x) Uniform Asymptotic Expansions for Large Orders

3.75 I x < = =.39894 228- .03988 024t-1 -.00362 018t-2+.00163 Bolt-' -.01031 555t-4+ .02282 967t-' -.02895 312t-'+.01787 654t-' -.00420 059t-8fe le1<2.2 10-1

x

9.7.8 4

See footnote 2, section 9.4.

379

BESSEL FUNCTIONS OF INTEGER ORDER

9.8.5

O<X52

1

K 0 ( x ) = -In (~/2)1~(2)-.57721 566 .42278 4 2 0 ( ~ / 2 ) ~.23069 + 756(~/2)~ +.03488 5 9 0 ( ~ / 2 ) ~.00262 + 698(~/2)~ +.oooio. ~ ~ o ( s / ~ ~ ~ o + . o740(x/2)12+~ oooo 1€1<1 x 10-8

+

2 sx<

9.8.6

Q)

Differential Equations

9.9.3 - .

x2wf‘+xwf- (ix*+v2)w=O, w=ber, x + i bei, 2, ber-, s+i bei-, 2, ker, x + i kei, z, ker-, x + i kei-, x 9.9.4 x4w‘O

x*ezKo(z)=1.25331 414-,07832 358(2/x) +.02189 568(2/x)*-.01062 4 4 6 ( 2 / ~ ) ~ ,00587 872(2/~)*-,00251 540(2/~)‘ +.00053 2 0 8 ( 2 / ~ ) ~ + e

+2 2 w f - (1+22) -xwf) +(2- 4va+ x4)w =0, w=ber*, x, bei*, x, ker*, x, kei*, x

+

lc1<1.9XlO-’ 9.8.7

o<xs2

(22w’f

I’

Relations Between Solutions

9.9.5

ber-, z=cos(va) ber, x+sin(vr) bei, x

+

(2/a) sin(va) ker, x xK1(x)=x In (x/2)11(s)+1+.15443 144(s/2)’ --.67278 579(~/2)~-.18156 897(~/2)’ bei-, x= -sin(va) ber, x+cos(vn) bei, x -.019 19 402 (2/2)’- .OO 110 404 ( ~ / 2 ) ” @/a) sin(vs) kei, x -.00004 6 8 6 ( ~ / 2 ) ’ ~ + ~

+

Iel <8

9.8.8

X 10-’

9.9.6

25x<(.

z*ezK1(~)=1.25331414-k.23498 619(2/z) --.03655 620(2/~)~+.015042 6 8 ( 2 / ~ ) ~ 9.9.7 -.00780 353(2/~)’+.00325 614(2/~)’ -.00068 2 4 5 ( 2 / ~ ) ~ + r 9.9.8

ker-, x=cos(va) ker, x-sin(va) kei, r kei-, z=sin(va) ker, Z + C O S ( V kei, ~) x her-,, z= (-)” ber,

ker-, x=(-)”

2,

bei-, r= (-)” bei, x

kern x , kei-, x=(-)“

1€1<2.2X 10-7 . . Ascending Series

For expansions of l0(x), Ko(x),I1(s),and K l ( 4 in series of Chebyshev polynomials for the ranges O S s 1 8 and O l S / a l l , see [9.37].

9,9,9 {(9V+3k>d ber, x= ( 3 ~ ) C ” cos k-0 k!r(V+k+l) OD

(+r2)k

Kelvin Functions

9.9. Definitions and Properties

9.9.10

In this and the following section v is real, x is

ber x=l-- (t.’)2+(W4 (2!)2 (4!)2

real and non-negative, and n is again a positive integer or zero. Definitions

9.9.1

bei

I

9.9.11

x,+x2,-+--

(t22>3 (W)’ (3!)2 (5!)*

.

, ,

* * *

keinz

380

BESSEL FUNCTIONS OF INTEGER ORDER n-1

kei, x=-3(4x)-”

k=0

X(n-k-l)!

k!

9.9.16

sin { ($n++k)*}

@ ber’ x=berl x+beil x

($.”)k-ln (3s) bei, x-+T ber, x

1/2 bei’ x=-berl

9.9.17

”.

1/2 ker’ x=kerl @ kei’ x=--erl If

where #(n) is given by 6.3.2.

x+beil x xfkei, x x+keil x

Recurrence Relations for Cross-Producta

9.9.18 9.9.12

ker z=-h

($2)

kei x=-ln

(32) bei x-+a ber x

ber

X + ~ T bei

p,=be$ x+beit x p,=ber, z bei: x-ber: 5 bei, z r,=ber, x ber: z+bei, x beil x sr=berL2z+beii2 z

x

then 9.9.19 4v p.+1=pv-1-;

Functions of Negative Argument

Tu

-; pv+r,= -p, - 1+ 2T” V

!&+I=

In general Kelvin functions have a branch point a t x=O and individual functions with arguments xe*ri are complex. The branch point is absent however in the case of ber, and bei, when v is an integer, and

ru+l=-- (v+l) X pu+,+q, 1 5.=g

1 3 Pu+l+g P u - l - ~ p,

and

9.9.13

ber,(-x) = (-)n ber, x,

bei,(-x) = (-)” bei,

2

9.9.20

Pu8u

=rz +d

The same relations hold with ber, bei replaced throughout by ker, kei, respectively.

Recurrence Relations

9.9.14

Indefinite Integrals

j,+l

+jw= -2 j:=-

v j:-;ju=-

1

uv+1+gY+l Y f u - 1 2 4 1 Uu+1+9u+1)

9.9.21

Jz

1 +; j u=-1/2 - -+

j:

9,-1)

In the followingf,, g, are any one of the pairs given by equations 9.9.15 and jf,gf are either the same pair or any other pair.

V

(fu

1

Q Y

-1)

9.9.22

where 9.9.15

j,=ber, x

jr=beis x

gr=beiv x

g,=

j,=ker, x

j,=kei, x

g,=kei, x

gY= -ker, z

9.9.23

-ber, x -gf(j,+1-

gu+J -f.(fl+1+ sf+J+S”

=g1 x (jlff-f,fl’

w+ 1 -

+g:g:-

st+J1 svs2’)

381

BESSEL FUNCTIONS OF INTEGER ORDER

Zeros of Functions of Order Zero 6

9.9.24

I

berz

1

2. 84892 7. 23883 11. 67396 16. 11356 20. 55463

1st zero 2nd zero 3rd zero 4th zero 5th zero

I

I

ber’ z

beiz

1

5. 02622 9. 45541 13. 89349 18.33398 22. 77544

I I

bei’ z

I

kerz 1. 71854 6. 12728 10. 56294 15.00269 19. 44381

1

ker’z

I

keiz 3. 91467 8. 34422 12. 78256 17. 22314 21. 66464

1

kei’ z

I

I

9.10. Asymptotic Expansions Asymptotic Expansions for Large Argumenta

When v is fixed and is large 9.10.1

--7r1 {sin ( 2 4 ker, z+cos ( 2 4 kei, z j

9.9.29

ber, z bei: z-ber: z bei, z =(3z)2v+1

OD

9.10.2 1

r(v+k+l)F(v+2k+2)

sin a-g,(z) cos a}

kl

+;1 {cos (2m) ker, z-sin

9.9.30

ber, z berl z+bei, z beil z

9.10.3

I

9.9.31

~

=(3z)a’-P k-0

(2vn) kei, z}

9.10.4

(2ka+2vk+tva) 1y~+k+i)r(v+2k+l)

kl

9.10.5

Expaneions in Series of Bessel Functions

9.9.32

9.9.33 6 From British Association for the Advancement of Science, Annual Report (J. R. Airey), 254 (1927) with permission. This reference also gives 5-decimal values of the next five zeros of each funotion.

382

BESSEL FUNCTIONS OF INTEGER ORDER

9.10.16

9.10.7

%(f4 (

-5 (F)k

1)(p-9)

k=l

. , .{p-(2k-1)2} k!(8x)k

sin

(2)

-)4.

9.10.17

The terms" in ker,x and kei,x in equations 9.10.1, and 9.10.2 are asymptotically negligible compared with the other terms, but their inclusion in numerical calculations yields improved accuracy. The corresponding series for ber: 2, b ei: x, ker: z and kei: x can be derived from 9.2.11 and 9.2.13 with z=xear'/4; the extra terms in the expansions of ber: x and beil 2 are respectively

- (l/r) Isin(2vr)ker: x+cos(2vr)kei:

Mi=MI COS (e,-eo-+r) ei =(Ml/Mo)sin (e, -e, d (xMfe;)/dx=sM,

9.10.18 N,=J(keI$r+kei:s),

&=arctan (kei,r/ker,z) kei, x= N, sin 4,

9.10.19 ker, z= N, cos ,$t

z}

0-

( l / ~{cos(2vr)ker: ) z-sin(2v~)kei: 2).

6-

and

x2M:'frMl -v ~ M , = ~ M ~ : ~

lkerx

, , , 1

I

4-

I

Modulus and Phase

9.10.8 M,=d(beI$z+bei: s),

&=arctan (bei,x/ber,r)

9.10.9 ber, x= M , cos e,,

M-n= Mn,

9.10.10

bei, x= M, sin e,

e-"=

9.10.11 berl i= M,+, cos (&+, -t.1= ( v / x ) ~COS , e,+M,+,

+

8-.

nr

3 M,-, cos ( L 1 - IT)

(e,+,-tr) = - ( v / ~ ) ~ , ~ ~ ~ e , -COS M (e,-,-tr) v-l

,

COS

\

FIGURE 9-10. ber z, bei x, ker s and kei 2.

9.10.12 bei: z=+M,+, sin (e,+,-tr) -+My-, sin @,-, -tr) = (v/x)M,sin e,+M,+, sin (e,+,-tr) = - (v/z)M,sin O,-M,-, sin (e,-,-)r) 9.10.13 ber' x=M,cos (e,-tr),

bei' 2=Ml sin (e,-+.)

-4

9.10.14

z' s

- -4

Ml= ( ~ / S ) M ~ + M COS ~+ (e,+,-e,-tr) , = - ( v / ~ ) ~ , - ~ M , -COS l (e,-,-e,-tr) 9.10.15

e:= (M,+,IM,)sin (OV+1-e,-4r) = - (M,-,/M,) sin (e,-,-e,-tr) 0 The coefficients of these terms given in [9.17]are incorrect. The present results are due to Mr. G. F. Miller.

x

FIGURE 9.11. In M,(z), Oo(x),In No(x)and 40(x). Equations 9.10.11 to 9.10.17 hold with the symbols b, M, e replaced throughout by k,N, 4, respectively. In place of 9.10.10

9.10.20

N-,= N,,

d-.=c$.+vr

383

BESSEL FUNCTIONS OF INTEUER ORDER

9.10.29

Asymptotic Expansions of Modulus and Phase

When v is fixed, x is large and p=4S 9.10.21

I

1. I

(~-1)~ 1 M,={ 1-Sx 842 x 256 x2 -( p - 1)(p2+ 14p-399) 1 614442 9.10.22 p-1 1 (p-l)(p-25) X In AI,=---$ In (2~x)-- 842 -42 x 38442 p-1

ez/v2

-(cce"-42 --+(L

2 v-i)

A+-

p-11

..

75

i+ . . .)

b e P x+beif2

2'

9.10.31

ker2 x+kei2

x4+0($)

*

x--e-*d2

22

I

p-11

1 2475

+zssJz ?+a92

1 -

9.10.23 X

--- -+--+

9.10.30

1)(cc- 13) 1_

128

---

ezd2 1 3 1 2*x (42 8 2 15 1 45 1 315 1 --6442xa 512 9 819242 x4

ber x ber' x+bei x bei' x--

(

1--

1 1 1 1 442 x+64 2

+----33 1 25642 x8

;+Is 3

24

1797 1 + 8192 x4

. . .)

9.10.32

x kei x---

ker x kei' x-ker' 9.10.24

A

2s e-zd2

1 3 9 -+--+ 1 75 +- 9 _--

1

.

6442 x2 5 1 2 9 8 1 9 2 4 2 ~ ~

+

(1.1-

1)(pz+14p-399) 614442

*

*)

9.10.33

1

ker x ker' x+kei x kei' x - - l 22 e-zda

9.10.25

1+- 315 -+ 1 . 15 -+1 45 2 --6442xa 512x 819242 x4 9.10.34 3 1 9 1 ker'z x+kei'% ~ -22~ e - " ~ ~ ( l + ~ ;

+a3

Asymptotic Expansions of Large Zeros

Let 9.10.35

Asymptotic Expansions of Cross-Products

If x is large 9.10.27

where p=43. 33 1 ----25642 xa

1797 1 8192 2 '

. . .)

9.10.28

ber x bei' x-ber'

x bei

9 1 3 9 1 75 1 +-6442 -++-+ . *) x2 512 2 x 819242 x4 *

Then if 8 is a large positive integer

9.10.36

Zeros of ber, x-Js{a-f(6)},

6=(8--)v-#)r

Zeros of bei, x-&'{

6=

S-j(S)},

(5-

$V+#>.

Zeros of ker, s-a{S+f(-S)},

6=(s-$v-%a

Zeros of kei, x-a{S+j(-S>},

S=(S-~Y-#)~

384

BESSEL F'TJNCTIONE OF INTEGER ORDER

For v=O these expressions give the sth zero of each function; for other values of v the zeros represented may not be the 6th.

9.11.3 ker z= -In

-60.60977 451(~/8)''+5.65539 121(~/8)" -.19636 347(~/8)'~+.00309699(~/8)"

When U is large and positive

ber,(vx) +i bei,(vx)

(3x) ber z + t ~bei x--.57721

566 -59.05819 744 ( ~ / 84)f 171.36272 I 33( ~ / 8 ) '

Uniform Asymptotic Expansions for Large Orders

9.10.37

O<x58

-.00002 4 5 8 ( ~ / 8 ) ~ ~ + c

-

Icl
9.11.4 kei r = -ln()x)bei

9.10.38 kerv ( v x ) + i kei, (vx)

x 10-8

O<x58

ber ~$6.76454 936(z/8)' - 142.91827 687(~/8)'+124.23569 650(~/8)'O -21.30060 904(2/8)l4+1.17509 064(~/8)" - ,02695 875(2/8)" f.00029 532(~/8)"+r

9.10.39 ber: (vx)+i bei: (vx)

2-4.

1c1<3x10-9

-85x58

9.11.5

ber' z=z[ -4(d8)*+ 14.22222 222(2/8)' -6.06814 810(2/8)' O + ,66047 849 ( ~ / 8 ) -.02609 253(~/8)'~+.00045 957(~/8)" -.OOOOO 394(~/8)'']+e

9.10.40 kerl (vx)+i h i ; (vz)

where

(€1 <2.1 x 10-8

9.10.41 E=JiT2 and U , @ , c,(t) are given by 9.3.9 and 9.3.13. All fractional powers take their principal values.

9.11.6 bei'

9.11. Polynomial Approximations

$=XI$-

-85x58 113.77777 7 7 4 ( ~ / 8 ) ~ -32.36345 652 ( ~ / 8''$2.64191 ) 397 ( ~ / 8'') -.08349 609(~/8)~0+.00122 552(~/8)~' -.OOOOO 901(~/8)~'+t lal
9.11.1 ber x= 1-64 ( 4 8 )

'+

-85x58

10.66666 666(3$3)' 11.37777 772(~/8)'-2.31167 514(~/8)" +.14677 204(~/8)''-.00379 386(~/8)~O +.00004 6 0 9 ( ~ / 8 ) ~ ~ ] + ~

+

lel<7x10-8

O<x58

ker' z= -In (3s)ber' X-Pber x+tr bei' z X[-3.69113 734 (~//8)'~+2 1.42034 0 17( ~ / 8 ) - 11.36433 272(~/8)"+1.41384 780(~/8)'* 72.81777 742 ( ~ / 8'O-) 10.56765 779 ( ~ / 8 ) " -.06136 358(~/8)'~$ .00116 1 3 7 (~/ 8 )~' ,52185 6 15(~/8) "-.O1103 667( ~ / 822) -.OOOO1 075( ~ / 8"1) f E ,00011 346 (x/8) c IC~<~X~O-~ Irl<8XlO-'

9.11.2 -85x58 bei z=16(~/8)~-113.77777774(x/8)'

+

+

+

+

"+

'

385

BESSEL FUNCTIONS OF ENTEGER ORDER

O
9.11.8

kei' z= -In (az) bei' z--2-' bei z - t ~ber' 3: +~[.21139217- 13.39858 846(~/8)' 19.41182 758(~/8)'- 4.65950 823(~/8)'* +.33049 424(~/8)'~-.00926707(~/8)~' .00011 997(x/8)'*]+a

where 9.11.11 e(z) = (.ooooo 00-.39269 91;)

+

+

+

(.01104 86- .01104 85i) ( 8 p ) (.OOOOO 00- ,00097 6%) (8/z)' (-.OOOO9 06- .00009 Oli) ( 8 / ~ ) ' (- .00002 52 .OOOOO OOi) ( 8 / ~')

la1<7X10-*

+(-.OOOOO

+

34+.00000 51i) ( 8 / ~ ) '

4-(.OOOOO 06+.00000 81x<-

9.11.12

+

ker' z+i kei' r= -f(z)+( -2) (1 ea)

81x<-

9.11.9

19i) ( 8 / ~ ) ~

lca1<2X10-7

ker z+i kei z=f(z)(l+al)

9.11.13 SSX<.D i ber' z+i bei' z-- (ker'z+i kei' J ) =g(z)+\z) (1+c4) 7r

81x<

9.11.10

-

ber x+i bei z-- i (ker z f i kei z)=g(z)(l+ta) r

1.52

I <3 x 10-7

(€41 <3 x 10-7 where 9.11.14 +(~)=(.7071068f.70710 68i) (-,06250 01-.OOOOO O l i ) (S/Z) (- .00138 13 .00138 1li)(8/;~)' (.OOOOO 05 .00024 52i) ( 8 / ~ ) ~ (.00003 46 .00003 38i) ( 8 / ~ ) ~ (.OOOOl 17- .OOOOO 24i) ( 8 / ~ ) ' (.OOOOO 16-.OOOOO 32i) (8/~)'

+ + + + +

+ +

+

+

Numerical Methods 9.12. Use and Extension of the Tables Example 1. To evaluate Jn(1.55), n=O, 1, 2, . ., each to 5 decimals. The recurrence relation Jn-1(2)

+ J , + I (= ~ )(2n/z)Jn(z)

can be u s 4 to compute J&), Jl(z),J&), . . ., successively provided that n<x, otherwise severe accumulation of rounding errors will occur. Since, however, Jn(z)is a decreasing function of n when n>x, recurrence can always be carried out in the direction of decreasing n. Inepection of Table 9.2 shows that J,(1.55) vanishes to 5 decimals when n>7. Taking arbitrary values zero for J9and unity for J8,wecompute by recurrence the entries in the second column of the following table, rounding off to the nearest integer at each step.

n

Trial value8

9 8 7 6 6 4 3 2 1 0

0 1 10 89 679 4292 21478 78829 181967 166964

Jn(1.66)

. ooooo o0000

*

.m

3 ,00028 .00211 .01331 ,06661 .24463 .66442 48376

.

We normalize the results by use of the equation gel.%, namely J0(z)+2J&)+2J4(z)+

. . . =1

This yields the normalization factor 1/322376=.00000 31019 7

BESSEL FUNCTIONS OF INTEGER ORDER

386

and multiplying the trial values by this factor we obtain the required results, given in the third column. As a check we may verify the value of J0(1.55) by interpolation in Table 9.1. Remarks. (i) In this example it was possible to estimate immediately the value of n=N, say, at which to begin the recurrence. This may not always be the case and an arbitrary value of Nmay have to be taken, The number of correct significant figures in the final values is the same as the number of digits in the respective trial values. If the chosen N is too small the trial values will have too few digits and insufficient accuracy is obtained in the results, The calculation must then be repeated taking a higher value. On the other hand if N were too large unnecessary effort would be expended. This could be offset to some extent by discarding significant figures in the trial values which are in excess of the number of decimals required in Jn. (ii) If we had required, say, J0(1.55), J1(1.55), , . ., Jl0(1.55), each to 5 significant figures, we would have found the values of J10(1.55) and Jll (1.55) to 5 significant figures by interpolation in Table9.3 and then computed by recurrence J8,Js,. . ., Jo,no normalization being required. Alternatively, we could begin the recurrence at a higher value of N and retain only 5 significant figures in the trial values for n110. (iii) Exactly similar methods can be used to compute the modified Bessel function I,@) by means of the relations 9.6.26 and 9.6.36. If x is large, however, considerable cancellation will take place in using the latter equation, and it is preferable to normalize by means of 9.6.37. Example 2. To evaluate Y,(1.55), n=O, 1, 2, . . ., 10, each to 5 significant figures. The recurrence relation ~

+

Ym-~(x> Yn+1(x)= (2nlz) Y*(z) can be used to compute Y,(z)in the direction of increasing n both for n<x and n>x, because in the latter event Y,(x) is a numerically increasing function of n. We therefore compute Y0(1.55) and Yl(1.55) by interpolation in Table 9.1, generate Ya(l.55), Ya(1.55), . . ., YlO(1.55)by recurrence and check YIO( 1.55) by interpolation in Table 9.3. n

Y.(1.66)

n

0 1 2 3 4 5

+O. 40225 -0.37970 -0. 89218 -1.9227 -6.6506 -31. 886

6 7 8 9 10

Y,(1.66)

- 1. 9917X 10* - 1. 5100X 10’

-1. 3440x10’ -1.3722XlS - 1. 5801X 10”

Remarks. (i) An alternative way of computing Yo(x), should Jo(z),Ja(x),J r ( z ) , . . ., be available (see Example l), is to use formula 9.1.89. The other starting value for the recurrence, Yl(z), can then be found from the Wronskian relation Jl(x) Yo(z)-Jo(z) Yl(z)=2/(m). This is a convenient procedure for use with an automatic computer. (ii) Similar methods can be used to compute the modified Bessel function K,(x) by means of the recurrence relation 9.6.26 and the relation 9.6.54, except that if x is large severe cancellation will occur in the use of 9.6.54 and other methods for evaluating KO@)may be preferable, for example, use of the asymptotic expansion 9.7.2 or the polynomial approximation 9.8.6. Example 3. To evaluate Jo(.36) and Yo(.36) each to 5 decimals, using the multiplication theorem. From 9.1.74 we have

We take 2=.4. Then X=.9, (X2-1)(#z)=-.038, and extracting the necessary values of J d . 4 ) and Yk(.4) from Tables 9.1 and 9.2, we compute the required results as follows: k ak akJd.4) a k y h (.I) 0 1 2 3 4 6

+ .96040 + .00745 + .00001

+1.0 $0.038 +0.7220X 10-* +0.914X10-6 +0.87x10-7 +0.7X10-0 Jo(.36) =

+ .96786

- .60602 - ,06767

- .00599 - -00074

- .00011 - .00002 Yo(.36) = - .68066

Remark. This procedure is equivalent to interpolating by means of the Taylor series

at z=.4, and expressing the derivatives 5 f i k ) ( 2 ) in terms of 5 f k ( 2 ) by means of the recurrence relations and differential equation for the Bessel functions. Example 4. To evaluate J&), J:(x),Yv(z) and Y:(z) for v=50, x=75, each to 6 decimals. We use the asymptotic expansions 9.3.35, 9.3.36, 9.3.43, and 9.3.44. Here z=x/v=3/2. From 9.3.39 we find

-2 ( - ~ ) ~ / a = 1~ 4-arccos 3

2-+.2769653.

3-

387

BESSEL FUNCTIONS OF INTEGER ORDER

we find

Hence {=-.5567724

Next,

and (T)L"=+l.155332. 1-22

Ai'(u5) = +.947336.

us= -7.944134,

Hence ~"~=3.684031 ,

v2/'S= -7.556562.

(=10-2/3~s= .21544347~5=- 1.7115118.

Interpolating in Table 10.11, we find that A ~ ( V ~ / ~+.299953, S)=

A i ' ( ~ ~ /=~+.451441, l)

Bi(~~/~f)=-.160565,

B ~ ' ( u ~ / ~ {+.819542. )=

Interpolating in the table following 9.5.26 we obtain

I

As a check on the interpolation, we may verify that Ai Bit-Ai'Bi=l/?r. Interpolating in the table following 9.3.46 we obtain

bo (f) = f .O136,

( l )= .1442.

CO

The contributions of the terms involving ul(t) and dl(c) are negligible, and substituting in the asymptotic expansions we find that

+2.888631, .fi (S) = f .O107,

&)=

h(O= .-. +.98259,

Fi (0 = - .001,

The bounds given at the foot of the table show that the contributions of higher terms to the asymptotic series are negligible. Hence

1

jlo,5=28.88631f.O0107f

. . =28.88738,

+.

. . .)=-.14381.

X(1--.OOOOl+

JL0(75)= -(4/3)(1.155332)-1(50-4/3X .299953 X ,1442+50-2/aX .451441) = - .038658, Ym(75) = - 1.155332(-50-"

X ,160565

+50-6mX.819542X .0136)= +.050335, zo(75) =

+ (4/3) (1.155332)-'( -50-4'

X .160565

X.l442+50-*/8X .819542)= +.069543.

As a check we may verify that JY'- J'Y=2/(75?r). Remarks. This example may also be computed using the Debye expansions 9.3.15,9.3.16, 9.3.19, and 9.3.20. Four terms of each of these series are required, compared with two in the computations above. The closer the argument-order ratio is to unity, the less effective the Debye expansions become. In the neighborhood of unity the expansions 9.3.23, 9.3.24, 9.3.27, and 9.3.28 will furnish results of moderate accuracy; for high-accuracy work the uniform expansions should again be used. Example 5. To evaluate the 5th positive zero of Jlo(x) and the corresponding value of Jio(x), each to 5 decimals. We use the asymptotic expansions 9.5.22 and 9.5.23 setting v=10, s=5. From Table 10.11

~ ~ ( x ) ~ ~ ( i xY) ~ - ( X ) J ~ ( ~for> A=# = O to 4 signifi-

cant figures. Let ail) denote the root. Direct interpolation in Table 9.7 is impracticable owing to the divergence of the differences. Inspection of 9.5.28 suggests that a smoother function is (h-1)ai1). Using Table 9.7 we compute the following values 1/A

(A-

1,ap

0. 4

3.110

0. 6

3.131

0. 8

1.0

3.140 3.142(n)

6

$21 $9

'6

-12 -7

$2

Interpolating for l/X=.667, we obtain (X-1)ai1)=3.134 and thence the required root ai!b=6.268. Example 7. To evaluate ber, 1.55, bei, 1.55, n=O, 1, 2, . , ., each to 5 decimals. We use the recurrence relation Jn-1(xe3rt/4)+J n f l (xeari14 1

taking arbitrary values zero for J9(zv3rt14) and l + O i for J8(xe3ri/4)(see Example 1).

388

BESSEL F"CT1ONS

OF INTEGER ORDER

14294989-2201 li)= (.337119f.025155i)X lO-', bei,,z

obtained from the relation 9 8

7 a

0

t: -i I

+89 I --.oooo3I +.00030

1 0

-197012 +281539

+123804 $195373

-.

Z

f106734

+207449

+.30763

69531 +.91004

+. 36781 $. 59461 +. 72619

Adequate checks are furnished by interpolating in Table 9.12 for ber 1.55 and bei 1.55, and the use of a simple s u m check on the normalization. Should kernz and kei,,z be required they can be computed by forward recurrence using formulas 9.9.14, taking the required starting values for n=O and 1 from Table 9.12 (see Example 2). If an independent check on the recurrence is required the asymptotic expansion 9.10.38 can be used.

References Texto

[9.l]E. E. Allen, Analytical approximations, Math. Tables Aids Comp. 8, 240-241 (1954). [9.2]E. E. Allen, Polynomial approximations t o some modified Bessel functions, Math. Tables Aids Comp. 10, 162-164 (1956). [9.3]H. Bateman and R. C. Archibald, A guide t o tables of Bessel functions, Math. Tables Aids Comp. 1, 205-308 (1944). [9,4]W. G. Bickley, Bessel functions and formulae (Cambridge Univ. Press, Cambridge, England, 1953). This is a straight reprint of part of the preliminaries t o [9.21]. [9.5]H. S. Carslaw and J. C. Jaeger, Conduction of heab in solids (Oxford Univ. Press, London, England, 1947). [9.6]E. T. Copson, An introduction t o the theory of functions of a complex variable (Oxford Univ. Press, London, England, 1935). [9.7]A. Erd6lyi et al., Higher transcendental functions, vol2, ch. 7 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [9.8]E. T. Goodwin, Recurrence relations for crossproducts of B w e l functions, Quart. J. Mech. Appl. Math. 2, 72-74 (1949). (9.91 A. Gray, G. B. Mathews and T. M. MacRobert, A treatise on the theory of Bessel functions, 2d ed. (Macmillan and Co., Ltd., London, England; 1931). [9.10]W. Magnus and F. Oberhettinger, Formeln und Satze fiir die speziellen Funktionen der mathematischen Physik, 2d ed. (Springer-Verlag; Berlin, Germany, 1948). [9.11]N. W. McLachlan, Bessel functions for engineers, 2d ed. (Clarendon Press, Oxford, England, 1955). (9.121 F. W. J. Olver, Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. Soc. 48, 414-427 (1952).

[9.13]F. W. J. Olver, The asymptotic expansion of Bessel functions of large order. Philos. Trans. Roy. Soc. London A287, 328-368 (1954). [9.14]G. Petiau, La th6orie des fonctions de Bessel (Centre National de la Recherche Scientifique, Paris, France, 1955). [9.15]G. N. Watson, A treatise on the theory of Bessel functions, 2d ed. (Cambridge Univ. Press, Cambridge, England, 1958). [9.16]R. Weyrich, Die Zylinderfunktionen und ihre Anwendungen (B. G. Teubner, Leipzig, Germany, 1937). [9.17]C. S. Whitehead, On a generalization of the functions ber 2, bei 2, ker x , kei 2 . Quart. J. Pure Appl. Math. 42, 316-342 (1911). [9.18]E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1962). Tables

[9.19]J. F. Bridge and S. W. Angris,t, An :xtended table of roots of JA(z) Yi(fix)-J,(Bx) Y,(x)=0, Math. Comp. 16, 198-204 (1962). [9.20]British Association for the Advancement of Science, Bessel functions, Part I. Functions of orders zero and unity, Mathematical Tables, vol. VI (Cambridge Univ. Press, Cambridge, England, 1950). [9.21] British Association for the Advancement of Science, Bessel functions, Part 11. Functions of positive integer order, Mathematical Tables, vol. X (Cambridge Univ. Press, Cambridge, England, 1952). (9.221 British Association for the Advancement of Science, Annual Report (J. R. Airey), 254 (1927). (9.23)E. Cambi, Eleven- and fifteen-place tables of Bessel functions of the first kind, t o all significant orders (Dover Publications, Inc., New York, N.Y., 1948).

BESSEL FUNCTIONS OF INTEGER ORDER

389

functions with imaginary argument and their [9.24] E. A. Chistova, Tablitsy funktsii Besselya ot integrals). deistvitel’nogo argumenta i integralov ot nikh [9.34] Mathematical Tables Project, Table of (Izdat. Akad. Neuk SSSR, MOSCOW,U.S.S.R., f,,(z)=nl(gz)-*J,(z). J . M a t h . P h y s . 23, 1958). (Table of Bessel functions with real 45-60 (1944). argument and their integrals). [9.35] National Bureau of Standards, Table of the Bessel [9.25] H. B. Dwight, Tables of integrals and other mathefunctions Jo(z) and JI(z)for complex arguments, matical data (The Macmillan Co., New York, 2d ed. (Columbia Univ. Press, New York, N.Y., N.Y., 1957). This includes formulas for, and tables of Kelvin 1947). [9.36] National Bureau of Standards, Tables of the Bessel functions. functions Yo(z)and Y l ( z )for complex arguments [9.26] H. B. Dwight, Table of roots for natural frequencies (Columbia Univ. Press, New York, N.Y., 1950). in coaxial type cavities, J. Math. Phys. 27, [9.37] National Physical Laboratory Mathematical Tables, 84-89 (1948). vol. 5, Chebyshev series for mathematical funcThis gives zeros of the functions 9.5.27 and 96.30 tions, by C. W. Clenshaw (Her Majesty’s Stafor n=0,1,2,3. tionery Office, London, England, 1962). [9.27] V. N. Faddeeva and M. K. Gavurin, Tablitsy funktsii Besselia J,(z) tselykh nomerov ot 0 [9.38] National Physical Laboratory Mathematical Tables, do 120 (Izdat. Akad. Nauk SSSR, MOSCOW, vol. 6, Tables for Bessel functions of moderate or U.S.S.R., 1950). (Table of J.(z) for orders 0 to large orders, by F. W. J. Olver (Her Majesty’s 120). Stationery Office, London, England, 1962). [9.28] L. Fox, A short table for Bessel functions of integer [9.39] L. N. Nosova, Tables of Thomson (Kelvin) functions orders and large arguments. Royal Society and their first derivatives, translated from the Shorter Mathematical Tables No. 3 (Cambridge Russian by P. Basu (Pergamon Press, New Univ. Press, Cambridge, England, 1954). York, N.Y., 1961). [9.29] E. T. Goodwin and J. Staton, Table of J3&,,,r), [9.40] Royal Society Mathematical Tables, vol. 7, Bessel Quart. J. Mech. Appl. Math. 1, 220-224 (1948). functions, Part 111. Zeros and associated values, [9.30] Harvard Computation Laboratory, Tables of the edited by F. W. J. Olver (Cambridge Univ. Press, Bessel functions of the first kind of orders 0 Cambridge, England, 1960). through 135, vols. 3-14 (Harvard Univ. Prew, The introduction includes many formulaa conCambridge, Mass., 1947-1951). nected with zeros. [9.31] K. Hayashi, Tafeln der Besselschen, Theta, Kugel[9.41] Royal Society Mathematical Tables, vol. 10, und anderer Funktionen (Springer, Berlin, GerBessel functions, Part IV. Kelvin functions, by ma.ny, 1930). A. Young and A. Kirk (Cambridge Univ. Press, [9.32] E. Jahnkz, F. Emde, and F. Losch, Tables of Cambridge, England, 1963). higher functions, oh. IX, 6th ed. (McGraw-Hill The introduction includes many formulas for Book Co., Inc., New York, N.Y., 1960). Kelvin functions. [9.33] L. N. Karmazina and E. A. Chistova, Tablitsy 19.421 W. Sibagaki, 0.01 % tables of modified Bessel funktsii Besselya ot mnimogo argumenta i functions, with the account of the methods used integralov ot nikh (Izdat. Akad. Nauk SSSR, in the calculation (Baifukan, Tokyo, Japan, 1955). Moscow, U.S.S.R., 1958). (Tables of Bessel

I

I I

I

iI I

I ~

I I

I

I

390

BESSEL FUNCTIONS OF INTEGER ORDER

Table 9.1

BESSEL FUNCTIONS-ORDERS

0, 1 AND 2

0.00000 0.00124 0.00498 0.01116 0. 01973

00000 89587 33542 58619 46631

0. 0 0.1 0.2 0. 3 0. 4

1.00000 0.99750 0.99002 0.97762 0.96039

00000 15620 49722 62465 82266

00000 66040 39576 38296 59563

0.00000 0.04993 0.09950 0.14831 0.19602

0. 5 0-._ 60. 7 0. 8 0. 9

0.93846 0.91200 OI88iZO 0.84628 0.80752

98072 48634 08886 73527 37981

40813 97211 07405 50480 22545

0.24226 84577 0.28670 09881 574i5 0.36884 20461 0.40594 95461

0: 05878 69444

1. 0 1.1 1.2 1.3 1.4

0.76519 0.71962 0.67113 0.62008 0.56685

76865 20185 27442 59895 51203

57967 27511 64363 61509 74289

0.44005 0.47090 0.49828 0.52202 0.54194

05857 23949 90576 32474 77139

0.11490 0.13656 0.15934 0.18302 0.20735

34849 41540 90183 66988 58995

1.5 1. 6 1.7 1.8 1.9

0.51182 76717 35918 0.45540 21676 39381 0.39798 48594 46109

0.55793 0.56989 0.57776 0.58151 0.58115

65079 59353 52315 69517 70727

0.23208 0.25696 0.28173 0.30614 0.32992

76721 77514 89424 35353 57277

00000 75260 08326 88163 65780

OI 12899

O:3i498 diiio 42558

0.28181 85593 14385

0.03060 40235 0.04366 50967 0.07581 77625 0. 09458 63043

2. 0 2.1 2.2 2; 3 2. 4

0.22389 0.16660 0,11036 0.05553 .I+O. 00250

07791 69803 22669 97844 76832

41236 31990 22174 45602 97244

0.57672 0.56829 0.55596 0.53987 0.52018

48078 21358 30498 25326 52682

0.35283 0.37462 0.39505 0.41391 0.43098

40286 36252 86875 45917 00402

2. 5 2.6 2.7 2. 8 2. 9

-0.04838 -0.09680 -0.14244 -0.18503 -0.22431

37764 49543 93700 60333 15457

68198 97038 46012 64387 91968

0.49709 0.47081 0.44160 0.40970 0.37542

41025 82665 13791 92469 74818

0.44605 0.45897 0.46956 0.47768 0.48322

90584 28517 15027 54954 70505

3. 0 3.1 3.2 3. 3 3. 4

-0.26005 -0.29206 -0.32018 -0.34429 -0.36429

19549 43476 81696 62603 55967

01933 50698 57123 98885 62000

0.33905 0.30092 0.26134 0.22066 0.17922

89585 11331 32488 34530 58517

0.48609 0.48620 0.48352 0.47803 0.46972

12606 70142 77001 16865 25683

3. - _-5 3. 6 3.7 3. 8 3.9

-0.38012 77399 -0; %i76 89837 -0.39923 02033 -0.40255 64101 -0.40182 60148

87263 00798 71191 78564 87640

0.13737 75274 0.05383 39877 +O. 01282 10029 -0.02724 40396

0.45862 0; 44480 0.42832 0.40930 0.38785

91842 53988 96562 43065 47125

4. 0 4.1 4.2 4. 3 4.4

-0.39714 -0.38866 -0.37655 -0.36101 -0.34225

98098 96798 70543 11172 67900

63847 35854 67568 36535 03886

-0.06604 -0.10327 -0.13864 -0.17189 -0.20277

33280 32577 69421 65602 55219

0.36412 0.33829 0.31053 0.28105 0.25008

81459 24809 47010 92288 60982

4. 5 4. 6 4.7 4. 8 4. 9

-0.32054 -0.29613 -0.26933 -0.24042 -0.20973

25089 78165 07894 53272 83275

85121 74141 19753 91183 85326

-0.23106 -0.25655 -0.27908 -0.29849 -0.31469

04319 28361 07358 98581 46710

0.21784 0.18459 0.15057 0.11605 0.08129

89837 31052 30295 03864 15231

5. 0

-0,17759 67713 14338

0; 09546 55472

-0.32757 91376

[(3!)6]

[(-;)51 Jn

0.04656 51163

[(-$PI

2n +1).( =y Jn ( r ) -J n - 1(x)

Compiled from British Association for the Advancement of Science, Bessel functions, Part 11. Functions of positive integer order, Mathematical Tables, vol. X (Cambridge Univ. Press, Cambridge, England, 1952) and Harvard Computation Laboratory, Tables of the Bessel functions of the first kindof orders 0 through 135, vols. 3-14 (Harvard Univ. Press, Cambridge, Mass., 1947-1951) (with permission).

391

BESSEL FUNCTIONS OF INTEGER ORDER

0, 1 AND 2

BESSEL FUNCTIONS-ORDERS X

0.0 0.1 0.2 0. 3 0.4

-

Yz(x)

Y1(x)

Yo(x) m

Table 9.1

-

-

m

m

-1.53423 -1.08110 -0.80727 -0.60602

86514 53224 35778 45684

-6.45895 -3.32382 -2.29310 -1.78087

10947 49881 51384 20443

-127.64478 - 32.15714 - 14.48009 - 8.29833

324 456 401 565

0. 5 0. 6 0. 7 0.8 0.9

-0.44451 -0.30850 -0.19066 -0.08680 +O. 00562

87335 98701 49293 22797 83066

-1.47147 -1.26039 -1.10324 -0.97814 -0.87312

23927 13472 98719 41767 65825

5.44137 3.89279 2.96147 2.35855 1,94590

084 462 756 816 960

1.0 1.1 1.2 1.3 1.4

0.08825 0.16216 0.22808 0.28653 0.33789

69642 32029 35032 53572 51297

-0.78121 -0.69811 -0.62113 -0.54851 -0.47914

28213 95601 63797 97300 69742

-

1.65068 1.43147 1.26331 - 1.13041 - 1.02239

261 149 080 186 081

1.5 1.6 1. 7 1.8 1.9

0.38244 0.42042 0.45202 0.47743 0.49681

89238 68964 70002 17149 99713

-0.41230 -0.34757 -0.28472 -0.22366 -0.16440

86270 80083 62451 48682 57723

0.93219 0,85489 0.78699 0.72594 0,66987

376 941 905 824 868

2. 0 2.1 2.2 21 3 2.4

0.51037 0.51829 0.52078 0: 51807 0.51041

56726 37375 42854 53962 47487

-

0.61740 Oi56751 0.51943 0.47261 - 0.42667

810 146 175 686 397

2.5 2. 6 2.7 2.8 2.9

0.49807 0.48133 0.46050 0.43591 0.40791

03596 05906 35491 59856 17692

0.14591 0.18836 0.22763 0.26354 0.29594

81380 35444 24459 53936 00546

0.38133 0.33643 0.29188 0.24766 0.20381

585 556 692 928 518

3. 0 3.1 3.2 3. 3 3. 4

0.37685 0.34310 0.30705 0.26909 0.22961

00100 28894 32501 19951 53372

0.32467 0.34962 0.37071 0.38785 0.40101

44248 94823 13384 29310 52921

---

0.16040 0.11753 0,07535 0.03402 + 0. 00627

039 548 866 961 601

3.5 3.6 3.7 3. 8 3. 9

0.18902 0.14771 0.10607 0.06450 +O. 02337

19439 00126 43153 32467 59082

0.41018 0.41539 0.41667 0.41411 0.40782

84179 17621 43727 46893 00193

0. 04537 0.08306 0.11915 0.15345 0.18576

144 319 508 185 256

4. 0 4.1 4.2 4.3 4.4

-0.01694 -0.05609 -0. 09375 -0.12959 -0.16333

07393 46266 12013 59029 64628

0.39792 0.38459 0.36801 0.34839 0.32597

57106 40348 28079 37583 06708

0.21590 0.24370 0.26899 0,29163 0.31150

359 147 540 951 495

4. 5 4. 6 4.7 4.8 4.9

-0.19470 -0.22345 -0.24938 -0.27230 -0.29205

50086 99526 76412 37945 45942

0.30099 0.27374 0.24450 0.21356 0.18124

73231 52415 12968 51673 66920

0.32848 0.34247 0.35343 0.36128 0.36603

160 962 075 928 284

5.0

-0.30851 76252

-0.10703 24315 -0: 05167 86121

+O. 00148 77893 0.05227 73158 0.10048 89383

0.14786 31434 2n

Yn+1(x)=; Y,(x)-Y,-l(X)

0.36766 288

'+

392

BESSEL FUNCTIONS O F INTEGER ORDER

BESSEL FUNCTIONS-ORDERS 0, 1 AND 2

Table 9.1

5. 0 5.1 5.2 5. 3 5.4

-0.17759 -0.14433 -0.11029 -0.07580 -0.04121

J o (4 67713 14338 47470 60501 04397 90987 31115 85584 01012 44991

-0.32757 -0.33709 -0.34322 -0.34596 -0.34534

91376 72020 30059 08338 47908

0.04656 +O. 01213 -0.02171 -0.05474 -0.08669

51163 97659 84086 81465 53768

5.5 5. 6 5.7 5; 8 5.9

-0.00684 +O. 02697 0.05992 0.09170 0.12203

38694 08846 00097 25675 33545

17819 85114 24037 74816 92823

-0.34143 -0.33433 -0.32414 -0.31102 -0.29514

82154 28363 76802 77443 24447

-0.11731 -0.14637 -0.17365 -0.19895 -0.22208

54816 54691 60379 35139 16409

6. 0 6.1 6.2 6.3 6.4

0.15064 0.17729 Oi20174 0.22381 0.24331

52572 14222 72229 20061 06048

50997 42744 48904 32191 23407

-0.27668 -0.25586 -0.23291 -0.20808 -0.18163

38581 47726 65671 69402 75090

-0.24287 -0.26118 -0.27688 -0.28987 -0.30007

32100 15116 15994 13522 23264

6. 5 6. 6 6; 7 6.8 61 9

0.26009 0.27404 0.28506 0.29309 0.29810

46055 33606 47377 56031 20354

81606 24146 10576 04273 04820

-0.15384 -0.12498 -0.09534 -0.06521 -0.03490

13014 01652 21180 86634 20961

-0.30743 -0.31191 -0.31352 -0.31227 -0.30821

03906 61379 50715 75629 85850

7. 0 7.1 7.2 7. 3 7.4

0.30007 0.29905 0.29507 OI28821 0.27859

92705 13805 06914 69476 62326

19556 01550 00958 35014 57478

-0.00468 +O. 02515 0.05432 0.08257 0.10962

28235 32743 74202 04305 50949

-0.30141 -0.29196 -0.27997 -0.26559 -0.24896

72201 59511 97413 49119 78286

7.5 7.6 7.7 7.8 7.9

0.26633 0,25160 0.23455 0.21540 0.19436

96578 18338 91395 78077 18448

80378 49976 86464 46263 41278

0.13524 0.15921 0.18131 0.20135 0.21917

84276 37684 27153 68728 93999

-0.23027 -0.20970 -0.18746 -0.16377 -0.13887

34105 34737 49278 78404 33892

8. 0 8.1 8.2 8.3 8.4

0.17165 0.14751 0.12221 0,09600 0.06915

08071 74540 53017 61008 72616

37554 44378 84138 95010 56985

0.23463 0.24760 01 25799 0.26573 0.27078

63469 77670 85976 93020 62683

-0.11299 -0.08637 -0.05928 -0.03197 -0.00468

17204 97338 88146 25341 43406

0.04193 92518 42935

-0.01252 27324 49665 -0.03923 38031 76542 -0.06525 32468 51244

0.27312 0.27275 0.26971 0; 26407 0.25590

19637 48445 90241 37032 23714

+O. 02232 0; 04880 0.07452 0.09925 0.12275

47396 83679 71058 05539 93977

X

8. 5 8.6 8.7 8.8 8. 9

+O. 01462 29912 78741

J2(4

Jl(4

9.0 9.1 9.2 9. 3 9.4

-0.09033 -0.11423 -0.13674 -0.15765 -0.17677

36111 92326 83707 51899 15727

82876 83199 64864 43403 51508

0.24531 0.23243 0.21740 0.20041 0.18163

17866 07450 86550 39278 22040

0.14484 0.16532 0.18401 0.20075 0.21541

73415 29129 11218 49594 67225

9.5 9.6 9.7 9.8 9.9

-0.19392 -0.20897 -0.22179 -0.23227 -0.24034

87476 87183 54820 60275 11055

87422 68872 31723 79367 34760

0.16126 0.13952 0.11663 0.09284 0.06836

44308 48117 86479 00911 98323

0.22787 0.23804 0.24584 0.25122 0.25415

91542 63875 46878 29849 31929

10. 0

-0.24593 57644 51348

0.04347 27462 [( 3 1 4 1

0.25463 03137 [( 7 1 4 1

,

393

BESSEL FUNCTIONS OF INTEGER ORDER

BESSEL FUNCTIONS-ORDERS 0, 1 AND 2 X

YO(4

Yl(X)

Table 9.1

Ydz)

5. 0 5.1 5.2 5.3 5.4

-0.30851 -0.32160 -0.33125 -0.33743 -0.34016

76252 24491 09348 73011 78783

0.14786 0.11373 0.07919 0.04454 +O. 01012

31434 64420 03430 76191 72667

0.36766 0.36620 0.36170 0.35424 0.34391

288 498 876 772 872

5. 5 5. 6 5.7 5. 8 5. 9

-0.33948 -0.33544 -0.32815 -0.31774 -0.30436

05929 41812 71408 64300 59300

-0. 02375 -0.05680 -0.08872 -0.11923 -0.14807

82390 56144 33405 41135 71525

0.33084 0,31515 0.29702 0.27663 0.25417

123 646 614 122 029

6. 0 6.1 6.2 6. 3 6.4

-0.28819 -0.26943 -0.24830 -0.22506 -0.19994

46840 49304 99505 17496 85953

-0.17501 -0.19981 -0.22228 -0.24224 -0.25955

03443 22045 36406 95005 98934

0.22965 0.20392 0.17660 0.14815 0.11863

790 273 555 715 613

6.5 6. 6 6.7 6.8 6.9

-0.17324 -0.14522 -0.11619 -0.08643 -0.05625

24349 62172 11427 38683 36922

-0.27409 -0.28574 -0.29445 -0.30018 -0.30291

12740 72791 93130 68758 76343

0.08890 0.05863 +0.02829 -0.00165 -0.03154

666 613 284 639 852

7. 0 7.1 7.2 7.3 7.4

-0.02594 +O. 00418 0.03385 0.06277 0.09068

97440 17932 04048 38864 08802

-0.30266 -0.29947 -0.29342 -0.28459 -0.27311

72370 88746 25939 43719 49598

-0.06052 -0,08854 -0.11535 -0.14074 -0,16449

661 204 668 495 573

7. 5 7. 6 7. 7 7.8 7.9

0.11731 0.14242 0.16580 0.18722 0.20652

32861 85247 16324 71733 09481

-0.25912 -0.24280 -0.22431 -0.20388 -0.18172

85105 10021 84743 50954 10773

-0.16641 -0.20632 -0.22406 -0.23950 -0.25252

422 353 617 540 628

8. 0 8.1 8.2 8.3 8.4

0.22352 0.23809 0.25011 0.25951 0.26622

14894 13287 80276 49638 18674

-0.15806 -0.13314 -0.10724 -0.08059 -0.05348

04617 87960 07223 75035 45084

-0.26303 -0.27096 -0.27627 -0.27693 -0.27695

660 757 430 605 627

8.5 8. 6 8.7 8.8 8. 9

0.27020 0.27145 0.26999 0.26587 0.25915

51054 77123 91703 49418 57617

-0.02616 +O. 00108 0.02801 0.05435 0.07986

86794 39918 09592 55633 93974

-0.27636 -0i27120 -0.26355 -0.25352 -0.24120

244 562 987 140 758

9.0 9.1 9.2 9.3 9.4

0. 24c '3 0.23833 0.22449 0.20857 0.19074

66983 59921 36870 00676 39189

0.10431 0.12746 0.14911 0; 16906 0.18713

45752 58820 27879 13071 56847

-0.22675 -0I21032 -0.19207 -0.17221 -0.15092

568 151 786 280 782

9.5 9.6 9.7 9.8 9.9

0.17121 0.15018 0.12787 0.10452 0.08037

06262 01353 47920 70840 73052

0.20317 0.21705 0.22866 0.23789 0.24469

98994 89660 00298 32421 24113

-0.12643 -0.10495 -0.06072 -0,05597 -0.03094

591 952 839 744 449

10. 0

0.05567 11673

0.24901 54242

-0.00586 808 [(-;)41

394

BESSEL FUNCTIONS OF INTEGER ORDER

BESSEL FUNCTIONS-ORDERS

Table 9.1 X

JO

(4

0, 1 AND 2

JI(4

J2(5)

10.0 10.1 10.2 10. 3 10.4

-0.24593 -0.24902 -0.24961 -0.24771 -0.24337

57644 96505 70698 68134 17507

51348 80910 54127 82244 14207

0.04347 +O. 01839 -0.00661 -0.03131 -0. 05547

27462 55155 57433 78295 27618

0.25463 0.25267 0.24831 0.24163 0.23270

03137 23269 98653 56815 39119

10.5 10. 6 10.7 10.8 10.9

-0.23664 -0.22763 -0.21644 -0.20320 -0.18806

81944 50476 27399 19671 22459

62347 20693 23818 12039 63342

-0. 07885 -0; 10122 -0.12239 -0.14216 -0.16034

00142 86626 94239 65683 96867

0.22162 0.20853 0.19356 0.17687 0.15864

91441 53000 43429 48248 02851

11. 0 11.1 11.2 11.3 11.4

-0.17119 -0.15276 -0.13299 -0.11206 -0.09021

03004 82954 19368 84561 45002

07196 35677 59575 09807 47520

-0.17678 -0.19132 -0.20385 -0.21425 -0.22245

52990 82878 31459 50262 05864

0.13904 0.11829 0.09658 0.07414 0. 05118

75188 47301 95894 72125 80816

11.5 11. 6 11.7 11.8 11.9

-0.06765 -0.04461 -0.02133 +O. 00196 0.02504

39481 56740 12813 71733 94416

11665 94438 88500 06740 99590

-0.22837 -0.23200 -0.23330 -0.23228 -0.22898

86207 04746 02408 47343 32497

0.02793 +O. 00461 -0.01854 -0.04133 -0.06353

59271 55923 91017 74673 40215

12. 0 12.1 12.2 12.3 12.4

0.04768 0.06966 0.09077 0.11079 0.12956

93107 67736 01231 79503 10265

96834 06807 70505 07585 17502

-0.22344 -0.21574 -0.20598 -0.19425 -0.18071

71045 89734 20217 88480 02469

-0.08493 -0.10532 -0.12453 -0.14238 -0.15870

04949 77609 76677 47549 78405

12.5 12.6 12.7 12.8 12.9

0.14688 0.16260 0.17658 0.18870 0,19884

40547 72717 78885 13547 24371

00421 45511 61499 80683 36331

-0.16548 -0.14874 -0.13066 -0.11143 -0.09124

38046 23434 22290 15593 82522

-0.17336 -0.18621 -0.19716 -0.20611 -0.21298

14634 71675 46175 25359 94530

13. 0 13.1 13.2 13.3 13.4

0.20692 0.21288 0,21668 0.21829 0.21772

61023 81975 59222 80903 51787

77068 22060 58564 19277 31184

-0. 07031 -0.04885 -0.02706 -0.00517 +O. 01659

80521 24733 67028 74806 90199

-0.21774 -0.22034 -0.22078 -0.21907 -0.21524

42642 65904 69378 66588 77131

13. 5 13. 6 13.7 13. 8 13.9

0.21498 0.21013 0.20322 0.19433 0.18357

91658 31613 08326 56352 98554

80401 69248 33007 15629 57870

0. 03804 0.05896 0.07914 0.09839 0.11652

92921 45572 27651 05167 48904

-0.20935 -0.20146 -0.19166 -0.18007 -0; 16681

22337 19030 71443 61400 36842

14. 0 14.1 14.2 14. 3 14.4

0.17107 0.15695 0.14136 0.12448 0.10648

34761 28770 93846 76852 41184

10459 32601 57129 83919 90342

0.13337 0.14878 0.16261 0.17472 0.18503

51547 43513 07342 90520 16616

-0.15201 -0.13584 -0.11846 -0.10005 -0.08078

98826 87137 64643 00556 52766

14. 5 14. 6 14.7 14. 8 14. 9

0.08754 0.06786 0.04764 0.02708 +O. 00639

48680 40683 18459 23145 15448

10376 23379 01522 85872 90853

0.19342 0.19985 0.20425 0.20659 0.20687

94636 26514 12683 55672 61718

-0.06086 -0.04048 -0.01985 +O. 00083 0.02137

49420 69928 25577 60053 70688

15. 0

-0.01422 44728 26781

0.20510 40386

0.04157 16780

395

BESSEL FUNCTIONS O F INTEGER ORDER

BESSEL FUNCTIONS-ORDERS X

0, 1 AND 2

Table 9.1

10. 0 10.1 10.2 10.3 10.4

YO(X) 0.05567 11673 0.03065 73806 +O. 00558 52273 -0.01929 78497 -0.04374 86190

Yl(X) 0.24901 54242 0.25084 44363 0.25018 58292 0.24706 99395 0.24155 05610

10.5 10. 6 10. 7 10. 8 10. 9

-0.06753 -0.09041 -0.11218 -0.13263 -0.15158

03725 51548 58897 83844 31932

0.23370 0.22362 0.21144 0.19728 0.18131

42284 92892 47763 90905 85097

0.11204 0.13260 0.15170 0.16917 0.18485

546 936 828 340 264

11. 0 11.1 11.2 11.3 11.4

-0.16884 -0.18427 -0.19773 -0.20910 -0.21829

73239 57716 28675 34295 37073

0.16370 0.14463 0.12431 0.10294 0.08074

55374 71102 26795 21889 39654

0.19861 0.21033 0.21993 0.22732 0.23245

197 651 156 329 932

11.5 11. 6 11.7 11.8 11.9

-0.22523 -0.22986 -0.23218 -0.23216 -0.22983

21117 97260 05930 17790 32139

0.05794 0.03476 +O. 01144 -0.01178 -0.03471

25471 64663 60113 90120 14983

0.23530 0.23586 0.23413 0.23016 0.22399

908 394 718 364 935

12.0 12.1 12.2 12. 3 12: 4

-0.22523 -0.21843 -0.20952 -0.19859 -0.18577

73126 83806 18128 30946 66153

-0.05709 -0.07873 -0.09941 -0.11894 -0.13714

92183 69315 84171 84033 43766

0.21572 0.20542 0.19322 0.17925 0.16365

078 401 371 189 655

12.5 12. 6 12.7 12. 8 12.9

-0.17121 -0.15506 -0.13749 -0.11870 -0.09887

43068 41238 83780 19463 03702

-0.15383 -0.16887 -0.18212 -0.19347 -0.20281

82565 79186 85528 38454 69743

0,14660 0.12825 0,10881 0.08847 0. 06742

019 810 672 166 588

13. 0 13.1 13. 2 13.3 13.4

-0.07820 -0.05692 -0.03523 -0.01336 +O. 00848

78645 52r68 7Ei771 34191 02072

-0.21008 14084 -0.21521 15060 -0.21817 29066 -0.21895 27145 -0,21755 94728

0. 04588 0.02406 +0.00218 -0.01956 -0.04095

765 854 138 180 177

13.5 13. 6 13. 7 13. 8 13. 9

0. 03007 0.05121 0. 07168 0.09129 0.10985

70090 50115 83040 90143 91895

-0.21402 -0.20839 -0.20074 -0.19115 -0.17975

29303 36044 21453 85095 09511

-0. 06178 -0.08186 -0.10099 -0.11900 -0.13572

411 113 373 315 264

14. 0 14.1 14. 2 14. 3 14.4

0.12719 0.14313 0.15754 0.17027 0.18123

25686 62286 20895 82640 02411

-0.16664 -0.15198 -0.13591 -0.11861 -0.10026

48419 13335 58742 65967 25924

-0.15099 -0.16469 -0.17668 -0.18686 -0,19515

897 386 517 800 560

14. 5 14.6 14.7 14. 8 14.9

0.19030 0: 19741 0.20251 0.20556 0.20654

18912 62858 63238 51604 64347

-0.08104 -0: 06115 -0.04078 -0.02016 +O. 00052

20909 05609 87536 07059 82751

-0.20148 -0.20579 -0.20806 -0I20828 -0.20647

011 307 581 958 553

15. 0

0.20546 4296G

-0.20265

448

0.02107 36280

Y2b) -0.00586 808 +0.01901 478 0. 04347 082 0.06727 260 0.09020 065

396

BESSEL FUNCTIONS OF INTEGER ORDER

Table 9.1

BESSEL FUNCTIONS-ORDERS

X

JOW

0, 1 AND 2 J2

J1(4

(4

15.0 15.1 15.2 15. 3 15.4

-0.01422 -0.03456 -0.05442 -0.07360 -0.09193

44728 18514 07968 75449 62278

26781 55565 44039 51123 62321

0.20510 0.20131 0.19554 0.18787 0.17840

40386 02204 54359 94498 02717

0.04157 0.06122 0.08015 0.09816 0.11510

16780 54568 04595 69502 50943

15.5 15.6 15.7 15.8 15.9

-0.10923 -0.12532 -0: i4ooi -0.15332 -0.16497

06509 59640 02i18 57477 04994

00050 22481

60686 85671

0.16721 0.15443 0; i402i 0.12469 0.10802

31804 95871 55469 13334 78901

0.13080 0.14512 0.15793 0.16910 0.17855

65451 59111 20904 94608 89133

16. 0 16.1 16.2 16.3 16. 4

-0.17489 -0.18302 -0.18927 -0.19360 -0.19597

90739 36924 49469 23723 48287

83629 65310 77945 28377 91007

0.09039 0.07197 0.05296 0.03353 +O. 01389

71757 94186 14991 50765 46807

0.18619 0.19196 0.19581 0.19771 0.19766

87209 52352 34037 71056 93020

16. 5 16. 6 16.7 16.8 16.9

-0.19638 -0.19482 -0.19134 -0.18597 -0.17878

06929 78558 35295 38653 33878

36861 05566 25189 47601 91219

-0.00576 -0.02524 -0.04436 -0.06292 -0.08074

42137 71116 24008 32177 92543

0.19568 0.19178 0.18603 0.17848 0.16922

20004 60351 06671 30061 72631

17. 0 17.1 17.2 17.3 17. 4

-0.16985 -0.15928 -0.14719 -0.13370 -0.11895

42521 53315 11467 06470 58563

51184 32265 66030 75764 36348

-0.09766 84928 -0.11351 88483 -0.12814 97057 . _ ~ -0.14142 33355 -0.15321 61760

0.15836 0.14600 ~. - . - .. 0.13229 0.11735 0.10134

38412 82733 -- .- 00182 11285 48016

17.5

-0.10311 03982 28686

-0.16341 99694

0.08443 38303

29049

BESSEL FUNCTIONS-MODULUS

Table 9.1

AND PHASE OF ORDERS 0 , l AND 2

Yn(x)=&(z) sin em(.)

Jn(.)=Mn(s) cos ).(no

0.79739 0.79748 0.79756 0.79764 0.79770

375 584 868 214 609

-0.79783 -0.79660 -0.79536 -0.79412 -0.79288

499 186 548 617 426

0.79936 0.79908 0,79883 0.79861 0.79842

575 654 586 398 116

-2.31885 -2.32256 -2.32627 -2.33000 -2.33372

508 201 732 016 965

ziM2 (4 0.80542 555 0.80398 367 0,80269 711 0.80156 472 0.80058 549

@z(x)---z -3.73985 605 -3.75850 527 -3.77717 539 -3.79586 377 -3.81456 786

<2>

0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

0.79776 0.79780 0.79783 0.79786 0.79787

040 498 975 463 957

-0.79164 -0.79039 -0.78914 -0.78789 -0.78664

009 402 641 764 810

0.79825 0.79812 0.79801 0.79794 0.79789

761 353 908 438 952

-2.33746 -2.34120 -2.34494 -2.34869 -2.35244

488 495 891 580 465

0.79975 0.79908 0.79855 0.79818 0,79795

-3.83328 -3.85201 -3.87075 -3,88949 -3.90824

20 25 33 50 100

0.00

0.79788 456

2-1

XtMO(X)

80

(4--z

-0.78539 816

X+Ml(X)

0.79788 456

@I (x)-2

-2.35619 449

=nearest integer to x.

851 299 829 387 937

0.79788 456

521 346 034 363 117

-3.92699 082

10 11 13 14 17

a

397

BESSEL FUNCTIONS OF INTEGER ORDER

BESSEL FUNCTIONS-ORDERS

Yo).(

091 AND 2

Table 9.1

15.0 15.1 15. 2 15. 3 15.4

0.20546 0.20234 0.19722 0.19018 0.18128

42960 32292 76821 15001 71741

0.02107 0.04127 0.06093 0.07985 0.09786

36280 35340 08736 51269 41973

Y2 (x) -0.20265 448, -0.19687 654 -0.18921 046 -0.17974 292 -0.16857 754

15. 5 15. 6 15. 7 15. 8 15. 9

0.17064 0: 15837 0.14459 0.12947 0.11315

49112 15368 92412 41833 49657

0.11478 0.13046 0.14474 0.15749 0.16860

61425 07959 12638 52835 64314

-0.15583 -0.14164 -0.12616 -0.10953 -0.09194

380 579 086 807 661

16. 0 16.1 16. 2 161 3 16. 4

0.09581 0: 07762 0.05876 0.03944 0.01985

09971 07587 99918 98249 48596

0.17797 0: 18551 0.19117 0.19490 0.19667

51689 97173 67538 19240 01648

-0.07356 -0IO5457 -0.03516 -0.01553 +0.00412

410 483 792 548 931

16. 5 16. 6 16. 7 16. 8 16. 9

+O. 00018 -0.01937 -0.03862 -0.05736 -0.07543

12325 53254 14147 78596 15476

0.19647 0.19433 0.19027 0.18434 0.17663

58378 26715 35142 99015 14431

0.02363 0.04278 0.06140 0.07931 0,09633

402 890 866 428 468

17. 0 17.1 17. 2 17. 3 17.4

-0.09263 -0.10881 -0.12382 -0.13750 -0.14973

71984 90473 24237 52134 91883

0.16720 0.15617 0.14365 0.12978 0.11470

50361 39131 65362 53467 53859

0.11230 0.12708 0.14052 0.15250 0.16292

838 500 667 930 372

17. 5

-0.16041 11925

7:

Y1(.)

[(-p]

0.17167 666

0.09857 27987

[(-64)2]

Table 9.1 BESSEL FUNCTIONS-AUXILIARY z

f2 ( 4

.fl(X)

TABLE FOR SMALL ARGUMENTS X

f 2 (4

f1( 4

0. 0 0.1 0. 2 0. 3 0.4

-0.07380 -0.07202 -0.06672 -0.05794 -0.04579

430 984 574 956 663

-0.63661 -0.63857 -0.64437 -0.65382 -0.66660

977 491 529 684 964

1. 0 1.1 1.2 1.3 1.4

0.08825 0.11849 0.15018 0.18296 0.21647

696 917 546 470 200

-0.78221 -0.79936 -0.81476 -0,82642 -0.83332

282 142 705 473 875

0. 5 0.6 0.7 0. 8 0.9

-0,03039 -0.01192 +0.00942 0.03341 0.05979

904 435 612 927 263

-0.68228 -0.70029 -0.71998 -0.74059 -0.76130

315 342 221 789 792

1.5 1.6 1.7 1.8 1.9

0,25033 0.28416 0,31758 0.35020 0.38166

233 437 436 995 415

-0.83449 -0.82895 -0,81583 -0.79427 -0.76356

074 780 036 978 508

-0.78121 282

2.0

0.41157 912

1.0

0.08825 696

-0.72304 896

(-311

(-415 2

Yo(s)=fi(z)+,Jo(x)

In z

1

2

yi(x)=-fz(z)+;J~(z) .2:

In x

398

BESSEL FUNCTIONS O F INTEGER ORDER

BESSEL FUNCTIONS-ORDERS

Table 9.2 X

0.0

J3

(4

0.0000

J4

(4

0.0000

J5

(4

0.0000

3-9 J7 (r ) 0.0000

J6

0.0000

-11 1.9816

- 91 2.5270 - 8 4.2907 - 713.1864

-

J g (z)

Jb(C)

lo*oOoo i

-13 -11 -9 8

I-

0.0000 -15 2.7530 -121 1.4053 -11 5.3755 -1017.1092

2.4774 6.3210 1.6110 1.5967

2.0 2.2 2.4 2.6 2.8

0.12894 Oil6233 0.19811 0.23529 0.27270

3.0 3.2 3.4 3.6 3.8

0.30906 0.34307 0.37339 0.39876 0.41803

0.13203 0.15972 0.18920 0.21980 0.25074

4.0 4.2 4.4 4.6 4.8

0.43017 0.43439 0.43013 0.41707 0.39521

0.28113 0.31003 0.3 3645 0.35941 0.37796

0.13209 0.15614 0.18160 0.20799 0.23473

5.0 5.2 5.4 5.6 5.8

0.36483 0.32652 0.28113 0.22978 0.17382

0.39123 0.39847 0.39906 0.39257 0.37877

0.26114 0.28651 0.31007 0.33103 0.34862

0.13105 0.15252 0.17515 0.19856 0.22230

6.0 6.2 6.4 6.6 6.8

0.11477 +0,05428 -0.00591 - 0,06406 -0.11847

0.35764 0.32941 0.29453 0.25368 0.20774

0.36209 0.37077 0.37408 0.37155 0.36288

0.24584 0.26860 0.28996 0.30928 0.32590

7.0 7.2 7.4 7.6 7.8

-0.16756 -0.20987 -0.24420 -0.26958 -0.28535

0.15780 0.10509 +0.05097 -0.00313 -0,05572

0.34790 0.32663 0.29930 0.26629 0.22820

0.33920 0.34857 0.35349 0.35351 0.34828

0.23358 0.25432 0.27393 0.29188 0.30762

8.0 8.2 8.4 8.6 8.8

-0.29113 -0.28692 -0.27302 -0.25005 -0.21896

-0.10536 -0.15065 -0.19033 -0.22326 -0.24854

0.18577 0.13994 0.09175 +0.04237 -0.00699

0.33758 0.32131 0.29956 0.27253 0.24060

0.32059 0.33027 0.33619 0.33790 0.33508

0.22345 0.24257 0.26075 0.27755 0.29248

0.12632 0.14303 0.16049 0.17847 0.19670

9.0 9.2 9.4 9.6 9.8

-0.18094 -0.13740 -0.08997 -0.04034 +0,00970

-0.26547 -0.27362 -0.27284 -0.26326 -0.24528

-0.05504 -0.10053 -0.14224 -0.17904 -0.20993

0.20432 0.16435 0.12152 0.07676 +0.03107

0.32746 0.31490 0.29737 0.27499 0.24797

0.30507 0.31484 0.32138 0.32427 0.32318

0.21488 0.23266 0.24965 0.26546 0.27967

10.0

0.05838

-0.21960

-0.23406

-0.01446

0.21671

0.31785

0.29186

4)1.7494

5.3376 I: 221 6.5447

Compiled from British Association for the Advancement of Science, Bessel functions, Part 11. Functions of positive integer order, Mathematical Tables, vol. X (Cambridge Univ. Press, Cambridge, England, 1952) and Mathematical Tables Project, Table of jn(.c) = n ! ( + z ) - n J n ( x ) . J. Math. Phys. 23, 45-60 (1944) (with permission).

399

BESSEL FUNCTIONS OF INTEGER ORDER

Table 9.2

BESSEL FUNCTIONS-ORDERS 3-9 5

0.0

Y3(2)

Ys(4

Y4(5)

-=

y6(5)

YdX)

Y7(5)

-ca

-ca

5 -7.6586 131111097 4 -1.9162 2 -2.4302 1 -7.8751

3 -3.2156 2 -7.7670

157/16-3.8274 :0163 4 -5.3351 3 -9.6300

171-1.8025 9 -2.2957 6 -1.0638 5 -1.4367

( 7)-4.9949

I 431 -3.0589 -8.6964

1.0 1.2 1.4 1.6 1.8

-5.8215 -3.5899 -214420 -1.7897 -1.3896

(11 -3.3278 1 -1.6686 -9.4432 -5.8564 -3.9059

121 -2.6041 2 -1.0765 1 -5.1519 1 -2.7492 (1)-1.5970

2.0 2.2 2.4 2.6 2.8

-1.1278 -0.94591 -0.81161 -0.70596 -0.61736

-2.7659 -2 -0603 -1.6 024 -1.2927 -1.0752

-9.9360 -6.5462 -4.5296 -3.2716 -2.4548

3.0 3.2 3.4 3.6 3.8

-0.53854 -0.46491 -0.39363 -0.32310 -0.25259

-0.91668 -0.79635 -0.70092 -0.62156 -0.55227

-1.9059 -1.5260 -1.2556 -1.0581 -0.91009

-5.4365 -3.9723 -2.9920 -2.3177 -1.8427

-9.3044 -6.6677 -4.9 090

4.0 4.2 4.4 4.6 4.8

-0.18202 -0.11183 -0.04278 +0.02406 0.08751

-0.48894 -0.42875 -0.36985 -0.31109 -0.25190

-0.79585 -0.70484 -0.62967 -0.56509 -0.50735

-1.5007 -1.2494 -1.0612 -0.91737 -0.80507

-3.7062 -2.8650 -2.2645 -1.8281 -1.5053

5.0 5.2 5.4 5.6 5.8

0.14627 0.19905 0.24463 0.28192 0.31001

-0.19214 -0.13204 -0.07211 -0.01310 +0.04407

-0.45369 -0.40218 -0.35146 -0.30063 -0.24922

-0.71525 -0.64139 -0.57874 -0.52375 -0.47377

-1.2629 -1.0780 -0.93462 -0.82168 -0.73099

-2.8209 -2.2608 -1.8444 -1.5304 -1.2 907

-7.7639 -5.8783 -4.5 302 -3.5510 -2.8295

6.0 6.2 6.4 6.6 6.8

0.32825 0.33622 0.33383 0.32128 0.29909

0.09839 0.14877 0.19413 0.23344 0.26576

-0.19706 -0.14426 -0.09117 0.03833 +0.01357

-0.42683 -0.38145 -0.33658 -0.29151 -0.24581

-0.65659 -0.59403 -0.53992 -0.49169 -0.44735

-1.1052 -0.95990 -0.844 50 -0.75147 -0.67521

-2.2907 -1.8831 -1.5713 -1.3301 -1.1414

7.0 7.2 7.4 7.6 7.8

0.26808 0.22934 0.18420 0.13421 0.08106

0.29031 0.30647 0.31385 0.31228 0.30186

0.06370 0.11119 0.15509 0.19450 0.22854

-0.19931 -0.1 5204 -0.10426 -0.05635 -0.00886

-0.4 0537 -0.36459 -0.32416 -0.28348 -0.24217

-0.61144 -0.55689 -0.50902 -0.46585 -0.42581

-0.99220 -0.87293 -0.77643 -0.69726 -0.63128

8.0 8.2 8.4 8.6 8.8

+0.02654 -0.02753 -0.07935 -0.12723 -0.16959

0.28294 0.25613 0.22228 0.18244 0.13789

0.25640 0127741 0.29104 0.29694 0.29495

+0.03756 0.08218 0.12420 0116284 0.19728

-0.20006 -0.15716 -0.11361 -0706973 -0.02593

-0.38767 -0.35049 -0.31355 -0.27635 -0.23853

-0.57528 -0.52673 -0.48363 -0.44440 -0.40777

9.0 9.2 9.4 9.6 9.8

-0.20509 -0I23262 -0.25136 -0.26079 -0.26074

0.09003 +0.04037 -0.00951 -0.05804 -0.10366

0.28512 0.26773 0.24326 0.21243 0.17612

0.22677 0.25064 0.26830 0.27932 0.28338

+0.01724 0.05920 0.09925 0.13672 0.17087

-0.19995 -0.16056 -0.12048 -0.07994 -0.03928

-0.37271 -0.33843 -0.30433 -0.26995 -0.23499

10.0

-0.25136

-0.14495

0.13540

0.2 8035

0.20102

+0.00108

-0.19930

-

I 321 -1.8539 -8.9196 . I

-7.6918

( 1)-1.1471

-8.3005 -6.1442 -4.6463 -3.5855

400

BESSEL FUNCTIONS O F INTEGER ORDER

BESSEL FUNCTIONS-ORDERS

Table 9.2

(4

J~(x)

(4

3-9

Js (z)

10.0 10.2 10.4 10.6 10.8

0.05838 0.10400 0.14497 0.17992 0.20768

-0.21960 -0.18715 -0.14906 -0.10669 -0.06150

-0.23406 -0.25078 -0.25964 -0.26044 -0.25323

-0.01446 -0.05871 -0; 10059 -0.13901 -0.17297

0.21671 0.18170 0.14358 0.10308 0.06104

0.31785 0.30811 0.29386 0.27515 0.25210

Jg(x) 0.29186 0.30161 0.30852 0.31224 0.31244

11.0 11.2 11.4 11.6 11.8

0.22735 0.23835 0,24041 0.23359 0,21827

-0.01504 +O. 03110 0.07534 0.11621 0.15232

-0.23829 -0.21614 -0.18754 -0.15345 -0.11500

-0.20158 -0.22408 -0.23985 -0.24849 -0.24978

+O. 01838 -0.02395 -0.06494 -0.10361 -0.13901

0.22497 0.19414 0.16010 0.12344 0.08485

0.30886 0.30130 0.28964 0.27388 0.25407

12.0 12.2 12.4 12.6 12.8

0.19514 0,16515 0,12951 0.08963 0,04702

0.18250 0.20576 0.22138 0.22890 0.22815

-0.07347 -0.03023 +O. 01331 0: 05571 0.09557

-0.24372 -0.23053 -0.21064 -0.18469 -0.15349

-0.17025 -0.19653 -0.21716 -0.23160 -0.23947

0.04510 +O. 00501 -0.03453 -0.07264 -0.10843

0.23038 0.20310 0.17260 0.13935 0.10393

13.0 13.2 13.4 13.6 13.8

+O. 00332 -0.03984 -0.08085 -0.11822 -0.15059

0.21928 0.20268 0.17905 0; 14931 0.11460

0.13162 0.16267 0.18774 0.20605 0.21702

-0.11803 -0.07944 -0.03894 +o. 00220 0. 04266

-0.24057 -0.23489 -0.22261 -0.20411 -0.17993

-0.14105 -0.16969 -0.19364 -0.21231 -0.22520

0.06698 +O. 02921 -0.00860 -0.04567 -0.08117

14.0 14.2 14.4 14.6 14.8

-0.17681 -0.19598 -0.20747 -0.21094 -0.20637

0.07624 +O. 03566 -0.00566 -0.04620 -0.08450

0.22038 0.21607 0.20433 0.18563 0.16069

0.08117 0.11650 0.14756 0.17335 0.19308

-0.15080 -0.11762 -0.08136 -0.04315 -0.00415

-0.23197 -0.23246 -0.22666 -0.21472 -0.19700

-0.11431 -0.14432 -0.17048 -0.19216 -0.20883

15.0 15.2 15.4 15.6 15.8

-0.19402 -0.17445 -0.14850

11723 --g..08188

-0.11918 -0.14901 -0.17296 -0.19021 -0.20020

0.13046 0.09603 0.05865 +O. 01968 -0.01949

0.20615 0.21219 0.21105 0.20283 0.18787

+O. 03446 0.07149 0.10580 0.13634 0.16217

-0.17398 -0.14634 -0.11487 -0.08047 -0.04417

-0.22005 -0.22553 -0.22514 -0.21888 -0.20690

16.0 16.2 16.4 16.6 16.8

-0.04385 -0.00461 +O. 03432 0.07146 0,10542

-0.20264 -0.19752 -0.18511 -0.16596 -0.14083

-0.05747 -0.09293 -0.12462 -0.15144 -0.17248

0.16672 0.14016 0.10913 0.07473 0.03817

0.18251 0.19675 0: 20447 0.20546 0.19974

-0.00702 +O. 02987 0.06542 0.09855 0.12829

-0.18953 -0.16725 -0.14065 -0.11047 -0.07756

17.0 17.2 17.4 17.6 17.8

0,13493 0.15891 0.17651 0.16712 0.19041

-0.11074 -0.07685 -0.34048 -0.00300 +O. 03417

-0.18704 -0.19466 -0.19512 -0.18848 -0.17505

+O. 00072 -0.03632 -0.07166 -0.10410 -0.13251

0.18755 0.16932 0.14570 0.11751 0.08571

0.15374 0.17414 0.18889 0.19757 0.19993

-0.04286 -0.00733 +O. 02799 0.06210 0.09400

18.0 18.2 18.4 18.6 18.8

0.18632 0.17510 0.15724 0,13351 0.10487

0.06964 0.10209 0.13033 0.15334 0.17031

-0.15537 -0.13022 -0.10058 -0.06756 -0.03240

-0.15596 -0.17364 -0.18499 -0.18966 -0.18755

0.05140 +O. 01573 -0.02007 -0.05481 -0.08731

0.19593 0.18574 0.16972 0;14841 0.12253

0.12276 0.14756 0.16766 0.18247 0.19159

19.0 19.2 19.4 19.6 19.8

0.07249 0,03764 +O. 00170 -0.03395 -0.06791

0.18065 0.18403 0.18039 0: 16994 0.15313

+O. 00357 0.03904 0.07269

-0.17877 -0.16370 -0.14292 -0.11723 -0.08759

-0.11648 -0.14135 -0.16110 -0.17508 -0.18287

0.09294 0.06063 +O. 02667 -0.00783 -0.04171

0.19474 0.19187 0.18309 0.16869 0.14916

20.0

-0.09890

0.13067

-0.05509

-0.18422

-0.07387

0.12513

X

J3

['-:"I

J5

J&)

[(-:)81

i

401

BESSEL FUNCTIONS O F INTEGER ORDER

BESSEL FUNCTIONS4RDERS 3-9

Y3(x)

YS(x)

Table 9.2

Y7b)

Y40)

Ys(x)

10.0 10.2 10.4 10.6 10.8

-0.25136 -0.23314 -0.20686 -0.17359 -0.13463

-0.14495 -0.18061 -0.20954 -0.23087 -0.24397

0.13540 0.09148 +O. 04567 -0.00065 -0.04609

0.28035 0.27030 0.25346 0.23025 0.20130

0.20102 0,22652 0,24678 0.26131 0.26975

0; 04061 0.07874 0.11488 0.14838

YQ@) -0.19930 -0.16282 -0.12563 -0. 08791 -0.04993

11.0 11.2 11.4 11.6 11.8

-0.09148 -0.04577 +O. 00082 0.04657 0.08981

-0.24851 -0.24445 -0.23203 -0.21178 -0.18450

-0.08925 -0.12884 -0.16365 -0.19262 -0.21489

0.16737 0.12941 0.08848 0.04573 +O. 00238

0.27184 0.26750 0.25678 0.23992 0.21732

0.17861 0; 20496 0.22687 0.24384 0.25545

-0.01205 +O. 02530 0.06163 0.09640 0.12906

12.0 12.2 12.4 12.6 12.8

0.12901 0.16277 0.18994 0,20959 0,22112

-0.15122 -0.11317 -0.07175 -0.02845 +O. 01518

-0.22982 -0.23698 -0.23623 -0.22766 -0.21163

-0.04030 -0.08107 -0.11875 -0.15223 -0.18052

0.18952 0.15724 0.12130 0.08268 0.04240

0.26140 0.26151 0.25571 0.24409 0.22689

0.15902 0.18573 0.20865 0.22728 0.24122

13.0 13.2 13.4 13.6 13.8

0.22420 0.21883 0,20534 0.18432 0,15666

0.05759 0.09729 0.13289 0.16318 0.18712

-0.18876 -0.15987 -0.12600 -0.08833 -0.04819

-0.20279 -0.21840 -0.22692 -0.22813 -0.22204

+O. 00157

0.20448 0.17738 0.14625 0.11185 0.07505

0.25010 0.25369 0.25184 0.24454 0.23190

14.0 14.2 14.4 14.6 14.8

0,12350 0,08615 0.04605 +O. 00477 -0.03613

0.20393 0.21308 0.21434 0.20775 0.19364

-0.00697 +O. 03390 0.07303 0.10907 0.14080

-0.20891 -0.18921 -0.16363 -0.13305 -0.09850

-0.17209 -0.19380 -0.20939 -0.21842 -0.22067

+O. 03682 -0.00186 -0. 03994 -0.07640 -0.11024

0.21417 0.19170 0.16501 0.13470 0.10149

15.0 15.2 15.4 15.6 15.8

-0. 07511 -0.11072 -0.14165 -0.16678 -0.18523

0.17261 0.14550 0.11339 0.07750 +O. 03920

0.16717 0.18730 0.20055 0.20652 0.20507

-0.06116 -0.02228 +O. 01684 0.05489 0.09059

-0.21610 -0.20489 -0.18743 -0.16430 -0.13627

-0.14053 -0.16644 -0.18723 -0.20234 -0.21134

0.06620 +O. 02969 -0.00710 -0.04322 -0.07775

16.0 16.2 16.4 16.6 16.8

-0.19637 -0.19986 -0.19566 -0.18402 -0.16547

-0.00007 -0.03885 -0.07571 -0.10930 -0.13841

0.19633 0.18067 0.15873 0.13135 0.09956

0.12278 0.15038 0.17250 0.18843 0.19767

-0.10425 -0.06928 -0.03251 +O. 00487 0.04164

-0.21399 -0.21025 -0.20025 -0.18432 -0.16297

-0.10975 -0.13838 -0.16286 -0.18253 -0.19685

17.0 17.2 17.4 17.6 17.8

-0.14078 -0.11098 -0.07725 -0.04094 -0.00347

-0.16200 -0.17924 -0.18956 -0.19265 -0.18846

0.06455 +O. 02761 -0.00990 -0.04663 -0.08123

0.19996 0.19529 0.18387 0.16616 0.14282

0.07660 0.10864 0.13671 0.15991 0.17752

-0.13688 -0,10686 -0.07387 -0.03895 -0.00320

-0.20543 -0.20805 -0.20464 -0.19533 -0.18039

18.0 18.2 18.4 18.6 18.8

+O. 03372 0.06920 0.10163 0,12977 0.15261

-0.17722 -0.15942 -0.13580 -0.10731 -0.07506

-0.11249 -0.13928 -0.16067 -0.17593 -0,18455

0.11472 0.08289 0.04848 +O. 01272 -0.02310

0.18897 0.19393 0.19229 0.18414 0.16980

+O. 03225 0.06629 0.09782 0.12587 0; 14955

-0.16030 -0.13566 -0.10722 -0.07586 -0.04252

19.0 19.2 19.4 19.6 19.8

0.16930 0,17927 0,18221 0,17805 0.16705

-0.04031 -0.00440 +O. 03131 0.06546 0.09678

-0.18628 -0,18111 -0.16930 -0.15134 -0.12794

-0.05773 -0.08993 -0.11857 -0.14267 -0.16139

0.14982 0.12490 0.09595 0.06399 +O. 03013

0.16812 0.18100 0; 18782 0.18838 0.18270

-0.00824 +O. 02593 0.05895 0. 08979 0.11750

20.0

0.14967

0,12409

X

-0.03868 -0.07722 -0.11296 -0.14489

0.17101

]8I:-([

402

BESSEL FUNCTIONS OF INTEaER ORDER

BESSEL FUNCTIONS-ORDERS

Table 9.3 z

1 0 , 1 1 , 2 0 AND 21

1O~oz-loJ10(z) 1011~-11J11(~) ~O-~Z~OYIO(~) 1025~-20~2~(z)1027~-21~2~(z)10-~3z~0~~0(z)

0.0 0.1 0.2 0.3 0.4

2.69114 2.69053 2.68869 2.68564 2.68137

446 290 898 500 477

1.22324 1.22299 1.22222 1.22095 1.21917

748 266 850 588 626

-0.11828 -0.11831 -0.11841 -0.11857 -0.11880

049 335 200 661 750

3.91990 3.91944 3.91804 3.91571 3.91244

9.33311 9.33205 9.32886 9.32357 9.31615

-0.406017 -0.406071 -0.406231 -0.406499 -0.406873

0.5 0.6 0.7 0.8 0.9

2.67589 2.66920 2.66132 2.65226 2.64201

362 838 738 043 878

1.21689 1.21410 1.21081 1.20703 1.20276

169 481 883 750 518

-0.11910 -0.11946 -0.11990 -0.12040 -0.12097

510 998 282 444 581

3.90825 3.90314 3.89710 3.89015 3; 88228

9.30663 9.29500 9.28128 9.26546 9.24758

-0.407355 -0.407945 -0.408644 -0.409452 -0.410369

1.0 1.1 1.2 1.3 1. 4

2.63061 2.61806 2.60437 2.58958 2.57368

512 358 963 012 323

1.19800 1.19276 1.18705 1.18087 1.17422

675 764 385 185 867

-0.12161 -0.12233 -0.12312 -0.12398 -0.12492

801 229 002 273 212

3.87350 3.86383 3.85325 3.84179 3.82945

9.22762 9.20562 9.18157 9.15550 9.12743

-0.411397 -0.412536 -0.413788 -0.415153 -0.416632

1.5 1.6 1.7 1.8 1.9

2.55670 2.53867 2.51960 2.49952 2.47846

842 639 907 955 207

1.16713 1.15958 1.15160 1.14320 1.13437

182 931 961 168 488

-0.12594 -0.12703 -0.12821 -0.12948 -0.13084

004 852 977 616 030

3.81624 3.80216 3.78723 3.77146 3.75485

9.09737 9.06534 9.03137 8.99546 8.95766

-0.418228 -0.419940 -0.421771 -0.423722 -0.425795

2.0 2.1 2.2 2.3 2.4

2.45643 2.43346 2.40959 2.38483 2.35922

192 545 000 384 612

1.12513 1.11550 1.10548 1.09508 1,08431

904 438 152 144 551

-0.13228 -0.13382 -0.13545 -0.13719 -0.13903

497 319 821 351 284

3.73742 3.71918 3.70015 3.68032 3.65973

8.91797 8.87643 8.83306 8.78790 8.74096

-0.427992 -0.430315 -0.432764 -0.435344 -0.438056

2.5 2.6 2.7 2.8 2.9

2.33279 2.30557 2.27759 2.24889 2.21948

682 673 732 074 976

1.07319 1.06173 1.04994 1.03783 1.02541

540 312 098 155 767

-0.14098 -0.14303 -0.14521 -0.14751 -0.14994

022 997 672 543 141

3.63837 3.61627 3.59344 3.56989 3.54564

8.69228 8.64189 8.58981 8.53609 8.48076

-0.440902 -0.443885 -0.447007 -0.450272 -0.453682

3.0 3.1 3.2 3.3 3.4

2.18942 2.15873 2.12745 2.09561 2.06325

770 836 598 517 085

1.01271 0.99972 0.98648 0.97298 0.95924

242 906 108 213 599

-0.15250 -0.15519 -0.15804 -0.16103 -0.16419

037 840 206 836 482

3.52071 3.49510 3.46885 3.44195 3.41444

8.42385 8.36539 8.30542 8.24397 8.18110

-0.457241 -0.460951 -0.464816 -0.468840 -0.473027

3.5 3.6 3. 7 3.8 3.9

2.03039 1.99709 1.96336 1.92926 1.89481

820 260 956 467 352

0.94528 0.93111 0.91675 0.90220 0.88749

659 794 415 939 785

-0.16751 -0.17102 -0.17470 -0.17859 -0.18268

951 110 889 286 376

3.38633 3.35763 3.32837 3.29855 3.26821

8.11682 8.05119 7.98424 7.91600 7.84653

-0.477379 -0.481902 -0.486600 -0.491476 -0.496537

4.0 4.1 4.2 4.3 4.4

1.86005 1.82501 1.78973 1.75425 1.71860

168 462 765 588 416

0.87263 0.85763 0.84250 0.82726 0.81193

375 130 469 806 548

-0.18699 -0.19153 -0.19631 -0.20136 -0.20667

314 346 812 159 950

3.23736 3.20601 3.17419 3.14192 3.10921

7.77586 7.70403 7.63108 7.55707 7.48202

-0.501786 -0.507229 -0.512872 -0.518719 -0.524777

4.5 4.6 4.7 4.8 4.9

1.68281 1.64692 1.61097 1.57498 1.53899

701 860 267 249 084

0.79652 0.78103 0.76550 0.74992 0.73431

093 829 130 357 852

-0.21228 -0.21820 -0.22445 -0.23105 -0.23802

873 757 582 498 840

3.07608 3.04256 3.00866 2.97440 2.93981

7.40598 7.32900 7.25112 7.17238 7.09282

-0.531051 -0.537549 -0.544276 -0.551240 -0.558448

5.0

1.50302 991

-0.24540 147

2.90490

7.01250

-0.565907

0.71869 942

(-4)1 2n =7Yn(z) -Yn-1(.) Compiled from British Association for the Advancement of Science, Bessel functions, Part 11. Functions of positive integer order, Mathematical Tables, vol. X (Cambridge Univ. Press, Cambridge, England, 1952),L. Fox, A short table for Bessel functions of integer orders and large arguments. Royal Society Shorter Mathematical Tables No. 3 (Cambridge Univ. Press, Cambridge, England, 1954),and Mathematical Tables Project, Table of fn(s) =n!(;z)-%Tn(r). J. Math. Phys. 23, 45-60 (1944) (with permission). 2n

Jn+l (z) = T J n ( z ) -Jn-1

(z)

L + l (z)

403

BESSEL FUNCTIONS OF INTEGER ORDER

BESSEL FUNCTIONS-ORDERS 10, 11, 20 AND 21 5.0 5.1 5.2 5.3 5.4

1.50302 1.46713 1.43132 1. 39564 1.36011

991 132 603 431 571

0.71869 0.70307 0.68747 0.67188 0,65634

942 931 104 722 019

10-9210 yl0(z) -0.24540 147 -0.25320 186 -0.26145 975 -0.27020 813 -0.27948 304

5.5 5.6 5.7 5.8 5.9

1.32476 1.28963 1.25473 1.22009 1.18574

904 229 264 642 907

0.64084 0.62540 0.61003 0.59475 0,57957

205 463 945 774 041

-0.28932 -0,29977 -0,31088 -0.32269 -0.33528

6. 0 6.1 6.2 6.3 6.4

1.15171 1.11801 1. 08468 1.05172 1.01917

513 822 098 510 129

0.56448 0.54952 0.53467 0.51997 0.50540

805 091 890 158 814

6.5 6.6 6.7 6.8 6.9

0.98703 0.95534 0.92411 0.89335 0.86308

926 769 427 563 740

0,49099 0.47674 0.46266 0.44876 0,43504

7.0 7.1 7.2 7.3 7.4

0.83332 0.80407 0.77536 0.74719 0,71957

414 941 570 450 626

7.5 7.6 7.7 7.8 7.9

0,69252 0.66603 0,64012 0.61480 0,59007

8. 0 8.1 8.2 8.3 8.4

z

10'0z-'0J10(z)

10"z-

"Jll(2)

1025z-2OJ20(2)

Table 9.3

1 0 2 7 ~ - 2 1J~~(2) 1 0 - 2 3 ~ 2 0 yzO(s)

2.90490 2.86969 2.83421 2.79846 2.76248

7. 01250 6.93145 6.84971 6.76734 6.68437

-0.565907 -0; 573626 -0.581612 -0.589875 -0.598423

400 431 154 795 105

2.72628 2.68988 2.65330 2.61656 2.57967

6.60085 6.51682 6.43233 6.34742 6.26213

-0.607266 -0.616414 -0.625876 -0.635663 -0.645788

-0.34869 -0,36300 -0.37829 -0.39464 -0.41215

413 693 631 698 232

2.54267 2.50556 2.46837 2.43111 2.39381

6.17651 6. 09059 6. 00443 5.91806 5.83152

-0.656261 -0.667094 -0.678301 -0.689895 -0.701890

740 781 745 400 477

-0.43091 -0.45104 -0.47267 -0.49594 -0.52098

524 907 855 084 648

2.35647 2.31913 2.28179 2.24448 2.20721

5.74485 5.65810 5.57131 5.48451 5.39775

-0.714300 -0.727140 -0.740427 -0.754178 -0.768410

0.42151 0.40818 0.39505 0.38214 0.36943

665 616 943 216 970

-0.54798 -0.57710 -0.60855 -0.64254 -0,67930

051 346 234 159 390

2.17000 2.13286 2.09582 2.05888 2.02206

5.31106 5.22448 5.13805 5.05181 4.96579

-0.783140 -0.798389 -0.814177 -0.830524 -0.847452

040 536 854 640 439

0.35695 0.34469 0,33266 0.32087 0,30930

696 850 845 058 826

-0,71909 -0.76217 -0.80884 -0.85940 -0.91419

088 356 258 807 914

1.98539 1.94887 1; 91252 1.87635 1.84038

4.88002 4.79455 4.70940 4.62461 4.54021

-0.864985 -0.883147 -0.901963 -0.921460 -0.941665

0.56593 0.54239 0.51945 0.49712 0,47539

704 791 967 408 201

0.29798 0.28690 0.27606 0.26546 0.25512

448 187 265 873 162

-0.97356 -1.03786 -1.10747 -1.18278 -1.26419

279 231 485 826 685

1.80462 1.76908 1; 73378 1.69874 1. 66395

4.45624 4.37272 4.28968 4.20716 4.12518

-0.962608 -0.984319 -1.006831 -1.030178 -1.054394

8.5 8.6 8.7 8.8 8.9

0.45426 0.43373 0.41381 0.39448 0.37575

352 779 323 748 740

0.24502 0,23517 0.22557 0.21621 0.20711

250 220 121 969 750

-1.35209 -1.44687 -1.54891 -1.65856 -1.77613

608 598 312 097 854

1.62944 1.59521 1.56128 1.52765 1.49434

4.04377 3.96296 3.88277 3.80323 3.72436

-1. 079518 -1.105589 -1.132647 -1.160736 -1.189902

9.0 9.1 9.2 9.3 9.4

0.35761 0,34006 0.32309 0,30670 0.29088

917 823 939 683 411

0.19826 0.18965 0.18130 0.17318 0.16532

418 897 082 839 010

-1.90191 -2.03610 -2.17882 -2.33011 -2.48986

706 452 801 366 396

1.46136 1.42872 1.39641 1.36447 1.33288

3.64619 3.56873 3.49201 ._ 3.41606 3.34088

-1.220192 -1.251657 -1.284351 -1; 318328 -1.353647

9.5 9.6 9.7 9.8 9.9

0.27562 0,26091 0.24676 0.23314 0.22005

422 963 227 362 470

0.15769 0.15030 0.14316 0.13624 0.12956

409 825 025 751 726

-2.65783 -2.83359 -3.01652 -3.20574 -3.40010

251 602 353 283 421

1.30166 1.27082 1.24036 1.21029 1.18061

3.26651 3.19294 3.12022 3.04834 2.97733

-1.390372 -1.428567 -1.468301 -1.509646 -1.552680

-3.59814 152

1.15134

2.90720

-1.597484

10.0

0.20748 611 [(

0.12311 653

-;PI

[( -;)31

[(-;U]

[( - p 5 1

404

BESSEL FUNCTIONS O F INTEGER ORDER BESSEL FUNCTIONS--ORDERS 10, 11, 20 AND 21

Table 9.3 X

Jio(x)

Jii(x)

YlO(X)

1 0 2 h - W a o ( x ) 10272-2' Jzl(2) 10- * 3 z 2 0 Y z 0 ( ~ )

-

10.0 10.1 10.2 10.3 10. 4

0.20748 0.21587 0.22413 0.23223 0.24011

611 417 707 256 699

0.12311 0.13041 0.13787 0.14549 0.15324

653 285 866 509 123

-0.35981 -0.34383 -0.32793 -0.31207 -0.29618

415 078 809 433 615

1.151337 1.122469 1.094012 1.065970 1.038347

2.907199 2.837961 2.769629 2.702215 2.635729

1.59748 1.64414 1.69275 - 1.74339 - 1.79618

10. 5 10. 6 10.7 10. 8 10. 9

0.24774 0.25507 0.26205 0.26863 0.27477

554 240 109 466 603

0.16109 0.16902 0.17701 0.18503 0.19304

407 861 780 266 230

-0.28022 -0.26416 -0.24795 -0.23159 -0.21505

819 276 949 513 324

1.011148 0.984374 0.958030 0.932118 0.906639

2.570182 2.505582 2.441939 2.379259 2.317550

-

11.0 11.1 11.2 11.3 11.4

0.28042 0.28554 0.29007 0.29398 0.29722

823 479 999 925 944

0.20101 0.20891 0.21670 0.22434 0.23181

401 340 446 974 048

-0.19832 -0.18140 -0.16429 -0.14700 -0.12955

403 409 620 917 753

0.881596 0.856989 0.832821 0.809092 0.785801

2.256817 21 197065 2.138299 2.080523 2.023738

- 2.16430

11.5 11. 6 11.7 11.8 11.9

0.29975 0.30153 0.30253 0,30270 0.30203

923 946 345 737 061

0.23904 0.24601 0.25268 0.25899 0.26492

680 789 218 761 183

-0.11196 -0.09424 -0.07644 -0.05858 -0.04071

142 628 263 580 566

0.762950 0.740539 0.718565 0.697029 0.675930

1.967947 1.913152 1.859352 1: 806548 1.754740

12. 0 12.1 12.2 12. 3 12.4

0.30047 0.29802 0.29464 0.29033 0.28507

604 036 445 357 771

0.27041 0.27542 0.27992 OI28386 0.28720

248 744 508 459 623

-0.02287 -0.00511 +O. 01251 0.02995 0.04716

631 577 441 946 182

0.655266 0.635035 0.615236 0.595866 0.576923

1.703925 1.654102 1.605267 1.557418 1.510551

12.5 12. 6 12. 7 12. 8 12.9

0.27887 0.27171 0.26361 0.25458 0.24462

175 575 509 064 889

0.28991 0.29194 0.29326 0.29385 0.29366

166 422 923 431 968

0.06406 0.08059 0.09670 0.11231 0.12737

154 668 381 845 554

0.558403 0.540305 0.522625 0.505359 0.488504

1.464660 1.419743 1.375791 1.332800 1.290762

13. 0 13.1 13.2 13. 3 13. 4

0.23378 0.22206 0.20952 0.19617 0.18208

201 793 032 859 776

0.29268 0.29088 0.28824 0.28474 0.28037

843 684 464 526 612

0.14180 0.15555 0.16855 0.18073 0.19204

995 698 286 529 392

0.472056 0.456011 0.440365 0.425114 0.410252

1.249671 1.209520 1.170299 1.132001 1.094617

13. 5 13. 6 13. 7 13.8 13. 9

0.16729 0.15186 0.13585 0.11932 0.10235

840 646 302 411 036

0.27512 0.26899 0.26198 0.25410 0.24534

884 942 851 149 866

0.20242 0.21181 0.22016 0.22743 0.23357

090 137 393 118 014

0.395776 0.381681 0.367961 0.354612 0.341628

1.058137 1.022552 0.987853 0.954028 0.921067

- 2.38746 - 2.46870 - 2.55367 -- 2.64257 2.73563 - 2.83307 - 2.93513 - 3.04208 - 3.15419 - 3.27175 - 3.39509 - 3.52453 - 3.66044 - 3.80321 - 3.95323 - 4.11095 - 4.27684 - 4.45140 - 4.63518 - 4.82874 - 5.03272 - 5.24778 - 5.47464 - 5.71407 -- 5.96691 6.23405 - 6.51646

14. 0 0.08500 14.1 0.06737 14. 2 0.04952 14. 3 0.03156 14. 4 +0.01356

671 200 862 199 013

0.23574 0.22531 0.21407 0.20206 0.18931

535 197 407 238 275

0.23854 0.24231 0.24486 0.24616 0.24620

273 614 329 313 100

0.329005 0.316736 0.304816 0.293240 0.282001

0.888960 0.857694 0.827260 0.797644 0.768835

--

14.5 14. 6 14. 7 14. 8 14.9

689 745 898 854 344

0.17586 0.16176 0.14707 0.13182 0.11609

611 836 028 729 931

0.24496 0.24246 0.23869 0.23367 0.22742

888 568 741 730 597

0.271095 0.260516 0.250257 0.240312 0.230676

0.740821 0.713590 0.687129 0.661426 0.636467

-

0.21997 141

0.221343

0.612240

-11.00239

-0.00438 -0.02218 -0.03974 -0.05697 -0.07378

15. 0 -0.09007 181

['-?'I

0.09995 048

[( -2'61

1 [' I:-

1

1.85121 1.90861 1.96848 2.03097 2.09619

-- 2.23544 2.30977

- 7.13138 6.81520 7.46624 7.82110 8.19739

- 8.59667 9.02062

9.47109 9.95006 -10.45971

405

BESSEL FUNCTIONS OF INTEGER ORDER

Table 9.3

BESSEL FUNCTIONS-ORDERS 10, 11, 20 AND 21 X

Jio(x)

Jii(x)

YlO(X)

~ O ~ ~ X - ' W ~ O ( 10272 X ) -21 J z

(5)

10- ' ~ x ~ O Y Z O ( Z )

-- 11.0024 11.5607 -- 12.1974 - 12.8555 13.5585

15. 0 15.1 15. 2 15. 3 15.4

-0,09007 -0.10575 -0.12073 -0.13494 -0.14826

181 330 964 535 828

0.09995 0.06344 0.06666 0.04967 0.03256

048 886 618 738 035

0.21997 0.21134 0.20160 0.19077 0.17693

141 904 159 902 834

0.22134 0.21230 0.20356 0.19510 0.16691

33 71 16 08 87

0.61224 0.58873 0.56593 0.54362 0.52239

04 25 06 12 14

15. 5 15.6 15. 7 15.8 15.9

-0.16069 -0.17207 -0,18238 -0.19154 -0.19949

032 791 269 204 958

+0.01539 -0.00173 -0.01674 -0.03555 -0.05207

539 513 731 621 632

0.16614 0.15246 0.13797 0.12276 0.10691

338 453 838 733 918

0.17900 0.17136 0.16396 0.15685 0.14997

91 62 38 60 67

0.50162 0,48151 0.46204 0.44319 0.42496

76 66 52 99 74

16. 0 16.1 16.2 16.3 16.4

-0.20620 -0.21161 -0.21570 -0.21842 -0.21978

569 797 160 977 394

-0.06622 -0.06390 -0.09905 -0.11357 -0.12738

215 874 224 046 344

0.09052 0.07368 0.05650 0.03907 0.02150

660 666 016 110 600

0.14334 0.13694 0.13077 0.12462 0.11910

00 00 08 65 14

0.40733 0.39026 0.37361 0.35769 0.34253

43 75 35 93 16

16.5 16. 6 16.7 16. 8 16.9

-0.21975 -0,21833 -0,21554 -0.21139 -0.20590

411 905 637 267 350

-0.14041 -0.15256 -0.16383 -0.17409 -0.16329

403 841 668 338 797

+0.00391 -0.01359 -0.03091 -0.04793 -0.06454

319 786 729 557 431

0,11358 0.10828 0,10318 0.09827 OIO9356

96 55 34 77 30

0,32769 0.31336 0.29957 0.26626 0.27343

75 39 78 66 76

17. 0 17.1 17.2 17.3 17. 4

-0.19911 -0.19106 -0.18181 -0.17141 -0.15993

332 538 155 203 505

-0.19139 -0.19633 -0.20407 -0.20856 -0.21182

539 646 831 485 701

-0.08063 -0.09610 -0.11086 -0.12479 -0,13782

696 960 170 683 343

0.08903 0.08468 OI0805l 0.07650 0.07266

37 45 02 53 49

0.26107 0.24917 0.23771 0.22669 0.21606

81 57 82 32 89

17.5 17. 6 17.7 17.8 17.9

-0.14745 -0.13405 -0.11983 -0.10467 -0. 08928

649 943 363 499 492

-0.21378 -0.21443 -0.21376 -0.21163 -0,20858

318 935 944 538 727

-0.14985 -0.16081 -0.17062 -0.17922 -0.16654

544 304 321 038 691

0.06898 0.06545 0.06207 0.05884 0.05575

37 69 96 68 39

0.20589 0.19609 0,18668 0.17764 0.16696

33 48 17 27 66

18. 0 18.1 18. 2 18.3 18, 4

-0.07316 -0.05663 -0.03960 -0iO2279 -0.00571

966 961 852 278 052

-0.20406 -0.19829 -0.19130 -0.18314 -0.17386

341 032 265 307 213

-0.19255 -0.19720 -0.20045 -0.20229 -0.20271

365 030 582 875 742

0.05279 0.04996 0.04726 0.04468 0.04222

63 93 85 96 83

0.16064 0.15265 0,14500 0.13767 0.13064

24 91 62 32 97

- 18.9460 - 20.0855 - 21.3104 - 22.6279 - 24.0462 - 25.5740 - 27.2209 - 28.9975 - 30.9150 - 32.9859 - 35.2237 - 37.6429 - 40.2594 - 43.0904 - 46.1543 - 49.4711 -- 53.0622 56.9506 - 61.1611 - 65.7197 - 70.6543 - 75.9946 - 81.7717 - 88.0182 - 94.7683

18. 5 18.6 18. 7 18. 8 18. 9

+0.01131 0.02817 0.04474 0.06090 0.07654

917 711 490 579 556

-0.16351 -0.15217 -0.13990 -0.12679 -0.11291

793 591 845 446 893

-0.20171 -0.19928 -0.19546 -0.19026 -0.18373

011 520 113 637 930

0.03986 0.03764 0.03550 0.03347 0.03154

04 17 84 64 21

0,12392 0,11749 0.11133 0110545 0,09982

57 14 69 28 98

-102.0574 -109.9219 -118.3992 -127.5270 -137.3432

19. 0 19.1 19.2 19.3 19.4

0.09155 0.10582 0.11925 0.13174 0.14321

333 247 134 416 168

-0,09837 -0.08325 -0.06765 -0.05168 -0.03544

240 039 283 334 863

-0.17592 -0.16688 -0,15669 -0.14540 -0.13312

797 985 143 785 231

0,02970 0.02795 0,02628 0.02470 0.02320

16 15 80 79 78

0.09445 0.06933 0,08443 0.07977 0.07532

89 10 76 01 03

-147.8850 -159.1885 -171.2882 -184.2155 -197.9980

19. 5 19.6 19. 7 19.8 19.9

0.15357 0.16275 0.17068 0.17731 0.18259

193 089 305 198 079

-0.01905 -0.00262 +0.01374 0,02994 0.04584

771 120 948 285 818

-0.11992 -0.10591 -0.09119 -0,07587 -0.06006

560 538 555 548 922

0.02176 0.02043 0.01915 0.01794 0,01679

44 46 54 37 67

0.07106 0.06704 0.06319 0.05953 0,05606

01 16 71 92 06

-212.6582 -228.2122 -244.6678 -262.0226 -280.2622

20.0

0.18646 256

0.05275 42

-299.3574

0.06135 630

-0.04369 465

0.01571 16

--

--

14.3098 15.1136 15.9742 16.8962 17.8849

BESSEL FUNCTIONS O F INTEGER ORDER

406

Table 9.3 BESSEL FUNCTIONS-MODULUS AND PHASE OF ORDERS 10, 11, 20 AND 21 Yn(r)=Hn(n(m) sin h ( r ) t T n ( ~ ) s & ( r ) COS h ( z ) h o ( x ) -T -13.94798 864 -14.05389 581 -14.15926 984 -14.26413 968 -14.36853 333

0.87222 0.86513 0.85857 0.85250 0.84689

790 271 314 587 281

-14.96758 -15.09771 -15.22701 -15.35552 -15.48330

686 672 466 901 635

20 21 22 23 24

419 012 472 403 546

-14.47247 -14.57600 -14.67912 -14.78187 -14.88428

807 035 589 967 611

0.84170 0.83689 0.83246 0.82836 0.82459

044 917 283 826 496

-15.61039 -15.73682 -15.86265 -15.98791 -16.11265

144 771 679 896 291

25 26 28 29 31

0.81683 0.81427 0.81191 0.80976 0.80780

775 076 546 370 825

-14.98636 -15.08815 -15.18965 -15.29090 -15.39191

880 085 477 253 569

0.82112 0.81794 0.81503 0.81237 0.80997

469 133 056 970 751

-16.23689 -16.36068 -16.48405 -16.60703 -16.72967

620 504 469 912 149

33 36 38 42 45

0.80604 0.80446 0.80305 0.80183 0.80077

267 127 902 156 512

-15.49271 -15.59332 -15.69375 -15.79403 -15.89418

527 192 598 741 589

0.80781 0.80588 0.80416 0.80267 0.80139

410 079 997 505 036

-16.85198 -16.97400 -17.09577 -17.21731 -17.33865

406 835 505 438 590

50 56 63 71 83

-15.99422 093

114 341 398 039 093

-17.45982 -17.58086 -17.70178 -17.82262 -17.94340

880 166 301 084 316

100 125 167 250 500

0.-050 0.048 0.046 0.044 0.042

r'lVi n(z) 0.85676'701 0.85136 682 0.84633 336 0.84164 245 0.83727 251

0.040 0.038 0.036 0.034 0.032

0.83320 0.82942 0.82590 0.82264 0.81962

0.030 0.028 0.026 0.024 0.022 0.020 0.018 0.016 0.014 0.012

P-

1

o. _.oin ___

.

0.799RR - . . .- - 647 -

P$Mll(T)

0.008 0.006 0.004 0.002

0.79916 297 0.79860 244 0.79820 323 0.19796 417

-16.19402 726 -16.29383 652 -16.39360 832

0.80031 Oi79943 0.79875 0.79827 0.79798

0.000

0.79788 456

-16.49336 143

0.79788 456

[(-56)51

[(-55)71

r--l

-ii; 09416 168

[)(-;

hi(%)--?.

-18.06415 776

Q

71

z.4 .lIZ1(.)

r~Alfzo(x) 1.474083 1.320938 1.211667 1.131459 1.070845

-21.047407 -21.606130 -22.149524 -22.676802 -23.188535

1.791133 1.525581 1.347435 1.224460 1.136653

-21.290925 -21.927545 -22.550082 -23.154248 -23.738936

20 21 22 23 24

0.038 0.036 0.034 0.032

..

1.023762 _ _ .__ 0.986284 0.955823 0.930635 0.909513

-23.685951 ~ . ~~. . .~ -24.170500 -24.643620 -25.106640 -25.560748

1.071741 1; 022171 0.983229 0.951902 0.926211

-24.304948 -24.853951 -25.387848 -25.908478 -26.417500

25 __

0.030 0.028 0. 026 0.024 0.022

0.891605 0.876293 0.863121 0.851743 0.841895

-26.006988 -26.446280 -26.879433 -27.307159 -27.730098

0.904821 0.886799 0.871483 0.858385 0.847145

-26.916369 -27.406346 -27.888527 -28.363869 -28.833211

33 36 38 42 45

0.020 0.018 0.016 0.014 0.012

0.833375 0.826019 0.819702 0.814321 0.809796

-28.148822 -28.563847 -28.975650 -29.384666 -29.791303

0.837487 0.829198 0.822114 0.816105 0.811069

-29.297299 -29.756800 -30.212318 -30.664405 -31.113569

50 56 63 71 83

0.010 0.008 0.006 0.004 0.002

0.806062 0.803071 0.800781 0.799165 0.798204

-30.195941 -30.598942 -31.000652 -31.401404 -31.801522

0.806925 0.803612 0.801081 0.799297 0.798237

-31.560285 -32.005000 -32.448139 -32.890109 -33.331307

100 125 167 250 500

0.000

0.797885

-32.201325

0.797885

-33.772121

Q

0.050 0.048 0.046 0.044 0.042 0.040 - ..

820 (z)

--I

821 ( T ) - r



26 28 29 31

[(-;)51 -nearest integer to z.

Compiled from L. Fox, A short table for Bessel functions of integer orders and large arguments. Royal Society Shorter Mathematical Tables No. 3(Cambridge Univ. Press, Cambridge, England, 1954) (with permission).

407

BESSEL FUNCTIONS OF INTEGER ORDER

BESSEL FUNCTIONS-VARIOUS n 0

1 2 3

Jn(2)

1 7.65197 1 4.40050 - 1 1.14903 2 1.95633 - 3)2.47663

6866 5857 4849 5398 8964

-

1)2.23890 1 5.76724 1 3.52834 - 1 1.28943 - 2 3.39957

7791 8078 0286 2495 1981

-

7302 3800 5817 4173 0180

-

9756 8972 0749 5229 3435

-

4)2.49757 5 2.09383 6 1.50232 8 9.42234 9 5.24925

i

10 11 12 13 14

Jd1)

I

-

I 1

3 7.03962 3 1.20242 4 1.74944 5 2.21795 6)2.49234

ORDERS

Table 9.4 Jn(5)

-

1 -1.77596 7713 1 -3.27579 1376 2 +4.65651 1628 - 1 3.64831 2306 ( - 1 3.91232 3605

Ij

2.63061 5124 I- 11101 1.19800 6746

15 16 17 18 19 20 30 40 50

100

(-189)8.43182 8790

n 0 1 2

(-158)1.06095 3112

I

(-119) 6.26778 9396

Jn(5O)

Jn

( 100)

-

2 +5.58123 2 -9.75118 2 -5.97128 2 +9.27348 2)+7.08409

-

1)-1.13847 8491

-2) -5.47321 7694

( -21) +l. 11592 7368

(-2)+9.63666 7330

2767 2813 0079 0406 7728

11

-2 +1.99858 5030 -2 -7.71453 5201

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 30 40 50 100

(-

3) 4.50797 3144

I- I 5 -12 -21 -30

1.15133 1.55109 6. 03089 1.78451

6925 6078 5312 3608

(-89) 6.59731 6064

BESSEL FUNCTIONS O F INTEGER ORDER

BESSEL FUNCTIONS-VARIOUS ORDERS

Table 9.4 n

0 1 2 3 4

Yn(1) ( - 2 +8.82569 6420 ( - 1)-7.81212 8213

(

Yn(2)

( - 1)-1.92142 2874

1) -3.32784 2303

5 6 7 8 9

(

10 ii 12 13 14

8 -1.21618 9)-2.42558 10 -5.32411 121 -1.27536 13)-3.31061

0143 0081 4376 1870 6748

15 16 17 18 19

14)-9.25697 16 -2.77378 171 -8.86684 19 -3.01195 21)-1.08341

3276 1366 3398 2974 6386

1 (

n

0 -9.93598 11 -4.69140 2 -2.71548 3) -1.85392 4)-1.45598

9128 0242 0254 2175 2938

5 -1.29184 6 -1.27728 i-1,39209 8 -1.65774 9 -2.14114

5422 5593 5698 1981 3619

0 -2.82086 9383 01 -7.76388 3188

7 -1.12865 9760 71 -7.95635 6938

20 30 40 50 100

Yn(5)

- 1 -3.08517 6252 - l]+l. 47863 1434 - 1 +3.67662 8826 - 11+1.46267 1627

62)-1.97615 0576 (185)-3.77528 7810

Yn(10)

(155)-3.00082 6049

-

177bt50)

2 -9.80649 21-5.67956 2 +9.57931 21+6.44591 ( - 2)-8.80580

0 1 2 3 4

(115)-5.08486 3915

Yn(100)

9547 6856 6873 2206 7408

,

( - 2) -7.85484 1391

10 11 12 13 14

-

2) -9.15700 8429

-2)+5.83315 -2 +6. 05863 -21 -4.50025 -2 -7.13869 -21 +2. 64419

7424 1093 8583 3153 8363

15 16 17 18 19 20

( 3) -1.59748 3848

40 50 100

(85)-4.84914 8271

(+18)-3.29380 0188

(-1)-1.66921 4114

409

BESSEL FUNCTIONS O F INTEGER ORDER Table 9.5 ZEROS AND ASSOCIATED VALUES OF BESSEL FUNCTIONS AND THEIR DERIVATIVES

J’o(jo, s)

J

jo, 2.40482 55577 5.52007 81103 8.65372 79129 11.79153 44391 14.93091 77086

-0; 20654 64331

3.83171 7.01559 10.17347 i 3 ; 12369 16.47063

6 7 8 9 10

18.07106 21.21163 24.35247 27.49347 30.63460

39679 66299 15308 91320 64684

+O. 18772 -0.17326 +O. 16170 -0.15218 +O. 14416

88030 58942 15507 12138 59777

19.61586 22.76008 25.90367 29.04683 32.18968

+O. 18006

11 12 13 14 15

33.77582 36.91709 40.05842 43.19979 46.34118

02136 83537 57646 17132 83717

-0.13729 +O. 13132 -0.12606 +O. 12139 -0.11721

69434 46267 94971 86248 11989

16 17 18 19 20

49.48260 52.62405 55.76551 58.90698 62.04846

98974 18411 07550 39261 91902

+O. 11342 -0.10999 +O. 10684 -0.10395 +O. 10129

91926 11430 78883 95729 34989

S

S

33, 8

6.38016 9.76102 13; 01520 16.22347 19.40942

10

22.58273 25.74817 28.90835 32.06485 35.21867

11 12 13 14 15

38.37047 41.52072 44.66974 47.81779 50.96503

16 17 18 19 20

1

J’3(~’3,s)

-0.29827

J‘l(j1,

I1. s

-0.51914 74973

h.s

J ‘ z ( ~ s)z ,

5. i3562 8.41724 11.61984 14.79595 17.95982

-0.33967 +O. 27138 -0.23244 +O. 20654 -0.18773

-0.16718 +O. 15672 -0.14801 +O. 14061

21.11700 24.27011 27.42057 30.56920 33.71652

+O. 17326 -0.16170 +O. 15218 -0.14417 +O. 13730

35.33231 38.47477 41.61709 44.75932 47.90146

-0.13421 +O. 12862 -0.12367 +O. 11925 -0.11527

36.86286 40.00845 43.15345 46.29800 49.44216

-0.13132 +O. 12607 -0.12140 +O. 11721 -0.11343

51.04354 54.18555 57.32753 60.46946 63.61136

+O. 11167 -0.10839 +O. 10537 -0.10260 +O. 10004

52.58602 55.72963 58.87302 62.01622 65.15927

+O. 10999 -0.10685 +O. 10396 -0.10129 +O. 09882

34, s

-0.40276

+O. 30012 -0.24970 +O. 21836 -0.19647

J’4Li4,

d

j5,s

J ‘ d j 5 , s)

-0; 18005

7.58834 11.06471 14.37254 17.61597 20.82693

-0.26836 +O. 23188 -0.20636 +O. 18766 -0.17323

8.77148 12.33860 15.70017 18.98013 22.21780

-0.24543 +O. 21743 -0.19615 +O. 17993 -0.16712

+O. 16718 -0.15672 +O. 14801 -0.14060 +O. 13421

24.01902 27.19909 30.37101 33.53714 36.69900

+O. 16168 -0.15217 +O. 14416 -0.13729 +O. 13132

25.43034 28.62662 31.81172 34.98878 38.15987

+O. 15669 -0.14799 +O. 14059 -0.13420 +O. 12861

-0.12862 -0.11925 +O. 11527 -0.11167

39.85763 43.01374 46.16785 49.32036 52.47155

-0.12607 +O. 12140 -0.11721 +O. 11343 -0.10999

41.32638 44.48932 47.64940 50.80717 53.96303

-0.12366 +O. 11925 -0.11527 +O. 11167 -0.10838

54.11162 57.25765 60.40322 63.54840 66.69324

+O. 10839 -0.10537 +O. 10260 -0.10004 +O. 09765

55.62165 58.77084 61.91925 65.06700 68.21417

+O. 10685 -0.10396 +O. 10129 -0.09882 +O. 09652

57.11730 60.27025 63.42205 66.57289 69.72289

+O. 10537

-0.10260 +O. 10003 -0.09765 +O. 09543

2 3 4 5

9.93611 13.58929 17.00382 20.32079 23.58608

-0.22713 +O. 20525 -0.18726 +O. 17305 -0.16159

11.08637 14.82127 18.28758 21.64154 24.93493

-0.21209 +O. 19479 -0.17942 +O. 16688 -0.15657

12.22509 16.03777 19.55454 22.94517 26.26681

-0.19944 +O. 18569 -0.17244 +O. 16130 -0.15196

6 7 8 9 10

26.82015 30.03372 33.23304 36.42202 39.60324

+O. 15212 -0.14413 +O. 13727 -0.13131 +O. 12606

28.19119 31.42279 34.63709 37.83872 41.03077

+O. 14792

29.54566 32.79580 36.02562 39.24045 42.44389

+O. 14404

-0.14055 +O. 13418 -0.12859 +O. 12365

11 12 13 14 15

42.77848 45.94902 49.11577 52.27945 55.44059

-0.12139 +O. 11721 -0.11343 +O. 10999 -0.10685

44.21541 47.39417 50.56818 53.73833 56.90525

-0.11924 +O. 11526 -0.11166 +O. 10838 -0.10537

45.63844 48.82593 52.00769 55.18475 58.35789

-0.11719 +O. 11342 -0.10998 +O. 10684 -0.10395

16 17 18 19 20

58.59961 61.75682 64.91251 68.06689 71.22013

+O. 10396 -0.10129 +O. 09882 -0.09652 +O. 09438

60.06948 63.23142 66.39141 69.54971 72.70655

+O. 10260

61.52774 64.69478 67.85943 71.02200 74.18277

+O. 09652

6 7 8 9

+O. 12367

-0.10003

+O. 09765 -0.09543 +O. 09336

-0.13722

+O. 13127 -0.12603 +O. 12137

+O. 10129 -0.09882 -0.09438

+O. 09237

410

BESSEL FUNCTIONS O F INTEGER ORDER Table 9.5 ZEROS AND ASSOCIATED VALUES OF BESSEL FUNCTIONS AND THEIR DERIVATIVES 9

Y’o(90,s)

YO, s

?I,

0.89357 3.95767 7.08605 10.22234 13.36109

697 842 106 504 747

87942 -0.40254 to. 30009 -0.24970 to. 21835

080 267 761 124 830

2.19714 5.42968 8.59601 11.74915 14.89744

6 7 8 9 10

16.50092 19.64130 22.78202 25.92295 29.06403

244 970 805 765 025

-0.19646 to. 18006 -0.16718 to. 15672 -0.14801

494 318 450 493 108

18.04340 21.18807 24.33194 27.47529 30.61829

11 12 13 14 15

32.20520 35.34645 38.48775 41.62910 44.77048

412 231 665 447 661

to.

14060 -0.13421 12861 -0.12366 to. 11924

578 123 661 795 981

33.76102 36.90356 40.04594 43.18822 46.33040

16 17 18 19 20

47.91189 51.05332 54.19477 57.33624 60.47772

633 855 936 570 516

-0.11527 11167 -0.10838 to. 10537 -0.10260

369 049 535 405 057

49.47251 52.61455 55.75654 58.89850 62.04041

1 2 3 4 5

4.52702 8.09755 11.39647 14.62308 17.81846

6 7 8 9 10

20.99728 24.16624 27.32880 30.48699 33.64205

11 12 13 14 15

36.79479 39.94577 43.09537 46.24387 49.39150

16 17 18 19 20

52.53840 55.68470 58.83049 61.97586 65.12086

1 2 3 4 5

7.83774 11.81104 15.31362 18.67070 21.95829

6 7 8 9 10

25.20621 28.42904 31.63488 34.82864 38.01347

11 12 13 14 15

41.19152 44.36427 47.53282 50.69796 53.86031

16 17 18 19 20

57.02034 60.17842 63.33485 66.48986 69.64364

to.

to.

to.

33256 -0.27080 to. 23232 -0.20650 to. 18771

5.64515 9.36162 12.73014 15.99963 19.22443

-0.17326 -0.15218 +O. 14416 -0.13730

22.42481 25.61027 28.78589 31.95469 35.11853

13132 -0.12607 to. 12140 -0.11721 to. 11343

38.27867 41.43596 44.59102 47.74429 50.89611

-0.10999 -0.10396 to. 10129 -0.09882

54.04673 57.19635 60.34513 63.49320 66.64065

23429 -0.21591 to. 19571 -0.17975 to. 16703

8.91961 13.00771 16.57392 19.97434 23.29397

-0.15664 14796 -0.14058 to. 13419 -0.12860

26.56676 29.80953 33.03177 36.23927 39.43579

+O. 12366

42.62391 45.80544 48.98171 52.15369 55.32215

to.

to. 16170

to.

to. 10685

to.

to.

-0.11924 to. 11527

-0.11167 to. 10838 -0.10537 to. 10260

-0.10003 to. 09765 -0.09543

58.48767 61.65071 64.81164 67.97075 71.12830

Y’1h1, s)

d

1 2 3 4 5

to. 52079

-0.34032 to. 27146

-0.23246 20655

to.

-0.18773 to. 17327

-0.16170 to. 15218 -0.14417 to. 13730

-0.13132 to. 12607

-0.12140 11721

to.

-0.11343 to. 10999

-0.10685 to. 10396

-0.10129

3.38424 6.79381 10.02348 13.20999 16.37897

Y’Z(?/2, *) 39921 -0.29992 to. 24967 -0.21835 to. 19646

19.53904 22.69396 25.84561 28.99508 32.14300

-0.18006 to. 16718 -0.15672 to. 14801 -0.14061

35.28979 38.43573 41.58101 44.72578 47.87012

to. 12367

51.01413 54.15785 57.30135 60.44464 63.58777

-0.11167 10839 -0.10537 to. 10260 -0.10004

?/2, 8

28909 -0.24848 to. 21805 -0.19635 to. 18001

6.74718 10.59718 14.03380 17.34709 20.60290

-0.16716

23.82654 27.03013 30.22034 33.40111 36.57497

to.

to. 15671

-0.14800 to. 14060 -0.13421 to. 12861

-0.12367 to. 11925 -0.11527 to. 11167 -0.10838 to. 10537

-0.10260 to. 10003 -0.09765

to. 21556

-0.20352 to. 18672 -0.17283 to. 16148 -0.15206

to.

to. 25795

-0.23062 20602 -0.18753 to. 17317

to.

-0.16165 15215 -0.14415 to. 13729 -0.13132 to.

55.54035 58.69393 61.84628 64.99759 68.14799

-0.10685 10396 -0.10129 to. 09882 -0.09652

9.99463 14.19036 17.81789 21.26093 24.61258

to. 20027

12138 -0.11720 to. 11342 -0.10999 to. 10684

44.04215 47.23298 50.41746 53.59675 56.77177

-0.10396 10129 -0.09882 +O. 09652 -0.09438

59.94319 63.11158 66.27738 69.44095 72.60259

to.

-0.11925 11527

to.

12606 -0.12140 to. 11721 -0.11343 to. 10999

27.91052 31.17370 34.41286 37.63465 40.84342

to.

13421 -0.12862

to.

39.74363 42.90825 46.06968 49.22854 52.38531

-0.13725 to. 13130 -0.12605

+O. 14409

to.

to.

to.

-0.19289 17880 -0.16662 to. 15643 to.

-0.14785 14051 -0.13415 to. 12857 -0.12364 to.

to. 11923

-0.11526 11166 -0.10838 to. 10537

to.

-0.10260

+O. 10003 -0.09765 09543 -0.09336

to.

41 1

BESSEL FUNCTIONS O F INTEGER ORDER Table 9.5 ZEROS AND ASSC)CIATED VALUES OF BESSEL FLXCTJONS AND THEIR DERIVATIVES

J o ( j’o,.v)

2 3 4 5

0.00000 3.83170 7.01558 10.17346 13.32369

00000 59702 66698 81351 19363

+1.00000 -0.40275 +0.30011 -0.24970 +0.21835

00000 93957 57525 48771 94072

1.84118 5.33144 8.53632 11.70600 14.86359

J l ( , j ’ l , .) +0.58187 -0.34613 ~0.27330 -0.23330 t0.20701

.j‘z,# 3.05424 6.70613 9.96947 13.17037 16.34752

J z ( j ’ 2 , .) +0.48650 -0.31353 +0.25474 -0.22088 +0.19794

6 7 8 9

10

16.47063 19i61585 22.76008 25.90367 29.04682

00509 85105 43806 20876 85349

-0.19646 +0.18006 -0.16718 +0.15672 -0.14801

53715 33753 46005 49863 11100

18.01553 21.16437 24.31133 27.45705 30.60192

-0.18802 ~0.17346 -0.16184 ~0.15228 -0.14424

19.51291 22.67158 25.82604 28.97767 32.12733

-0.18101 +0.16784 -0.15720 +0.14836 -0.14088

11 12 13 14 15

32.18967 35.33230 38.47476 41.61709 44.75931

99110 75501 62348 42128 89977

+0.14060 -0.13421 +0.12861 -0.12366 +0.11924

57982 12403 66221 79608 98120

33.74618 36,88999 40.03344 43.17663 46.31960

+0.13736 -0.13137 +0.12611 -0.12143 +0.11724

35.27554 38.42265 41.56893 44.71455 47.85964

+0.13443 -0.12879 +0.12381 -0.11937 +0.11537

16 17 18 19 20

47.90146 51.04353 54.18555 57.32752 60.46945

08872 51836 36411 54379 78453

-0.11527 +O.lll67 -0.10838 +0.10537 -0.10260

36941 04969 53489 40554 05671

49.46239 52.60504 55.74757 58.89000 62.03235

-0.11345 +0.11001 -0.10687 ~0.10397 -0.10131

51.00430 54.14860 57.29260 60.43635 63.57989

-0.11176 +0.10846 -0.10544 +0.10266 -0,10008

Y

1

j’n,

j ’ l . *.

3 4 5

4.20119 8.01524 11.34592 14.58585 17.78875

+0.43439 -0.29116 +0.24074 -0.21097 +0.19042

5.31755 9.28240 12.68191 15.96411 19.19603

+0.39965 -0.27438 +0.22959 -0.20276 +0.18403

6.41562 10.51986 13.98719 17.31284 20.57551

+0.37409 -0,26109 +0.22039 -0.19580 +0.17849

6 7 8 9 10

20.97248 24.14490 27.31006 30.47027 33.62695

-0.17505 +0.16295 -0.15310 +0.14487 -0.13784

22.40103 25.58976 28.76784 31.93854 35.10392

-0.16988 +0.15866 -0.14945 +0.14171 -0.13509

23.80358 27.01031 30.20285 33.38544 36.56078

-0.16533 +0.15482 -0.14616 ~0.13885 -0.13256

11 12 13 14 15

36.78102 39.93311 43.08365 46.23297 49.38130

+0.13176 -0.12643 +0.12169 -0.11746 +0.11364

38.26532 41.42367 44.57962 47.73367 50.88616

+0.12932 -0.12425 +0.11973 -0.11568 +0.11202

39.73064 42.89627 46.05857 49.21817 52.37559

t0.12707 -0.12223 +0.11790 -0.11402 +0.11049

16 17 18 19 20

52.52882 55.67567 58.82195 61.96775 65.11315

-0.11017 +0.10700 -0.10409 +0.10141 -0.09893

54.03737 57.18752 60.33677 63.48526 66.63309

-0.10868 +0.10563 -0.10283 +0.10023 -0.09783

55.53120 58.68528 61.83809 64.98980 68.14057

-0.10728 ~0.10434 -0.10163 +0.09912 -0.09678

Y

j’ti 7.50127 _. -~ 11.73494 15.26818 18.63744 21.93172

,j’;,

+0.35414 .._. -0.25017 +0.21261 -0.18978 +Oil7363

8.57784 12.93239 16.52937 19.94185 23.26805

J i (,j’i,.*j +0.33793 -0124096

10

25.18393 28.40978 31.61788 34.81339 37.99964

-0.16127 +o,i5137 -0.14317 +0.13623 -0.13024

11 12 13 14 15

41.17885 44.35258 47.52196 50.68782 53.85079

16 17

57.01138 60.16995 63.32681 66.48221 69.63635

1

2

1

2 3 4

5 6

7

8 9

18 19 20

-J~j(,i’fi, s)

.i’h,

A ( , i ’ s , d)

-0.18449 +Oil6929

9.64742 14.11552 17.77401 21.22906 24.58720

+0.32438 -0.23303 +0.19998 -0.17979 +0.16539

26.54503 29.79075 33.01518 36.22438 39.42227

-0.15762 +0.14823 -0.14044 +0.13381 -0.12808

27.88927 31.15533 34.39663 37.62008 40.83018

-0.15431 t0.14537 -0.13792 +0.13158 -0.12608

+0.12499 -0.12035 ~0.11620 -0.11246 +0.10906

42.61152 45.79400 48.97107 52.14375 55.31282

+0.12305 -0.11859 +0.11460 -0.11099 +0.10771

44.03001 47.22176 50.40702 53.58700 56.76260

+0.12124 -0.11695 +0.11309 -0.i0960 +O.10643

-0.10596 +0.10311 -0.10049 +0.09805 -0.09579

58.47887 61.64239 64.80374 67.96324 71.12113

-0.10471 +0.10195 -0.09940 +0.09704 -0.09484

59.93454 63.10340 66.26961 69.43356 72.59554

-0.10352 +0.10084 -0.09837 ~0.09607 -0.09393

~

+0.20588

412

BESSEL FUNCTIONS O F INTEGER ORDER Table 9.5 ZEROS AND ASSOCIATED VALUES OF BESSEL FUNCTIONS AND THEIR DERIVATIVES S

Y’O,

8

YO(Y’O,*) 641

5

2.19714 5.42968 8.59600 11.74915 14.89744

133 104 587 483 213

+OS2078 -0.34031 +0.27145 -0.23246 +0.20654

6 7 8 9 10

18.04340 21.18806 24.33194 27.47529 30.61828

228 893 257 498 649

11 12 13 14 15

33.76101 36.90355 40.04594 43.18821 46.33039

16 17 18 19 20

49.47250 52.61455 55.75654 58.89849 62.04041

1 2 3 4

1 2 3 4

?/‘l,s

Y’Z, s

Yl(?/’1,8)

YZ(Y’2,R) +0.36766 -0.27928 +0.23594 -0.20845 ~0.18890

988 177 711

3.68302 6.94150 10.12340 13.28576 16.44006

+0.41673 -0.30317 +0.25091 -0.21897 +0.19683

8.35072 11.57420 14.76091 17.93129

-0.18772 +0.17326 -0.16170 +o.i5218 -0.14416

909 604 163 126 600

19.59024 22.73803 25.88431 29.02958 32.17412

-0.18030 +0.16735 -0.15684

21.09289 24.24923 27.40215

-0.17405 +0.16225

lok,ZSI

33.70159

+o.i4448 -0.13754

780 532 464 810 925

+0.13729 -0.13132 +0.12606 -0.12139 +0.11721

696 464 951 863 120

35.31813 38.46175 41.60507 44.74814 47.89101

+0.13427 -0.12866 +0.12370 -0.11928 +0.11530

36.84921 39.99589 43.14182 46.28716 49.43202

+0.13152 -0.12623 +OX153 -0.11732 +0.11352

568 077 488 617 115

-0.11342 +0.10999 -0.10684 +0.10395 -0.10129

920 115 789 957 350

51.03373 54.17632 57.31880 60.46118 63.60349

-0.11169 +0.10840 -0.10539 +0.10261 -0.10005

52.57649 55.72063 58.86450 62.00814 65.15159

-0.11007 +0.10692 -0.10402 t0.10135 -0.09887

805

+o;i48io -0.14067

5.00258

-0.15259 __-_-_

+0.33660 -0.26195 t0.22428 -0.19987 t0.18223

7.46492 11.00517 14.33172 17.58444 20;80106

+0.31432 -0.24851 +0.21481 -0.19267 +0.17651

8.64956 12.28087 15.66080

5

6.25363 9.69879 12.97241 16.19045 19.38239

22119284

+0.29718 -0.23763 ~0.20687 -n 1 R i m t0.17151

6 7 8 9 10

22.55979 25.72821 28.89068 32.04898 35.20427

-0.16867 +0.15779 -0.14881 +0.14122 -0.13470

23.99700 27.17989 30.35396 33.52180 36.68505

-0.16397 +0.15384 -0.14543 +0.13828 -0.13211

25.40907 28.60804 31.79520 34.97389 38.14631

-0.15980 +0.15030 -0.14236 +0.13559 -0.12973

11 12 13 14 15

38.35728 41.50855 44.65845 47.80725 50.95515

+0.12901 -0.12399 +0.11952

-0.11550 +0.11186

39.84483 43.00191 46.15686 49.31009 52.46191

+0.12671 -0.12193 +0.11765 -0.11380 +0.11031

41.31392 44.47779 47.63867 50.79713 53.95360

+0.12458 -0.12001 ~0.11591 -0.11221 ~0.10885

16 17 18 19 20

54.10232 57.24887 60.39491 63.54050 66.68571

-0.10855 +0.10552 -0.10273 +0.10015 -0.09775

55.61257 58.76225 61.91110 65.05925 68.20679

-0.10712 +0.10420 -0.10151 +0.09901 -0.09669

57.10841 60.26183 63.41407 66.56530 69.71565

-0.10578 ~0.10295 -0.10035 t0.09793 -0.09568

v’7

I ? i ( l / ’ i “) +0.27194 -0.22077 +0.19414 -0.17634 ~0.16311

12.10364 15.98284 19.51773 22.91696 26.24370

YdY‘Y .) +0.26220 -0.21402 +0.18891 -0.17207 +0.15953

“

18.94974

-.---_-

5

v‘ll

1 2 3 4

9.81480 13.53281 16.96553 20.29129 23.56186

17G(f/’li, A ) t0.28339 -0.22854 +0.20007 -0.18111 ~0.16708

10.96515 14.76569 18.25012 21.61275 24.91131

9 10

26.79950 30.01567 33.21697 36.40752 39.59002

-0.15607 +0.14709 -0.13957 +0.13313 -0.12753

28.17105 31.40518 34.62140 37.82455 41.01785

-0.15269 +0.14417 -0.13700 +0.13085 -0.12549

29.52596 32.77857 36.01026 39.22658 42.43122

-0.14962 +0.14149 -0.13463 +0.12874 -0.12359

11 12 13 14 15

42.76632 45.93775 49.10528 52.26963 55.43136

+0.12260 -0,11822 +0.11428 -0.11072 +0.10748

44.20351 47.38314 50.55791 53.72870 56.89619

t0.12076 -0.11654 +0.11275 -0.10931 t0.10618

45.62678 48.81512 51.99761 55.17529 58.34899

t0.11904 -0.11497 +0.11131 -0.10798 +0.10494

16 17 18 19

58.59089 61.74857 64.90468 68.05943 71.21301

-0.10451 t0.10177 -0.09925 +0.09690 -0.09471

60.06092 63.22331 66.38370 69.54237 72.69955

-0.10330 +0.10065 -0.09820 +0.09592 -0.09379

61.51933 64.68681 67.85185 71.01478 74.17587

-0.10216 t0.09958 -0.09720 +0.09498 -0.09291

5

I/‘\

I

I

6 7 8

20

413

BESSEL FUNCTIONS O F INTEGER ORDER BESSEL FUSCTIONS-JU~,, ,p)

0. 00 0.02 0. 04 0. 06 0. 08

Joijo,iX) 1.00000 0.99942 0.99769 0.99480 0.99077

Jo(jo,4 1.00000 0.99696 0.98785 0.97276 0.95184

0.10 0.12 0.14 0.16 0.18

0.98559 0.97929 0.97186 0.96333 0.95370

0.20 0. 22 0. 24 0. 26 0. 2 8 0. 0. 0, 0. 0.

Table 9.6

0.99253 0.97027 0.93373 0.88372

1.00000 0.98614 0.94515 0.87872 0.78961

Jo( j o 5x) , 1.00000 0.97783 0.91280 0.80920 0.67388

0.92526 0,89328 0.85617 0.81429 0.76800

0.82136 0.74804 0.66537 0.57518 0.47943

0. 68146 0.55871 0.42632 0.28958 0.15386

0.51568 0.34481 0.17211 +O. 00827 -0.13693

0.94300 0; 93124 0. 91844 0.90463 0.88982

0.71773 0.66392 0.60706 0.54766 0.48623

0.38020 0.27960 0.17976 +O. 08277 -0.00942

+O. 02438

-0,09404 -0.19716 -0.28155 -0.34466

-0.25533 -0.34090 -0.39013 -0.40225 -0.37917

30 32 34 36 38

0.87405 0.85734 0.83972 0.82122 0.80187

0.42333 0.35950 0.29529 0.23126 0.16795

-0.09498 -0.17226 -0.23986 -0.29664 -0.34171

-0.38498 -0,40207 -0.39653 -0.36998 -0.32493

-0.32527 -0.24698 -0.15223 -0. 04980 +O. 05137

0. 0. 0. 0. 0.

40 42 44 46 48

0.78171 0.76077 0.73908 0.71669 0.69362

0.10590 +O. 04562 -0.01240 -0.06769 -0.11983

-0.37453 -0.39482 -0.40264 -0.39835 -0.38259

-0.26467 -0.19304 -0.11431 -0.03289 +O. 04684

0.14293 0.21767 0.27011 0.29684 0.29671

0. 0. 0. 0, 0.

50 52 54 56 58

0.66993 0.64565 0.62081 0.59547 0.56967

-0.16840 -0.21306 -0,25349 -0.28941 -0.32062

-0.35628 -0.32056 -0.27678 -0.22648 -0.17130

0.12078 0.18527 0.23725 0.27445 0.29541

0.27086 0.22252 0.15667 +O. 07960 -0.00168

0. 60 0. 62 0. 64 0. 66 0.68

0.54345 0.51685 0.48992 0.46270 0.43524

-0.34692 -0.36821 -0.38441 -0.39551 -0.40152

-0.11295 -0.05320 +O. 00622 0.06363 0.11745

0.29959 0.28731 0.25977 0.21892 0.16735

-0.08007 -0.14891 -0.20259 -0.23697 -0.24965

0. 70 0.72 0. 74 0. 76 0. 78

0.40758 0.37977 0.35186 0.32389 0.29591

-0.40255 -0.39871 -0. 39019 -0.37721 -0.36003

0.16625 0.20878 0.24399 0.27107 0.28945

0.10814 +O. 04470 -0.01945 -0.08082 -0.13618

-0.24019 -0.21003 -0.16237 -0.10179 -0.03389

0. 80 0, 82 0. 84 0.86 0. 88

0.26796 0.24009 0.21234 0.18476 0.15739

-0.33896 -0.31433 -0.28652 -0.25591 -0.22293

0.29882 0.29915 0.29063 0.27374 0.24914

-0.18270 -0.21808 -0.24067 -0.24957 -0.24461

+O. 03525

0.90 0. 92 0.94 0. 96 0. 98

0.13027 0.10346 0.07698 0.05089 0.02521

-0.18800 -0.15157 -0.11411 -0.07605 -0. 03787

0.21774 0.18059 0.13891 0.09399 0.04722

-0.22637 -0.1961? -0.15580 -0.10779 -0.05486

0.21694 0.20021 0.16630 0.11854 0.06138

1.00

0.00000

0.00000

0.00000

0.00000

0.00000

X

JO(.I’O,.~;C)

0.09960 0.15369 0.19306 0.21464

From E. T. Goodwin and J. Staton, Table of .Jo(,jo,,,~), Quart. J. Mech. Appl. Math. 1, 220-224 (1948) (with permission).

I

414

BESSEL FUNCTIONS O F INTEGER ORDER

BESSEL FUNCTIONS-MISCELLANEOUS

Table 9.7

stil

x\s

1

ZEROS

Zero of xJl(x)- xJo (x)

2

0. 00 0. 02 0. 04 0. 06 0. 08 0.10

0.0000 0.1995 0.2814 0.3438 0.3960 0.4417

3.8317 3.8369 3.8421 3.8473 3.8525 3.8577

3 7. 0156 7. 0184 7. 0213 7.0241 7.0270 7.0298

4 10.1735 10.1754 10.1774 10.1794 10.1813 10.1833

5 13.3237 13.3252 13.3267 13.3282 13.3297 13.3312

0. 20 0.40 0. 60 0. 80 1. 00

0.6170 0.8516 1.0184 1.1490 1.2558

3.8835 3.9344 3.9841 4.0325 4. 0795

7.0440 7. 0723 7.1004 7.1282 7.1558

10.1931 10.2127 10.2322 10.2516 10.2710

13.3387 13.3537 13.3686 13.3835 13.3984

x -- '\s 1. 00 0. 80 0. 60 0. 40 0.20 0.10 0. 08 0. 06 0. 04 0: 02 0. 00

1

1.2558 1.3659 1.5095 1.7060 1.9898

4.0795 4.1361 4.2249 4.3818 4.7131

3 7.1558 7.1898 7.2453 7.3508 7.6177

4 10.2710 10.2950 10.3346 10.4118 10.6223

13.3984 13.4169 13.4476 13.5079 13.6786

1 1 2 3 5

2.1795 2.2218 2.2656 2.3108 2.3572 2.4048

5. 0332 5.1172 5.2085 5.3068 5.4112 5. 5201

7.9569 8.0624 8.1852 8.3262 8.4840 8.6537

10.9363 11.0477 11.1864 11.3575 11.5621 11.7915

13.9580 14.0666 14.2100 14.3996 14.6433 14.9309

10 13 17 25 50

2

stll

x\s

1

2

0. 5 0.6 0. 7 0.8 0.9 1.0

0.0000 1.1231 1.4417 1.6275 1.7517 1.8412

5.1356 5.2008 5.2476 5.2826 5.3098 5.3314

A- '\s

1

CO

Zero of J I (2) - A d o (z)

3 8.4172 8.4569 8.4853 8.5066 8.5231 8.5363 3

2

5

5

4 11.6198 11.6486 11.6691 11.6845 11.6964 11.7060 4

14.7960 14.8185 14.8346 14.8467 14.8561 14.8636 5



1.00 0. 80 0.60 0. 40 0.20

1.8412 1.9844 2.1092 2.2192 2.3171

5.3314 5.3702 5.4085 5.4463 5.4835

8. 5363 8.5600 8.5836 8.6072 8.6305

11.7060 11.7232 11.7404 11.7575 11.7745

14.8636 14.8771 14.8906 14.9041 14.9175

1 1 2 3 5

0.10 0. 08 0. 06 0. 04 0. 02 0. 00

2.3621 2.3709 2.3795 2.3880 2.3965 2.4048

5.5019 5.5055 5.5092 5. 5128 5.5165 5.5201

8.6421 8.6445 8.6468 8.6491 8.6514 8.6537

11.7830 11.7847 11.7864 11.7881 11.7898 11.7915

14.9242 14.9256 14.9269 14.9282 14.9296 14.9309

10 13 17 25 50



CO

=nearest integer to A.

Compiled from H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids (Oxford Univ. Press, London, England, 1947) and British Association for the Advancement of Science, Bessel functions, Part 1. Functions of orders zero and unity, Mathematical Tables, vol. VI (Cambridge Univ. Press, Cambridge, England, 1950)(with permission).

415

BESSEL FUNCTIONS OF INTEGER ORDER

BESSEL FUNCTIONS-MISCELLANEOUS ~ t l kZero

*

*

Table 9.7

of Jo (z) YO(XZ) - Y&) J o (XZ) 3

4

5

0.80 0.60 0.40 0.20

12.55847 4.69706 2.07322 0.76319

031 410 886 127

25.12877 9.41690 4.17730 1.55710

37.69646 14.13189 6.27537 2.34641

50.26349 18.84558 8.37167 3.13403

62.83026 23.55876 10.46723 3.92084

1 2 3 5

0.10 0. 08 0. 06 0. 04 d. 02 0.00

0.33139 0,25732 0.18699 0.12038 0.05768 0.00000

387 649 458 637 450 000

0.68576 0.53485 0.39079 0.25340 0.12272 0.00000

1.03774 0.81055 0.59334 0.38570 0.18751 0.00000

1.38864 1.08536 0.79522 0.51759 0.25214 0.00000

1.73896 1.35969 0.99673 0.64923 0.31666 0.00000

10 13 17 25 50

1

X-l\S

2

1



12.59004 4.75805 2.15647 0.84714

151 426 249 961

2 25.14465 9.44837 4.22309 1.61108

3 37.70706 14.15300 6.30658 2.38532

4 50.27145 18.86146 8.39528 3.16421

62.83662 23.57148 10.48619 3.94541

1 2 3 5

0.10 0. 08 0. 06 0. 04 0.02 0. 00

0.39409 0.31223 0123235 0.15400 0.07672 0.00000

416 576 256 729 788 000

0.73306 0.57816 0.42843 0.28296 0.14062 0.00000

1.07483 0.84552 0.62483 0.41157 0.20409 0.00000

1.41886 1. 11441 0.82207 0.54044 0.26752 0.00000

1.76433 1.38440 1.02001 0.66961 0.33097 0.00000

10 13 17 25 50

A-l\S

X-'\S

Zero of 2

1

0.80 0.60 0. 40 0.20

6.56973 2.60328 1.24266 0.51472

310 138 626 663

18.94971 7.16213 3.22655 1.24657

0.10 0. 08 0. 06 0. 04 0.02 0. 00

0,24481 0.19461 0.14523 0.09647 0.04813 0.00000

004 772 798 602 209 000

0.57258 0.45251 0.33597 0.22226 0.11059 0.00000

J l ( z ) Yo (Az)

- YI (Z)Jo

5

(U) 4

5

56.58224 21.23751 9.45462 3.56157

1 2 3 5

1.25099 0.90956 0.71635 0.98327 0.72594 0.53005 0.47768 0.34957 0.23666 0.17353 0.00000 0.00000 =nearest integer to A.

1.59489 1. 25203 0.92301 0.60634 0.29991 0.00000

10 13 17 25 50



OD

Compiled from British Association for the Advancement of Science, Bessel functions, Part I. Functions of orders zero and unity, Mathematical Tables, vol.VI (Cambridge Univ. Press, Cambridge, England, 1950) (with permission). *See page 11.

*

0

44.02544 16.53413 7.36856 2.78326

3

31.47626 11.83783 5.28885 2.00959

*

rn

0.80 0. 60 0. 40 0.20

5th

* *

ZEROS

*

416

BESSEL FUNCTIONS O F INTEGER ORDER Table 9.8 r

MODIFIED BESSEL FUNCTIONS-ORDERS

(x)

(x)

0, 1 AND 2

(x)

0.1 0.2 0. 3 0. 4

e-do 1.00000 0.90710 0.82693 0. 75758 0.69740

00000 09258 85516 06252 21705

0.00000 0.04529 0.08228 0.11237 0.13676

00000 84468 31235 75606 32243

0.12500 0.12510 0.12541 0.12594 0; iZk67

00000 41992 71878 01407 50222

0. 5 0. 6 0. 7 0. 8 0. 9

0.64503 0.59932 0.55930 0.52414 0.49316

52706 72031 55265 89420 29662

0.15642 0.17216 0.18466 0.19449 0.20211

08032 44195 99828 86933 65309

0.12762 0.12879 0.13018 0.13180 0.13365

45967 24416 29658 14318 39819

1. 0 1.1 1.2 1.3 1. 4

0.46575 0.44144 0.41978 0.40042 0.38306

96077 03776 20789 49127 25154

0.20791 0.21220 0.21525 0.21729 0.21850

04154 16132 68594 75878 75923

0.13574 0.13809 0.14069 0.14356 0.14670

76698 04952 14455 05405 88837

1. 5 1. 6 1.7 1.8 1.9

0.36743 0.35331 0.34051 0.32887 0. 31824

36090 49978 56880 19497 31629

0.21903 0.21901 0.21855 0.21772 0.21661

93874 94899 28066 62788 19112

0.15014 0; i5389 0.15795 0.16235 0; i 6 7 i i

87192 34944 79288 80900 1472

2. 0 2.1 2.2 2.3 2.4

0.30850 0.29956 0.29131 0.28369 0.27662

83225 30945 73331 29857 23231

0.21526 0.21374 0,21208 0.21032 0.20848

92892 76721 77328 30051 10887

0.17223 0.17775 0.18368 0.19006 0.19690

71119 56370 94251 26964 16460

2.5 2.6 2.7 2. 8 2. 9

0.27004 0.26391 0.25818 0.25280 0.24775

64416 39957 01238 55337 57304

0.20658 0.20465 0.20269 0.20073 0.19877

46495 22544 90640 74113 72816

0.20423 0.21209 0.22050 0.22951 0.23915

45837 20841 71509 53938 52213

3. 0 3.1 3. 2 3.3 3. 4

0.24300 0.23851 0.23426 0.23024 0.22643

03542 26187 88316 79845 14011

0.19682 0.19489 0.19297 0.19109 0.18922

67133 21309 86229 01727 98511

0.24946 0.26049 0.27229 0.28490 0.29839

80490 85252 47757 86686 61010

3. 5 3. 6 3. 7 3. 8 3.9

0.22280 0.21934 0.21604 0.21290 0,. 20988

24380 62245 94417 01308 75279

0.18739 0.18560 0.18383 0,18210 0.18041

99766 22484 78580 75810 18543

0.31281 0.32823 0.34472 0.36235 0.38121

73100 72078 57467 83128 61528

4. 0 4.1 4. 2 4. 3 4. 4

0.20700 0.20423 0.20157 0.19902 0.19656

19211 45274 73840 32571 55589

0.17875 0.17712 0.17553 0.17397 0.17245

08394 44763 25260 46091 02337

0.40138 0.42296 0.44605 0.47075 0.49719

68359 47539 16629 72701 98689

5 6 7 8 9

0.19419 0.19191 0.18971 0.18758 0.18552

82777 59151 34330 62042 99721

0.16807 22681 0.16667 57058 0.16530 92936

0.55581 0.58827 0.62304 0; 66030

0.18354 08126

0.16397 22669

0.70022 45988

0. 0

4. 4. 4. 4. 4.

5. 0

[(-93121

r-zI1

0.17095 88223 0; i6949

973ii

.1:-212

0.52550 70212 .~ _-_ . _.

63319 61978 67409 07270

[(-;I31

2n

In+l(x)=--In(x)+In-l(l)

Compiled from British Association for the Advancement of Science, Bessel functions, Part I. Func tions of orders zero and unity, Mathematical Tables, vol. VI , Part 11. Functions of positive integer order, Mathematical Tables, vol. X(Cambridge Univ. Press, Cambridge, England, 1950, 1952)and L. Fox, A short table for Be& functions of integer orders and large arguments.Roga1 Society Shorter Mathematical Tables NO. 3 (Cambridge Univ. Press, Cambridge, England, 1954) (with permission).

417

BESSEL FUNCTIONS OF INTEGER ORDER

MODIFIED BESSEL FUNCTIONS-ORDERS X

e z K o (4

ezKi (5)

0. 0 0.1 0.2 0. 3 0. 4

2.68232 2.14075 1.85262 1.66268

61023 73233 73007 20891

10.89018 5.83338 4.12515 3.25867

0.5 0. 6 0. 7 0. 8 0.9

1.52410 1.41673 1.33012 1.25820 1.19716

93857 76214 36562 31216 33803

1.0 1.1 1.2 1.3 1.4

1.14446 1.09833 1.05748 1.02097 0.98806

1.5 1.6 1.7 1. 8 1.9

m

0, 1 AND 2

Table 9.8 .r2K2 (x)

2683 6037 7762 3880

2.00000 1.99503 1.98049 1.95711 1.92580

0000 9646 7172 6625 8202

2.73100 2.37392 2.11501 1.91793 1.76238

97082 00376 13128 02990 82197

1.88754 1.64330 1.79405 1.74067 1,68401

5888 9881 1681 2762 1992

30797 02828 45322 31613 99961

1.63615 1.53140 1.44289 1.36698 1.30105

34863 37541 75522 72841 37400

1.62483 1.56385 1.50167 1.43886 1,37590

8899 0953 3576 2011 4446

0.95821 0.93094 0.90591 0.88283 0.86145

00533 59808 81386 35270 06168

1.24316 1.19186 1.14603 1.10480 1.06747

58736 75654 92462 53726 09298

1.31322 1,25119 1.19011 1.13026 1.07184

5917 2681 6819 0897 2567

2. 0 2.1 2.2 2.3 2.4

0.84156 0.82301 0.80565 0.78935 0.77401

82151 71525 39812 61312 81407

1.03347 1.00236 0.97377 0.94737 0.92291

68471 80527 01679 22250 36650

1.01503 0.95999 0.90680 0,85556 0.80633

9018 1226 7952 9487 1113

2.5 2. 6 2.7 2.8 2.9

0.75954 0.74586 0.73290 0.72060 0.70890

86903 82430 71515 41251 49774

0.90017 0.87896 0.85913 0.84053 0.82304

44239 72806 18867 00604 20403

0.75912 0.71396 0,67085 0.62977 0.59070

6289 9565 9227 9698 3688

3. 0 3.1 3.2 3. 3 3. 4

0. 69776 0.68713 0.67697 0.66725 0.65795

15980 11010 51139 91831 22725

0.80656 0.79100 0.77628 0.76232 0.74907

34800 30157 02824 42864 20613

0.55359 0.51640 0.48508 0.45358 0.42362

4126 5885 7306 1550 7789

3. 5 3.6 3.7 3.8 3.9

0.64902 0.64045 0.63221 0.62429 0.61665

63377 59647 80591 15812 73147

0.73646 0.72446 0.71300 0.70206 0.69159

75480 06608 65010 46931 88206

0.39576 0.36931 OI34443 0.32103 Oi29905

2241 9074 1194 0914 0529

4. 0 4.1 4.2 4.3 4.4

0.60929 0.60219 0.59533 0.58871 0.58230

76693 65064 89889 14486 12704

0.68157 0.67196 0.66274 0.65387 0.64535

59452 61952 24110 98395 58689

0.27642 0.25908 0.24096 0,22399 0.20813

2808 1398 1165 8474 1411

4.5 4. 6 4. 7 4. 8 4.9

0.57609 0.57008 0.56426 0.55861 0.55313

67897 72022 24840 33194 10397

0.63714 0.62924 0.62161 0.61425 0.60714

97988 26383 69312 66003 68131

0.19329 0,17944 0.16651 0,15445 0.14320

9963 6150 4127 0249 3117

5. 0

0.54780 75643

m

0.60027 38587

0.13272 3593

[(PI

418

BESSEL FUNCTIONS O F INTEGER ORDER

Table 9.8 X

MODIFIED BESSEL FUNCTIONS-ORDERS e-=Zo(x)

e - ~ 1 (x) 1

0, 1 AND 2 e -212 (x)

5. 0 5.1 5.2 5.3 5. 4

0.18354 0.18161 0.17974 0.17794 0.17618

08126 51021 94883 08646 63475

0.16397 0.16266 0.16138 0.16012 0.15890

22669 38546 32850 97913 26150

0.11795 0.11782 0.11767 0,11751 0.11733

1906 5355 8994 4528 3527

5. 5 5.6 5. 7 5. 8 5.9

0.17448 0.17282 0.17122 0.16965 0.16813

32564 90951 15362 84061 76726

0.15770 0.15652 0.15537 0.15424 0.15313

10090 42405 15922 23641 58742

0.11713 0.11692 0.11670 0.11647 0.11622

7435 7581 5188 1384 7207

6. 0 6.1 6.2 6. 3 6. 4

0.16665 0.16521 0.16381 0.16244 0.16110

74327 59021 14064 23718 73175

0.15205 0.15098 0.14994 0.14892 0.14792

14593 84754 62978 43212 19595

0.11597 0.11571 0.11544 0.11516 Oil1488

3613 1484 1633 4809 1705

6. 5 6.6 6. 7 6. 8 6. 9

0.15980 0.15853 0.15729 0.15608 0.15489

48490 36513 24831 01720 56090

0.14693 0.14597 0.14502 0.14409 0.14318

86457 38314 69866 75991 51745

0.11459 0.11429 0.11400 0.11369 0.11339

2958 9157 0845 8525 2660

7. 0 7.1 7. 2 7. 3 7. 4

0.15373 0.15260 0.15149 0.15041 0.14935

77447 55844 81855 46530 41371

0.14228 0.14140 0.14054 0.13969 0.13886

92347 93186 49809 57915 13353

0.11308 0.11277 0.11245 0.11214 0.11182

3678 1974 7913 1833 4046

7. 5 7. 6 7. 7 7. 8 7. 9

0.14831 0.14729 0.14630 0.14532 0.14436

58301 89636 28062 66611 98642

0.13804 0.13723 0.13644 0.13566 0.13489

12115 50333 24270 30318 64995

0.11150 0.11116 0.11086 0.11054 0.11021

4840 4481 3215 1268 8852

8. 0 8.1 8.2 8. 3 8.4

0.14343 0.14251 0.14160 0.14072 0.13985

17818 18095 93695 39098 49027

0.13414 0.13340 0.13267 0.13195 0.13124

24933 06883 07705 24362 53923

0.10969 0.10957 0.10925 0.10892 0.10860

6158 3368 0645 8142 6000

8. 5 8.6 8.7 8.8 8. 9

0.13900 0.13816 0.13734 0.13653 0.13573

18430 42474 16526 36147 97082

0.13054 0.12986 0.12918 0.12852 0.12786

93551 40505 92134 45873 99242

0.10828 0.10796 0.10764 0.10732 0.10700

4348 3305 2983 3481 4894

9.0 9.1 9.2 9.3 9.4

0.13495 0.13419 0.13343 0.13269 0.13196

95247 26720 87740 74691 84094

0.12722 0.12658 0.12596 0.12534 0.12473

49839 95342 33501 62139 79145

0.10668 0.10637 0.10605 0.10574 0.10542

7306 0796 5437 1294 8428

9.5 9. 6 9. 7 9. 8 9.9

0.13125 0.13054 0.12985 0.12916 0.12849

12609 57016 14223 81248 55220

0.12413 0.12354 0.12296 0.12238 0.12182

82477 70154 40258 90929 20364

0.10511 0.10480 0.10449 0.10419 0.10388

6893 6740 8015 0759 5010

10.0

0.12783 33371

"

--6!Y

0.12126 26814 [I(

-;)"3

0.10358 0801

[(-:)21

419

BESSEL FUNCTIONS OF INTEGER ORDER

MODIFIED BESSEL FUNCTIONS- -ORDERS 0, 1 AND 2 X

exKo(x)

exKl(x)

Table 9.8 eZKz(x)

5. 0 5.1 5.2 5.3 5.4

0.54780 0.54263 0.53760 0.53271 0.52795

75643 53519 73540 69744 80329

0.60027 0.59362 0.58718 0.58095 0.57490

38587 50463 86062 36085 98871

0.78791 0.77542 0.76344 0.75194 0.74088

5.5 5.6 5. 7 5, 8 5.9

0.52332 0,51881 0.51441 0.51012 0.50594

47316 16252 35938 58183 37583

0.56904 0.56335 0.55783 0.55246 0.54725

79741 90393 48348 76495 02639

0.73025 127 0.72001 128 0,71014 511 0.70063 190 0.69145 232

6. 0 6.1 6.2 6. 3 6.4

0.50186 0.49787 0.49399 0.49019 0.48647

31309 98929 02237 05093 73291

0.54217 0.53723 0.53243 0.52774 0.52318

59104 82386 12833 94344 74101

0.68258 0.67402 0.66574 0.65773 0.64997

843 358 225 001 339

6. 5 6.6 6.7 6. 8 6.9

0.48284 0.47929 0.47582 0.47242 0.46910

74413 77729 54066 75723 16370

0.51874 0.51440 0.51017 0.50604 0.50200

02336 32108 19097 21421 99471

0.64245 0.63517 0.62811 0.62126 0.61461

982 753 553 350 177

7. 0 7.1 7. 2 7.3 7.4

0.46584 0.46265 0.45953 0.45646 0.45346

50959 55657 07756 85618 68594

0.49807 0.49422 0.49046 0.48678 0.48318

15749 34737 22755 47842 79648

0,60815 0.60187 0,59577 0.58983 0.58405

126 345 030 426 820

7.5 7.6 7. 7 7. 8 7.9

0.45052 0.44763 0.44480 0.44202 0.43930

36991 71996 55636 70724 00819

0.47966 0.47622 0.47285 0.46955 0.46631

89336 49486 33995 18010 77847

0.57843 0.57295 0.56762 0.56242 0.55735

541 955 463 497 522

8. 0 8.1 8. 2 8.3 8.4

0.43662 0.43399 0.43141 0.42887 0.42638

30185 43754 27084 66329 48214

0.46314 0.46004 0.45699 0.45401 0.45108

90928 35709 91615 39001 59089

0.55241 0.54758 0.54287 0.53827 0,53378

029 538 592 757 623

8. 5 8.6 8. 7 8.8 8.9

0.42393 0.42152 0.41916 0.41683 0.41454

59993 89433 24781 54743 68462

0.44821 0.44539 0.44262 0.43991 0.43724

33915 46295 79775 18594 47648

0.52939 0.52510 0,52091 0.51681 0.51280

797 909 604 544 410

9.0 9.1 9.2 9.3 9.4

0.41229 0. 41008 0.40790 0.40575 0.40364

55493 05783 09662 57809 41245

0.43462 0.43205 0.42952 0.42703 0.42459

52454 19116 34301 85204 59520

0.50887 0.50503 0.50127 0.49759 0.49398

894 704 562 202 369

9.5 9. 6 9.7 9. 8 9.9

0.40156 0.39951 0.39750 0.39551 0.39355

51322 79693 18313 59416 95506

0.42219 0.41983 0.41751 0.41522 0.41297

45430 31565 06989 61179 84003

0.49044 0.48698 0.48358 0.48025 0.47698

819 321 651 597 953

10. 0

0.39163 19344

0.41076 65704

711 949 913 475 762

0.47378 525

,

I

420

BESSEL FUNCTIONS OF INTEGER ORDER

Table 9.8 X

MODIFIED BESSEL FUNCTIONS-ORDERS e -zZo (x)

e-LZl(x)

0, 1 AND 2 e-zIz(x)

10.0 10.2 10.4 10. 6 10. 8

0.12783 0.12653 0.12528 0.12406 0.12288

33371 91639 35822 47082 07840

0.12126 0.12016 0.11909 0.11805 0.11704

26814 64024 89584 91273 57564

0.10358 0.10297 0.10237 0.10178 0.10120

0801 7124 9936 9401 5644

11.0 11.2 11.4 11; 6 11. 8

0.12173 0.12061 0.11952 0.11846 0.11743

01682 13250 28165 32942 14923

0.11605 0.11509 0.11415 0.11323 0.11233

77582 41055 38276 60059 97710

0,10062 0.10005 0.09949 0.09893 0.09839

8758 8806 5829 9845 0853

12. 0 12.2 12.4 12. 6 12. 8

0.11642 0.11544 0.11449 0.11355 0.11264

62212 63616 08594 87206 90074

0.11146 0.11060 0.10977 0.10895 0.10815

42993 88096 25611 48501 50080

0.09784 0,09731 0.09678 0.09626 0.09574

8838 3770 5608 4300 9787

13. 0 13. 2 13. 4 13. 6 13. 8

0.11176 0.11089 0.11004 0.10921 0.10840

08338 33621 57995 73954 74378

0.10737 0.10660 0.10585 0.10512 0.10440

23993 64190 64916 20685 26267

0.09524 0.09474 0.09424 0.09375 0.09327

2003 0874 6323 8268 6622

14. 0 14. 2 14. 4 14. 6 14. 8

0.10761 0.10684 0.10608 0.10533 0.10461

52517 01959 16613 90688 18671

0.10369 0.10300 0.10232 0.10166 0.10101

76675 67148 93142 50311 34506

0,09280 1299 0.09233 2208 0.09186 9257 0.09141 2352 0.09096 1401

15. 0 15.2 15. 4 15.6 15. 8

0.10389 0.10320 0.10251 0.10184 0.10118

95314 15618 74813 68351 91887

0.10037 0.09974 0.09913 0.09852 0.09793

41751 68245 10348 64572 27574

0.09051 0.09007 0.08964 0.08921 0.08879

6308 6980 3321 5238 2637

16. 0 16. 2 16. 4 16.6 16. 8

0.10054 0.09991 0.09929 0.09868 0.09808

41273 12544 01906 05729 20539

0.09734 0.09677 0.09621 0.09566 0.09511

96147 67216 37828 05145 66444

0.08837 0.08796 0,08755 0,08715 0.08675

5426 3511 6802 5210 8644

17. 0 17.2 17. 4 17. 6 17. 8

0.09749 0.09691 0. 09634 0.09579 0.09524

43005 69938 98277 25085 47546

0.09458 0.09405 0.09353 0.09303 0.09252

19107 60614 88542 00560 94423

0.08636 0.08598 0,08559 0.08522 0.08484

7017 0242 8235 0911 8188

18. 0 18.2 18. 4 18. 6 18. 8

0.09470 0.09417 0.09365 0.09314 0.09264

62952 68703 62299 41336 03503

0.09203 0.09155 0.09107 0.09060 0.09014

67968 19113 45848 46237 18411

0.08447 0.08411 0.08375 0.08340 0.08305

9984 6221 6819 1701 0793

19. 0 19. 2 19.4 19. 6 19. 8

0.09214 0.09165 0.09117 0.09070 0.09023

46572 68400 66923 40151 86167

0.08968 0.08923 0.08879 0.08835 0.08792

60569 70968 47929 89829 95099

0.08270 0.08236 0.08202 0.08168 0.08135

4020 1309 2590 7792 6848

20.0

0.08978 03119

r( - 6 x 4

0.08750 62222

0.08102 9690

I

421

BESSEL FUNCTIONS O F INTEGER ORDER

MODIFIED BESSEL FUNCTIONS-ORDERS X

exKo(x)

0, 1 AND 2

exK1( 2 )

Table9.8 eZKz(x)

10.0 10.2 10.4 10.6 10. 8

0.39163 0.38786 0.38419 0.38063 0.37716

19344 02539 55846 29549 77125

0.41076 0.40644 0.40225 0.39819 0.39425

65704 68479 98277 88825 78391

0.47378 0.46755 0.46155 0.45576 0.45017

525 571 324 482 842

11.0 11.2 11.4 11.6 11.8

0.37379 0.37051 0.36731 0.36419 0.36115

54971 22156 40243 73076 86616

0.39043 0.38671 0.38309 0.37958 0.37616

09362 27920 83725 29618 21391

0.44478 0.43956 0.43452 0.42964 0.42491

294 807 427 265 496

12.0 12.2 12.4 12.6 12.8

0.35819 0.35530 0.35247 0.34972 0.34703

48784 29318 99643 32746 03081

0.37283 0.36958 0.36642 0.36334 0.36033

17534 79032 69191 53438 99192

0.42033 0.41589 0.41158 0.40739 0.40333

350 111 108 714 342

13. 0 13. 2 13.4 13. 6 13.8

0.34439 0.34182 0.33931 0.33684 0.33444

86455 59943 01806 91405 09142

0.35740 0.35454 0.35175 0.34902 0.34635

75702 53922 06397 07143 31558

0.39938 0.39554 0.39181 0.38817 0.38463

443 499 028 572 702

14. 0 14.2 14.4 14.6 14. 8

0.33208 0.32977 0.32751 0.32530 0.32312

36383 55402 49332 02091 98364

0.34374 0.34119 0.33870 0.33626 0.33387

56322 59314 19539 17039 32858

0.38119 0.37783 0.37455 0.37136 0.36824

016 131 687 346 785

15. 0 15. 2 15.4 15.6 15.8

0.32100 0.31891 0.31687 0.31486 0.31289

23534 63655 05405 36051 43424

0.33153 0.32924 0.32700 0.32480 0.32264

48949 48132 14043 31080 84361

0.36520 0.36223 0.35933 0.35650 0.35373

701 805 826 503 592

16. 0 16.2 16. 4 16. 6 16.8

0.31096 0.30906 0.30720 0.30537 0.30357

15880 42269 11919 14592 40487

0.32053 0.31846 0.31643 0.31443 0.31248

59682 43471 22766 85164 18807

0.35102 0.34838 0.34579 0.34325 0.34077

858 081 049 562 427

17. 0 17. 2 17.4 17; 6 17. 8

0.30180 0.30007 0.29836 0.29668 0.29504

80193 24678 65276 93657 01817

0.31056 0.30867 0.30682 0.30500 0.30321

12340 54888 36027 45765 74518

0.33834 0.33596 0.33363 0.33134 0.32910

464 497 361 898 956

18. 0 18. 2 18.4 18. 6 18.8

0.29341 0.29182 0.29025 0.28870 0.28718

82062 26987 29472 82654 79933

0.30146 0.29973 0.29803 0.29637 0.29472

13089 52642 84697 01096 94003

0.32691 0.32476 0.32264 0,32057 0.31854

391 064 843 602 218

19. 0 19. 2 19. 4 19.6 19.8

0.28569 0.28421 0.28276 0.28133 0.27993

14944 81554 73848 86117 12862

0.29311 0.29152 0.28996 0.28842 0.28691

55877 79458 57766 84068 51886

0.31654 0.31458 0.31266 0.31077 0.30891

577 565 076 008 262

20. 0

0.27854 48766

0.28542 54970

"-:)21

0.30708 743

[( 2 3 1

422

BESSEL FUNCTIONS O F INTEGER ORDER MODIFIED BESSEL FUNCTIONS-AUXILIARY

Table 9.8

r-lr*eZKz(z)

0.40150 0.40140 0.40129 0.40119 0.40108

9761 4058 8619 3443 8526

0.39133 0.39164 0.39195 0.39226 0.39257

9722 8743 7336 5502 3245

0.36237 0.36380 0.36523 0.36667 0.36811

579 578 854 408 237

0.39651 0.39661 0.39670 0.39680 0.39689

5620 0241 5057 0069 5278

0.40631 0.40601 0.40572 0.40543 0.40514

0355 9771 8854 7604 6017

0.43714 0.43558 0.43403 0.43247 0.43092

666 814 211 858 754

23 24

0.040 0.038 0.036 0.034 0.032

0.40098 0.40087 0.40077 0.40067 0.40056

3868 9466 5319 1424 7781

0.39288 0.39318 0.39349 0.39380 0.39410

0567 7470 3958 0032 5695

0.36955 0.37099 0.37244 0.37389 0.37534

342 722 375 302 502

0.39699 0.39708 0.39718 0.39727 0.39737

0686 6293 2101 8110 4322

0.40485 0.40456 0.40426 0.40397 0.40368

4094 1832 9230 6286 2998

0.42937 0.42783 0.42628 0.42474 0.42321

901 299 949 850 003

25 26 28 29 31

0.030 0.028 0.026 0.024 0.022

0.40046 0.40036 0.40025 0.40015 0.40005

4387 1241 8340 5684 3270

0.39441 0.39471 0.39502 0.39532 0.39562

0950 5798 0243 4286 7929

0.37679 0.37825 0.37971 0.38118 0.38264

973 716 729 012 564

0.39747 0.39756 0.39766 0.39776 0.39785

0738 7359 4186 1221 8465

0.40338 0.40309 0.40280 0.40250 0.40221

9365 5386 1058 6380 1349

0.42167 0.42014 0.41860 0.41708 0.41555

410 070 984 153 576

33 36 38 42 45

0.020 0.018 0.016 0.014 0.012

0.39995 0.39984 0.39974 0.39964 0.39954

1098 9164 7469 6009 4785

0.39593 0.39623 0.39653 0.39683 0.39714

1176 4028 6487 8556 0236

0.38411 0.38558 0.38705 0.38853 0.39001

385 474 830 453 342

0.39795 0.39805 0.39815 0.39824 0.39834

5918 3583 1460 9551 7857

0.40191 0.40162 0.40132 0.40102 0.40073

5965 0226 4130 7674 0858

0.41403 0.41251 0.41099 0.40947 0.40796

256 191 383 833 540

50 56 63 71 83

0.010 0.008 0.006 0.004 0.002

0.39944 0.39934 0.39924 0.39914 0.39904

3793 3033 2503 2202 2128

0.39744 0.39774 0.39804 0.39834 0.39864

1530 2440 2968 3116 2886

0.39149 0.39297 0.39446 0.39595 0.39744

496 915 599 546 756

0.39844 0.39854 0.39864 0.39874 0.39884

6379 5119 4077 3256 2657

0.40043 0.40013 0.39983 0.39953 0.39924

3679 6136 8226 9949 1300

0.40645 0.40494 0.40344 0.40193 0.40043

505 730 214 958 962

100 125 167 250 500

0.000

0.39894 2280

xf e - l

~

10 (2)

3

0.39894 2280 1

z$e-ZIz(z)

r-1s+ezKo(x)

<x>

r-lxielKl(z)

0.050 0.048 0.046 0.044 0.042

2-1

z+e-ZIl(s)

TABLE FOR LARGE ARGUMENTS

0.39894 228

0.39894 2280

[(-;)31

[(-:)51

~

3

0.39894 2280 1

22

0.39894 228

[( I-:

[(-:)51

20 21

m

31

1

For interpolating near 2-1 = O note that if fn(z-l) =x2e-ZIn(z) then fn(

-2-l)

=T-

'd@K,(z).

<x> =nearest integer to x.

Compiled from L. Fox, A short table for Bessel functions of integer orders and large arguments. Royal Society Shorter Mathematical Tables No. 3 (Cambridge Univ. Press, Cambridge, England, 1954) (with permission). MODIFIED BESSEL FUNCTIONS-AUXILIARY

TABLE FOR SMALL ARGUMENTS

Oil1593 0.11872 0.12713 0.14124 0.16121

152 387 128 511 862

1.00000 0.99691 0.98754 0.97158 0.94852

000 180 448 819 090

1. 0 1.1 1.2 1.3 1.4

0. 6 0. 7 0. 8 0. 9

0.18726 0.21967 0.25879 0.30504 0.35892

857 734 579 682 957

0.91759 0.87784 0.82804 0.76669 0.69201

992 980 659 810 997

1.5 1.6 1.7 1.8 1.9

0.88149 1.01045 1.15456 1.31536 1.49454

1.0

0.42102 444

0.60190 723

2.0

1.69398 200

X

0o:

0.1 0.2 0. 3 0. 4 0. 5

Ko(r)+Io(z) In z 0.42102 444 0.49199 896 0.57261 444 0.66373 364 0.76632 938

[(-;)31

436 200 879 786 429

z [ K i ( z ) - I i ( z ) In 51

0.60190 0.49390 0.36514 0.21236 +0.03176

723 093 944 381 677

-0.18096 -0.43076 -0.72326 -1.06486 -1.46281

553 964 976 242 214

-1.92535 914

423 p-zf.3(s)

r-.rf,(s)

0.0000 ( - 4 ) 1.3680 (-4)9.0273 ( . - 3 ) 2.5257 ( - 3 ) 4.9877

0.0000 ( -6) 3.4182 (-5)4.5047 (-4)1.8858 ( - 4 ) 4.9483

0.0000 ( -8) 6.8341 (-6)1.7995 ( - 5 ) 1.1281 ( -5) 3.9377

3) 1.0069

( - 5 9.9866 (-412.0719 -4) 3.7459 -4)6.1288 -4) 9.2978

1.0 1.2 1.4 1.6 1.8

I- 1

3 3.9110 3 5.3023

I- 1

-2) 2.8791 2 3.2978 -2 3.7001 ( - 2 4.0823 (-214.4421

2.0 2.2 2;4 2.6 2.8

3.0 3.2 3.4 3.6 3.8

( - 2 ) 5.3795 ( - 2 ) 5.6454 ( - 2 ) 5.8893

I1

4.0 4.2 4.4 4.6 4.8

-2 6.1124 -2 6.3161 -2 6.5015 I -21 6.6699 1-21 6;8227

5.0 5.2 5.4 5.6 5.8

-2) 6.9611 -2) 7.0861 -2) 7.1989 -2 7 3005 -2]7:3917

6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8

*

I1

0.0000 ( -13) 2.0328 (-11 4.2740 (-101 9.0201 ( - 9 ) 7.4343

0.0000 ( -15) 2.2585 (-13)9.4957 ( -11) 3.0037 ( -10) 3.2983

( - 6 ) 8.2731 5) 2.0544 5) 4.3203 f -51 8.0504 I-ij 113686

I - 71 5.8832

( - 8) 3.66.13 ( - 77/3:7225 13058

( - 9) 2.0301

1:

I- 1

( - 4)4.9334 ( - 4)6.1640

( - 3 ) 2.8291 (-3)3.2785 ( - 3 ) 3.7566 ( - 3 ) 4.2609 ( - 3 ) 4.7890

( - 4)7.5698 4 9 1545 311:0919 3) 1.2864 3)1.4986

I- 1

- 2 ' 1 4540 -2\ 1:5605 -231.6662 ( -2)1.7707 ( - 2 ) 1.8738

3) 5.3384 3) 5.9065 ( - 3 ) 6.4909

11 311.9747

(-2) 1.9752

I - 3 ) 8.3181

3) 3.1156

2 1.2402 2 1.3471

-3 9 5763 -211:0212 -2) 1.0849

-2) 2.3608

-2)4.6509 -2) 4.7401 -2) 4.8244 2 4.9040 2 4.9791

(-2)2.4516 1-212.6261 (-2 2.5401

1-211.2122 211.2756

-2 2.7096 -2)2.7907

-2 1.3387 -2 1.4012

I

( - 2 1.1486

10.5 11.0 11.5 12.0 12.5

I

1-I/

( - 3 4.1105 ( - 31 4.4637

11::

3 3 3 3 3

-

6)3.7487 6)6.7325

/

4.8261 5.1969 5.5750 5.9596 6.3499

( - 2 ) 3.5284

( - 2 ) 2.0398

-2 7.8067 -21 7.7635 -2)7.7183 -2 7 6716 -2) 716236

-2) 5.9211 -2 5 9497 -21 5:9723 -2) 5.9896 (-2) 6.0022

-2)4.4225

2 ) 6.0106

-2)4.4726

I- 1

4 1.0698 4 1.3965 4) 1.7968

11 4) 2.2703

( - 4)2.8224 ( - 4 ) 3.4578 ( - 4)4.1806

1; 1 -

4 4 4 4 4

I- 1

4.9939 5.9005 6.9020 7,9996 9.1937

(-2)6.0119 2 4.6571 \-214.6831

I- 1

6 6.5905 6 9.9425

( - 5)1.4507 ( - 5) 2.0556 ( - 5)2.8380 ( - 5) 3.8284 ( - 5)5.0587 ( - 5) 6.5607

( - 5) 8.3667 ( - 4) 1.0508

( - 4)1.9240

I-

4)2.3010 4) 2.7249

( - 4)3.1978

-

3) 1.3351 ( - 3) 1.4924 ( - 3 ) 1.6587 ( - 3)1.8337

-

4)3.7214 4)4.2971 4)4.9261 4)5.6094

( - 4) 6.3475

( - 4) 7.1409 ( - 4 7.9897 4 8.8937 4 9.8527

i

:I

( - 3 ) 1.0866

I- 1

3 1.1933

3 1.3053 ( - 3 1.4224 ( - 311.5446

I-

3) 1.6716 3) 1.8035 3)1.9399 3)2.0808 3)2.2260

(((((-

( - 2)1.0806

( - 3) 5.2694

( - 3)2.3753

I-

( - 3 ) 2.7653 ( - 3) 3.1769 ( - 3) 3.6073

-

i

( - 7)4.1199 ( - 7 81206 ( - 6) 114883

( - 3)4.4908 ( - 3) 4.7470 ( - 3 ) 5.0066

-

-2)2.7188 -2) 2.8106 -2) 2.8969 -2 2 9779 -2) 310538

9) 8.6707 8) 2.8797 8) 7.9596 7)1.9131

3 8.7663 3 9.1750 3)9.5839 3)9.9924 2)1.0400

-

-2) 3.6602 -2 3.7804 -21 3.8900 -2 3 9898 -214:0805

(-21 5.6549 5;7284 -2 5.7905 (-21 5.8425 1 -2j5i8857

I-

5) 2.7609

6.7449 7.1440 3 7.5464 3 7.9513

-

2 6.0059 I1215.9978 (-2 5.9880 2 5.9767 1145.9640

7 9.0178 6 1.9302

1- 3)3 I- 1

-2) 7.9687 -2) 7.9465 -2) 7.9182 2) 7.8848 2) 7.8474

I

3 1.7282

3) 2.2374 3 2.5157 11 312.8087

1-- 1 ( - 2 ) 5.5683

7.3263 7.2768 7.2275 7.1785 7.1300

-

-38;9445

( - 2 ) 4.5567

( -2) 7.9830

(-2 (-2 (-2 2 112

1

( - 3 ) 9.2443

10.0

15.5 16.0 16.5 17.0 17.5

5)3.0402 5 4.9776 5 7.7080 4)1.1395 4) 1.6197

(-2 2.5940 ( -2) 2.7761 ( - 2 ) 2.9523 ( - 2 ) 3.1221 ( - 2 ) 3.2854

(-2 7 9808 -2) 7:9852 -2) 7.9875 -2 7.9878 (-21 7.9862

I-

I-I-

- 3 ) 2.4106

( - 3 ) 7.2431 (-3)8.2288

-2 5 0500 -21511169 -2) 5.1800 .-2 5 2395 -215:2954

8.8

6)4.2831 I - 619.0974 1 - 5j i ; 7 m

1;

( - 514.0512 ( - 5) 5.7482 ( - 5) 7.9208

I: 1

r-.vf!,(.v)

0.0000 -11) 1.6265 - 9)1.7109 - 8) 2.4084 ( - 7) 1.4902

-2 1.6216 -2 1.8206 - 2 2.0188

2 3.4419 2 3.5916 ( - 2 ) 3.7346

r-.cf,(.v)

P-,Y~~(.Y)

0.0000 ( -9) 1.1388 ( - E ! 5.9925 (-7,5.6286 ( - 6 ) 2.6152

-

9.0 9.2 9.4 9.6 9.8

18.0 18.5 19.0 19.5 20.0

r-.~

-4)4.6097 4 6.3166 4 8.3747

2) 1.0386 ( - 2 ) 1.2283 ( - 2 ) 1.4234

I- 1

8.0 8.2 8.4 8.6

13.0 13.5 14.0 14.5 15.0

r,,(s)

P - ~ I: ( . Y )

s 0.0 0.2 0.4 0.6 0.8

-

I1-

2 11814 211:2805 2)1.3775 2)1.4722 2)1.5642 2 1 6533 211:7394 2)1.8225 211.9025 2j119794

-- 2212:1240 2 0532 - 2 2.1918 - 2 2.2567 - 2 2.3187

3) 3.9907 3) 4.2386

3) 5.9380 3)6.6192 ( - 3) 7.3082 3 8.0010 3 8.6939

I- 1 -

1-

3)4.0537 3)4.5134

3)9.3836 2 1 0068 211:0744 2)1.1410 2)1.2064

( - 3) 4.9837

2)1.2705 2)1.3333 2)1.3946 - 2)1.4543 ( - 2)1.5125

( - 3) 7.4171 ( - 3)7.9071

-

I1 I

-

3)5.4622

- 3)5.9469 - 3 6.4354 - 316.9260 3) 8.3947

-2 3.4186

\-;\::;;:: -2 4.7444

Compiled 'from British Association for the Advancement of Science, Bessel functions, Part 11. Functions of positive integer order, Mathematical Tables, vol. X (Cambridge Univ. Press, Cambridge, England, 1952) (with permission). *See page 11.

424

BESSEL FUNCTIOKS O F INTEGER ORDER

MODIFIED BESSEL FUNCTIONS-ORDERS

Table 9.9 x

ex K j(x)

@K3(%)

0.0

@KS(X)

I

6 1.4620 4 5.5388 318.7987 3)2;5064

(

2) 1.2024

2 9.8119 214.6886 f 2 2.5675

@K9(X) m

m

m

m

3-9

Ii \1

7 7.3138 6 1.3875 5 1.4730 ( 4j3.1578

II

6 1.6923 5 4.7326

3 9.9322 3 3.9756 3 1.8772

I I

2 3.6466

2 2.4157

I 1

3 2.2576 3 1.3680

f 7) 2.7197

f 4)1.6168

2.4

2.6 ~

2.8

- .

( 0)2;7276 f 1)6.8929

I

3) 1.3252

0) 8.4268 0) 7.4295

f 0) 2.2646

1 9.9723

6.0 6.2

I111 9.6194

7.0

f -1) 8.4559

10.0

(-1)6.0028

(-1)8.3395

:I I

1 7.8717 1 7.4597

1 1.0455 0 9.5723

f 1) 2.5628

I 0) 6.9684

f 1)1.5610

II

0 2.6213 0 2.5002

I 0)4.9998

( 0) 1.2674

( 0) 2.1014

8:;/0 1.1747

I

( 0) 3.7891

0 1.9059 0011.6008 1.7411

0 3.3529 0012.6956 2.9941

-1)9.4941

(-1)8.5449

I I

1

-1 9.6415 -1 9.1031

0 1.4803

( 0)1;3758

1

( 0) 7.4062

( 1)1.5639

II

0 6.3764 0 5.5518

0 2.4444 0 2.2310

I -1) 6.2106

15.5 16.0 16.5 17.0 17.5

(-1 4.1826 (-114.0829 (-1 3.9895 (-113.9017 (-1)3.8191

18.0

(-1) 3.7411

!I

0 1.4319 0 1.3480

0) 2.9275 0) 2.7150

0) 1.1442

I - 1 ) 4.5162 ( -1)4.4055 -1)4.1114

(-1 5.2559 (-11 5.1130

-1) 7.7097 -1) 7.4176 -1) 7.1482 -1)6.8990 (-1) 6.6679

I 0:1; (-1)9.1137

0) 2.5269

0) 1.3048

425

BESSEL FUNCTIONS O F INTEGER ORDER

MODIFIED BESSEL FUNCTIONS-ORDERS C

~O~S-~OZIO(Z)

1011x-11111 (.r)

10, 11, 2 0 AND 21

Table 9.10

IO-h1OKin(.c) 1024r-zoZzo(~)1026r-21121 (z) IO-~~.EZOKZO(T)

0.0 0.2 0.4 0.6 0.8

0.26911 0.26935 0.27009 0.27132 0.27305

445 920 468 457 504

1.22324 1.22426 1.22733 1.23245 1.23965

748 724 125 366 820

1.85794 1.85588 1.84970 1.83947 1.82524

560 251 867 021 326

0.391990 0.392177 0.392738 0.393674 0.394988

0.933311 0.933736 0.935008 0.937136 0.940123

6.37771 6.37435 6.36429 6.34757 6.32424

1.0 1.2 1.4 1.6 1.8

0.27529 0.27805 0.28135 0.28519 0.28961

480 517 012 648 396

1.24897 1.26045 1.27414 1.29011 1.30843

831 740 918 798 932

1.80713 1.78527 1.75981 1.73095 1.69887

290 169 781 297 992

0.396684 0.398766 0.401239 0.404112 0.407392

0.943974 0.948703 0.954321 0.960843 0.968285

6.29437 6.25807 6.21545 6.16665 6.11184

2.0 2.2 2.4 2.6 2.8

0.29462 0.30025 0.30653 0.31350 0.32118

538 682 784 170 565

1.32920 1.35250 1.37845 1.40718 1.43883

036 061 262 285 260

1.66381 1.62600 1.58569 1.54314 1.49861

982 944 822 529 645

0.411087 0.415209 0.419768 0.424778 0.430253

0.976669 0.986016 0.996351 1.007703 1.020101

6.05118 5.98488 5.91314 5.83620 5.75428

3.0 3.2 3.4 3.6 3.8

0.32963 0.33888 0.34899 0.36002 0.37203

121 455 681 459 039

1.47355 1.51153 1.55295 1.59803 1.64700

907 657 782 551 388

1.45238 1.40471 1.35587 1.30613 1.25574

126 020 192 075 432

0.436209 0.442662 0.449632 0.457139 0.465205

1.033581 1.048178 1.063935 1.080893 1.099102

5.66764 5.57655 5.48128 5.38210 5.27932

4.0 4.2 4.4 4.6 4.8

0.38508 0.39925 0.41464 0.43132 0.44940

316 889 125 237 362

1.70012 1.75766 1.81995 1.88733 1.96016

064 896 978 435 700

1.20496 1.15402 1.10314 1.05255 1.00243

150 052 736 442 944

0.473853 0.483111 0.493006 0.503569 0.514832

1.118613 1.139481 1.161768 1.185538 1.210861

5.17321 5.06408 4.95224 4.83797 4.72159

5.0 5.2 5.4 ._ 5.6 5.8

0.46899 0.49022 0.51322 0.53813 0.56513

655 387 061 536 169

2.03886 82 2.12388 83 2.21572 08 2.31490 7 1 2.42204 09

0.95298 0.90435 0.85670 0.81016 0.76484

465 626 405 129 483

0.526830 0.539601 0.553186 0.567630 0.582979

1.237813 1.266475 1.296933 1.329281 1.363622

4.60339 4.48367 4.36272 4.24084 4.11830

6.0 6.2 6.4 6.6 6.8

0.59438 0.62610 0.66050 0.69781 0.73832

965 759 400 972 033

2.53777 2.66282 2.79796 2.94406 3.10208

36 00 48 93 00

0.72085 0.67827 0.63718 0.59762 0.55964

532 767 161 235 137

0.599284 0.616599 0.634984 0.654501 0.675219

1.400061 1.438715 1.479709 1.523176 1.569259

3.99537 3.87234 3.74945 3.62695 3.50507

7.0 7.2 7.4 7.6 7.8

0.78229 0.83007 0.88201 0.93850 0.99998

881 854 663 764 773

3.27303 3.45808 3.65847 3.87560 4.11098

69 34 74 29 38

0.52326 0.48851 0.45539 0.42389 0.39401

729 672 529 854 295

0.697210 0.720554 0.745333 0.771639 0.799570

1.618113 1.669904 1.724808 1.783016 1.844734

3.38405 3.26411 3.14543 3.02821 2.91264

8.0 8.2 8.4 8.6 8.8

1.06693 1.13989 1.21945 1.30625 1.40103

936 641 007 534 829

4.36629 4.64339 4.94432 5.27132 5.62688

90 88 35 42 64

0.36571 0.33898 0.31377 0.29004 0.26776

690 159 202 783 418

0.829231 0.860735 0.894204 0.929769 0.967571

1.910180 1.979593 2.053225 2.131351 2.214264

2.79887 2.68705 2.57733 2.46983 2.36466

9.0 9.2 9.4 9.6 9.8

1.50460 1.61784 1.74175 1.87744 2.02612

429 713 933 369 620

6.01375 6.43496 6.89386 7.39417 7.93999

48 31 57 36 51

0.24687 0.22732 0.20905 0.19202 0.17616

251 134 690 382 568

1.007764 1.050510 1.095988 1.144389 1.195919

2.302281 2.395741 2.495011 2.600488 2.712593

2.26193 2.16172 2.06411 1.96916 1.87692

10.0

2.18917

062

8.53588

02

0.16142 553

1.250800

2.831786

1.78744

(-4)5 [ 6 1

(-4)4 [ 4 1

(-4)9 [ 5 1

(-418 [ 4 1

I I

2n

I n t 1 (x)= -- Zn(x) t z n - 1 (z)

2n

Kn+l(x)=+(x)

+ Kn-1 (z)

Compiled from British Association for the Advancement of Science, Bessel functions, Part 11. Functions of positive integer order, Mathematical Tables, vol. X (Cambridge Univ. Press, Cambridge, England, 1952) and L. Fox, A short table for Ressel functions of integer orders and larg? arguments. Royal Society Shorter Mathematical Tables No. 3 (Cambridge Univ. Press, Cambridge, England, 1954) (with permission).

426

BESSEL FUNCTIONS O F INTEGER ORDER

Table 9.10

MODIFIED BESSEL FUYCTIONS-ORDERS 10, 11, 20 AND 21

10.0 10.2 10.4 10.6 10.8

e-zIlo(x) 0.00099 38819 0.00107 29935 0.00115 52835 0.00124 06973 0.00132 91744

e -zI 11 (x) 0.00038 75284 0.00042 45861 0.00046 37417 0.00050 50080 0.00054 83934

35.55633 32.60759 29.98423 27.64297 25.54714

91 68 91 29 23

1.25080 1.30927 1.37160 1.43806 1.50895

2.83179 2.95856 3.09345 3.23703 3.38992

1.787443 1.700753 1.616873 1.535814 1.457578

11.0 11.2 11.4 11.6 11.8

0.00142 0.00151 0.00161 0.00171 0.00181

06490 50508 23051 23339 50559

0.00059 0.00064 0.00069 0.00074 0.00079

39013 15309 12768 31298 70766

23.66558 21.97172 20.44277 19.05917 17.80405

79 20 46 72 56

1.58462 1.66540 1.75169 1.84390 1.94249

3.55278 3.72634 3.91139 4.10876 4.31937

1.3821 60 1.309546 1.239714 1.172637 1.108279

12.0 12.2 12.4 12.6 12.8

0.00192 0.00202 0.00213 0.00225 0.00236

03870 82412 85303 11650 60548

0.00085 0.00091 0.00097 0.00103 0.00109

31003 11805 12937 34132 75097

16.66281 15.62277 14.67293 13.80364 13.00649

24 97 16 34 01

2.04795 2.16080 2.28162 2.41105 2.54975

4.54421 4.78434 5.04093 5.31521 5.60856

1.046601 0.987556 0.931095 0.877164 0.825703

13.0 13.2 13.4 13.6 13.8

0.00248 0.00260 0.00272 0.00284 0.00297

31086 22347 33415 63375 11314

0.00116 0.00123 0.00130 0.00137 0.00144

35512 15035 13301 29926 64509

12.27407 11.59989 10.97821 10.40394 9.87258

71 74 07 07 79

2.69846 2.85799 3.02921 3.21306 3.41058

5.92244 6.25845 6.61832 7.00393 7.41731

0.776652 0.729947 0.685520 0.643305 0.603230

14.0 14.2 14.4 14.6 1418

0.00309 0.00322 0.00335 0.00348 0.00361

76327 57518 53999 64894 89341

0.00152 0.00159 0.00167 0.00175 0.00183

16634 85870 71776 73898 91776

9.38015 8.92308 8.49819 8.10265 7173392

52 36 79 95 53

3.62289 3.85121 4.09686 4.36131 4164613

7.86068 8.33644 8.84722 9.39585 9.98543

0.565225 0.529218 0.495137 0.462910 0.432464

15.0 15.2 15.4 15.6 15.8

0.00375 0.00388 0.00402 0.00416 0.00429

26491 75510 35583 05908 85705

0.00192 0.00200 0.00209 0.00218 0.00227

24942 72921 35235 11403 00942

7.38971 7.06797 6.76684 6.48467 6.21995

31 04 87 94 46

4.95305 5.28394 5.64087 6.02608 6.44202

10.61932 11.30119 12.03503 12.82520 13.67643

0.403728 0.376630 0.351101 0.327070 0.304470

16.0 16.2 16.4 16.6 16.8

0.00443 0.00457 0.00471 0.00485 0.00500

74209 70675 74378 84612 00690

0.00236 0.00245 0.00254 0.00263 0.00273

03366 18192 44936 83118 32259

5.97130 5.73750 5.51741 5.31001 5.11438

87 35 43 78 19

6.89137 7.37705 7.90228 8.47055 9.08571

14.59389 15.58322 16.65059 17.80271 19.04691

0.283235 0.263299 0.244598 0.227071 0.210658

17.0 17.2 17.4 17.6 17.8

0.00514 0.00528 0.00542 0.00557 0.00571

21947 47735 77427 10418 46119

0.00282 0.00292 0.00302 0.00312 0.00322

91884 61523 40709 28982 25887

4.92965 4.75506 4.58989 4.43349 4.28527

63 40 42 60 20

9.75197 10.47392 11.25663 12.10562 13.02697

20.39124 21.84444 23.41611 25.11674 26.95781

0.195301 0.180944 0.167532 0.155012 0.143336

18.0 18.2 18.4 18.6 18.8

0.00585 0.00600 0.00614 0.00629 0.00643

83964 23403 63909 04971 46098

0.00332 0.00342 0.00352 0.00362 0.00373

30977 43808 63948 90969 24450

4.14467 4.01119 3.88437 3.76378 3.64903

40 75 85 89 41

14.02734 15.11406 16.29515 17.57946 18.97668

28.95188 31.11272 33.45541 35.99648 38.75407

0.132454 0.122321 0.112891 0.1 04124 0.095978

19.0 19.2 19.4 19.6 19.8

0.00657 0.00672 0.00686 0.00701 0.00715

86817 26672 65226 02059 36768

0.00383 0.00394 0.00404 0.00415 0.00425

63982 09161 59590 14885 74667

3.53974 3.43559 3.33626 3.24146 3.15093

93 74 62 65 00

20.49749 22.15363 23.95803 25.92489 28.06989

41.74804 45.00024 48.53460 52.37745 56.55768

0.088414 0.0 81397 0.074892 0.068865 0.063285

20.0

0.00729 68965

3.06440 75 [( 3 1 4 1

30.41029

61.10706

0.058124 -;)41

2

0.00436 38567

e W 1 o(2)

10 2 4-~2oI2 0 ( 2 ) 1026x- 2112I ( $ ) 1.0- 2 h 2 0 K 2 o ( s )

[(-;PI[( -52151

[I(

427

BESSEL FUNCTIONS OF INTEGER ORDER

MODIFIED BESSEL FUNCTIONS-AUXILIARY 1

1

TABLE FOR LARGE ARGUMENTS 1

1

1

Table 9.10 1

In [zze-zZ~o(z)] In [zge-zZll(z)] In [ ~ - 1 z z e z K l o ( z ) ] In [zge-zZ2o(z)] In [zze--zZ21(z)] In [r-lzzezKzo(z)]

0.050 0.049 0.048 0.047 0.046

-3.42244 -3.37316 -3.32386 -3.27447 -3.22501

002 689 306 055 139

-3.93653 -3.87762 -3.81861 -3.75949 -3.70026

292 888 524 454 938

1.47299 1.42771 1.38232 1.33681 1.29118

048 939 785 644 575

-10.434749 -10.263511 -10.091302 - 9.918126 - 9.743983

-11.346341 -11.160467 -10.973471 -10.785351 -10.596108

8.250182 8.088946 7.926737 7.763551 7.599386

<x> 20 20 21 21 22

0.045 0.044 0.043 0.042 0.041

-3.17548 -3.12590 -3.07625 -3.02655 -2.97678

766 147 496 033 979

-3.64094 -3.58151 -3.52199 -3.46237 -3.40267

242 639 408 835 211

1.24543 1.19956 1.15358 1.10748 1.06126

642 910 449 332 635

-

-10.405744 -10.214259 -10.021658 - 9.827944 - 9.633121

7.434240 7.2 68110 7.100994 6.932893 6.763806

22 23 23 24 24

0.040 0.039 0.038 0.037 0.036

-2.92697 -2.87711 -2.82719 -2.77723 -2.72722

559 002 539 405 837

-3.34287 -3.28300 -3.22304 -3.16300 -3.10288

833 006 039 246 949

1.01493 0.96848 0.92192 0.87525 0.82847

437 822 874 686 349

6.593733 6.422673 6.250630 6.077603 5.903597

25 26 26 27 28

0.035 0.034 0.033 0.032 0.031

-2.67718 -2.62709 -2.57696 -2.52681 -2.47661

076 365 948 074 992

-3.04270 -2.98245 -2.92213 -2.86175 -2.80131

472 146 308 298 461

0.78157 0.73457 0.68746 0.64024 0.59291

961 624 441 520 975

5.728614 5.552659 5.375732 5.197843 5.018998

29 29 30 31 32

0.030 0.029 0.028 0.027 0.026

-2.42639 -2.37615 -2.32588 -2.27558 -2.22527

955 216 032 659 356

-2.74082 -2.68027 -2.61968 -2.55904 -2.49837

147 709 504 894 243

0.54548 0.49795 0.45031 0.40257 0.35474

920 475 764 915 059

4.839203 4.658466 4.476796 4.294202 4.110696

33 34 36 37 38

0.025 0.024 0.023 0.022 0.021

-2.17494 -2.12460 -2.07424 -2.02388 -1.97351

384 002 475 063 031

-2.43765 -2.37691 -2.31613 -2.25533 -2.19451

918 291 733 620 329

0.30680 0.25876 0.21063 0.16241 0.11409

331 871 822 332 551

- 9.392809 - 9.215785 - 9.037810 - 8.858889 - 8.679029 -- 8.498236 8.316519 - 8.133888 - 7.950352 - 7.765923 - 7.580613 - 7.394434 - 7.207403 - 7.019533 - 6.830842 - 6.641348 - 6.451070 -- 6.260027 6.068243 - 5.875738 - 5.682539

3.926290 3.740995 3.554826 3.367799 3.179929

40 42 43 45 48

0.020 0.019 0.018 Oi017 0.016

-1.92313 -1.87276 -1.82238 -1.77201 -1.72165

643 162 853 979 806

-2.13367 -2.07281 -2.01195 -1.95107 -1.89020

239 731 186 986 514

0.06568 +0.01718 -0.03139 -0.08007 -0.12883

636 745 959 306 128

2.991233 2.801730 2.611440 2.420383 2.228582

50 53 56 59 63

0.015 0.014 0.013 0.012 0.011

-1.67130 -1.62096 -1.57064 -1.52033 -1.47004

595 610 113 365 626

-1.82933 -1.76846 -1.70760 -1.64675 -1.58592

153 286 295 564 472

-0.17767 -0.22659 -0.27559 -0.32467 -0.37383

247 485 659 581 061

2.036059 1.842840 1.648949 1.454415 1.259264

67 71 77 83 91

0.010 0.009 0.008 0.007 0.006

-1.41978 -1.36954 -1.31933 -1.26914 -1.21900

154 207 040 908 063

-1.52511 -1.46432 -1.40356 -1.34284 -1.28214

400 725 824 072 841

-0.42305 -0.47235 -0.52172 -0.57116 -0.62066

904 911 881 608 881

1.063526 0.867231 0.670412 0.473099 0.275328

100 111 125 143 167

0.005

-._

754

0.003 0.002 0.001

-1.16888 -1.11881 - -- - - -1.06877 -1.01878 -0.96883

735 514 808

-1.22149 499 -1.16088 414 - _ - - _. -1.10031 949 -1.03980 463 -0.97934 314

-0.67023 -0.71986 -0i76954 -0.81929 -0.86908

489 215 839 138 886

200 250 333 500 1000

0.000

-0.91893 853

-0.91893 853

-0.91893 853

+0.077133 -0.121451 -0.320388 -0.519640 -0.719170 -0.918939

z-1

0.004 -

229 --

(-511

-

9.568876

5.488669 5.294155 5.099025

-- 4.903309 4.707035 -- 4.312943 4.510235 - 4.115190 - 3.917011 -- 3.718443 3.519520 - 3.320281 - 3.120763 - 2.921004 - 2.721043 - 2.520921 - 2.320676 - 2.120350 - 1.919982 - 1.719613 - 1i519284 - 1.319036 - 1.118907 -

0.918939

- 9.437195 - 9.240173 --- 9.042063 8.842873 8.642612 - 8.441293 - 8.238927 - 8.035529 - 7.831113 - 7.625695 - 7.419294 - 7.211929 - 7.003620 -- 6.794389 6.584261 - 6.373261 - 6.161416 - 5.948754 - 5.735305 -

5.521102

- 5.306177 - 5.090565

-- 4.874302 4.657427

- 4.439978 - 4.221995 --- 4.003521 3.784599 3.565272

- 3.345586

- 3.125587 - 2.905322 - 2.68483.8 - 2.464184 - 2.243408 - 2.022558 - 1.801685 - 1.580838 - 1.360065 - 1.139416 - 0.918939

00

(-4)1

=nearest integer to :c. Compiled from L. Fox, A short table for Bessel functions of integer orders and large arguments. Royal Society Shorter Mathematical Tables No. 3 (Cambridge Univ. Press, Cambridge, England, 1954) (with permission).

BESSEL FUNCTIONS OF INTEGER ORDER

Table 9.11

MODIFIED BESSEL FUNCTIONS-VARIOUS

n

ORDERS

Zn(1)

0 1 2 3 4

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 30 40 50 100

(-189)8.47367 4008

(-158)l. 08217 1475

(-119)7.09355 1489

I n (100)

I n ( 10)

(42)1.07375 171

5 6 7 8 9

I

217.77188 2864

10 11 12 13 14 15 16 17 18 19 20 30 40 50 100

(-88)1. 08234 4202

(+lO)1.76508 024

(36b4.82195 809

( -16)2.72788 795

(21)4.64153 494

429

BESSEL FUNCTIONS O F INTEGER ORDER

MODIFIED BESSEL FUNCTIONS-VARIOUS Kn(1)

Kn(2)

ORDERS

Table 9.11 Kn(5)

10 11 12 13 14 15 16 17 18 19 20 30 40 50 100

(185)5.90033 3184

( 155)4.61941 5978

( 115)7.03986 0193

Kn(100)

Kn( 10)

8 9 10 11 12 13 14 15 16 17 18 19 20 30 40 50 100

I1 2 9 17 27

1.78744 2.03024 5.93822 2.06137

2782 7813 4681 3775

(85)4.59667 4084

(+13)1.63940 352

(-25)7.61712 963

430

BESSEL FUNCTIONS O F INTEGER ORDER Table 9.12

KELVIN FUNCTIONS-ORDERS

ber

r

bei x

R'

0 AND 1

berl

beil x

1'

0. 0 0.1 0.2 0.3 0.4

1.00000 0.99999 0.99997 0.99987 0.99960

00000 84375 50000 34379 00044

0.00000 0.00249 0.00999 0.02249 0.03999

00000 99996 99722 96836 82222

0.00000 -0.03539 -0.07106 -0.10725 -0.14423

00000 95148 36418 47768 08645

0.00000 0.03531 0.07035 0.10486 0.13857

00000 11265 65360 83082 41359

0.5 0.6 0. 7 0. 8 0.9

0.99902 0.99797 0.99624 0.99360 0.98975

34640 51139 88284 11377 13567

0.06249 0.08997 0.12244 0.15988 0.20226

32184 97504 89390 62295 93635

-0.18224 -0.22153 -0.26233 -0.30485 -0.34931

31238 37177 33470 87511 01000

0.17119 0.20244 0.23202 0.25962 0.28491

51797 39824 24623 00070 16898

1. 0 1.1 1.2 1.3 1.4

0.98438 0.97713 0.96762 0.95542 0.94007

17812 79732 91558 87468 50567

0.24956 0.30173 0.35870 0.42040 0.48673

60400 12692 44199 59656 39336

-0.39586 -0.44469 -0.49591 -0.54964 -0.60594

82610 19268 45913 13636 56099

0. 30755 0.32719 0.34345 0.35593 0.36421

66314 65305 43903 34649 64560

0.92107 21835

00623 56770 72924 19548 23406

-0.66486 -0; 72639 -0.79050 -0.85709 -0.92601

54180 98786 51846 05470 39357

0.36786 0.36641 0.35939 0.34630 0.32660

49890 93986 88584 18876 72722

1-. .5 -

1.6 1.7 1.8 1.9

0.86997 12370 0.83672 17942 0.79752 41670

0.55756 0; 63272 0.71203 0.79526 0.88212

2.0 1 2.2 2.3 2.4

0.75173 0; 69868 0.63769 0.56804 0.48904

41827 50oi4 04571 89261 77721

0.97229 1.06538 1.16096 1.25852 1.35748

16273 81608 99438 89751 54765

-0.99707 -1.07002 -1.14452 -1.22020 -1.29657

76519 37462 92997 15903 31717

0.29977 0.26525 0.22246 0.17082 0.10976

54370 03092 17120 83322 13027

2.5 2.6 2.7 2.8 2.9

0.39996 0.30009 0.18870 +O. 06511 -0. 07136

84171 20903 63040 21084 78258

1.45718 1.55687 1.65574 1.75285 1.84717

20442 77737 24073 05638 61157

-1.37309 -1.44914 -1.52398 -1.59680 -1.66670

68976 09315 37854 94413 26139

+O. 03866 -0.04304 -0.13594 -0.24062 -0.35762

84440 07916 96285 74875 26713

3.0 3.1 3.2 3.3 3.4

-0.22138 -0.38553 -0.56437 -0.75840 -0.96803

02496 14550 64305 70121 89953

1.93758 2.02283 2.10157 2.17231 2.23344

67853 90420 33881 01315 57503

-1.73264 -1.79350 -1.84805 -1.89492 -1.93265

42211 71373 23125 53482 36306

-0.48745 -0.63060 -0,78750 -0.95851 -1.14396

41770 25952 00586 92089 11510

3.5 3.6 3.7 3.8 3.9

-1.19359 -1.43530 -1.69325 -1.96742 -2.25759

81796 53217 99843 32727 94661

2.28324 2.31986 2.34129 2.34543 2.33002

99669 36548 77145 30614 18823

-1.95964 -1.97418 -1.97443 -1.95842 -1.92410

41313 19924 00262 92665 07174

-1.34404 -1.55888 -1.78847 -2.03271 -2.29130

23731 06139 96677 31257 70630

4.. 0 .

4.1 4.2 4... 3 4.4

-2.56341 -2; 88430 -3.21947 -3.56791 -3.92830

65573 57320 98323 08628 66215

2.29269 2.23094 2.14216 2.02364 1.87256

03227 27803 79867 70694 37958

-1.86924 -1.79156 -1.68863 -1.55794 -1.39689

84590 42730 39648 55649 95997

-2.56382 -2.84963 -3.14790 -3.45759 -3.77737

16886 19932 74393 07560 59182

4.5 4.6 4.7 4.8 4.9

-4.29908 -4.67035 -5.06388 -5.45307 -5.84294

65516 69372 55867 61749 24419

1.68601 1.46103 1.19460 0.88365 0.52514

72036 68359 07968 68537 68109

-1.20282 -0.97297 -0.70458 -0.39486 -0.04099

16315 72697 98649 10961 46681

-4.10568 -4.44064 -4.78006 -5.12141 -5.46179

54084 68813 99721 92170 58790

5.0

-6.23008 24787

z:

0: 89789 ii186

"-p]

0.11603 43816

[(-;I21

KELVIN FUNCTIONS-AUXILIARY J

0. 0 0.1 0.2 0. 3 0.4 0. 5

ker z+ber I In 0.11593 0.11789 0.12374 0.13339 0.14669

kei t +bei I In

I

1516 2485 5076 8210 9682

-0.78539 -0.78260 -0.77421 -0.76019 -0.74045

0.16343 5574

~

5

I

8163 7108 9267 0919 0212

-0.71489 8693

1

[(-y]

[(-p]

+O. 35977 66668

-5.79790 79018

[( 5 1 2 1

TABLE FOR SMALL ARGUMENTS

(ken 1 +ben -0.70710 -0.70651 -0.70486 -0.70248 -0.69994

L

In I )

6781 7131 2164 3157 6658

-0.69804 1049

[

( 6 ) ;

13

1

Peii ztbeii -0.70710 -0.70215 -0.68733 -0.66272 -0.62851

1

1 )

6781 4903 0339 8003 1738

['-"'I

-0.58492 2770

ComDiled from National Bureau of Standards, Tables of the Bessel functions Jo(z) and J d z ) for complex arguments, 2d ed. (Columbia Univ. Press, New York, N.Y., 1947) and National Bureau of Standards, Tables of the Bessel functions Yo(z) and Y I ( z for ) complex arguments (Columbia Univ. Press, New York, N.Y., 1950) (with permission).

431

BESSEL FUNCTIONS O F INTEGER ORDER KELVIN FUNCTIONS-ORDERS 7:

ker

0. 0

m

z

kei .r

0 AND 1

ker,

Table 9.12

kei, n.

2

-

-7.14668 -3.63868 -2.47074 -1.88202

1711 3342 2357 4050

-6.94024 -3.32341 -2,08283 -1.44430

2153 7218 4751 5150

-

0. 3 0.4

2.42047 1.73314 La33721 1.06262

3980 2752 8637 3902

-0.78539 -0.77685 -0.75812 -0.73310 -0.70380

0. 5 0. 6 0. 7 0.8 0. 9

0.85590 0.69312 0.56137 0.45288 0.36251

5872 0695 8274 2093 4812

-0.67158 -0.63744 -0.60217 -0.56636 -0.53051

1695 9494 5451 7650 1122

-1.52240 -1.27611 -1.09407 -0.95203 -0.83672

3406 7712 2943 2751 7829

-1.05118 -0.78373 -0.59017 -0.44426 -0.33122

2085 8860 5251 9985 6820

1.0 1.1 1.2 1.3 1.4

0.28670 0.22284 0.16894 0.12345 0.08512

6208 4513 5592 5395 6048

-0.49499 -0.46012 -0.42616 -0.39329 -0.36166

4636 9528 3604 1826 4781

-0.74032 -0.65791 -0.58627 -0.52321 -0.46718

2276 0729 4386 5989 3076

-0,24199 -0.17068 -0.11325 -0.06683 -0.02928

5966 4462 6800 2622 3749

0.05293 4915 0.02602 9861

5562 5474 8834 7069 2235

-0.41704 -0.37195 -0.33125 -0.29442 -0.26105

4285 1238 0485 5803 9495

+O. 00100 0,02530 0.04461 0.05974 0.07137

8681 6776 5190 7779 3592

-0.23080 5929

9398 3202 1619 2940 7161

0.1 0; 2

8163 0646 4933 1912 0212

m

(D

1.5 1. 6 1.7 1.8 1.9

+OS 00369 1104 -0.01469 6087 -0.02966 1407

-0.33139 -0.30256 -0.27522 -0.24941 -0.22514

2.0 2.1 2.2 2.3 2.4

-0.04166 -0.05110 -0.05833 -0.06367 -0.06737

4514 6500 8834 0454 3493

-0.20240 -0.18117 -0.16143 -0.14313 -0.12624

0068 2644 0701 5677 1488

-0.17850 9812 -0.15599 6054 -0.13563 6638

0.08004 0;08624 0.09035 0.09271 0.09361

2. 5 2. 6 2.7 2. 8 2.9

-0.06968 -0.07082 -0.07097 -0.07029 -0.06893

7972 5700 3560 6321 9052

-0.11069 -0.09644 -0.08342 -0.07157 -0.06082

6099 2891 1858 0646 5473

-0.11725 -0.10069 -0.08580 -0.07246 -0.06052

6136 5314 8451 1339 9755

0.09331 0;09201 0.08991 0.08716 0.08391

3788 8037 5810 7762 2666

3. 0 3.1 3.2 3. 3 3. 4

-0.06702 -0.06467 -0.06198 -0.05903 -0.05589

9233 8610 4833 2916 6550

-0.05112 -0.04239 -0.03458 -0.02761 -0.02144

1884 5446 2313 9697 6287

-0.04989 -0.04045 -0.03211 -0.02476 -0.01832

8308 9533 3183 5662 9556

0.08027 0.07634 0,07222 0.06797 0.06367

0223 3451 0724 7529 7999

3. 5 3. 6 3. 7 3. 8 3. 9

-0.05263 -0.04931 -0.04597 -0.04264 -0.03937

9277 5556 1723 6864 3608

-0.01600 -0.01123 -0.00707 -0.00348 -0.00041

2568 1096 6704 6665 0809

-0.01272 -0,00787 -0.00370 -0; 00014 +O. 00285

3249 0585 0576 7138 1155

0.05937 0.05511 0. 05093 0;04687 0. 04294

6256 7592 9514 2681 1728

4.0 4.1 4.2 4. 3 4.4

-0.03617 -0.03308 -0.03010 -0.02726 -0.02455

8848 4395 7574 1764 6892

+O. 00219 8399

4.5 4. 6 4.7 4.8 4. 9

-0.02199 -0.01959 -0.01734 -0.01524 -0.01330

9875 5024 4409 8188 4899

5.0

-0.01151 1727

0.00438 0,00619 0.00766 0.00882

5818 3613 1269 5624

0.00535 0.00740 0.00906 0. 01037 0.01136

1296 6063 4226 0752 6998

0. 03916 0.03556 0,03213 0.02889 0.02585

6011 0272 5235 8142 3229

0,00972 0.01037 0.01082 0,01109 0.01120

0918 8865 8725 7399 9526

0,01209 0.01257 0.01285 0.01296 0.01291

0904 7182 7498 0651 2753

0.02300 0.02034 0.01787 0.01559 0.01349

2160 4409 7607 7847 9960

0.01118 7587

2

ker z+ber z In .Z 0.16343 0.18332 0.20604 0.23116 0.25823

5574 9435 1279 6407 4099

1.0

0.28670 6208

[(-;)41

kei .u+bei c In -0.71489 -0,68341 -0.64584 -0.60204 -0.55182

0.01273 7390

0.01157 7754

TABLE FOR SMALL ARGUMENTS

KELVIN FUNCTIONS-AUXILIARY 0. 5 0.6 0.7 0.8 0.9

-0; 20337 3135

,I’

8693 3456 9920 5231 2327

-0,49499 4636

.t’(kerl a+berl x In .I,) .r(keil a+beil r In .r) -0.69804 -0. b9777 -0.70035 -0.70720 -0.71993

1049 1567 3648 4389 1903

-0.58492 2770 -0.53229 1460 -0.47105’2294 -0.40176 2012 -0.32512 0736

-0.74032 2276

-0.24199 5966

432

BESSEL FUNCTIONS OF INTEGER ORDER Table 9.12

KELVIN FUNCTIONS-MODULUS

berl

ber x=Mo(z) cos OO(F) bei ~ = X o ( T )sin On).(

844

c =Jfi(r)

AND PHASE

cos ~ I ( I ) beil z =XI(z) sin 81 ( T )

0.0 0.2 0.4 0. 6 0.8

lTfn(x) 1.000000 1.000025 1. 000400 1.002023 1.006383

0.000000 0.010000 0.039993 0.089919 0.159548

,If 1 ( T ) 0.000000 0.100000 0.200013 0.300101 0.400427

2.356194 2.361194 2.376194 2.401189 2.436166

1.0 1.2 1.4 1.6 1.8

1.015525 1.031976 1.058608 1.098431 1.154359

0.248294 0.354999 0.477755 0.613860 0.759999

0.501301 0.603235 0.706982 0.813585 0.924407

2.481086 2.535872 2.600386 2.674406 2.757605

2.0 2.2 2. 4 2. 6 2.8

1.229006 1.324576 1.442891 1.585536 1.754059

0.912639 1,068511 1.225011 1.380379 1.533667

1.041167 1.165949 1.301211 1.449780 1.614838

2.849536 2.949617 3.057139 3.171285 3,291160

3. 0 3.2 3. 4 3.6 3.8

1.950193 2.176036 2.434210 2.127979 3.061341

1.684559 1.833156 1.979784 2.124854 2.268771

1.799908 2.008844 2.245840 2.515453 2.822653

3.415839 3.544415 3.676044 3.809981 3.945601

4.0 4. 2 4. 4 4; 6 4. 8

3.439118 3.867032 4.351791 4.901189 5.524209

2.411887 2.554483 2.696771 2.838893 2.980942

3.172896 3.572227 4.027393 4.545990 5.136619

4.082407 4.220023 4.358179 4.496691 4.635441

5.0 5.2 5. 4 5; 6 5.8

6.231163 7.033841 7.945700 8. 982083 10.160473

3.122970 3.265002 3.407044 3.549094 3.691142

5.809060 6.574474 7.445618 8.437083 9.565568

4.774362 4.913417 5.052589 5.191872 5.331267

6. 0 6.2 6.4 6. 6 6. 8

11.500794 13.025757 14.761257 16.736836 18.986208

3.833179 3.975197 4.117190 4.259152 4.401083

10.850182 12.312791 13.978402 15.875614 18.037122

5.470772 5.610390 5.750117 5.889950 6.029884

7. 0

21.547863

4.542982

20.500302

6.169913

7741

(-312 [ 8 1

(-2)4 [ 6 1

T

KELVIN FUNCTIONS-MODULUS

81 (1.)

AND PHASE FOR LARGE ARGUMENTS

(4- (.NZ)

<.('>

-0.40758 -0.40644 -0.40534 -0.40427 -0.40323

0.38359 0.38457 0.38556 0.38655 0.38755

1.22254 1.21922 1.21598 1.21280 1.20968

7

0.40246 0.40211 0.40176 0.40141 0.40106

-0.40221 -0.40119 -0.40019 -0.39921 -0.39824

0.38856 0.38957 0.39060 0.39162 0.39266

1.20660 1.20356 1.20057 1.19762 1.19471

10 11 13 14 17

0.05 0. 04 0. 03 0. t2 0.01

0.40071 0.40035 0.40000 0.39965 0.39930

-0.39728 -0.39634 -0.39541 -0.39449 -0.39359

0.39369 0.39474 0.39578 0.39683 0.39789

1.19184 1.18901 1.18622 1.18348 1,18077

20 25 33 50 100

0. 00

0.39894

-0.39270

0.39894

1.17810

m

.c- 1

0.15 0.14 0.13 0.12 0.11

0.40418 0.40383 0; 40349 0.40315 0.40281

0.10 0. 09 0. 08 0. 07 0. 06

8 o ( d - (xi421

81

(-511

<x>=nearest integer to 2.

7

a

8 9

433

BESSEL FUNCTIONS O F INTEGER ORDER

Table 9.12 KELVIN FUNCTIONS-MODULUS AND PHASE kerl .r =NI(c) cos $1 (z) ker x=No (z) cos $0 (x) kei, c =NI (z) sin $1 (3) kei z=No (x) sin $0 (z) z ’ No(x) +o (4 311 (z) $1 (4 n n m 0.000000 m -2.356194 __.. ~... 0. 2 1.891702 -0.412350 4.927993 -2.401447 1.274560 -0.584989 2.312347 -2.487035 0. 4 1.497572 -2.590827 -0.743582 0.941618 0.6 1.050591 -2.704976 -0.896284 0.725172 0. 8 1 .

-

1. 0 1.2 1.4 1. 6 1.8

0.572032 0.458430 0.371548 0.303683 0.249850

-1.045803 -1.193368 -1.339631 -1.484977 -1.629650

0.778870 0.597114 0.468100 0.372811 0.300427

-2.825662 -2.950763 -3.078993 -3,209526 -3.341804

2.0 2.2 2.4 2. 6 2. a

0.206644 0.171649 0.143095 0.119656 0.100319

-1.173813 -1.917579 -2. 061029 -2.204225 -2.347212

0.244293 0.200073 0.164807 0.136407 0.113353

-3.475437 -3.610143 -3.745715 -3. 881994 -4.018860

3. 0 3. 2 3.4 3.6 3. a

0.084299 0.070979 0.059870 0.050578 0. 042789

-2.490025 -2.632692 -2.775236 -2.917672 -3.060017

0.094515 0.079039 0.066264 0.055677 0.046873

-4.156217 -4.293990 -4.432118 -4.570551 -4.709250

4.0 4.2 4. 4 4. 6 4.8

0.036246 0.030738 0.026095 0.022174 0.018859

-3.202283 -3.344478 -3.486612 -3.628692 -3.770724

0.039530 0.033389 0.028242 0.023918 0.020280

-4.848179 -4.987312 -5.126623 -5.266093 -5.405705

5.0 5.2 5.4 5. 6 5. a

0.016052 0.013674 0.011656 0.009942 0.008485

-3.912712 -4.054662 -4.196576 -4.338460 -4.480314

0.017213 0.014624 0.012435 0.010583 0.009013

-5.545443 -5.685295 -5.825250 -5.965298 -6.105430

6. 0 6. 2 6.4 6. 6 6. a

0.007246 0.006191 0.005292 0.004526 0.003872

-4.622142 -4.763947 -4.905730 -5.047493 -5. 189238

0.007682 0.006551 0.005590 0.004773 0.004077

-6.245638 -6.385917 -6.526260 -6.666662 -6. 807119

7.0

0.003315

-5.330966

0.003485

-6.947625

KELVIN FUNCTIONS-MODULUS 2-1

AND PHASE FOR LARGE ARGUMENTS

z f e ~ ’ ~ 2 N o ( x ) $o(x)+(x/iJz)

.fez/JZiV1

(z)

7 7

$1 (Z)+(Z/&)

0.15 0.14 0.13 0.12 0.11

1.23695 1.23802 1.23909 1.24017 1.24125

-0.38070 -0.38142 -0.38217 -0.38291 -0.38367

1.30377 1.30039 1.29701 1.29363 1.29024

-1.99943 -1.99725 -1.99505 -1.99281 -1.99055

0.10 0. 09 0.08 0.07 0.06

1.24233 1.24342 1.24451 1.24560 1.24670

-0.38444 -0,38522 -0.38600 -0.38680 -0.38761

1.28687 1.28349 1.28012 1.27675 1.27339

-1.98825 -1.98592 -1.98357 -1.98118 -1.97876

0.05 0.04 0. 03 0.02 0.01

1.24779 1.24889 1.25000 1.25110 1.25221

-0.38843 -0.38926 -0.39010 -0.39096 -0.39182

1.27002 1.26667 1.26332 1.25998 1.25664

-1.97630 -1.97381 -1.97128 -1.96872 -1.96613

20 25 33 50 100

0. 00

1.25331

-0.39270

1.25331

-1.96350

m

=nearest integer to 2.

a

a

9

.#

10 11 13 14 17

10. Bessel Functions of Fractional Order H. A. ANTOSIEWICZ'

Contents Mathematical Properties . . . . . . . . . 10.1. Spherical Bessel Functions . . . . 10.2. Modified Spherical Bessel Functions 10.3. Riccati-Bessel Functions . . . . . 10.4. Airy Functions . . . . . . . . . Numerical Methods . . . . . . . . . . 10.5. Use and Extension of the Tables . References . . . . . . . .

.

.. .. ...

Table 10.1. Spherical Bessel Functions-Orders j n ( 4 , Yn(x)

. . . . . . . . . . . . . . . ... . . . ...

. . . . . .

. . . .

.. . . . . .. . . .

.. . . . . .. . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . .

Page

437 437 443 445 446

.

452 452

. . 0, 1, and 2 (0 51:5 10) .

455

..

457

n=O, 1, 2; ~ = 0 ( . 1 ) 5 , 6-8S, ~=5(.1)10, 5s

Table 10.2. Spherical Bessel Functions-Orders 3-10 (05x_, Yn(x)

..

459

n=3(1)8; ~=0(.1)10, 5s

~ - ~ j n ( xn+'yn(x) ~), n=9, 10; ~=0(.1)10, 7-8s 463

Table 10.3. Spherical Bessel Functions-Orders 20 and 21 (0 5 x 1 2 5 ) exp (2/(4n+2))jn(x) xn+' e ~ (-z2/(4n+2))yn(x) p n=20, 21; ~=0(.5)25, 6-8s Table 10.4. Spherical Bessel Functions-Modulus 9 , 1 0 , 2 0 and 2 1 . . . . . . . . . . . . . &Mn++ (2) on+i (2)-X where j n ( x > =dGFMn++ - (2) COS on++ (2) y n ( x ) = . \ 1 b / X M n + + ( x ) sin on++ (XI n=9, 10; ~"=.1(-.005)0, 8D n=20, 21; ~-'=.04(-.002)0, 8D

and Phase-Orders

..........

464

.

465

Table 10.6. Zeros of Bessel Functions of Half-Integer Order (0 ,
467

Table 10.5. Spherical Bessel Functions-Various jn(4, yn(4 n=0(1)20, 30, 40, 50, 100 x=1, 2, 5, 10, 50, 100, 10s

~ = n + & ,n=0(1)19,

Orders (0 Ln 5 100)

6-7D

Table 10.7. Zeros of the Derivative of Bessel Functions of Half-Integer Order (O
.

1

National Bureau of Standards.

468

(Presently, University of Southern California.) 435

BESSEL FUNCTIONS OF FRACTIONAL ORDER

436

Page

Table 10.8. Modified Spherical Bessel Functions-Orders 0, 1 and 2 (05x55).

..........................

469

dmIn+t(5), dG&Kn+i (2) n=O, 1, 2; ~=0(.1)5, 4-9D

Table 10.9. Modified Spherical Bessel Functions-Orders 9 and 10

- . . . . . . . . . . . . . . . . . . . . . . . . . .

(0 2x5 a).

470

x-nJ+n/xIn+t(2) xn+1J3?r/zKn+i(2) n=9, 10; ~=0(.1)5, 7-8s e-’In+t (x), ( 2 / ~8)K n + t (2) n=9, 10; ~=5(.1)10, 6 s 4% exp [-x+n(n+ 1)/(2Wn+,(4 4G exp [x-n(n+ 1>/(2x>IKn+t(x) n=9, 10; ~”=.1(--.005)0, 7-8s Table 10.10. Modified Spherical Bessel Functions-Various Orders (O
473

(x), M x K n + t(2) n=0(1)20, 30, 40, 50, 100 x=1, 2, 5, 10, 50, 100, 10s Table 10.11. Airy Functions (05x5a) .

475

&+In+i

. . . . . . . . . . . . .

Ai(z), Ai’(x), Bi(z), Bi’(x) ~=0(.O1)1, 8D

Ai( -x), Ai’( -x), Bi( -x), Bi’( -z) x=O( .Ol) 1(.1)10, 8D Auxiliary Functions for Large Positive Arguments Ai (2)=+~-‘/~e-y( -4) ; Bi (2)= ~ - ‘ / ~ e y4)( Ai’(z) = -3x1/4ee-4g(-4); Bi’(x) erg([) f (f4) , g ( f4) ; &=$$I2, [-’=1.5(-.1) .5( -.05)0, 6 D Auxiliary Functions for Large Negative Arguments Ai(--2)=x-”4rf44) cos S + f i ( O sin 41 Bi( -2) =~-‘/~lf~(t) cos [-fi(t) sin t] _Ai’(-X) __ = . 1 1 / 4 [ ~ a , ( f . ) sin E-$:(:) c,ng [j Bi’( -2) =x’/4[g2([) sin t+gl([) cos 41 fl (41, f2 “1, 91 (E), $72 (4) ; 4 =P$/2

[-I= .05( - .01)0, 6-7D Table 10.12. Integrals of Airy Functions (0 5x510).

J Ai(t)dt, s=0(.1)7.5; J Ai(-t)dt, 0

s,” Bi(t)dt, x=0(.1)2;

z=0(.1)10,

. . . . . . . .

478

7D

0

l

Bi(-t)dt, x=0(.1)10,

7D

Table 10.13. Zeros and Associated Values of Airy Functions and Their Derivatives (1 5 s 5 10) . . . . . . . . . . . . . . . . . . . . Zeros a,, a:, b,, b: of Ai(z), Ai’(z), Bi(z), Bi’(z) and values of &‘(a8), Ai(a:), Bi’(b,), Bi(b:) s=1(1)10, 8D Complex Zeros and Associated Values of Bi(z) and Bi’(2)

478

(19 5 5 )

Modulus and Phase of e-*f/3p,,e-*f/3pL,Bi’(f18),Bi(fli) s= 1(1)5, 3D The author acknowledges the assistance of Bertha H.,Walter and Ruth Zucker in the preparation and checking of the tables and graphs.

10. Bessel Functions of Fractional Order Mathematical Properties wronskians

10.1. Spherical Bessel Functions

Definitions DifferentialEquation

10.1.7

10.1.1

W { h t ) ( z )h, f ) ( ~ ) } = - 2 i z - ~

z2w”+2zw’+[z2-n(n+l)]w=0 (n=o,&1,

&2,.

. .)

(n=O, 1, 2,

. . .)

Representations by Elementary Functions

Particular solutions are the Spherical Bessel functions of the Jirst kind jn(Z)=AGJn+i(Z),

the Spherical Bessel functions of the second kind Yn( 2 ) =W 2 Y n + i( 2 )

and the Spherical Bessel functions of the third

kind hkl)( 2 ) = j n ( z )

+iYn(2) =AGHLYt(2)

hk2)(2) = j n ( z ) -iYn(z) =dWZHLY+(z). The pairs j n ( 2 ) , Yn(z) and hA1’(z), hi?(z) are linearly independent solutions for every n. For general properties see the remarks after 9.1.1.

=E

(-l)k(n++,

2k)(2~)-~’

0

Ascending Series (See 9.1.2, 9.1.10)

10.1.2

f

-n

’n(2)=103.5

P..

1 a-m2

(2n+l)i1-1!(2n+3)

(b2)*

+2! (2n+3) (2n4-5)

. .)

I.

10.1.3 1.395.. yn(2) =

. (2n-1)

2n+ 1

)n2- -1( !2+

4zf

(n=O, 1 , 2 ,

1! (1-2n) (422>2

(3-2n)

. . .)

- . . .} 1 2 3 4 5

2 6 12 20 30

12 60 180 420

120 840 3360

1680 15120

30240

437

438

BESSEL FUNCTIONS OF FRACTIONAL ORDER

10.1.10

Yn ( x )

jn(z)=fn(z) sin ~ + ( - l ) ~ + ' f - ~ - ~ cos ( z )2 fo(z)=z-', fn- 1 (2) + f n +

1

f'(2)=2-2

+1)

( 4=(2n

2- 'fn (2)

(n=O, f l , f2, The Functionsj,(z), y,(z)

10.1.11

. . .)

for n=O, 1, 2

sin z j o ( z ) =2 -.I

j2(z)=($-i)

sin z--$

-.2

cos z

-7.43

10.1.12

~

yo(Z)=--j-~(z)=--

cos z 2

cos z sin z y1(z) =j-* (2)=-- -22

y2(2>= - j + ( z )

=(-,?+;) 3 1

FIGURE 10.2. gn(x). n=O(lj3.

2

cos z--$

sin z

i

.5

*

-.20 -*I5

t

FIGURE 10.3. j,,(z),

g,,(z>.

z=10.

Po;qson's Integral and Gegenbauer's Generalization

2nzn! s,'

10.1.13 jn(z) =-

cos (2 cos O) sin2n+18 de

(See 9.1.20.) 10.1.14 -.3 L

FIGURE 10.1. j,,(z).

*See page U.

n=0(1)3.

(n=O, 1, 2, . . .)

439

BESSEL FUNCTIONS OF FRACTIONAL ORDER

Modulus and Phase

Spherical Bessel Functions of the Second and Third Kind

10.1.15

(n=O, fl, f 2 ,

y,(z>=(-l)"+'j~,_,(z)

. . .)

10.1.16 n

(n+$,k ) ( - 2 i z )

h',"(z)=i-n-lz-le**>:

0

-k

10.1.17 n

h A 2 ) ( ~ ) = i n + 1 z - 1 e -(n+$, io~ k) ( 2 i ~ ) - ~

*

0

10.1.18

h!!i-1 ( 2 ) =i (- 1) "hA1)( 2 ) hl",-1(2)=-;(-1)"hA2tI"'(z)

(n=O, 1, 2,

* *

.)

Elementary Properties Recurrence Relations

10.1.20 nfn-1(z)-(n+l)fn+l(z)=(2n+l)

d

& fn(4

d n2+lf n ( 2 ) +& fn(Z)=fn-l(z)

10.1.21

(See 10.1.23.) n

10.1.22

-fn 2

d

(4-&fn

(4= f n + 1 ( 4 (n=O, 1, 2,

(See 10.1.24.)

Analytic Continuation

Differentiation Formulas

( m = 1 , 2, 3,

. . .)

10.1.34

jn(zem*')=emnrfj&)

10.1.37

h f ) ( z e ( 2 " + 1 ) r f )(-l)"h:)(z) =

10.1.38

Rayleigh's Formulas

h~z)(zemrf)=h~z)(z) (Z=l, 2 ; m ,n=O, 1 , 2 , . . .) Generating Functions

10.1.25

10.1.39

10.1.26

~

*See page U.

. . .)

440

BESSEL FUNCTIONS OF FRACTIONAL ORDER

Fresnel Integrals

Derivatives With Respect to Order

10.1.53

10.1.41

10.1.42

10.1.43

10.1.44

( i n / z ){Ci(2z) sin z-[Si(2z)-n]

cos z)

Addition Theorems and Degenerate Forms T , p , 8, h

10.1.45

(See also 11.1.1, 11.1.2.)

arbitrary complex; R=4(r2+p2--2rp cos 8)

sin h R -AR

-5(2n+1) o

Zeros and Their Asymptotic Expansions j n ( A T ) j n (A p ) pn (cos 0)

The zeros of j,(z) and yn(z) are the same as the zeros of Jn+t(z)and Yn+,(z)and the forniulas for j u , sarid yy,s given in 9.5 are applicable with v=n++. There are, however, no simple relations connecting the zeros of the derivatives. Accordingly, we now give formulas for bk,s, the s-th positive zero of j ; ( z ) , yk( z ) , respectively; z=O is counted as the first zero of j i ( z ) .

Uupiication Formuia

10.1.49

j n(22) =

Some Infinite Series Involving jl,(z)

.fI: (2n+l) j i ( 2 ) =1

10.1.50

0

>: (-1)"(2n+l)ji(z)=QD

10.1.51 10.1.52 *See page U.

0

sin 22

fn(z)

= j , ( ~ )COS at+yn(z) sin at

If

is a zero of fk

Tn

(2)

(t a real parameter, 0 I t 5 1) then

10.1.55 f n ( T n > = [ T n / ( n + (See 10.1.21.) 10.1.56 = (Tn/n)fn+l (See 10.1.22.)

22

10.1.57

l)]fn-l(Tn)

(Tn)

441

BESSEL FUNCTIONS OF FRACTIONAL ORDER, McMahon's Expansions for n Fixed and s Large

10.1.58 U:,,s,

Uniform Asymptotic Expansions of Zeros and Associated Values for n Large

10.1.63

b:,,s- 8- b + 7 ) (88) -I 4 -(7p2+ 154~4-95)(88)-3 3

-32 (85p3+3535p2+356 1p+61 33)(8#1)-6 15 --64

105

(6949p4+474908p3+330638p2

...

+9046780p-5075147)(8~)-'~ = ~ ( s + + n - $for ) a:,,s,~ = ~ ( s + + nfor )

10.1.65

b;,8;

p =. (2n+

1)2

Asymptotic Expansions of Zeros and Associated Values for n Large

j n(a;, 8 ) -&Ai (a:) (n+ 3) -'I8 h[(n+6)-2l3a;]( z [(n+ 4) -2/3a:])

10.1.59 a;,

-

(n+ 3)

+.8U86165(n+ 3)

'I3-

3) -'I3

.236680(n+

10.1.66

...

--.20736(n+4)-'+.0233(n+$)-5/3+

10.1.60

b:,

-

+ +1.8210980(n+ 3) +.802728(n+ a) 11740(n+3) -'

(n $)

-'I3-.

...

+.0249(n++)-5/3+

10.1.61

jn(a:, 1) r ~ . 8 4 5 8 4 3 0 ( n + # ) - ~{/1-.566032(n+$)-2/3 ~

. . .]

+.38081(n+4)-4/3-.2203(n+4)-2+

10.1.62

yn( b:, 1)

+3)-5/5{ 1-1.274769(n+

.7183921(n

+1.23038 (n+ 3)

+)-*I3

-'I3-

1.0070 (n+ 4) -2+

...

See [ 10.311 for corresponding expansions for s=2, 3.

h ( [ ) ,z ( [ ) are defined as in 9.5.26, 9.3.38, 9.3.39. a:, b,' s-th (negative) real zero of Ai'(z), Bi'(z) (see 10.4.95, 10.4.99.) Complex Zeros of

z ) , hi1)'(Z )

- " (z> and hL1)( ~ e ~ m~ any ~ ~ integer, ) : have the same zeros. h$I)( z ) has n zeros, symmetrically distributed with respect to the imaginary axis and lying approximately on the finihe arc joining z=-n and z=n shown in Figure 9.6. If n is odd, one zero lies on the imaginary axis. I$)'( z) has n+ 1 zeros lying approximately on the same curve. If n is even, one zero lies on the imaginary axis. I&

442

BESSEL FUNCTIONS OF FRACTIONAL ORDER

0.0 0.2 0.4 0.6 0.8

-. -. -. -. -.

4409724 4572444 4702250 4802184 4875705

-. -. -. -. -.

122500 114201 107243 101318 096159

1.0

-.

4926355

-.

091561

-. -. -. -. -. -.

-. -. -.

09156 05056 03043 01950 01310

-. 037 -. 014 -. 006 -. 003 -. 001

1. 0 1. 2 1. 4 1. 6 1. 8 2. 0 2. 2 2. 4 2. 6 2. 8 3.0 3. 2 3. 4 3. 6 3. 8 4.0 4. 2 4. 4 4. 6 4. 8 5. 0 5. 2 5. 4 5. 6 5. 8

-. -.

4926355 4131280 -. 3551700 -. 3108548 -. 2757704

-. -. -. -.

-.

2472521 2235898 2036314 1865701 1718217

-. -. -. -.

1589519 1476304 1376005 1286601 -. 1206469

-. -. -. -. -. -. -.

1134296 1069004 1009699 0955634 0906180

0860804 0819049 -. 0780523 -.0744888 -. 0711850

-.

6.0 6.2 6.4 6.6 6.8

-. -. -.

-.

0681152 0652570 0625905 0600985 0577653

7.0

-.

0555773

-. -.

. 000000

06806 05986 05279 04674 04160

.027518 .049069 .065677 .078255

03725

.087587

-. -. -. -. -. -.

-.

-. -. -. -. -. -.

-. -. -. -. -.

-. -. -. -. -.

.00575 .01118 .01592 .01983 .02290

I

.087587 .065507 .050524 .039890 .032085

.0229 .0121 .0070 .0042 .0027

00485 00366 00280

,026206 .021682 .018141 .015326 .013061

.0018 .0012 .0008 .0006 .0004

00219 00173 00138 00112 00091

.011217 .009701 . 008443 .007391 .006505

.0003 .0002 .0002 . 0001 . 0001

00075 00062 00052 00044 00037

.005753 .005111 .004560 .004085 .003672

00032 00027 00023 00020 00018

.003313 .002998 .002722 .002478 .002262

00015 00013 00012 00010 00009

.002070 ,001899 .001746 .001609 .001486

00008

.001375

~

-.-. 00914 00658 -.

. 00000

I

0.40 . 36 .32 .28 .24 .20 . 16 . 12 .08 .04

.oo

I

-. -. -. -.

-.

I

I

I

0645731 0487592 0352949 0242415 0155683

-. 0091416 -. 0047276 -. 0020068 -. 0005965 -. 0000747 -.ooooooo

I

-. -. -. -.

00013 00005 00002 00001

I

.001859 .001056 .000551 .000259 .000106

. 000037

.000010 .000002

I

I

. 0000

.0023 . 0043 .0061 .0075 .0085

443

BESSEL FUNCTIONS OF FRACTIONAL ORDER

10.2. Modified Spherical Bessel Functions

Wronskiane

10.2.7

Definitions Differential Equation

10.2.1 22w'

10.2.8

'+2zw '-[ z2+

+

n (n 1)]w=0 (n=O, f l , f 2 ,

. . .) Representations by Elementary Functions

Particular solutions are the ModiJied Spherical Bessel functions of the jirst kind,

10.2.9

10.2.2

of the second kind, 10.2.3

=A(n+4,4 (22)

-k

0

(n=O, 1 , 2 ,

(See 10.1.9.)

. . .)

10.2.12

The pairs M z ~ n + t ( z >A&zI-n-t(z) ,

and

d Z z I n + i ( z )9 A & z ~ n + t ( z ) are linearly independent solutions for every n. Most properties of the Modified Spherical Bessel functions can be derived from those of the Spherical Bessel functions by use of the above relations.

The Functions

(Z),

n=O, 1, 2

10.2.13

Ascending Series

10.2.5

=($+:)

4w1512(2)

sinh x--

3 Z2

*

cosh z

10.2.14 10.2.6 J$TJZLl,&)

1 . 3 - 5 . . . (2n-1) J-$qiLn-+(2) = (- 1)nzn+'

{

4Z2

'+1!(1-2n)

cosh z

=-

Z

sinh z cosh z

- J ~ I - ~ /=------~ (2 z )

+ (n=O, 1, 2,

. . .)

,/$&1-5/2(z)=-*See page

11.

3 z2 sinh

Z2

z+($+$)

cosh

2

*

444

BESSEL FUNCTIONS OF FRACTIONAL ORDER

ModSed Spherical Bessel Functions of $he Third Kind

Y

10.2.15

10.2.16

(n=o,I , 2, . . .)

K,+&)=K-,-&)

Elementary Properties ,-

Recurrence Relations

10.2.23

(k $y[ nfn 2-

(2)] =z - n+""n

+ m (2)

(m=l, 2 , 3 , . . .)

7

n c>/rt u o

/

445

BESSEL FUNCTIONS OF FRACTIONAL ORDER

Addition Theorems and Degenerate Forms

Formulas of Rayleigh's Type

d " sinh z dmIn+f (2) = zn ( ;1 z )7

10.2.24

T, p,

8, h arbitrary complex; R = 4 r 2 + p 2 - - 2 ~ p cos

10.2.35

10.2.25

e-XR 2

-=-

AR

JrnI-,-,(Z)

e

T

2 (%+I) 0

[JrnIn+,(hT)l

=zn (n=O, 1 , 2 , . . . )

(2n-k) ! (2n-2k)! k![ (n-k) !I2

>:

l n =(-1)k+' z2 0

(2z)2k-2n

(n=O, 1 , 2 , . . .)

Duplication Formula

10.2.38 Kn+t(2z) =

( $ T / z ) [ ~ ;(~)--12_~,~ ,~ (z)]=z-~-z-~

10.2.28

10.3. Riccati-Bessel Functions Differential Equation

10.3.1

Generating Functions

9w"+ [9-n(n+ l)]w =0

10.2.30

(n=O, f l , f 2 ,

..

.)

Pairs of linearly independent solutions are Zjfl(4,

(it)" 1 coshJz2+2izt =En! [ 4$&In-*(z)] Z OD

ZYfl(4

ZhL1' ( 2 ) , ZhL2'(2) All properties of these functions follow directly from those of the Spherical Bessel functions.

0

The Functions z ~ ~ ( zzYn(Z>, ), n=o, 1, 2 Derivatives With Respect to Order

10.3.2

10.2.32

zj,(z)=sin z,

zjl(z)=z-'sin z-cos z sin z-3z-l

cos z

*

zyo(z)=-cos z, zyl(z)=-sin Z-Z-'COS z zy2(z)=-3z-'sin z - - ( 3 ~ - ~ - 1 ) cos z

*

zj2(z)=(3z-2-1)

10.3.3 10.2.33

[

I"(z)]

vp-$

=L [Ei (22) e -'+E1 (2lrx

10.2.34 [ g K v ( 2 ) ]

"=*t

-

= TJ?r/2xEi(-2z)ez

For El(z) and Ei(z), see 5.1.1, 5.1.2. ~

*See page U.

22) e']

10.3.5

W {z ~ : " ( z ) , Z ~ ? ' ( Z ) ) =-2i (n=O, 1, 2,

. . .)

446

BESSEL FUNCTIONS OF FRACTIONAL ORDER

10.4. Airy Functions

10.4.12 W(Ai ( z ) , Ai ( ~ e - ~ *) =$?r-le*r/s ~/~)

Definitions and Elementary Properties

10.4.13 W(Ai ( ~ e ~ * Ai ~ / (~~) e, - ~ *1 ~ / ~ )

Differential Equation

w” -zw=o

10.4.1

Pairs of linearly independent solutions are Ai ( z ) , Bi ( z ) ,

I .o

I

n

Ai ( z ) , Ai ( z P / ~ ) , Ai ( z ) , Ai ( ~ e - 2 * ~ / ~ ) .

.8 .6

Ascending Series .4

10.4.2 Ai ( z ) =clf(z)-c2g(x) 10.4.3 Bi

( 2 )=

.2

& [cJ(z) +c2g(z)]

0 -.2 -.4

-.6

-8 U

(a+

a>,=

3k (a+;>,=

-1.0

FIGURE10.6. Ai 1

( 3 4 - 1 ) (34-4)

. . . (3a+3k-2)

(a! arbitrary; k=1,

(See 6.1.22.)

2,3, 2.2

10.4.4

2.0

. 1.8

cl=Ai (O)=Bi (0)/@=3-2/~/r(2/3) =.35502 80538 87817

I .6

10.4.5 I .4

c2=-Ai’ (@)=Bi’(0)/,6=3-1/3/r (1/3) =.25881 94037 92807

1.2 1.0

Relations Between Solutions

10.4.6 Bi ( z ) =e*r/6 Ai (ze2*fB)+e+l6

8

Ai

.6 .4

.2 0

-.2

-.4 -.6

Wronskians

10.4.10

W(Ai ( z ) , Bi

10.4.11 W (Ai

(2)

, Ai (

(2)) = T - ~

~ e ~ }* = ~ +?r-1e-*t/6 /~)

-.a -1.0

1

(ktz),Ai’ (hts).

\I

447

BESSEL FUNCTIONS OF FRACTIONAL ORDER Representations in Terms of Bessel Functions

Integral Representations

10.4.32

(3a)-%

Ai [ f(3a)-"%]=

10.4.33

10.4.34 10.4.35

s,'Ai (t)dt=i s,' [I-ii3(t) -IlI3(t)]dt Ai (-t)dt=i s,' [J-i/3(t) +J,/,(t)]dt

Ascending Series for

Representations of Bessel Functions in Terms of Airy Functions

n = L3 --

10.4.22 90.4.23

J*1/3(r)=im[fi

Ai (-z)FB i

(-z)-i

(-41 Bi ( - z ) ]

H!3,3(C) = e * l f / " m [ A i (- z) +i Bi (- z)]

10.4.25

I*1/3(f)+@[T

10.4.27

&i

& Ai (z)+Bi n(6) =rdG Ai ( 2 )

J*2&)=(fl/2z)[&fl

Bi ( ft)&

T2& ( - t : d t = - C I ~ ~ - ) ) S t . i ~ ( - . ? ) Jo

10.4.24

.

sos

(See 10.4.2.) I.LV.-r.r)l n 4 Qn

H$!],3(6)=e~rf16mz[Ai

10 4.26

Ai ( ft)dt,

s,'Ai (t)dt=clF(z)-cPG(z)

10.4.38

?\"3

\2 *)

L'

(z)]

10.4.40

l'

Bi ( t ) d t = f l [ c l F ( z )+czG(z)l

(See 10.4.3.) 10.4.41

L'Bi (-t)dt=-fi[clF(-z)+c2G(-z)]

Ai'(-z)+Bi'(-z)]

10.4.28

(0=e-2~':/3H-(21/)3 w = e r t / " ( f l / z )[Ai' (- z) -i Bi' (- z)] 10.4.29

=e-rf/6(fl/z)[Ai'(-z)+i *See page

n.

Bi' (-z)]

T h e constants c1,c2 are given in 10.4.4, 10.4.5.

448

BESSEL FUNCTIONS OF FRACTIONAL ORDER Differential Equations for Gi ( z ) , Hi ( z )

Tbe Functions Gi(z), Hi(z)

10.4.42

w’’-zw=

10.4.55

Gi (z)=?r-‘lmsin(5ta+zt)dt [Ai (2) Bi (t)-Ai ( t ) Bi (z)]dt

w(O)=L Bi 3

(O)=LAi (0)=.20497 6

w’(O)=-Bi’ 1 (O)=--

10.4.43 1

6”

w(z) =Gi(z)

[Ai’(z)Bi (t)-Ai ( t ) Bi’(z)]dt

10.4.44

Hi(z)= n - l l - exp

10.4.56

(-5

zw=n-’

W”-

w(O)=32 Bi (O)=-

2 Ai (0)=.40995 10849 56

6

t3+ z t ) dt 2

[Ai ( t ) Bi (2)-Ai(z) Bi(t)]dt

w’(0)=3 Bi’

(0) =-- Ai’ (0)=.29885 89048 98

6

w(z)=Hi

10.4.45 2

Hi’(z)=3Bi’ (z)+

6”

Representations of

(2)

Differential Equation for Products of Airy Functions

[Ai ( t ) Bi’(z)-Ai’(z) Bi (t)]dt

Gi (z)+Hi (z)=Bi ( z )

10.4.46

55424 78

Ai’ (0) =.14942 94524 49

G

3

Gi’ (z)=3Bi’ ( z ) +

-*-I

i’

Ai( ft)dt,

is

Bi( ft)dt

10.4.57

w”’--4zw’-2w=0

Linearly independent solutions are Ai2 ( z ) , Ai ( z ) Bi ( z ) ,Bi2 (2). Wronskian for Products of Airy Functions

.

10 4.47

b

1 i (t )dt =3+ n[Ai ’ ( z ) Gi ( z ) -Ai ( z ) Gi’( z ) ]

10.4.58 W(Ai2( z ) , Ai ( z ) Bi ( z ) , Bi2 ( z ) ] = 2 ~ - ~ Asymptotic Expansions for Iz I Large

10.4.48

=-!-n[Ai’

(2) Hi

3

(2)-Ai ( z ) Hi’ (z)] do= 1,dk=

10.4.49

1 -6k+ 6k-1

(k=l, 2 , 3 , . . .)

ck 9

-Ai

Gi’

(-2)

r=S

(-41 10.4.59

10.4.50

1

=z+n[Ai’ 3

(-2)

Hi

-Ai (-z)Hi’(-z)]

l’

Bi (t)dt=n[Bi’ ( z ) Gi (2)-Bi ( z ) Gi’ =-n[Bi’

( z ) Hi (2)-Bi

(2)

(-2)

Gi

Hi’(z)]

(-2)

0

10.4.54

=n[Bi’

(-2)

-Bi (-2) Hi ( - 2 ) -Bi

(-2)

10.4.60

(41

10.4.53

J Bi (-t)dt=-n[Bi’

m

(-2)

10.4.51

10.4.52

23/2

Gi’ ( - z ) ] Hi’ ( - z ) ]

-cos

((+-3, >: (-l)~c2k+,s-=-1]

449

BESSEL FUNCTIONS OF FRACTIONAL ORDER

10.4.62

Ai‘ (- z )

-

-t - ) z i

[COS

({+a)

10.4.70

2 (- 1)’dzk

Ai’

0

(-2) =N ( z ) cos

(-2) =N(x) sin $(x)

4 (x), Bi’

N(x)=J[Ai” (-x) +Bi’2 (-x)], 4 ( x ) =arctan [Bi’ (-x)/Ai’ (-x)l Differential Equations for Modulus and Phaee

Primes denote differentiation with respect to x

10.4.72

N2=M’2+M2e‘2=Mi2 + T - ~ M- 2

10.4.73

NN’= -xMM’

*

10.4.74

tan (#-O)=MO’/M’= -(rMM‘)-l, MN sin (4-e)=r-1

M” +x

10.4.75 10.4.76 10.4.77

~

9

-

~

-

=30

(M2)‘”+4x(M2)’-2M2=O

et2+ +(el”/e’) -p (e”/e’) 2

= ~

Asymptotic Expansions of Modulus and Phase for Large x

10.4.66

*

10.4.79

aD

Bi’ ( z ) -r-)z*er

>:dkr-k (larg zl<$

t)

0

10.4.67

Bi’ ( - z ) - t - ) z f

--82825 (2x) i2S

-9+

12820 31525 (2x)-12id336

. . .A1

10.4.80

1 (-1)&+’6k+l N2(x); xf >:-23k aD

o

12&k! 6k-1

3k

10.4.81

96

+495271(2x)-Q640

2048

Asymptotic Forms o f p i ( ft ) Modulus and Phase

10.4.82

10.4.69

Ai (-z)

=M(z) COS

e(x), Bi ( -z)

=M(x) sin

e(x)

M(x) =J[Ai2 (-2) +Bi2 (-41,

e(x)=arctan [Bi (--s)/Ai (-41 *See page U.

10.4.83

LAi

(Qdt-1-L 3

&,pi

( ft ) dt for Large z

2 t - 1 / 2 ~ - 3 / ~exp

(-32 x312)

450

BESSEL FUNCTIONS OF FRACTIONAL ORDER

10.4.104 Bi’ (/3,)=(-l)8fie-*f/efi

[s 31r ( 4 s - l ) + ~ In 2 3i

1

10.4.105 Asymptotic Forms of Gi ( fz), Gi’ ( fz), Hi ( fz), Hi’ ( fz) for Large z

I ZI

Gi (x)- l r - l ~ - ~

10.4.86

sufficiently large 5 36

2-2,-

-1080 56875 2 - 8

--

Gi’ (2)

10.4.88 10.4.89 Gi’

77125 z-B 82944

2-4+-

69 67296

7 R-~X-~ 96

(-2) -7r-1/2x1/4sin

16 23755 96875 z-lo+ 3344 30208

(: x3/2+:)

. . .)

Hi (x)-7r-1/2x-1/4 exp (3z3i2)

10.4.90

Hi (-x)

10.4.91

18683371 + 12 44160

-lr-lx-l

-9 11458 84361 z-lo+ . . .)

Hi’ (x)-7r?r-1/2x1/4 exp (3s3l2)

10.4.92

Hi’ (-s)---

10.4.93

z-8

1911 02976

3 lr-1x-2 2

fl(z) m r - 1 / 2 2 1 / 6

Zeros and Their Asymptotic Expansions

(1 +-4“s

1525 z - 4 4608

2-2,-

23 97875

Ai ( z ) , Ai’ ( z ) have zeros on the negative real axis only. Bi ( z ) , Bi’ ( z ) have zeros on the negative real axis and in the sector i n < / arg zl<$lr. a,, a:; b,, b: s-th (real) negative zero of Ai ( z ) , Ai’ (2); Bi ( z ) , Bi’ ( z ) , respectively. p,, pi; p,, 5: s-th complex zero of Bi ( z ) , Bi’ ( z ) in the sectors Q?r<arg z < m ; -+
U,=

10.4.95

U: = -g [ 3 ~ (4 3)/8] ~

- f [ 3 ~ ( 4 ~ - 1)/8]

10.4.97 Ai (a:)= (- l)*-’g1[3~(4s-3)/8]

b, = -f[3 T (4s -3) /8]

10.4.99

b:= - g [ 3 ~ ( 4 ~ -1)/8]

10.4.100 Bi’ (b,) = (- 1)”-’f,[3~(4~-33)/8]

(4s-l)+-

10.4.103

(4s-3)+-

+Seepage XI.

3i 4 3i 4

I--

(

2-2+-

9’6

1673 2 - 4 6144

-843 94709

* z-6+

265 42080

. . .)

end -Agymptnti~Snliitinnn nf Ordinary Differential Equations of Second Order With Turning Points

Fcrm-a!

In 2

10.4.106 w’+a(z, A)W’+b(z, A)w=o

in which A is a real or complex parameter and, for fixed A, a(z, A) is analytic in z and b ( z , A) is continuous in z in some region of the z-plane, may be reduced by the transformation 10.4.107 W(z)=w(z) exp

(-f

Jz

a ( t , A)&)

to the equation

10.4.101 Bi (b:) = (- 1),g1[3r(4s- 1)/8] 10.4.102 &=e*f/3f

g1(z)-T-1/2z-1/6

. .,)

An equation

10.4.96 Ai’ (a,)= (- 1),-y1[3~(4s- 1)/8]

10.4.98

z-6-

+ 6 63552

1 1

In 2

10.4.108

w”+cp(z, A)w=O p(z, A)=&,

A)-z

1

I d a+, A)-- 2 d z a(z, A).

451

3ESSEL FUNCTIONS OF FRACTIONAL ORDER

10.4.114

If p(z, X) can be written in the form 10.4.109

q(z,

where q(z, X) is bounded in a region R of the zplane, then the zeros of p(z) in R are said to be turning points of the equation 10.4.108. The Special Case w ” + [ X 2 z + q ( z , X)]w=O

Let X = I X l e f U vary over a sectorial domain S: IXl?Xo(>O), w 1 < w < u 2 , and suppose that. q(z, X) is continuous in z for Izl
-2an(z>x-n as

in S.

0

y1(2)

Formal Series Solution W

The General Case w ” + [ X 2 p ( z ) +q(z, X)]w=O W

>:+n(z)X-n

pn(z)X-s+X-lu’(z) 0

U‘

0

’+X2zu=O

I

( XI -+a)

For further representations and details, we refer to [10.4]. When z is complex (bounded or unbounded), conditions under which the formal series 10.4.110 yields a uniform asymptotic expansion of a solution are given in [10.121 if q(z, A) is independent with fixed U, and in 110.141 if X of X and [XI-... lies in any region of the complex plane. Further references are [10.2; 10.9; 10.101.

10.4.110 w(z)=u(z)

+ =Bi ( -X2/”.) [1+O(X-l)]

yo(x) =Ai ( -X2%) [ 1 O(X-91

X)=X2p(z)+q(z, A)

Let X=JXleiUwhere lA.l>&(>O) and - - a < o ~ n ; suppose that p(z) is analytic in a region R and has a zero z=zo in R, and that, for fixed X, q(z, A) is analytic in z for z in R. The transformation [ = [ ( z ) , u=[p(~>/[]~l~ww(z), where [ is defined as the (unique) solution of the equation 10.4.115

yields the special case (n-0, 1, 2,

,

. .)

10.4.116

d2v --+ dt2

[X2[+f([,

X)]V = 0,

*

Uniform Asymptotic Expansions of Solutions

For z real, i.e. for the equation 10.4.111 ,L A”, w uui

y”

+IX2x+a(x,

Exumple:

IY=0

. iii . ii b~iliictdi i i t e i - d ii <;i;5 6 that

a ---.An v ai icjp N

includes the origin and where, for each fixed X in S, q(x, X) is continuous in x for a < x
+O(X- 91 X2%) [1+ O(x-l)]

y0(z) =Ai ( -X2%) [1 y1(s)=Bi (-

and, uniformly in x on 0

(A+-

Consider the equation 10.4.117

whence

yo(x)=Ai ( -X2%) [ 1 0(X-l)]+ Bi ( -X 2 ~ x0)(A- l) , ~2%)

O(A-’) Thus

(ii) If g X > O , A # O , there are solutions yo(x), yl(z) such that, uniformly in x on a<x
x‘~]~=O Q)

10.4.113

+ y1(x)=Bi ( -X2’%) [ 1+O(X-91 +Ai ( -

(A2-$)

for which the points x=O, are singular points and x=1 is a turning point. I t has the functions ~+J,(xz), d Y A ( ~ zas ) particular solutions (see 9.1.49). The equation 10.4.115 becomes

)

sx
y”+[X2-

10.4.118

452

BESSEL F U N C T I O N S O F FRACTIONAL ORDER

satisfies the equation

which is of the form 10.4.111 with x replaced by 4 and q ( [ , A) independent of X. Suppose B X 2 0 , Y X # O . By the f i s t equation of 10.4.114 there is a solution vo(f) of 10.4.119, i.e., a solution yo(x) of 10.4.117 for which the representation 10.4.120

I t follows from 10.4.120 that uniformly in x on o<x< O3 10.4.125

Numerical Methods 10.5. Use and Extension of the Tables Spherical Bessel Functions

To computej,(x), y,(x), n=O, 1, 2, for values of holds uniformly in x on O<x<m as lXl+m. x outside the range of Table 10.1, use formulas To identify yo(x) in terms of ztJk(Xx),x+YA(XX),10.1.11, 10.1.12 and obtain values for the circular restrict x to O<x_
-

- -.0434O 7356.

To compute j,(x), 11 5 n 5 2 0 , for a value of x within the range of Table 10.3, obtain from Table 10.3, directly or possibly by linear interpolation, &(x) and use these as starting values in the &(x), Let now X be fixed and x+O in 10.4.121. There recurrence relation 10.1.19 for decreasing n. results An alternative procedure which often yields b e t t e r t~ccuri~cy a i d ~ l i i d &jzl i iipplies t~ ZGZ-,Utations of j,(x) when both n arid x are outside the range of Table 10.1 is the following device essenOn the other hand, yo(x) is a solution of 10.4.117 tially due to J. C. P. Miller [9.20]. and therefore it can be written in the form At some value N larger than the desired value n , assume tentatively FN+,=0, FN=l and use recurrence relation 10.1.19 for decreasing N to obtain the sequence FN-l, . . ., Fo. If N was where, from 9.1.7 for X fixed and x+O chosen large enough, each term of this sequence up to F, is proportional, to a certain number of significant figures, to the corresponding term in the sequence jN-l(x), . . ., jo(x) of true values. The factor of proportionality, p , may be obtained by comparing, say, Fo with the true value jo(x) computed separately. The terms in the sequence Thus, letting x-+O in 10.4.123 and comparing the pFo, . . . p F , are then accurate to the number resulting relation with 10.4.122 one finds that of significant figures present in the tentative c2=0 and values. If the accuracy obtained is not sufficient, the process may be repeated by starting from a 10.4.124 y0(x)= ~ - 1 / 2 X - - X - 1 / 6 e A l ?(A+ ~ ) X ~ / ~ J ~ ( X Z ) . larger value N.

453

BESSEL FUNCTIONS OF FRACTIONAL ORDER

Example 2. Compute j15(x) for x=24.6. Interpolation in Table 10.3 yields for x=24.6 1 (S)= ( -28) 3.9346 16 (2) = ( -27) 9.48 683 whence. j,1(24.6)=.05604 29, j20(24.6) = .03896 98. From the recurrence relation 10.1.19 results jlQ(24.6)= .00890 67660 [.00890 j1,(24.6)= - .02484 93173 [- .02485 j17(24.6)=-.04628 17554 [--.04628 j16(24.6) = - .04099 87086 [- .04099 jl5(24.6)=--.00871 65122 [-.00871

there 701 901 161 881 671

For comparison, the correct values are shown in brackets. To compute j 1 5 ( ~ ) for x=24.6 by Miller's device, take, for example, N = 3 9 and assume F40=0, F3Q=1. Using 10.1.19 with decreasing N , i.e., FN_1=[(2N+l)/x]:]FN-:]FN+1, N-39, 38, . . ., 1, U, generate the sequence F38, Fa,, . . . ,' F1: Fo, compute from Table 4.6, j0(24.6)= (sin 24.6)/24.6 - -.02064 620296, and obtain the factor of proportionality p=j0(24.6)/F0= .OOOOO 03839 17642.

Modified Spherical Bessel Functions

To compute 44n/xIn++(x),&&&++(x), n=o, . . . for values of x outside the range of Table 10.8, use .formulas 10.2.13, 10.2.14 together with 10.2.4 and obtain values for the hyperbolic and exponential functions from Tables 4.4 and 4.15. In those cases when J$?r/xIn+*(x) and 4+?r/zI-,-+(x) are nearly equal, i.e., when x is sufficiently large, compute - J m ~ ~ + t ( from x) formula 10.2.15, for which the coefficients (n+ 4, Ic) are given in 10.1.9. Example 3. Compute J=I5/2(x), 4 3 K 5 / 2 ( x ) for x= 16.2. From 10.2.13, dmI5,2(~) = (3+x2) sinh x / 2 3 cosh x/x2;from Table 4.4, cosh 16.2=(6)5.4267 59950 and this equals the value of sinh 16.2 to the same number of significant figures. Hence 1, 2,

& ~ / 1 6 . 2 I ~ / ~ ( 1 6= . 2(.06243 ) 402371 -.01143 118427)[(6)5.4267 599501 =338814.4594- 62034.29298 =276780.1664.

To compute 4 + ~ / 1 6 . 2 K16.2) ~ ~ ~ use ( 10.2.17 and obtain

The value pF15 equals j15(24.6) to 8 decimals. The final part of the computations is shown in the following table, in which the correct values are given for comparison. N

FN

15 - 22704. 71 107 78178. 88236 14 13 f 114866. 8081 1 47894. 44353 12 11 -66193. 59317 10 - 109782. 76234 9 -27523. 39903 88524. 85252 8 7 +88699. 11017 6 - 34440.02929 5 - 106899. 12565 4 - 13360. 39272 3 4- 102011. 17704 2 42387. 96341 1 -93395. 73728 0 - 53777. 68747

+ +

= (-7)2.8945

38069[.036932 604001

= (-8)1.0690

28283.

To compute d@ln++(x), 3 5 ~ 5 5 8 , for a r wit,h_in&p r ~ n of g ~ Tahlc?10.9: oht,nin from Table 10.9, dmIlQ/2(x), 4-&7&2,/2(x) for the desired value of x and use these as starting values in the recurrence relation 10.2.18 for decreasing n. To compute &$Kn++(x) for some integer n outside the range of Table 10.9, obtain from 10.2.15 or from Table 10.8, &&K+(x), 4 m K 3 , 2 ( xfor ) the desired value of x and use these as starting values in the recurrence relation 10.2.18 for increasing n. If x lies within the range of Table 10.9 and n>10, the recurrence 4&7&&/2(x) may be started with 4+7r/5K19/2(x), obtained from Table 10.9. Example 4. Compute m K l l D ( x for ) x=3.6. Obtain from Table 10.8 for x=3.6 vg1-g~ cf

-.

00871 67391 -.00871 674

+. 03001 42522 f . 03001 425 +. 04409 93941 +. 04409 939 +. 01838 75218 +. 01838 752 -. 02541 28882 -. 02541 289 -.

04214 01056 03398 03405 -. 01322 -. 04104 -. 00512 .03916 01627 -. 03585 -. 02064

-.

+

+. +.

+

+ +.

75392 67185 62526 31532 21348 04602 92905 38905 34870 62712 62030

-.04214 -. 01056 03398 +.03405 -.01322 -. 04104 -.00512 +.03916 01627 -. 03585 -. 02064

+.

+.

754 672 625 315 213 046 929 389 349 627 620

It may be observed that the normalization of the sequence FN, FN-1, . . ., Fo can also be obtained from formula 10.1.50 by computing the sum m a=x (2k+l)F: and finding p=1/4; This 0

yields, in the case of the example, p=l/&= .OOOOO 03839 177.

&*/zK1,2(x) =.01192 222

J + T / x K ~= ~ (.01523 x ) 3952

454

p BESSEL FUNCTIONS 01 FRACTIONAL ORDER

The recurrence relation 10.2.18 yields successively

-&~/3.6&/~(3.6)= - .01192 222 I -

--.02461 -

3.6 718

(.01523 3952)

y’(z), respectively, the following formulas may be used, in which d, d’ denote approximations to c, c’ and u= y(d)/y’(d), v= y’ (d’)/df2y(d’).

c=d-~-2d

u3 -+2 3!

5! +88d g-(88+720d3) US

5 +r6 (.02461 718) =.04942*4480

+5856d2 c‘=d’ {l-v---

$- (16640d+40320d4)3+ ug

3.6 034

.. .

+

a

-(945+756d’3+272d’e)$-.

.

+

(3 2d ’3) U33 (15 1Od ’3) V 4 2! -(105 76d’3 24d”) v-g 5 V2

+

+

718

--

--.12072 -

U7

7!

J+~/3.6&/2(3.6) =.01523 3952

-&G7K6&/2(3.6)=--.02461

U6

--24d2 u4 4!

(.04942 4480)

4+~/3.6K11/2(3.6)=.049424480

9

+% (.12072 034) =.35122 533.

As a check, the recurrence can be carried out until n = 9 and the value of 4 3 ~ / 3 . 6 K ~ ~ / ~ (so3 .obtained 6) can be compared with the corresponding value from Table 10.9. To compute v&&dn+*(x) when both n and x are outside the range of Table 10.9, use the device described in [9.20].

u2 u3 1-d -+--3d2 2! 3!

u4 a+ 14d 5!

US -

US u7 - (14+45d3)g+471d2 g

-(1432d+1575d4)g!+ U8 1-d”

...

v5j-d’3 z v3-(3d’3+3d’6)3 v4 4!

-(15d’3+14d’S)gjV 6 -((105d’3+101d’6+45d’4)$-

Airy Functions

.}

..

.}

To compute Ai(x), Bi(x) for values of x beyond 1, use auxiliary functions from Table 10.11. Example 5. Compute Ai(x) for x=4.5. Pirst, for x=4.5,

Example 6. Compute the zero of y(x)=Ai(z) -Bi(z) near d = -.4. From Table 10.11,

4=fx3’2=6.36396 1029, [-’= .15713 48403.

y( - .4) z.02420 467, y’( - .4) = - .71276 627

Hence, from Table 10.11, f(-[)=.55848 thus

24 and

Ai (4.5)

~=-.4+.03395

=*(4.5)-lI4(.55848 24) exp (-6.36396 =it.68658

whence u=y(-.4)/y’(-.4)= the above formulas

1029)

905) (.55848 24) (.00172 25302)

= .00033 02503.

To compute the zeros c, c’ of a solution y(z) of the equation y”-xy=O and of its derivative

-.03395 8776. From

8776-.OOOOO 5221 .ooooo 0111 .ooooo 0001 = - .36604 6333. y’(~)=(-.71276 627) (1+.00023 0640 - .OOOOO 6527- .OOOOO 0027 .OOOOO 0002) = (- .71276 627) (1.00022 4088) --.71292 599.

+

+

+

455

BESSEL FUNCTIONS O F FRACTIONAL ORDER

References Texts

Tables

[10.1] H . Bateman and R. C. Archibald, A guide to tables of Bessel functions, Math. Tables Aids Comp. 1, 205-308 (1944), in particular, pp. 229-240. [10.2] T. M. Cherry, Uniform asymptotic formulae for functions with transition points, Trans. Amer. Math. Soc. 68, 224-257 (1950). [10.3] A. Erdglyi et al., Higher transcendental functions, vol. I, 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [10.4] A. Erd6lyi, Asymptotic expansions, California Institute of Technology, Dept. of Math., Technical Report No. 3, Pasadena, Calif. (1955). [10.5] A. Erdklyi, Asymptotic solutions of differential equations with transition points or singularities, J. Mathematical Physics 1, 16-26 (1960). [10.6] H . Jeffreys, On certain approximate solutions of linear differential equations of the second order, Proc. London Math. Soc. 23, 428-436 (1925). [10.7] H . Jeffreys, The effect on Love waves of heterogeneity in the lower layer, Monthly Nat. Roy. Astr. Soc., Geophys. Suppl. 2, 101-111 (1928). [10.8] H . Jeffreys, On the use of asymptotic approximations of Green’s type when the coefficient has zeros, Proc. Cambridge Philos. Soc. 52, 61-66 (1956). [10.9] R. E. Langer, On the asymptotic solutions of differential equations with an application t o the Bessel functions of large complex order, Trans. Amer. Math. Soc. 34, 447-480 (1932). [10.10] R. E. Langer, The asymptotic solutions of ordinary linear differential equations of the second order, with special reference t o a turning point, Trans. Amer. Math. Soc. 67, 461-490 (1949). [10.11] W. Magnus and F. Oberhettinger, Formeln und Satee fur die speziellen Funktionen der mathematischen Physik, 2d ed. (Springer-Verlag, Berlin, Germany, 1948). [10.12] F. W. J. Olver, The asymptotic solution of linear differential equations of the second order for large values of a parameter, Philos. Trans. Roy. Soc. London [A] 247, 307-327 (1954-55). 110.131 F. W. J. Olver, The asymptotic expansion of Bessel functions of large order, Philos, Trans. Roy. Soc. Lbndon [A] 247, 328-368 (1954). [10.14] F. W. J. Olver, Uniform asymptotic expansions of solutions of linear second-order differential equations for large values of a parameter, Philos. Trans. Roy. Soc. London [A] 250,479-517 (1958). [10.15] W. R. Wasow, Turning point problems for systems of linear differential equations. Part I : The formal theory; Part 11: The analytic theory. Comm. Pure Appl. Math. 14, 657-673 (1961); 15, 173-187 (1962). [10.16] G. N. Watson, A treatise on the theory of Bessel functions, 2d ed. (Cambridge Univ. Press, Cambridge, England, 1958).

[10.17] H . K. Crowder and G. C. Francis, Tables of spherical Bessel functions and ordinary Bessel functions of order half and odd integer of the first and second kind, Ballist,ic Research Laboratory Memorandum Report No. 1027, Aberdeen Proving Ground, Md. (1956). [10.18] A. T. Doodson, Bessel functions of half integral order [ Riccati-Bessel functions], British Assoc. Adv. Sci. Report, 87-102 (1914). [10.19] A. T. Doodson, Riccati-Bessel functions, British Assoc. Adv. Sci. Report, 97-107 (1916). [10.20] A. T. Doodson, Riccati-Bessel functions, British Assoc. Adv. Sci. Report, 263-270 (1922). [10.21] Harvard University, Tables of the modified Hankel functions of order one-third and of their derivatives (Harvard Univ. Press, Cambridge, Mass., 1945). [10.22] E. Jahnke and F. Emde, Tables of functions, 4th ed. (Dover Publications, Inc., New York, N.Y., 1945). [10.23] C. W. Jones, A short table for the Bessel functions In++(z),( 2 / ~K,+)(z) ) (Cambridge Univ. Press, Cambridge, England, 1952). [10.24] J. C. P. Miller, The Airy integral, British Assoc. Adv. Sci. Mathematical Tables, Part-vol. B (cambridge Univ. Press, Cambridge, England, 1946). [ 10.251 National Bureau of Standards, Tables of spherical Bessel functions, vols. I, I1 (Columbia Univ. Press, New York, N.Y., 1947). [ 10.261 National Bureau of Standards, Tables of Bessel functions of fractional order, vols. I, I1 (Columbia Univ. Press, New York, N.Y., 1948-49). [10.27] National Bureau of Standards, Integrals of Airy functions, Applied Math. Series 52 (U.S. Government Printing Office, Washington, D.C., 1958). [10.28] J. Proudman, A. T. Doodson and G. Kennedy, Numerical results of the theory of the diffraction of a plane electromagnetic wave by a conducting sphere, Philos. Trans. Roy. SOC. London [A] 217, 279-314 (1916-18), in particular pp. 284288. [10.29] M. Rothman, The problem of an infinite plate under an inclined loading, with tables of the integrals of Ai (hz),Bi (hz),Quart. J. Mech. Appl. Math. 7, 1-7 (1954). [10.30] M. Rothman, Tables of the integrals and differential coefficients of Gi (+z), Hi (-z), Quart. J. Mech. Appl. Math. 7, 379-384 (1954). E10.311 Royal Society Mathematical Tables, vol. 7, Bessel functions, Part 111. Zeros and associated values (Cambridge Univ. Press, Cambridge, England, 1960).

456 [10.32] R.

BESSEL FUNCTIONS OF FRACTIONAL ORDER

S.Scorer, Numerical evaluation of integrals of the form

and the tabulation of the function Gi

sin (uz+ 1/3u3)du,

( 2 ) = (l/7r)Lm

Quart. J. Mech. Appl. Math. 3, 107-112 (1950).

[10.33] A. D. Smirnov, Tables of Airy functions (and special confluent hypergeometric functions). Translated from the Russian by D. G. Fry (Pergamon Press, New York, N.Y., 1960). [10.34] I. M. Vinogradov and N. G . Cetaev, Tables of Bessel functions of imaginary argument (Iadat . Akad. Nauk SSSR., Moscow, U.S.S.R., 1950). [10.35] P. M. Woodward, A. M. Woodward, R . Hensman, H. H . Davies and N. Gamble, Four-figure tables of the Airy functions in the complex plane, Phil. Mag. (7) 31, 236-261 (1946).

457

BESSEL FUNCTIONS OF FRACTIONAL ORDER

SPHERICAL BESSEL FUNCTIONS-ORDERS 71

j , (4

j , (4

Table 10.1

0 . 1 AND 2

0.0 0.1 0.2 0.3 0.4

1.00000 0.99833 0.99334 0.98506 0.97354

000 417 665 736 586

0.00000 0.03330 0.06640 0.09910 0.13121

0000 0012 0381 2888 215

j, (4 0.00000 000000 0.00066 619061 0.00265 90561 0.00596 15249 0.01054 5302

-9.95004 -4.90033 -3.18445 -2.30265

17 29 50 25

0.5 0.6 0.7 0.8 0.9

0.95885 0.94107 0.92031 0.89669 0.87036

108 079 098 511 323

0.16253 0.19289 0.22209 0.24998 0.27639

703 196 828 551 252

0.01637 0.02338 0.03153 0.04075 0.05094

1107 8995 8780 0531 5155

-1.75516 -1.37555 -1.09263 -0.87088 -0.69067

51 94 17 339 774

-4.46918 -3.23366 -2.48121 -1.98529 -1.63778

13 97 34 93 29

- 14.792789

1.0 1.1 1.2 1.3 1.4

0.84147 0.81018 0.77669 0.74119 0.70389

098 851 924 860 266

0.30116 0.32417 0.34528 0.36438 0.38137

868 490 457 444 537

0.06203 0.07392 0.08651 0.09968 0.11334

5052 4849 2186 8571 028

-0.54030 -0.41236 -0.30196 -0.20576 -0.12140

231 011 480 833 510

-1.38177 -1.18506 -1.02833 -0.89948 -0.79061

33 13 66 193 059

-3.60501 -2.81962 -2.26887 -1.86995 -1.57276

76 54 66 92 05

1.5 1.6 1.7 1.8 1.9

0.66499 0.62473 0.58333 0.54102 0.49805

666 350 224 646 268

0.39617 0.40870 0.41892 0.42679 0.43228

297 814 749 364 539

0.12734 0.14159 0.15595 0.17029 0.18450

928 426 157 628 320

-0.04715 +0.01824 0.07579 0.12622 0.17015

8134 9701 0879 339 240

-0.69643 -0.61332 -0.53874 -0.47090 -0.40849

541 744 937 236 878

-1.34571 -1.16823 -1.02652 -0.91106 -0.81515

27 87 51 065 048

2.0 2.1 2.2 2.3 2.4

0.45464 0.41105 0.36749 0.32421 0.28144

871 208 837 966 299

0.43539 0.43614 0.43454 0.43065 0.42451

778 199 522 030 529

0.19844 0.21200 0.22506 0.23749 0.24920

795 791 330 812 113

0.20807 0.24040 0.26750 0.28968 0.30724

342 291 051 523 738

-0.35061 -0.29657 -0.24590 -0.19826 -0.15342

200 450 723 956 325

-0.73399 -0.66408 -0.60282 -0.54829 -0.49902

142 077 854 769 644

2.5 2.6 2.7 2.8 2.9

0.23938 0.19826 0.15828 0.11963 0.08249

886 976 884 863 9769

0.41621 0.40583 0.39346 0.37923 0.36326

299 020 703 606 136

0.26006 0.26999 0.27889 0.28668 0.29328

673 585 675 572 784

0.32045 0.32957 0.33484 0.33650 0.33481

745 260 153 798 316

-0.11120 588

-0.45390 -0.41208 -0.37292 -0.33592 -0.30072

450 537 316 641 380

3.0 3.1 3.2 3.3 3.4

0.04704 t0.01341 -0.01824 -0.04780 -0.07515

0003 3117 1920 1726 9148

0.34567 0.32662 0.30626 0.28475 0.26224

750 847 652 092 678

0.29863 0.30267 0.30536 0.30666 0.30655

750 895 678 620 336

0.32999 0.32230 0.31196 0.29923 0.28435

750 166 712 629 241

0.06295 0.09055 0.11573 0.13847 0.15879

9164 5161 164 939 221

-0.26703 -0.23466 -0.20346 -0.17334 -0.14424

834 763 870 594 164

3.5 3.6 3.7 3.8 3.9

-0.10022 -0.12292 -0.14319 -0.16101 -0.17635

378 235 896 523 030

0.23892 0.21495 0.19051 0.16577 0.14091

369 446 380 697 846

0.30501 0.30205 0.29766 0.29189 0.28474

551 107 961 179 912

0.26755 0.24909 0.22921 0.20814 0.18613

905 956 622 940 649

0.17666 0.19211 0.20514 0.21579 0.22407

922 667 929 139 760

-0.11612 - 0.08900 - 0.06287 0.03778 -0.01376

829 2337 8964 7773 9102

4.0 4.1 4.2 4.3 4.4

-0.18920 -0.19957 -0.20751 -0.21306 -0.21627

062 978 804 185 320

0.11611 0.09152 0.06731 0.04365 +0.02069

075 2967 9710 9843 5380

0.27628 0.26654 0.25560 0.24352 0.23038

369 781 355 220 368

0.16341 0.14020 0.11672 0.09320 0.06984

091 096 877 9110 8380

0.23005 0.23377 0.23531 0.23473 0.23214

335 514 060 838 783

+ 0.00912

0.03085 0.05135 0.07056 0.08843

9107 4018 0236 1855 4232

4.5 4.6 4.7 4.8 4.9

-0.21722 -0.21601 -0.21274 -0.20753 -0.20050

892 978 963 429 053

-0.00142 -0.02257 -0.04262 -0.06146 -0.07898

95812 9838 9993 5266 2225

0.21627 0.20129 0.18553 0.16911 0.15214

586 380 900 850 407

0.04684 0.02438 + 0.00263 - 0.01822 - 0.03806

3511 0984 5886 8955 3749

0.22763 0.22132 0.21331 0.20373 0.19273

858 000 046 659 242

0.10491 0.11995 0.13351 0.14556 0.15606

554 814 972 433 319

5.0

-0.19178 485

[( -;)“I

-0.09508 9408

0.13473 121

[(-;PI

Y,(4

Y&>

Y1(4

-cm

-cm

- 0.05673 2437

-00

-200.49875 -25.495011 -11.599917 --6.73017 71

- 0.07151 1067 - 0.03427 3462

+ 0.00054 2796 0.03295 3045

0.18043 837

- 3005.0125

- 377.52483 - 112.81472

- 48.173676 - 25.059923 -9.54114 00 -6.57398 92 -4.76859 87

-

0.16499 546

I--

Compiled from National Bureau of Standards, Tables of spherical Bessel functions, vols. I, 11. Columbia Univ. Press, New York, N.Y., 1947 (with permission).

458

BESSEL FUNCTIONS O F FRACTIONAL ORDER SPHERICAL BESSEL FUNCTIONS-ORDERS

Table 10.1 j o (4

X

5.0 5.1 5.2 5. 3 5.4

y,

ji

Yz (4

(4

-1 -1 -1 -1

-1.9178 -1.8153 -1.6990 -1.5703 -11 -1.4310

5.5 5.6 5. 7 5.8 5.9

-2 -8.0104 -21 -6.3369

6. 0 6.1 6. 2 6.3 6.4

-2) -2 -2) -3 -21

-4.6569 -2.9863 -1.3402 +2.6689 1.8211

6. 5 6.6 6. 7 6. 8 6.9 7. 0 7.1 7.2 7.3 7.4

0 , l AND 2

-1 -1.2435 -11-1.2995

-1

1.1650 -2)-5.9263

(-11 1.2145

7.5 7. 6 7.7 7.8 7.9

-1 -1.2437 -11-1.1816

ji -1 -1 -2 -2 -21

8. 0 8.1 8.2 8. 3 8.4

-1.1105 -1.0313 -9.4473 -8.5177 -7.5334

8. 5 8. 6 8. 7 8.8 8. 9

I1 It -2 -2 -2 -2 -2

1.8188 3.0067 4.1360 5.1973 6.1820

-1 -1 -2 -2 -2

It

1.0497 1.0257 9.9213 9.4941 8.9817

(-2)

8.3907

-1 -1 -1 -1

-1.2140 -1.1603 -1.0968 -1.0243 -2 -9.4378

9. 0 9.1 9.2 9. 3 9.4 9.5 9.6 9. 7 9. 8 9.9

10. 0

( -2) -5.4402

(-2)

7.8467

(-2)

7.7942

(-2)

(-2) -6.5069

6.2793 /--

459

BESSEL FUNCTIONS O F FRACTIONAL ORDER SPHERICAL BESSEL FUNCTIONS-ORDERS 2

j , (4

j , (4

0.0 0.1 0.2 0.3 0.4

0.0000 -6 9.5185 1-51 7.6021 -4 2.5586 -4 6.0413

0.0000 -7 1.0577 1-61 1.6900 -6 8.5364 -5 2.6894

0.5 0.6 0.7

-3 1.1740 1-313.1787 -3 2.0163

-5 6.5390 -412.4847 -4 1.3491

0.8 0.9

-3 4.7053 -3 6.6361

I

-4 4.2098 1-4 6.6912

1

j , (4

-

( - 2 2.8325 -213.3774 1-2)3.9754 -2 4.6252 1-215.3249

(-3)4.8324 -3 6.1667 [-317.7397 3 9.5709 [1211.1679

[ --- 4)6.6962 4 9.1354 311.2212

:2;2!

-2 6.0722 1-2 6.8639 ( - 2 7.6962 ((-18.5650 - 2 9.4654

( - 2 1.4079 [-211.6788 -2 1.9817 [-212.6872 -2 2.3176

3 2.6352 [-( - 313.3094 3 4.1059 [-- 3)6.1171 3)5.0375

-2 8.8610 -219.5706 1-111.1022 -1 1.0292

3,9

,-1, 2,2414

, -1,1:175h

-

[I

1:

6 1.8989

- 613.8277

5 5.5059

1.5 1.6 1.7 1.8 1.9

-1 1.9681 -1 2.0456 -1 2.1174 1-42.1829

- 7 1.1467 - 748.5649 3.4113

- 5 3.0755

-3 2.7976 -31 3.7164

3.5 3.6 3.8 3.7

11

6 2.9775 511.5866 6 7.3776

1.9033 2.3411

2.4 2.3

[

0.0000 -12 7.3975 -101 4.7297 9 5.3784 813.0149

1- 1 -

j , (4

&(x)

0.0000 -10 9.6163 - 81 3.0737 - 7)2.3296 - 7)9.7904

-

3 1 6031 312:0705

1-[I

5 7.8174 4 1.1395 411.6212 4 2.2577 4 3.0840

1:

( - 4 4.1404 - 4 5.4720 - 4 7.1289 j9.1665 3)1.1645

1:

-

2 3.1042 2 3.4705 44.2765 2 3.8614

8.9491 [-- 32241.3569 1.0336 1.1873

-

2, 4,7151

2, 1,5429

0.0000 -14)4.9319 -12) 6.3072 -10 1.0761 [-1018.0448

109~-~j~(.) 1 0 1 1 -lojlo(x) ~ 1.52734 1.52698 1.52589 1.52407 1.52154

93 56 53 96 09

7.27309 7.27151 7.26677 7.25887 7.24783

19 10 00 47 46

1.51828 1.51430 1.50962 1.50423 1.49815

26 88 48 66 12

7.23366 7.21637 7.19599 7.17254 7.14605

29 65 61 61 44

1.49137 1.48392 1.47579 1.46700 1.45757

65 11 48 80 18

7.11655 7.08407 7.04866 7.01035 6.96919

26 57 21 39 61

6 7.8875 5 1.2279 41.8587 5 2.7444 513.9632

1.44749 1.43680 1.42549 1.41358 1.40109

84 05 17 63 93

6.92523 6.87852 6.82912 6.77708 6.72246

71 85 49 37 53

5 5.6097 517.7975 4 1.0661 411.4358 4 1.9071

1.38804 1.37444 1.36030 1.34565 1.33050

63 35 78 67 81

6.66533 6.60575 6.54379 6.47951 6.41301

28 19 07 98 19

1.31488 1.29879 1.28226 1.26531 1.24796

05 28 44 50 48

6.34434 6.27358 6.20082 6.12613 6.04960

22 74 63 95 91

1.23023 1.21214 1.19371 1.17496 1.15592

41 38 48 82 54

5.97131 5.89135 5.80979 5.72674 5.64226

85 26 75 00 82

1.13660 1.11703 1.09723 1.07722 1-05702

79 73 52 33

5.55647 5.46943 5.38125 5.29201

31

05 61 47 62 5-20181 05

1.03665 1.01614 0.99550 0.97477 0.95395

63 44 88 06 10

5.11072 5.01885 4.92629 4.83311 4.73942

78 80 07 51 00

0.93307 0.91215 0.89120 0.87026 0.84934

06 01 97 94 88

4.64529 4.55082 4.45609 4.36119 4.26620

34 25 35 18 13

[

1:

0.0000 (-16) 2.9012

9 3.8259 44.0046 8 1.3665 7 1.0153

[I 712.3040

-

[I [I -

1-

-

-

5)1.9754

- 5)2.7420

3 2.1980 3 2.6195 43.6544 3 3.1030

- 3, 4,2801

I-

- 3 2.8229 3 - 3 - 3 - 3

I 1 5.0

(-1)2.2982

(-1)1.8702

( - 1)1.0681

( - 2) 4.7967 __

Table 10.2

3-10

( - 2) 1.7903

3.2814 3.7976 4.3763 5.0226

( - 3) 5.7414

0.82846 70

4.17120 50

jntxi =&/XJ.,,

Compiled from National Bureau of Standards, Tables of spherical Be:ssel functions, vols. I, 11. Columbia Univ. Press, New York, N.Y., 1947 (with permission).

460

BESSEL FUNCTIONS OF FRACTIONAL ORDER SPHERICAL BESSEL FUNCTIONS-ORDERS

Table 10.2 /'

0.0 3.1 0.2 0.3 0.4

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

( ( ( (

5 -1.5015 31-9.4126 3)-1.8686 2)-5.9544

- 0 0

( 4)-1.0372

( 5)-2.3278

11) 8) 7) 6)

-1.0400 -8.1359 -4.7726 -6.3910

Y, (4

Y,(4

- 0 0

( 8)-2.0747

( 9)-7.7739

7) -3.4929

( 9)-1.0465

( 4)-6.1328

l o - s P y , ).(

10-gxr"l,lo(..)

-0.34459 -0.34469 -0.34499 -0.34550 -0.34622

42 56 99 77 02

-0.65472 -0.65490 -0.65541 -0.65628 -0.65749

90 14 86 18 23

86 48 12 04 56

-0.65905 -0.66096 -0.66323 -0.66586 -0.66885

23 47 28 06 29

( 1)-2.4854

( 2) -1.8854

-2.2552

5) -3.2389

-0.34713 -0.34826 -0.34960 -0.35115 -0.35291

( 0 -6.2927 ( 01-4.8264

( 2)-1.1290 1 -7.1198 11-4.6879 1 -3.2014 11-2.2559

4) -1.0881 3 -5.6378 31 -3.0988 3) -1.7901 3) -1.0790

5) -1.4045 4 -6.6058 41 -3.3227 4 -1.7686 ( 31-9.8790

-0.35490 -0.35710 -0.35954 -0.36221 -0.36512

04 89 56 57 46

-0.67221 -0.67595 -0.68007 -0.68458 -0.68949

50 30 37 47 42

-0.36827 -0.37168 -0.37534 -0.37928 -0.38348

87 46 96 17 96

-0.69481 -0.70054 -0.70670 -0.71331 -0.72036

14 60 90 20 75

-0.38798 -0.39277 -0.39786 -0.40327 -0.40901

26 08 50 71 97

-0.72788 -0.73589 -0.74439 -0.75340 -0.76294

93 19 11 38 81

-0.41510 -0.42155 -0.42837 -0.43558 -0.44320

62 14 10 18 20

-0.77304 -0.78371 -0.79497 -0.80685 -0.81937

34 06 18 08 31

-0.45125 -0.45975 -0.46872 -0.47818 -0.48818

11 01 14 95 03

-0.83256 -0.84645 -0.86108 -0.87646 -0.89265

59 82 11 78 39

-0.49872 -0,50984 -0.52158 -0.53396 -0.54704

20 49 17 75 05

-0.90967 -0.92757 -0.94640 -0.96619 -0.98699

72 84 10 15 97

-0.56084 -0.57541 -0.59081 -0.60708 -0.62428

19 63 20 14 15

-1.00887 -1.03188 -1.05608 -1.08153 -1.10831

91 69 44 78 79

-0.64247 43 -0.66172 73 -0.68211 42 -0.70371 55 -0.72661 94

-1.13650 -1.16616 -1.19741 -1.23032 -1.26500

10 90 05 08 29

-0.75092

-1.30156

80

-1.2907

( 0)-2.4804 ( 0)-2.0598 ( 0)-1.7366

2.0 2.1 2.2 2.3 2.4

( 0 -1.4844 ( 01-1.2846 ( 0)-1.1242 ( -1)-9.9368 (-1)-8.8622

2.5 2.6 2.7 2.8 2.9

( - 1 -7.9660 ( -11-7.2096 ( -1) -6.5632 ( -1) -6.0041 ( -1) -5.5144

3.0 3.1 3.2 3.3 3.4

(-1 -5.0802 ( - 1 -4.6905 (-1 -4.3365 (-1 -4.0112 (-1) -3.7091

4.0 4.1 4.2 4.3

?I&)

.U&)

I

( 2)-2.4613

1.5 1.6 1.7 1.8 1.9

3.5 3.6 3.7 3.8 3.9

Y4 (1.)

!/.<(.I,,

3-10

i

I- 1

-1 -2.6551 1 -2.4173

2) -6.7473 2) -4.3572 -2.8948 -1.9724 2) -1.3747

-2.2859 -1.7812 -1.1189 -9.0069

-1) -7.4514 -1 -6.7752 (-11 -6.1940

0 -1.4467 01 -1.2687

-2.9528

-5.6501 [(-1 -11 -5.2492 -1 -4.8600

I

-11 -4.5131 -1) -4.2011

( -1) -2.1864 (-1) -1.9615

(-1)-1.7418

4.4

I- i

-1) -3.1913 (-1) -2.9788

-1 -5.1525 -11 -4.7765

-1) -4.4430 -1 -4.1450 -11 -3.8766 -1 -3.6331 -11 -3.4102

4.5 4.6 4.7 4.8 4.9

(-1 -2 2 (-2) ( -2)

-1.1107 -9.0931 -7.1268 -5.2107 -3.3484

-2.7768

5.0

(-2) -1.5443

(-1) -1.8662

1 -4.8802 11-3.7729

( - 1 ) -3.2047 !in(x)-

I

0 -9.8471 01 -8.1040 0) -6.7182 0 -5.6086 01 -4.7139

( 1)-1.6498

0)-5.4951

( 0 -1.2915 ( 01-1.1491

(-1) -5.1841

( 0)-1.0274

( 0)-2.5638

23

461

BESSEL FUNCTIONS O F FRACTIONAL ORDER SPHERICAL BESSEL FUNCTIONS-ORDERS j , (4

T

j, (4 -3 5.7414 -31 6.5379 -3 7.4172 -3 8.3843 -3 9.4443

&(x)

il I

-1 -1 -1 -1 -1

il

2.0078 2.0150 2.0147 2.0069 1.9913

0.72528 0.70501 0.68490 0.66497 0.64523

47 58 78 60 54

3.69892 3.60555 3.51270 3.42045 3.32886

98 18 30 23 66

(-2 9.3796 -219.8696 1-111.0356 -1 1.0837 -1)1.1309

0.62570 0.60638 0.58729 0.56845 0.54987

01 37 93 94 57

3.23801 3.14794 3.05873 2.97043 2.88309

06 66 50 34 73

0.53155 0.51352 0.49577 0.47831 0.46116

94 10 04 68 a9

2.79677 2.71153 2.62739 2.54443 2.46266

98 12 98 09 76

1.7217 1.7036 1.6793 1.6486 1.6117

0.44433 0.42782 0.41163 0.39578 0.38026

45 11 52 30 97

2.38215 2.30291 2.22500 2.14844 2.07326

03 70 27 05 03

(-1) 1.5685 ;-l) 1.5193

0.36510 0.35027 0.33580 0.32169 0.30793

02 86 85 28 39

1.99948 1.92715 1.85627 1.78687 1.71897

99 45 66 63 14

(-1)1.5312 (-1)1.2227 (-1 1.5212 -1 1.2524 (-1 1.5060 -1 1.2795 (-1 1.4857 -1 1.3039 (-2) 9.3440 ( -1)1.4601 (-1)1.3252

0.29453 0.28149 0.26881 0.25649 0.24453

36 30 29 33 39

1.65257 1.58770 1.52436 1.46257 1.40232

72 64 97 53 92

0.23293 0.22169 0.21080 0.20027 G.i3009

38 16 54 29 il

1.34363 1.28649 1.23090 1.17687 1.1233s

53 51 84 25 32

0.18025 0.17076 0.16161 0.15280 0.14432

78 84 93 62 46

1.07343 1.02401 0.97612 0.92973 0.88485

42 72 24 83 16

0.13616 0.12833 0.12081 0.11360 0.10670

93 53 68 83 35

0.84144 0.79950 0.75902 0.71996 0.68231

75 99 10 20 26

-1 -1 -1 -1 -1

il [

i-ij -i.i558

[

-21 5.1815 -2 5.5543

-2 5.9379

(-2) 3.2552

i

8.9

II

I1!-

(-1 1.4292 (-1)1.3434 -111.3932 -1 1.3581 t.3520 1-11 1.3693 -1 1.3059 (-1)1.3767 \ -1) 1.2546 -i) i.380i

(-2)-3.3894 -2 -4.3664 -21-5.2894 -2)-6.1529 \ -2) -6.9520

9.0 ( -1)-1.1207 9.1 (-1 -1.0770 9.2 -11-1.0252 9.3 -2 -9.6572 9.4 (-2)-8.9931

[

-1)-1.0170 -1)-1.0419

10.0

(-2)-3.9496

(-1)-1.0559

11 -10

109z29j9(s) 10 z jl,,(x) 0.82846 70 4.17120 50 0.80764 29 4.07628 42 0.78689 50 3.98151 88 0.76624 10 3.88698 72 0.74569 86 3.79276 59

-2)6.9660

I

7.5 -2 -6.1713 7.6 -21-7.1719 7.7 -2 -8.0904 7.8 1-2 -8.9217 7.9 ( -2)-9.6613

Table 10.2

3-10

(-2)-5.5535

(-2)4.4501 (-1)1.1339

(-1)1.2558 0.10009 64 0.64605 15

[ (-515 4 3

462

BESSEL FUNCTIONS O F FRACTIONAL ORDER

Table 10.2

SPHERICAL BESSEL FUNCTIONS-ORDERS

.?/&9 (-1) -1.8662

:::5.4

?j6 ( J )

-1)-3.2047 -1) -3.0134

(-1)-5.1841

-2 -2.7936 -2 -1.3619 +2.9727 (-2 1.3770 (-21 2.6754

I 1

0 -1.0274 -1 -9.2298 -1) -8.3305 -1 -7.5528 -1 -6.8777

I-4j

-1) -2.1672

( -1)-2.0428 (-1)-1,9220 -1)-2.4985

7 5 7:6

{:I]

1.5273 1.4956 1.4535 1.4016 1.3404

10-8210y9 ( x )

10-9

-0.75092 -0.77673 -0.80415 -0.83333 -0.86440

23 01 92 74 56

-1.30156 -1.34013 -1.38083 -1.42381 -1.46922

80 68 98 86 70

-0.89751 -0.93284 -0.97058 -1.01093 -1,05412

9C 85 31 09 18

-1.51723 -1.56801 -1.62178 -1.67873 -1.73913

25 75

97 16

-1) -6.0416 -1) -5.5598

-1.10040 -1.15007 -1.20342 -1.26079 -1.32256

93 32 16 38 26

-1.80321 -1.87128 -1.94363 -2.02062 -2.10262

67 02 49 45 69

-5.1344 -4.7576 -4.4227 -4.1239 -3.8565

-1.38913 -1.46096 - 1.53853 -1.62238 -1.71309

71 57 78 69 24

-2.19005 -2.28337 -2.38308 -2.48973 -2.60393

78 46 14 26 95

-3.6163 -3.3996 -3.2032 -3.0246 -2.8613

-1.81128 -1.91762 -2.03285 -2.15774 -2,29311

11 85 95 75 31

-2.72637 -2.85777 -2.99896 -3.15082 -3.31433

44 73 17 08 45

-2.43982

-3.49057

53

-3.88603 -4.10791 -4.34788

37 96

Y&

Y&)

0 -2.5638 01 -2.2343

1 1

1-21 3.4698 -2 4.9908

-1 7.0 7.1 1-11 -1 7.2 -1 7.3 7.4 (-1

3-10

(-2) (-1 (-11 (-1 (-11

1 1.2705 1.1925

I1

9.1846 1.0016 1.0764 1.1427 1.2001

-1 -2.3703 -1 -2.2489

1 1.2485

I

11 J:

Y &)

oa

-1 -2.7112 -11 -2.5726 -1 -2.4439 -1) -2.3236 -1) -2.2106

-2.77091 -2.95720 -3.15862

13 67 77 73 24

-2.1038 -2.0022 -1.9050 -1.8115 -1.7211

-3.37613 -3.61071 -3.86327 -4.13466 -4.42566

93 67 49 98 30

-4.60754 -4.88866 -5.19312 -5.52293 -5.88021

55 85 95 51 45

09 69 35 68 88

- 2.59877

- 3.68071 56

05

8.0 -2 8.1577 8.1 1-21 7.0941 8.2 -2 5.9992 8.3 -2 4.8821 8.4 ( - 2 3.7517

-1 1.3509 -11 1.3435

8.5 ( - 2 2.6172 8.6 (-21 1.4876 8.7 -3 +3.7160 8.8 -7.2210

-1 1.2259 -11 1.1762 -1 1.1191 -11 1.0551 ,-2, 3.8432

-1 -1.6331 -11 -1.5471 -1.4627 -1.3795 -1.2973

-4.73689 -5.06881 -5.42169 -5.79550 -6.18991

-6.26721 -6.68628 -7.13987 -7.63051 -8.16074

41 70 95 13 96

( - 2 9.0898 -21 8.2794

-1 -1.2156 -11 -1.1345 -1.0536 -2 -9.7298 -2 -8.9243

-6.60420 33 -8.73317 -7.03717 50 -9.35034 -7.48710 95 -10.01475 -7.95166 19 -10.72873 -8.112777 38 -11.49443

65 96 2 2 4

8.9

1-?I

-

-2 -2 -2 -2 -2

I1

7.0400 7.8334 8.5654 9.2329 9.8330

. . . . , I

(-f)-l.J03L

I-lj -2) -2) -2) -2 -31 10.0 (-2) -9.5327

4.6623 3.6995 2.7280 1.7550 +7.8793

( -3)- 1.6599

(-1) 1.1522

(-2)

9.3834

-8.1193 -8.91157 -7.3150 -9.39828 -6.5114 -9.88210 -5.7090-10.35610 -4.9088 .- 10.81210 (-1) 1.0488

(-2)

4.2506

56 63 58 3 4

(-2) -4,1117 - 11.24057 9

-12.31371 -13.18805 -14.11841 -15.10518 -16.14793

9 2 9

-17.24536

7

5

o

BESSEL FUNCTIONS O F FRACTIONAL ORDER

SPHERICAL BESSEL FUNCTIONS-ORDERS

20 AND 21

Table 10.3

7.62597 7.62705 7.63028 7.63560 7.64293

90 91 29 15 25

1.77348 1.77371 1.77439 1.77552 1.77707

35 23 56 32 85

1 0 ~20~(XI~ 9 -0.31983 10 -0.31988 11 -0.32003 25 -0.32028 86 -0.32065 49

1 0 - ~ -1.31130 -1.31149 -1.31205 -1.31300 -1.31436

(3c> ~ 9

0.0 0.5 1.0 1.5 2.0

2.5 3.0 3.5 4.0 4.5

7.65215 7.66313 7.67566 7.68952 7.70444

99 22 19 28 90

1.77903 1.78137 1.78403 1.78699 1.79018

78 03 80 49 73

-0.32113 -0.32175 -0.32250 -0.32342 -0.32450

96 30 82 08 98

-1.31616 -1.31842 -1.32121 -1.32457 -1.32856

11 87 43 29 95

5.0 5.5 6.0 6.5 7.0

7.72013 7.73621 7.75231 7.76795 7.78264

23 95 00 28 38

1.79355 1.79702 1.80050 1.80392 1.80717

29 05 95 94 91

-0.32579 -0.32730 -0.32907 -0.33112 -0.33350

69 79 24 44 34

-1.33328 -1.33879 -1.34521 -1.35264 -1.36123

02 33 03 77 89

7.5 8.0 8.5 9.0 9.5

7.79582 7.80686 7.81509 7.81976 7.82005

23 80 84 53 32

1.81014 1.81270 1.81472 1.81605 1.81653

64 77 70 56 14

-0.33625 -0.33943 -0.34309 -0.34731 -0.35216

47 07 23 02 70

-1.37113 -1.38251 -1.39557 -1.41055 -1.42771

69 67 96 73 82

10.0 10.5 11.0 11.5 12.0

7.815076 7.803876 7.785428 7.758627 7.722309

1.815979 1.814208 1.811016 1.806185 1.799482

-0.35776 -0.36420 -0.37164 -0.38023 -0.39019

04 59 20 59 23

-1.447374 -1.469891 -1.495697 -1.525305 -1.559325

12.5 13.0 13.5 14.0 14.5

7.675238 7.616116 7.543601 7.456316 7.352841

1.79 0664 1.779472 1.765639 1.748885 1.728929

-0.40176 -0.41527 -0.43113 -0.44987 -0.47223

53 46 22 76 40

-1.598497 -1.643728 -1.696143 -1.757166 -1.828625

15.0 15.5 16.0 16.5 17.0

7.231764 7.091 689 6.931 265 6.74 9220 6.544411

1.705481 1.678251 1.64 6956 1.611324 1.571 096

-0.49918 -0.53209 -0.57279 -0.62378 -0.68821

70 15 98 79 72

-1.912922 -2.013273 -2.134049 -2.281228 -2.462936

17.5 18.0 18.5 19.0 19.5

6.315851 6.062784 5.784739 5.481584 5.153621

1.526041 1.475960 1.420698 1.360155 1.294299

-0.76981 49 -0.87240 01 -0.99883 14 -1.149171 -1.317987

-2.689957 -2.975953 -3.336925 -3.789188 -4.344958

20.0 20.5 21.0 21.5 22.0

4.801647 4.427041 4.031843 3.61 8830 3.191590

1.223178 1.146936 1.065826 0.98022 63 0.89065 46

22.5 23.0 23.5 24.0 24.5

2.754567 2.313103 1.873442 1.442686 1.028721

0.79777 0.70243 0.60561 0.50849 0.41242

52 25 45 80 27

- 1.024223 - 0.274630 2.072631 3.508629

-7.504782 -6.640003 -4.7 17888 -1.52185 +3.01816

0.640055

0.31888 30

4.901591

+8.74251

5

25.0

1026f2 0 ( 4

[(-;)31

1027f21 (2)

- 1.490982 -1.641599 - 1.728777 - 1.697442 - 1.483467

+ 0.773430

70 33 61 70 61

-5.0 04711 -5.745922 -6.5 08927 -7.182333 -7.592679

[(-;)73

Compiled from National Bureau of Standards, Tables of spherical Bessel functions, vols. I, 1I.ColumbiaUniv. Press, New York, N.Y., 1947(with permission).

~ ~

464

BESSEL FUNCTIONS O F FRACTIONAL ORDER

Table 10.4 SPHERICAL BESSEL FUNCTIONS-MODULUS

2-1

JEiM196 (x)

elyi(x)- x

AND PHASE-ORDERS 9, 10, 20 AND 21 G M 2 3 4

(x)

<x>

e 2 3 w -2

0.100 0.095 0.090 0.085 0.080

1.50513 1.41043 1.33509 1.27462 1.22560

630 073 121 197 809

1.72311 1.44562 1.17232 0.90378 0.64017

121 029 718 457 615

1.84157 1.65174 1.50947 1.40190 1.31955

799 534 539 550 792

1.35401 1.00196 0.65310 +0.30984 -0.02643

461 372 249 705 915

10 11 11 12 13

0.075 0.070 0.065 0.060 0.055

1.18548 1.15231 1.12467 1.10147 1.08190

011 423 134 221 340

0.38142 +0.12729 -0.12255 -0.36849 -0.61090

613 416 277 087 826

1.25559 1.20514 1.16476 1.13202 1.10519

223 049 186 416 883

-0.35524 -0.67664 -0.99107 -1.29911 -1.60143

574 889 278 571 947

13 14 15 17 18

0.050 0.045 0.040 0.035 0.030

1.06534 1.05133 1.03949 1.02956 1.02130

781 389 892 235 658

-0.85018 -1.08669 -1.32077 -1.55274 -1.78293

673 229 114 891 175

1.08304 1.06466 1.04939 1.03675 1.02635

588 562 746 104 931

-1.89870 -2.19155 -2.48055 -2.76627 -3.04920

678 009 907 814 936

20 22 25 29 33

0.025 0.020 0. 015 0. 010 0.005

1.01456 1.00920 1.00512 1.00226 1.00056

304 210 574 240 327

-2.01160 -2.23905 -2.46552 -2.69127 -2.91655

832 224 469 701 326

1.01794 1. 01130 1.00628 1.00276 1.00068

637 529 277 864 866

-3.32981 -3.60853 -3.88577 -4.16191 -4.43732

737 532 070 106 935

40 50 67 100 2 00

0.000

1.00000 000

-3.14159

265

1.00000 000

-4.71238

898

Q)

2-1

0.040 0.038 0. 036 0.034 0.032

[(-;PI

,/)?rzMa>$(x) 1.31126 1.25741 1.21433 1.17917 1.15001

605 042 612 949 033

[(;a91

x

8 4 $ $ ( ~ )-

1.12909 0.61321 +0.11048 -0.38066 -0.86163

d G M 4 %(x)

207 135 098 745 915

1.37979 1.30763 1.25205 1.20806 1.17245

868 025 767 627 178

-x

64$/2(~)

<x>

+0.54348 -0.04056 -0,60729 -1.15885 -1.69717

547 472 830 172 688

25 26 28 29 31

-2.22398

514

33

-2.74W5 480

36

-3.24876 -3.74910 -4.24275

024 503 239

38 42 45

561 092 154 423 560

-4.73055 -5.21325 -5.69154 -6.16604 -6.63731

105 651 843 479 350

50 56 63 71 83

351 093 153 654 253

-7.10588 -7.57224 -8.03687 -8.50021 -8.96270

196 522 285 498 770

100 125 167 250 500

1.00000 000

-9.42477

796

06

'3.'331!

1.12549 756

0.028 0.026 0.024 0.022

1.10467 1.08687 1.07157 1.05838

736 488 283 371

-1.33366 819 -1.79783 172 -2.25507 118 -2.70621 373 -3.15199 149

1.14310 1.11857 1.09787 1.08027 1.06523

153 851 629 122 083

0.020 0.018 0.016 0.014 0.012

1.04700 1.03721 1.02883 1.02170 1.01571

987 972 137 104 485

-3.59305 -4.03000 -4.46335 -4.89362 -5.32124

805 220 928 072 187

1.05235 1.04134 1.03195 1. 02400 1.01735

0.010 0.008 0.006 9.004 0.002

1.01078 1.00683 1.00381 1.00168 1.00042

282 452 592 705 044

-5.74664 -6.17024 -6.59240 -7.01351 -7.43392

872 356 995 707 365

1.01189 1.00753 1.00420 1.00185 1.00046

0.000

1.00000 000

-7.85398

164

<x> = nearest integer to X. Compiled from National Bureau of Standards, Tables of spherical Bessel functions, vols. I, 11. Columbia Univ. Press , New York, N.Y., 1947 (with permission).

465

BESSEL FUNCTIONS O F FRACTIONAL ORDER

SPHERICAL BESSEL FUNCTIONS-VARIOUS

x=2

x=l

n

0 1 2 3 4

8.41470 3.01168 6.20350 9.00658 1.01101

9848 6789 5201 1117 5808

5 6 7 8 9

I- 1 I-

(((-

-

-

-

x=5

2 6.07220 9766 2 1.40793 9276

I--

3)2.63516 4 4.14040 515.60965 6) 6.68320 7)7.10679

9770 9734 5703 4324 7192

I- 1

5.74143 4675 1.61809 9715

0865 1612 8901 1402 9119

- 6 - 71

3.69320 6998 6.55454 3131

iI j

10 11 12 13 14

8 6.82530 9 5.97687 1 0 4.81014 11 3.58145 1 2 2.48104

15 16 17 18 19

5 6 7 8 9

( -190) 7.44472 7742

20 7.63264 341 5.83661 49) 1.66097 66)4.01157

i

( - 1)-1.05589

2851

( - 2)-5.55345

1162

-

( - 46) 2.85747 (-120)

~ = 5 0

+7.84669 4180 +7.79421 9363

-

1101 7888 8779 5290

( -160) 9.36783 2591

x=10

10 11 12 13 14 15 16 17 18 19

3 3

I-

n

0 1 2 3 4

1) 2.29820 6182 1) 1.87017 6553

9 2.44802 0198 1 0 3.36741 6303 11 4.38678 6630

20 30 40 50

100

Table 10.5

ORDERS

5 3.23884 7439 61 8.89662 7269

((( -'

I

5.53565 0303

x = 100

7074 7051 0843 9460 7600

-3) -5.06365 -3 -8.67382 -31 +4.80344 -3) +8.91399 -3) -4.17946

-2.00483

0056

-3 j -$~.ZrOir8435

-3.10114 +1.92420 +8.87374 -1.62249

8524 0195 9108 2725

-3 +3.15754 -31+9.70062 -3) -1.70245 -3) -9.99004

5454 9844 0977 6510

+ 95971 l. 1 0 4 1

-4) -1.95657 -3) +9.94895 -3 +2.48391 -31 -9.32797 -3) -5.00247

8597 8359 8282 8789 2555

-3)+7.87726

1748

3) -5.24749 2)-1.94042 3)+4. 08324 2)+1.98125 3) -1.30947

(- 2 - 3 - 2 - 3 (- 2

9350

i j

+ 26561 l. + 96438 l. ( - 3 +1.09459 ( - 21 -1.88338

6411 5287 1652 7370 1837

3175 9234 2888 9360

I

20 30 40 50

(-31)

2.23069 6023

( - 2)+1.88291 0737

( - 2 +1.01076 -31 +8. 70062 -2)+1.04341 ( -4) +5.79714

100

(-90)

5.83204 0182

(-22)+1.01901

( -2) + 08804 l. 7701

2263

7128 8514 0851 0882

466

BESSEL FUNCTIONS OF FRACTIONAL ORDER

Table 10.5

SPHERICAL BESSEL FUNCTIONS-VARIOUS

ORDERS

Y,(4 n 0 1 2 3 4

x=2

x=l

( (

x=5

0 -1.48436 6557 01-4.46129 1526

( -2 -5.67324 ( - 11+l. 80438 ( - 1 +1.64995 ( - 2 -1.54429 ( - 11-1.86615

3709 3675 4576 0991 5315

8 -9.26795 181-7.76071 30)-2.05575 ( 42)- 6.9641 0

1403 7570 8716 9188

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

( 11)-3.23191 3629 ( 12)-8.06570 3047 ( 14)-2.17450 7909

i

( 15 -6.29800 7233 ( 17 -1.95020 7734 ( 18 -6.42938 7516

20 30 40 50

(

100

( 186)-6.68307 9463

n

x=10 -2)+8.39071 5291 -2 C6.27928 2638 -21-6.50693 0499 -2)-9.53274 7888 (-3)-1.65993 0220

0 1 2 3 4

5 6 7 8 9

(156)-2.65595 5830

~=50

(-2)+9.38335 4168

(116)-1.79971 3983

x= 100

-3)+3.72067 8486

-4)+8.07441 4285

10 11 12 13 14

( -3)-5.61681 8446

-2) tl. 71231 -2)Cl. 62332 -3)-6.40928 -2 -2.07197 -31 -8.92329

15 16 17 18 19 20 30 40 50

27)-4.52822 7272

100

(85)-8.57322 6309

9725 0074 4759 0007 3294

(-3)+8.72020 2503 (-3 +6.25864 1510 ( -31-6.78002 3635

( -2)tl. 07478 2297

(+18)-1.12569 2891

(-2)-2.29838 5049

467

BESSEL FUNCTIONS O F FRACTIONAL ORDER ZEROS OF BESSEL FUNCTIONS OF HALF-INTEGER ORDER JY(.L, J=O ,>=o v

s

1/2

1 2 3 4 5 6 7 8

3/2

5/2

7/2

9/2

-0.36741 +0.28469 -0.24061 +0.21220 -0.19194 +0.17656 -0.16437

35 20 69 57 77 64 44

2.798386 6.121250 9.317866 12.486454 15.644128 18.796404 21.945613

+0.44914 -0.31827 +0.25989 -0.22507 +O. 20131 -0.18377 +0.17014

84 37 33 76 63 61 37

1 2 3 4 5 6 7

5.763459 9.095011 12.322941 15.514603 18.689036 21.853874

-0.31710 +0.25973 -0.22503 +O. 20130 -0.18376 +0.17014

58 30 59 14 96 05

3.959528 7.451610 10.715647 13.921686 17.103359 20.272369 23.433926

-0.36184 +0.28430 -0.24053 +O. 21218 -0.19193 +0.17656 -0.16437

68 75 93 15 81 19 21

1 2 3 4 5 6 7

6.987932 10.417119 13.698023 16.923621 20.121806 23.304247

-0.28223 71 +O. 24019 23 -0.21208 02 +0.19189 90 -0.17654 40 +O. 16436 28

5.088498 8.733710 12.067544 15.315390 18.525210 21.714547 24.891503

+O. 30882 36 -0.25896 77 +0.22485 68 -0.20124 01 +0.18374 36 -0.17012 77 +O. 15914 62

1 8.182561 -0.25620 49 6.197831 -0.27236 2 11.704907 +0.22432 53 9.982466 +0.23908 3 15.039665 -0.20107 12 13.385287 -0.21179 4 18.301256 +0.18367 44 16.676625 +0.19179 5 m4f8 -O.!,op! !! A.! pu?? -0,. yb4’!

;:.

L Y . I L f 3 0 0 +U. 1 3 7 L l 00

25 76 27 35

f3:

un+

11.657032 15.431289 18.922999 22.295348

-0.20550 +0.19008 -0.17582 +0.16402

46 87 99 38

,yu,,s (V V ” , ,sl 9.457882 +0.20754 83 13.600629 -0.19801 01 17.197777 +0.18264 01 20.619612 -0.16964 44 23.955267 +O. 15890 14

1 2 3 4

12.790782 16.641003 20.182471 23.591275

-0.19382 +0.18155 -0.16922 +0.15870

82 15 10 04

10.529989 14.777175 18.434529 21.898570

-0.19361 +0.18810 -0.17517 +0.16373

38 92 27 75

19/2

1 2 3 4

13.915823 17.838643 21.428487 24.873214

-0.18376 12 +0.17398 80 -0.16326 17 +0.15383 84

11.597038 15.942945 19.658369 23.163734

+O. 18186 -0.17944 +0.16849 -0.15837

42 10 33 45

21/2

1 15.033469 -0.17496 82 2 19.025854 +O. 16722 59 3 22.662721 -0.15785 09 4

12.659840 17.099480 20.870973 24.416749

-0.17179 22 +O. 17176 97 -0.16247 13 +O. 15347 56

23/2

1 16.144743 -0.16720 39 2 20.203943 +0.16113 25 3 23.886531 -0.15290 87

13.719013 +O. 16304 06 18.247994 -0.16491 86 22.073692 +0.15700 50

25/2

1 17.250455 -0.16028 44 2 21.373972 +0.15560 47 3

14.775045 -0.15534 97 19.389462 +0.15875 20 23.267630 -0.15201 34

27/2

1 18.351261 -0.15406 88 2 22.536817 +0.15056 00 3

15.828325 +0.14852 56 20.524680 -0.15316 36 24.453705 +O. 14743 15

29/2

1 19.447703 -0.14844 69 2 23.693208 +0.14593 21

16.879170 -0.14242 04 21.654309 +0.14806 91

31/2

1 20.540230 -0.14333 12 2 24.843763 +0.14166 70

17.927842 +0.13691 88 22.778902 -0.14340 05

s

JU!

s

.J: ( j y ,

L 2 . I L 0 0 . t ~ +U. 1o.t>2 0 7

1 9.355812 -0.23580 60 2 12.966530 +0.21109 29

7.293692 +0.24538 14 11.206497 -0.22293 49

5 22.904551 -0.16428 83 6

21.283249 +O. 17002 38 24.518929 -0.15909 15

~;:::;:;:;it ~ ~

~ T: ~:i:!$g : :f~ 33/2 ~ ~21 ~21.629221 ~f

~~

~

~

-0.13865 11

18.974562 -0.13192 99 23.898931 +O. 13910 20

35/2

1 22.715002 -0.13434 93

20.019515 +0.12738 05

;::;:;::zi

37/2

1 23.797849 -0.13037 81

21.062860 -0.12321 13

15.945983 -0.19106 59 19.324820 +O. 17619 60 22.628417 -0.16419 26

39/2

1 24.878005 -0.12669 81

22.104735 +0.11937 34

2’ i::zi;!iz ~~:~~~~~ ft: l;:il;$ 3 17.647975 -0.18321 82 4 20.983463 +O. 16988 82 5 24.262768 -0.15902 21

J

1 2 3 4 5

Yu,s (-l)n+ly:(.?/v,,J 1.570796 -0.63661 98 15/2 4.712389 +0.36755 26 7.853982 -0.28470 50 10.995574 +O. 24061 97 14.137167 -0.21220 66 17.278760 +0.19194 81 20.420352 -0.17656 66 23.561945 +O. 16437 45 17/2

4.493409 7.725252 10.904122 14.066194 17.220755 20.371303 23.519452

:

13/2

82 99 89 91 68 63 38

1 2 3 4 5 6 7

6

11/2

J X i , s>

L,s

3.141593 -0.45015 6.283185 +0.31830 9.424778 -0.25989 12.566370 +O. 22507 15,707963 -0.20131 18.849556 +0.18377 21.991149 -0.17014

Table 10.6

Values to greater accuracy and over a wider range are given in [10.31]. From National Bureau of Standards, Tables of spherical Bessel functions, vols. I, 11. Columbia Univ. Press, New York, N.Y., 1947 (with permission).

468

BESSEL FUNCTIONS O F FRACTIONAL ORDER Table 10.7

v

s

1/2

1 2 3 4 5 6 7 8

3/2

5/2

7/2

ZEROS OF THE DERIVATIVE OF BESSEL FUNCTIONS OF HALF-INTEGER ORDER I*,5s

1.165561 4.604217 7.789884 10.949944 14.101725 17.249782 20.395842 23.540708

J,

(L,,>

+O. 679192 -0.369672 +O. 285287 -0.240870 +O. 212340 -0.192029 +O. 176620 -0.164412

2.975086 -0.456186 6.202750 +O. 319331 9.371475 -0.260267 12.526476 +O. 225258 15.676078 -0.201419 18.822999 +O. 183841 21. 968393 -0.170188

1 2 3 4 5 6 7

2.460536 6.029292 9.261402 12.445260 15.611585 18.769469 21.922619

+O. 525338 -0.328062 +O. 263295 -0.226711 +O. 202245 -0.184363 +O. 170542

4.354435 7.655545 10.856531 14.029845 17.191285 20.346496 23.498023

+O. 388891 -0.290138 +O. 242910 -0.213417 +O. 192678 -0.177046 +O. 164709

1 2 3 4 5 6 7

3.632797 7.367009 10.663561 13.883370 17.072849 20.246945 23.412100

+O. 457398 -0.301449 +O. 247304 -0.215670 +O. 194015 -0.177917 +O. 165314

5.634297 9.030902 12.278863 15.480655 18.661309 21.830390 24.992411

-0.350669 +O. 270006 -0.229783 +O. 203956 -0.185432 +O. 171262 -0.159953

1 2 3 4 5 6 7

4.762196 8.653134 12.018262 15.279081 18.496200 21.690284 24.870602

+O. 415533

-0.282237 +O. 234875 -0.206685 +O. 187103 -0.172377 +O. 160741

6.863232 10.356373 13.656304 16.891400 20.095393 23.281796

+O. 324651 -0.254849 +O. 219318 -0.196124 +O. 179270 -0.166245

1 2 3 4 5

17/2

1 2 3 4

12. 669130 16.586323 20.145940 23.563314

1 2 3 4

13.793646 17.784362 21.392422 24.845689

+O. 249935

14.910648 18.971857 22.627032

-0.242951 +O. 200296 -0.178048

16.021196 20.150142 23.851147

+O. 236710 -0.195742 +O. 174383

17.126125 21.320300

+O. 191594

18.226109 22.483219

+O. 225965 -0.187792

19/2

21/2

1 2 3 4

12.299124 17.025072 20.828186 24.385974

+O. 298179

23/2

1 2 3

13.353045 18.173567 22.031181

+o. 289825 -0.211893 +o. 183813

25/2

1 2 3

27/2

9/2

1 2 3 4

5 6

11/2

1 2

3

4 5 6

13/2

1 2 3 4 5

5.868420 9.904306 13.337928 i6.6417~7 19.888934 23.105297

6.959746 11.129856 14.630406 17.977886 21.256291 24.496327

8.040535 12.335631 15.901023 19.291967 22. 602185

+O. 386006 -0.267385 +O. 224788 -n. 199151 +O. 181169 -0.167534

+O. 363557 -0.255385 +O. 216349 -0.192692 +O. 175987 -0.163244

+O. 345649 -0.245384 +O. 209127 -0.187058 +O. 171399

8.060030 11.646354 14.999624 18.270330 21.500029 24.705942

9.234274 12.909478 16.315912 19.623229 22.879980

10.391621 14.151399 17.610124 20.954335 24.238863

-0.305246 +O. 242810 -0.210673 +O. 189472 -0.173929 +O. 161826

+O. 289946 -0.232895 +O. 203344 -0.183714 +O. 169229

-0.277420 +O. 224513 -0.197009 +O. 178651 -0.165043

Y;+ (-l)n+lYy(Y:,s) 11.535731 +O. 266883 15.376058 -0.217283 18.885886 +O. 191447 22.266861 -0.174147

15/2

1 2 3

29/2

1 2

31/2

1 2

-0.217118 +O. 187870 -0.169950

-0.257833

+O. 210950 -0.186505 +O. 170098

-0.205332

+O. 182067 -0.166427

-0.231081

15.452196 20.450018 24.411571

+o.

17.542024 22.703832

+O. 263833 -0.195187

20.413362 24.790191

+O. 216981

19.624460 24.939214

+o.

22.586374

+O. 209303

275596 -0.202830

+O. 176690

-0.181040

33/2

1 2

35/2

1 2

37/2

1

23.668335

-0.205855

39/2

1

24.747606

+O. 202629

253871 -0.188612

Values to greater accuracy and over a wider range are given in [10.31]. From National Bureau of Standards, Tables of spherical Bessel functions,.vols. I, 11. Columbia Univ. Press, New York, N.Y., 1947 (with permission).

BESSEL FUNCTIONS O F FRACTIONAL ORDER

MODIFIED SPHERICAL BESSEL FUNCTIONS-ORDERS 2

i o (2)

il(2)

i 2 (2)

Table 10.8

0, 1 AND 2

ko (4

kl (2)

k2

00

(4

0.1 0.2 0.3 0.4

1.00000 1.00166 1.00668 1.01506 1.02688

000 750 001 764 081

0.00000 0.03336 0.06693 0.10090 0.13547

000 668 370 290 889

0.00000 0.00066 0.00267 0.00603 0.01078

0000 7143 4294 8668 9114

14.21315 6.43029 3.87891 2.63234

293 630 513 067

0.5 0.6 0.7 0.8 0.9

1.04219 1.06108 1.08369 1.11013 1.14057

061 930 100 248 414

0.17087 0.20729 0.24496 0.28412 0.32501

071 319 858 808 361

0.01696 0.02462 0.03382 0.04465 0.05719

6360 3348 5678 2156 5452

1.90547 1.43678 1.11433 0.88225 0.70959

226 550 482 536 792

5.71641 3.83142 2.70624 1.98507 1.49804

679 801 170 456 005

1.0 1.1 1.2 1.3 1.4

1.17520 1.21422 1.25788 1.30644 1.36021

119 497 446 803 536

0.36787 0.41299 0.46064 0.51112 0.56477

944 416 259 785 365

0.07156 0.08787 0.10627 0.12692 0.14998

2871 7251 7995 2227 6112

0.57786 0.47533 0.39426 0.32930 0.27668

367 880 230 149 115

1.15572 0.90746 0.72281 0.58261 0.47431

735 4974 4219 0332 0537

4.04504 2.95024 2.20129 1.67378 1.29306

57 33 78 69 09

1.5 1.6 1.7 1.8 1.9

1.41951 1.48472 1.55625 1.63454 1.72008

964 997 408 127 574

0.62192 0.68295 0.74827 0.81829 0.89349

665 906 140 550 778

0.17566 0.20418 0.23577 0.27071 0.30929

6332 1728 5138 5433 9770

0.23366 0.19821 0.16879 0.14425 0.12365

136 144 918 049 360

0.38943 0.32209 0.26809 0.22438 0.18873

5596 3595 2818 9655 4440

1.01253 0.80213 0.64190 0.51823 0.42165

25 693 415 325 535

2.0 2.1 2.2 2.3 2.4

1.81343 1.91516 2.02595 2.14650 2.27759

020 988 690 513 551

0.97438 1.06149 1.15543 1.25683 1.36639

274 681 247 283 653

0.35185 0.39874 0.45036 0.50715 0.56959

6089 5868 7165 7959 9849

0.10629 0.09159 0.07911 0.06847 0.05937

208 719 327 227 476

0.15943 0.13521 0.11507 0.09824 0.08411

8124 4906 3847 2824 4246

0.34544 0.28476 0.23603 0.19661 0.16451

927 135 215 508 757

2.5 2.6 2.7 2.8 2.9

2.42008 2.57489 2.74306 2.92568 3.12398

179 701 041 513 658

1.48488 1.61311 1.75200 1.90251 2.06572

308 877 304 546 335

0.63822 0.71360 0.79639 0.88727 0.98703

2102 6125 0365 5704 1387

0.05157 0.04487 0.03909 0.03411 0.02980

553 256 858 437 354

0.07220 0.06213 0.05357 0.04629 0.04008

5736 1241 9539 8067 0625

0.13822 0.11656 0.09863 0.08371 0.07126

241 246 140 944 626

3.0 3.1 3.2 3.3 3.4

3.33929 3.57304 3.82683 4.10238 4.40157

164 872 875 723 747

2.24279 2.43498 2.64368 2.87041 3.11681

012 437 983 631 153

1.09650 1.21661 1.34837 1.49291 1.65144

152 224 954 787 965

0.02606 0.02282 0.02000 0.01755 0.01541

845 681 910 635 841

0.03475 0.03019 0.02626 0.02287 0.01995

7931 0302 1944 6452 3243

0.06082 0.05204 0.04462 0.03835 0.03302

638 323 967 312 422

3.5 3.6 3.7 3.8 3.9

4.72646 494

3.38467 421

1.82531 562

0.01355 255

0.01742

0.02848 802

3.67596 83i

2.01598 623

0.0iiz 222

n

4712

5.07929 3i6

U.UIJL3

2731

V.UL-tO1

5.46251 092 5.87879 128 6.33105 220

3.99283 865 4.33762 799 4.71289 572

2.22507 418 2.45434 813 2.70574 780

0.01049 611 0.00924 735 0.00815 280

0.01333 2903 0.01168 0862 0701024 3262

0.02130 658 0.01846 908 0.01603 223

4.0 4.1 4.2 4.3 4.4

6.82247 7.35655 7.93706 8.56816 9.25438

930 060 374 571 538

5.12143 5.56631 6.05085 6.57872 7.15390

838 208 704 451 628

2.98140 3.28363 3.61502 3.97835 4.37672

051 932 300 791 200

0.00719 O.CO634 0.00560 0.00495 0.00438

253 934 833 661 300

0.00899 0.00789 0.00694 0.00610 0.00537

0668 7961 3650 9316 9136

0.01393 0.01212 0.01056 0.00921 0.00805

554 834 808 893 059

4.5 4.6 4.7 4.8 4.9

10.00066 10.81241 11.69554 12.65647 13.70227

914 998 012 789 889

7.78076 8.46407 9.20906 10.02142 10.90741

689 908 250 620 515

4.81349 5.29236 5.81741 6.39308 7.02426

122 840 513 652 961

0.00387 0.00343 0.00303 0.00269 0.00238

777 248 975 318 716

0.00473 0.00417 0.00368 0.00325 0.00287

9498 8666 6506 4257 4331

0.00703 0.00615 0.00539 0.00472 0.00414

744 769 284 709 695

5.0

14.84064 212

U. U

11.87386 128

7.71632 535

00

0.00211 679

00

156.344682 38.58177 78 16.80863 22 9.21319 233

A - - - -

4704.5536 585.15696 171.96524 71.731283 36.203973 20.593926 12.712514 8.32628 49 5.70306 48

c~nr-

A

=

4;

a/xK

n+3

1 1 0

0.00364 088

0.00254 0146

k,(x)

A C I I I I ~ - 3 r 1

(x)

470

BESSEL FUNCTIONS O F FRACTIONAL ORDER

Table 10.9

MODIk'IED SPHERICAL BESSEL FUNCTIONS-ORDERS

10gz-gig( 2 )

2

lol~X-'oilo(x) 0.72730 92 0.72746 73 0.72794 1 9 0.72873 35 0.72984 30

10-7.~10kg(x)

9 AND 10

10- r1 k, (X)

5.41287 5.41128 5.40650 5.39856 5.38746

38 21 99 70 92

1.02844 1.02817 1.02736 1.02601 1.02412

60 54 41 35 59

18 17 47 33 04

5.37323 5.35590 5.33549 5.31206 5.28564

85 33 79 23 31

1.02170 1.01875 1.01527 1.01128 1.00678

47 42 95 67 27

0.74327 0.74667 0.75040 0.75448 0.75891

93 38 79 62 37

5.25629 5.22406 5.18902 5.15123 5.11078

13 45 48 93 01

1.00177 0.99627 0.99028 0.98382 0.97689

53 31 56 30 61

30 02 13 38 60

0.76369 0.76883 0.77434 0.78023 0.78649

58 83 76 05 43

5.06772 5.02215 4.97414 4.92379 4.87119

38 07 57 68 57

0.96951 0.96169 0.95345 0.94478 0.93572

68 72 03 97 94

1.67930 1.69563 1.71292 1.73118 1.75044

73 90 33 39 59

0.79314 0.80019 0.80765 0.81552 0.82381

68 63 17 21 79

4.81643 4.75961 4.70083 4.64019 4.57780

66 72 65 67 09

0.92628 0.91646 0.90629 0.89579 0.88495

41 88 89 04 95

2. 5 2.6 2.7 2. 8 2.9

1.77073 1.79208 1.81451 1.83806 1.86277

63 32 64 76 03

0.83254 0.84172 0.85136 0.86147 0.87206

94 78 49 30 54

4.51375 4 1 4.44816 23 4.38113 22 4.31277 1 0 4.24318 63

0.87382 0.86239 0.85069 0.83874 0.82655

25 63 78 39 20

3. 0 3.1 3. 2 3.3 3. 4

1.88865 1.91577 1.94414 1.97382 2.00485

96 24 79 74 39

0.88315 0.89475 0.90688 0.91956 0.93279

57 86 95 42 97

4.17248 4.10077 4.02816 3.95475 3.88064

53 50 19 12 76

0.81413 0.80152 0.78872 0.77574 0.76262

92 28 01 83 45

3, 5

3. 6 3. 7 3.8 3. 9

2-03727 2.07113 2.10648 2.14337 2.18187

33 33 43 94 40

0.94661 0.96102 0.97605 0,99171 1.00804

40 55 38 97 44

3.80595 3.73076 3.65519 3.57933 3.50326

33 99 70 16 88

0.74936 0.73598 0.72250 0.70894 0.69531

56 84 95 53 19

4. 0 4.1 4.2 4. 3 4.4

2.22202 2.26389 2.30755 2.35306 2.40049

68 90 54 35 43

1,02505 1.04276 1.06120 1.08040 1.10038

08 26 45 28 47

3.42710 3.35091 3.27481 3.19885 3.12314

13 95 07 96 76

0.68162 0.66790 0.65415 0.64039 0.62664

50 02 25 66 70

4.5 4. 6 4. 7 4. 8 4.9

2.44992 2.50142 2.55508 2.61099 2.66924

27 71 99 74 03

1.12117 1.14281 1.16532 1.18874 1.21310

91 58 63 39 29

3.04775 2.97275 2.89821 2.82421 2.75081

39 34 88 90 98

0.61291 0.59922 0.58557 0.57198 0.55846

75 16 24 25 39

5. 0

2.72991 40

0. 0 0.1 0.2 0. 3 0. 4

1.52734 1.52771 1.52880 1.53062 1.53317

93 30 46 54 79

0. 0. 0. 0. 0.

5 6 7 8 9

1.53646 1.54049 1.54526 1.55078 1.55706

54 23 36 57 60

0.73127 0.73302 0.73509 0.73749 0.74022

1. 0 1.1 1.2 1.3 1.4

1.56411 1.57193 1.58054 1.58994 1.60016

27 49 32 87 42

1.5 1. 6 1.7 1. 8 1.9

1.61120 1.62308 1.63581 1.64941 1.66390

2. 0 2.1 2. 2 2. 3 2. 4

[(-;)33

1.23843 97

[(-:)I]

2.67808 38

0.54502 82

[(-;)41

Compiled from C. W. Jones, A short table for the Bessel functions Zn++(z),( Z / T ) K ~ + : ( ~ U ) . Cambridge Univ. Press, Cambridge, England, 1952 (with permission).

BESSEL FUNCTIONS O F FRACTIONAL ORDER

MODIFIED SPHERICAL BESSEL FUNCTIONS-ORDERS X

5. 0 5.1 5.2 5.3 5. 4

5.5 5. 6 5. 7 5.8 5. 9 6. 0 6.1 6.2 6. 3 6.4 6. 5 6. 6 6. 7 6.8 6. 9

2

7

( 2) 4.62276 ( 2) 4.11899

-5) 2.38413

2) 2.96863

-5) 2.67535 -5 2.99285 -51 3.33809 -5) 3.71252 -5) 4.11760

2j2.19449

-5) 4.55480 -5 5.02559 -5 5.53143

I i

dKzl(x) r 2

( 3) 1.88159 3 1.64774 31 1.44818 3 1.27719 31 1.13013

2) 1.39236

-4) 2.35684 ( 2 ) 1.08697

(-4) 2.54221

2) 2.71019 2) 2.48857

7. 0 7.1 7.2 7. 3 7.4 7.5 7. 6 7.7 7. 8 7.9

2

-5) 6.40961 -5) 7.16216

-4 1.43285 -41 1.56545

9 AND 10 Table 10.9

e-vz (x)

e-=IE(X)

471

-4) 1.48492 -4) 4.93867

ii

-4 -4 -4 -4 ( -4)

1.59365 1.70773 1.82727 1.95236 2.08311

8. 0 8- 1 8. 2 8. 3 8.4 8.5 8. 6 8. 7 8. 8 8. 9

-4) 8.38228

1 2.55287 ( 1 1 2.42979

9. 0 9.1 9.2 9. 3 9.4 9.5 9. 6 9. 7 9. 8 9.9 10. 0

(-3) 1.55045

-4) 4.77560

1 2. 01109 111.92209

-4 5.73844 ( -41 5.99571

( 1 ) 1.83843 1 1.75973 111.68563 1 1.61578 111.54991

( -4) 6.25963

1 4.98356 ( 1 1 4.72722

I

( 1 ) 1.48772

( 1 ) 3.50537

472

BESSEL FUNCTIONS OF FRACTIONAL ORDER

Table 10.9 MODIFIED SPHERICAL BESSEL FUNCTIONS-ORDERS

f9 (4

(4

9 AND 10

<x>

0.100 0.095 0.090 0.085 0.080

1.10630 1.08238 1.06167 1.04394 1.02899

573 951 683 741 406

1.21411 1.17260 1.13650 1.10534 1. 07872

149 877 462 464 041

0,65502 0.68557 0.71563 0.74502 0.77352

364 030 676 124 114

g10(x) 0.56777 303 0.60351 931 0.63926 956 0.67473 612 0.70961 813

0.075 0.070 0.065 0.060 0.055

1.01661 1.00662 0.99883 0.99304 0.98907

895 998 728 985 251

1.05626 1.03762 1.02248 1.01055 1.00151

085 412 982 159 009

0,80093 0.82707 0.85175 0.87480 0.89608

667 483 354 587 425

0.74360 0.77639 0.80768 0.83717 0.86461

745 538 018 510 675

13 14 15 17 18

0.050 0.045 0.040 0.035 0.030

0.98670 0.98573 0.98593 0.98707 0,98892

320 080 357 842 100

0.99506 0,99091 0.98874 0.98822 0.98900

643 634 519 421 824

0.91546 0.93284 0.94817 0.96140 0.97253

455 978 344 216 769

0.88977 0.91245 0.93251 0.94985 0.96444

340 301 041 358 830

20 22 25 29 33

0.025 0.020 0.015 0.010 0.005

0.99120 0.99367 0.99605 0.99807 0.99947

680 323 259 595 760

0.99073 0.99302 0.99549 0.99774 0.99937

519 746 538 259 316

0.98161 0.98871 0.99394 0.99744 0.99939

804 764 654 863 894

0.97632 0.98556 0.99231 0.99679 0.99925

121 077 623 434 415

40 50 67 100 200

0.000

1 ~ 0 0 0 0 0000

X-1

f10

99 ( X I

1.00000 000

1.00000 000

4-K

1.00000 000

19(x) = g9 (x)e-x+45x2

-.

4 2 2 / a K2 1 f ~= )g 2

<x>

= nearest

10(~)e-~+~~~-~

integer to x.

10 11 11 12 13

ab

473

BESSEL FUNCTIONS O F FRACTIONAL ORDER

MODIFIED SPHERICAL BESSEL FUNCTIONS-VARIOUS .-

x=2

x=l

n

Table 10.10 ORDERS

x=5 (

0 1 2 3 4

(

1)1.48406 4212 1)1.18738 6128

0) 1.89577 5037

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

( - 11)8.07224 1852

20 30 40 50

100

ri 0 1 2 3 4

( -190) 7.48149 1755

(-160) 9.55425

1030

~=50

x= 10

(-120) 6.26113

2=

6933

100

3 1.10132 3287 21 9.91190 9633

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 30 40 50 100

5) 2.37154 -12 1.22928 -21 2.81471 -31 5.88991

I- I

3577 4325 5830 6154

(-90) 9.54463

8661

1

1717.90430 1 5 5.67659 1 2 7.34905 + 9 2.00489

4104 3929 8082 8633

401 1.64074 ( 3 9 1.30147 37 3.95371 35 4.74095

7551 2327 9716 0959

(-17) 2.34189

3740

( 2 0 ) 3.73598 8741

474

BESSEL FUNCTIONS OF FRACTIONAL ORDER

Table 10.10 MODIFIED SPHERICAL BESSEL FUNCTIONS-VARIOUS

n

x=2

x=l

ORDERS

x=5

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14

11 1.38508 0704 121 2.15637 9105

15 16 17 18 19 20 30 40 50

( 23)4.95991 7633

100

( 87)1.04451 3645

n

40)4.55045 5450

.%'=

10

( ( ( (

17)2.27598 31)2.06581 46 5.55624 631 1.86314

6819 6824 8963 7755

156)4.08894 4237

X=

50

0 1 2 3 4

1

( 1817.97979 8)7.66744 6235 3303

30 2.35318 1718 42 8.49795 8757 (116)2.49323 8041

x=

100

(-46)6.02053 9173 -46 6.20234 3454 -461 6.45474 5215

5 6 7 8 9 1Q 11 12 13 14

-23 3.63628 4300 -231 4.78207 7170

15 16 17 18 19

-45 1.44325 8856 -451 1.65826 2396

-23)6.40988 9058

20 30 40 50

( 2)1.45175 0001

(-22)3.67748 3017

(-45)4.68935 4218 -44)5.77221 5084

100

( 85)8.14750 7624

(+12)5.97531 1344

(-25)1.48279 6529

475

BESSEL FUNCTIONS O F FRACTIONAL ORDER

AIRY FUNCTIONS Ai(x)

x

0.35502 0.01 0.35243 0.02 0.34985 0.03 0.34726 0.04 0.34467

Ai'(x)

Bi(x)

Bi'(x)

x

805 992 214 505 901

-0.25881 -0.25880 -0.25874 -0.25866 -0.25854

940 174 909 197 090

0.61492 0.61940 0.62389 0.62837 0.63286

663 962 322 808 482

0.44828 0.44831 0.44841 0.44856 0.44878

836 926 254 911 987

0.50 0.51 0.52 0.53 0.54

0.23169 0.22945 0.22721 0.22499 0.22279

0.05 0.06 0.07 0108 0.09

0.34209 435 0.33951 139 0.33693 047 0133435 191 0.33177 603

-0.25838 -0.25819 -0.25797 -0.25772 -0.25744

640 898 916 745 437

0.63735 409 0.64184 655 0.64634 286 OI65084 370 0.65534 975

0.44907 0.44942 0.44984 0145033 0.45088

570 752 622 270 787

0.55 0.56 0.57 0158 0.59

0.10 0.11 0.12 0.13 0.14

0.32920 0.32663 0.32406 0.32150 0.31894

313 352 751 538 743

-0.25713 042 -0.25678 613 -0.25641 200 -0.25600 854 -0.25557 625

0.65986 0.66438 0.66890 0.67343 0.67798

169 023 609 997 260

0.45151 0.45220 0.45297 0.45381 0.45472

263 789 457 357 582

0.15 0.16 0.17 0.18 0.19

0.31639 0.31384 0.31130 0.30876 0.30623

395 521 150 307 020

-0.25511 -0.25462 -0.25411 -0.25356 -0.25300

565 724 151 898 013

0.68253 0.68709 0.69167 0.69625 0.70085

473 709 046 558 323

0.45571 0.45677 0.45791 0.45912 0.46041

0.20 0.21 0.22 0.23 0.24

0.30370 0.30118 0.29866 0.29615 0.29365

315 218 753 945 818

-0.25240 -0.25178 -0.25114 -0.25047 -0.24977

547 548 067 151 850

0.70546 0.71008 0.71472 0.71938 0.72405

420 928 927 499 726

0.25 0.26 0727 3.28 0.29

0.29116 0.28867 0.28619 0.28372 0.28126

395 701 757 586 209

-0.24906 -0.24832 -0.24756 -0.24677 -0.24597

211 284 115 753 244

0.72874 0.73345 0.73818 0.74292 0.74769

0.30 0.31 0.32 0.33 0.34

0.27880 0.27635 0.27392 0.27149 0.26906

648 923 055 064 968

-0.24514 -0.24429 -0.24343 -0.24254 -0.24164

636 976 309 682 140

0.35 0.36 0.37 OI38 0.39

0.26665 787 0.26425 540 0.26186 243 0125947 916 0.25710 574

0.40 0.41 0.42 0.43 0.44

0.25474 0.25238 0.25004 0.24771 0.24539

0.45 0.46 0.47 0.48 0.49

0.24308 0.24078 0.23849 0.23621 0.23394

0.00

Table 10.11

Ai(x) 361 031 872 894 109

Bi(x)

Ai'(x) -0.22491 -0.22374 -0.22257 -0.22138 -0.22018

Bi'(x)

053 617 027 322 541

0.85427 0.85974 0.86525 0.87081 0.87641

704 431 543 154 381

0.54457 0.54890 0.55334 0.55789 0.56257

256 049 239 959 345

0.22059 527 0.21841 158 0.21624 012 0721408 099 0.21193 427

-0.21897 720 -0.21775 898 -0.21653 112 -0.21529 397 -0,21404 790

0.88206 0.88776 0.89350 0.89930 0.90515

341 152 934 810 902

0.56736 0.57227 0.57730 0.58246 0.58774

532 662 873 311 120

0.60 0.61 0.62 0.63 0.64

0.20980 0.20767 0.20556 0.20347 0.20138

006 844 948 327 987

-0.21279 -0.21153 -0.21025 -0.20898 -0.20769

326 041 970 146 605

0.91106 0.91702 0.92303 0.92910 0.93524

334 233 726 941 011

0.59314 0.59867 0.60433 0.61012 0.61603

448 447 267 064 997

223 373 125 572 808

0.65 0.65 0.67 0.68 0.69

0.19931 0.19726 0.19521 0.19318 0.19116

937 182 729 584 752

-0.20640 -0.20510 -0.20380 -0.20248 -0.20117

378 500 004 920 281

0.94143 0.94768 0.95399 0.96037 0.96681

066 241 670 491 843

0.62209 0.62827 0.63460 0.64106 0.64766

226 912 222 324 389

0.46178 0.46324 0.46477 0.46638 0.46808

928 026 197 539 147

0.70 0.71 0.72 0.73 0.74

0.18916 0.18717 0.18519 0.18322 0.18127

240 052 192 666 478

-0.19985 -0.19852 -0.19719 -0.19585 -0.19451

119 464 347 798 846

0.97332 0.97990 0.98655 0.99327 1.00006

866 703 496 394 542

0.65440 0.66129 0.66832 0.67549 0.68282

592 109 121 810 363

690 477 170 857 624

0.46986 0.47172 0.47367 0.47571 0.47783

119 554 549 205 623

0.75 0.76 0.77 0.78 0.79

0.17933 0.17741 0.17549 0.17360 0.17171

631 128 975 172 724

-0.19317 -0.19182 -0.19047 -0.18912 -0.18777

521 851 865 591 055

1.00693 1.01387 1.02088 1.02798 1.03516

091 192 999 667 353

0.69029 0.69792 0.70571 0.71365 0.72174

970 824 121 062 849

0.75248 0.75729 0.76213 0.76699 0.77187

559 752 292 272 782

0.48004 0.48235 0.48474 0.48722 0.48980

903 148 462 948 713

0.80 0.81 0.82 0.83 0.84

0.16984 0.16798 0.16614 0.16431 0.16249

632 899 526 516 870

-0.18641 -0.18505 -0.18369 -0.18232 -0.18096

286 310 153 840 398

1.04242 1.04976 1.05719 1.06470 1.07230

217 421 128 504 717

0.73000 0.73842 0.74701 0.75576 0.76468

690 795 380 663 865

-0.24071 730 -0.23977 495 -0.23881 481 -0123783 731 -0.23684 291

0.77678 0.78172 0.78669 0.79169 0.79671

917 770 439 018 605

0.49247 0.49524 0.49810 0.50106 0.50412

861 501 741 692 463

0.85 0.86 0.87 0.88 0.89

0.16069 0.15890 0.15713 0.15536 0.15362

588 673 124 942 128

-0.17959 -0.17823 -0.17686 -0.17549 -0.17413

851 223 539 823 097

1.07999 1.08778 1.09566 1.10363 1.11170

939 340 096 385 386

0.77378 0.78304 0.79249 0.80211 0.81191

215 942 282 473 759

235 916 630 395 226

-0.23583 -0.23480 -0.23376 -0.23270 -0.23163

203 512 259 487 239

0.80177 0.80686 0.81198 0.81714 0.82233

300 202 412 033 167

0.50728 0.51053 0.51389 0.51736 0.52092

168 920 833 025 614

0.90 0.91 0.92 0.93 0.94

0.15188 0.15016 0.14845 0.14676 0.14508

680 600 886 538 555

-0.17276 -0.17139 -0.17003 -0.16866 -0.16730

384 708 090 551 113

1.11987 1.12814 1.13651 1.14499 1.15357

281 255 496 193 539

0.82190 0.83207 0.84243 0.85298 0.86373

389 615 695 891 470

135 139 250 482 848

-0.23054 -0.22944 -0.22833 -0.22720 -0.22606

556 479 050 310 297

0.82755 0.83282 0.83812 0.84346 0.84885

920 397 705 952 248

0.52459 0.52837 0.53225 0.53625 0.54035

717 457 956 338 729

0.95 0.96 0.97 0.98 0.99

0.14341 0.14176 0.14012 0.13850 0.13689

935 678 782 245 066

-0.16593 -0.16457 -0.16321 -0.16185 -0.16050

797 623 611 781 153

1.16226 1.17106 1.17998 1.18901 1.19815

728 959 433 352 925

0.87467 0.88581 0.89716 0.90871 0.92046

704 871 253 137 818

0.50 0.23169 361 -0.22491 053 0.85427 704 0.54457 256 1.00 0.13529 242 -0.15914 744 1.20742 359 0.93243 593

[(-46''1

AIRY FUNCTIONS-AUXILIARY FUNCTIONS FOR LARGE POSITIVE ARGUMENTS 2-

'

d-4

0.619954 0.617156 0.614275 0.611305 9.608239

0.478728 0.479925 0.481658 0.484018 0.487107

0.50 0.45 0.40 0.35 0.30

2.080084 2.231443 2.413723 2.638450 2.924018

0.548230 0.549584 0.550980 0.552421 0.553912

f(r)

0.619912 0.620335 0.620327 0.619799 0.618649

0.593015 0.589451 0.585855 0.582330 0.578985

0.587245 0.585235 0.583174 0.581056 0.578878

0.526011 0.530678 0.535345 0.539902 0.544235

1.310371 1.405721 1.520550 1.662119 1.842016

0.536489 0.616764 0.538618 0.614022 0.540844 0.610309 0.543180 0.605543 0.545636 0.599723

0.605068 0.601782 0.598372 0.594823 0.591120

0.491037 0.495921 0.501859 0.508909 0.517032

0.25 0.20 0.15 0.10 0.05

3.301927 3.831547 4.641589 6.082202 9.654894

0.555456 0.557058 0.558724 0.560462 0.562280

0.575908 0.573135 0.570636 0.568343 0.566204

0.576635 0.574320 0.571927 0.569448 0.566873

0.548255 0.551930 0.555296 0.558428 0.561382

2.080084

0.548230

0.587245

0.526011

0.00

0.564190 0.564190

0.564190

0.564190

f(-4

X

1.5 1.4 1.3 1.2 1.1

1.000000 0.527027 1.047069 0.528783 1.100099 0,530601 1.160397 0.532488 1.229700 0.534448

1.0 0.9 0.8 0.7 0.6 0.5

-_

'

f(r,

0.593015

A(-O

'

__

f(-r)

r-l

2 5 - r ) Bi(x)=x ecf(r) Ai'(n)=-ix'e-'g(-r> Bi'(x)=x'erg(r) r= , ! From J. C. P. Miller, The Airy integral, British Assoc. Adv. Sci. Mathematical Tables, Part-vol. B. Cambridge Univ. Press, Cambridge, England, 1946 (with permission).

Ai(x)=ix

e-rf(

476

BESSEL FUNCTIONS OF FRACTIONAL ORDER

Table 10.11 x

Ai(-x)

Ai’(-x)

Bi(-x)

AIRY FUNCTIONS Bi’(-x) x Ai(-x)

Ai’(-x)

Bi( -x)

Bi’(-x)

0.00 0.01 0.02 0.03 0.04

0.35502 0.35761 0.36020 0.36279 0.36537

805 619 397 102 699

-0.25881 -0.25880 -0.25874 -0.25865 -0.25852

940 157 771 731 986

0.61492 0.61044 0.60596 0.60147 0.59698

663 364 005 524 863

0.44828 0.44831 0.44841 0.44856 0.44877

836 896 015 104 074

0.50 0.51 0.52 0.53 0.54

0.47572 0.47775 0.47976 0.48174 0.48369

809 692 138 089 487

-0.20408 -0.20167 -0.19920 -0.19668 -0.19410

167 409 846 449 192

0.38035 0.37528 0.37019 0.36508 0.35996

266 379 579 853 193

0.50593 0.50784 0.50976 0.51169 0.51363

371 166 123 132 080

0.05 0.06 0.07 0.08 0.09

0.36796 0.37054 0.37312 0.37570 0.37827

149 416 460 243 725

-0.25836 -0.25816 -0.25792 -0.25763 -0.25731

484 173 001 918 872

0.59249 0.58800 0.58351 0.57901 0.57450

963 767 218 261 841

0.44903 0.44936 0.44974 0.45017 0.45066

833 293 364 955 976

0.55 0.56 0.57 0.58 0.59

0.48562 0.48752 0.48939 0.49124 0.49306

274 389 774 369 115

-0.19146 -0.18875 -0.18600 -0.18318 -0.18030

050 999 016 078 166

0.35481 0.34965 0.34446 0.33926 0.33403

589 033 520 043 599

0.51557 0.51753 0.51949 0.52145 0.52342

853 339 424 991 927

0.10 0.11 0.12 0.13 0.14

0.38084 0.38341 0.38597 0.38853 0.39109

867 628 967 843 213

-0.25695 -0.25655 -0.25611 -0.25563 -0.25510

811 685 443 033 406

0.56999 0.56548 0.56096 0.55643 0.55189

904 397 268 466 940

0.45121 0.45180 0.45245 0.45315 0.45390

336 945 712 546 355

0.60 0.61 0.62 0.63 0.64

0.49484 0.49660 0.49833 0.50003 0.50170

953 821 659 408 007

-0.17736 -0.17436 -0.17130 -0.16818 -0.16500

260 341 392 399 345

0.32879 0.32352 0.31824 0.31294 0.30761

184 796 435 101 795

0.52540 0.52737 0.52934 0.53132 0.53329

115 438 780 022 046

0.15 0.16 0.17 0.18 0.19

0.39364 0.39618 0.39871 0.40124 0.40376

037 269 868 789 987

-0.25453 -0.25392 -0.25326 -0.25256 -0.25182

511 297 716 716 250

0.54735 0.54280 0.53824 0.53367 0.52909

642 523 536 634 771

0.45470 0.45554 0.45643 0.45737 0.45835

047 530 713 503 806

0.65 0.66 0.67 0.68 0.69

0.50333 395 0.50493 511 0.50650 295 0.50803 685 0.50953 620

-0.16176 -0.15846 -0.15509 -0.15167 -0.14818

218 007 701 290 768

0.30227 0.29691 0.29153 0.28612 0.28070

521 282 084 932 835

0.53525 0.53721 0.53917 0.54112 0.54306

733 964 618 575 714

0.20 0.21 0.22 0.23 0.24

0.40628 0.40879 0.41128 0.41377 0.41625

419 038 798 653 557

-0.25103 -0.25019 -0.24931 -0.24838 -0.24741

267 720 559 737 206

0.52450 0.51990 0.51529 0.51067 0.50604

903 986 977 835 518

0.45938 0.46045 0.46156 0.46272 0.46391

529 578 860 279 740

0.70 0.71 0.72 0.73 0.74

0.51100 040 0.51242 882 0.51382 087 0.51517 591 0.51649 336

-0.14464 -0.14103 -0.13736 -0.13363 -0.12984

129 366 479 464 322

0.27526 0.26980 0.26432 0.25883 0.25331

801 840 964 185 516

0.54499 0.54692 0.54883 0.55072 0.55260

912 048 000 642 852

0.25 0.26 0.27 0.28 0.29

0.41872 0.42118 0.42363 0.42606 0.42849

461 319 082 701 126

-0.24638 -0.24531 -0.24419 -0.24303 -0.24181

919 828 888 053 276

0.50139 0.49674 0.49207 0.48738 0.48268

987 203 127 722 953

0.46515 0.46642 0.46773 0.46908 0.47046

148 408 423 095 327

0.75 0.76 0.77 0.78 0.79

0.51777 0.51901 0.52021 0.52137 0.52249

258 296 390 479 501

-0.12599 -0.12207 -0.11810 -0.11406 -0.10996

055 665 157 538 815

0.24777 973 0.24222 571 0.23665 329 0.23106 265 0.22545 398

0.55447 0.55632 0.55815 0.55996 0.56176

506 480 647 884 063

0.30 0.31 0.32 0.33 0.34

0.43090 0.43330 0.43568 0.43805 0.44041

310 200 747 900 607

-0.24054 -0.23922 -0.23785 -0.23643 -0.23496

513 719 851 865 718

0.47797 0.47325 0.46851 0.46375 0.45898

784 181 112 543 443

0.47188 022 0.47333 081 0.47481 405 0.47632 895 0.47787 450

0.80 0.81 0.82 0.83 0.84

0.52357 0.52461 0.52560 0.52655 0.52746

395 101 557 703 479

-0.10580 -0.10159 -0.09731 -0.09297 -0.08857

999 101 134 113 055

0.21982 751 0.21418 345 0.20852 204 0.20284 354 0.19714 820

0.56353 0.56527 0.56699 0.56869 0.57036

059 745 994 679 671

0.35 0.36 0.37 0.38 0.39

0.44275 0.44508 0.44739 0.44968 0.45196

817 477 535 937 631

-0.23344 -0.23186 -0.23023 -0.22855 -0.22682

368 773 893 687 116

0.45419 0.44939 0.44457 0.43974 0.43488

784 534 667 156 973

0.47944 970 0.48105 354 0.48268 500 0.48434 307 0.48602 670

0.85 0.86 0.87 0.88 0.89

0.52832 0.52914 0.52991 0.53064 0.53132

824 678 982 676 700

-0.08410 -0.07958 -0.07500 -0.07036 -0.06566

979 904 854 852 925

0.19143 0.18570 0.17996 0.17420 0.16842

630 813 399 419 906

0.57200 0.57362 0.57520 0.57675 0.57826

845 071 220 165 777

0.40 0.41 0.42 0.43 0.44

0.45422 0.45646 0.45868 0.46089 0.46307

561 675 918 233 567

-0.22503 -0.22318 -0.22128 -0.21Y33 -0.21732

141 723 826

0.48773 0.48946

486 652 062

0.90 0.91

0.53195 0.53254

995 502

-0.06091 -0.05609

100 407

0.16263 0.15683

895 420

0.57974 926 0.58119 484

U . ~ Y L Y Y 611

U.Y>

u . 3 ~ ~ 3Y0L U

-U.U+OLO

3.t7

U.LWLO

LLO

u . ~ o JJ u ~7 ~

447

0.43002 094 0.42513 495 0.42023 153 u.4i5ji u 4 i 0.41037 154

0.49479

193

0.94

0.53400

715

-0.04129

452

0.13933 585

0.58530 317

0.45 0.46 0.47 0.48 0.49

0.46523 0.46738 0.46950 0.47159 0.47367

864 066 119 965 548

-0.21525 -0.21313 -0.21095 -0.20872 -0.20643

894 721 893 379 147

0.40541 0.40043 0.39544 0.39043 0.38540

457 934 570 348 251

0.49660 0.49844 0.50029 0.50215 0.50403

702 031 070 713 850

0.95 0.96 0.97 0.98 0.99

0.53439 0.53473 0.53501 0.53525 0.53543

490 189 754 129 259

-0.03624 -0.03114 -0.02597 -0.02076 -0.01548

628 116 957 197 880

0.13347 0.12760 0.12171 0.11582 0.10991

634 415 971 346 587

0.58659 0.58783 0.58904 0.59019 0.59131

217 879 174 973 145

0.50

0.47572

803

-0.20408

167

0.38035 266

0.50593

371

1.00

0.53556

088

-0.01016

057

0.10399

739

0.59237

563

(-613

412

(-617

(-612

0.49:2?

(-618

1).!1 0.53308 i.3 -!.O,Tf$l,

(-617

(-618

,O.,l::li

23:

2i8 O, ?!:.b,!

(-612

(-616

477

BESSEL FUNCTIONS OF FRACTIONAL ORDER

Ai(-x)

AIRY FUNCTIONS Bi’(-x) x Ai(-x)

Bi(-x)

Ai’(-x)

Table 10.11 Bi(-x) Bi’(-x)

Ai’(-x)

X 1.0 1.1 1.2 1.3 1.4

0.53556 0.53381 0.52619 0.51227 0.49170

088 -0.01016 057 +0.10399 739 051 +0.04602 915 +0.04432 659 437 0.10703 157 -0.01582 137 201 0.17199 181 -0.07576 964 018 0.23981 912 -0.13472 406

0.59237 0.60011 0.60171 0.59592 0.58165

563 970 016 975 624

5.5 5.6 5.7 5.8 5.9

+0.01778 -0.06833 -0.15062 -0.22435 -0.28512

154 070 016 192 278

0.86419 0.85003 0.78781 0.67943 0.52962

1.5 1I6 1.7 1.8 1.9

0.46425 0.42986 0.38860 0.34076 0.28680

658 298 704 156 006

0.30918 0.37854 0.44612 0.50999 0.56809

697 219 455 763 172

-0.19178 -0.24596 -0.29620 -0.34140 -0.38046

486 320 266 583 588

0.55790 0.52389 0.47906 0.42315 0.35624

810 354 134 137 251

6.0 6.1 6.2 6.3 6.4

-0.32914 -0.35351 -0.35642 -0.33734 -0.29713

517 168 107 765 762

0.34593 +0.13836 -0.08106 -0.29899 -0.50147

549 -0.14669 838 -0.81289 879 394 -0.06182 255 -0.87622 530 856 +0.02679 081 -0.88697 896 161 0.11373 701 -0.84276 110 985 0.19354 136 -0.74461 387

2.0 0.22740 743 2.1 0.16348 451 2.2 0.09614 538 2.3 +0.02670 633 2.4 -0.04333 414

0.61825 0.65834 0.68624 0.70003 0.69801

902 069 482 366 760

-0.41230 -0.43590 -0.45036 -0.45492 -0.44905

259 0.27879 517 235 0.19168 563 098 +0.09622 919 823 -0.00581 106 228 -0.11223 237

6.5 6.6 6.7 6.8 6.9

-0.23802 -0.16352 -0.07831 +0.01210 0.10168

030 646 247 452 800

-0.67495 -0.80711 -0.88790 -0.91030 -0.87103

249 925 797 401 106

266 995 774 418 784

-0.59717 -0.40856 -0.19009 +0.04437 0.27926

067 734 878 678 391

2.5 2.6 2.7 2.8 2.9

-0.11232 -0.17850 -0.24003 -0.29509 -0.34190

507 243 811 759 510

0.67885 0.64163 0.58600 0.51221 0.42118

273 799 720 098 281

-0.43242 -0.40500 -0.36709 -0.31929 -0.26258

247 828 211 389 500

-0.22042 -0.32739 -0.42989 -0.52445 -0.60751

015 717 534 040 829

7.0 7.1 7.2 7.3 7.4

0.18428 0.25403 0.30585 0.33577 0.34132

084 633 152 037 375

-0.77100 -0.61552 -0.41412 -0.18009 +0.07027

817 0.29376 207 879 0.23425 088 428 0.15821 739 580 +0.07087 411 632 -0.02159 652

0.49824 0.68542 0.82650 0.90998 0.92812

459 058 634 427 809

3.0 3.1 3.2 3.3 3.4

-0.37881 -0.40438 -0.41744 -0.41718 -0.40319

429 0.31458 377 222 0.19482 045 342 +0.06503 115 094 -0.07096 362 048 -0.20874 905

-0.19828 -0.12807 -0.05390 +0.02196 0.09710

963 165 576 800 619

-0.67561 -0.72544 -0.75412 -0.75926 -0.73920

122 957 455 518 163

0.32177 7.5 7.6 0.27825 7.7 0.21372 7.8 0.13285 7.9 +0.04170

572 023 037 154 188

0.31880 0.54671 0.73605 0.87115 0.94004

951 882 242 540 300

-0.11246 -0.19493 -0.26267 -0.31030 -0.33387

349 0.87780 376 0.76095 007 0.58474 057 0.36122 856 +0.10670

228 509 045 930 215

3.5 3.6 3.7 3.8 3.9

-0.37553 -0.33477 -0.28201 -0.21885 -0.14741

382 748 306 598 991

343 397 780 257 809

0.16893 0.23486 0.29235 0.33904 0.37289

984 631 261 647 058

-0.69311 -0.62117 -0.52461 -0.40581 -0.26829

628 283 361 592 836

8.0 8.1 8.2 8.3 8.4

-0.05270 -0.14290 -0.22159 -0.28223 -0.31959

505 0.93556 815 0.85621 945 0.70659 176 0.49727 219 +0.24422

094 859 870 679 089

-0.33125 -0.30230 -0.24904 -0.17550 -0.08751

158 331 019 556 798

050 664 293 044 958

4.0 -0.07026 553 -0.79062 858 4.1 +0.00967 698 -0.80287 254 4.2 0.08921 076 -0.78221 561 4.3 0.16499 781 -0.72794 081 4.4 0.23370 326 -0.64085 018

0.39223 0.39593 0.38346 0.35494 0.31122

471 -0.11667 057 974 +0.04347 872 736 0.20575 691 906 0.36320 468 860 0.50858 932

8.5 8.6 8.7 8.8 8.9

-0.33029 -0.31311 -0.26920 -0.20205 -0.11726

024 245 454 445 631

335 +0.00775 444 -0.96296 917 027 0.10235 647 -0.91547 918 685 0.18820 363 -0.78882 623 301 0.25778 240 -0.59221 371 276 0.30483 241 -0.34136 475

-0.34344 -0.46986 -0.58272 -0.67688 -0.74755

4.5 4.6 4.7 4.8 4.9

0.29215 0.33749 0.36736 0.38003 0.37453

278 598 748 668 635

-0.52336 -0.37953 -0.21499 -0.03676 +0.14695

5.0 5.1 5.2 5.3 5.4

0.35076 0.30952 0.25258 0.18256 0.10293

101 600 034 793 460

0.32719 0.49458 0.63990 0.75457 0.83122

5.5

0.01778 154

253 0.25387 266 391 0.18514 576 018 0.10794 695 510 +0.02570 779 743 -0.05774 655

0.63474 0.73494 0.80328 0.83508 0.82721

282 600 517 542 307

0.77841 177 0.68948 513 (3.56345 8S8

-0.13836 913 -0.21208 913 -0.27502 704 -0.32311

608

U.4U333

-0.35531 708

477 444 926 976 903

bY4

0.22307 496

0.86419 722 -0.36781 345

-0.03231 -0.30933 -0.56297 -0.77061 -0.91289

722 256 722 152 857

-0.36781 -0.36017 -0.33245 -0.28589 -0.22282

0.26101 0.31159 0.34172 0.34908 0.33283

9.0 -0.02213 372 -0.97566 398 9.1 +0.07495 989 -0.95149 682 9.2 0.16526 800 -0.84067 107 9.3 0.24047 380 -0.65149 241 9.4 0.29347 756 -0.39986 237 9.5 9.6 9.7 Y.8 9.9

0.02511 158 10.0

0.32494 0.31603 0.27858 0.21570 0.13293

0.31910 325 -0.10809 532 0.31465 158 +0.19695 044 0.48628 629 0.28013 250 U . L l 6 8 b 143 6.73‘153 456 0.13623 503 0.90781 333 0.04024 124

345 223 825 021 969

fz:

JDL

9.654894 11.203512 13.572088 17.784467 28.231081

0.00

CO

0.39752 0.39781 0.39809 0.39838 0.39866

21 14 83 24 38

0.39894 23

0.40028 0.40002 0.39975 0.39949 0.39921

87 58 97 03 79

0.39894 23

0.40092 0.40052 0.40012 0.39972 0.39933

r= -3 x

11.t

0.99626 504 -0.31467 983

0.11941 411

31 06 11 48 19

0.39894 23

0.39704 0.39741 0.39779 0.39817 0.39855

87 99 49 37 62

0.39894 23

(-715

3

5

a!!?.!u . u i 7 ~ 0 3,8,!

0.42147 209

(-714

2

0.98471 407 0.97349 918

-0.28738 356

AIRY FUNCTIONS- AUXILIARY FUNCTIONS FOR LARGE NEGATIVE ARGUMENTS rX f&) f2N g#) g2(0 0.05 0.04 0.03 0.02 0.01

-0.15945 -0.41615 -0.64232 -0.81860 -0.92910

158 760 903 203 952

732 -0.05740 051 471 +0.23484 379 425 0.50894 402 835 0.73928 028 876 0.90348 537

+0.03778 543 -0.06091 293

-!.1_.?!9 -U.L>LOO

+0.02511 -0.17783 -0.37440 -0.55300 -0.70247

<{>=nearestinteger to r.

20 25 33 50 100 Q)

478

BESSEL FUNCTIONS OF FRACTIONAL ORDER INTEGRALS OF AIRY FUNCTIONS

Table 10.12 x

Ji Ai(t)dtJi

A i ( - t ) d t Y Bi(t)dt -

0

Bi(-t)dt

Oi03421 0.06585 0.09497 0.12164

30 01 15 09 06

0.00000 -0.03679 -0.07615 -0.11802 -0.16229

00 54 70 51 44

0.00000 C.06373 0.13199 0.20487 0.28256

00 " 0.00000 00 -0.05924 87 67 -OX398 i o 45 -0.16411 57 68 -0.20952 89 70

0.5 0.6 0.7 0.8 0.9

0.14595 0.16801 0.18795 0.20589 0.22196

33 79 52 45 97

-0.20880 -0.25736 -0.30768 -0.35944 -0.41225

95 07 05 15 56

0.36533 0.45356 0.54773 0.64845 0.75649

85 50 36 82 64

-0.25006 -0.28553 -0.31575 -0.34052 -0.35966

1.0 1.1 1.2 1.3 1.4

0.23631 0.24907 0.26037 0.27034 0.27910

73 33 12 09 66

-0.46567 -0.51918 -0.57224 -0.62421 -0.67447

40 94 05 79 31

0.87276 0.99838 1.13466 1.28318 1.44579

91 41 38 00 42

1.5 1.6 1.7 1.8 1.9

0.28678 Oi29349 0.29932 0.30438 0.30876

67 24 75 82 29

-0.72232 88 -0.76709 26 -0.80807 24 -0.84459 41 -0.87602 06

1.62470 1.82252 2.04231 2.28772 2.56304

2.0 2.1 2.2 2.3 2.4

0.31253 0.31577 0.31854 0.32091 0.32292

28 11 43 19 74

-0.90177 28 -0.92135 09 -0.93435 56 -0.94050 97 -0.93967 67

2.5 2.6 2.7 2.8 2.9

0.32463 0.32608 0.32730 0.32833 0.32919

80 57 74 55 83

-0.93187 -0.91730 -0.89633 -0.86951 -0.83758

3.0 3.1 3.2 3.3 3.4

0.32992 0.33052 0.33102 0.33144 0.33178

04 31 49 15 65

3.5 3.6 3.7 3.8 3.9

0.33207 0.33230 0.33249 0.33265 0.33278

4.0 4.1 4.2 4.3 4.4

Ji Ai

x

( t ) dt

E

.

6

5.5 5.6 5.7 5.8 5.9

0.33331 0.33332 0.33332 0.33332 0.33332

97 27 50 69 83

-0.82151 82 -0.ei897 90 -0.80797 96 -0.78914 06 -0.76354 19

-0.01617 +0.02038 0.05518 0.08625 0.11181

86 99 54 18 25

-0.37300 -0.38042 -0.38185 -0.37726 -0.36673

50 77 43 99 34

6.0 6.1 6.2 6.3 6.4

0,33332 0.33333 0.33333 0.33333 0.33333

95 03 10 16 20

-0.73267 53 -0.69836 93 -0.66268 96 -0.62781 93 -0.59592 62

0.13038 0.14086 0.14262 0.13555 0.12011

11 00 05 73 15

81 33 52 12 90

-0.35038 -0.32847 -0.30132 -0.26939 -0.23325

81 24 67 97 04

6.5 6.6 6.7 6.8 6.9

0.33333 0.33333 0.33333 0.33333 0.33333

23 25 27 29 30

-0.56902 -0.54883 -0.53667 -0.53334 -0.53906

35 59 65 74 98

0.09726 0.06847 0.03562 +0.00088 -0.03340

08

2.87340 83

-0.19354 -0.15106 -0.10667 -0.06132 -0.01603

74 46 18 23 45

7.0 7.1 7.2 7.3 7.4

0.33333 0.33333 0.33333 0.33333 0.33333

31 31 32 32 33

-0.55345 -0.57549 -0.60365 -0.63593 -0.66999

17 72 96 60 96

-0.06491 -0.09147 -0.11121 -0.12273 -0.12521

67 36 47 90 80

78 54 20 37 77

+0.02812 0.07009 0.10878 0.14317 0.17234

94 01 06 88 20

7.5 7.6 7.7 7.8 7.9

0.33333 33

-0.70336 -0.73355 -0.75830 -0.77575 -0.78453

19 34 99 13 65

-0.11847 -0.10300 -0.07997 -0.05114 -0.01872

31 57 85 35 22

-0.80146 -0.76220 -0.72100 -0.67915 -0.63602

29 32 37 91 56

0.19544 0.21180 0.22092 0.22252 0.21655

25 21 49 61 57

8.0 8.1 8.2 8.3 8.4

-0.78398 -0.77413 -0.75578 -0.73041 -0.70011

26 57 55 93 70

+0.01475 0.04664 0.07440 0.09577 0.10902

64 84 43 87 22

15 63 93 76 70

-0.59897 -0.56335 -0.53242 -0.50730 -0.48892

71 61 25 05 77

0.20321 0.18296 0.15652 0.12485 0.08914

50 47 33 43 28

8.5 8.6 8.7 8.8 8.9

-0.66739 21 -0.63499 08 -0.60566 32 -0.58192 70 -0.56584 22

0.11303 0.10749 0.09285 0.07039 0.04205

86 35 98 64 63

0.33289 0.33297 0.33304 0.33310 0.33315

27 86 84 50 07

-0.47800 -0.47496 -0.47992 -0.49268 -0.51269

75 79 95 51 28

+0.05076 +0.01121 -0.02786 -0.06494 -0.09837

01 78 79 00 02

9.0 9.1 9.2 9.3 9.4

-0.55881 -0.56148 -0.57358 -0.59403 -0.62093

97 12 51 00 76

+0.01033 -0.02196 -0.05192 -0.07682 -0.09439

04 26 24 93 87

4.5 4.6 4.7 4.8 4.9

0.33318 0.33321 0.33324 0.33326 0.33327

76 73 11 02 54

-0.53908 -0.57068 -0.60606 -0.64358 -0.68146

35 59 63 51 70

-0.12673 -0.14876 -0.16347 -0.17018 -0.16857

04 50 66 59 74

9.5 9.6 9.7 9.8 9.9

-0.65181 -0.68375 -0.71373 -0.73889 -0.75680

01 25 85 84 07

-0.10300 27 -0.10183 70 -0.09101 44 -0.07157 33 -0.04539 57

5.0

0.33328 76

-0.71788

22

-0.15873

09

10.0

-0.76569

84

-0.01504

~

6

L

J

S

~~

~~

+0.53565 -0.41901 +0.38040 -0.35790 +0.34230

281 012

- 8.48848 673 - 9.53544 905

065 386

-10.52766 040 -11.47505 663 -12.38478 837

-0.33047 +0.32102 - -- --0.31318 +0.30651 -0.30073

+0.70121 -0.80311 +0.86520 -0.91085 +0.94733

082 137 403 074 571

9.02265 085

-0.97792 +1.00437 -i;oi773 +1.04872 -1.06779

- i n n4ni7 434

430 -11.93601 556 -12.82877 675

~1.01879 3.24819 4.82 009 6.16330 7.37217

297 758 921 736 726

741 944 983 809 359

E69

~~

~

623

539 729 083

L 7 J

J

Bi(b',)

322 330 784 213 208

+0.60195 -0.76031 +0.83699 -0.88947 +0.92998

789 014 101 990 364

2.29443 4.07315 5.51239 6.781 29 7.94017

968 509 573 445 869

-0.45494 +0.39652 -0.36796 +0.34949 -0.33602

438 284 916 912 624

8.49194 - 9.53819 -10.52991 -11.47695 -12.38641

885 438 351 355 714

-0.96323 +0.99158 -1.01638 +Is03849 -1.05847

443 637 966 429 184

- 9.01958 -10.03769 -11.00646 -11.93426 -12.82725

336 633 267 165 831

+0.32550 -0.31693 +0.30972 -0.30352 +0.29810

974 465 594 766 491

-

AUXILIARY TABLE-COMPLEX ZEROS AND ASSOCIATED VALUES OF Bi(z) AND Bi'(z) e-*i/3ps

s 1 2 3 4 5

Modulus Phase 2.354 4.093 5.524 6.789 7.946

0.095 0.042 0.027 0.020 0.015

Bi' (pS) Modulus Phase 0.993 1.136 1.224 1.288 1.340

+2.641 -0.513 +2.625 -0.519 +2.622

04

1.17371 3.27109 4.83073 6.16985 7,37676

666 548 647 794 124

-779 -,

29 42 80 40

r ( - 3 ~ 1 rL(-3111 7 1

-ZEROS AND ASSOCIATED VALUES OF AIRY FUNCTIONS AND THEIR DERIVATIVES Bi' (b,) b'S Ai (a',) Ai' (a,) b, a's

as

2.33810 4.08794 5.52055 6.78670 7.94413

k -ii:io852

9 10

dt

09 -0.14113 39 -0.11667 30 -0.08660 41 -0.05250 03

28 62 56 58 27

L

7

(-t)

76 73 50 11 59

Table 10.13 -

fiBi

., -0.15873

0.33328 0.33329 0.33330 0.33331 0.33331

0.00000

((-7)31

S 1 2 3 4 5

Ai ( - t ) dt

-0.71788 22 -0.75103 62 -0.77926 27 -0.80111 58 -0.81545 49

5.05.1 5.2 5.3 5.4

0.0 Oil 0.2 0.3 0.4

e-

*i/3pts

Modulus Phase 1.121 3.257 4.824 6.166 7.374

0.331 0.059 0.033 0.023 0.017

Frorii J. C. P. Miller, The Airy integral, British Assoc. Adv. Sci. Mathematical Tables Part-vol. B. Cambridge Univ. Press, Cambridge, England, 1946 and F. W. J. Olver. The asymptotic expansion of Bessel functions of large order. Philos. Trans. R o j . Soc. London [A] 247, 328-368,1954(with permission).

Bi (dS) Modulus Phase 0.750 0.592 0.538 0.506 0.484

+0.466 -2.632 +0.515 -2.624 +0.519

11. Integrals of Bessel Functions YUDELLL. LUKE' Contents Mathematical Properties . . . . . . . . . . . . . 11.1. Simple Integrals of Bessel Functions . . . . 11.2. Repeated Integrals of Jn(z)and Ko(z) . . . 11.3. Reduction Formulas for Indefinite Integrals 11.4. Definite Integrals . . . . . . . . . . . . Numerical Methods . . . . . . . . . . . . . . . 11.5. Use and Extension of the Tables . . . . . References . . . . . . . . . . . . . . . . . . . Table 11.1 Integrals of Bessel Functions . , . . . .

Table 11.2 Integrals of Bessel Functions

Lrn

Yo(t)dt,

e - z s ' [lo(t)-lldt, 0

t

. . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

Page

480 480

482 483 485

488 488 490 492

494

8D

t

8D; 2ez

. . . . . . . . . . . . . . . . . . . . . . . . . . .

lm

x=O (.1)5

6D

The author acknowledges the assistance of Geraldine Coombs, Betty Kahn, Marilyn Kemp, Betty Ruhlman, and Anna Lee Samuels for checking iormuias and deveiopirig numerical examples, only a portion of which could be accommodated here.

Midwest Research Institute. of Standards.)

(Prepared under contract with the National Bureau

479

11. Integrals of Bessel Functions Mathematical Properties 11.1. Simple Integrals of Bessel Functions

It U,

( t )dt

11.1.1

A and B are constants.

HY(x) and L,(x) are Struve functions (see chapter 12). 11.1.9

(v+2k+l)r (++k) -P+l X k =5 O

r (++"+k)

11.1.2 y (Euler's constant)=.57721 56649

n-1

...

In this and all other integrals of 11.1, x is real and positive although all the results remain valid for extended portions of the complex plane unless stated to the contrary. 11.1.10

Recurrence Relations

I"

K,(t)dt=Z

11.1.5

l

Jo(t)dt+i;

Asymptotic Expansions

11.1.11

Jo-JI( t ) d t=1-Jo(z)

11.1.6

11.1.13

A and B are constants. 480

so"

Y,(t)dt

481

INTEGRALS O F BESSEL FUNCTIONS

11.1.18

75x5..

K 0 ( t ) d t = 2(-)kek(x/7)-k+€(x)

2tej

where the a k are defined as in 11.1.12.

k =O

2

lE(x)15 2 x 1 0 - 7 k

where the

ak

ek

0 1 2 3

are defined as in 11.1.12.

Polynomial Approximations 2

1.25331 0. 11190 .02576 .00933 .00417 .00163 . 00033

4

11.1.16

5 6

8525a3

11.1.19 1

7

k 0 1 2 3 4 5

6

7

ak

.06233 .00404 . 00100 . 00053 .00039 .00027 .00012 .00002

=1-22-

bk

. 79788

47304 03539 89872 66169 92825 55037 70039 68482

.01256 .00178 . 00067 .00041 . 00025 . 00011 .00002

45600 42405 70944 40148 00676 43955 07299 26238

For $ ( z ) , see 6.3. 11.1.20

85~5.3

11.1.17

(x)

'J1

03

=-E k=l

11.1.21

k

d k

0

.39894 23

1

.03117 34

2

.00591 91

3

.00559 56

4

--.01148 58

5

6

.01774 40 -.00739

95

Approximation 11.1.16 is from A. J. M. Hitchcock. Polynomial approximations to Bessel functions of order zero and one and t o related functions, Math. Tables Aids Comp. 11, 86-88 (1957) (with permission). 2

11.1.22

11.1.23

414 289 646 994 454 271 934

482

INTEGRALS OF BESSEL FUNCTIONS

.

11 1.30

Asymptotic Expansions

4 5 x 5 03

where

g1(x) 11.1.25

-2

(-)

k=O

go(x)=2Xf

(;)

k 0 1 2 3 4 5 6

+1)!

-2kk!(k

*g

11.1.31

dk

1.25331 0.50913 .32191 .26214 .20601 . 11103 .02724

41 39 84 46 26 96 00

5 5 x 5 03

where 11.1.27

co=1, c1=-

13 8

[3(k+1 i] (k+;ye,

+

2 (k 1)Ck+ 1=

ck-

)2+

IC 0 1 2 3 4 5 6 7 8

fk

0.39893 14 . 13320 55 -.04938 43 1.47800 44 -8.65560 13 28.12214 78 -48.05241 15 40.39473 40 -11.90943 95 -3.51950 09 9 2.19454 64 10 -11.8. Repeated integrais of J,(zj and K,,(zj

where ck is defined as in 11.1.27. Polynomial Approximations

11.1.29

-1

5 5 x 5 03

where

Repeated Integrals of Jn(z)

Let 11.2.1

I

k

ak

0 1 2 3 4

5 6 7 8 9

1. 0 0. 15999 2815 . 10161 9385 . 13081 1585 . 20740 4022 .28330 0508 .27902 9488 . 17891 5710 . 06622 8328 . 01070 2234

-

bk

1. 0 0. 31998 . 30485 . 52324 1. 03702 1. 69980 1.95320 1. 43132 0. 59605 . 10702

5629 8155 6341 0112 3050 6413 5684 4956 2336

11.2.3 (2- t)'

- 'J,(t)dt

(9?r>O)

483

INTEGRALS O F BESSEL FUNCTIONS

Recurrence Relations

11.2.5

Repeated Integrals of K&)

Let 11.2.8

11.3.5

K i l ( z ) = l m Ko(t)dt, . . ., Ki,(z)=Lm Kir-l(t)dt 11.2.9

Ki-,

(2)

= (-4‘

d’

Ko(z>

11.3.6

Then 11.2.10 Case 1:

p2+ab=0,

U=

& (P-1)

11.3.9 7r

11.2.13

Ki2r+1(0)=

2 r(r+3) r (3)r (r+ 1)

11.3.10

11.2.14

11.3. Reduction Formulas for Indefinite

Integrals Let 11.3.1

g,

,V (2)

J e-

=

(t)dt

PttfiZ,

where Z,(z) represents any of the Bessel functions of the first three kinds or the modified Bessel functions. The parameters a and b appearing in the reduction formulae are associated with the particular type of Bessel function as delineated in. the following table.

11.3.12

s,’e*ctvI,(t)dt=-e

*ZZV+1

2u+ 1

Ffl”+l(Z)I

(9v>3)

484

INTEGRALS OF BESSEL FUNCTIONS

11.3.13

11.3.26

+n[e-Zlo(z)-1]+2e-z~ (n-k)l,(z) k=l

11.3.27

11.3.14

11.3.28

Lrn

t - vKv+l( t ) d t =2 - %(z)

Indefinite Integrals of Products of Bessel Functions

11.3.15

Let q p ( z ) and 9 ” ( z ) denote any two cylinder functions of orders p and Y respectively. 11.3.29

King’s integral (see [ 11.51) 11.3.16 lze‘Ko(t)dt= zez[[Ko(z)+Kl(z)]-1 11.3.17

11.3.30

11.3.31

P Z

11.3.32

1

11.3.21

lz

tJ9-,(t)dt=2

11.3.22

11.3.33 o

11.3.23

=E z

m

>: k-”+l

(2k-l)J2k-1(2)

(n>O)

11.3.35 11.3.24

s,’

+-

tYYV-l(t)dt=ZYYv (2) 2yr(v) 7r

(9Y>O)

+See page

11.

2 (v+2k)J9+&) k =O

(9v>O)

INTEGRALS OF BESSEL FUNCTIONS

485

2. There must exist numbers k, and k2 (both not zero) so that for all n

I n connection with these formulae, see 11.3.29. If a=O, the above is valid provided B=O. This case is covered by the following result. Convolution Type Integrals

11.4.5

11.3.37

(%p>-l, %v>-l) (m=n, b=o, v > - I )

11.3.38

I’

J , (t ) Jl-

y

( 2 - t ) dt =

Jo(2 ) -cos 2

<

(-1 %V<2)

(m=n, b zO, CY~,a2,

V>

- 1)

. . . are the positive zeros of

aJy(z)+bscJ:(z) =O, where a and b are real const ants.

11.3.40

11.4.6

(m=n)(v+n+m>-l) Definite Integrals Over a Finite Range

11.4. Definite Integrals Orthogonality Properties of Bessel Functions

Let % ( z ) be a cylinder function of order I n particular, let

V.

11.4.8 11.4.9

where A and B are real constants.

so‘ s,”

J0(2zsin t ) cos 2ntdt=lrJi(z) YO(2z sin t ) COS %hit=- n- J n ( z ) Y n ( z ) 2

Then 11.4.10

11.4.2

11.4.11

provided the following two conditions hold : 1. An is a real zero of

s,”

Jp(z sin2 t)J,(z cos2 t ) csc 2tdt

486

INTEGRALS OF BESSEL FUNCTIONS

Infinite Integrals Integrals of the Form

I KO( t ) d t =2-

11.4.23

1-

e-pttpZ,(t)dt

11.4.12

=0(w2>1)

where T,(u)is the Chebyshev polynomial of the first kind (see chapter 22).

(W"
W(P+V)>O)

11.4.25

11.4.13

-2 (- i)"( 1 -U2)*U,- 1(U) (U2< 1 ) n

=o (U2>

1)

where U,(U) is the Chebyshev polynomial of the second kind (see chapter 22). 11.4.26

11.4.15

where P,(u) is the Legendre polynomial (see chapter 22). 11.4.27

11.4.1'7

lm J,(t)dt =1

(9V>-1)

where ?(a, z ) is the incomplete gamma function (see chapiJer 6). Integrals of the Form

11.4.28

som

e-a2 12tPZy (bl)02

11.4.19

11.4.20 l m Y , ( t ) d t = - t a n VP 2 11.4.21 11.4.22

l m Y (t)dt=O o

(I 94<1)

where the notation M(a, b, z ) stands for the confluent hypergeometric function (see chapter 13). 11.4.29

e

--aW

tY+'Jv(bt)dt

487

INTEGRALS OF BESSEL FUNCTIONS,

I

11.4.30

11.4.37

:]

im

cos [ p arc sin J,(at) cos bt dt=-----(a2- b2)f -

11.4.31

(s)

h2 v-

(b2- a2)f[ b

(0Ib
+(b2-a2) ]r f

lme-a2i21u(bt)dt=7rf eW2 -I f v 2a

lm

J p ( a t )sin bt dt=

11.4.32

($)

l m e - u 2 t 2 K o ( b t ) d t =7r)z eh’- KO

(9a2>0)

-

Weber- Schafheitlin Type Integrals

( 0 2 b
(a2- b2)f apcos w 2

(b2-a2)f [ b

+(b2-a2)

f ] ~

11.4.33 11.4.39

J 0

eibtJo(at)dt= (a2- b2)f

( 0 2 b
-- a ( b2-a2)f

(O
b arc sin a

(O
11.4.40

lmeibzYo

2i

(at)dt= 7r(a2- b2)f

11.4.34

-1 2; ( b -a2)f +T (b2- a2)4

--

x 2 qp+v--X+l 2

’

p--Y--X+l

2

(9 (p+v--X+l)

>0, 9 X > -1,O
For 2F1,see chapter 15. Special Cases of the Discontinuous Weber- Schafheitlin Integral

11.4.35 JOm

J p ( a t )sin bt dt

t

1

=- sin [ p c1

arc sin

L]

11.4.42

w sin w 2

-

p[ b

+(b2-a2)f]p

lmJr

t

1

=- cos [ p I.1

arc sin

!]

1

(O
-1 -_

(O
2b

(%>-U

J,,(at) cos bt dt

br-

(at)Jp-l(bt)dt=-w

(b

11.4.36 JOm

( 0 5b s a )

( 0 5b l a )

11.4.43

laJ*

(BP>O)

{ l-J,(bt)}dt=O

=In

(Oa>O) a

-

488

INTEGRALS O F BESSEL FUNCTIONS

Hankel-Nicholson Type Integrals

11.4.47

11.4.44

(u>O, 92>0,-

l<wV
11.4.45 (a>o, 9 2 > 0 , 9Y>-l)

11.4.49

.

_ .

(a,>o, 9 2 > 0 ,

11.4.46

Numerica Methods 11.5. Use and Extension of the Tables P Z

For moderate values of z, use 11.1.2 and 11.1.711.1.10 as appropriate. For z sufficiently large, use the asymptotic expansions or the polynomial approximations 11.1.11-11.1.18.

Example 1.

Compute

L ’ O 6

J,(t)dt to 5D.

Using 11.1.2 and interpolating in Tables 9.1 and 9.2, we have

This value is readily checked using 2-3.1 a’nd h=-.05. Now IJo(z)J 51 for all 2 find IJn(z)J <2-*, n > l for all x. In Table 11.1, we can always choose Ihl<.O5. Thus if all terms of O(h4) and higher are neglected, then a bound for the absolute error is 2%4/48<.3.10-6 for all s if Ihl 5 . 0 5 . Similarly, the absolute error for quadratic interpolation does not exceed h3(2* 2) /24 .2 10-4.

+

+.

Example 2.

Compute

*

sb;.(

Example 3. Interpolation of t)dt using Sirripson’s rule. We have r zJo(tjdt+J, y Z + h Jo(tjdt Jo(tjdt=Jo

Jo(t)dt=2[.32019 09+.31783 69+.04611 52 00283 19 =1.37415

<

2

+,00009 72 +.OOOOO 211

s,”””

Jo(t)dt to 5D by

interpolation of Table 11.1 using Taylor’s formula. We have

and with Ihl L.05, it follows that IR1<.9

Then with s=3.0 and h=.05,

10-’O

Thus if s=3.0 and h=.05

3.05

J0(t)dt=1.387567+(.05)(-.260052)

- (.00125) (.339059)

-l-(.OOOOlO) (.746143) =1.37415

So3’05J0(t)dt=1.38’7567 2 5 2 0 + 0 [ - . 2 6 0 0 5 19549 6 +4( -26841 13883)-,27653 495991 =1.37414 86481

489

INTEGRALS OF BESSEL FUNCTIONS

which is correct to 10D. The above procedure gives high accuracy though it may be necessary to

(

interpolate twice in Jo(x)to compute Jo x+-

9

and Jo(z+h). A similar technique based on the trapezoidal rule is less accurate, but at most only one interpolation of Jo(x)is required.

The recurrence formula gives Similarly,

f2=2Cfi+f-1)=1.69809

10

When r>>r, it is convenient to use the auxiliary function

Example 4. C o r n p u t e l J o ( t ) d t a n d l Y o ( t ) d t to 5 D using the representation in terms of Struve functions and the tables in chapters 9 and 12. For x = 3 , from Tables 9.1 and 12.1 Jo=-. 260052 J1= .339059 Yo= .376850 Y1= .324674

Ho=

-574306

H1=1.020110

Using 11.1.7, we have Jo(t)dt=3(-.260052)

This satisfies the recurrence relation

+f

[ (.574306)(.339059)

Example 6. Compute g7(x) to 5 D for r = 1 0 and r=0(1)6. We have for s=10,

- (1.020110) (- .260052)1 =1.38757

Similarly,

P3

J

and the forward recurrence formula gives

Yo(t)dt=.19766 0

Using 11.1.8 and Tables 9.8 and 12.1, one can For tables of 2-'f,(a), see [11.16].

c o m p u t e 1 I o ( t ) d t a n d 1 Ko(t)dt.

Repeated Integrals of KO( x )

For moderate values of x, use 11.1.19-11.1.23. For x sufficiently large, use the asymptotic expansions or the polynomial approximations 11.1.24-11.1.31. Repeated Integrals of Jo(z)

K i 1 ( z ) =Kl(x), Kio(rc)=Ko(x), Kil(r)=lmK,(f)dt

For moderate values of x and r, use 11.2.4. If r=1, see Example 1. For moderate values of x, use the recurrence formula 11.2.5. If x is large and x>r, see the discussion below. Example 5. Compute fr, o(r)=.f,(~) to 5 D for x=2 and r=0(1)5 using 11.2.6. We have rfr+l(x)

=.f7~x)-(~-1)fr-i(~)

f-l ( 4 = -J1 (4,f 0 W = J o ( 4 ,

+~j7-2(~) fl

=l

(x)

Jo (t)dt

and the terns on this last line are tabulated. Thus for x=2, f-l

For moderate values of x, use the recurrence formula 11.2.14 for all r. Example 7. Compute Ki,(z) to 5 D for x = 2 and r=0(1)5. We have

= -37672 48, fo= .22389 08, f l= 1.42577 03

and the functions on this last line are tabulated Thus for ~ = 2 , K0=.11389 39, K1=.13986 59, Kil=.09712 06

and Ki2=-2Kil+2K1=.08549

06

Similarly, Ki3=.07696 36, I(ip=.07043 17, Ki5=.06525 22

If Z/T is not large the formula can still be used provided that the starting values are sufficiently accurate to offset the growth of rounding error. For tables of Ki,(x), see [ 11.1I].

490

INTEGRALS O F BESSEL FUNCTIONS

Apart from round-off error, the value of r needed to achieve a stated accuracy for given x and m can be determined a priori. Let Then In Let

the latter following from 11.3.27 with v = l . 11.3.5, put a = l , b=-1, p=O and v=O. p=m. Then

x2keT

e r -2k=

( ~ - 1 ) ~ ( r - 3 ) ~. .. ( ~ - 2 k + 1 ) ~

since for x fixed,fi(x) is positive and decreases as T increases. Using tabular values of f o and f l ,one can compute in successionf2,f3, . . . provided that m/x is not large. Example 8. Computef,(z) to 5D for 2 = 5 and m=0(1)6. We have, retaining two additional decimals

KO= .00369 11 56738 74

fo=1.

K1=.00404 46 fi=. 19595 54

Computef,(x) to 5D for x=3 and We have

Example 9.

m=O(2)1O.

Taking g16=0, we compute the following values of 914, g12, . . ., go by recurrence. Also recorded are the required values of f m to 5D.

Thus f2=.O5791 27,f4=.01458 93,f~=.OO685 36 Similarly starting withf,, we can computef3 andf5. If m>z, employ the recurrence formula in backward form and write fm-2(x)= [2fm(z) +2&(2) + 4 m -

m

14 12 10 8 6 4 2 0

1)Ko(x)l/(m--

In the latter expression, replacef, by g,. Fix x. Take r>m and assume g,=O. Compute g7-2, gr-4, etc. Then

!irE y"T--2i(z) --jfT'(Z), ?E=r-22k r+

. 00855 .01061 .01325 . 01751 .02548 .04447 . 11936 1. 53994

42 09 05 39 09 31 90 71

.01325 .01751 .02548 .04447 . 11937 1.53995 I

For tables off,(z), see L11.211.

Q)

References Texts [11.1] H. Bateman and R. C. Archibald, A guide to tables of Bessel functions, Math. Tables Aids Comp. 1, 247-252 (1943). See also Supplements I, 11, IV, same journal, 1, 403-404 (1943); 2, 59 (1946); 2, 190 (1946), respectively. [11.2] A. ErdBlyi e t al., Higher transcendental functions, vol. 2, ch. 7 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [ 11.31 A. ErdBlyi e t al., Tables of integral transforms, vols. 1, 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1954). [11.4] W. Grobner and N. Hofreiter, Integraltafel, I1 Teil (Springer-Verlag, Wien and Innsbruck, Austria, 1949-1950). [11.5] L. V. King, On the convection of heat from small cylinders in a stream of fluid, Trans. Roy. SOC. London 214A, 373-432 (1914). [11.6] Y. L. Luke, Some notes on integrals involving Bessel functions, J. Math. Phys. 29, 27-30 (1950).

[11.7] Y. L. Luke, An associated Bessel function, J. Math. Phys. 31, 131-138 (1952). [11.8] F. Oberhettinger, On some expansions for Bessel integral functions, J. Research NBS 59, 197-201 (1957) R P 2786. [11.9] G. Petiau, La t h b r i e des fonctions de Bessel (Centre National de la Recherche Scientifique, Paris, France, 1955). [11.10] G. N. Watson, A treatise on the theory of Bessel functions, 2d ed. (Cambridge Univ. Press, Cambridge, England, 1958).

Tables [11.11] W. G. Bickley and J. Nayler, A short table of the functions Ki,(z), from n = l to n=16. Philos.

Mag. 7, 20,343-347 (1935).

=dm

Kin@)

(.1)2, 3,

Ki,-l(t)dt,

9D.

Kil(z) = l Q ) K o ( t ) d t ,

n=1(1)16, z=0(.05).2

491

INTEGRALS O F BESSEL FUNCTIONS

[11.12] V. R. Bursian and V. Fock, Table of the functions

Akad.

Nauk,

Leningrad, Inst.

Fiz.

Mat.,

Trudy (Travaux) 2, 6-10 (1931). JmKo(t)dt,

z=0(.1)12, 7D; ez

1"

lm

Ko(t)dt,z= 0(.1) 16, 7D;

lo(t)dt, z=0(.1)6,

7D; e-z

1'

~ o ( t ) d tz= ,

0 (.1)16, 7D. [11.13] E. A. Chistova, Tablitsy funktsii Besselya o t deistvitel' nogo argumenta i integralov o t nikh (Izdat. Akad. Nauk SSSR., Moscow, U.S.S.R.,

[11.18] H. L. Knudsen, Bidrag til teorien for antennesystemer med he1 eller delvis rotations-symmetri. I (Kommission Has Teknisk Forlag, Copenhagen, Denmark,

1953).

0(.01)10, 5D.

1"

J n ( t ) d l , n=0(1)8, z=

Also

Jz

J,(t)eiadt, a=t, a=z-t.

[11.19] Y. L. Luke and D. Ufford, Tables of the function

E&)

=J'Ko(t)dt.

Math. Tables Aids Comp.

U M T 129. ~ o ( z ) = - [ r + l n (z/2)lA1(z)+A~(z), Ai(z), Az(z). z=0(.01).5(.05)1, 8D.

[11.20] C. Mack and M. Castle, Tables of

J=

lo(z)& and

JamKo(z)&,Roy. Soc. Unpublished Math. Table n=O, 1; z=0(.001)15(.01)100, 7D. Also tabulated are auxiliary expressions to facilitate interpolation near the origin. [11.14] A. J. M. Hitchcock, Polynomial approximations to Bessel functions of order zero and one and to related functions, Math. Tables Aids Comp. 11, 86-88 (1957). Polynomial approximations

for

l'

Jo(t)dt and JmKo(t)dt.

[11.15] C. W. Horton, A short table' of Struve functions and of some integrals involving Bessel and Struve functions, J. Math. Phys. 29, 56-58

so"

(1950). C,(z) =

tnJ,(t)dt,n=1(1)4,~=0(.1)10,

so"

tnH,(t)dt, n=0(1)4, ~ = 0 ( . 1 )10,

4D;

Dn(Z)=

4D, where H,(z) is Struve's function; see chapter 12. [11.161 J. C. Jaeger, Repeated integrals of Bessel functions and the theory of transients in filter circuits, J. Math. Phys. 27, 210-219 (1948). fi(z)=

r"

File No. 6. a=0(.02)2(.1)4, 9D. [11.21] G. M. Muller, Table of the function K j ,,(z)= r n J z u n & ( u )du, Office of Technical Services, U.S. Department of Commerce, Washington, D.C. (1954). n=O( 1)31, z=O(.Ol) 2( .02)5, 85. [ 11.221 National Bureau of Standards, Tables of functions and zeros of functions, Applied Math. Series 37 (U.S. Government Printing Office, Washington,

z=O(.Ol)lO, 10D.

(2) pp. 33-39:

z=0(.1)10(1)22,

10D;

Jzm

F(z)=Jsm

Jo(t)dt/t, Jo(t)dt/t

+In (z/2), z=0(.1)3, 10D; F(n)(z)/n!, z= 10(1)22, n=0(1)13, 12D. [ 11.231 National Physical Laboratory, Integrals of Bessel functions, Roy. Soc. Unpublished Math. Table File No. 17. 10D.

1

Jo(t)dt,

Yo(t)dt, z=0(.5)50,

r"lo(z)dZ for 0(.1)20(1)25, 1 ~ 7 , [11.24] M. Rothman, Table of JO Quart. J. Mech. Appl. Math. 2, 212-217 (1949). ~ = 0 ( 1 ) 2 48D. , Also @,(z) = Jo[2(zt)']Jn(t)dt, 8s-9s. Jo(t)dt for large z, [11.25] P. W. Schmidt, Tables of @pn(z), @:(z), n=1(1)7, ~ = 0 ( 1 ) 2 4 , 4D. j11.171 L. N. Karmazina and E. A. Chistova, Tablitsy J. Math. Phys. 34, 169-172 (1955). z=10(.2)40, funktsii Besselya ot mnimogo argumenta i 6D. integralov o t nikh (Izdat. Akad. Nauk SSSR., E11.263 G. N. Watson, A treatise on the theory of Bessel Moscow, U.S.S.R., 1958). e-zlo(z), e-=I1(z), functions, 2d ed. (Cambridge Univ. Press, Cambridge, England, 1958). Table VIII, p. 752: ezKo(z), eZKI(z), ez, e - z i ' ~ o ( t ) d tezJzm , ~o(t)dt, JO

~ ~ ( t ) df rt( z, > = f " f 7 - w t , 2-rfr(z), JO

r

T=

7D except for eE which is 75. Also tabulated are auxiliary expressions t o facilitate interpolation near the origin.

z=0(.001)5(.005)15(.01)100,

1"

k i z J o ( t ) d t , ~ i z Y o ( 1 ) d lz=0(.02)1, ,

7D,

with

the first 16 maxima and minima of the integrals to 7D.

492

INTEGRALS OF BESSEL FUNCTIONS

Table 11.1

INTEGRALS OF BESSEL FUNCTIONS PX

Jo J o W t

X

yo ( 2 ) dt

e - z c

I o ( t ) dt

e z l m~ ~ ( t ) d t

0.0 0.1 0.2 0. 3 0. 4

0.00000 0.09991 0.19933 0.29775 0.39469

00000 66979 43325 75802 85653

0.00000 -0.21743 -0.34570 -0.43928 -0.50952

00000 05666 88380 31758 48283

0.00000 0.09055 0.16429 0.22391 0.27172

00 92 28 79 46

1.57079 1.35784 1.25032 1.17280 1.11171

63 82 54 09 28

0. 0. 0. 0. 0.

5 6 7 8 9

0.48968 0.58224 0.67193 0.75834 0.84106

05066 12719 68094 44308 59149

-0.56179 -0.59927 -0.62409 -0.63786 -0.64184

54559 15570 96341 88991 01770

0.30964 0.33929 0.36206 0.37910 0.39137

29 99 71 05 42

1.06127 1.01836 0.98109 0.94821 0.91885

17 48 70 80 56

1. 0 1.1 1.2 1.3 1.4

0.91973 0.99399 1.06355 1.12813 1.18750

04101 71082 76711 83885 20495

-0.63706 -0.62447 -0.60490 -0.57911 -0.54783

93766 91607 26964 12548 19295

0.39970 0.40479 0.40721 0.40745 0.40593

88 52 52 78 39

0,89237 0.86829 0.84626 0,82596 0,80719

52 97 10 89 04

1.5 1. 6 1.7 1. 8 1.9

1.24144 1.28982 1.33249 1.36939 1.40048

95144 09734 68829 85727 85208

-0.51175 -0.47156 -0.42788 -0.38136 -0.33260

90340 13039 62338 24134 04453

0.40298 0.39891 0.39394 0.38828 0.38211

85 09 29 68 11

0.78973 0.77344 0.75819 0.74386 0.73037

57 80 62 97 44

2. 0 2.1 2. 2 2. 3 2.4

1.42577 1.44528 1.45912 1.46740 1.47029

02932 81525 63387 80303 39949

-0.28219 -0.23071 -0.17871 -0.12672 -0.07526

28501 32490 50399 97284 50420

0.37555 0.36873 0.36174 0.35467 0.34757

57 67 98 38 2p

0.71762 0,70556 0,69412 0.68324 0.67288

95 50 02 16 26

2.5 2. 6 2.7 2.8 2.9

1.46798 1.46069 1.44871 1.43231 1.41181

09446 96081 25408 16899 57386

-0.02480 +O. 02420 0.07132 0.11617 0.15839

29261 24953 69288 78353 62206

0.34049 0.33349 0.32659 0.31981 0.31319

93 48 30 99 59

0.66300 0.65356 0.64452 0.63587 0.62757

15 16 98 68 60

3. 0 3.1 3.2 3. 3 3.4

1.38756 1.35992 1.32928 1.29602 1.26056

72520 96508 4nM6 59125 17835

0.19765 82565 0.23367 66986 0.29502 36222 0.31996 99576

0.30673 0.30045 0.29435 0.28843 0.28271

62 18 04 67 31

0.61960 0.61193 0.60455 0.59744 0.59059

34 74 84 84 11

3. 5 3.6 3. 7 3. 8 3. 9

1.22330 1.18467 1.14509 1.10496 1.06471

57382 59706 13136 78009 52877

0.34090 0.35775 0.37044 0.37896 0.38335

94657 03989 06831 74266 61369

0.27718 0.27183 0.26668 0.26170 0.25691

02 70 11 94 78

0.58397 0.57757 0.57139 0.56540 0.55961

14 57 13 66 09

4. 0 4.1 4.2 4. 3 4.4

1.02473 0.98541 0.94712 0.91021 0.87502

41595 21560 13375 52175 60866

0.38366 0.38000 0.37250 0.36131 0.34665

96479 67672 06552 69475 16398

0.25230 0.24785 0.24357 0.23945 0.23548

18 61 56 46 74

0.55399 0.54854 0.54326 0.53812 0.53314

42 72 15 91 27

5 6 7 8 9

0.84186 0.81100 0.78271 0.75721 0.73468

25481 72858 50802 10902 94106

0.32872 0.30779 0.28413 0.25802 0.22977

87513 77892 10351 06786 58227

0.23166 0.22799 0.22445 0.22104 0.21775

83 15 13 21 83

0.52829 0.52358 0.51899 0.51452 0.51017

52 03 19 43 24

0.21459 46

0.50593

10

4. 4. 4. 4. 4.

5. 0

0.71531 19178

0 , 7 h h 2 0 20748

0.19971 93876

493

INTEGRALS OF BESSEL FUNCTIONS

INTEGRALS OF BESSEL FUNCTIONS 2

JZJ0

( t ) dt

yo (0 dt

exso'

Table 11.1

Io(t)dt

exJxw

KO( t )dt

5. 0 5.1 5. 2 5.3 5.4

0.71531 0.69920 0.68647 0.67716 0.67131

19178 74098 10457 40870 39407

0.19971 0.16818 0.13551 0.10205 0.06814

93876 49405 34784 01932 12463

0.21459 0.21154 0.20860 0.20577 0.20303

46 58 68 28 89

0.50593 0.50179 0.49776 0.49382 0.48998

10 55 16 50 19

5.5 5. 6 5.7 5. 8 5.9

0.66891 0.66992 0.67427 0.68187 0.69257

44989 67724 98068 18713 19078

0.03413 +O. 00035 -0.03284 -0.06517 -0.09630

05806 67983 98697 04775 01348

0.20040 0.19785 0.19539 0.19301 0.19072

08 40 44 81 13

0,48622 0.48256 0.47897 0.47547 0,47204

86 16 75 34 60

6. 0 6.1 6. 2 6.3 6.4

0.70622 0.72263 0.74160 0.76290 0.78628

12236 54100 64692 51256 33012

-0.12595 -0.15385 -0.17975 -0.20344 -0.22470

06129 27646 87372 39625 89068

0,18850 0.18635 0.18427 0.18225 0.18030

02 16 20 84 78

0.46869 0.46541 0.46219 0.45905 0.45596

29 11 83 20 99

6.5 6. 6 6.7 6. 8 6.9

0.81147 0.83820 0.86618 0.89512 0.92470

67291 76824 77897 09137 60635

-0.24338 -0.25931 -0.27239 -0.28252 -0.28966

05692 37161 18447 78684 45218

0.17841 0.17658 0.17480 0.17308 0.17140

74 44 64 09 55

0.45294 0.44998 0.44708 0,44424 0.44144

98 97 76 15 97

7. 0 7.1 7.2 7. 3 7.4

0.95464 0.98462 1.01435 1.04354 1.07190

03155 17153 21344 00558 32638

-0.29377 -0.29486 -0.29295 -0.28811 -0.28043

44843 02239 35658 49927 26862

0.16977 0.16819 0.16665 0.16516 0.16370

82 68 93 39 89

0.43871 0.43602 0.43338 0.43079 0.42824

05 22 34 23 76

7.5 7.6 7.7 7. 8 7.9

1.09917 1.12508 1.14941 1.17192 1.19243

14142 84628 49299 99830 33198

-0.27002 -0.25702 -0.24159 -0.22392 -0.20421

13202 06208 37080 52368 93575

0.16229 0.16091 0.15956 0.15825 0.15698

24 30 91 93 21

0.42574 0.42329 0.42087 0.41850 0.41617

81 20 86 63 40

8.0 8.2 8.3 8.4

1.21074 1-2 2 6 7 1 1.24021 1.25112 1.25939

68348 60587 13565 88778 12520

-0.18269 -0.15959 -0.13516 -0.10966 -0.08335

75150 61109 40494 01934 07540

0.15573 0.15452 0.15333 0.15217 0.15104

64 08 42 55 36

0.41388 0.41162 0.40940 0.40722 0.40507

07 52 65 37 56

8.5 8. 6 8.7 8.8 8.9

1.26494 1.26777 1.26787 1.26528 1.26005

80243 58297 83120 57796 46162

-0.05650 -0.02940 -0.00230 +O. 02451 0.05078

66385 07834 54965 01664 29664

0.14993 0.14885 0.14779 0.14676 0.14575

74 61 88 44 23

0.40296 0.40088 0.39883 0.39681 0.39482

15 04 15 40 69

9.0 9.1 9.2 9. 3 9.4

1.25226 1.24202 1.22946 1.21473 1.19799

64460 70675 51666 08237 38314

0.07625 0.10069 0,12385 0.14552 0,16550

79635 08937 04194 02334 09969

0.14476 0.14379 0.14284 0.14191 0.14099

16 16 16 08 87

0.39286 0.39094 0.38904 0.38716 0.38532

97 15 17 95 41

9.5 9.6 9.7 9.8 9.9

1.17944 1.15927 1.13772 1.11499 1.09134

18392 83464 05614 71504 58985

0.18361 0.19969 0.21360 0.22523 0.23448

20962 32017 56169 34059 42919

0.14010 0.13922 0.13836 0.13752 0.13669

46 78 79 43 65

0.38350 0.38171 0.37994 0,37820 0.37648

53 20 39 03 06

8; 1

10. 0

1.06701 13040

0.24129 03183

0.13588 40

0.37478 43

494

INTEGRALS OF BESSEL FUNCTIONS

INTEGRALS OF BESSEL FUNCTIONS

Table 11.2

lo(t)-1

X

dt

+K

067 832 037

0 ~ 0 0 0 0 0000 0.00113 1 4 0 0.00409 877 0.00835 768 0.01347 363

0.000000 0.368126 0.460111 0.506394 0.532910

0.26968 0.33839 0.37689 0.39543 0.40022

854 213 807 866 301

0.01910 0.02497 0.03088 0.03667 0.04222

285 622 584 383 295

0.548819 0.558366 0.563828 0.566545 0.567355

525 721 240 570 610

0.39527 0.38332 0.36633 0.34572 0.32256

290 909 694 398 701

0.04744 0.05229 0.05672 0.06070 0.06425

889 376 080 995 420

0.566811 0.565291 0.563058 0.560302 0.557163

0.26227 0.29557 0.33031 0.36636 0.40360

724 796 288 308 666

0.29769 0.27176 0.24529 0.21871 0.19235

696 713 896 360 409

0.06735 0.07002 0.07228 0.07414 0.07563

663 797 458 688 806

0.553745 0.550126 0.546364 0.542506 0.538587

2.0 2.1 2.2 2.3 2.4

0.44191 0.48117 0.52124 0.56200 0.60333

940 541 775 913 248

0.16650 0.14138 0.11719 0.09408 0.07218

135 594 681 798 365

0.07678 0.07760 0.07813 0.07839 0.07841

298 744 746 884 674

0.534635 0.530670 0.526711 0.522768 0.518854

2.5 2. 6 2.7 2.8 2.9

0.64509 0.68716 0.72942 0.77174 0.81402

164 194 081 836 795

0.05158 0.03235 +0.01457 -0.00174 -0.01655

229 987 248 144 931

0.07821 0.07781 0.07724 0.07652 0.07566

544 809 664 168 245

0.514976 0.511139 0.507350 0.503610 0.499924

3. 0 3: 1 3.2 3.3 3.4

0.85614 0.89799 0.93947 0.98047 1.02091

669 596 188 571 428

-0.02987 -0.04168 -0.05201 -0.06088 -0.06833

272 613 554 740 756

0.07468 0.07361 0.07245 0.07121 0.06993

681 124 090 963 006

0.496292 0.492717 0.489198

3. 5 3. 6 3. 7 3.8 3.9

1.06070 1.09975 1.13799 1.17536 1.21179

032 277 707 536 667

-0.07441 -0.07915 -0. 08263 -0.08491 -0.08605

025 722 683 323 553

0.06859 0.06722 0. 06582 0.06440 0.06297

360 060 033 109 029

0.478984 0.475694 0.472459 0.469280 0.466155

4. 0 4.1 4. 2 4. 3 4.4

1.24723 1.28163 1.31496 1.34718 1.37826

707 975 504 044 060

-0.08613 -0.08523 -0.08342 -0. 08079 -0.07742

706 459 762 769 769

0.06153 0.06009 0.05867 0.05725 0.05584

450 952 042 166 708

0.463085 0.460067 0.457100 0.454185 0.451320

4. 5 4. 6 4. 7 4. 8 4.9

1.40818 1.43694 1.46454 1.49096 1.51622

716 870 052 446 864

-0.07340 -0.06880 -0.06371 -0.05821 -0.05239

123 199 317 690 371

0.05446 0.05309 0.05174 0.05042 0,04913

000 325 921 989 691

0.448503 0.445734 0.443012 0.440335 0.437703

5. 0

1.54034 722

-0.04632

205

0.04787 1 6 1

0.435114

0. 0 0.1 0.2 0. 3 0.4

0.00000 0.00124 0.00499 0.01121 0,01990

000 961 375 841 030

-1.34138 -0.43423 -0.05107 +0.15238

0. 5 0. 6 0. 7 0. 8 0.9

0.03100 0.04449 0.06032 0.07841 0.09872

699 711 057 882 519

1. 0 1.1 1.2 1.3 1.4

0.12116 0.14565 0.17211 0.20043 0.23052

1.5 1.6 1.7 1.8 1.9

~ 0 3 1

-03

382

0.485736 0.482332

dt

12. Struve Functions and Related Functions MILTONABRAMOWITZ

Contents Page

Mathematical Properties . . . . . . . . . . 12.1. Struve Function H Y ( z ) . . . . . . 12.2. Modified Struve Function Ly(z) . . 12.3. Anger and Weber Functions . . . Numerical Methods . . . . . . . . . . . . 12.4. Use and Extension of the Tables . References . . . . . . . . . . . . . . . . Table 12.1. Struve Functioqs (0 L 2 . 5 03 ) . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

496 496 498 498

Table 12.2. Struve Functions for Large Arguments.

. . . . . . . .

502

499 499

500 501

sw

t-'H0(t)dt, x=0(.1)5, 5 D to 7D

(2/~)

2

The author acknowledges the assistance of Bertha H. Walter in the preparation and checking of the tables.

1

National Bureau of Standards.

(Deceased.) 495

12. Struve Functions and Related Functions Mathematical Properties 12.1. Struve Function H Y ( z ) Differential Equation and General Solution

12.1.11 12.1.12

12.1.1 22

d2w 7+z dz

dw -+(22-v2)W= dx

Hb=(2/~)-Hl d z’H”)= z ~ H ~ - ~ dz

-(

4( 3 2 ) ” + I

tl;;r(v+4)

The general solution is

n ”(X)

12.1.2 w=aJ,(z) + b Y , ( z ) + H , ( x ) (a$, constants)

where z - ” H y ( z )is an entire function of z. Power Series Expansion

12.1.3

12.1.5

-.4

L FIGURE 12.1. Struve functions.

Integral Representations

E,(%), n=0(1)3

If 9 v > - # , 12.1.6

12.1.8

=Y, (z)

Recurrence Relations

FIGURE 12.2. Struve functions. H,(z), - ~ ~ = 1 ( 1 ) 3

497

STRUVE FUNCTIONS AND RELATED FUNCTIONS Struve’s Integral

12.1.25 t

12.1.26 23

12.1.27

FIGURE 12.3. Struve functions. H,(z), 2=3, 5 Special Properties

12.1.14

H,(z) 2 0

Aeymptotic Expansions for Large 1.1

(z>O and

3)

12.1.29

12.1.15 H-,,+t,(z)=(-l)”J,+i(z)

12.1.16

Hl(z)=(;Y

(n a n integer>())

(I-cos z)

12.1.17

12.1.18

(larg z l < 4

If v is real, z positive where R,,,=O(IZ~~-~~-*). and m+$-v>,O, the remainder after m terms is of the same sign and numerically less than the first term neglected.

HY( zemrf)=em(v+i)rf H,(z) ( man integer)

12.1.19

Integrals (See chapter 11)

12.1.22 12.1.23

JW

t-’ &(t)dt=2

U U k=l

(-l)kf’(2k)!(2k-l)! (k!)2(2 2)*

where -;=.57721 56649

12.1.33

*See page 11.

(18%

zl
. . . is Euler’s constant.

*

498

STRUVE FUNCTIONS AND RELATED FUNCTIONS Asymptotic Expansion for Large I a

Asymptotic Expansions for Large Orders

12.1.34

12.2.6

1

Lv(~)-I-v(~)

Integrals

12.1.35

12.2.1

12.2. Modified Struve Function Lv(z) 12.2.8

Power Series Expansion

L’

[Io(t) -Lo(t)]dt-;2 [In (22)+7]

f V r

12.2.1 Lv(z)=- - i e - T

H,(iz)

12.2.9

Integral Representations

12.2.2

Lv(2) = 2 ( z / 2 ) ’ fsinh ( z COS 0) sin2v0d0

J;;r (v+ 3)

n- z

Relation to Modified Sphericd Bessel Function

12.2.10 L-(,++l (z)=I(,++,(z)

@v>-4)

12.2.3

s,’Ll(t)dt=L,(z)--2

(n an integerro)

12.3. Anger and Weber Functions Anger’s Function

12.3.1

Recurrence Relations

12.3.2

Jv(z ) =:

J“cos ( ~ 6 - z sin 0) d6

n o

J,( Z ) =Jn(Z )

(n an integer)

Weber’s Function

12.3.3

E&) -1

sin

(ye-

z sin 0) d0

Relations Between Anger’s and Weber’s Function

12.3.4 sin (m) J,(z)=cos (vn) E,(z)-E-,(z) 12.3.5 sin (m) E,(z)=J-,(z)-cos (m) J,(z) Relations Between Weber’s Function and Function

‘:i

+X /-

0

/i:

-3

I I

If n is a positive integer or zero, rn-11

0

0

/

I 1’1-4

I

I

I

12.3.1

FIGURE 12.4. Moda3ed Struve functions. Ln(z), f n = 0 ( 1 ) 5 *See page II.

I51

Struve’s

499

STRUVE FUNCTIONS AND RELATED FUNCTIONS

12.3.10

Numerical Methods 12.4. Use and Extension of the Tables

Example 1. Compute L0(2) to 6D. From Table 12.1 10(2)-L0(2)=.342152; from Table 9.11 we have 10(2)=2.279585 so that L0(2)= 1.937433. Example 2. Compute Ho(lO) to 6D. From Table 12.2 for x-’=.l. Ho(10)-Yo(10)=.063072; from Table 9.1 we have ~0(10)=.055671. Thus, Ho ( 10)=.I 18743. Example 3. Compute

s,” H o ( t ) d t for s = 6

to

5D. Using Tables 12.2, 11.1 and 4.2, we have

s,”Ho t>dt (

l6

=

Y o(t )dt

+

lr

+

ln 6 fi(6)

=-.125951+(.636620)(1.791759) ,816764

+

=1.83148

Example 4. Compute H,(z) for 2=4, - n = 0(1)8 to 6s. From Table 12.1 we have H0(4)= .1350146, H1(4)= 1.0697267. Using 12.1.9 we find H-1(4) = - ~ 3 3 1 0 7 &(4) = .240694 H-3(4) = .152624 H-4(4) = -.439789

H-5(4) = .689652 H-6(4) = - 1.21906 H-7(4) = 2.82066 H 4 ( 4 ) = -8.24933

Example 5. Compute H,(z) for 2=4, n= 0(1)10 to 7 s . Starting with the values of H0(4) and H1(4) and using 12.1.9 with forward recurrence, we get H,(4)= .13501 H1(4)=1.06972 H2(4)=1.24867 H3(4)= .85800 H4(4)= .42637 H5(4)= .16719

46 67 51 95 41 87

He(4) =.OM33 H7(4) z.01510 Hs(4) z.00367 Hg(4) =.00080 H10(4)=.00018

54 37 33 02

25

We note that for n > 6 there is a rapid loss of significant figures. On the other hand using 12.1.3 for 2=4 we find H9(4)=.0007935729, H10(4)= .00015447630 and backward recurrence with 12.1.9 gives Hs(4) =.OO367 H,(4)=.01510 &(4)=.05433 H5(4)=.16719 H4(4) =.42637

1495 315 519 87 43

H3(4) = .85800 H2(4)=1.24867 H1(4)=1.06972 H0(4)= .13501

94 6 7 4

Example 6. Compute L,(.5) for n=0(1)5 to 8s. From 12.2.1 we find L5(.5)=9.6307462 X 10-7, L4(.5)=2.1212342X 10-5. Then, with 12.2.4 we get L3(.5) =3.82465 03X10-4 L1(.5)=.05394 2181 L2(.5)=5.36867 34X10-3 Lo(.5)=.32724 068 Example 7. Compute L,(.5) for -n=0(1)5 to 6s. From Tables 12.1 and 9.8 we find L0(.5)= .327240, L1(.5) = .053942. Then employing 12.2.4 with backward recurrence we get

Example 8. Compute L,(z) for 2=6 and --n=0(1)6 to 8s. From Tables 12.2 and 9.8 we find Lo(6) =67.124454, Li(6) =60.725011. Using 12.2.4 we get L-1(6)=61.361631 L-,(6) ~ 4 6 . 7 7 6 6 8 0 L-3(6) =30.159494

L-4(6)=16.626028 L-5(6) = 7.984089 L-g(6) = 3.32780

We note that there is no essential loss of accuracy until n=-6. However, if further values were necessary the recurrence procedure becomes unstable. To avoid the instability use the methods described in Examples 5 and 6.

500

STRUVE FUNCTIONS AND RELATED FUNCTIONS

References Texts [12.1] R. K. Cook, Some properties of Struve functions, J. Washington Acad. Sci. 47, 11, 365-368 (1957). [12.2] A. Erdklyi et al., Higher transcendental functions, vol. 2, ch. 7 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [12.3] A. Gray and G. B. Mathews, A treatise on Bessel functions, ch. 14 (The Macmillan Co., New York, N.Y., 1931). [12.4] N. W. McLachlan, Bessel functions for engineers, 2d ed. ch. 4 (Clarencion Press, Oxford, England, 1955). [12.5] F. Oberhettinger, On some expansions for Bessel integral functions, J. Research NBS 59 (1957) RP2786. [12.6] G. Petiau, La theorie des fonctions de Bessel, ch. 10 (Centre National de la Recherche Scientifique, Paris, France, 1955). 112.71 G. N. Watson, A treatise on the theory of Bessel functions, ch. 10 (Cambridge Univ. Press, Cambridge, England, 1958).

I

Tables [12.8] M. Abramowitz, Tables of integrals of Struve functions, J. Math. Phys. 29, 49-51 (1950). [12.9] C. W. Horton, On the extension of some Lommel integrals to Struve functions with a n application t o acoustic radiation, J. Math. Phys. 29, 31-37 (1950). [12.10] C. W. Horton, A short table of Struve functions and of some integrals involving Bessel a n d Struve functions, J. Math. Phys. 29,56-58 (1950). [ 12.1 I] Mathematical Tables Project, Table of t h e Struve functions Ly(z) and HY(z), J. Math. Phys. 25, 252-259 (1946).

501

STRUVE FUNCTIONS AND RELATED FUNCTIONS

Table 12.1

STRUVE FUNCTIONS X

Hi (x)

HO(4

To(x>-

b(4

I1).(

-Ll(X)

fo(4

"s

rn

How dt

0.06359 0.12675 0.18908 0.25014

00 13 90 29 97

0.00000 0.00212 0.00846 0.01898 0.03359

00 07 57 43 25

0.000000 0.003181 0.012704 0.028505 0.050479

1.000000 0.938769 0.882134 0.829724 3.781198

0.000000 0.047939 0.091990 0.132480 0.169710

0.00000 0.09690 0.18791 0.27347 0.35398

t 1.000000 0.959487 0.919063 0,878819 0.838843

0.5 0. 6 0.7 0.8 0. 9

0.30955 0.36691 0.42184 0.47399 0.52303

59 14 24 44 50

0.05217 0.07457 0.10063 0.13012 0.16281

37 97 17 25 75

0.078480 0.112322 0.151781 0.196597 0.246476

0.736243 0.694573 0.655927 0.620063 0.586763

0.203952 0.235457 0.264454 0.291151 0.315740

0.42982 0.50134 0.56884 0.63262 0.69294

0.799223 0.760044 0.721389 0.683341 0.645976

1. 0 1.1 1.2 1.3 1. 4

0.56865 0.61057 0.64855 0.68235 0.71179

66 87 00 03 25

0.19845 0.23675 0.27742 0.32012 0.36452

73 97 18 31 80

0.301090 0.360084 0.423074 0.489655 0.559399

0.555823 0.527058 0.500300 0.475391 0.452188

0.338395 0.359276 0.378530 0.396290 0.412679

0.75005 0.80418 0.85553 0.90430 0.95066

0.609371 0.573596 0.538719 0.504803 0.471907

1. 5 1,6 1.7 1. 8 1.9

0.73672 0.75702 0.77261 0.78345 0.78952

35 55 68 23 36

0.41028 0.45704 0.50444 0.55210 0.59966

85 72 07 21 45

0.631863 0.706590 0.783111 0.860954 0.939643

0.430561 0.410388 0.391558 0.373970 0.357530

0.427810 0.441783 0.454694 0.466629 0.477666

0.99479 1.03082 1.07691 1.11518 1.15174

0.409388 0.440D86

2. 0

2.1 2.2 2. 3 2. 4

0.79085 0.78752 0.77961 0.76726 0.75064

88 22 35 65 85

0.64676 0.69304 0.73814 0.78174 0.82351

37 18 96 98 98

1.018701 1.097659 1.176053 1.253434 1.329364

0.342152 0.327756 0.314270 0.301627 0.289765

0.487877 0.497329 0.506083 0.514194 0.521712

1.18672 1.22020 1.25230 1.28309 1.31265

0.298634 0.274109 0.250891 0.228992 0.208417

2. 5 2.6 2.7 2.8 2.9

0.72995 0.70542 0.67729 0.64586 0.61142

77 23 77 46 64

0.86315 0.90036 0.93489 0.96649 0.99496

42 74 57 98 63

1.403427 1.475227 1.544392 1.610577 1.673465

0.278627 0.268162 0.258319 0.249056 0.240332

0.528685 0.535156 0.541164 0.546746 0.551933

1.34106 1.36840 1.39472 1.42008 1.44455

0.189168 0.171238 0.154618 0.139293 0.125242

3. 0 3.1 3.2 3. 3 3.4

0.57430 0.53484 0,49339 0.45032 0.40600

61 44 57 57 80

1.02010 1.04177 1.05983 1.07418 1.08477

96 30 03 63 74

1.732773 1.788248 1.839675 1.886873 1.929699

0.232107 0.224348 0.217022 0.210099 0.203553

0.556757 0.561246 0.565426 0.569319 0.572948

1.46816 1.49098 1.51305 1.53440 1.55508

0.112439 0.100857 0.090460 0.081212 0.073071

3. 5 3. 6 3. 7 3. 8 3. 9

0.36082 0.31514 0.26935 0.22382 0.17893

08 40 59 98 12

1.09157 1.09457 1.09380 1.08934 1.08127

23 16 77 44 62

1.968046 2.001847 2.031071 2.055726 2.075858

0.197357 0.191488 0.185924 0.180646 0.175634

0.576333 0.579492 0.582442 0.585199 0.587776

1.57512 1.59456 1.61343 1.63176 1.64957

0.065992 0.059928 0.054829 0.050642 0.047311

4. 0 4.1 4. 2 4. 3 4. 4

0.13501 0.09242 0.05147 +0.01247 -0.02427

46 08 40 93 98

1.06972 1.05484 1.03681 1.01584 0.99214

67 79 86 22 51

2.091545 2.102905 2.110084 2.113265 2.112655

0.170872 0.166343 0.162032 0.157926 0. 154012

0.590187 0.592445 0.594560 0.596542 0.598402

1.66689 1.68375 1.70017 1.71616 1. 73176

0.044781 0.042994 0.041891 0.041414 0.041502

4. 5 4. 6 4. 7 4. 8 4. 9

-0.05854 -0.09007 -0.11867 -0.14415 -0.16637

33 71 42 67 66

0.96597 0.93759 0.90729 0.87535 0.84208

44 56 01 28 90

2.108492 2.101037 2.090574 2.077406 2.061852

0.150279 0.146714 0.143309 0.140053 0.136938

0.600147 0.601787 0.603328 0.604777 0.606142

1.74697 1.76182 1.77632 1.79049 1.80434

0.042096 0.043139 0.044571 0.046335 0.048376

5. 0

-0.18521 68

0.80781 19

2.044244

0.133955

0.607426

1.81788

0.050640

0. 0 0. 1 0.2 0. 3 0.4

0.00000

A

X

0.379857 0.351533 0.324450

Ho(x), HI(x), Lo(x), L1 (z), compiled from Mathematical Tables Project, Table of the Struve functions LV(z)and HJX),J. Math. Phys. 25,252-259, 1946 (with permission). JozH0(l)dt, Joz [ & ( t ) - I 4 t ) ] d t ,

zJzw$H

dt, compiled from M. Abramowitz, Tables of integrals of Struve

functions, J. Math. Phys. 29,49-51, 1950 (with permission).

502

STRUVE FUNCTIONS AND RELATED FUNCTIONS

Table 12.2

STRUVE FUNCTIONS FOR LARGE ARGUMENTS

Hi(x)- Yi(x> 0.659949 0.657819 0.655774 0.653818 0.651952

lo(x)-Low

-L1@)

0.18 0.17 0. 16

Hob9 - Y d X ) 0.123301 0.117449 0.111556 0.105625 0.099655

0.819924 0.818935 0.817981 0.817062 0.816182

0.133955 0.126683 0.119468 0.112319 0.105242

0.607426 0.610467 0.613348 0.616060 0.618598

0.793280 0.794902 0.796448 0.797910 0.799279

0.125868 0.119694 0.113505 0.107299 0.101079

5 5 6 6 6

0.15 0.14 0.13 0.12 0.11

0.093647 0.087602 0.081521 0.075404 0.069254

0.650180 0.648504 0.646927 0.645452 0.644081

0.815341 0.814541 0.813785 0.813074 0.812411

0.098241 0.091318 0.084474 0.077706 0.071010

0.620955 0.623129 0.625119 0.626927 0.628558

0.800551 0.801721 0.802787 0.803750 0.804611

0.094843 0.088593 0.082328 0.076051 0.069761

7 7 8 8 9

0.10 0. 09 0. 08 0.07 0. 06

0.063072 0.056860 0.050620 0.044354 0.038064

0.642817 0.641663 0.640622 0.639696 0.638888

0.811796 0.811232 0.810722 0.810266 0.809866

0.064379 0.057805 0.051279 0.044793 0.038340

0.630018 0.631315 0.632457 0.633450 0.634302

0.805374 0.806047 0.806634 0.807140 0.807572

0.063460 0.057147 0.050824 0.044492 0.038152

10 11 13 14 17

0. 05 0. 04 0.03 0.02 0. 01

0.031753 0.025425 0.019082 0.012727 0.006366

0.638200 0.637634 0.637191 0.636874 0.636683

0.809525 0.809244 0.809023 0.808865 0.808770

0.031912 0.025506 0.019116 0.012738 0.006367

0.635016 0.635596 0.636045 0.636365 0.636556

0.807933 0.808225 0.808450 0.808611 0.808706

0,031805 0.025451 0.019093 0.012731 0.006366

20 25 33 50 100

0.00

0.000000

0.636620

0.808738

0.000000

0.636620

0.808738

0.000000

a l

2-'

0. 20 0. 19

fdx)

"

-

" :'"I

f&>

f3(4

<x>

-

< x > = nearest integer to x. Starting with Ho(z) and HI(%), recurrence formula 12.1.9 may be used to generate Hn(x)for nx/2, forward recurrence is unstable. To avoid the h ~ t ~ h i l i tcheese y, z> >z, cerr,pcte Hk(z)2nd HL+:(T)wit.h 12-13; rrmd then use backward recurrence with 12.1.9. If n>O, Ln(X) must be generated by backward recurrence. If n
See Examples 4- 8.

.

13 Confluent Hypergeometric Functions LUCYJOANSLATER'

Contents Mathematical Properties . . . . . . . . . . . . . . . . 13.1. Definitions of Kummer and Whittaker Functions . . 13.2. Integral Representations . . . . . . . . . . . . 13.3. Connections With Bessel Functions . . . . . . . . 13.4. Recurrence Relations and Differential Properties . . 13.5. Asymptotic Expansions and Limiting Forms . . . . 13.6. Special Cases . . . . . . . . . . . . . . . . . . . 13.7. Zeros and Turning Values . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . .

Page

504 504 505 506 506 508 509 510

Numerical Methods . . . . . . . . . . . . . . 13.8. Use and Extension of the Tables . . . 13.9. Calculation of Zeros and Turning Points 13.10. Graphing M(a. b. 2) . . . . . . . . .

511 511 513 513

References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

514

. . . . .

516

. . . . .

535

Table 13.1. Confluent Hypergeometric Function M(a. b. 2) 2= .1 (.1) l ( 1 ) 10. a= - 1 (.1) 1. b =.1 (. 1) 1. 8 s Table 13.2. Zeros of M(a. b. 2) . . . . . . . . . . . . . a=- 1(.1)-.1, b=.I(.I)I, 7D

The tables were calculated by the author on the electronic calculator EDSACI in the Mathematical Laboratory of Cambridge University. by kind permission of its director. Dr . M . V . Wilkes . The table of M(a. b. z) was recomputed by Alfred E . Beam for uniformity t o eight significant figures .

1 University Mathematical Laboratory. Cambridge . (Prepared under contract with the National Bureau of Standards.)

503

13. Confluent Hypergeometric Functions Mathematical Properties 13.1. Definitions of Kummer and Whittaker Functions

Logarithmic Solution

Kummer's Equation

z d2w -+(b-z)

13.1.1

dz2

U(a,b, z ) is a many-valued function. Its principal branch is given by -a<arg z 5 T. 13.1.6

dw ---uw=O dz

It has a regular singularity a t z=O and an irregular singularity at W . Independent solutions are for n=O, 1, 2 , . . ., where the last function is the sum to n terms. It is to be interpreted as zero when n=O, and $ ( u ) = I " ( a ) / r ( a ) .

Kummer's Function

13.1.2

13.1.1

where

U(a, 1-n, z)=znU(a+n, l + n , z )

As 9 z - + ~

( u ) ~ = u ( u + ~ ) ( u .+.~. )(u+n-I), (a)o=I,

13.1.8

U(a, b, z ) =z-'[ 1

and

+O(I z ~ - ' ) ]

Analytic Continuation

13.1.9

13.1.3

U ( a ,b , z e f r f

)-=

7r

e-

M(b-a, b , Z ) { r (1 +a- b ) r ( b )

fri (1 - b) 21 - b

M ( l -U, 2 - b , Z )

r ( U ) r (2- b ) Parameters ( m ,n positive integers)

b#-n

a#-m

b#-n

a=--m

b=-n a+-m b = - n a=-m, m>n

b,

2)

a convergent series for all values of a, b and z a poiynomiai of degree m in z

where either upper or lowksigns are to be taken throughout. t

\

13.1.10

U(a,b , ze2*fn)=[l-e-2nibnr(1-b) M ( a , b , z ) r(l+a-b)

a simple pole at b= -n

b=-n a=-na, undefined min U(u,b , z ) is defined even when b + f n

+e-2ribnU(a, b, z) Alternative Notations

,Fl(a; b; z ) or @(a;b; z ) for M(a, b, z ) z-'$,(a, 1 +a- b; ;- l / z ) or *(a; b; z ) for U(a,b, z )

As Izl+m,

Complete Solution

13.1.11

Y = A M ( u , b, z)+BU(a, b,

Z)

where A and B are arbitrary constants, b # -n. Eight Solutions

13.1.12 yl=M(a, b, z ) 13.1.13 y,=zlVbM(1+a-b, 2 - b , z) 13.1.14 y3=ezM(b-a, b, 504

1

-2)

505

CONFLUENT HYPERGEOMETRIC FUNCTIONS

13.1.15

ya= z1- bezM(1 -a, 2 -b , - z )

13.1.34

13.1.16 y5=U(a, b, z ) 13.1.17

y,=z'-bU(l+a-b, b,

2-b,

13.1.18

y,=ezU(b-a,

13.1.19

yB=zl-bezU(l-a, 2-b,

Z)

General Confluent Equation

13.1.35

-2)

-2)

Wronskians

h"

If W ( m ,n }=ymyL- y,yh and e=sgn, ( Y z ) = 1 if Y z > O , =-I if YzSO

A(A-I)

+-Z+f"+f'2-T]w=0 2Aj ' ah'

13.1.20

S o h tions:

W ( 1 ,2)=W{3, 4 ) = U 7 { 1 , 4}=-W{2, 3 ) = ( 1 -b) z- bez 13.1.37

13.1.21

W ( 1 ,3}=W(2,4 ) = W ( 5 , 6}=W(7,S)=O 13.1.22

13.2. Integral Representations

W (1 , 5)=-I'(b)z-bez/ir(a)

13.1.23

U7{1 , 7 ) = r(b)et?r"bz-bez//r(b--a)

13.1.24

U'( 2, 5 } = - r ( 2 -b) z - be'/r ( 1 a -3)

13.1.25

W (2, 7 ) = - r(2- b)z- bez//r(l-a)

13.1.26

W (5, 7 } =ec*i(b-a)z - b e z

13.2.1

+

13.2.2

Kummer Transformations

13.1.27

M(a, b, z)=eZM(b-a, b,

-2)

13.1.28

zl-bM(l+a-b,

2-b, z ) = z l - b e z M ( l - a , 2-b)

13.1.29 U ( a , b, z)=zl-bU(l+a-b,

-2)

2-b, z )

13.1.30

Whittaker's Equation

Solutions:

13.2.6 Whittaker's Functions

13.2.7

*See page 11.

Z-Ae-f(z)U(a, b, h ( 2 ) )

9b>9a>O

506 13.2.8

CONE'LUENT HYPERGEOMETRIC FUNCTIONS

13.3.7

r(a)U(U,b, Z )

Similar integrals for M K , r (and ~ ) WK,,,(z)can be deduced with the help of 13.1.32 and 13.1.33. Barnes-type Contour Integrals

13.2.9

for larg ( - z ) / < + a , a, b # o , -1, -2, contour must separate the poles of those of r(a+s); c is finite.

.... r(--s)

The from

13.2.10

where

r (-

S)

r (a+s) r ( 1+a-

3a for larg zl<-, a#O, -1, 2

-2,

b +s) z -"s

. . ., b - u f l ,

2,

. . . . The contour must separate the poles of r(--s)from those of r(a+S) and r(i+a-b+~).

3,

+ +

CO= 1 , C1= -bh, Ca = -3 (Zh- 1)U 3b (b 1) h2, (n+ 1 ) Cn+l=[( 1 -2h)n-bh]C, [ ( 1-2h)u-h(h1) (b+n- l)]Cn-l -h(h- ~ ) u C , - ~ (h real)

+

13.3. Connections With Bessel Functions (see chapters 9 and 10) Bessel Functions as Limiting Cases

If b and z are fixed,

13.4. Recurrence Relations and Differential Properties 13.4.1

(b-u)M(a- 1, b,

2)

+(2a-b+z)M(a,

b, 2) -aM(a+l, b, z)=O

13.4.2

b(b- 1)M(a,b - I , 13.3.4

Z)

+b(I - b - ~ ) M ( a , b, Z ) +z(b-a)M(a, b + l , z)=O

13.4.3

(1+a-b)M(a, b, z)-uM(a+l, b, 2) +(b-l)M(a, b-1, z)=O 13.4.4

bM(a, b, 2)-bM(u-1,

b, z)-zM(a, b + l , z)=O

13.4.5

b(a+z)M(a, b, z ) + z ( u - ~ ) M ( ub,+ l , Z ) -ubM(a+l, b, z)=O *See page U.

507

CONFLUENT HYPERGEOMETRIC FUNCTIONS

13.4.19

13.4.6

(a-l+z)M(a, b, z)+(b-a)M(u-l, b, 2) +(1-b)M(a, b-1,

z)=O

(a+z)U(a, b, z ) - - U a , b + l , z> +a(b-U- 1) U(a+ 1, b, a)=O 13.4.20

13.4.7

b(1-b+z)M(a, b, z)+b(b-l)M(a-1,

b-1,

(a+z-l)U(a, b, z)-U(a-l, b, 2) +(l+a-b)U(a,

2)

-azM(a+l, b + l , z)=O 13.4.8

M'(a, b, z ) =% M(a+ 1, b+ 1, Z)

13.4.21

U'(a, b, z)=-uU(a+l,

b-1,

z)=O

b + l , z)

13.4.22 d" - {U(a,b , z))=(-l)"(a).U(a+n, dz"

b+n,

4

13.4.23 13.4.10 aM(a+l, b, ~ ) = a M ( ab,, z)+zM'(a, b, z )

a(l+a-b)U(a+l,

b, z)=aU(a, b,

2)

+zU'@, b,

13.4.11

4

13.4.24

(b-u)M(a-l, b, z)=(b-a-z)M(a,

b, 2) +zM'(a, b,

2)

13.4.12

(l+a-b)U(a, b - 1 , z)=(l-b)U(a, b, 2) -zU'(a, b, z ) 13.4.25 U(a, b + l , z)=U(a, b, z)-U'(a, b,

(b-a)M(a, b + l , z)=bM(a, b, 2)-bM'(a, b,

Z)

2)

13.4.26 13.4.13

U(a-1, b, z)=(a-b+z)U(a, b, z)-zU'(a, b,

(b-l)M(a, b - 1 , z)=(b-l)M(a, b,

2)

2)

13.4.27

+zM'(a, b,

4

U(a-1, b - 1 , 2)=(1-b+z)U(a,

b,

2)

-zU'(a, b, z )

13.4.15

U(a-1, b, z)+(b-2a-z)U(a, b, 2) +a(l +a-b) U(a+ 1 , b,

2)

=o

13.4.16

(b-a-l)U(n, b-1,

z)+(l-b-z)U(a, b, 2) +zU(a, b + l , z)=O

13.4.17

U(a, b, z)-uU(a+l, b, 2)-U(a, b-1,

z)=O

13.4.18

(b--a)U(a, b, z)+U(a-1, b,

2)

-zU(a, b + l , z)=O

13.4.31

508

CONFLUENT HYPERGEOMETRIC FUNCTIONS

13.5. Asymptotic Expansions and Limiting Forms For 1.1

13.5.10 U(U,b, z)=

r(1-b) r(1+a-

b)

+O(lzl'-

g b )

large, (a,b fixed)

13.5.1

13.5.11 13.5.12

(b=O)

. .

( m l o ,bz0)

For large a ( b , a fixed)

13.5.13

the upper sign being taken if-+n<arg -in. lower sign if -+n<arg

z<#T,

the

13.5.2

13.5.14

M ( u ,b , 2)=r(b)etz(3bz-uaz)t-tba-t

Converging Factors for the Remainders

13.5.3

COS

( d ( 2 b z - 4 a ~-+bn+t~) ) [1

as a+--03

+O(1 3b -al -9 1

for b bounded, x real.

and 13.5.4

where

Q

is defined in 13.5.13.

where the R'th and S'th terms are the smallest in the expansions 13.5.1 and 13.5.2. For small a (a,b fixed)

13.5.5

as a+-a

AS Iz/+O, M(a,b , O)=I, b f-n

for b bounded, x real. For large real a, b, x

If Cosh28 = ~ / ( 2 b - 4 a )SO that x>2b-a>l, r(b-1)

13.5.1

=-

13.5.8

r(b-1) --

r (4

zl-"+O(lln

21)

(b=2)

r (a)

Z'-b+

O(1)

13.5.17 M(U,b, ~)=r(b) sin (aa) exp [(&-a)(+ sinh ~ O - O + C O S ~ O~ ) ] [(b-zu) cosh O]1-b[a(3b-a)sinh 281-4 [ 1 0(13b -4 -9 1 13.5.18 U(a, b, x)=exp [(b-2a)(3 sinh ~ o - - B + c o O)] s~~ [(b-zu) cosh 0]~-~[((3b--a)sinh 283-4 [1+O(l4b-al-')l

+

509

CONFLUENT HYPERGEOMETRIC FUNCTIONS

If x= (2b-4u)[l+t/(b-2~)~], so that x-2b-4a 13.5.19 M(a, b, x) =etz(b-2a)?-”(b)[Ai(t) cos (an) Bi ( t ) sin (aa) 0( 1&b-Q 1 -3) J 13.5.20 U(a, b, 2)=eJz+a-Jbr(+)n-fx6-* { I--tr(+)(bs-2&x)-t3tn-t+O(lgb-al-~) 1

+

+

If ~os28=~/(2b-4a) so that 2b--4a>z>O,

13.5.21

M(a, b, s) = r (b) exp { (b-24 cos2 81 [(b-za) COS 8]1-b[a(+-a) sin 281-4 [sin (an)+sin { (ib-a) (28-sin 2e) + i n )

+

O(I*b - I--I) 1 13.5.22 U(a,b, s)=exp [@--a) cos2 e][(b-%) cos [(+a) sin 2e]-J{sin [(+&a) (20-sin 2 e ) ~ i ~ ] + O ( l 4 b - a l - ~ ) )

13.6. Special Cases

M(a, b, 4 b

a ~~

Relation

Function

2

--

13.6.1

2u+ 1

2iz

Bessel

13.6.2

-2u+l

2iz

Bessel

13.6.3

2u+ 1

22

Modified Bessel

13.6.4

2n+2

2iz

Spherical Bessel

13.6.5

-2n

2iz

Spherical Bessel

13.6.6

2n+2

22

Spherical Bessel

13.6.7

2n+ 1

-2 6 u

Kelvin

13.6.8

2L+2

2iz

Coulomb Wave

13.6.9

-n

a+l

2

Laguerre

13.6.10

U

a+l

-2

Incomplete Gamma

13.6.11

-n

l+u-n

2

Poisson-Charlier

13.6.12

U

a

Z

Exponential e-i.

- sin z

Trigonometric

e# -sinh z

Hyperbolic

13.6.13

1

2

-2iz

13.6.14

1

2

22

13.6.15

-3 y

3

3z’

13.6.16

3-3y

4

32’

Weber or Parabolic Cylinder

13.6.17

-n

3

w

Hermite

13.6.18

-n

s

tzl

n! (2n+ l)! (-+I-”

13.6.19

3

6

-d

lr4 erf 22

13.6.20

3m+3

l+n

P

*See page

11.

1

;r

*

H er mite Error Integral

2

*

Toronto

510

CONF’LUENT HYPERGEOMETRIC FUNCTIONS

13.6. Special Case-Continued

Relation

b

a

Function

Z

13.6.21

y+3

zv+ 1

22

Modified Bessel

13.6.22

U++

2u+ 1

-2iz

Hankel

13.6.23

U+$

2u+ 1

2iz

Hankel

13.6.24

n+l

2n+2

22

Spherical Bessel

13.6.25

4

Q

62312

Airy

13.6.26

n+3

2n+ 1

&

Kelvin

13.6.27

-n

a+l

X

Laguerre

13.6.28

1-U

1-a

X

Incomplete Gamma

13.6.29

1

1

-X

Exponential Integral

13.6.30

1

1

X

Exponential Integral

13.6.31

1

1

--In x

Logarithmic Integral

13.6.32

3m-n

l+m

X

Cunningham

13.6.33

-+U

0

22

Bateman

13.6.34

1

1

iz

Sine and Cosine Integral

13.6.35

1

1

--iz

Sine and Cosine Integral

13.6.36

-3 y

3

3 2’

13.6.37

3-tu

.s

t

Weber or Parabolic Cylinder

13.6.38

3-3n

4

39

Hermite

13.6.39

3

3

XS

Error Integral

13.7. Zeros and Turning Values

If j b - l , r is the r’th positive zero ofJb--l(z), then a first approximation X o to the r’th positive zero of M(a, b, x) is

For the derivative, 13.7.4

M’(a, b, XI)=

13.7.1 XO=&,,, { 1/(2b-4a)+ O(l/(+b-a)2) 1 13.7.2

XO =

r2(r+

3b-

$)2

2b-4a

A closer approximation is given by 13.7.3 X,=Xo-M(a, b , Xo)/M’(a,b, Xo) +See page U.

If XA is the first approximation to a turning value of M(a, b, x), thtit is, to a zero of M’(a, b, z) then a bet,ter approximation is

51 1

CONFLUENT HYPERGEOMETRIC FUNCTIONS

The self-adjoint equation 13.1.1 can also be written 13.7.6 The Sonine-Polya Theorem

The maxima and minima of IwI form an increasing or decreasing sequence according as

is an increasing or decreasing function of x, that is, they form an increasing sequence for M(a, b, x) if a>O, xb---, and a decreasing sequence if a>O and x>b-+ or if a
e -22

-ax2b-l

Numerical Methods 13.8. Use and Extension of the Tables Calculation of M(a, b, x)

In this way 13.4.1-13.4.7 can be used together with 13.1.27 to extend Table 13.1 to the range -10

Kummer’s Transformation

Example 1. Compute M(.3, .2, -.1) to 7 s . Using 13.1.27 and Tables 4.4 and 13.1 we have a=.3, b=.2 so that

M(.3, 2, - .I) =e-*’M( - .1, .2, .1) =.85784 90.

Thus 13.1.27 can be used to extend Table 13.1 to negative values of x. Kummer’s transformation should also be used when a and b are large and nearly equal, for x large or small. Example 2. Compute M(17, 16, 1) to 7 s . Here a=17, b=16, and M(17, 16, l)=e’M(-l,

16, -1)

=2.71828 18X1.06250 00 =2.88817 44. Recurrence Relations

.2, .1)-.3M(.7, .2, .1)]

=.35821 23.

By 13.4.5 when a= - 1.3 and b= .2, -.24 M(-1.3,

.2, .1)]/.15

=.89241 08.

1.2, .1)=.97459 52.

=A9241 08.

b+-I

for all x, when a=b.

Hence lim M(b, b, x)=ez. b+-I

In the first case b+- 1 along the line a= - 1, and along the line a=b. in the second case b+-1 Example 5. To evaluate M’(--7, -.6, .5) to 7 s . By 13.4.8, whena=-.7 andb=-.6, we have -7 - .6

M (.3, .4, .5)

= 1.724128.

Asymptotic Formulas

Check, by 13.4.6, 1.2, .1)=[.2 M(-.3, .2, .I) +1.2 M(-.3,

.?:

When a=-l, hf(-l, 6 , zj=l---ior afi x. b Hence lim M ( - I , b,x)=I tx. ButM(b, b,x)=tY

M’( - .7, - .6, .5)=-

Similarly when a= - .3 and b = .2

M(-1.3,

Example 4. At the point (- 1. - 1,x), M(a, b, x) is undefined.

Derivatives

M ( - 1.3,1.2, .1) = [.26 M ( - .3, .2, .1)

M(-.3,

iX
This extension of ten units in any direction is possible with the loss of about 1s. All the recurrence relations are stable except i) if a ~x>O, b ~ , or ii) blbl, x
_.

Example 3. Compute M ( - 1.3, 1.2, . l ) to 7s. Using 13.4.1 and Table 13.1 we have a=-.3, b=.2 so that M(-1.3, .2, .1)=2[.7M(-.3,

La< 10, -10
1.2, .1)]/1.5

For 2 2 1 0 , a and b small, M(a, b, x) should be evaluated by 13.5.1 using converging factors 13.5.3 and 13.5.4 to improve the accuracy if necesmry .

512

CONFLUENT HYPERGEOMETRIC FUNCTIONS

Example 6. Calculate M(.9, .l, 10) to 7S, using 13.5.1.

Hence

U( .l, 2 , 1) ~ 5 . 3 4 4 7 9 (.371765 9 - .194486) = .94752.

Similarly U(-.9,

+1237253(.99190 285) +O(l> = 1227235.23-.17 +O(1) --.198(.869) -

Check, from Table 13.1, M(.9, .l, 10)=1227235. To evaluate M(a, b, x) with a large, x small and b small or large 13.5.13-14 should be used. Example 7. Compute M(-52.5, .l, 1) to 3s, using 13.5.14. .I, 1 ) = r ( . i ) e. ~ ( . 0 5+52.5).25-.05

.5642 COS [ ( . 2 - 4 ( - 5 2 . 5 ) ) . 5 - . 0 5 ~ + . 2 5 ~ ] [ l +0((.05+52.5)-*5)]= - 16.34+0(.2)

By direct application of a recurrence relation, M(-52.5, .1, 1) has been calculated as -16.447. To evaluate M(a, b, x) with x, a and/or b large, 13.5.17,19 or 21 should be tried. Example 8. Compute M(-52.5, .1, 1) using 13.5.21 to 3s, COS e=Ji/210.2. M(-52.5,

.l, 1)

- r(.l)do5.l

Hence by 13.4.15 u(1.1, .2, 1)=[U(.l, .2, l)-U(-.9, = .38664.

Example 10. To compute U'(-.9, 5s. By 13.4.21

=1227235+0(1)

M(-52.5,

.2, 1)=.91272.

.2, 1)]/.09 - 3 , 1) to

U'(--.9, -.8, 1)=.9U(.l, .2, 1) = (.9) (.94752) =35276. Asymptotic Formulas

Example 11. To compute U(1, .l, 100) to 5 s . By 13.5.2 1 1.9 1.9 2.9 U(1, .l, loo)==fiG{l--+100 100100

1.9 2.9 ---

3.9 0(10-9)>). 100 100 100

=.01{ 1-.019+.000551-.000021 +O(lO-g) =.00981 53.

1 9

Example 12. To evaluate U(.l, .2, .Or). For x small, 13.5.6-12 should be used.

[105.1 COS 8]'-*'.5641 52.55-3 sin 28-i[sin ( - 5 2 . 5 ~) +sin y - 5 m I - s i n q + b )

+0((52.55)-')]=-

16.47+ 0(.02)

A full range of asymptotic formulas to cover all possible cases is not yet known. Calculation of U(a, b, x)

For -1O<x
= 1.09 to 3S, by 13.5.10.

To evaluate U(a, b, x) with a large, x small and b small or large 13.5.15 or 16 should be used. To evaluate U(a, b, x) with x, a and/or b large 13.5.18, 20 or 22 should be tried. In all these cases the size of the remainder term is the guide to the number of significant figures obtainable. Calculation of the Whittaker Functions

U(.l, .2, 1)=

Example 13. Compute M.o,-.4(l)and W.o,-.4(l) to 5s. By formulas 13.1.32 and 13.1.33 and Tables 13.1, 4.4

But M(.9, 1.8, 1)=.8[M(.9, .8, 1)-M(-.l, = 1.72329,

using 13.4.4.

.8, l)]

M.o,-. 4 ( 1) =e- .5M(.1, .2, 1) = 1.10622, Ws0,. 4 ( 1) =e-.5U(.l, .2, 1) = .57469.

513

CONFLUENT HYPERGEOMETRIC FUNCTIONS

Thus the values of M,,,(x) and W,,,(x) can always be found if the values of M(a, b, x) and U(a, b, 2) are known.

[l-M(-2,1.6,X;)/.6M(--3, .6,Xi)]

13.9. Calculation of Zeros and Turning Points Example 14. Compute the smallest positive zero of M ( -4, a . 6 , ~ ) . This is outside the range of Table 13.2. Using 13.7.2 we have, as a fist

.6, X;) .6, X ; )1

M’(-3, x;=x,!, [ 1-3M(-3,

=.9715X1.0163=.9873 to 4 s.

This process can be repeated to give as many significant figures as are required.

approximation I

I

I

\ I

I

I

I

Using 13.7.3 we have

But, by 13.4.8, M’(-4,

.6, Xo)=-(.15)-’M(-3,

1.6, Xo)

Hence

=.174+ (-15)(.030004) =,17850 as a

second approximation. b

If we repeat this calculation, we find that X2=X1+.00002 99=.17852 99 to 7 s .

FIGURE13.1.

Figure 13.1 shows the curves on which M(a,b , 2) = O in the a,b plane when x = I . The function is positive in the unshaded areas, and negative in the Example 15. Compute the value of x at which shaded areas. The number in each square gives 9 A 0 ;“I(i .8, - -2,,j ilttstt iui:niiig v-al-ce. TUT -DI -l l l-g 1Ic).-J!.W the nuniber of real positive zeros of M ( a , 6 , x) as a and Table 13.2, we find that M’(-1.8, - 2 , ~ ) function of x in that square. The vertical =9M(-.8, .8, s)=O when s=.94291 59. boundaries to the left are to be included in each Also M”(-1.8, -.2, ~ ) = 9 M ’ ( - . 8 , .8, x)= square. -9M(.2, 1.8, x) and M(.2, 1.8, .94291 59)>0. 13.10. Graphing M(a, b, x ) Hence M(- 1.8, - 2 , ~has ) a maximum in x when s=.94291 59. Example 17. Sketch M(-4.5, 1, 2). Firstly, from Figure 13.1 we see that the function has Example 16. Compute the smallest positive five real positive zeros. From 13.5.1, we find value of x for which M(-3, .6, x) has a turning that M-+--m, M’-+-m as x++m and that value, X : . This is outside the range of Table 13.2. Using 13.4.8 we have M-++m, M ’ + + m as x 4 - m . By 13.7.2 we have 6s first,approximations to the zeros, .3,1.5,3.7, M’(-3, .6, ~ ) = - 3 M ( - 2 , 1.6, ~ ) / . 6 . 6.9, 10.6, and by 13.7.2 and 13.4.8 we find as first approximations to the turning values .9, 2.8, 5.8, By 13.7.2 for M(-2, 1.6, x), 9.9. From 13.7.7, we see that these must lie near the curves Xo = (1 . 0 5 42/ (11.2) = .97 15. y = fe,”(5x)-t(l-x/ll)-t.n-). This is a first approximation to X,!,for M(-3, .6, x). Using 13.7.5 and 13.4.8 we find a second approxiFrom these facts we can form a rough graph of mation the behavior of the function, Figure 13.2. Calculation of Maxima and Minima

514

CONFLUENT HYPERGEOMETRIC FUNCTIONS M

M

10

80

t

I I

9 8 7 6

5

xi I

\

\

FIGURE 13.2. M(-4.5, 1,

l 4

I:

3

O L 5).

0

(From F. Q . Tricomi, Funzioni, ipergeometriche confluenti, Edizioni. Cremonese, Rome, Italy, 1954, with permission.)

1

I

I

I

I

.5

I

I .5

2

2.5

M

4 10

M

8 n

10

6

9

4

8

2 I

7

0

6

X

-2

5 4

3

-8

2

- 10

I

0

I

-5 M

1

1

I

I .5

1

1

2

2.5

cx

//.. / I

8

FIGURE13.4. M (.a,, .5,, 5) .. (From E. Jahnke and F. Emde Tables of functions Dover Publications, Inc., New York, &.Y., 1945, with per&sion.)

References Texts

-2

6 4

2 I 0

-2 -4 -6

-8 -10

FIGURE13.3. M ( a , 1,

3).

(From E . Jshnke and F Emde Tables of functions Dover Publications, Inc., New York, A ,.,. 1945, with penhsion.) \

[ 13.11 H . Buchholz, Die konfluente hypergeometrische

Funktion (Springer-Verlag, Berlin, Germany, 1953). On Whittaker functions, with a large bibliography. [13.2] A. Erd6lyi e t al., Higher transcendental functions, vol. 1, ch. 6 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). On Kummer functions. [13.3] H . Jeffreys and B. S.Jeffreys, Methods of mathematical physics, ch. 23 (Cambridge Univ. Press, Cambridge, England, 1950). On Kummer functions. [13.4] J. C. P. Miller, Note on the general solutions of t h e confluent hypergeometric equation, Math. Tables Aids Comp. 9,97-99 (1957). [13.5] L. J. Slater, On the evaluation of the confluent hypergeometric function, Proc. C a m b r i d g e Philos. Soc. 49, 612-622 (1953).

X

CONFLUENT HYPERGEOMETRIC FUNCTIONS

[13.6] L. J. Slater, The evaluation of the basic confluent hypergeometric function, Proc. Cambridge Philos. Soc. 50, 404-413 (1954). [13.7] L. J. Slater, The real zeros of the confluent hypergeometric function, Proc. Cambridge Philos. SOC.52, 626-635 (1956). [13.8] C. A. Swanson and A. ErdBlyi, Asymptotic forms of confluent hypergeometric functions, Memoir 25, Amer. Math. Soc. (1957). [13.9] F. G. Tricomi, Funzioni ipergeometriche confluenti (Edizioni Cremonese, Rome, Italy, 1954). On Kummer functions. [13.10] E. T. Whittaker and G. N. Watson, A course of modern analysis, ch. 16, 4tb ed. (Cambridge Univ. Press, Cambridge, England, 1952). On Whittaker functions.

Tables [13.1I] J. R. Airey, The confluent hypergeometric function, British Association Reports, Oxford, 276-294 (1926), and Leeds, 220-244 (1927). M ( a , b, z), ~ = - 4 ( . 5 ) 4 , b=$, 1, Q, 2, 3, 4, ~=.1(.1)2(.2)3 (.5)8, 5D.

515

[13.12] J. R. Airey and H. A. Webb, The practical importance of the confluent hypergeometric function, Phil. Mag. 36, 129-141 (1918). M ( a , b, z), U= - 3(.5)4, b= 1(1)7, z= 1(1)6(2)10, 4s. [13.13] E. Jahnke and F. Emde, Tables of functions, ch. 10, 4th ed. (Dover Publications, Inc., New York, N.Y., 1945). Graphs of M ( a , b, z) based on the tables of [13.11]. [13.14] P. Nath, Confluent hypergeometric functions, Sankhya J. Indian Statist. Soc. 11, 153-166 (1951). M ( u , b, s), ~ = 1 ( 1 ) 4 0 b=3, , 2=.02(.02) ,1(.1)1(1)10(10)50, 100, 200, 6D. [13.15] S. Rushton and E. D. Lang, Tables of the confluent hypergeometric function, Sankhya,J, Indian Stntist. SOC.13, 369-411 (1954). M ( u , b, z),~=.5(.5)40, b= .5 (.5)3.5, z= .02 (.02).1(. 1)1(1)10(10)50, 100, 200, 7s. [13.16] L. J. Slater, Confluent hypergeometric functions (Cambridge Univ. Press, Cambridge, England, 1960). M ( u , b, z), ~ = - l ( . l ) l ,b = . l ( . l ) l , z=.l(.l)lO, 8s; M(u, b, l), ~=-11(.2)2, b= -4(.2) 1, 8s; and smallest positive values of s for which Mfa, b, z)=O, u=-4(.1)-.1, 6=.1(.1)2.5, 8s.

516

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION M ( a , b, x)

Table 13.1

ab\ -1.0 -0.9 -0.8 -0.7 -0.6

0.1 -2) [-1) (-1) (-1)

0.00000 9.58364 1.92586 2.90253 3.88843

0.3

0.2 00 34 25 86 71

(-1 -11 [-1 (-1 (-1

i

5.00000 5.48093 5.96605 6.45537 6.94891

00 23 00 25 92

(-1)

0.5

0.4

7.50000

00

6.66666

67

(-1)

( 0) 1.00000

00

( 0) 1.00000 00

(-1 8.96573 73 ( - 1 1 9.48069 7 8 0.0

nb\ -1.0 -0.9 -0.8 -0.7 -0.6

( 0) 1.00000 00

( 0) 1.00000 00

0.6 -1 -11 -1) -1) -1)

8.33333 8.49524 8.65820 8.82221 8.98727

0.8

0.7 33 54 31 06 18

(-1) (-1 (-11 (-1) (-1)

8.57142 8.71045 8.85031 8.99103 9.13259

-0.2 -0.1

-1) 9.27501 22 -1 9.41828 4 7 - l \ 9-56341 h 4 (-ij 9.70741 0 8 ( - 1 ) 9.85327 09

0.0

( 0) 1.00000 00

( 0) 1.00000 00

5.3

i

0.1 0.2 0.3 0.4 0.5

0) 0 01 0) 0)

1.01725 1.03461 1.05209 1.06967 1.08736

53 94 25 52 79

[[

0.6 0.7 0.8 0.9 1.0

0) 0 0 0 0)

1.10517 1.12308 1.14110 1.15924 1.17749

09 48 98 65 53

( 0) 1.08988 06

I

0) 0 01 0 ( 0)

1.01476 1.02960 1.04454 1.05956 1.07467

01 78 34 71 94

9.36616 9.49150 9,61755 9.74432 9.87180

( 0) 1.00000

1.o

0.9

86 21 91 26 59

( - 1 ) 9.15339 1 0 (-1) 9.32057 22 ( - 1 ) 4.48881 96 ( - 1 ) 9.65813 72 ( - 1 ) 9.82852 93

-0.5 -0.4

( 0) 1.00000 00

18 52 81

32 29 00

I-{

(-1 -1

9.43700 9.54840 9.66040 -1 9.77300 -1 9.88619

64 68 42 09 88

-1 9.79591 86 -1 9.89770 1 6

( 0) 1.00000 00

( 0) 1.00000 00

1.01144 1.02294 1.03450 1.04612 1.05781

07 21 45 80 30

0 0 0 0 ( 0

[ [

( 0) 1.15158 03

For O l z l 1 , linear interpolation in a , 2, or x provides 3-43. Lagrange four-point interpolation gives 7s in a,b or 2 over most of the table, but the Lagrange six-point formula is needed over the range 1 1 x 1 1 0 . Any interpolation formula can be reapplied to give two dimensional interpolates in a and b , a and x or b and x. This calculation can be checked by being repeated in a different order.

517

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b, x) 2

ab\

0.3

0.r30000 00

-0.8 -0.7 -0.6

-1)-6.30239 72 -1)-4.39817 97 1-1)-2.45653 39 -2)-4.77093 -1)+1.54050 -1) 3.59664 -1) 5.69168 -1) 7.82601

96 87 50 81 37

0.0

( 0) 1.00000 00

0.1 0.2

( 0 1.22140 28 ( 0’ 1.44684 80

a\b

-1.0

-0.5 -0.4 -0.3 -0.2 -0.1 0.0

=0.2

0.2

0.1

0.6

-2) I-1) (-1 (-11

9.22415 1.86164 2.81785 3.79118

48 63 03 64

i’

-1 -1 -1 -11 I-1)

3.33333 3.95232 4.58166 5.22143 5.87174

0.7

0.5

0.4

33 64 34 72 11

(-1)

( 0) 1.00000 00

Table 13.1

( 0) 1.00000 00

0.8

(-1) 5.00000 5.46684 5.94088 6.42219 (-1) 6.91083

00 38 89 72 10

-1 1-11 (-1 -1 -1

6.00000 6.37527 6.75592 7.14199 7.53352

00 43 38 30 62

7.40685 28

(-1) 7.93056 84

( 0) 1.00000 00

( 0) 1.00000 00

0.9

1.o

(-1) 6.66666 67 7.96353 68 8.24393 73

-1 8.21990 25 1-11 8.46551 94

51 89 91 97 51

-1 9.40043 88 I-11 9.69842 13

-1 9.47678 92 t-11 9.73691 22

-1 9.53603 91 I-11 9.76677 40

-1 9.58333 69 1-11 9.79060 58

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

-1 1-11 -1 -1 [-11

8.27917 8.61453 8.95426 9.29839 9.64696

518

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b, x)

Table 13.1

x =0.3 0.I

ab\

0.2

-1.0

0)-2.00000 00

-0.5 -0.4

-1 -6.10043 44 1-11-3.06158 84

-1 1.99731 93 1-11 3.51517 11

( 0) 1.00000 00

( 0) 1.00000 00

0.3

0.4

0.5

-0.1 0.0

0.4

ab\

0.6

0.7

0.0

( 0) 1.00000 00

0.1

( 0 1.05560 11

0.6

( 0) 1.00000 00

( 0) 1.00000 00

1.34985 1.70931 2.07850 2.45757 2.84665

0.1

0.4 0.5

( 0) 1.00000 00

II 0 0 0 0

1.11226 1.17001 1.22885 1.28879

90 62

51 84

( 0) 1.34985 88

0) 1.41204 93

0.9

0.8

1.o

9.07807 88 9.53485 19

-1 9.19659 93 [-11 9.59485 17

[-11 9.64133 99

-1 9.36172 12 t-11 9.67839 44

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

-1 9.28846 71

1I

0 1.03645 08 01 1.07349 27

Ol 1.11113 16 0 1.14937 40 0 1.18822 61

519

CONFLUENT HYF'ERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b , x)

Table 13.1

x =0.4 ab\

0.2

0.1

0.3

90 0)-1.20063

-0.1

(-1) 5.25850

(-1)

1.46463 65

(-1

19

66

(-1)

7.66207

0.0

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

0.1

( 0) 1.49182 47

[

0 1.24182 32 01 1.49182 47

( 0 1.15892 34 ( 01 1.32283 59

( 0 1.11772 8 1 ( 01 1.23890 28

1.0

I;

( 0) 6.77113 1 2

0.7

0.9

0.8

-U.J

-0.2 -0.1

I-:\

-1 6.44112 7.11565 3 anar3 -1 8.52000 9.25039

I;\

.UVU>L

32 94 14 13 46

0.0

( 0) 1.00000 00

0.1 0.2

[ [(

0":;

0.5

1.59818 1.70731 1.81925 1.93404 2.05174

0 01 0 01 0)

1.07691 1.15580 1.23671 1.31966 1.40469

20 59 28 37 04

1.49182 1.58109 1.67254 1.76619 1.86208

47 90 59 84 99

6.96144 7.53923

1-1 'LI

iI:{

64 92 Q 1 9 1 7 L 35 8.73924 56 9.36191 40 ".*/A.

I U

( 0) 1.00000 00

1I 0 0 0 0 0

1.06537 1.13233 1.20091 1.27112 1.34299

37 62 13 31 62

1.41655 1.49182 1.56883 1.64759 1.72815

50 47 03 75 18

[I:\ [-ij -1)

Q 97117 I /.. I

u . /

9.02958 86 9.50948 02

( 0) 1.00000 00

( 0) 1 . 0 ~ 0 0 0 00

7.65220 44 8.10109 70

41

( 0) 1.00000 00

00 28 89 21 78

7.89289 2 1 8.29670 27 n.?ncrls 82 9.13049 86 9.56072 5 1

7.35049 77 7.85576 88

8.90289 30 9.44510 72

80 73 64 94 12

1.o 6.00000 6.36214 6.73238 7.11085 7.49764

-1.0 -0.9 -0.8 -0.7 -0.6

n q

( 0) 1.00000 00

0 01 0 0 0

0.6

-0.5 -0.4

0.5

0.4

[I:\

( 0

0 0 0 ( 0

[ (

( 0) 1.23364 74

520

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b, x )

Table 13.1

x =0.5 ab\

-1.0 -0.9 -0.8 -0.7 -0.6

0.0

0.1 0.2 0.3 0.4 0.5 0.6

:0.:9 1.0

( 0)-4.00000 00

(-1 -8.71196 18 (-1 -6.39608 65

(-1)-6.66666 -1 -5.31342 1-11-3.89475 (-1 -2.40912 (-21-8.54965

( 0) 1.00000 00

( 0) 1.00000 00

1

I

1

1 1

0 1.64872 01 2.32717 0 3.03607 0 3.77614 ( 0) 4.54811

13 78 92 69 35

0) 5.35272 38 0 6.19073 40 01 7.06291 26 0 7.97004 04 ( 01 8.91291 03

-Oo5 0.4

-1) 5.47357 40 1-1) 6.31897 nn 89

1.66666 2.37390 3.10765 3.86848 4.65693

1 00 (

I n-7-

1-131 i:i~i; G 9.03363 78

0.0

( 0) 1.00000 00

9

(-2)+7.35066 66 (-1) 1.90722 60

( 0) 1.00000 00

( 0) 1.00000 00

3.93567 68

( 0 4.36993 59 ( 01 4.82026 39

00

( 0) 1.00000 00

0 1.20798 34

0 2.37449 45 0 2.63441 32 01 2.90364 98 0 3.18240 09 0 3.47086 63

3.11399 83

0.00000

01 1.42416 39

0 1.64872 13 0 1.88183 81 ( 0 2.12369 98

01 3.51714 35

0.5

0.8

1

0 2.00868 23 0 2.19799 70 01 2.39370 49 0 2.59593 60 0 2.80482 21

0.9

1.o

67 35 94 36 33

-0.2 -0.1

-U.3

0 1.99359 02 0 2.35254 68 ( 0 2.72590 86

0.4

67 47 90 78 30

0.7

-0.8 -0.7 -0.6

-

0 1.31762 72 01 1.64872 13

0.6

1 [Ill (-1 1-11 -1)

A

( 0)-1.50000 00 ( 0)-1.30112 70 ( 0 -1.09161 33

86 89 85 47

0 -3.61201 01-3.20079 ( 0)-2.76573 ( 0)-2.30622

ab\

1:;

0.3

0.2

0.1

!-I

-1 6.13943 38 6.86349 09 7

1’1’1’1’1

/ I

( 0) 1.00000 00

1.08465 1.17189 1.26178 1.35435 1.44966

27 67 10 51 91

1.54777 40 1.64872 13

( 0) 1.00000 00

0 0 0 0 ( 0

[

0 1.47291 64 01 1.55963 60

( 0) 1.00000 00

( 0) 1.00000 00

52 1

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION Rf(a, b, x)

Table 13.1

x =0.6 a\b

( 0 -2.00000 ( 01-1.77497

-? .s -0.7 -0.6

( 0)-5.00000 0)-4.56442 0 -4.09525 ( 51-3.59141 ( 0)-3.05183

00 36 03 57 34

[ (

-0.1

0 -2.47539 01-1.86097 0 -1.20740 -11-5.13527 (-1)+2.21866

54 11 73 80 89

-1 1-1 -2 [-1 (-1)

-1.0

0.0

( 0) 1.00000 00 1.82211 88

6.63788

-0.5 -0.4 -0.3 -0.2 -0.1

0.0

00 51 09 76 35

( 0) 1.40083 1.82211 2.26441 2.72828 3.21432

55 88 16 58 45

[ [ (

( 0) 3.72312

11

0.7 -1 [-1 (-1 (-1 (-1

-1 4.47097 1-11 5.48800 (-1) 6.54835 -1 7.65309 1-11 8.80327

05 20 72 05 45

-1 5.28944 -11 6.16039 [(-1) 7.06609

( 0) 1.00000 00

( 0)-1.00000 ( - 1 -8.45926 -11-6.82397 1-1)-5.09139 (-1)-3.25877

( 0) 1.00000 00

00 80 03 99 62

-2) -1) -1 -11

00 83 51 65 96

0.5

0.4

( 0) 1.00000 00

0.00000 8.15612 1.66954 2.56274 3.49622

-1.0 -0.7

04

0)-1.53457 0 -1.27832 01-1.00575

0.6

nb\

0.3

0.2

0.1

(-1)-5.00000 (-1 -3.81848 (-1 -2.57117 (-1 -1.25627

00 50 79 00 81

( 0) 1.00000 00

0 0 0 0 0

(-1)-2.00000 -1)-1.03687 1-3)-2.46606 -1)+1.03792 1-1) 2.15219

00 14 50 44 91

(-1) 8.54662

21

( 0) 1.00000 00

( 0) 1.82211 88 0) 2.00672 5 1

1.o

0.9

0.8

00 69

-1) 3.33333 33 1-1) 3.89744 84

-1 4.00000 00 1-11 4.51255 49

63 00 56 -1 8.00737 79 (-1) 8.98506 53

-1 8.27114 95 1-11 9.12029 84

-1 8.47485 87 1-11 9.22465 40

I-11

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

2.50000 3.12786

-1 8.63672 59 9.30751 06

522

CONFLUENT HYPERGEOMETRIC FUNCTIONS

COYFLUENT HYPERGEOMETRIC FUNCTION M(a, b,

Table 13.1

x)

x =0.7 ab\

-1.0 -0.9 -0.8 -0.7 -0.6

[

0)-6.00000 00 0)-5.52819 79

( 0 -2.50000 00 ( 01-2.25396 47

-0.5 -0.4 -0.3 -0.2 -0.1 0.0

( 0) 1.00000 00

( 0) 1.00000 00

0.4

0.3

0.2

0.1

0 -1.33333 01-1.16362 -1)-9.81007 -1 -7.85028 (-11-5.75241

33 83 11 60 82

0.5

-7.50000 00 -6.19090 30

(-1 -3.51185 70 -1 -1.12388 92 [-l/+1.41630 28 -1 4.11364 25 (-11 6.97316 13

I-1 +1.76203 27 -11 3.65553 75

( 0) 1.00000 00

( 0) 1.00000 00

-1 -4.00000 00 1-11-2.92768 78

(-3 -2.63083 59 -1 5.65746 78 [-1 7.77115 48 ( 0) 1.00000 00

8.10395 56 "0:'; [ 010 9.54222 25

ab\

0.6

1;:;

-1 1.22710 86 [-11 2.30054 51

[

-0.5 -0.4 -0.3 -0.2 -0.1

-1) 3.43109 4.62042 -1 5.87022 1-11 -1) 7.18224 [-1) 8.55823

1-d

0.0

52 36 82 16 13

( 0) 1.00000 00

1.15093 1.30882 1.47385 1.64621 1.82611

0.8

0.7

86 66 50 90 74

1.o

0.9

1-d

1-d

-1 4.40977 87 -1 5.42816 47 6.49502 91 1 7.61170 97 [Ill 8.77956 99

-1 5.13967 66 -1 6.02994 98 6.96004 90 1 7.93103 40 [Ill 8.94398 42

1-d

1-11 9.07068 09

I-1) 9.17114 12

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

10

1

1

1

1

1.12744 17

01 1.26042 67

1.39910 20 0 1.54361 79 0 1.69412 73 01

0 0 0 01 ( 0)

1.11002 1.22457 1.34377 1.46774 1.59660

02 33 57 58 44

-1 5.70431 32 -1 6.49501 40 7.31906 85 -1

0 0 0 01 ( 0)

8.17733 33

1.09661 1.19701 1.30129 1.40953 1.52184

96 89 20 43 32

-1 6.15369 36 -1 6.86479 13 7.60426 03 -1) 8.37280

0 0 0) 0) 01

1.08601 1.17522 1.26772 1.36357 1.46286

46

24 70 07 19 04

523

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b, x)

Table 13.1

x =0.8 0.2

0.1

d'\6

0.3

0.4

0.5

-0.6

0.0

( 0)+1.0000(1 00

0.4 0.5

[

n\b -1.0 -0.9 -0.8 -0.7 -0.6

0.6

I

-1 -3.33333 -1 -2.33826 ( - 1 -1.27465 -2 -1.40115 [-j +1.06779

33 62 48 64 15

-1 6.68394 1 0 1-11 8.29734 28

( 0) 1.00000

0.8

0.7

-0-3

0.0

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

0) 6.45617 50 0) 8.06281 37

-0.5 -0.4 -0.2 -0.1

( 0) 1.00000 00

00

I1-d' -1 -2 -2 -1

-1.42857 -5.57356 +3.69102 1.35264 2.39517

I1

-1 3.49860 -1 4.66490 [-11 5.89611 -1 7.19428 -1 8.56152

14 94 15 99 31 15 92 50 36 59

( 0) 1.00000 00

0.9

1.o

-2 -1

(-1

fIi/

-1 8.09652 62

7.57323 29 8.75747 79

-1) 7.86514 37 1-1) 8.90826 3 1

I-11 9.02766 05

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

[I:)

524

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b, x)

Table 13.1

x =0.9 a\b

0.1

-1.0 -0.9

( 0)-8.00000 ( 0)-7.49259 0)-6.90878

-4.65980 -3.73067 -2.70466 -1.57731 -3.44010 0.0

( 0)+1.00000

0.3

0.2

00 77 25

55 11 65 62 11 00

( 0 -3.50000 ( 01-3.22852

00 60

-2.00000 -1.80907

0.5

0.4 00 26

[

0 -1.25000 0{-1.10046

00 05

(-1)-8.00000 (-1)-6.76001

00 98

1

( 0 0 -1 -1 (-1

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

0 1.45345 52 01 1.93955 77

5.07749

a\b -0.9 -0.8 -0.7 -0.6

0.6 -1) -5.00000 -1 -3.93506 -11-2.78312 1-1)-1.54071 -2) -2.04284

00

0.8

0.7

1

0.0

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

0.1 0.2 0.3 0.4 0.5

( 0

0) 1.17668 1.36339 1.56347 0 1.76826 ( 01 1.98713

( 0)

1

1 .o

00 44 29 44 74

-0.1

1.21023 01 1.43307 1.66896 O\ 0 1.91836 ( 0) 2.18175

0.9

31 07

10 37 01

1

82 71 09 25 34

2.45960 31 2.98095 2 1 3.26095 72

-1) 7.80706 95

7.53728 29 8.73679 1 4

[-1) 8.87657 20

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.64871 85

525

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b, x)

Table 13.1

x=l.0

ab\

0.0

0.1

0.3

0.2

-7.19487 27 -6.38931 44

( 0 -3.04197 32 ( 01-2.62968 42

( 0)+1.00000 00

( 0) 1.00000 00

0.5

0.4

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

0) 2.71828 18 0 8.84990 62 1) 1.12452 68

:; [

ab\

0.6

0.7

-2.50000 -1.60990 -6.48146 +3.88236 1.50229

1;:

A

-

\-!I

\-?I

-3 +6.30910 70 1.77225 36 3.614G3 67

-1

-1 -1 1-11

-0.1 0.0

0.1 0.2 0.3 0.4 0.5

( 0) 1.00000 00

1 I/ [

0 1.24339 88 0 1.50311 03 1.77978 05 0 2.07407 40 0 2.38667 38

( 0) 1.00000 00

1.o

0.9

0.8

00 29 54 65 88

2.69717 87 3.97610 c 1a3-n nn 35 J.JVLJ7 U0 6.79945 04 8.35078 67

( 0) 1.00000 00

II

-1)-1.11111 -2 -3.01549 -2 +5.68299 -1 1.50083 -1 2.49853

\-?I -1

\-A1

-1 -1

11 81 01 68 18

3.56392 05 4.69960 88 IZ . 7 nUnU oL- 7 J I

9U 0 J

7.19266 55 8.55560 76

( 0) 1.00000 00

Ii

-2) -1 -1 -1

7.43386 1.53827 2.38663 o*ooooo 3.29050

23 23 42 15 O0

4.25195 5.27314 I - A j 6.35625 -1 7.50355 -1 8.71734

83 45 70 07 01

\-:I

-1

( 0) 1.00000 00

526

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b, x )

Table 13.1

x =2.0 ab\

1:; -0.8 -0.7 -0.6

1

0.1

1 -1.90000 11-1.94803 1 -1.95774 11-1.92363 ( 1)-1.83976

00 05 57 39 09

( 0)+1.00000

00

0) 1) ( 1) ( 1 ( 1,'

61 73 96 44 43

0.2

0.3

80

67 46 85 69 57

[

( 01-8.30798

-5.66666 -5.67351 -5.57239 -5.34952 ( 0)-4.99011

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000 00

3.94227 7.38905 1.13864 1.59833 2.12317

09 61 24 25 23

(i

65 25 61 79 54

0.5

0.4

0 -4.00000 01-3.96130 0 -3.84746 0 -3.64939 0 -3.35738

ii

00 19 13 40 15

-0.2 0.0

:"0 [ 0.3 0.4 0.5

ab\

7.38905 1.49320 2.37378 3.39223 4.56085

0.6

1I 0 0 1 1 ( 1

0.7

2.82379 4.94472 7.38905 1.01846 11 1.33611

( 0)-2.33333

33

( 0)-1.85714

29

-0.2 -0.1

(-1)+3.33718

60

(-1)+4.51527

65

01 ( 0) 1.00000 00

( 0) 1.00000 00

(-1) 5.37263

1.o

0.9

0.8

-1.0

0.0

0 0 0 1

41

( 0) 1.00000 00

(-1)+2.51516 (-1) 6.02027

( 0) 1.00000 00

76

13

1-1) 6.52400 38

( 0) 1.00000 00

( 0) 1.00000 00

527

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION M(u, b, X)

Table 13.1

x =3.0 ab\

00 11 78 55 65 -0.5 -0.4

[

1 -3.97869 11-3.72604

0.3

0.2

0.1

1

1 -1.40000 11-1.57397 1 -1.71028 1 -1.79849 11-1.82694

00 85 23 94 57

[

( 0)-5.60854

66

(

( 0)+1.00000

00

( 1))+1.0000

[( [

0 -9.00000 0 -9.93407 -1.06346 -1.10419 -1.10887

0.4 00 08 98 34 39

( 0 0 0 0 0

00

( 0)+1.00000

0.5

[ [

07 95

-0.2 0.0

( 0)+1.00000 00

0.1 0.2 0.3 0.4 0.5

1) 1 11 ( 2 ( 21

[(

2.00855 4.41540 7.38953 1.10064 1.53485

37 99 06 09 39

00

( 0)+1.00000

00

( 0) 9.47722 60 ( 1 ) 2.00855 37

2 2.05059 1 4 21 2.65765 56 1.0

( 2) 5.13805 80

nb\

0.6

-1.0 -0.9

( 0)-4.00000 ( 0 -4.22698

( 21 2.22239 0 1

0.7

0.8

01-4.35776 01-4.37205 0 -4.24734

00 22 62 21 55

[ (

0 -3.95879 01-3.47899 0 -2.77784 ( 0)-1.82229 (-1) -5.76188

09 58 38 72 60

0)-3.06922 0 -2.65319 0 -2.06432 ( 0)-1.27772 ( - 1 ) -2.66178

11

34 12 89 88 30

0.0

( 0)+1.00000

00

( 0)+1.00000

00

0.1

( 0) 2.94937 02

0.5

5.31885 8.15947 1.15266 ( 1) 1.54802

1;:; -0.6

-Oo5 -0.4 -0.3 -0.2 -0.1

1

34 04

06 96

( 0) 2.55311 64 0 4.42829 20 01 6,66364 6 1 ( 0 9.30049 38 ( 11 1.23835 54

[

1.o

0.9

( 0)+1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 0) 0) ( 0) ( 1)

( 0) 0 01 ( 0 ( 0

( 0 5.69204 1 8 ( 01 7.38010 1 3

I

2.27097 3.79559 5.60309 7.72517 1.01960

84 01 84 18 38

I

2.06241 3.32891 4.82245 6.56784 8.59185

49 38 42 35 66

528

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b, x )

Table 13.1

X

a\b

0.1

1

-1.0 -0.9

0.2

1 -3.90000 1 -5.28985

00 40

0.0

( 0)+1.00000

00

0.1 0.2 0.3 0.4 0.5

( 1) 2 21 2 2

5.45981 1.25936 2.18189 3.34927 4.80147

50 21 72 25 67

2 21 3 3) 3)

6.58320 8.74427 1.13401 1.44322 1.80888

17 45 20 61 49

=4.0

0.3

0.5

0.4

-0.6

a\b

1

I

( 0)+1.00000 2.40818 5.45981 9.38520 1.43304 2.04591

00

08 50 09 83 31

0.7

0.6

1

( 0)+1.00000

00

1 1.44217 35

11 3.20473 65 1 5.45981 50 1 8.28815 42 2 1.17799 11

1

( 0)+1.00000 00 0

00

9.87867 7 1

11 2.14598 1 8 1 3.61972 65 1 5.45981 50 1 7.72277 23

1.o

0.9

0.8

( 0)+1.00000

-1.0

0.0

( 0)+1.00000 5.73952 1.18390 1.85174 2.90181 4.06117

00

( 0)+1.00000

56 73 11 11 30

I1

4.68094 9.38676 1 1.52787 1 2.25363 10 3.13582 0

00

79 76

90 21 01

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

529

CONFLUENT HYPERGEOMETRIC FUNCTIONS

Table 13.1

CONFLUENT HYPERGEOMETRIC FUNCTION M ( a , b, x)

x =5.0 ab\

0. I

0.2

0.3

0.5

0.4

-0.6

0.0

( 0)+1.00000' 00

ab\

0.6

1

-0.1

1 -1.73534 19 1 -1.72563 11 1 -1.57953 99 -1.2586e 94 ij-7.15818 24

0.0

( 0)+1.00000 00

-0.5 -0.4 -0.3

-'"[

( 0)+1.00000 00

0.7

( 0)+1.00000 00

( 0)+1.00000 00

0.8

( 0)+1.00000 00

1.0

0.9

1 ( 0)+1.00000 00

( 0)+1.00000 00

0 -8.54492 0 -8.28701 4-7.38548 -5.671% 01-2.95155

28 58 98 55 22

( 0)+1.00000 00

6.43982 1.36614 2.29989 3.48308 4.95851

88 90 34 09 46

0 -7.01437 97 0 -6.74333 16

1

i-?.W!$?

!?

U

u.4.

-Y.~uuYO

01-2.24261 78

( 0)+1.00000 00

1

0 5.42870 1.12729 1 1.87930 1 2.82840 1 11 4.00784

50 02 66 13 46

530

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b,

Table 13.1

x)

x=6.0

ab\

0.1

0.2

0.3

0.4

0.5

1 -1.90000 00 11-3.88390 81

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

ab\

( 0)+1.00000 00

( 0)+1.00000 00

0.7

0.6

( 0)+1.00000 00

0.8

( 0)+1.00000 00

0.9

( 0)+1.00000 00

1.o

-0.8 -0.6

0.0

( 0)+1.00000

00

( 0)+1.00000

2.19683 5.13440 9.10486 1.43316 2.10737

00

( 0)+1.00000 00

( 0)+1.00000 00

( 0)+1.00000 00

531

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b,

Table 13.1

x)

x = 7.0 ab\

0.1

0.3

0.2

0.5

0.4

-0.2 0.0

ab\

0.0

( 0)+1.00000 00

0.6

( 0)+1.00000 00

( 0)+1.00000

00

0.7

( 0)+1.00000

( 0)+1.00000

00

0.8

00

-7.75000 -1.53927 -2.33863 -3.12617 -3.83826

00 06 78 60 01

( 0)+1.00000

00

( 0)+1.00000

I

00

( 0)+1.00000

1.0

0.9

0 -6.77777 I1-1.27O12 1 -1.88526 1 -2.48676 1)-3.02562

78 46 21 78 11

( 0)+1.00000 00

4.47895 6.17802 8.31248 1.09663 1.42364

00

79 12 87 32 54

( 0)+1.00000

2) 2 2 2 3

/I

3.47456 4.78318 6.42409 8.46076 1.09663

00

13 84 85 16 32

532

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYF'ERGEOMETRIC FUNCTION M(a, b, x)

Table 13.1

x =8.0 ab\

0.1

0.2

0.3

0.5

0.4

-0.5

-0.1 0.0

ab\

( 0)+1.00000 00

0.6

( 0)+1.00000 00

0.7

( 0)+1.00000 00

( 0)+1.00000 00

( 0)+1.00000 00

0.8

-9.38477 -1.03097 -1.03347 -3.iXijb3 t\-5.76904

-0.2 0.0

( 0)+1.00000 00

( 0)+1.00000 00

1.o

0.9

69 46 63 22

74

( 0)+1.00000 00

12

1 -7.22077 10 1 -7.89678 13 d-7.88488 72 I

-6.84858

28

I !j-B'!~~~ 1 -5.66582 71 1 -6.16743 32

3;:

L

-J.iJUJJA

>U

1j-4.36332 11

1j-3.36181 13

( 0)+1.00000 00

( 0)+1.00000 00

533

CONFLUENT HYPERGEOMETRIC FUNCTIONS

CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b, x)

Table 13.1

5 =9.0

ab\

0.3

0.2

0.1

0.4

0.5

-0.1 0.0

ab\

-1.0 -0.9 -0.8 -0.7 -0.6

( 0)+1.00000 00

1

( 0)+1.00000 00

0.6

0.7

1 -1.40000 00 21-1.53831 1 -7.88310 87 88 2 -2.35259 85 2 -3.17089 67

i

1 -1.18571 43 21-1.1*401 1 -5.86101 35 55 2 -1.70516 69 2 -2.28631 95

( 0)+1.00000 00

( 0)+1.00000 00

( 0)+1.00000 00

1

I

1

0.8

1 -1.02500 00 1 11-8.46300 -4.49394 77 10 2 -1.27296 76 2 -1.69747 84

[ 0.0

( 0)+1.00000 00

( 0)+1.00000 00

0.6 0.7 0.8 0.9 1.0

1

1

3

8.10308 1.15389 4 1.60085 4 2.17532 441 2.90602

39 32 54 51 06

3

5.69778 8.10308 4 1.12277 4 1.52385 431 2.03337

22 39 41 32 24

( 0)+1.00000 00

0.9

0 -9.00000 00 l1j -3.53363 b r j 0 0 7 44 88

1 -9.73476 07 2 -1.29066 47

2 -1.59711 34 21-1.04195 05

( 0)+1.00000 00

[

1.o

0 -8.00000 00 11-5.14354 1 -2.83797 81 17

1 -7.59652 04 2 -1.00113 60

2 -1.22131 75 11-7.94021 75

( 0)+1.00000 00

534

CONFLUENT HYPERGEOMETRIC FUNCTIONS

Table 13.1

CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b, x) 2

ab\

=10.0

0.3

0.2

0.1

1)-4.90000 3 -1.04774 31-2.26606 3 -3.65315 31-5.11412

00 98 51 21 18

4 -1.66086 4 -1.93829 4 -0.4:; 4!-2.05153 4 -1.88191 -0.1 4 -1.26894

1I 1

19 90 93 87 82

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000 00

2.20264 5.69563 1.09330 1.84869 2.90713

66 19 93 24 00

I

30 66 41 87 64

[

-0.5

0.0

a\b

-1.0 -0.9 -0.8 -Oo7 -0.6

0.0

1 1

0.6 1 -1.56666 21-1.59656 2 -3.32180 21-5.25566 2)-7.26224

67 19 59 60 96

( 0)+1.00000 00

4) 4 41 4 41

2.20264 3.17106 4.44649 6.10528 8.23940

66 89 42 43 35

3 41 4) 4 51

11

8.52983 2.20264 4.22272 7.13160 1.12016

0.5

0.4

I

3 4 41 4 4

0.7

4.46140 1.15043 2.20264 3.71537 5.82887

89 71 66 68 58

0.8

40 23 45 83 59

II !

( 0)+1.00000

00

( 3 31 41 4 4

71 74 73 66 55

1

2.65569 6.83804 1.30747 2.20264 3.45147

3 -1.33814 3 -1.54831 3{-1.62617 3 -1.48115 2 -9.91916

36 94 57 94

( 0)+1.00000

00

3 3 31 4 4

1.70399 4.38084 8.36496 1.40739 2.20264

35

66 00 74 54 66

1.o

0.9

1

1 -1.32857 2 -1.15824 2 -2.38103 2 -3.74603 ( 2)-5.15669

14 17 41 08 48

1 1 2 2 ( 2

( 0)+1.00000

00

( 0)+1.00000

1.53174 58 2.20264 66

-2.06370 -2.39329 -2.51877 -2.29844 -1.54205

00

4 1.09611 92 41 1.57436 46

( 0)+1.00000

00

( 0)+1.00000

00

14. Coulomb

Wave Functions

MILTON ABRAMOWITZ l

Contents Mathematical

Properties

Page 538 538 539 539 539 540 542

....................

14.1. Differential Equation, Series Expansions ......... .......... 14.2. Recurrence and Wronskian Relations ................ 14.3. Integral Representations 14.4. Bessel Function Expansions ............... ................ 14.5. Asymptotic Expansions 14.6. Special Values and Asymptotic Behavior ......... ...................... Numerical Methods 14.7. Use and Extension of the Tables .......................... References Table 14.1. Coulomb Wave .......................... l
q=.5(.5)20,

of

Order

Zero

(.5$&20,

5s +iq) 1 . . . . .

l

. . . . . . . . . .

l

554

6s

The author wishes to acknowledge formulas and tables.

1 National

544 546

p=1(1)20,

Table 14.2. CO(q)=e-@q(r(l n=0(.05)3.

Functions

543 543

............

Bureau of Standards

the assistance of David S. Liepman

in checking the

(deceased).

537

I

546

COULOMB WAVE FUNCTIONS

COULOMB WAVE FUNCTIONS OF ORDER ZERO

Table 14.1

Fo(~,P) 2

1

ll\p

0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 3. 5 4. 0 4. 5 5. 0

4

3

- 4) 2.8622

5. 5 6. 0 6. 5 7. 0 7. 5 8. 0 8. 5 9.0 9. 5 10. 0

-5 -5 -6

0. 5 1. 0 1. 5 2. 0 2. 5

I

3.9115 1.4023 4.9481 - 6 1.7207

- 9) 7.8306

10. 5 11.0 11.5 12. 0 12.5 13. 0 13. 5 14. 0 14. 5 15. 0 15. 5 16. 0 16. 5 17. 0 17. 5 18. 0 18. 5 19. 0 19. 5 20. 0

-

9 -10 -10

-18

I

2.3567 7.0332 2.0826

9 4708

-181 2:3372

- 1) 5.9292

3. 0

3. 5 4. 0 4. 5 5. 0

( - 5j 6.7615

-

5. 5 6. 0

3) 1.1690

b. 5

7. 0 7. 5 8. 0 8. 5 9. 0 9. 5 10.0 10. 5 11.0 11.5 12. 0 12.5 13. 0 13. 5 14. 0 14. 5 15. 0 15. 5 16. 0 16. 5 17. 0 17. 5 18.0 18. 5 19. 0 19. 5 20.0

5

-11 -111 -12 -13 -13 -14 -14 -15) -161 (-16)

4 5150 1:lSol 3.0676 7.9334 2.0420 5.2322 1.3350 3.3929 8.5905 2.1673 -15) 4.0861

-19 1 0461 -20) 217464

For use of this table see Erarnplrs 1-3.

-18 4 3698 -181 112090

-12 -13 -13 -14 -14 -15 -15 -16 17 17

1.6705 5.0433 1.5132 4.5133 1.3386 3.9490 1.1590 3.3848 9.8388 2.8470

547

COULOMB WAVE FUNCTIONS

COULOMB WAVE FUNCTIONS OF ORDER ZERO

Talde 14.1

GdriP 1 .

1

cl\p 0.5 1. 0 1. 5 2.0 2.5 3. 0 3. 5 4.0 4.5 5. 0

2

I

3

4

-

5 1 9 3493 1118: 9841

1

1 1.0423 1 2.1964 1 4.9434

5. 5 6.0 6.5 7. 0 7. 5 0.0 8.5 9. 0 9.5 10.0

3 8 2458 41 212097

10. 5 11.0 11. 5 12. 0 12.5 13.0 13.5 14. 0 14.5 15. 0 15. 5 16. 0 16.5 11.0 17.5 18.0 18.5 19.0 19. 5 20. 0

( 1 5 j 1;1464

0. 5 1.0 1.5 2.0 2.5 3.0 3. 5 4.0 4.5 5.0

4 -8 0354 51 -2: 9409

5. 5 6.0 6. 5 7. 0 7.5 8.0 8.5 9.0 9.5 10.a 10. 5 11.0 11. 5 12. 0 12. 5 13. 0 13. 5 14.0 14.5 15.0 15. 5 16.0 16.5 17. 0 17.5 18.0 18. 5 19.0 19. 5 20.0

15 -3 0026 161-1: 2434 1 6 -5.0296 171 -2.0399

*see page XI.

13 -1 4017 131 -4: 9720

548

COULOMB WAVE FUNCTIONS

Table 14.1

COULOMB WAVE FUNCTIONS OF ORDER ZERO

?Ap 0. 5 1.0 1. 5 2. 0 2. 5 3. 0 3. 5 4. 0 4. 5 5. 0

6

-

0

11 1) - 1 - 1 -2

I

15. 5 16. 0 16. 5 17. 0 17. 5 18.0 18. 5 19. 0 19. 5 20.0

0. 5 1.0 1.5 2. 0 2.5 3. 0 3. 5 4. 0 4. 5 5. 0

15. 5 16. 0 16. 5 17. 0 17.5 18.0 18.5 19. 0 19. 5 20. 0

-

1 1 9214 11 1:0100

- 5) 1.5930

-10 6 4858 -101 2:1306 (-11) -11 -12 -12 -13 -13 -14 -14 -15 1-15!

I

6.9438 2.2461 7.2135 2.3009 7.2918 2.2965 7.1900 2.2382 6.9296 2;1342

(-101 5.0935

I

-11 3.2487 -11 1.1173 -12 3.8154 -12 1.2942

- 1 3. 0193

-

I

1 2.1173 1 1. 3148

5. 5 6. 0 6. 5 7. 0 7. 5 8. 0 8. 5 9. 0 9. 5 10.0 10.5 11.0 11.5 12. 0 12.5 13. 0 13. 5 14. 0 14. 5 15. 0

1)+1.0351

1 1633 8:3763 5.2251 2.9445 1. 5362 7.5384

5. 5 6. 0 6.5 7. 0 7. 5 8. 0 8. 5 9. 0 9. 5 10. 0 10. 5 11. 0 11.5 12. 0 12. 5 13. 0 13. 5 14. 0 14. 5 15. 0

9

-

-

-

21 7:9960

-

9) 5.4422

1 1 3640

9) 2.7940

-

10 1)+9.3919

549

COULOMB WAVE FUNCTIONS

COULOMB WAVE FUNCTIONS OF ORDER ZERO “P 0.5 1. 0 1. 5 2. 0 2. 5 3. 0 3. 5 4. 0 4. 5 5.0 5. 5

6.0 6. 5 7. 0 7.5 8.0 8.5

9.0 9. 5 10. 0 10. 5 11.0 11.5 12.0 12. 5 13.0 13. 5 14.0 14. 5 15. 0 15. 5 16. 0 16.5 17. 0 17.5 18. 0 18. 5 19. 0 19. 5 20.0

0. 5 1. 0 1.5 2. 0 2. 5 3.0 3.5 4. 0 4. 5 5.0 5.5 6.0

6.5

7. 0

1. 5 8. 0 8. 5

9. 0 9. 5

10.0

10. 5 11.0 11.5 12. 0 12.5 13. 0 13. 5 14.0 14.5 15. 0 15.5 16. 0 16.5 11.0 17.5 18.0

18.5 19.0 19.5

20.0

6

7

Gn(vrP)

8

9

Table 14.1

10

550

COULOMB WAVE FUNCTIONS

Table 14.1 s\p

COULOMB WAVE FUNCTIONS OF ORDER ZERO

11

14

0. 5

1. 0 1.5 2. 0 2. 5 3. 0 3. 5 4. 0 4.5 5.0 5. 5 b. 0 6.5 7.0 7.5 8.0

8.5 9. 0

9.5 10.0 10. 5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14. 5 15.0 15. 5 16. 0 16.5 17.0 17.5 18.0 18.5 19.0 19. 5 20.0

0. 5

1. 0 1.5 2.0 2.5 3. 0 3. 5 4. 0 4.5 5.0 5.5 6. 0 6. 5 7. 0 7.5 8. 0 8.5 9.0 9.5 10.0 10. 5 11.0 11.5 12. 0 12.5 13. 0 13.5 14. 0 14. 5 15. 0 15. 5 16. 0 16.5 17. 0 17.5 18. 0 18.5 19. 0 19.5 20.0

f - 31 5.3087

( - 6) 3; 0731

-

9) 1.0052

I

- I\+& 7670

15

551

COULOMB WAVE FUNCTIONS

COULOMB WAVE FUNCTIONS OF ORDER ZERO ?\p

Table 14.1

15

14

11

0.5 1.0 1.5 2.0 2.5 3. 0 3.5 4. 0 4.5 5. 0

I

- 1 -4.6254

0 -1.1612

0 -1.2413 f

5. 5 6. 0 6. 5 7. 0 7. 5 8. 0 8. 5 9.0 9. 5 10.0

.

1 8 0595 . 1111: 2194

0) 1.1100

10. 5 11. 0 11.5 12. 0 12.5 13.0 13. 5 14. 0 14. 5 15. 0 4) 9.7988

15. 5 16. 0 16. 5 17. 0 17.5 18.0 18. 5 19. 0 19. 5 20. 0

0. 5 1. 0 1.5 2.0 2.5 3. 0 3. 5 4.0 4.5 5. 0 5. 5 6. 0

6. 5 7. 0 7.5 8. 0 8.5 9. 0 9.5 10.0 10.5 11. 0 11.5 12. 0 12.5 13. 0 13. 5 14. 0 14. 5 15. 0 15. 5 16. 0 16.5 17.0 17.5 18. 0 18. 5 19.0 19. 5 20.0

I

6 4.0107 6 9.7253 7 2.3033

8) 3.4512

-

1) -4.6958

I - 11-4.4342

- 1 -5.5863 - 1 -4.9764 - 1 -5.7431

I

( - 1)-6.8002

I:

1 -6 4050 1\-5:4165

I

1 1 2 2 2

-3.6758 -7.5239 -1.5749 -3.3666 -7.3407

I

4 -1.1531 4 -2.6329 4 -6.0946 5 -1.4291 5 -3.3924 5 8 1473 6111: 9785 6 -4.8557 7 -1.2038 7 -3.0133

I

552

COULOMB WAVE FUNCTIONS

COULOMB WAVE FUNCTIONS OF ORDER ZERO

Table 14.1 \I 0.5

17

16

p

(

&(%P)

-

0)t1.0105

1. 0 1.5 2. 0 2. 5 3. 0 3. 5 4.0 4.5 5. 0

20

19

18 1)-2.6356 0 +1.0298

I

- 2 -4.2659 0 -1.0610

5. 5 6. 0 6. 5 7. 0 7. 5 8. 0 8.5

9. 0 9. 5 10.0

( - 1) 1.8899

( - 1) 3.2134

( - 1) 5.0960

( - 1) 7.5308

0) 1. 0343

- 1 5 2163 - 1) 314376 - 1) 2.1696

10.5 11.0 11. 5 12. 0 12.5 13. 0 13. 5 14. 0 14.5 15. 0 15. 5 16. 0 16. 5 17. 0 17. 5 18.0 18.5 19.0 19. 5 20.0

0. 5 1.0 1. 5 2.0 2.5 3. 0 3. 5 4. 0 4.5 5. 0

( - 1)-7.4641

I

5. 5 6. 0 6.5 7.0 7. 5 8. 0 8. 5 9. 0 9.5 10.0 10.5 11.0 11.5 12.0 12. 5 13. 0 13. 5 14.0 14. 5 15. 0 15. 5 16. 0 16. 5 17. 0 17. 5 18.0 18. 5 19. 0 19. 5 20.0

-1

i:- I - I

2.9470 1 2.6893 1 2.1715

-1 -- ;{ - 21 L

1: :I

- 4 1.1789

(-

I-

(-

:I

((-

2.4992 I 5233 5 ) 1. '101 6) 5.!'69 6 2 -120 7 9 ,588 7 4.14C9 7 1.7529 8 7.3379

j

-

1 0912 7:0640 4.3620 2.5860

1 1.6072 1 1.1118 2 7.2792

-

2)-8.5571

- 1 t4.2976 - 1 -1.7601 - 1 -6.1873 - 1 7 4462 - 1116: 0113 - 1 -3.2623 - 2 -4.9686 - 1 t1.5282 - 1 2.6076 - 1 2.8881 - 1 2.6513 - 1 2.1673 - 1 1.6300 - 1 1.1487 -- 22 47.6757 9026 - 21 3:0112 - 2 1.7867 - 2 1.0279 - 3 5.7512 - 3 3.1370 - 3 1.6717

i

I

I - I

3 1 1861 4) 5:9443 4 2 9194

- 41 1:4071

4 8.7182

( - 7) 7.1576

553

COULOMB WAVE FUNCTIONS

COULOMB WAVE FUNCTIONS OF ORDER ZERO q \p 0.5 1.0 1.5 2.0 2. 5 3. 0 3. 5 4. 0 4. 5 5. 0

16

19

Table 14.1 20

I: 1

1 -7.7111 1 +8.3065

( - 1)-7; 6338

5. 5 6. 0 6.5 7. 0 7. 5 8.0 8. 5

9. 0 9. 5 10.0 10.5 11.0 11. 5 12. 0 12. 5 13. 0 13. 5 14. 0 14. 5 15. 0 15. 5 16. 0 16. 5 17. 0 17. 5 18.0 18. 5 19. 0 19. 5 20. 0

0. 5

1.0 1. 5 2. 0 2. 5 3. 0 3. 5 4. 0 4. 5 5. 0

+6.6241

5. 5 6. 0

6. 5 7. 0 7. 5 8. 0 8. 5 9. 0 9. 5 10. 0 10.5 11. 0 11. 5 12. 0 12. 5 13. 0 13. 5 14. 0 14. 5 15. 0 15. 5 16. 0 16. 5 17. 0 17.5 18. 0 18. 5 19. 0 19. 5 20. 0

3) -4.0483

554

COULOMB WAVE FUNCTIONS

Table 14.2

CO(?)

=e-+"o

lW+h)l

17

CO (1)

1.00 1. 05 1.10 1.15 1.20

-1)l. 08423

0.10 0.15 0. 20

CO (1) 1.000000 0.922568 0.847659 0.775700 0.707063

0. 25 0. 30 0. 35 0.40 0.45

0.642052 0.580895 0.523742 0.470665 0.421667

1.25 1. 30 1.35 1.40 1. 45

2.25 2.30 2.35 2.40 2.45

0. 50 0.55 0. 60 0. 65 0. 70

0.376686 0.335605 0.298267 0.264478 0.234025

1. 50 1.55 1. 60 1.65 1.70

2. 50 2.55 2.60 2. 65 2.70

0. 75 0. 80 0. 85 0. 90 0.95

0.206680 0.182206 0.160370 0.140940 0.123694

1.75 1.80 1.85 1.90 1.95

2.75 2.80 2.85 2.90 2.95

1. 00

0.108423

2.00

17

0. 00 0. 05

For In r(l+iy), see Table 6.7.

(-2)6.33205

(-3)6.61992

1

2. 00 2. 05 2.10 2.15 2.20

3.00

CO (1)

-3)3.70402

(-4)3.50366

15. Hypergeometric Functions FRITZ OBERHETTINGER

Contents

..

.. . . . theArgument. . . . . . . . . . . . . . . . . . . . Differentiation Formulas and Gauss’ Relations for Contiguous Functions. . . . . . . . . . . . . . . . . . . . . .

Mathematical Properties . . . . . . . . . . . . . 15.1. Gauss Series, Special Elementary Cases, Special Values of 15.2.

15.3. Integral Representations and Transformation Formulas . . . 15.4. Special Cases of F(a, 6; c; z ) , Polynomials and Legendre

. .., .. . . . . . . . . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .

Functions.

15.5. The Hypergeometric Differential Equation 15.6. Riemann’s Differential Equation , . 15.7. Asymptotic Expansions . . . . .

References

1

National Bureau of Standards.

. . . . .

. . . . .

,

. . . .

. . . . .

. . . . .

Page

556 556 557 558 56 1 562 564 565 565

(Presently, Oregon State University, Corvallis, Oregon.) 565

15. Hypergeometric Functions Mathematical Properties 15.1. Gauss Series, Special Elementary Cases, Special Values of the Argument

15.1.7

F ( 3 , 3; Q ; -z')=(l+z2)'F(l,

1; Q ; -2)

Gauss Series

=z-l

The circle of convergence of the Gauss hypergeometric series

15.1.8

15.1.1

15.1.9 F(u, ++U;

F ( a , b;c; z ) = z F i ( ~ ,b; C;

In [z+(1+22)+]

F(a, b; b ; z ) = ( l - Z ) - a

3; z ~ ) = ~ [ ( ~ + z ) - ~ ~ + ( ~ - z ) - ~ ~ ]

15.1.10

Z)

F ( a , ++a; Q; z*)= *2-'(1-2a)

- r(c) -

.gr ( a +r(c+n) n)r(b+n)

r(a)r(b)n - ~

zn

15.1.11

n!

F (- a,a ; 4; -2') =3 { [ (1

.-

is the unit circle 1z1=1. The behavior of this series on its circle of convergence is: (a) Divergence when W (c-a-b) 4 -1. (b) Absolute convergence when 92 (c-a-b)>O. (c) Conditional convergence when - 1<9(c-a -b) SO; the point z = 1 is excluded, The Gauss series reduces to a polynomial of degree n in z when a or b is equal to -n, (n=O, 1, 2, . . . ) . (For these cases see also 15.4.) The series 15.1.1 is not defined when c is equal to -m, (m=O, 1, 2, . . .), provided a or b is not a negative integer n with n<m. For c=-m

lim

c+-m

1 F(a, b; c; 2)' r(c)

22)

(For cases involving higher functions see 15.4.) In (1-2)

*

(23

15.1.4

F(+, 1; 3; Z ~ ) = + Z - ~ In -

15.1.5

F ( 3 , 1; Q; -z2>=z-'

arct,an z

15.1.6

F(4, 4;4;z2)=(1-zz2)fF(1, 1; 8; zP)=z-l arcsin z ~

*See page II.

556

21")

F ( a , 1-a; 3; - z 2 ) =

4( 1+ 22) -4

+ +z y a - '+ [ (1+

{ [ (1

22)f-

Z2)f

I

z]2a-'

15.1.13

F ( a , ++a; 1+2a;

+ (1-

2 ) =22"[ 1

2)']-2°

= ( l - z ) ~ F ( l + a , $+a; 1+2a;

2)

15.1.14

F (a, 3 +a; 2a;

2 ) =220- 1( 1- 2 )

-" 1+ (1 -2 ) ~ ] l - z a

sin [ (2a- 1) z)=w-mz z]

sin [ (2a-2) 21 15.1.16 F(a, 2-U; Q ; sin2 z ) = ( ~ - ~ )sin (22)

Special Elementary Cases of Gauss Series

F(1, 1 ; 2 ; z)=-z-l

f

15.1.12

15.1.17

15.1.3

(1-2)1-za]

+ i +z p + [ (1+z2> -

15.1.15 F (a, 1-a; Q ; sin2

15.1.2

[ (1+z)--

-1

F(--a, a; 3; sin2 z)=cos (2az)

cos [ (2a-1)zl 15.1.18 F(a, 1-a; 4; sin2 z ) = cos z

15.1.19 F(a, ++a;

3; -tan2 z)=cos"

z cos (2az)

Special Values of the Argument

15.1.20

F(a, b; c; 1)=

r (c) r (c-a- b) r (c--a> r (c-b)

(CZO, -1, -2,

. . ., 9 ( c - u - b ) > O )

-ERGEOMETRIC

557

FUNCTIONS

15.2. Differentiation Formulas and Gauss' Relations for Contiguous Functions (l+a-b#O,

15.1.22 F(u, b; a-b+2;-1) 1

-1, -2,

..

Differentiation Formulas

.)

=2-ad/2(b-i)-1r(~-b+2) 1

(a-b+2#0,

-1, -2,

15.2.2

1

[r (+a>r (g+aa-b>- r (3+ia>r (1 *

15.1.23 F(1,U ;a + l ; -1)=4~[r1(3+3a>-r1(3a)]

.

15.2.1 d F(a, b; C; z)=- ab F(a+l, b+l; ~ + 1 ;Z ) dz C

a"

- F(a, b; c; z)==

.)

F(a+n, b+n; c+n; z)

(c>n

dz"

15.2.3

dz" a" [zQ+"-'F(a, b; C;

~)]=(a)nz"-'F(afn,b; C;

Z)

15.2.5

d" [Zc-a+n-l (l-Z)"fb-cF(a, dz" = (c-&z"-"-'

b; c; z ) ]

(l-Z)"+b-C-nF(a-n, b; c; 2 )

15.2.6 d" - [ (1-Z)"+b-CF(a, b; c; z ) ] d z" a (c-a) n (c- b ) n (l-z)"+*-C-"F(a, b; c+n;

(c) n

2)

15.2.7 d" - [(l-Z)"+"-'F(a, b ; c ; z)] dz" (--l)"(a)n(c---b)n (l-z)"-'F(a+n, (c>n

b; c+n;

2)

15.2.8 dn - [ Z ~ - ' ( I - Z ) ~ - ~ + "b;c; F ( ~z,) ] dz" = (c-~,),z'-"-' (l-~)~--"F(a-n,b; c-n; z ) 15.2.9 dn - [Z'-'(l-Z)"+a-cF(a, dz" =(C-n)"Zc-"-'

b; c; z)]

(l-z)"+~-C-"F(a-n,b-n; c-n;

2)

Gauss' Relations for Contiguous Functions

ThesixfunctionsF(af1, b; c; z ) , F(a, bzkl; c; z ) , F(a, b; c f l ; z ) are called contiguous to F(a, b; c; a). Relations between F(a, b; c; z) and

558

HYPERGEOMETRIC FUNCTIONS

any two contiguous functions have been given by Gauss, By repeated application of these relations the function F(a+m, b+n; c+l; z ) with integral m, n, Z(c+l#O, - 1 , -2, . . .) can be expressed as a linear combination of F(a, b; c; z) and one of its contiguous functions with coefficients which are rational functions of U , b, c, z. 15.2.10

(c-a)F(a-I,b;

C;

C;

[a-l-(c-b-l)z]F(a, b; c; 2 ) +(c--a)F(a-1, b; c; 2) - ( c - l ) ( l - z ) F ( a , b; C-1; z ) = O 15.2.22

[c-2b+(b-a)z]F(a, 0; c; 2) +b(l-z)F(a, b + l ; c; 2) -(c-b)F(a, b-1; c; z)=O

z ) + ( ~ u - c - u z + ~ z ) F ( u ,b; C ; Z ) +a(z-I)F(a+l, b; C ; z ) = O

15.2.23

z)+ ( ~ ~ - c - ~ z + u z ) F ( b; u ,C ; Z )

c[b-(c--a)z]F(a, b; c; z)-bc(l-z)F(a, b + l ; c; 2) +(c-a)(c-b)zF(a, b; c + l ; z)=O

15.2.11

(c-b)F(a, b-1;

15.2.21

+b(z-l)F(a, b+l; c; z)=O

15.2.24

(C-b-l)F(a,

15.2.12

c(c-l)(z-l)F(a, b; c-1; 2) +c[c-1-(2c-a-b-l)z]F(a, +(c-a)(c-b)zF(a,

b; c; 2) b; c + l ; z)=O

15.2.25

c(1-z)F(a, b; c; 2)-cF(a, b-1; c; 2 ) * ( C - U ) x F ( a , b; c + l ; z)=O

+

15.2.13

[c-2a-(b--a)z]F(a, b; c; 2 ) + a ( l - ~ ) F ( a + I , b; C ; Z ) -(c-a)F(a-I,

b; c; z)+bF(a, b + l ; c; 2 ) -(C-I)F(a, b; C-1; z ) = O

15.2.26

b; C ; z ) = O

15.2.14

(b-a)F(a, b; c; z)+aF(a+l, b; c; 2) -bF(u, b+ 1 ; c; 2) =o 15.2.15

(c-U-b)F(a, b; c; z)+a(l-z)F(a+l, b; c; 2 ) -(C-b)F(a, b-1; c; z)=O

[b-l-(c-a-l)z]F(a, b; c; 2 ) +(c-b)F(a, b-1; c; 2 ) -(c-l)(l-z)F(u, b; c-1; z)=O 15.2.27

c[c- 1 - (2c-a-bl ) z ] F ( a ,b; c; 2 ) +(c--a)(c-b)zF(a, b; c + l ; 2) -c(c-l)(l-z)F(u, b; c-1; z)=O

15.2.16

c[u-(c-b)z]F(a, b; c; z)-uc(l-z)F(a+1, b; c; 2 ) +(c-a)(c-b)zF(a, b; c + l ; z ) = O 15.2.17

(c-a-l)F(a,

b; c; z)+aF(a+l, b; c; 2 ) - (c- l ) F ( a , b; c- 1 ;

Integral Representations 2)

=o

15.2.18

(c-a-b)F(a,

15.3. Integral Representations and Transformation Formulas

15.3.1

F ( a , b ; c; z)= b; c; z>-(c-a)F(a-l,

b; c;

2)

+b(l-z)F(a, b + l ; c; z)=O 15.2.19

(b--a)(l-z)F(a, b; c; z)-(c--a)F(a-l, b; c; 2 ) +(c-b)F(a, b-1; c; z)=O 15.2.20

C(l-Z)F(U, b; c; 2)-cF(a-1, b; c; 2 ) +(c-b)zF(u, b; c+l; z)=O 'See pagc U.

(9?c>9'b>O) The integral represents a one valued analytic function in the zplane cut along the real axis from 1 to m and hence 15.3.1 gives the analytic continuation of 15.1.1, F(a, b; c; z ) . Another integral representation is in the form of a Mellin-Barnes integral

559

HYPERGEOMETRIC FUNCTIONS

Here -?r<arg(-z)
From 15.3.1 and 15.3.2 a number of transformation formulas for F(u, b; c; z ) can be derived. 15.3.3

F(u, b; C; z)=(l-z)c-a-bF(c-a,

c-b;

C; Z )

15.3.4

15.3.7

15.3.8

15.3.9

Furthermore for m = l , 2, 3,

...

(larg (l--z)l
11--21<1)

HYPERGEOMETRIC FUNCTIONS

560

-9(n

+m +1)++(a +n)++(b +4 1

(I&%(1- z> I<*,I 1- Z l < U

Similarly each term of 15.3.7 has a pole when b = a f m or b-a= f m and the case is covered by

The case b-a=m, ( m = l , 2, 3,

. , .) is covered by

15.3.14 F(a,a+m; C; z)=F(a+m,~;c; Z )

(larg(-z)l<.,

r

lzl>l, (c-a)#O, f l , 3%.

The case c-a=O, -1, -2, . . . becomes elementary, 15.3.3, and the case c-a=l, obtained from 15.3.14, by a limiting process (see [15.2]).

2, 3,

..>

. . . can be

Quadratic Trannformation Formulae

If, and only if the numbers f(1-c), &(a-b), f(a+b-c) are such, that two of them are equal or one of them is equal to 3, then there exists a quadratic transformation. The basic formulas are due to Kummer [15.7] and a complete list is due to Goursat [15.3]. See also [15.2]. 15.3.15 15.3.16

F(a,b;2b;z)=(l-z)-rF(3a, =(1

b-ia; b+3;

4x) Za

-4 2 ) -OF($a,3++a;b +3; 2*(2-2 )-

2,

15.3.17

15.3.18

15.3.20

(2a, 2c-2u-1;

15.3.21

=(l-z)-"F

15.3.23

=(+++Fz)-@

c;

-)-2

F(%, a-b++; a+b+t;+z-l) Ji--z+l

561

HYPERGEOMETRIC FUNCTIONS

15.3.24

F(u, b; U+ b-4;

$-3JG)

2)= ( l - ~ ) - * F ( 2 ~ - 1 , 2 b - - l ; ~ ~ + b - $ ;

=(1-2)-*

15.3.25

(+++U'1_2)L-"F(2u-1,

a-b++; a+&+;

111-2-1)

G + l

15.3.27 15.3.28

= F ( ~ u$b; , i a f i b + $ ; 42-4z2)

15.3.30 15.3.31 15.3.32

F(u, 1-U;

C;

~ ) = ( l - ~ ) ~ - ' F ( i ~ - 3 a~,c + $ U - $ ; =( 1 - ~ ) ~ - ' ( 1 - 2 2 ) ~ - ~ F ($C-$U,

C;

42-42')

iC-3U+i;

C;

(4Z2-4Z) (1-22)-')

Cubic transformations are listed in [15.2] and [15.3]. In the formulas above, the square roots are defined so that their value is real and positive when 0 4 z < l . All formulas are valid in the neighborhood of z=O. 15.4. Special Cases of F(a, b; c;

2)

Polynomial.

When a or b is equal to a negative integer, then

This formula is also valid when c=-m-Z;

m, Z=O,

1, 2,

...

Some particular cases are 15.4.3

F ( - n , n; 3; s ) = T , ( ~ - ~ z )

15.4.4

F(-n, n+l; 1; ~ ) = P n ( l - 2 ~ )

15.4.5

1 F (-n, n + h ; a+z; x

15.4.6

F(-n, a + l + @ + n ; a + l ; x)=O, PF88)(1-22)

n!

Here Tn, P,, CF),P$. 8 ) denote Chebyshev, Legendre's, Gegenbauer's and Jacobi's polynomials respectively (see chapter 22). Legendre Functions

Legendre functions are connected with those special cases of the hypergeometric function for which a quadratic transformation exists (see 15.3).

15.4.8

562

HYPERGEOMETRIC FUNCTIONS

15.4.10

15.4.11 15.4.12

15.4.13 15.4.14

15.4.15 15.4.16

15.4.17 15.4.18

15.4.19 15.4.20

15.4.21 15.4.22

15.4.23 15.4.24

* 15.4.25 15.4.26 ~

~~~~~~~

15.5. The Hypergeometric Differential Equation

The hypergeometric differential equation 15.5.1 'See page

d2W dz

~(1-Z) -+[c-(~+b+l)~] n.

d- - wu ~ w = O dZ

563

HYPERGEOMETRIC FUNCTIONS

has three (regular) singular points z=O, 1, a . The pairs of exponents at these points are 15.5.2

pl'th=O,~-~-b,

piPh=O, l-C,

pipD2)=~,

b

respectively. The general theory of differential equations of the Fuchsian type distinguishes between the following cases. A. None of the numbers c, c-a-b; a-b is equal to an integer. Then two linearly independent solutions of 15.5.1 in the neighborhood of the singular points 0, 1, 03 are respectively

The second set of the above expressions is obtained by applying 15.3.3 to the first set. Another set of representations is obtained by applying 15.3.4 to 15.5.3 through 15.5.8. gives 15.5.9-15.5.14. 15.5.9

W1(0)=(l-z)-@F

(a, c - b ;

c; - =(l-z)-*F

2-1

15-5-11 w~(~)=z-'F(u,a-C+l; a + b - c + l ;

(b , c-a;

This

">

c; 2-1

1-z-I) =z-*F(b, b - ~ + l ; a f b - c f l ;

1-z-l)

15.5.12 WZ(',=z'-'( 1 - 2 ) '-'-bF(c

15.5.13

-a, 1 -a;

(

c- a -b

w ~ ( ~ ) = ( z - ~ ) - ' B ' U , c-b; a-b+l;

15.5.14 wZ(m)=Z'-C(2-1)C-Q-IF

(a-c+l,

+1; 1 -

2-1)

= 2a-q 1

- 2) C-@-bF(c-b,

'>

= ( z - ~ ) - ~ F b , c-a; 1-2 (

'1

1-b; a-b+l; -1--z

=Z'-'(Z-l)'-b-'F

-

1 b ; c -a

b-~+l;

-b + 1 ; 1 - 2-1)

'>

1-2

(b-c+l,

'>

l--a; b-a+l; 1-2

15.5.3 to 15.5.14 constitute Kummer's 24 solutions of the hypergeometric equation. The analytic continuation of wl,z(o)(z)can then be obtained by means of 15.3.3 to 15.3.9. B. One of the numbers a, b, c-a, c - b is an kteger. Then one of the hypergeometric series for instance w1,a(o), 15.5.3, 15.5.4 terminates and the corresponding solution is of the form 15.5.15

w=P(~-z)~~P~(z)

where p,(z) is a polynomial in z of degree n. This case is referred to as the degenerate case of the hypergeometric differential equatidn and its solutions are listed and discussed in great detail in [15.2]. C. The number c-a-b is an integer, c nonintegral. Then 15.3.10 to 15.3.12 give the analytic continuation of w1,2(0) into the neighborhood of z=1. Similarly 15.3.13 and 15.3.14 give the analytic continuinto the neighborhood of z= w in case a-b is an integer but not c, subject of ation of course to the further restrictions c-a=O, 51, f2 . . . (For a detailed discussion of all possible cases, see [15.2)). D. The number c = l . Then 15.5.3, 15.5.4 are replaced by 15.5016 w ~ ( ~ ) = F b; ( u1;, Z)

(Iz1<1 and a, b#O, -1, -2,

. . . -(m-1))

15.6. Riemann’s Differential Equation

Special Cases of Riemann’s P Function

The hypergeometric differential equation 15.5.1 with the (regular) singular points 0, 1 , P) is a special case of Riemann’s differential equation with three (regular) singular points a, b, c

(a) The generalized hypergeometric function

15.6.1 2-c

aa’(a-b)(a-c)

BB’(b-C)(b-a) z-b

rr’(c--a)(c-b)

+

fg

1 -Y-”]

+ z-b

15.6.4

(b) The hypergeometric function F(a, b; c; z) 15.6.5

w=P{

o

w

1

0

a

0

W

] (z-a)( z-

Z-C

b

I-c

b)( z-c)”

c-a-b

(c) The Legendre functions The pairs of the exponents with respect to the singul.ar points a; b; c are a, a’;p, p‘; y , y‘ respectively subject to the condition 15.6.2

15.6.6

0

a+a’+8+8’+Y+Y’=l

The complete set of solutions of 15.6.1 is denoted by the synibol

b

c

1

0)

3+3v

4

-3p

(d) The confluent hypergeometric function 15.6.7

0 15.6.3

Pt(z),Q!!(z>

w=P{

$+U

3-U

provided lim c+w.

-

C

c-k

-c

O

k

z }

565

HYPERGEOMETRIC FUNCTIONS

Transformation Formulas for Riemann's

a

15.6.8

(Ey(Ep{1, z-b

2-b

b

c

;t

-rt

.)-.I

P Function

U

a+k

a'+k

b

8-k--l 8'-k-l

C

7+l

Y'+l

where 15.6.10

z=-

Azl+B Aal+B Abl+B Acl+B Czl+D' a=Ca,+D' b'Cb,+D' '=CclfD

and A, B, C, D are arbitrary constants such that AD-BC#O. Riemann's P function reduced to the hypergeometric function is

{1, a

15.6.11

P

b

c

,". ;, ; } = ( E y ( E y 2-b

2-b

P

{

0

1

W

0

a+8+y

&'-a

&+8'+Y

0 7'-y

'I (2-a)(c-b) (2-b)

(c-a)

t

J

The P function on the right hand side is Gauss' hypergeometric function (see 15.6.5). If it is replaced by Kummer's 24 solutions 15.5.3 to 15.5.14 the complete set of 24 solutions for Riemann's differential equation 15.6.1 is obtained. The first of these solutions is for instance by 15.5.3 and 15.6.5

15.7. Asymptotic Expansions

The behavior of F(a, b ; c ; z ) for large IzI is described by the transformation formulas of 15.3.

For fixed a, b, z and large

(CI one has [15.8]

15.7.1

For fixed a, c, z, (c#O, -1, -2, and large Ibl one has 115.21

. . . , O<(zl
(15.11 P. Appell and J. KampB de FBriet, Fonctions hyper-

g6omBtriques et hyperspheriques Villars, Paris, France, 1926).

[15.4] E. Goursat, PropriBtBs gen6rales de 1'6quation

(Gauthiers-

d'Euler et de Gauss (Actualit& scientifiques et industri6lles 333, Paris, France, 1936).

[15.2] A. ErdBlyi et al., Higher transcendental functions, vol. 1 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953).

[ 15.51 J. KampB de FBriet, La fonction hypergbmbtrique (Gauthiers-Villars, Paris, France, 1937).

[15.3] E. Goursat, Ann. Sci. Ecole Norm. Sup(2)10, 3-142( 1881).

[15.6] F. Klein, Vorlesungen iiber die hypergeometrische

Funktion (B. G. Teubner, Berlin, Germany, 1933).

566

HYPEROEOMETRIC FUNCTIONS

[15.7] E.E.Kummer, u b e r die hypergeometrische Reihe, J. Reine Angew. Math. 15,39-83, 127-172(1836). [15.8]T.M. MacRobert, Proc. Edinburgh Math. Soc. 42, 84-88(1923). [15.9]T. M. MacRobert, Functions of a complex variable, 4th ed. (Macmillan and Co., Ltd., London, England, 1954). [15.10]E. G.C. Poole, Introduction to the theory of linear differential equations (Clarendon Press, Oxford, England, 1936).

[15.11]C. Snow, The hypergeometric and Legendre functions with applications to integral equations of potential theory, Applied Math. Series 19 (U.S. Government Printing Office, Washington, D.C., 1952). [15.12]E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed. (Cambridge Univ. Press, Cambridg., England, 1952).

16. Jacobian Elliptic Functions and Theta Functions L. M. MILNE-TEOMSON

Contents

.

Mathematical Properties . . . . . . . . . . . . . . . . . . . 16.1. Introduction . . . . . . . . . . . . . . . , 16.2. Classification of the Twelve Jacobian Elliptic Functions 16.3. Relation of the Jacobian Functions to the Copolar Trio . . 16.4. Calculation of the Jacobian Functions by Use of the Arith-

... . .

16.5. 16.6. 16.7. 16.8. 16.9. 16.10. 16.11. 16.12. 16.13. 16.14. 16.15. 16.16. 16.17. 16.18. 16.19. 16.20. 16.21. 16.22. 16.23. 16.24. 16.25. 16.26. 16.27. 16.28. 16.29. 16.30. 16.31. 16.32.

. .

metic-Geometric Mean (A.G.M.) . . . . . . . . . . . Special Arguments . . . . . . . . . . . . . . . . . . Jacobian Functions when m=O or 1 . . . . . . . . . . . Principal Terms , , . . , . , , , . , . . . . . . . . Change of Argument . . . . . , . . , , , . . . . . . Relations Between the Squares of the Functions . . . . . Change of Parameter,. . , , , . , . . . . . . . . . . Reciprocal Parameter (Jacobi’s Real Transformation) . . . Descending Landen Transformation (Gauss’ Transformation) . . . . . . . . . . . . . . . . . . . . . . . Approximation in Terms of Circular Functions . . . . . . Ascending Landen Transformation . . . . . . . . . . . Approximation in Terms of Hyperbolic Functions . . . . Derivatives , . . . . . . , . . . . . . , . . . . . . Addition Theorems , , , . . . . . . , . . . . . . . . Double Arguments . . . . . . , . . . . , , , . . . . Half Arguments , , , . . , . . . . . . . . . . . . . Jacobi’s I~maginaryTransformation. , . . . . . . . . . Complex Arguments . . . . , . . , . . . . . . . . . Leading Terms of the Series in Ascending Powers of U . . . Series Expansion in Terms of the Nome p . . . . . . . . Integrals of the Twelve Jacobian Elliptic Functions. . . . Notation for the Integrals of the Squares of the Twelve Jacobian Elliptic Functions . . . . . . . . . . . . . Integrals in Terms of the Elliptic Integral of the Second Kind.. . . . . . . . . . . . . . . . . . . . . . Theta Functions; Expansions in Terms of the Nome p . . Relations Between the Squares of the Theta Functions . . Logarithmic Derivatives of the Theta Functions . . . . . Logarithms of Theta Functions of Sum and Difference . . Jacobi’s Notation for Theta Functions . . . . . . . . . Calculation of Jacobi’s Theta Function O(ulm) by Use of the Arithmetic-Geometric Mean . . . . . . . . . . . I

Page

569 569 570 570 571 571 571 572 572 573 573 573 573 573 573 574 574 574 574 574 574 575 575 575 575 576 576 576 576 576 577 577 577

1 University of Arizona. (Prepared under contract with the National Bureau of Standards.)

507

568

JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS

Page

16.33. Addition of Quarter-Periods to Jacobi’s Eta and Theta Functions . . . . . . . . . . . . . . . . . . . . . 16.34. Relation of .Jacobi’s Zeta Function to the Theta Functions. 16.35. Calculation of Jacobi’s Zeta Function Z(u(m)by Use of the Arithmetic-Geometric Mean . . . . . . . . . . . . . 16.36. Neville’s Notation for Theta Functions . . . . . . . . . 16.37. Expression as Infinite Products . . . . . . . . . . . . . . 16.38, Expression as Infinite Series . . . . . . . . . . . . . .

Numerical Methods . . . . . . . . . . . . 16.39. Use and Extension of the Tables . References . . . . . . . . . . . . . . . . Table 16.1. Theta Functions S,(eO\aO),

6,( e O \ a O )

57 7 578 578 578 579 579

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

579 579

. . . . . . . . . . . . . . . . . . .

582

581

liseccy 8c(€;\ao)

, Jsecll8,j(E;\ao)

c ~ = 0 ~ ( 5 ~ ) 8e , 5el=0°(50)900, ~,

9-10D

Table 16.2. Logarithmic Derivatives of Theta Functions

. .

,

584

d - In 8,(u)=f(eo\ao) du

d In 8Ju)=-f(e;\ao) du d In tYn(u)=g(eo\ao) du

d - In 8d(~)=-g(~y\ao) du

The author wishes to acknowledge his great indebtedness to his friend, the late Professor

E. H. Neville, for invaluable assistance in reading and criticizing the manuscript. Professor Neville generously supplied material from his own work and was responsible for many improvements in matter and arrangement. The author’s best thanks are also due t o David S. Liepman and Ruth Zuoker for t h e preparation and checking of t h e tables and graphs.

16. Jacobian Elliptic Functions and Theta Functions Mathematical Properties Jacobian Elliptic Functions

16.1. Introduction

A doubly periodic meromorphic function is called an elliptic function. Let m, ml be numbers such that

m+ml=l. We call m the parameter, m1 the complem’entary

parameter. In what follows we shall assume that the parameter m is a real number. Without loss of generality we can then suppose that O l m l l (see 16.10, 16.11). We define quarter-periodsK and iK‘ by 16.1.1

so that K and K’ are real numbers. K is called the real, iK’ the imaginary quarter-period. We note that 16.1.2

Let p, q be any two of the letters s, c, d, n. Then p, q determine in the lattice a minimum rectangle whose sides are of length K and K’ and whose vertices s, c, d, n are in counterclockwise order. Definition

The Jacobian elliptic function pq U is defined by the following three properties. (i) pq U has a simple zero a t p and a simple pole a t q. (ii) The step from p to q is a half-period of pq U . Those of the numbers K , iK’, K+iK’ which differ from this step are only quarter-periods. (iii) The coefficient of the leading term in the expansion of pq U in ascending powers of U about u=O is unity. With regard to (iii) the leading term is U , l/u, 1 according as u=O is a zero, a pole, or an ordinary point. Thus the functions with a pole or zero at the origin (i.e., the functions in which one letter is s) are odd, and the others are even. Should we wish to call explicit attention to the value of the parameter, we write pq (ulm)instead of pg U . The Jacobian elliptic functions can also be defined with respect to certain integrals. Thus if

K(m)=K’(ml)=K’ ( 1-m).

We also note that if any one of the numbers m, ml, K(m), K’(m), K’(m)/K(m)is given, all the rest are determined. Thus K and K’ can not both be chosen arbitrarily. I n the Argand diagram denote the points 0, K , K+iK’, iK’ by s, c, d, n respectively. These points are a t the vertices of a rectangle. The translations of this rectangle by XK, piK’ , where X, p are givQn all integral values positive or negative, will lead to the lattice .S

.C

.n

.C .d

.n

.S

.C

.S

.n

.d

.n

.d .c .d

.S

the pattern being repeated indefinitely on all sides.

the angle

cp

is called the amplitude

16.1.4 cp=am

U

and we define 16.1.5

sn u=sin

cp,

cn u=cos cp, dn u=(l-m sin2 ~ p ) ’ / ~ = A ( ( p ) .

Similarly all the functions pq U can be expressed in terms of cp. This second set of definitions, although seemingly different, is mathematically equivalent to the definition previously given in terms of a lattice. For further explanation of notations, including the interpretation, of such expressions assn (cp\a), cn (ulm),dn (U, k),see 17.2. 569

570

JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONB

16.2. Classification of the.Twelve Jacobian Elliptic Functions According to Poles and Half-Periods

I

1

I

Pole iK'

K Y k

I

Pole K

-1-1

Half period iK'

sn

U

I cd

U

1 do

U

Half period K + i K '

cn

U

sd

U

nc

U

-

c

Half period K

i

n

d

1

I

1

I

I

I

Pole 0

I

I ns

U

I Periods 2iK', 4K+4iK', 4K

ds U

Periods 4iK', 2K+2iK', 4K

l ~ - l ~ l

Periods 4iK', 4K+4iK', 2K

u

The three functions in a vertical column are copolar. The four functions in a horizontal line are coperiodic. Of the periods quoted in the last line of each row only two are independent.

I.5

I0

.s 0

- .5 -1.0

-1.5

FIQURE 16.3. Jacobian elliptic functions

4-

sc U , ca

FIQURE6.1. Jacobian elliptic functions an U , cn U , dn U

1

The curve for an (ult) is the boundary between those which have an inflexion and those which have not.

16.3. Relation of the Jacobian Functions to the Copolar Trio sn U, cn U , dn U 16.3.1

cdu=- cn U dn u

16.3.2 Sdu=-

-

an U dn u

16.3.3 ndu==

5-

0

U

K

3K

2K

cd U , dc U

m=Z

1 m=$

-5

U,

1

&U=---

dn u on U

na U=- 1 sn U

nc

U=-

1 on U

da U=-

sc U=-

sn U on u

c9

dn u an U

u=-

cn u sn U

4K

And generally if p, q, r are any three of the letter a ' 1 ' 1

FIQURE 16.2. Jacobian elliptic junctions ns U , nc U , nd

"=z

1

U

dJ

nJ

16.3.4 provided that when two letters are the same, e.g., pp U , the corresponding function is put equal to unity.

571

JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS

16.4. Calculation of the Jacobian Functions by Use of the Arithmetic-Geometric Mean (A.G.M.) For the A.G.M. scale see 17.6. To calculate sn (ulm),cn (ulm), and dn (ulm) form the A.G.M. scale starting with

and then compute successively ( P ~ - ~(, P ~ - ~ ., from the recurrence relation sin

16.4.3

(~(P,-~-(P~)=-

Cn

an

sin (on.

Then 16.4.4

16.4.1 ao= 1, bo= fil, co= f i , terminating at the step N when cN is negligible to the accuracy required. Find pN in degrees where

sn (ulm)=sin

'po,

cii (ulm)=cos

cos (PO

dn (ulm)=

(po

,

cos (n-(Po) From these all the other functions can be determined. 16.5. Special Arguments

16.4.2

cn

sn U

U

dn

U ~

*

. ,,

(PI, (o0

16.5.1

0

16.5.2

IK

16.5.3

K

16.5.4

i(iK')

16.5.5

1 (K +iK') 2

16.5.6

K

16.5.7

iK'

16.5.8

aK+iK' 1

16.5.9

K+iK'

1

1

m

m

2

+ 1(iK ' )

rn-l/4

m

0

m-i/a

16.6. Jacobian Functions when m=O or 1 =

m=O

m=l

16.6.1 16.6.2 16.6.3

sin U cos U

tanh U sech U sech U

16.6.4 16.6.5 16.6.6

cos U sin U 1

1 sinh U cosh U

16.6.7 16.6.8 16.6.9

sec U sec U tan U

1 cosh U sinh U

16.6.10 16.6.11 16.6.12

csc U csc U cot U

coth csch csch

16.6.13

U

gd U

1

U U

U

U

16.7. Principal T e r m s

When the elliptic function pq U is expanded in ascending powers of (U-K,), where K, is one of 0, K, iK’, K+iK‘, the first term of the expansion is called the principal term and has one of the forms A, BX (U-K,), C+(u-K,) according as K, is an ordinary point, a zero, or a pole of pq U . The following list gives these forms, where x means that the factor (U--K,)has to be supplied and + means that the divisor (U-K,) has to be supplied. K 16.7.1 16.7.2 16.7.3 16.7.4 16.7.5 16.7.6

sn U cn U dnU cd u sd U nd u

1x

16.7.7 16.7.8 16.7.9 16.7.10 16.7.11 16.7.12

dc u nc U sc U ns U ds U cs U

1

iK’

K+iK’

1 1 1

1x 1

1

1x l+ l+ l+

16.8. Change of Argument

I

u-2K

2K-U

cd U -m1%d U m11/2nd U

-sn U -cn U dn U

-sn U -cn U dn U

sn U -cn U dn U

sn -cn -dn

U U

-cd U --sd U nd U

-cd U -Sd U nd U

-cd U sd U nd U

cd U -sd U -nd U

-Cd U sd U -nd u

U U

-dc -nc sc

U U U

-dc U --nc U sc U

-dc -nc -sc

U U U

dc U -nc U -sc U

-dc U nc U -sc U

-ns -ads cs

U

-ns U -ds U cs U

ns U ds U -cs U

ns U -ds U

-ns

16.8.1 16.8.2 16.8.3

sn cn dn

sn U cn U dn U

-

16.8.4 16.8.5 16.8.6

cd sd nd

cd U sd U nd U

cdu -sd U nd U

-mu ml-112cn ml-lndn

16.8.7 16.8.8 16.8.9

dc nc sc

dc U nc U sc U

dc U nc U -sc U

-mu

16.8.10 16.8.11 16.8.12

ns

ns

- ns

U

-ds

U U

dc U mllhc U -mllfisc U

ds cs

U &U cs U

sn U cn U dn U

u+2K

-cs

-ml-l/2ds

- ml-%s

u+iK’

u+2iK’

K-u

U U

--cs

U

U U

U

U+

K+iK’

u+2K f2iK’ -sn cn -dn

U U

U

U

dsU -cs U

573

JACOBIAN ELLIPTIC FTJNCTIONS AND THETA FUNCTIONS

16.9. Relations Between the Squares of the Functions

16.9.1 -dn2u+ml=-m 16.9.2

cn2u=m sn2u-m

-mlnd2u+ml= -mm1sd2u=m cd2u-m

16.9.3

mlsc2u+ ml =mlnc2u=dc2u-m

16.9.4

cs2u+ ml =ds2u= ns%- m

In using the above results remember that m+ml=l. If pq U, rt U are any two of the twelve functions, one entry expresses tq2u in terms of pqau and another expresses qt2u in terms of rt2u. Since tq2u.qt2u=l, we can obtain from the table the bilinear relation between pq2u and rt2u. Thus for the functions cd U, sn U we have 16.9.5

Note that successive applications can be made conveniently to find sn (ulm) in terms of sn ( v l p ) and dn (ulm) in terms of dn (VIP), but that the calculation of cn (ulm) requires all three functions. 16.13. Approximation in Terms of Circular Functions

When the parameter m is so small that we may neglect m2 and higher powers, we have the approximations

and therefore 16.9.6

then

(1-m cd2u)(l-m sn2u)=ml. 16.10. Change of Parameter

16.13.1

sn (ulm) =sin

1

m(u-sin

U

cos U) cos U

1 cn (ulm) =msu+- m(u-sin

U

cos U) sin U

U--

4

Negative Parameter

If m is a positive number, let m 16.10.1 p = - 1+ m'

1 Ccl=l+m)

U V=K*

16.10.2

sn (ul-m)=plfsd

16.10.3

cn (U/-m)=cd

16.10.4

dn (ul-m)=nd

(VIP) (DIP) (VIP).

16.11. Reciprocal Parameter (Jacobi's Real Traneformation)

16.11.1 16.11.2

m>O, p=m-l, v=um1/,2 sn (ulm) =p%n

(VIP)

16.11.3

cn (ulm>=dn (VIP)

16.11.4

dn (ulm)=cn

(VIP)

Here if m > l then m-l=p
To decrease the parameter, let

16.13.2 4

16.13.3

1

dn (ulm) = 1-- 2 m sin2U

16.13.4 am (ulm) =U--

1

4

m(u-sin

U

cos U).

One way of calculating the Jacobian functions is to use Landen's descending transformation to reduce the parameter sufficiently for the above formulae to become applicable. See also 16.14. 16.14. Ascending Landen Transformation

To increase the parameter, let

574

JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTION&

Note that, when successive applications are to be made, it is simplest to calculate dn (ulm) since this is expressed always in terms of the same function. The calculation of cn (ulm) leads to that of dn ( V I P ) . The calculation of sn (ulm) necessitates the evaluation of all three functions.

16.17. Addition Theorems 16.17.1

sn(u+v) -sn

16.17.2

u.cn v-dn v+sn vacn uadn 1-m sn2u sn2v

cn(u+v)

-cn u . cn w-sn

Functions

16.17.3

When the parameter m is so close to unity that m12and higher powers of ml can be neglected we have the approximations

-

u . dn u . sn w dn w

1-m sn2u'sn2w

16.15. Approximation in Terms of Hyperbolic dn(u+v)=

U

dnu. dnv- msnu. cnu. snv cna 1- msnau sn2v

sechu

Addition theorems are derivable one from another and are expressible in a great variety of forms. Thusns(u+v) comes from l/sn(u+u) in the form (1-msn2u sn2u)/ (sn u cn w dn v+sn v cn U dn U) from 16.17.1. Alternatively ns(u u)=m'%n { (iK'-U)-U} which againErom.16.17.1 yields the form (ns U cs v dsu -ns v cs u ds 2i)/(ns2u- ns2w) . The function pq(u+v) is a rational function of the four functions pq U, pq v, pq'u, pq'v.

1 +i ml (sinh U cosh u+u) tanh U sech U

16.18. Double Arguments sn 2u -2snu.cnu-dnu -2 s n u . c n u . d n u cn2u+sn2u dn2u 1- msn4u

16.15.1 1

sn (ulm) ST tanh u s z m1 (sinh

16.15.2 cn (ulm) =sech

U

cosh

U-U)

sech2 U

U

-z1 ml (sinh 16.15.3 dn (ulm) =sech

U

cosh U-U)

tanh

U

+

16.18.1

U

a

16.18.2

16.15.4 1

am (ulm) =gdu+z m, (sinh U cosh U-U) sech

U.

Another way of calculating the Jacobian functions is to use Landen's ascending transformation to increase the parameter sufficiently for the above formulae to become applicable. See also 16.13.

16.16. Derivatives Function

sn U cn U dn U

cn U dn U -sn U dn U - m sn U cn

16.16.4 16.16.5 16.16.6

cd U sd U nd U

-ml

16.16.7 16.16.8 16.16.9

dc U nc U sc U

16.16.10 16.16.11 16.16.12

ns

U

ds U cs U

-cn2u-sn2u dn2u-cn2u-sn2u dn2u cn2u+sn2u . dn2u 1-msn'u 16.18.3 dn 2u -dn2u- msn2u.cn2u-dn2u+cn2u(dn2u-1) dn2u- cn2u(dn2u- 1) 1 msn4u 1-cn 2u-sn2u. dn2u 16.18.4 1 cn 2ucn2u

-

+

1-dn 2u- msn2u. cn2u

Derivative

16.16.1 16.16.2 16.16.3

16*18*5

Pole n

16.19.1

sd U nd U cd U nd U m sd U cd U

Pole d

16.19.2

sc U nc dc U dc U no U

Pole c

16.19.3

ml

sc

-ds -cs -ns

U

U U

U

cs U ns U ds u

cn2u

U

l+dn 2u-

dn2u

16.19. Half Arguments 1-cn U sn2?p =l+dn u dn u+cn U cn2@= l +d n U

U

dn2&U=

ml+dn ufmcn l +d n U

U

16.20. Jacobi's Imaginary Transformation Pole

8

Note that the derivative is proportional to the product of the two copolar functions.

16.20.1

sn (iul m) =isc (ul m 1)

16.20.2

cn(iuIm)=nc(u( ml)

16.20.3

dn(iulm)=dc(ul ml)

575

JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONB

16.23.6

16.21. Complex Arguments

With the abbreviations 16.21.1 s=sn(z1 m), c=cn(zIm), d=dn(zlm), sl=sn(ylml), ci=cn(ylmi), di=dn(YIml)

s* di+k * d * ~ 16.21.2 sn(z+iylm) = c; ms2. s:

+

16.23.7

dc

(U

R I m) =-2K sec w

2n 1 ~1 -

cos (2nfl)u

n-0

16.23.8 R

c * c1-is d * 81 * d1 16.21.3 cn(z+iyl m) = c: ms2. s:

nc (ulm)=-2m:/2K sec

+

2n -m:I2K c ( n..~

16.22. Leading Terms of the Series in Ascending Powers of U 16.22.1 U3

sn(ul m)=u- (l+m> 3+(1+14m+m2) -(l+135m+135m2+m3)

R

+- 2R

+

R

U7

%+ . . .

cn ( U [m)=1-- (1+4 m) 4? 2!

U4

-(1+44m+16m2)

( - I ) ~ ~q2" f qsin ~ 2nv "

16.23.10

ns (ulm)=- csc v - 2 2K K 16.23.11

zn

n

U2

OD

m:'2K n - ~

U6

3

16.22.2

gU6+ . . .

16.22.3

5 l-pZnfl -sin (2nfl)v p + l

n-0 m

$n+l

ds (u]m)=- csc v-C -sin (2nfl)v K n-0 2K 16.23.12 n 2n q2" sin2nv cs (uIm)=cot v-2K K n-1 1+q2"

5

16.24. Integrals of the Twelve Jacobian Elliptic Functions

-m(16+44m+m2)

U6

.

No formulae are known for the general coefficients in these series. 16.23. Series Expansions in Terms of the Nome Q = ~ - * ~ ' J ~and the Argument V=HU/(2EI) 16.23.1

16.23.9 sc m l(')="K 2't,

- 1 l P2"+l n W cos (2n+l)v

2n m p + 1 / 2 sn (ulm)=C sin (2n+l)v m1IzKn-O 1- pZn+l ~

2n 16.23.2 cn (ul m)= -C pn+l12 cos (2nfl)v m1I2Kn=O 1+pZn+1 ~

H 2H 16.23.3 dn (ul m)=-+Q" cos 2nv 2K K n-1 I+$" 16.23.4

n n+1/2

cd (uIm)=- 27r (-l) cos (2n+l)v m112Kn.IO 1- p2"+l 16.23.5

16.24.1

J sn U du=rn-ll2 In (dn u-rn1l2cn

16.24.2

J cn U du=m-'I2 arccos (dn U ) Jdn

16.24.3

U

U)

du=arcsin (sn U )

Jcd

U

du=rn-ll2 In (nd u+mlJ2sd U )

16.24.5 Jsd

U

du= (mml)-112arcsin (-m112cd U )

16.24.4

16.24.6 16.24.1 16.24.8

J nd U du=ml-1/2 arccos (cd U ) J dc U du=ln (nc u+sc U ) J nc U du=m;ll2 In (dc ~ + r n i / ~ sUc)

16.24.9 J s c

U

du=m;lJ2 In (dc u+m:I2nc

16.24.10

Jns

U

du=ln (ds U-CS

U)

16.24.11

Jds

U

&=In (ns U-cs

U)

16.24.12

f cs U du=h (ns u-ds

U)

U)

In numerical us4 of the above table certain reitrictions must be put on U in order to keep the trguments of the logarithms positive and to avoid

576

JACOBIAN ELLIPTIC FUNCTION,S AND THETA FUNCTIONS

trouble with many-valued inverse circular functions. 16.25. Notation for the Integrals of the Squares of the Twelve Jacobian Elliptic Functions 16.25.1

Pqu=

16.25.2

Ps

pq2t dt when q f s

u=l'

(pq't-$)

dt-; 1

Examples

cd2t d t , Ns a=l(ns2t-;)dt-;

1

16.26. Integrals in Terms of the Elliptic Integral of the Second Kind (see 17.4)

mSn ~ = - E ( u ) + u

16.26.1

mCn u=E(u)-mlu

16.26.2

Pole n

D n u=E(u)

16.26.3

16.26.4 mCd u=-E(u)+u+msn

U

cd

cd

U

mlNd u=E(u)-msn

16.26.6

Dc

16.26.7

U=

-E(u)+u+sn

U

U

dc

U

16.26.8

mlNc u=-E(u)+mlu+sn

U

16.26.9

mlSc U= -E(u) +sn

16.26.10

Ns U= -E(u) +zl-cn

16.26.11 Ds u=-E(u)+m,u-cn 16.26.12

U

Cs u=-E(u)-cn

Pole c

dc U

U

dc U

U

ds

E K

Theta functions aye important because every one of the Jacobian elliptic functions can be expressed as the ratio of two theta functions. See 16.36. The notation shows these functions as depending on the variable z and the nome p, 1q1<1. I n this case, here and elsewhere, the convergence is not dependent on the trigonometrical terms. I n their relation to the Jacobian elliptic functions, we note that the nome p is given by P=e-rK'IK,

where K and iK' are the quarter periods. Since p=q(m) is determined when the parameter m is given, we can also regard the theta functions as dependent upon m and then we write a)=8dZIm), a=1, 2, 3, 4

but when no ambiguity is to be feared, we write 8,(z) simply. The above notations are those given in Modern Analysis [16.6]. There is a bewildering variety of notations, for example the function 8,(z) above is sometimes denoted by & ( z ) or $ ( z ) ; see the table given in Modern Analysis [ 16.61. Further the argument u=2Kz/lr is frequently used so that in consulting books caution should be exercised. 16.28. Relations Between the Squares of the Theta Functions

ds U U

Pole s

ds U

where E = E ( K )

--U,

cos 2n'z

n=l

U

All the above may be expressed in terms of Jacobi's i e t a function (see 17.4.27). Z(u)=E(u)

&(z, p)=S4(z)=1+25 (-l)nqn2

gd2,

Pole d

U

cd

U

n=l

16.27.4

U

16.26.5

mmlSd u=E(u) -mlu-msn

16.27.3 &(z, p ) = 8 3 ( ~ ) = 1 + 2 2 pn2cos 2nz

16.27. Theta Functions; Expansions in Terms of the Nome Q

16.28.1 8:( Z ) 8:(0) =O ; ( Z ) 8:(0)

-a:(

16.28.2 8i(~)8:(0)=8~(~)t9~(0)-8~(~)8;(0) 16.28.3 8;(z) 8:(0) =8:( Z ) @.(U)

-8;( Z)Oi(0)

16.28.4 8: (Z)8: (0) =8; (Z)8$(0)-8; (Z ) 8: (0) 16.28.5

W O ) +8:(0)=8m)

Note also the important relation 16.28.6 $(O) =82(0)8,(0)84(0) or

16.27.1

~)8;(0)

81-828384

m

i+(z, p)=O1(z) =2p1/4C(-l)npn(n+1)sin (2n+l)z n=O

16.27.2 m

8&, ~ ) = 8 ~ ( ~ ) = 2 q p* /+ ~l C ) COS ( 2 n f i ) z n=O

16.29. Logarithmic Derivatives of 'the Theta Functions

577

JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS

16.31. Jacobi's Notation for Theta Functions

16.29.2

@(U

16.31.2

0 1( U /

m )= @ I

16.31.3 16.31.4 16.30. Logarithms of Theta Functions of Sum and Difference 16.30.1

lrU

I m )= @ ( ~ ) = 8 4 ( ~ ) ,

16.31.1

U=%

=O(u+K)

( U ) =&(U)

H(ul m)=H(u) =&(U) Hl(U1 m) = H ~ ( ~ ) = ~ ~ ( v ) = H ( u + K )

16.32. Calculation of Jacobi's Theta Function @ ( U /m ) by Use of the Arithmetic-Geometric Mean

Form the A.G.M. scale starting with 16.32.1

uo= 1, bo=

JK,CO= fi

terminating with the Nth step when cN is negligible to the accuracy required. Find 'pN in degrees, where

16.30.2

and then compute successively (pN-1, (ol, cpo from the recurrence relation 16.30.3

16.32.3

'pN-2,

. . .,

Cn sin (2'p,-l-'pn) =sin qn.

an

Then 16.32.4

16.30.4

In

1

@ ( U [ m)=s

In

2m:'2K(m) lr

1

+Z In

cos (Cpl--%) cos cpo

The corresponding expressions when @ =iy are easily deduced by use of the formulae 4.3.55 and 4.3.56. 16.33. Addition of Quarter-Periods to Jacobi's Eta and Theta Functions U

U+

-U

2K +2iK'

16.33.1 H (U) 16.33.2 HI (4 16.33.3 @l(4 16.33.4 @(U)

where *-i,

P-

H(u) and Hl(u) have the period 4K. @(U) and Ol(u) have the period 2K. 2iK' is a quasi-period for all four functions, that is to say, increase of the argument by 2iK' multiplies the function by a factor.

JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS

578

16.34. Relation of Jacobi’s Zeta Function to the Theta Functions Z(u)=-

16.34.

n-

Z(u)=-

a bU

In

If A, p are any integers positive, negative, or zero the points u0+2kK+2piK‘ are said to be congruent t o uo.

@(U)

”(g)c nsnu d n u

8$(u)has zeros at the points congruent to 0 #,(U) has zeros at the points congruent to K an(u)has zeros at the points congruent to iK‘

U

bd(u)has zeros at the points congruent to

K+iK’ Thus the suffix secures that the function S,(u) has zeros at the points marked p in the introductory diagram in 16.1.2, and the constant by which Jacobi’s function is divided secures that the leading coefficient of #,(U) at the origin is unity. Therefore the functions have the fundamentally important property that if p, q are any two of the letters s, c, n, d, the Jacobian elliptic function pqu is given by

16.34.2

16.34.3

--n- 0;

-2K 83

(g) ($)-

m

sn U cn U dn U

16.34.4

16.36.3

These functions also have the property 16.35. Calculation of Jacobi’s Zeta Function Z(ulmJ by Use of the Arithmetic-Geometric Mean

Form the A.G.M. scale 17.6 starting with 16.35.1

Q,=

1, bo=

&G,

CO=

J;;L

terminating at the Nth step when cN is negligible to the accuracy required. Find (pN in degrees where 16.35.2

(pN=2NaNU

180’

7r

and then compute successively ( p N - 1 , cpo from the recurrence relation

(pN-2,

....

(pl,

16.35.3

C

sin (2(pn-1-(pn)=2 sin (pn. an

Then 16.35.4

Z(ulm)=clsin ( p l + c 2 sin cpa+

. . . +cN

sin (pN.

16.36. Neville’s Notation for Theta Functions

These functions are defined in terms of Jacobi’s theta functions of 16.31 by

16.36.4

m;1’48c(K-u)=#,(U)

16.36.5

m;1’48,(K-~)=S,(U),

for complementary arguments U and K-U. In terms of the theta functions defined in 16.27, let v=m/(2K),then

579

JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS

'.

16.38. Expression as Infinite Series

\. \

Let v=m/(2K) %\

16.38.1

'.

i.

\.

'*

\salnes(u)

\""

FIGURE16.5. Logarithm+ derivatives of theta d

functions d

du lnO,(U),

16.38.3 6,Ju)

=[&I

'I2 { 1+2

2 pn2cos 2nv}

n-1

16.38.4

In 6,(4

m = -1

6n(u)=[*K]

2

lI2{1+2

2 (-l)"qn2cos 2nvj

n=l

16.37. Expression as Infinite Products

a= a(@ , 0 = 4 ( 2 K ) 16.37.1 16.37.2 16.37.3 16.37.4

Numerical Methods Example 1. Calculate nc (1.996501.64) to 4 s . From Table 17.1,1.99650=K+.001. From the table of principal terms

nc u=-m;112/(u-K)+

- (.36)-'12+ nc ( K + .0011.64) = .ool

...

+...

6

=-1667+.

pz=

.

and since the next term is of order .001 this value -1667 is correct to a t least 4 s. Example 2. Use the descending Landen transformation to calculate dn (.20(.19) to 6D. Here m=.19, m:I2=.9 and so from 16.12.1

=10- X 7.67

which is negligible. From 16.12.4 dn2[.19 ($Y]-(l-A)

1

dn(.201.19) =

., ,

=--10000

(A)

4

16.39. Use and Extension of the Tables

(l+$)-dn2[ Now from 16.13.3

1 (A>']

I

dn [.I9 (;)1=.999951 whence dn (.201.19)=.996253. Example 3. Use the ascending Landen transformation to calculate dn (-201.81) to 5D. From 16.14.1 I/=--

4(.9)

360

20

Also

.I9

pi

is negligible to 4D. Thus

JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS

580

1 g)+i

19 dn2(.I9 dn (,201.81)=-X2o dn(.19/g)

Example 4. Use the ascending Landen transformation to calculate cn (.20(.81) to 6D. Using 16.14.4, we calculate dn (.20(.81) and deduce cn (.201.81) from 16.14.3 settling the sign from Figure 16.1. As in the preceding example, we reduce the

'

From 1.6.15.3 1

1

=sech ( . 1 9 ) + - X - - tanh .19 sech .19 4 361 [sinh .19 cosh .19+.19]

calculation of dn (.20/.81) to that of dn when dn .19 - =.982267

() : : 1

1 1 =.982218+-X--(.187746) (.982218) 4 361 [(.191145)(1.01810)+.19] 1 1 =.982218+-X(.184408)[.384605] 4 361

dn (.201.81)=.984056 cn (.201.81) =.980278.

= .982218$ .000049= .982267.

Thus dn (.20l .8l) = ,98406.

n

an

Cn

bn

pn

sin ( 2 ~ n - 1 - d

an

-

~

0 1 1 .9 2 . 89721 3 . 89721

1. 0780 2. 1533

sin

Pn

Cn

I

. .

. 55472

.88101 83509 -. 91879

~

. 65546 1. 2069 2. 4117 4.8234

.8 .6 .6 . 11111 89443 . 1 00311 89721 . 00279 0 . 89721 0

.

2~n-l-qn

. 60952 . 93452

-.

. 66679 99384

. 10383 . 00207

0

~

. 10402 ,

0

00207

I

I

I

. 09789

,

0'

00260

. 09805 . 00260

0

and then complete the calculation outlined in 16.32 to give In @(ulm)= -.05734+.02935+.00120 = -AI2679 O(ulm)=.97357. The series expansion for 0 is preferable.

1. 0048 1. 1.

.00120 0 0

581

JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS

0,(30'\45')=

Example 7. Use the q-series to compute cs (.53601 621.09).

Here we use the series 16.23.12, K= 1.60804 862, *U

?r

q= .00589 414 ,v=-=-radians 2K 6

(sec 45')h~,(30~\45~) = 1.02796.

Therefore

or 30'.

sc (.6 1802/.5)=l.o279s *59128 (sec 450)i

Since $ is negligible to 8D, we have to 7 D cs (.53601 621.09) I

-2K* --

.59128

= .68402.

.

"m sin GO'}

cot 30'-2"(K

=(.97683 3852) (1.73205 081)

-3.90733 541[(.00003 4740)(.86602 5404)] =1.69180 83.

Example 10. Find sn (.753421.7) by inverse interpolation in Table 17.5. This method is explained in chapter 17, Example 7. Example 11. Find U , given that cs (4.5) =.75. From 16.9.4 we have

Example 8. Use theta functions to compute sn (.61802).5) to 5D.

Here K(3) =1.85407

Sn2u=--.

1

l+cs

2 u

Thus sn2 ( U [-5)=.64 and sn (u(.5)=.8.

sin2 a= 112. a=45'

We have therefore replaced the problem by that of finding U given sn (ulm),where mis known.

If =--'59128

1.04729-

p=am U

sin p=sn

and so (~'~9272952 radians or 53.13010O.

.56458

~~

from Table 16.1. Example 9. Use theta functions to compute sc (.61802(.5) to 5D. As in the preceding example r0=3O0. a0=45'

so that 9, (30'\45') sc (.618021 '5)=9,(300\450)*

We use Table 16.1 to give Texts [16.1] A. Erdblyi e t al., Higher transcendental functions, vol. 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [16.2] L. V. King, On t h e direct numerical calculation of elliptic functions and integrals (Cambridge Univ. Press, Cambridge, England, 1924). [16.3] W. Magnus and F. Oberhettinger, Formulas and theorems for the special functions of mathematical physics (Chelsea Publishing Co., New York, N.Y., 1949). [16.4] E. H. Neville, Jacobian elliptic functions, 2d ed. 1951)* (Oxford Univ* Press, London, [ 16.51 F. Tricomi, Elliptische Funktionen (Akademische Verlagsgesellschaft, Leipzig, Germany, 1948). L16.6) E. T. Whittaker and G. N. Watson, .4 course of modern analysis, chs. 20, 21, 22, 4th ed. (Cam-

U

From Table 17.5, ~=F(53.13010'\45')=

.99391.

Alternatively, starting with the above value of can use the A.G.M. scale to calculate F(p\ar) as explained in 17.6. This method is to be preferred if more figures are required, or if a differs from a tabular value in Table 17.5. p we

Tables HiPPiSleJ', Smithsonian [16.71 E- p. k h m s and R. mathematical formulae and tables of elliptic functions, 3d reprint (The SmithRonian Institution, Washington, D.C., 1957). [16.8] J. Hoiiel, Recueil de formules e t de tables numbriques (Gauthier-Villars, Paris, France, 1901). [16.9] E. Jahnke and F. Emde, Tables of functions, 4th ed. (Dover Publications, Inc., New York, N.Y., 1945). 16.101 L. M. Milne-Thomson, Die elliptischen Funktionen von Jacobi (Julius Springer, Berlin, Germany, 1931). 16.111 L. M. Milne-Thomson, Jacobian elliptic function tables (Dover Publications, Inc., New York, N.Y., 1956). [16.12] G. W. and R. M. Spencoley, Smithsonian elliptic function tables, Smithsonian Miscellaneous col-

582

JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS THETA FUNCTIOIVS

'I'nlllc~ 16.1

& (€\a) 0"

5'

5 10 15 20

0.03000 0.08715 0.17364 0.25881 0.34202

0000 5743 8178 9045 0143

Oi08732 0.17397 0.25931 Oi34267

25 30 35 40 45

0.42261 0.50000 0.57357 0.64278 0.70710

8262 0000 6436 7610 6781

50 55 60 65 70

0.76604 0.81915 0.86602 0.90630 0.93969

75 80 85 90

0.96592 0.98480 0,99619 1.00000

E\ff 0' 5 10 15 20

10"

0.00000 0000

20"

15'

25"

Q/c,

1966 9362 2677 2476

0.00000 0.08782 0.17497 0.26080 0.34464

0000 4152 9967 4191 3695

0.00000 0.08867 0.17667 0.26332 0.34797

0000 3070 1584 6099 7361

0.00000 0.08988 0.17909 0.26693 0.35274

0000 7414 1708 4892 9211

0.00000

0000

0.09149 0.18229 0.27171 0.35907

5034 6223 4833 2325

'90' 85 80 75 70

0.42342 0.50095 0.57467 0.64401 0.70845

4343 3708 0526 3768 5688

0.42586 0.50383 0.57797 0.64772 0.71253

0446 6358 7994 1085 4820

0.42998 0;50871 0.58357 0.65399 oi71544

1306 3952 6134 8067 3681

0.43588 0.51570 0.59159 0.66299 0.72935

2163 1435 9683 9145 6053

0.44370 0.52497 0.60225 0.67495 0.74253

5382 0857 0597 6130 3161

65 60 55 50 45

4443 2044 5404 7787 2621

0.76750 0.82071 0.86767 0.90803 0.94148

5843 4821 7668 6964 5546

0.77192 0.82544 0.87267 0.91326 0.94691

5893 2256 6562 9273 1395

0.77941 Oi83345 0.88115 0.92214 0.95611

4712 4505 1505 2410 4956

0.79016 0.84496 0.89332 0.93489 0.96935

4790 1783 9083 7610 0025

0.80446 0.86028 Oi90955 0.95189 0.98700

5863 0899 1166 9199 0216

40 35 30 25 20

5826 7753 4698 0000

0.96776 0.98668 0.99809 1.00190

8848 6836 5528 8098

0.97334 0.99237 1.00384 1.00768

6839 4367 9133 3786

0.98281 1.00202 1.01361 1.01748

0311 5068 2807 5224

0.99642 1.01591 1.02766 1.03158

3213 0350 2527 9925

1.01458 1.03444 1.04641 1.05041

4761 0908 6011 7974

15 10 5 0

0.03000 0.09353 0.18636 0.27778 0.36710

0000 4894 3367 4006 5393

0.00000 0.09606 0.19139 0.28530 0.37706

0000 0073 9811 3629 5455

0.00000 0.09914 0.19754 0.29449 0.38924

0000 2353 9961 2321 7478

0.00000 0.10287 0.20501 0.30564 0.40405

0000 9331 0420 8349 4995

0.00000 0.10740 0.21405 0.31918 0.42204

0000 5819 3194 5434 9614

0.00000 0.11291 0i22506 0.33569 0.44403

0000 2907 4618 3043 4769

90" 85 80 75 70

25 30 35 40 45

0.45365 0.53676 021581 0.69019 0.75934

1078 4494 3814 6708 4980

0.46599 0.55141 0.63268 0.70917 0.78030

3521 5176 1725 3264 3503

0.48110 0.56937 0.65339 0.73250 0.80611

6437 7735 2178 7761 4729

0.49950 0;59127 0.67868 0.76106 Oi83776

2749 8602 8658 3101 1607

0.52189 0.61799 0.70961 0.79606 0;87664

9092 6720 8904 0581 1114

0.54932 0.65080 0.74770 0.83928 Oi92480

5515 1843 4387 2749 2089

65 60 55 50 45

50 55 60 65 70

0.82272 0.87986 Oi93030 0.97366 1.00961

9031 2121 4365 6431 2870

0.84552 0.90433 0.95626 1.00092 1.03795

4503 1298 6326 3589 2481

0.87364 Oi93455 0.98837 1.03467 1.07308

0739 6042 8598 8996 5074

0.90817 0.97175 1.02796 1.07635 1.11651

9128 1955 3895 2410 4503

0.95071 1i01765 1.07692 1.12798 1.17041

1025 9399 1759 8100 0792

1.00355 1.07485 1.13807 1.19262 1.23801

1297 2509 1621 9342 2299

40 35 30 25 20

75 80 85 90

1.03786 1.05820 1.07047 1.07456

5044 3585 0366 9932

1.06706 1.08801 1.10066 1.10488

1179 9556 1511 6686

1.10328 1.12503 1.13815 1.14254

6100 6391 8265 4218

1.14811 2152 1.17087 7087 1.18461 4727

1.20381 1.22789 1.24242 1.24728

2008 0346 6337 6586

1.27378 1.29959 1.31518 1.32039

3626 2533 2322 6454

15 10 5 0

0000 1778 4577 4091 6153

0.00000 0.12814 0.25558 Oi38160 0.50544

0000 8474 9564 3032 4270

0.00000 0.13904 0.27747 0.41467 0.54994

0000 1489 6571 2740 7578

0.00000 0.15372 0.30706 0.45960 0.61082

0000 0475 5715 9511 7702

0.00000 0.17522 0.35063 0.52633 0.70219

0000 3596 9262 5260 9693

0.00000 0000

5 10 15 20

0.00000 0.11968 0.23861 0.35604 0.47120

25 30 35 40 45

0.58332 0.69160 0.79525 0.89344 0.98538

3727 6043 0355 6594 4972

0.62633 0.74345 0.85596 Oi96294 1.06350

5361 9784 1570 9380 5669

0.68254 0.81164 0.93630 1i05553 1.16824

9331 3704 8263 5305 3466

0.76005 0.90647 1.04907 1.18666 1.31788

8920 6281 2506 0037 6740

50 55 60 65 70

1.07026 1.14731 1.21579 1.27502 1.32438

6403 5349 4546 0900 1718

1.15670 1.24161 1.31733 1.38304 1.43795

0687 0747 9855 3549 3601

1.27329 1.36953 1.45580 1.53099 1.59408

7730 6895 7011 8883 7380

1.44126 1.55522 1.65814 1.74846 1.82467

75 80 85 90

1.36335 1.39150 1.40851 1.41421

0417 0813 9209 3562

1.48140 1.51284 1.53187 1.53824

2159 3876 4716 6269

1.64417 1.68050 1.70253 1.70991

0149 3336 2036 3565

1.88545 1.92971 1.95660 1.96563

0"

35"

30"

65"

60" 0"

45"

40"

50"

85"

80"

75"

70"

90' 85 80 75 70

0.21321 0.42844 0.64743 0.87146

7690 3440 4941 4767

0.87783 8622 1.05251 4778 i A i i 1680 1.39412 6403 1.55769 2334

1.10111 1.33612 1i57526 1.81633 2.05616

6239 3616 8297 9939 7815

65 60 55

6644 4175 9352 0610 1332

1.71363 1.85953 1i99285 2.11103 2.21162

1283 2258 2358 3523 7685

2.29072 2.51529 2.72469 2.91357 3.07668

3417 0558 4161 4159 6743

40 35 30 25 20

5864 0721 6998 0511

2.29242 2.35155 2.38762 2.39974

2061 6149 2438 3837

3.20921 3.30704 3.36705 3.38728

2227 7313 9918 7004

15

In calculating elliptic functions from theta functions, when the modular angle exceeds about 60", use the descending Landen transformation 16.12 to induce dependence on a smaller modular angle. Compiled from E. P. Adams and R. L. Hippisley, Smithsonian mathematical formulae and tables of elliptic functions, 3d reprint (The Smithsonian Institution, Washington, D.C., 1957) (with permission).

50 45

10

5 0

JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS

583 Table 16.1

THETA FUNCTIONS 9Jf \a)

5 10 15 20

1.00000 1.00001 1.00005 1.00012 1,00022

00000 44942 75362 78184 32051

1.00000 1.00005 1.00023 1.00051 1.00089

00000 83670 16945 47160 88322

1.00000 00000 1.00013 28199 1.00052 72438 iiOOii7 iZ8iS 1.00204 53820

1.00000 1.00023 1.00095 1.00211 1.00369

00000 99605 25510 61200 53131

1.00000 1.00038 1.00152 1.00337 1.00589

00000 29783 02770 73404 77438

a c, 90 85 80 75 70

25 30 35 40 45

1.00034 1.00047 1.00062 1.00078 1.00095

07982 70246 77451 83803 40492

1.00137 1.00192 1.00252 1.00317 1;00384

23717 09464 78880 47551 18928

1.00312 1.00437 1.00575 1.00722 1.00874

29684 13049 24612 44718 26104

1.00564 1.00789 1.01039 1.01305

i;Oii79

21475 74700 27539 21815 49474

1.00900 1.01260 1.01658 1.02083 1.02520

49074 44231 69227 14013 88930

65 60 55 50 45

50 55 60 65 70

1.00111 1.00128 1.00143 1.00156 1.00168

97181 03532 10738 73002 48932

1.00450 1.00515 1.00576 1.00631 1.00678

90305 58975 28392 14139 49535

1.01026 1.01173 1.01311 1.01436 1.01543

07491 27599 39167 22536 98405

1.01853 1.02119 1.02369 1.02594 _.-1.02789

77143 71444 24323 77596 45992

1.02958 1.03383 1.03781 1.04141 1.04452

63905 08852 34098 29561 01522

40 35 30 25 20

75 80 85 90

1.00178 1.00185 1.00189 1.00190

02800 05621 36042 80984

1.00716 1.00745 1.00762 1.00768

90696 20912 54187 37857

1.01631 1.01695 1.01735 1.01748

39354 79795 24037 52237

1.02947 1.03063 1.03134 1.03158

37972 73701 99632 99246

1.04704 1.04889 1.05003 1.05041

05862 76746 49895 79735

15 10 5 0

c\a

0"

O0

1

eli! a

5"

I

1

30"

45"

40"

35"

20"

15"

10"

25"

55"

50"

10 15 20

1.00000 1.00056 1.00224 1.00499 1.00872

00000 64294 85079 51300 28461

1.00000 1.00079 1.00316 1.00702 1.01226

00000 66833 25308 56701 87413

1.00000 1.00108 1.00429 1.00954 1.01667

00000 26253 76203 73402 23379

1.00000 1.00143 1.00570 1.01267 1,02212

00000 67802 35065 06562 67193

1.00000 1.00187 1.00745 1.01655 1.02891

00000 71775 17850 47635 00179

1.00000 1.00243 1.00964 1.02143 1i03743

00000 05914 88003 61311 56974

a e1 90' 85 80 75 70

25 30 35 40 45

1.01331 1.01864 1.02453 1.03081 1.03728

83978 21583 23143 00797 45330

1.01873 1.02622 1.03450 1.04333 1.05244

24599 04548 52308 50787 17208

1.02545 1.03563 1.04689 1.05889 1.07126

62012 21191 09786 07481 68617

1.03378 1.04729 1.06223 1.07816 1.09458

46028 03271 37524 10137 82886

1.04414 1.06179 1.08131 1.10213 1.12360

27466 07561 84270 29153 23058

1.05716 1.08002 1.10531 1.13227 1.16009

29130 00285 40947 78297 27802

65 60 55 50 45

50 55 60 65 70

1.04375 1.05003 1.05592 1.06125 1.06584

90125 67930 71242 10260 67280

1.06154 1.07037 1.07866 1.08615 1.09261

84606 85902 37978 23221 66042

1.08364 1.09564 1.10690 1.11708 1.12586

32917 39724 42279 18582 75438

1.11101 1.12694 1.14189 1.15540 1.16706

64844 63970 38846 45920 77783

1.14507 1.16589 1.18543 1.20309 1.21834

37802 54205 40490 54999 25328

1.18791 1.21489 1.24021 1.26310 1.28287

40899 61356 82552 97835 36204

40 35 30 25 20

75 80 85 90

1.06957 1.07232 1.07400 1.07456

45853 13226 34764 99318

1.09786 1.10172 1.10408 1.10488

02047 37756 99048 66859

1.13299 1.13824 1.14146 1.14254

42539 53698 12760 42177

1.17652 1.18350 1.18776 1.18920

88244 00363 94140 71150

1.23071 1.23982 1.24540 1.24728

12287 51648 69243 65857

1.29890 1.31072 1.31795 1.32039

75994 29838 95033 64540

15 10 5 0

a c1

00 5

&a

~

'0 5 10 15 20

1.00000 1.00313 1.01245 1.02768 1.04834

00000 85295 94672 16504 57003

1.00000 1.00406 1.01615 1.03589 1.06269

00000 92257 50083 51569 75825

1.00000 1.00534 1.02121 1.04715 1.08238

00000 44028 95717 56657 38086

1.00000 1.00720 1.02862 1.06363 1.11122

00000 88997 79374 90673 86903

1.00000 1.01026 1.04076 1.09068 1.15864

00000 06485 43440 07598 11101

85" 1.00000 00000 1.01663 88247 1.06618 38299 1.14751 59063 1.25875 62174

25 30 35 40 45

1.07382 1.10335 1.13604 1.17088 1.20684

76019 71989 11010 93642 51910

1.09575 1.13408 1.17651 1.22176 1.26848

73598 00433 06705 77148 10938

1.12585 1.17627 1.23214 1.29176 1.35335

71388 97795 31946 91861 85717

1.17001 1.23826 1.31398 1.39491 1.47863

24008 96285 80140 71251 07744

1.24276 1.34068 1.44960 1.56636 1.68752

19421 05139 33094 90138 66770

1.39725 1.55957 1.74151 1i93815 2.14389

25218 26706 57980 19599 95792

65 60 55 50 45

50 55 60 65 70

1.24281 1.27771 1.31046 1.34007 1.36565

67937 04815 39783 89457 16965

1.31523 1.36060 1.40320 1.44173 1.47501

31927 17261 31647 53793 81348

1.41504 1.47494 1.53123 1.58218 1.62620

43413 78592 64694 06891 90720

1.56259 1.64425 1.72108 1.79070 1.85094

67789 25175 41609 70015 39670

1.80942 1.92833 2.04054 2.14249 2.23090

88493 82823 54606 29245 12139

2.35264 2.55792 2.75309 2.93165 3.08742

71220 12198 84351 25995 47870

40 35 30 25 20

75 80 85 90

1.38640 1.40169 1.41105 1.41421

11169 28947 92570 35624

1.50203 1.52194 1.53413 1.53824

00916 10514 83232 62687

1.66195 1.68832 1.70447 1.70991

87940 00831 27784 35651

1.89989 1.93602 1.95816 1.96563

92030 35909 92561 05108

2.30289 2.35609 2.38873 2.39974

04563 12550 86793 38370

3.21489 91220 3.30946 52989 3.36764 82512

15 10 5 0

60"

65"

70"

75"

80"

In calculating elliptic functions from theta functions, when the modular angle exceeds about 60",use the descending h n d e n transformation 16.12 to induce dependence on a smaller modular angle.

90° 85 80 75 70

584

JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS

LOGARITHMIC DERIVATIVES OF THETA FUNCTIONS

'I'uhle 16.2

10"

15"

20"

25"

m

m

m

m

m

m

5 10 15 20

11.43005 5.67128 3.73205 2.74748

11.40829 5.66049 3.72495 2.74225

11.34306 5.62812 3.70365 2.72658

11.23449 5.57427 3.66823 2.70051

11.08275 5.49902 3.61876 2.66414

10.88811 5.40253 3.55536 2.61756

90 O 85 80 75 70

25 30 35 40 45

2.14451 1.73205 1.42815 1.19175 1.00000

2.14043 1.72875 1.42543 1.18949 0.99810

2.12820 1.71888 1.41729 1.18270 0.99240

2.10787 1.70248 1.40378 1.17143 0.98296

2.07952 1.67962 1.38497 1.15577 0.96985

2.04325 1.65043 1.36096 1.13581 0.95315

65 60 55 50 45

50 55 60 65 70

0.83910 0.70021 0.57735 0.46631 0.36397

0.83750 0.69888 0.57625 0.46542 0.36328

0.83273 0.69489 0.57297 0.46277 0.36121

0.82481 0.68830 0.56754 0.45839 0.35779

0.81383 0.67915 0.56001 0.45232 0.35306

0.79987 0.66754 0.55047 0.4 4 4 6 4 0.34708

40 35 30 25 20

75 80 85 90

0.26795 0.17633 0.08749 0.00000

0.26744 0.17599 0.08732 0.00000

0.26592 0.17499 0.08683 0.00000

0.26340 0.17334 0.08600 0.00000

0.25992 0.17105 0.08487 0.00000

0.25553 0.16816 0.08344 0.00000

15 10 5 0

'\a O0

0"

5"

55"

a11

ale,

30"

35"

40"

45"

m

m

m

m

5 10 15 20

10.65083 5.28496 3.47816 2.56090

10.37113 5.14645 3.38730 2.49430

10.04914 4.98711 3.28290 2.41789

9.68479 4.80696 3.16502 2.33179

9.27764 4.60585 3.03365 2.23605

8.82657 4.38332 2.88859 2.13062

90 85 80 75 70

25 30 35 40 45

1.99919 1.61498 1.33189 1.11167 0.93301

1.94749 1.57348 1.29791 1.08352 0.90958

1.88828 1.52607 1.25919 1.05154 0.88302

1.82172 1.47292 1.21591 1.01592 0.85355

1.74793 1.41419 1.16828 0.97687 0.82139

1.66695 1.35001 1.11647 0.93462 0.78679

65 60 55 50 45

50 55 60 65 70

0.78307 0.65359 0.53902 0.43543 0.33992

0.76355 0.63743 0.52579 0.42482 0.33169

0.74151 0.61923 0.51093 0.41292 0.32248

0.71714 0.59918 0.49462 0.39991 0.31242

0.69066 0.57749 0.47705 0.38595 0.30168

0.66232 0.55441 0.45846 0.37125 0.29042

40 35 30 25 20

75 80 85 90

0.25028 0.16471 0.08173 0.00000

0.24424 0.16076 0.07977 0.00000

0.2 3.7 5 1 0.15634 0.07759 0.00000

0.23017 0.15155 0.07522 0.00000

0.22235 0.14645 0.07270 0.00000

0.21419 0.14114 0.07009 0.00000

15 10 5 0

(\a O0

50" m

m

65"

70"

75"

80"

85"

m

m

m

m

m

5 10 15 20

8.32941 4.13843 2.72935 2.01530

7.78200 3.86930 2.55490 1.88950

7.17654 3.57238 2.36323 1.75208

6.49756 3.24056 2.15026 1.60057

5.71 041 2.85790 1.90678 1.42943

4.71263 2.37760 1.60605 1.22261

90 O 85 80 75 70

25 30 35 40 45

1.57876 1.28047 1.06066 0.88940 0.75000

1.48308 1.20552 1.00096 0.84142 0.71131

1.37931 1.12492 0.93737 0.79086 0.67101

1.26603 1.03795 0.86969 0.73784 0.62941

1.13996 0.94288 0.79715 0.68225 0.58682

0.99169 0.83453 0.71737 0.62344 0.54358

65 60 55 50 45

50 55 60 65 70

0.63242 0.53023 0.43911 0.35605 0.27885

0.60125 0.50526 0.41932 0.34063 0.26719

0.56918 0.47987 0.39943 0.32532 0.25574

0.53662 0.45454 0.37992 0.31054 0.24484

0.50411 0.42988 0.36140 0.29684 0.23497

0.47247 0.40690 0.34488 0.28513 0.22685

40 35 30 25 20

75 80 85 90

0.20584 0.13572 0.06742 0.00000

0.19749 0.13034 0.06478 0.00000

0.18935 0.12512 0.06224 0.00000

0.181 70 0.12026 0.05988 0.00000

0.17490 0.11601 0.05784 0.00000

0.16949 0.11272 0.05628 0.00000

15 10 5 0

Aa 00

60" m

ah

In calculating elliptic functions from theta functions, when the modular angle exceeds about Go", use the descending Landen transformation 16.12 to induce dependence on a smaller mod-

ular angle.

17. Elliptic Integrals L. M. MILNE-THOMSON

'

Contents tical Properties . . . . . . . . . . . . . . . . . . . . Definition of Elliptic Integrals . . . . . . . . . . . . Canonical Forms . . . . . . . . . . . . . . . . . . Complete Elliptic Integrals of the First and Second Kinds . Incomplete Elliptic Integrals of the First and Second Kinds Landen's Transformation . . . . . . . . . . . . . . . The Process of the Arithmetic-Geometric Mean . . . . . Elliptic Integrals of the Third Kind . . . . . . . . . .

Mathem 17.1. 17.2. 17.3. 17.4. 17.5. 17.6. 17.7.

Page

. .

.

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

589 589 589 590 592 597 598 599

Numerical Methods . . . . . . . . . . . 17.8. Use and Extension of the Tables

600 600

References

606

Table 17.1. Complete Elliptic Integrals of the First and Second Kinds and the Nomeq With Argument the Parameter m . . . . . . . . .

608

K ( m ) ,K'(m), 15D; d m ) , ql(m>, 15D; E ( m ) ,E'(m), 9D m =O( . O l ) 1 Table 17.2. Complete Elliptic Integrals of the First and Second Kinds and the Nome q W t h Argument the Modular Angle a . . . . . .

K ( 4 , K ' ( 4 , a(.>,

"

a =0 ( 1") 90

"

nl(4'

610

Eb), E ' ( 4 , 15D

Table 17.3. Parameter m With Argument K'(m)/K(m) .

. . . . . .

612

Table 17.4. Auxiliary Functions for Computation of the Nome q and the Parameterm. . . . . . . . . . . . . . . . . . . . . . . . .

612

K'(m)/K(m)=.3(.02)3, 10D

Q(m)=pi ( m ) h , 15D

ml= O( .Ol). 15 Table 17.5. Elliptic Integral of the First Kind F((P\LY)

. . . . . . .

613

. . . . . .

6 16

"

a =0 (2") 90"' 5"(10") 85", (P= 0 "(5") 90", 8D

Table 17.6. Elliptic Integral of the Second Kind E(cp\a) 0 r = 0 ~ ( 2 ~ ) 95"(10°)85", 0~, ( ~ = 0 ~ ( 5 ~ ) 98D 0~,

University of Arizona. (Prepared under contract with the National Bureau of Standards.) 587

588

ELLIPTIC INTEGRALS

Table 17.7. Jacobian Zeta Function Z('p\a)

. . . . . . . . . . . .

Page

619

Values of K(a)Z('p\a) (Y

"(2") 90°, 5"(10") 85", 'p= 0"(5") 90

=0

O,

6D

Table 17.8. Heurnan's Lambda Function A,,('p\cr) . Ao('p\4=

. . . . . . .

,

,

622

F('p\g00-(Y)+2K(~) K'@ ?r Z('p\9O0--(u), 6 D

( ~ = 0 ~ ( 2 ~ ) 95"(10°)85", 0~, 'p=0°(50)900 Table 17.9. Elliptic Integral of the Third Kind rI(n,; q\a) n=0(.1)1,

'p,

. . . . . .

625

a=0°(150)900, 5D

The author acknowledges with thanks the assistance of Ruth Zucker in the computation of the examples, Ruth E. Capuano for Table 17.3, David S. Liepman for Table 17.4, and Andreas Schopf for Table 17.9.

589

17. Elliptic Integrals Mathematical Properties 17.1. Definition of Elliptic Integrals

If R(z, y) is a rational function of z and y, where y2 is equal to a cubic or quartic polynomial in x, the integral

17.1.5

(2-s)bds-a+

+3

3 bi(3-28)Js-z+bz(l-s)J,-,

-28)Ja- Sb4Ja+1=y(x-c)-a

b3( 1

(s=l, 2, 3,

J R b , Y)dX

17.1.1

is called an elliptic integral. The elliptic integral just defined can not, in general, be expressed in terms of elementary functions, Exceptions to this are (i) when R(z,y) contains no odd powers of y. (ii) when the polynomial y2 has a repeated factor. We therefore exclude these cases. By substituting for y2 and denoting by p,(z) a polynomial in z we get

..

.)

By means of these reduction formulae and certain transformations (see Examples 1 and 2) every elliptic integral can be brought to depend on the integral of a rational function and on three canonical forms for elliptic integrals. 17.2. Canonical Forms Definitions

17.2.1

m=sin2 a ; m is the parameter, a is the modular angle 17.2.2

x=sin (p=sn

17.2.3

cos p=cn

U

U

17.2.4 (1-m sin2 (p)f=dn u=A((p), the delta amplitude

where RI($)and R2(z)are rational functions of z. Hence, by expressing R,(z) as the sum of a polynomial and partial fractions

S R (x,y)dx=SRl(z)dx+ZsA,Sxay-ldx

17.2.5

S

17.2.7

Reduction Formulae

Let 17.1.2

U,

the amplitude

Elliptic Integral of the First Kind

17.2.6

+ZsBs [ (x-c)'Y]-'~x

(p=arcsin (sn u ) = m

F((p\a)=F((plm)=S*(l-sinz~ 0 sin2e ) - W

=lz[ =lu

(1 -t 2, (1-m t z, ]-+dt

dw=u

yz= ag4+al$ +a2sC2+a3x+ u4

(laol+Ia1l#O)

=bo(s-c)~+bl(z-C)~+b2(z-C)~+b3(s-C)

Elliptic Integral of the Second Kind

+b4

(/~OI+I~l/~O)

17.1.3

Is=

S

z'y-'dz,

S

J,= [Y(z-C) s ] - l d ~

By integrating the derivatives of yxs and Y(Z-C)-~we get the reduction formulae 17.1.4

17.2.10

= i u d n 2 w dw

17.2.11

=mlufmSucnz 0 wdw

590

ELLIPTIC INTEGRALS

17.2.12 E(q\\a)=u- m

1’

I n computation the modulus is of minimal importance, since it is the parameter and its complement which arise naturally. The parameter and the modular angle will be employed in this chapter to the exclusion of the modulus.

sn2w dw

(For theta functions, see chapter 16.)

The Characteristic n

Elliptic Integral of the Third Kind

17.2.14

=J‘( 1 -n

n(n;p\a>

sin2 q - 1 [l-sinz

Ly

sin2 e

1 - 1 ~

0

If z=sn (ulm), 17.2.15

l

n(n;U [m)= 17.2.16

=sou

( 1 -nt2) -I[ ( 1 -t’) ( 1 -mt2)]-’/2dt

(l-nsn2 ( w l m ) ) - ~ w The Amplitude q

17.2.17

q=am u=arcsin (sn u)=arcsin

2

can be calculated from Tables 17.5 and 4.14. The Parameter m

Dependence on the parameter m is denoted by a vertical stroke preceding the parameter, e.g., F(dm). Together with the parameter we define the complementary parameter ml by 17.2.18

m+ml= 1

When the parameter is real, it can always be arranged, see 17.4, that 0 I m 5 1 .

The elliptic integral of the third kind depends on three variables namely (i) t h e parameter, (ii) the amplitude, (iii) the characteristic n. When real, the characteristic may be any number in the interval (-a, a). The properties of the integral depend upon the location of the characteristic in this interval, see 17.7. 17.3. Complete Elliptic Integrals of the First and Second Kinds

Referred to the canonical forms of 17.2, the elliptic integrals are said to be complete when the amplitude is i n and so x = 1 . These complete integrals are designated as follows 17.3.1

[ K ( m ])=K=

17.3.2

s,’

[ ( 1 -t 2, ( 1 - m t’) ]-‘%t

=S,

*I2

(1-m sin2 e)-1/2de

K=F($alm)=F(+a\a)

17.3.3 1

E[K(~)~=E= ( 1J - t 2 ) - 1 / 2 ( 1 -mt2)1’2dt 0

The Modular Angle LY

Dependence on the modular angle a,defined in terms of the parameter by 17.2.1, is denoted by a backward stroke \ preceding the modular angle, thus E(q\a). The complementary modular angle is ~ 1 2 - a or 90’-a according to the unit and thus ml=sin2 (90’-a)=cos2 a. The Modulus k

In terms of Jacobian elliptic functions (chapter 16), the modulus k and the complementary modulus are defined by

We also define 17.3.5 */ ! 0

K’=K(m,)=K(I-m)=J 17.3.6

K’ =F(

( l - m l sin2 e)-1/2de

I m ,) =F($a\ &r -a)

17.3.7

E’ =E(ml)=E(I-m)=

( 1 -m,sin2e) 1/2d8

S,=I2

17.2.19

k=ns (K+iK’), k’=dn K.

They are related to the parameter by kz=m, k‘2=ml. Dependence on the modulus is denoted by a comma preceding it, thus n(n; U , k).

17.3.8 E ’ = E I K ( m l ) ] = E ( m l ) = E ( ~ \ ~ ~ - a )

K and iK’ are the “real” and “imaginary” quarter-periods of the corresponding Jacobian elliptic functions (see chapter 16).

ELLIPTIC INTEGRALS

Relation to the Hypergeometric Function (see chapter 15)

17.3.9

K=$ n.F(+,3; 1; m)

17.3.10

E=&n.F(-+, 3; 1; m)

591 2

=;

+ + (7r/W2[1/12-2 5

(1 m,)

17.3.24 am u = v + C 8=l

m + (1m. 3)

2m2

m3+

2@ sin 2sv where V=HU/(211) 41+qZS)

m-10

lim [K-i In (16/m1)]=0

17.3.26

m-11

lim m-’(K--E)=lim

17.3.27

lim q/m=lim pl/ml=l/16

17.3.28

-(-)1 . 3 - 5 2 5-. m 3 . .]

m-0

2.4.6

ml+l

Alternative hvaluations of K and E (see also 17.5)

17.3.29

Legendre’s Relation

K(m)=2[1 +mt’2]-1K[ (1-m:lz)/(l

EK’+ E’K- KK‘= .4

17.3.30

Auxiliary Function

+

+

+

m1

n.

exp [-n.(K(m)+L(m))/K’(m)]

K(a)=2F(arctan

a ) \ . )

17.3.33

+[bo+bi 773.1 In (l/ml)+4m)

7711 +azm:l

+bm:l

1 c (m)I 5 3 x 10-6 ao= 1.38629 44 l = .11197 23 UZ= .07252 96

q=q(m) =exp [ - s K ’ / w

~

17.3.19

ql=q(ml) =exp [-aK/K’] 1 P‘

In - In

1 -=+

CY

Polynomial Approximations 3 (O<m< 1)

The Nome q and the Complementary Nome p1

17.3.18

-1 +cos

~ ( C Y )

K(m)=[ao+ai

q-Series

+

=2E(arctan (sec1IZa)\.)

17.3.32

m=16 exp [-a(K’(m)+L(ml))/K(m)]

The function L(m) is tabulated in Table 17.4.

17.3.17

+m~lz)]2

E(m)= (1 m y )E([(1 -mt/2)/(1 m y )12) -2 mt/2(1 m i/z) - ‘K([ (1-m:/2)/( 1 m:9] a)

16 L ( m ) = K x l n --K(m)

17.3.31 17.3.16

m-’(E--m,K)=?r/4

m-10

m-10

17.3.15. m=1-16

-21

lim K’(E-K) =O

17.3.25

17.3.12

17.3.14

(1-@a)

Limiting Values

+(F6.) . .] Z

17.3.13

$8

ail

Infinite Series

[l+(f)’

1+p

8-1

17.3.23

17.3.11

K(m)=i*

PE

K=-1 T + 2 H

17.3.22

bo= .5 bl=.12134 78 bz=.O2887 29

17.3.34

.

K(m)=~[ao+alml+ . . +a4m:l+[bo+blml+ +b*m?l In (l/mi>+e(m>

P1

17.3.20 1 1 log10- log10-=(r logloe)’=1.86152 28349 to 10D P P1

17.3.21

q=exp [-n.K’/K]=g+8 ($>’+84

(Ey

+ .. .

I e (m)I 5 2 x 10-8

uO=

1.38629 436 112

al= .09666 344259 az= .03590 092383 .03742 563713 a4= .01451 196212 ~ 3 =

16

(Iml
...

bo= .5 b1=.12498 bz= .06880 b3= .03328 b,=.00441

593597 248576 355346 787012

a The approximations 17.3.33-17.3.36 are from C. Hastings, Jr., Approximations for Digital Computers, Princeton Univ. Press, Princeton, N. J. (with permission).

592

ELLIPTIC INTEGRALS

17.3.35

F

t

+

E( m) =[I +alml +azm:l +[bl ml b m f l In (Urnl>

+ (m) e

I e(m) 1 <4X .46301 51 az= .10778 12

bl=

uI=

10-’

.24527 27

bz= .04124 96

17.3.36 E(m)=[l+alml+.

. . +a4m:]+[blml+.

..

+b4m:l In (l/mi)+o(m) I e(m) 1<2 X 10-’ a1=.44325 U2= .O626O a3= .04757 a d = .01736

141463 601220 383546 506451

b1=.24998 bz= .O92OO b3= .04069 b4= .00526

368310 180037 697526 449639

17.4. Incomplete Elliptic Integrals of the First and Second Kinds Extension of the Tables Negative Amplitude

FIGURE 17.1. Complete elliptic integral of the first kind.

17.4.1

F(- cplm) = --F(cplm)

17.4.2

E(-cplm) =--E(ullm) Amplitude of Any Magnitude

17.4.3

F(sT&vim) =2sKfF(cplm)

17.4.4

E ( u + ~ K )=E(u) + 2 E

17.4.5

E(uf2iK’) =E(u) +2i(K’-E’)

17.4.6

E(u+ 2mK+2niK‘) 17.4.7

+

=E(u) 2mE+ 2ni(K’ -E’)

E(K-U) =E--(U)

+msn

U

cd

U

Imaginary Amplitude

If tan O=sinh 17.4.8

cp

~(i++) =iF(e\3T-

a)

17.4.9

13(iCPjOL) = -iE(e\gT-a)

+iF(e\+T-a)

+i tan e(i--cos2

OL

sin2O),

Jacobi’s Imaginary Transformation

17.4.10

E(iu1m) =i[u+ dn (ulml)sc (ulmd -E ( u b 1 ) 1 Complex Amplitude

FIQURE 17.2. Complete elliptic integral of the second kind.

17.4.11

F(cp+ i#lm)=F(him) 4-iF(~ l m i )

593

ELLIPTIC INTEGRAL6 Parameter Greater Than Unity

where cot2 X is the positive root of the equation x2- [cot2 cp+m sinh2$ csc2cp-ml]x-ml cotZcp=O and m tan2 p= tan2cpcot2X- 1.

+

by which a parameter greater than unity can be replaced by a parameter less than unity.

+

a)

bi ibz bs

+iF(p\9OO-a)+-

Negative Parameter

where

17.4.17

bl=sin2 a sin X cos X sin2 p(l-sin2 a sin2 X)*

F( cpI -m) = (1 m)-)K(m( 1 m)-')

b2= (1-sin2 a sin2 X) (1 -cos2

sin2 p)* sin

a

p

p

- (1+rn)-*F

b3=cos2 p+sin2 a sin2 X sin2 p

cp

E(ul -m)

tan $ = l

+m)-*sn

-m(1

+m)t Im(1+m)-')

( U (1

1

whereby computations can be made for negative parameters, and therefore for pure imaginary modulus.

E(cp\a) +E($\a) =E(a/2\a) +sin2a sincp sin$ Values when cp is near to 1r/2 and m is near to unity can be calculated by these formulae.

"I

I m(1 +m)-1)

cd(u(1+m)+lm(l+m)-')

17.4.14

FlQ\\al

2.2

//

2.0

-

1.8

-

1.6

-

1.4

-

1.2 -

1.2

-

1.0 -

1.0

-

2.0

--cp

= (1+m)* { E(u(1+m)+lm(m+

=F(~/2\a) =K

17.4.13 F(cp\a) +F($\a)

c

17.4.18

Amplitude Near to 7r/2 (see also 17.5)

If cos a tan

+

+

cos

cp

17.4.16 E(u1m)=m$E(um*Im-') - (m- 1)u

17.4.12

E(cp i$\a) =E( X\a) -iE(p\900-

F(cplm)=m-iF(Blm-l), sin O=m* sin

17.4.15

l.8-

0 -

.8 4 0"

.6 -

6 -

3 0" 4 -

42' 0

.2 I'0 l

I

l

/

"

'

~

~

I

~

'

~

~

'

'

210.

'

;oo

I

4100

o:.

I

$0'

;on

'

Bloo

9'0. -'p

FIGURE 17.4. Incomplete elliptic integral of the first kind. F(~\u),

U

constant

594

ELLIPTIC INTEGRALS Elcp\Ql 4

Qif 9 ' -'90' b

I

1.6

-

1.4

-

1.2

-

1.0

-

.e

-

Q.=O'

24'-

22O-

zooI s"-

16"-

.6 -

4 -

FIGURE 17.7. Incomplete elliptic integral of the second kind. E(9\a), 30..

a constant

u'py- 'p'

t

F('\cu), K

FIGURE 17..5. cp-90'

a

constant.

E(cp\Q 1

t -

1.6

1.0

-

8-

5 00 4' 0

6-

-

30'

4-

200 2-

I

I

I

I

I

I

I

I

I

I

I

I

I

1

I

I

1

cx

constant.

17.4.21

I00

-

FIGURE 17.8. 90' 7 E(cp\4 -cp,

-,

~((p\900)=In (sec cp+tan .cpj =ln ban

17.4.22

F(icp\90°) =i arctan (sinh

17.4.23

E(cp\O) =(0

Special Cases

17.4.24

E(icp\O) =icp

17.4.19

F(v\O) =lp

17.4.25

E(p\90°) =sin

17.4.20

F(ip\O) =ip

17.4.26

E(icp\90°) = i sinh

p cp

cp)

595

ELLIPTIC INTEGRALS

Heuman's Lambda Function

Jacobi's Zeta Function

17.4.27

17.4.39

Z(cp\a) =E(cp\a) --E(a)F(cp\a)/K(a)

Z(ulm)=Z(u)=E(u)-uE(m)/K(m)

17.4.28

Z(-u)= -Z(u)

17.4.29

17.4.40

Z(u++K)=Z(u)

17.4.30

=2 { K(a)E(cp\90°--) lr

-Z(K+U)

17.4.31

Z(K-uu)=

17.4.32

Z(u)=Z(z(--K)

-msn (u--K)cd(u-K)

Aorcp\a J

t

Special Values

17.4.33

Z(ul0)=o

17.4.34

Z(U11) =tanh u Addition Theorem

17.4.35

Z(u+v)=Z(u)+Z(u)-msn

U

sn

+

sn (u v )

,'t

Jacobi's Imaginary Transformation

17.4.36

4-

TU

iZ(iulm)=Z(ul m,)+-,-dn(ulmJsc(ul 2KK

ml)

Relation to Jacobi's Theta Function

17.4.37

17.4.38

d Z(u)=O'(u)/@(u)=du In

2n

-

l 0

@(U)

q-Series

Z ( u ) - - x Q"(1-qp2a))-1 sin ( ~ s u / K ) -K&

KlolZ(~\nJ

A

.s e7-

a-

FIGURE 17.9. Jacobian zeta function K(a)Z(cp\a). 'See page lI.

. I \

5' ~

~

l

1

*

~

Equivalent Inverse Jacobian Elliptic Function

t Substitution

(P

117.4.41

COS

X

tan

q=-

tan

q=-

b

t = b sc

U

t = a cs

U

a= bla

a>b

m=(a'-P)/a*

a X

a q 9 - P) sin2 = ~

xZ(a2-P) d-39

sinz q=-

a2-

P

t = b nd

U

t = a dn

U

t = b sn

U

117.4.45

sin

dt [(a' - F) (P-F

)p

q=-

sin2

Q = -

sin2

(P=-

X

b

d ( P -22) P(d-9)

t = b cd U

1 17.4.47

sin

b cot a=a m=a*/(al+P)

b tan a=-

cos

x2-

U2

X2-

P

q=-

q=-

a X

b

t = a dc

dt

Tp

t = a ns

U

t = b nc

U

t= (a'+

P)112 ds U

t = b cn

U

X

sin2 q=-

a*+P d+XZ

xf(aZ+P) bya' +z2)

sin2 q=-

U

m=P/(af+P)

X

co s q=-

b

$J

U

aSirn(%9 dt

[(F- d )(F-P)]"f

597

ELLIPTIC INTEGRALS S o m e Immrtant Special Cases

1

cos p

31/'F (v\4

ff

-

2

2

45'

2's 1

37.4.58 z dt

17.4.54

Li$-v iz&*

1-22

1+22

45O

-1

45'

43+ 1-2

15'

&-l+z 17.4.59

17.4.55

*U2

a

17.4.57

17.4.53

JZm-

cos p

2

17.4.60

17.4.56

=

45'

2

Reduction of Sat/@ where P=P(t) is a cubic polynomial w i t h t h r e e r e a l factors P = (t-81) (t-82) (t-83) where 8i>h>Ba. Write

at

1 4 - 2

Reduction of s d t / @ when P=P(t)=P+alt2 a cubic polynomial with only one real root t=8. We form the h t and second derivatives P'(t), P"(t) with respect to t and then write +azt+h is

17.4.61

1 1 P''(8) 2 8 [P'(/3)]1'2

17.4.70 X2=[P'(@)]1'2, m=sin2 a=---

x=21 (@1-8a)1/2J m=sin2 a=- 8 2 -83, 81 -8a

ml=cos2 a=- 81-8 2

81-8a

17.4.71

xJ"

6

17.4.62

sin4 p=-

750

2-83

dt 4P

17.4.72

-

82 83

cos 9-

17.4.63 17.4.73 dt

CO8 p=

17.4.64 17.4.74 dt

17.4.65

cos

(2-8) -A'

(2-8) +A)

(8-2) - P (8-2)

+

v-

p= 4

(8-2)

+ (8-2)

17.5. Landen's Transformation 17.4.66

Descending Landen Transformation

Let a,,,a,,+lbe two modular angles such that 17.4.67

dt 17.4.68

17.5.1

(a"+~
and let pn, P,,+~be two corresponding amplitudes such that

* dt

17.5.2 17.4.69

( l + s h afi+l)(l+c~s aJ=2

tan ((o,,+l-d=cos a,,tan v,,

((P.+I>~

I The emphasis here is on the modular angle since this is an argument of the Tables. All formulae concerning hnden's transformation m y also be expreeeed in terms of the modulus k=mt=mn a and its complement k f = d =cos a.

ELLIPTIC INTEGRALS

598

Thus the step from n to n+ 1 decreases the modular angle but increases the amplitude. By iterating the process we can descend from a given modular angle to one whose magnitude is negligible, when 17.4.19 becomes applicable. With ao=a we have

With a0=a we have 17.5.13

n=1

17.5.14 F(cp\a) =2" II (1+sin at0

17.5.15

17.5.3

F(p\a) =2(1 +sin a)-'F(cpl\al)

F(~\CY)=

as)-lF(Vn\an)

Tl

(1 +COS as)F(qn\an) s=1 m

17.5.16 F(cp\a)=[csc a ll sin as]* In tan

(txfia)

8-1

17.5.17

@=lim (P, n+

Neighborhood of a Right Angle (see also 17.4.13)

m

F(cp\a)=+

17.5.5

II (1Ssin

a,)

When both cp and a are near to a right angle, interpolation in the table F(cp\a) is difficult. Either Landen's transformation can then be used with advantage to increase the modular angle and decrease the amplitude or vice-versa.

8=1

17.5.6

1 9 @=lim -F(cp,\a,,)=lim 2 n+m 2" n-+m 2" m

17.5.7

K=F(&r\a) =hII (1+sin

as>

17.6. The Process of the Arithmetic-Geometric Mean

8=1

17.5.8

m

F(p\a) =27r-lK@

Starting with a given number triple (ao,bol co) we p r o c e e d t o d e t e r m i n e n u m b e r t r i p l e s (alj bl, cl,), (a2, bZt cZ)t * * (UN, bN, CN) according to the following scheme of arithmetic and geometric means

17.5.9

*)

)]+sin a [a

1

fZ? sin a1sin a,+ . . .

(sin a 1 > 1 / 2 sin cpl

1 +3 (sin al sin a 2 ) 1 sin /2

2

cp2+

,

1

.,

17.5.10

sin al+- 1 sin a1sin 22

a2

+$1 sin al sin a2sin a3+ . . . Ascending Landen Transformation

Let an,an+lbe two modular angles such that 17.5.11

( I f s i n an)(l+cos an+l)=2

and let qn, such that

be two corresponding amplitudes

17.5.12 sin (29,+,-9,J

=sin ansin cpn

a

CN = (aN-1-bN- 1)

(an+l>an)

(Cpn+l
Thus the step from n to n + l increases the modular angle but decreases the amplitude. By iterating the process we can ascend from a given modular angle to one whose difference from a right angle is so small that 17.4.21 becomes applicable.

*

We stop a t the Nth step when aN=bNl i.e. when cN=O to the degree of accuracy to which the numbers are required. To determine the complete elliptic integrals K(a),E(a) we start with 17.6.2

whence 17.6.3

ao= 1 bo=cos a,c,,=sin a

599

ELLIPTIC INTEGRALS

17.7.4

T o determine K'(a), E'(a) we start with a;=1, b;=sin a,c;=cos a

17.6.5

17.7.5

whence a

K'(a)=-

17.6.6

2a&

In the above we can also use Neville's theta functions 26.36.

17.6.7

n(n\a) =K(a)+61K(a)Z(t\a)

17.7.6

Case (ii) Hyperbolic Case n> 1

To calculate F(cp\a), E(cp\a) start from 17.5.2 which corresponds to the descending Landen transformation and determine cpl, cpz, . . . , cpN successively from the relation

The case n>l can be reduced to the case O
17.6.8 tan (cpn+,-cpn)=(bn/an) tan

17.7.8

(On, VO=P

Then to the prescribed accuracy 17.6.9

F(cp\a)=(PN/(2%v)

n(n; (0\4=-U(N; +-In

cpa+

. . . +cNsin 'P,V

17.7.9

so*

n(v; cp\a)=

Case (iii) Circular Case sin2 a
=arcsin

Olc54?r

q=d4

n

= (n\a)

(n/sin2a)*,

I

,8=+?rF(c\SOO-a)/K(a)

(1-n sin2 8)-1(1-sin2 a sin2 8)%8

Case (i) Hyperbolic Case O < n < t

+')-'I

= K ( 4 -n(N\4

€=arcsin [(1-n)/cos2 a]&

n (n;3 ..\a)

17.7.2

[(A (v )+ p 1tan c p > ( A ( c p ) - ~tan ~

U@\.)

17.7. Elliptic Integrals of the Third Kind 17.7.1

cp\4+F(cp\4

where A ( q ) is the delta amplitude, 17.2.4.

acp\4 =E(cp\4 - (E/K)F(cp\4 =cI sin cpl+c2 sin

1 2Pl

*

17.6.10

*

17.7.7 N=n-l sinZa,p l = [(n-1 )(1 -nn-' sinZa)])

sin* a

17.7.10 'U=

0 5 e 5 3 ?r

+ ~F(cp\a) /K(a),62=[n(1 -n) n(n;(p\a)

17.7.11

-l

(n-sin2 a)7'

=6z(x-4@')

17.7.12

A=arctan (tanh B tan v) +2

5 (-l)a-1s-1q2s(l-q2J)-1 sin 2su sinh 2sB

8-1

[n(I-n) -'(sin' a-n)

-l])

17.7.13

sp2sinh ZsB]

[I1+ 2 5 q" cosh 2sB 8-1

17.7.14 II(n\a) =K(cY) +3di[l-A0(t\a)]

where A. is Heuman's Lambda function, 17.4.39. * S e e page

11.

600

ELLIPTIC INTEGRALS

Special Cases

17.7.18

n=O

cp\4 =F((P\a) n=o, a=o

17.7.19

cp\O)=cp

17.7.20 n = l,y= 45'

a=O

n(n;cp\O)=(l-n)-f

arctan [(1-n)+ tan

*

cp],

n< 1 = (n-1)-+ 15'

00

30'

45'

15'

60'

arctanh [(n- l ) +tan

cp],

n> 1

90'

=tan

FIGURE17.11. Elliptic integral of the third kind

n(n;cp\a,.

n=l

cp

a=n/2 n(n;(~\n/2)=(1-n)-'[ln (tan cp+sec 17.7.21

Case (iv) Circular Case n < O

The case n
-3 nt In

( l + n + sin cp)(I-ni sin

cp)

n#I

cp)-l]

n=&sin a

17.7.22

( 1 ~ s i na){2n(&sin a; cp\a)-F(cp\a)} =arctan [ ( l T s i n a) tan cp/A(cp)]

N = (sin2 a-n) (1 -n)-I p 2 = [-n(1 --)-'(sin2

n = l &cos a

17.7.23

a-n)]4

2 cos d ( l & c o s a; cp\a)=&$ln[(1+tan

17.7.16

A(cp))(l-tan cp.AL(d)-'1+3 ln [(Ncp) $cos a . t a n cp)(A(cp)-cos a tan cp)-']

*

[(l-n)(l-n-'

sin2 a)]4II(n; cp\a)

=[(l-N)(l-N-'

sin2 a)]+II(N; q\a)

F (1 f c o s

+p;' sin2aF(cp\a)+arctan [3p2 sin 2cp/A(cp)]

a)F(cp\a)

%=sin2 a

17.7.24

17.7.17

n(sin2a;cp\a)=sec* aE(cp\a)-(tan2 a sin 2cp)/(2A(cp))

II(n\a)=(--ncos2 a)(1-nn)-'(sin2 a-n)-'n(N\a)

17.7.25

+sin2 a(sin2 a-n)-lK(a)

n=l

II (1 ;p\a)

+secZa tan cpA(cp)

=F(cp\a) -sec2 aE(cp\a)

Numerical Methods 17.8. Use and Extension of the Tables Example 1. Reduce to canonical form where y~=-3x~+34~-119x2+1725-90

s

y-ldz,

B y inspection or by solving an equation of the fourth degree we find that y2=Q1Q2where Q1=3x2-105+9, Q2=--22+8x-io &1-X&2=(3+X)2~-(~0+8X)5+9$1OX ' S e e page U.

i.e., if X=-3 2 or 51

and then

Solving for Q1= (5-

Q1

and Q2 we get

1)2+2(5-2)2, Q2 =2

(5-

The substitution t= (5-1)/(5-2)

First Method

fect square if the discriminant

(10+8X)2-4(3+X)(9+10X)=0;

is a per-

S[

Sy-'dz= f

1)~--3 (s-2)'

then gives

(t2+2) (2t2-3)]-+dt

601

ELLIPTIC INTEGRALS

If the quartic y2=0 has four real roots in x (or in the case of a cubic all three roots are real), we must so combine the factors that no root of Ql=O lies between the roots of Q2=0 and no root of Q2=0 lies between the roots of Ql=O. Provided this condition is observed the method just described will always lead to real values of A. These values may, however, be irrational. Second Method

The discriminant of Q2w-Ql=zzw- (2w+1)2+1 is 4 (2w+ 1)2-4w=4w2+1 so that 1 W=Aw2+Bw+C where A=1, B=O, C=4 and if we write z2=W/w and

w=

z2=(B-Z2)z-4AC=

Write

(22)2-1=(22-1)(22+1),

~y-'dx=*~[(z~-l)(z~+l)]-l~~dz

and let the discriminant of Q2t2-Q1 be 4T2= (8t2+ 10)2-4(t2+3) (10t2$9) =4(3t2+2) (2t2--1) Then

The first method of Example 1 fails with the above values of Q1 and Q2 since the root of Ql=O lies between the roots of Q2=0, and we get imaginary values of A. The method succeeds, however, if we take Q1=x, Q2=(x-l)(2-2), for then the roots of Ql=O do not lie between those of Q 2 = O . Example 3. Find K(80/81). First Method

This method will succeed if, as here, TZ as a function of t 2 has real factors. If the coefficients of the given quartic are rational numbers, the factors of T 2 will likewise be rational.

Use 17.3.29 with m=80/81, m1=l/81, m:/2=1/9. Since [(l- m:") (1 f m:12)-1]2=.64, K(80/81) = 1.8 K(.64)=3.59154 500 to 8D, takingK(.64) from Table 17.1.

Third Method Second Method

Write w=--Q1- 3 ~ * - 1 0 ~ + 9 Qz -z2+8z- 10 and let the discriminant of Qzw-Ql be 4 W = 4 ( 3 ~ + 2 ) ( 2 ~ - 1 )= 4 ( A d + B w + C ) Then if z2=

W/w and ~ 2 = ( ~ - z 2 ) 2 - 4 ~ ~ = ( z 2 - 1 ) 2 ~ 4 8

However, in this case the factors of 2 are complex and the method fails. Of the second and third methods one will always succeed where the other fails, and if the coefficients of the given quartic are rational numbers, the factors of T2 or Z2, as the case may be, will be rational. Example 2. Reduce to canonical form y-ldz where y2=x(z-l)(z-2). We use the third method of Example 1 taking Q=(z-l), Q2=5(x-2) and writing

Table 17.4 giving L(m) is useful for computing K(m) when m is near unity or K'(m) when m is near zero. K(80/8l)=l K'(80/81) In (16X81)--L(80/81). ?r

By interpolation in Tables 17.1 and 17.4, since Sol81 =.98765 43210, K'(80/81) =1.57567 8423 L(80/81) =,00311 16543 K(80/81) =n-'(1.57567 8423) (7.16703 7877) -.00311 16543 =3.59154 5000 to 9D. Third Method

The polynomial approximation 17.3.34 gives to 8D K(80/81) =3.59154 501 Fourth Method, Arithmetic-Geometric Mean

Here sin2 a=80/81 and we start with

giving

602

0

3

4

I

I

ELLIPTIC INTEGRALS

I

I

I

I

1. 00000 00000 . 11111 1 1 1 1 1 . 99380 79900 .44444 44444 .55555 55555 .33333 33333 . 44444 44444 . 43033 14829 . 1 1 1 1 1 11111 . 43738 79636 . 43733 10380 .00705 64808 . 43735 94999 .00002 84628 . 43735 95003 0

% E: :!: 1

Thus K(80/81)=$ ~a,'=3.59154 5001. Example 4. Find E(80/81). First Method

The computation could also be made using common logarithms with the aid of 17.3.20. The point of this procedure is that it enables us to calculate p1 without the loss of significant figures which would result from direct interpolation in Table 17.1. By this means In (l/ql>can be found without loss of accuracy. Example 6. Find m to 10D when K'/K=.25 and when K'/K= 3 -5. From 17.3.15 with K'/K=.25 we can write the iteration formula m(n+l)

Use 17.3.30 which gives, with m=80/81

= 1- 1ee-4"

exp [-*L(m("))/K'(m("))].

Then by iteration using Tables 17.1 and 17.4

E ( 8 0 / 8 1 ) 10 = ~E(.64)-5 1 K(.64) =1.01910 6047 0 1 2 3

taking E(.64) and K(.64) from Table 17.1. Second Method

Polynomial approximation, 17.3.36 gives E(80/81)=1.01910 6060. The last two figures must be dropped t o k e e p within the limit of accuracy of the method. Third Method

Arithmetic-geometric mean, 17.6. The numbers were calculated in Example 3, fourth method, and we have K(80/81) -E(80/81) K(80/81)

=z1 [C2,+2C:+224+ . .

,

+25CE]

=z1 [ 1.43249 712981 =.71624 85649.

Using the value of K(80/81) found in Example 3, fourth method, we have E(80/81) =1.01910 6048 to 9D.

Example 5. Find p when m=.9995. Here ml=.0005 and so from Table 17.4

&(m)=.06251 563013 ql=mlQ(m)=.00003 12578 15.

From 17.3.19

1. .99994 42026 .99994 42041 .99994 42041

Thus m=.99994 42041. From 17.3.16 with K'/K=3.5 we can write the iteration formula, m(n+1)=16e-a*6"exp [-*L(m,'"))/K(m("))l m(n)

n 0 1 2 3

0 .(3)26841 25043 .(3)26837 68 .(3)26837 66

Thus m= ,00026 83765. The above methods in conjunction with the auxiliary Table 17.4 of L(m)enable us to extend Table 17.3 for K'/K>3, and for K'lKC.3. Example 7. Calculate to 5D the Jacobian elliptic function sn (.753421.7) using Table 17.5. Here

m=sin2 a=.7, a=56.789089O. Thus, sn (.75342[.7)=sh from

cp

where

cp

is determined

F(cp\56.789089°) =.75342. = ,95144 84701

q=.38618 125.

Inspection of Table 17.5 shows that cp lies between 40' and 45O. We have from the table of F(cp\a)

603

ELLIPTIC INTEGRALS

I

56' 63803 73914 .84450 .95479

35O 40' 45' 50'

, ,

58'

1

Second Method, Numerical Quadrature 60'

Simpson's formula with 11 ordinates and interval . I gives ,141117. Example 9. Evaluate

. 64085 . 74358 . 85122

, 63945 .74138 . 84788 .95974

l4

.96465

[(t2-2)(t2-4)]-W.

~~

From this we form the table of F(cp\56.789089')

. 63859

35O

45'

. 74003 . 84584

50'

.95674

40'

Here we can use 17.4.48 noting that a2=4, b2=2, and that 10144 10581

437

72

509

A rough estimate now shows that

cp

lies between

in the foregoing table

[(P-2) (t2-4)]-+dt=

Elrn

1 =- [F(~1\45') -F(cp2\45') 1 2 =1 [1.854075-.535623]=.659226 2

where sin

2

. sm

2 . 2 sin2a=--' 4 4

(02=-l

Thus (~=45', cpl=90°, 'p2=3Oo.

F

'p

.74003 .75040 .76082

Second Method, Numerical Integration

whence by linear inverse interpolation 9=40.5'+ .5'

l4

11090

40' and 41'. We therefore form the following table of F(cp\56.789089') by direct interpolation

40.0' 40.5' 41.0'

First Method, Reduction to Standard Form and Bivariate Interpolation

[.75342- .75040]=40.

64490

If we wish t o use numerical integration we must observe that the integrand has a singularity at t = 2 where it behaves like [8(t-2)]-6. We remove the singularity at t =2, by writing

.76082- .75040

and so sin cp=.65137=sn (.75342).7). This method of bivariate interpolation is given merely as an illustration. Other more direct methods such as that of the arithmetic-geometric mean described in 17.6 and illustrated for the Jacobian functions in chapter 16 are less laborious. Example 8. Evaluate

where

f(t)=[ (t2-

2) ( t 2 - 4) ] -4-[8(t -2) ] -t .

If we define f (2) =O, P 4

ra

can be calculated by numerical quadrature. Also First Method, Bivariate Interpolation

From 17.4.50 we have

&

s'

and thus we calculate the integral as [(2t2+l)(t2-2)1-"2dt=F(cpl\or)--F(~~\\u,

2

where 1 cos cpl=-, sin2a=--,

5

Jz cos p2=-JZ 3

2

Thus c~=26.5650512', ~1=61.8744943', (02=45', F(a\a)= 1.115921 and F(~\LY)=.go0380 and therefore the integral is equal to .141114.

l4

+

1

f ( t ) d t = 1- .340773= .659227.

Example 10. Evaluate

u=l:

(z3-7z+6)-tds.

9-7s+6=(s-l)(z-2)(~+3) with &=2, p2=1, pS=-3,

and we use 17.4.65

604

ELLIPTIC INTEGRAL8

m=sinz a=4/5, X=-@2, Thus a=63.434949',

COS'

p=3/4.

the integral=F(9O0\3O0) =? (1.071797) (1.001292) 2

p=3Oo and

=1.68575 to 5D.

2 (5)-*F(30°\63.434949') =2(5)-*(.543604)=.486214 from Table 17.5.

U=

The above integral is of the Weierstrass type and in fact 17= @(+U; 28, -24) (see chapter 18). Example 11. Evaluate

r3

(24-12t+2t2-t3)-1/2

Second Method, Ascending Transformation

We use 17.5.11 to give I+cOs c~n+l=2/(1+~ina n ) 11.1

dt.

1

- (t-2)(t2+

24-12t+2t2-&

12)= -P(t).

There is only one real zero and we therefore use 17.4.74 with P(t)=ta-2tZ+ 12t-24, 8 = 2 so that P'(2)=16, P"(2)=8, X=2 and therefore m=sin2

1 (Y=z)

a=30'

I

EI

We have

sin a n

cos un

. 94280 904 . 99956 663

.33333 333. ..__ ,02943 725 ,00021 673 ~~~

sin (2q1-9O')=sin 30°, sin (2qz-p1)=sin a1sin sin (2p3-(pz)=sin a2sin sin (2p4-p3)=sin a3sin

. 99999

998

p1=60' cpl,

(pz=57.367805'

(pz,

~ 3 =

p3,

(P= ,

57.348426'

57.348425'=9.

From 17.5.16

Therefore the given integral'is

2 2 2 F(900'300)=~ 1.94280 904 1.99956 663 1.99999 998

where

In tan

(4jo+i@)

= 1.37288 050 In tan 73,674213'

=1.37288 050(1.22789 30) F(9O0\30')= 1.68575 to 5D. Example 13. Find the value of F(89.5'\89.5').

and the integral= $[ 1.510344- 1.2125971=.148874. Example 12. Use Landen's transformation to evaluate

s,'"(1 -a sinZ8)-'"

d8 to 5D.

First Method, Descending Transformation

al=

2 =1.071797 l+cos 30'

cos al= [ (1-sin 1+sin az=

1+sin

a3=

1+cos a1

Thus from 17.5.7,

F(89.5'\89.5')

=F(90°\89 .5')

-F(#\89.5')

where

a l )(1 +sin a1)]1/2= -997419

(.5O)

#=45.00109 084'

and F(~\SS.5')=.881390 from Table 17.5. F(90°\89.5') =K(sin2 89.5') =K(.99992 38476) =6.12777 88

2

1+cos

This is a case where interpolation in Table 17.5 is not possible. We use 17.4.13 which gives

cot #=sin (3')cot (.5')=cos

We use 17.5.1 to give 1+sin

First Method

~ 1 . 0 0 1 2 9 2COS ; CYZ= ,999999

a2

=1.000000

Thus F(89.5'\89.5')=5.246389. Second Method

Landen's ascending transformation, 17.5.11, gives

605

ELLIPTIC INTEGRALS

cos al=(l-sin 89,5')/(1+sin 89.5') sin a1=[(l-cos a1)(l+cos a1)]4=.99999 99997 cos a2=0 sin a*=l.

n=$j p=45', a=3Oo,

t = a r ~ s h (n/sh2 a)4=3O0,

17.5.12 then gives sin (2pI-89.5')=sin

p= ~~(30'\30')/K(30') =,49332 60

89.5'sin 89.5'

= ,99992 38476

2p1-89.5'=89.2929049',

sin (2~p2-49~)=sinalsin sin (293-'Pz)=sin Cp2,

This is case (i) of integrals of the third kind, O
p1=89.39645 245' (PI,

~p2=89.39645 602'

pa=Cp2=@.

~=&rF(45'\30')/K(30')=.74951 61= (16/45)+

51,

and so from 17.7.3

II (k;45'\30°)

=

Thus 17.5.16 giverr

F (89.5'\89.5')

=

>'

(.99996 19231

p=.O1797 24. In (tan 89.69822 801')=5.24640.

Example 14. Evaluate

Using the q-series, 16.27, for the 6 functions we get,

II (&;45'\3Oo)=(16/45)i{ --.02995 89

la

+(l.86096 21)(.74951 51)) =.813845.

[(9-ta)(16+ta)3]-'dt t05D.

From 17.4.51 the given integral

Table 17.9 gives .81385 with 4 point Lagrangian interpolation.

where

Example 16. Evaluate the complete elliptic integral II (I$\30°) to 6D. sin a=& Sin Pi=+

sin p2=-

a=36.86990°

4,

p1=48.18968' p2d3.84264'.

3 Ji7'

By bivariate interpolation in Table 17.6 we find that the given integral

=A[ ,80904-

.41192]=.00496.

Simpson's rule with 3 ordinates gives

8 [ .00504+.01975+

s,

rtl

II (&\3O0)=K(30')

=

(l-~siIP8)-1(1-+ sin'8)-*& t 6D.

+(16/45)1/2K(a)Z(t\300)

where s=arcsin(n/sin2a)~=300. Thus using Table 17.7 II (&\30°) =1.743055. Table 17.9 gives 1.74302 with 5 point Lagrangian interpolation. Example 17. Evaluate

II (g; 45'\30')

=S,

I 4/

.OO5]= .00496.

Example 15. Evaluate

II ($ 45'\30')

From 17.7.6 we have

(l-gsina e)-l(i-+sinZe)-1/2&

to 6D. This is case (iii) of integrals of the third kind, sina a
606

ELLIPTIC INTEGRALS

arcsin [ (1 -n) /cos2 a]*=45O f l = $ ~ F ( 4 5 ~ \ 6 0 ~ ) / K ( 3=.70317 0~) 74 ~ = $ r F ( 4 5 ~ \ 3 O ~ ) / K ( 3=0.74951 ~) 51 8 2 = (40/9)4 q=.01797 24 and so from 17.7.11

Here n=-75 (p=45O, a=3Oo and since the character4 istic is greater than unity we use 17.7.7

e=

N=n-1 sin2 a=,2, pl=(1/5)+ II ($; 45O\3Oo) =-11( 2; 45°\300)$F(450\300)

Iz (8; 45O\3Oo)= (40/9)'/2(X-4p~)

--.83612+.80437 -

=2.10818 51 {.55248 32-4(.03854 26) (.74951 51) }, =.921129.

Table 17.9 gives ,92113 with 4 point Lagrangian interpolation, Example 18. Evaluate the complete elliptic integral II (#\30°) to 5 D .

=1.13214.

Numerical quadrature gives the same result. Example 20. Evaluate ll

From 17.7.14 we have

II (#\3O0)=K(3Oo)+i

(-a;

45O\3Oo)

=S,

+I4

8

[1-&(e\30°)]

(I+* sin2 e ) - y i - a

sin2 e1-m

to 5 D .

where €=arcsin [ (1-n)/cos2 a]1/2=450. Thus using Table 17.8 Iz (#\30°)=2.80099.

Here the characteristic is negative and we therefore use 17.7.15 with n=--

Table 17.9 gives 2.80126 by 6 point Lagrangian interpolation, The discrepancy results from interpolation with respect to n for cp=90° in Table 17.9. Example 19. Evaluate

1

4'

1

sin2a=4

N = (1 -n) -'(sing a-n) =.4, p,=

fi

and therefore (5/2)* ll

(-a;

45O\3Oo) =(9/4O)+II

(8; 45O\3Oo)

+k(5/2)*F(45°\300) +arc tan (35) -5

II (+; 45O\3Oo)

=S,

+I4

(1 -9 sin2 e) -1(1 -a sin2 e) -112 de

Using Tables 4.14, 17.5, and 17.9 we get

to 5 D .

II (-+; 45O\3Oo)

.76987

References Texts [17.1] A. Cayley, An elementary treatise on elliptic functions (Dover Publications, Inc., New York, N.Y., 1956). [17.2] A. Erdklyi e t al., Higher transcendental functions, vol. 2, oh. 23 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [17.3] L. V. King, On the direct numerical calculation of elliptic functions and integrals (Cambridge Univ. Press, Cambridge, England, 1924). [17.4] E. H. Neville, Jacobian elliptic functions, 2d ed. (Oxford Univ. Press, London, England, 1951). [17.5] F. Oberhettinger and W. Magnus, Anwendung der elliptischen-Funktionen in-Physik und Technik (Springer-Verlag, Berlin, Germany, 1949).

[17.6] F. Tricomi, Elliptische Funktionen (Akademische Verlagsgesellschaft, Leipzig, Germany, 1948). [17.7] E. T. Whittaker and G. N. Watson, A course of modern analysis, chs. 20, 21, 22, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952).

Tables

I

(17.81 P. F. Byrd and M. D. Friedman, Handbook of elliptic integrals for engineers and physicists (Springer-Verlag, Berlin, Germany, 1954). [17.9] C. Heuman, Tables of complete elliptic integrals, J. Math. Phys. 20, 127-206 (1941). [17.10] J. Houel, Recueil de formules e t de tables numbriques (Gauthier-Villars, Paris, France, 1901).

ELLIPTIC MTEGRALS

[17.11]E. Jahnke and F. Emde, Tables of functions, 4th ed. (Dover Publications, Inc., New York, N.Y., 1945). [17.12]L. M. Milne-Thomson, Jacobian elliptic function tables (Dover Publications, Inc., New York, N.Y., 1956). [17.13]L. M. Milne-Thomson, Ten-figure table of the complete elliptic integrals K , K ’ , E, E’ and a table of

1 9 Proc. London Math. 8:(0 17) Ot”(0 17)

-

Soc. 2, 33 (1931).

607

[17.14]L. M. Milne-Thomson, The Zeta function of Jacobi, Proc. Roy. Soc. Edinburgh 52 (1931). [17.15]L. M. Milne-Thomson, Die elliptischen Funktionen von Jacobi (Julius Springer, Berlin, Germany, 1931). [17.16]K. Pearson, Tables of the complete and incomplete elliptic integrals (Cambridge Univ. Press, Cambridge, England, 1934). [17.17]G. W. and R. M. Spenceley, Smithsonian elliptic function tables, Smithsonian Miscellaneous Collection, vol. 109 (Washington, D.C., 1947).

608

ELLIPTIC INTEGRALS Table 17.1 COMPLETE ELLIPTIC INTEGRALS OF THE FIRST AND SECOND KINDS AND THE NOME q WITH ARGUMENT THE PARAMETER rn -I l i ( m ) = f (1-?)Asin* 8) ' ( 1 8

~ ' ( n i ) = ~ ( m , )

I

7

E ( ~ ) = J (1-m : sin'

J;'(?n) =E(7tq)

s)*t/s

q(m)=exp [ - n K ' ( m ) K ( m ) ] K(m)

711

q1 ( m ) = q ( m i )

'I 04

K' (m)

ml

00000 81456 26665 36912 13525

00000 60383 23204 69025 13689

1. 00

89875 99160 91841 77648

0.00000 0.00062 0.00126 0.00190 0.00255

72484 24967 30040 14063 33320

44552 55872 24667 15229 84344

0.00320 0.00386 0.00453 0.00521 0,00589

57869 71356 55438 11618 41444

70686 22010 98018 66885 34269

0.95 0.94 0. 93 0. 92 0.91

2.57809 _ _ _ -. 2.53333 2.49263 2.45533 2.42093

21133 45460 53232 80283 29603

48173 02200 39716 21380 44303

0.00658 0.00728 0.00798 0.00870 0.00942

46515 28484 89058 30002 53141

53858 49518 49815 35762 02678

0. 90 0. 89 0. 88 0. 87 0. 86

64580 64511 98881 94514 22527

2.38901 2i35926 2.33140 2.30523 2.28054

64863 35547 85677 17368 91384

25580 45007 50251 77189 22770

0.01015 0.01089 0.01164 0.01240 0.01316

60362 53620 34936 06407 70202

37153 10173 87540 58856 86392

0. 85 0. 84 0. 8 3 0.82 0. 81

1.65962 1.66470 Lib985 1.67507 1.68037

35986 10528 07858 45692 o o k o 83368 34293 77219 28228 48361

2.25720 2.23506 2.21402 2.19397 2.17482

53268 77552 24978 09253 70902

20854 60349 46332 19189 46414

0.01394 0.01472 0.01552 0.01632 0.01714

28572 83850 38457 94906 55806

75318 66891 56320 37206 74605

0. 80 0.79 0. 78 0. 77 0. 76

0. 25 0. 26 0. 27 0.28 0.29

1.68575 1.69120 1.69674 1.70237 1.70808

03548 81991 86201 39774 67311

12596 86631 96168 10990 34606

0.01797 23870 0;oissi 01914 0.01965 92872 0.02051 99793 OiO2139 25853

08967 93399 66940 66788 82708

0. 75 0. 74 0.73 0.72 0. 71

0. 30 0. 31 0. 32 0.33 0.34

1.71388 i;7i$78 1.72577 1.73186 ii73805

94481 48080 56096 47782 53734

78791 56405 29320 52098 56358

2.07536 2.06088 2.04689 2.03336 2.02027

31352 16467 40772 94091 94286

92469 30131 10577 52233 03592

0.02227 0.02317 0.02408 0.02500 0.02594

74361 48765 52661 89803 64110

57154 35013 67250 73177 66576

0. 0. 0. 0. 0.

0.35 0. 36 0.37 0. 38 0. 39

1.74435 1.75075 1.75726 1.76389 1.77064

05972 38029 85048 83888 73233

25613 15753 82456 83731 33534

2.00759 -. 1.99530 1.98337 1.97178 1i96052

83984 27776 09795 31617 10441

24376 64729 27821 25656 65830

0.02689 0.02786 0.02884 0.02984 0.03085

79677 40785 51915 17757 43225

51443 93729 76181 44138 51033

0.65 0.64 0. 63 0. 62 0. 61

0. 40

0. 41 0. 42 0.43 0. 44

1.77751 1.78451 1.79165 1.79891 1.80632

93714 88046 01166 80391 75591

91253 81873 52966 87685 07699

1.94956 1.93890 1.92852 __ 1.91841 1.90854

77498 76652 63181 02691 70162

06026 34220 14418 09912 81211

0.03188 0.03292 0103399 0.03507 0.03617

33473 93907 30208 48344 54594

13363 86003 70043 66773 93133

0. 60 0. 59 0. 58 0. 57 0. 56

0.45 0. 46 0.47 0. 48 0. 49

1.81388 1.82159 ii82945 1.83749 1.84569

39368 27265 97985 13633 39983

16983 56821 64730 55796 74724

0.03729 55570 75822

0. 55 0.54 0. 53 0. 52 0. 51

0. 50

1.85407 46773 01372

0.04321 39182 63772

0. 50

1.57079 1.57474 ii57873 1.58278 1.58686

63267 55615 99120 03424 78474

94897 17356 07773 06373 54166

3.69563 3.35414 3.15587 3.01611

73629 14456 49478 24924

0. 08 0. 09

1.59100 1. 59518 1.59942 1.60370 1.60804

34537 82213 32446 96546 86199

90792 21610 58510 39253 30513

2.90833 2.82075 2.74707 2.68355 2.62777

0.10 0.11 0.12 0.13 0.14

1.61244 1161688 1.62139 1.62595 1i63057

13487 90905 31379 48290 55488

20219 05203 80658 38433 81754

0.15 0.16 0.17 0.18 0.19

1.63525 1.63999 1.64480 1.64967 1.65461

67322 98658 64907 82052 66675

0.20 0. 21 0.22 0.23 0. 24

0.00

0. 01 0. 02 0.03 0. 04 0. 05 0. 06 0. 07

111 1

m

~~~~

~~~

1.85407 46773 01372

K (ni)

(711)

qi(7n)

0.99 0.98 0. 97 0.96

70 69 68 67 66

m

E(m) and 'E'(m) from L. M. Milne-Thomson, Ten-figure table of the complete elliptic integrals

1

1

K , R',E, E' and a table of 71 Proc. London Math. Soc.(2)33,1931(Gth permission). *:(0,4 8: (017) -9

609

ELLIPTIC INTEGRALS

COMPLETE ELLIPTIC INTEGRALS OF THE FIRST AND SECOND KINDS AND THE NOME q WITH ARGUMENT THE PARAMETER m

Table 17.1

K’ (n1)= R(m1)

m 0. 0. 0. 0. 0.

00 01 02 03 04

1.00000 0.26219 0.22793 0.20687 0.19149

00000 62679 45740 98108 63082

00000 17709 67492 47842 09940

E(ni) 1.57079 6327 1.56686 1942 1.56291 2645 1.55894 8244 1.55496 8546

1.00000 1.01599 1.02859 1.03994 1.05050

0000 3546 4520 6861 2227

1.00 0.99 0. 98 0. 97 0. 96

0. 0. 0. 0. 0.

05 06 07 08 09

0.17931 0.16920 0.16055 0.15298 0.14624

60069 75311 42010 14810 42694

55723 46133 73011 09741 73236

1.55097 1.54696 1154293 1.53889 1.53483

3352 2456 5653 2730 3465

1.06047 1. 06998 1.07912 1.08793 1.09647

3728 6130 1407 7503 7517

0.95 0. 94 0. 93 0. 92 0.91

0.10 0.11 0.12 0.13 0.14

0.14017 0.13464 0.12957 0.12488 0.12051

31269 58847 14695 01223 71957

54262 92091 20553 52049 28729

1.53075 1.52666 1.52255 1.51842 1.51428

7631 5017 5369 8454 4027

1.10477 1.11285 1.12074 1.12845 1.13599

4733 5607 1661 0735 7843

0.90 0. 89 0. 88 0. 87 0. 86

0.15 0.16 0.17 0.18 0.19

0.11643 0.11261 0.10900 0.10558 0.10235

90607 03164 18330 93457 24235

17472 23363 23834 98477 13544

1.51012 1.50594 1.50174 1.49752 1.49329

1831 1612 3101 6026 0109

1.14339 1.15065 1.15778 1.16479 1.17169

5792 5629 6979 8293 7053

0. 0. 0. 0. 0.

0. 20 0.91 0.22 0. 23 0. 24

0.09927 0.09633 0.09353 0.09084 0.08827

36973 82749 32888 75434 12359

38825 65990 80648 60707 87862

1.48903 1.48476 1.48046 1.47615 1.47181

5058 0581 6375 2126 7514

1.17848 1.18518 1.19178 1.19829 1.20471

9924 2883 1311 0087 3641

0. 80 0.79 0.78 0.77 0. 76

0. 25 0.26 0.27 0. 28 0. 29

0.08579 0.08341 0.08111 0.07890 0.07676

57337 33938 74173 17281 08740

02195 83117 41165 26084 04317

1.46746 I;46308 1.45868 1.45426 1.44982

2209 5873 8155 8698 7128

1.21105 1.21732 1.22351 1.22963 1.23568

6028 0955 1839 1828 3836

0. 75 0.74 0.73 0. 72 0.71

0. 30 0. 3 1 0. 32 0.33 0. 34

0. 07468 0.07268 0.07074 0.06885 0.06702

99435 44965 05053 43052 25515

37179 37110 87511 47167 69108

1.44536 1.44087 1.43636 1.43183 1.42727

3064 6115 5871 1919 3,821

1.24167 1.24759 1.25345 1.25926 1.26501

0567 4538 8093 3421 2576

0. 70 0.69 0. 68 0. 67 0. 66

0. 35 0. 36 0.37 0. 38 0. 39

0.06524 0.06351 0.06182 0.06018 0.05858

21836 03934 45979 24161 16483

78738 00746 15898 79938 56838

1.42269 1.41808 1.41345 1,40879 1,40410

1133 3394 0127 0839 5019

1.27070 1.27634 1.28194 1.28748 1.29297

7480 9943 1668 4262 9239

0. 0. 0. 0. 0.

65 64 63 62 61

0. 40 0. 4 1 0. 42 0.43 0.44

0.05702 0,05549 0.05400 0.05255 0.05113

02578 63553 81850 41123 26127

14610 09081 43499 42653 21764

1.39939 1.39465 1.38988 1.38508 1.38025

2139 1652 2992 5568 8774

1.29842 1.30383 1.30919 1.31451 1.31978

8034 2008 2448 0576 7557

0. 0. 0. 0. 0.

60 59 58 57 56

0. 45 0.46 0. 47 0. 48 0. 49

0.04974 0.04838 0.04704 0.04574 0.04446

22621 17284 97634 51959 69259

64574 53289 16424 80149 25028

1.37540 1.37051 1.36559 1.36064 1.35566

1972 4505 5691 4814 1135

1.32502 1.33022 1.33538 1.34050 1.34559

4498 2453 2430 5388 2245

0. 0. 0. 0. 0.

55 54 53 52 51

0. 50

0.04321 39182 63772

ml

Pl

9 (4

1.35064 3881

E’(m)

1.35064 3881

85 84 83 82 81

0. 50 111

610

ELLIPTIC INTEGRALS

Table 17.2 COMPLETE ELLIPTIC INTEGRALS OF THE FIRST AND SECOND KINDS AND THE NOME q WITH ARGUMENT THE MODULAR ANGLE Q

-- I

5?

~ ( a ) = s , 2(l-sin2 a

2

sin2 e)

K'(~)=K(~wJ-~)

1

77

E ( ~ ) = J (l-sinz ~'

a

sin2 0)2de

Q (ff)=exp [ -TK ' (a)lK (0) 1

E' (ff)= E ( 90"-a)

Ql(ff)=Q(90"-a)

a

O0

1.57079 1.57091 1.57127 1.57187 1.57271

63267 59581 49523 36105 24349

94897 27243 72225 14009 95227

m

25564 78886 99725 49437

0.00000 0.00001 0,00007 0.00017 0.00030

00000 90395 61698 14256 48651

00000 55387 24680 42257 48814

90' 89 88 87 86

1.58054 1*57848 65776 09338 88648 95721

3.83174 3.65185 3.50042 3.36986 3.25530

98296 72652 39759 81695 19997 59694 24991 80266 29421

84146 78752 71838 68445 43555

0.00047 0.00068 0.00093 0.00122 0.00154

65699 66451 52197 24470 85045

16867 27305 97816 64294 16579

85 84 83 82 81

10 11 12 13 14

1.58284 1.58539 1.58819 1.59125 1.59456

28043 41637 72125 43820 83409

38351 75538 27520 13687 31825

3.15338 3. 06172 2.97856 2.90256 2.83267

52518 86120 89511 49406 25829

87839 38789 81384 70027 18100

0.00191 0.00231 0.00276 0.00324 0.00376

35945 79450 18093 54674 92262

90170 15821 29252 43525 86978

80 79 78 77 76

15 16 17 18 19

1.59814 1. 60197 1.60608 1.61045 1.61510

20021 85300 13494 41537 09160

12540 86952 10364 89663 67722

2.76806 2.70806 2.65213 2.59981 2.55073

31453 76145 80046 97300 14496

68768 90486 30204 61099 27254

0,00433 0.00493 0.00558 0.00627 0.00700

34205 84132 45970 23946 22602

09983 64213 58517 95994 97383

75 74 73 72 71

20 21 22 23 24

1.62002 1.62523 1.63072 1.63651 1.64260

58991 36677 91016 74093 41437

24204 58843 30788 35819 12491

2.50455 2.46099 2.41984 2.38087 2.34390

00790 94583 16537 01906 47244

01634 04126 39137 04429 46913

0.00777 0.00859 0.00944 0.01035 0.01130

46804 01752 92999 26461 08432

16442 53626 75082 44729 78049

70 69 68 67 66

25 26 27 28 29

1.64899 1.65569 1.66271 1.67005 1.67773

52184 69263 59584 94262 48840

78530 10344 91370 69580 80745

2.30878 2.27537 2.24354 2.21319 2.18421

67981 64296 93416 46949 32169

67196 11676 98626 79374 49248

0.01229 0.01333 0.01442 0.01555 0.01673

45605 45085 14412 61584 95077

27181 07947 80638 97708 33023

65 64 63 62 61

30 31 32 33 34

1.68575 1.69411 1.70283 1.71192 1.72139

03548 43573 59363 46951 08313

12596 05914 12341 55678 74249

2.15651 2.13002 2.10465 2.08035 2.05706

56474 14383 76584 80666 23227

99643 99325 91159 91578 97365

0.01797 0.01925 0.02059 0.02197 0.02341

23870 57475 05967 80013 90910

08967 39635 10437 16901 88188

60 59 58 57 56

35 36 37 38 39

1.73124 1.74149 1.75216 1.76325 1.77478

51756 92344 52364 61840 59091

57058 26774 68845 59342 05608

2.03471 2.01326 1.99266 1.97288 1.95386

53121 65652 97557 22662 48092

85791 05468 34209 74650 51663

0.02491 0.02646 0.02807 0.02974 0.03147

50625 71830 67957 53239 42771

23981 76961 17219 19583 20286

55 54 53 52 51

40 41 42 43 44

1.78676 1.79922 1.81215 1.82560 1.83956

91348 15440 98536 18981 67210

85021 49811 62126 35889 93652

1.93558 1.91799 1.90108 1.88480 1.86914

10960 75464 30334 86573 75460

04722 36423 63664 80404 26462

0.03326 0.03511 0.03704 0.03902 0.04108

52566 99625 02001 78889 50703

95577 22096 87133 26607 79885

50 49 48 47 46

45 90"-

1.85407 46773 01372

0.04321 39182 63772

45

1 2 3 4 5 6 7 8 9

1.57379 21309 24768 1.57511 36077 77251 1.57667 79815 92838

5.43490 4.74271 4.33865 4.05275

1.85407 46773 01372 K(n)

(Y

Compiled from G. W. and R. M. Spenceley, Smithsonian elliptic function tables, Smithsonian Miscellaneous Collection, vol. 109, Washington, D.C., 1947 (with permission).

,

611

ELLIPTIC INTEGRALS

COMPLETE ELLIPTIC INTEGRALS OF THE FIRST AND SECOND KINDS AND THE NOME g WITH ARGUMENT THE MODULAR ANGLE a

a

Table 17.2

900-ff

1 2 3 4

1.00000 0.40330 0.35316 0.32040 0.29548

00000 93063 56482 03371 83855

00000 38378 96037 34866 58691

1.57079 1.57067 1.57031 1.56972 1.56888

E(4 63267 67091 79198 01504 37196

5 6 7 8 9

0.27517 0.25794 0.24291 0.22956 0.21754

98048 01957 29743 71598 89496

73563 66337 06665 81194 99726

1.56780 1.56649 1.56494 1.56316 1.56114

90739 67877 75629 22295 17453

77622 60132 69419 18261 51334

1.01266 1,01723 1.02231 1.02784 1.03378

35062 69183 25881 36197 94623

34396 41019 67584 40833 90754

85 84 83 82 81

10 11 12 13 14

0.20660 0.19656 0.18728 0.17865 0.17059

97552 76611 51836 56628 45383

00965 43642 10217 04653 49477

1.55688 1.55639 1.55368 1.55073 1.54755

71966 97977 08919 19509 45758

01596 70947 36509 84013 69993

1.04011 1.04678 1.05377 1,06105 1.06860

43957 64993 69204 93337 95329

06010 44049 07046 53857 78401

80 79 78 77 76

15 16 17 18 19

0.16303 0.15591 0.14919 0.14283 0.13680

35348 66592 73690 65198 08474

21581 65792 67429 36280 28619

1.54415 1.54052 1.53666 1.53259 1.52830

04969 15741 97975 72877 62960

14673 27631 68556 45636 54359

1.07640 1.08442 1.09265 1.10106 1.10964

51130 52193 03455 21687 34135

76403 72543 37715 57941 42761

75 74 73 72 71

20 21 22 23 24

0.13106 0.12559 0.12037 0.11539 0.11062

18244 47852 82455 33684 35386

99858 09819 07894 49987 78854

1.52379 1.51907 1.51414 1.50900 1.50366

92052 85300 69174 71479 21353

59774 25531 93342 16775 53715

1.11837 1.12724 1.13624 1.14534 1.15454

77379 96377 43646 78566 66775

69864 57702 84239 80849 24465

70 69 68 67 66

25 26 27 28 29

0.10605 0.10167 0.09746 0.09342 0.08953

40201 16783 47524 26672 58769

85996 93444 70352 88483 52553

1.49811 1.49236 1.48642 1.48029 1.47396

49284 87111 68037 26638 98872

22116 24151 44253 27039 41625

1.16382 1.17317 1.18258 1.19204 1.20153

79644 93826 90849 56765 81841

93139 83722 45384 79886 13662

65 64 63 62 61

30 31 32 33 34

0.08579 0.08219 0.07872 0.07537 0.07215

57337 43773 46415 99738 43668

02195 66408 92073 58803 98737

1.46746 1.46077 1.45390 1.44686 1.43966

22093 35062 77960 92406 21471

39427 13127 65210 95183 15459

1.21105 1.22058 1.23012 1.23966 1.24918

60275 89957 72241 11752 16206

68459 54247 85949 88672 07472

60 59 58 57 56

35 36 37 38 39

0.06904 0.06603 0.06313 0.06033 0.05763

22996 86859 88302 83890 33361

09032 10861 96461 33716 79494

1.43229 1.42476 1.41707 1.40923 1.40125

09693 03101 49233 97160 97507

06756 24890 71952 46096 85523

1.25867 1.26814 1.27757 1.28695 1.29627

96247 65310 39482 37387 80079

79997 65206 50391 83001 94134

55 54 53 52 51

40 41 42 43 44

0.05501 0.05249 0.05005 0.04769 0.04541

99336 47051 44121 60340 67490

98829 04844 29953 17056 83529

1.39314 1.38488 1.37650 1.36799 1.35937

02485 65913 43257 91658 69972

23812 75413 72082 73159 75008

1.30553 1.31472 1.32384 1.33286 1.34180

90942 95602 21844 99541 60581

97794 64623 81263 17179 29911

50 49 48 47 46

45

0.04321 39182 63772

1.35064 38810 47676

45

O0

90"-.

da)

E'(a) 1.00000 1.00075 1.00258 1.00525 1.00864

15777 40855 85872 79569

01834 27552 09152 07096

90' 89 88 87 86

1.35064 38810 47676

E'(a)

00000 00000

94897 27960 97448 23979 07763

44

a

612

ELLIPTIC INTEGRALS PARAMETER rn WITH ARGUMENT K‘(rn)/K(rn) Table 17.3 K’ h7‘ m 711 111 h7 K h2.10 0.02158 74007 1.20 0.30866 25998 0. 30 0.99954 69976 2.12 0.02028 61803 1.22 0.29292 52811 0.99912 85258 0. 32 2.14 0.01906 26278 1. 24 0.27782 39170 0.99844 79307 0. 34 2.16 0.01791 21974 0.26335 17107 1. 26 0. 36 0.99740 80762 2.18 0.01683 05990 0.24949 94512 1.28 0. 38 0.99590 01861 0. 40 0. 42 0. 44 0. 46 0.48

0.99380 0.99101 0.98739 0; 98284 0.97726

79974 23521 58502 72586 54540

1. 30 1. 32 1. 34 1. 36 1.38

0.23625 0.22360 0.21154 0.20003 0.18908

58558 78874 10467 96393 70181

2.20 2.22 2.24 2. 26 2. 28

0.01581 0.01485 0.01395 0.01311 0.01232

37845 79356 94517 49385 11967

0. 50 0. 52 0. 54

0.56 0. 58

0.97056 0.96266 0.95352 0.94310 0.93138

27485 75125 60602 38029 57063

1.40 1. 42 1.44 1.46 1. 48

0.17866 0.16875 0.15934 0.15040 0.14193

58032 80773 55603 97635 21249

2. 30 2. 32 2.34 2. 36 2.38

0.01157 0.01087 0,01021 0.00959 0.00901

52117 41433 53165 62118 44574

0. 60 0. 62 0. 64 0. 66 0. 68

0.91837 0.90409 0.88859 0.87191 0.85413

61134 80105 18214 38254 42916

1. 50 1. 52 1. 54 1.56 1.58

0.13389 0.12627 0.11906 0.11223 0.10577

41273 73987 38004 54993 50300

2. 40 2. 42 2.44 2.46 2.48

0.00846 0.00795 0.00747 0.00701 0.00659

78199 41974 16117 82011 22140

0. 70 0. 72 0. 74 0. 76 0.78

0.83533 0.81560 0.79505 0.77377 0.75188

54217 91841 51193 81814 66711

1. 60 1.62 1.64 1. 66 1. 68

0.09966 0.09388 0.08843 0.08327 0.07841

53447 98538 24583 75739 01486

2. 50 2.52 2.54 2. 56 2.58

0.00619 0.00581 0.00546 0.00513 0.00481

20026 60167 27984 09763 92610

0.80 0. 82 0. 84 0. 86 0. 88

0.72949 0.70669 0.68361 0.66035 0.63700

03078 84707 86358 50204 74395

1.70 1. 72 1.74 1. 76 1.78

0.07381 0.06948 0. 06539 0. 06153 0.05789

56747 01950 03054 31533 64327

2.60 2.62 2. 64 2. 66 2.60

0. 90 0.92 0. 94 0. 96 0.98

0.61367 0.59043 0.56737 0.54457 0.52209

03730 22404 48621 30994 46531

1.80 1.82 1. 84 1.86 1.88

0.05446 0; 05123 0.04819 0.04532 0.04262

83767 77481 38272 63995 57408

2. 70 2. 72 2.74 2.76 2,78

0.00330 0.00310 0.00291 0.00274 0.00257

81448 69966 80610 05988 39151

1.00 1 02

0.50000 0.47834 0.45716 0.43651 0.41642

00000 24497 83054 71048 19278

1.90 1. 92 1.94 1. 96 1.98

0.04008 0.03768 0.03543 0.03331 0.03131

26022 81947 41720 26147 60134

2.80 2.82 2. 84 2. 86 2.88

0.00241 0.00227 0.00213 0.00200 0. 00188

73568 03103 21990 24811 06475

1.10 1.12 1.14 1.16 1.18

0.39690 97552

0; 32503 98919

2. 00 2. 02 2. 04 2. 06 2. 08

0.02943 0.02766 0.02600 0.02444 0.02297

72515 95892 66464 23873 11038

2. 90 2. 92 2.94 2.96 2.98

0.00176 0.00165 0. 00155 0.00146 0. 00137

62198 87487 78119 30127 39785

1.20

0.30866 25998

2.10

0.02158 74007

3. 00

0.00129 03591

1. 04 1. 06 1.08

For

“-p]

[(-y]

0.00452 64398 0; 00425 13725

0.00399 29873 0.00375 02764 0.00352 22924

“-“‘3

$ >3.0,$ < 0.3, see Example 6.

Table 17.4 AUXILIARY FUNCTIONS FOR: COMPUTATION OF THE NOME p AND THE P’ARAMETER m R’(m) 16 c, ( V l ) == W) I,(m)=-K(m)+In ml

ml

0 @)

m1 0.00

0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.06250 0.06281 0.06313 0.06345 0.06378 0.06411 0.06445 0.06479

00000 45660 33261 63756 38128 57394 22603 34842

[(-2’61

L (m) 00000

38302 60188 34180 42217 13714 66828 57396

See Examples 3, 5 and 6.

0.00000 0.00251 0.00506 0. 00765 0.01 027 0.01292 0.01561 0.01834

00000 65276 66040 09870 04595 58301 79344 76360

mI 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15

6, (4 0.06513 0.06549 0.06584 0.06620 0.06657 0.06694 0.06732 0.06770

95233 04937 65155 77131 42154 61556 36721 69082

36060 14101 38584 77434 15123 59704 61983 47689

I,(111)

0.02111 0.02392 0.02677 0.02966 0.03259 0.03556 0.03858 0.04165

58281 34345 14110 07472 24670 76342 73466 27452

613

ELLIPTIC INTEGRALS

ELLIPTIC INTEGRAL OF THE FIRST KIND ~(q\a)=J:

5"

(l-sinz a sin2 s)-+ds 15"

10"

Table 17.5

F(p\a)

a\p

0"

O0 2 4 6 8

0 0 0 0 0

0.08726 0.08726 0.08726 0.08726 0.08726

646 660 700 767 860

0.17453 0.17453 0.17453 0.17454 0.17454

293 400 721 255 999

0.26179 0.26180 0.26181 0.26183 0.26185

939 298 374 163 656

0.34906 0.34907 0.34909 0.34914 0.34919

585 428 952 148 998

0.43633 0.43634 0.43639 0.43647 0.43659

231 855 719 806 086

0.52359 0.52362 0.52370 0.52384 0.52403

878 636 903 653 839

10 12 14 16 18

0 0 0 0 0

0.08726 0.08727 0.08727 0.08727 0.08727

980 124 294 487 703

0.17455 0.17457 0.17458 0.17459 0.17461

949 102 451 991 714

0.26188 0.26192 0.26197 0.26202 0.26208

842 707 234 402 189

0.34927 0.34936 0.34947 0.34959 0.34972

479 558 200 358 983

0.43673 0.43691 0.43711 0.43735 0.43761

518 046 606 119 496

0.52428 0.52458 0.52493 0.52533 0.52578

402 259 314 449 529

20 22 24 26 28

0 0 0 0 0

0.08727 -.---. 0.08728 0.08728 0.08728 0.08729

940 199 477 773 086

0.17463 -.- - 0.17465 0.17467 0.17470 0.17472

611 --675 895 261 762

0.26214 . - --0.26221 0.26228 0.26236 0.26245

-568 -511 985 958 392

0.34988 0; 35004 0.35022 0.35040 0.35060

016 395 048 901 870

0.43790 0: 43822 0.43856 0.43893 0.43932

635 422 733 430 365

0.52628 0.52682 0.52741 0.52804 0.52872

399 887 799 924 029

30 32 34 36 38

0 0 0 0 0

0.08729 0.08729 0.08730 0.08730 0.08730

413 755 108 472 844

0.17475 0.17478 0.17480 0.17483 0.17486

386 119 950 864 848

0.26254 0.26263 0.26273 0.26282 0.26293

249 487 064 934 052

0.35081 0.35103 0.35126 0.35150 0.35174

868 803 576 083 218

0.43973 0.44016 0.44060 0.44107 0.44154

377 296 939 115 622

0.52942 0.53017 0.53094 0.53174 0.53257

863 153 608 916 745

40 42 44 46 48

0 0 0 0 0

0.08731 0.08731 0.08731 0.08732 0.08732

222 606 992 379 765

0.17489 0.17492 0.17496 0.17499 0.17502

887 967 073. 189 300

0.26303 0.26313 0.26324 0.26335 0.26345

369 836 404 019 633

0.35198 0.35223 0.35249 0.35274 0.35300

869 920 254 748 280

0.44203 0.44252 0.44302 0.44353 0.44404

247 769 960 584 397

0.53342 0.53429 0.53517 0.53606 0.53696

745 546 761 986 798

50 52 54 56 58

0 0 0 0 0

0.08733 0.08733 0.08733 0.08734 0.08734

149 528 901 265 620

0.17505 0.17508 0.17511 0.17514 0.17517

392 448 455 397 260

0.26356 0.26366 0.26376 0.26387 0.26396

191 643 936 020 842

0.35325 0.35350 0.35375 0.35400 0.35424

724 955 845 269 101

0.44455 0.44505 0.44555 0.44604 0.44652

151 593 469 519 487

0.53786 0.53876 0.53965 0.54053 0.54139

765 438 358 059 069

60 62 64 66 68

0 0 0 0

0.08734 0.08735 0.08735 0.08735 0.08736

962 291 605 902 182

0.17520 0.17522 0.17525 0.17527 0.17529

029 690 232 640 903

0.26406 0.26415 0.26424 0.26432 0.26440

355 509 258 556 362

0.35447 0.35469 0.35490 0.35511 0.35530

217 497 823 081 160

0.44699 0.44744 0.44787 0.44828 0.44867

117 153 348 459 252

0.54222 0.54304 0.54382 0.54456 0.54527

911 111 197 704 182

70 72 74 76 78

0 0 0 0 0

0.08736 0.08736 0.08736 0.08737 0.08737

442 681 898 092 262

0.15r532 0.17533 0.17535 0.17537 0.17538

010 949 712 289 672

0.26447 0.26454 0.26460 0.26465 0.26470

634 334 428 883 671

0.35547 0.35564 0.35579 0.35592 0.35604

959 377 326 721 488

0.44903 0.44936 0.44967 0.44994 0.45019

502 997 538 944 046

0.54593 0.54654 0.54710 0.54760 0.54804

192 316 162 364 587

80 82 84 86 88

0 0 0 0 0

0.08737 0.08737 0.08737 0.08737 0.08737

408 528 622 689 730

0.17539 0.17540 0.17541 0.17542 0.17542

854 830 594 143 473

0,26474 0.26478 0.26480 0.26482 0.26483

766 147 795 697 842

0.35614 0.35622 0.35629 0.35634 0.35636

560 881 402 086 908

0.45039 0.45056 0.45070 0.45079 0.45085

699 775 168 795 596

0.54842 0.54873 0.54898 0.54916 0.54927

535 947 608 348 042

90

0

0.08737 744

0.17542

583

0.26484 225

0.35637 851

0.45087 533

0.54930 614

5 15 25 35 45 55 65 75 85

0 0 0 0 0 0 0

0.08726 0.08727 0.08728 0.08730 0.08732 0.08734 0.08735 0.08736 0.08737

0.17453 0.17459 0.17469 0.17482 0.17497 0.17512 0.17526 0.17536 0.17541

962 198 061 397 630 935 454 525 895

0.26182 0.26199 0.26232 0.26277 0.26329 0.26382 0.26428 0.26463 0.26481

0.34911 0.34953 0.35031 0.35138 0.35261 0.35388 3.35501 0.35586 0.35631

0.43643 0.43722 0.43874 0.44083 0.44328 0.44580 0.44808 0.44981 0.45075

0.52377 0.52512 0.52772 0.53134 0.53562 0.54009 0.54419 0.54735 0.54908

0

0

0

730 387 623 289 185 084 756 998 659

20"

180 739 912 965 709 007 466 238 840

25"

842 092 330 244 989 123 092 223 976

30"

361 998 792 848 233 113 179 645 457

095 754 849 425 273 391 926 991 352

The table can also be used inversely to find v=amu where u=F('~\a) and so the Jacobian elliptic functions, for example sn u=sin 'P, cn u=cos 'P, dn U = (l-sin2 o sin2 V)W. See Examples 7-11. Compiled from K. Pearson, Tables of the complete and incomplete elliptic integrals, Cambridge Univ. Press, Cambridge, England, 1934 (with permission). Known errors have been corrected.

614

ELLIPTIC INTEGRALS

ELLIPTIC INTEGRAL OF THE FIRST KIND F(v\a)

Table 17.5

de

35" O0

40"

45"

50"

55"

60"

2 4 6 8

0.61086 0.61090 0.61103 0.61125 0.61155

524 819 691 108 010

0.69813 0.69819 0.69838 0.69869 0.69913

170 436 220 484 161

0.78539 0.78548 0.78574 0.78617 0.78678

816 509 574 974 644

0.87266 0.87278 0.87312 0.87370 0.87451

463 045 784 649 593

0.95993 0.96008 0.96052 0.96127 0.96231

109 037 821 450 911

1.04719 1.04738 1.04794 1.04888 1.05019

755 465 603 194 278

10 12 14 16 18

0.61193 0.61239 0.61294 0.61357 0.61428

318 927 707 504 140

0.69969 0.70037 0.70117 0.70209 0.70313

159 358 608 730 511

0.78756 0.78851 0.78963 0.79091 0.79236

494 403 221 768 827

0.87555 0.87682 0.87832 0.88004 0.88199

545 412 076 389 174

0.96366 0.96530 0.96723 0.96947 0.97200

180 224 998 438 462

1.05187 1.05394 1.05638 1.05919 1.06239

911 160 099 813 384

20 22 24 26 28

0.61506 0. 61592 0.61684 0.61784 0.61890

406 071 871 515 682

0.70428 0.70555 0.70692 0.70839 0.70997

706 037 183 788 451

0.79398 0.79575 0.79768 0.79976 0.80199

143 422 324 461 389

0.88416 0.88655 0.88915 0.89198 0.89501

214 254 992 071 076

0.97482 0.97794 0.98135 0,98505 0.98904

960 790 773 681 227

1.06596 1.06992 1.07425 1.07897 1.08407

891 405 976 628 347

70 32 34 36 38

0.62003 0.62121 0. 62244 0.62373 0.62505

018 138 622 019 840

0.71164 0.71341 0.71526 0.71719 0.71919

728 124 098 052 335

0.80436 0.80687 0.80951 0.81228 0.81516

610 558 599 024 039

0.89824 0.90167 0.90530 0.90911 0.91310

524 852 415 465 148

0.99331 0.99785 1.00267 1.00776 1.01311

059 743 749 438 039

1.08955 1.09540 1.10163 1.10824 1.11521

067 656 899 474 933

40 42 44 46 48

0.62642 0.62782 0.62925 0.63070 0.63216

563 630 446 385 783

0.72126 0.72338 0.72556 0.72778 0.73003

235 982 741 615 640

0.81814 0.82123 0.82440 0.82764 0.83095

765 227 346 941 712

0.91725 0.92156 0.92601 0.93059 0.93528

487 370 535 558 835

1.01870 1.02454 1.03060 1.03687 1.04333

633 127 230 427 948

1.12255 1.13024 1.13828 1.14665 1.15533

667 880 546 369 731

50 52 54 56 58

0.63363 0.63511 0.63657 0.63802 0.63945

947 150 639 636 343

0.73230 0.73458 0.73687 0.73913 0.74137

789 970 028 751 870

0.83431 0.83770 0.84110 0.84450 0.84788

247 010 344 468 483

0.94007 0.94493 0.94985 0.95479 0.95973

568 756 177 381 682

1.04997 1.05676 1.06367 1.07067 1,07772

735 412 248 128 516

1.16431 1.17356 1.18305 1.19275 1.20261

637 652 833 650 907

60 62 64 66 68

0.64084 0.64220 0.64351 0.64476 0,64595

944 613 521 839 751

0.74358 0.74572 0.74781 0.74981 0.75172

071 998 266 471 208

0.85122 0.85450 0.85769 0.86077 0.86373

375 024 220 677 057

0.96465 0.96950 0.97426 0.97889 0.98336

156 647 773 946 406

1.11226 392

1.21259 1.22263 1.23265 1.24259 1.25236

661 139 660 576 238

70 12 74 76 78

0.64707 0.64811 0.64906 0.64991 0.65067

458 189 209 829 415

0.75352 0.75519 0.75673 0.75813 0.75936

078 716 800 076 376

0.86652 0.86915 0.87157 0.87316 0.87572

996 135 159 830 037

0.98762 0.99163 0.99536 0.99876 1.00180

253 507 166 287 067

1.11864 1.12471 1.13039 1.13561 1.14031

920 530 401 610 304

1.26185 1.27098 1.27961 1.28763 1.29492

988 218 482 696 436

80 82 84 86 88

0.65132 0.65186 0.65228 0.65259 0.65277

394 270 622 116 510

0.76042 0.76130 0.76200 0.76250 0.76280

640 931 457 582 846

0.87740 0.87881 0.87992 0.88072 0.88121

833 481 495 675 143

1.00443 1.00664, 1.00839 1.00966 1.01042

942 678 470 028 658

1.14441 1.14787 1.15062 1.15261 1.15382

892 262 010 652 828

1.30135 1.30680 iIjiii7 1.31436 1.31630

321 495 166 170 510

90

0.65283 658 (-;)52]

0.76290 965

0.88137 359

1.01068 319

1.15423 455

1.31695 790

5 15 25 35 45 55 65 75 85

0.61113 0.61325 0.61733 0.62308 0.62997 0.63730 0.64414 0.64950 0.65245

0.69852 0.70162 0.70764 0.71621 0.72667 0.73800 0.74882 0.75745 0.76227

0.78594 0.79025 0.79870 0.81088 0.82601 0.84280 0.85924 0.87269 0.88036

0.87338 0.87915 0.89054 0.90718 0.92829 0.95232 0.97660 0.99710 1.00908

0.96086 0.96832 0.98317 1.00518 1.03371 1.06716 1.10223 1.13306 1.15171

1.04836 1.05774 1.07657 1.10489 1.14242 1.18788 1.23764 1.28370 1.31291

[

335 114 857 236 691 374 930 235 368

295 198 702 617 222 634 464 364 978

111 416 514 311 788 548 936 924 502

828 412 388 679 036 094 210 535 899

405 014 128 803 296 268 077 645 457

~~~

715 229 042 545 906 407 210 993 870

615

ELLIPTIC INTEGRALS

ELLIPTIC INTEGRAL OF THE FIRST KIND ~(r\a)=J:

65"

a\q

70"

Table 17.5

F(q\a)

(l-sin2 a sin2 s)-'ds

75"

85"

80"

90"

1.30899 1.30931 1.31028 1.31190 1.31417

694 959 822 491 314

1.39626 1.39663 1.39775 1.39962 1.40225

340 672 763 909 598

1.48352 1.48395 1.48523 1.48736 1.49036

986 543 342 769 470

1.57079 1.57l27 1.57271 iI575ii 1.57848

633 495 244 361 658

2 4 6 8

1.13446 1.13469 1.13537 1.13652 1.13813

401 294 994 576 158

1.22173 1.22200 1.22282 1.22420 1.22612

048 477 810 180 810

10 12 14 16 18

1.14019 1.14273 1.14572 - - .- . 1.14919 1.15313

906 032 789 .-. 471 409

1.22861 1.23165 1.23525 1.23943 1.24418

010 180 808 470 827

1.31709 1.32068 1.32494 1.32988 1.33550

778 514 296 047 840

1.40564 1.40980 1.41474 1.42048 1.42703

522 577 871 728 700

1.49423 1.49898 1.50463 1.51120 1.51870

361 627 742 474 904

1.58284 1.58819 1.59456 1.60197 1.61045

280 721 834 853 415

1.15754 967

O0

-.

---

1.16782 525 1.17369 362 1.18005 472

1.24952 1.25545 i126198 1.26913 1.27690

627 700 957 385 045

1.34183 1.34888 iI35666 1.36519 1.37448

901 616 531 359 981

1.43441 1.44264 i.4ii74 1.4 174 1.41266

578 399 466 360 958

1.52717 1.53662 1.54710 1.55863 1.57125

445 865 309 334 942

1.62002 1.63072 1.64260 1.65569 1.67005

590 910 414 693 943

30 32 34 36 38

1.18691 1.19427 1.20213 1.21050 1.21938

274 162 489 542 520

1.28530 1.29434 1.30404 1.31442 1.32547

059 605 906 210 772

1.38457 1.39547 1.40720 1.41979 1.43327

455 013 064 198 179

1.48455 1 49743 1.51134 1,52633 1.54244

455 384 644 523 734

1.58502 1.59998 1.61618 1.63370 1.65259

624 406 906 398 894

1.68575 1.70283 1.72139 1.74149 1.76325

035 594 083 923 618

40 42 44 46 48

1.22877 - .- - - . . 1.23867 1.24907 1.25998 1.27138

499. 392 904 475 210

1.33722 1.34968 1.36286 1.37676 1.39139

824 545

Oil

148 640

1.44766 1,46301 1.47934 1.49668 1.51507

938 565 287 437 416

d. 55973 1.57825 1.59806 1.61923 1.64184

441 301 493 762 453

1.67295 1.69485 1.71839 1.74369 1.77086

226 156 498 264 836

1.78676 1.81215 1.83956 1.86914 1.90108

913 985 672 755 303

50 52 54 56 58

1.28325 1.29559 1.30836 1.32154 1.33507

798 414 604 149 910

1.40676 1.42287 1.43971 1.45726 1.47551

855 717 560 935 372

1.53454 1.55513 1.57686 1.59977 1.62387

619 354 709 378 409

1.66596 1.69168 1.71910 1.74830 1.77941

542 665 125 880 482

1.80006 -_.. ~ 1.83143 1.86515 1.90143 1.94050

176 068 414 591 873

1.93558 1.97288 2Io1326 2.05706 2.10465

110 227 657 232 766

60 62 64 66 68

1.34892 - _i.363oI 1.37727 1.39159 1.40586

643 323 384 195

1.49441 087 1.51390 - _.-_ 609 1.53392 332 1.55435 972 1.57507 940

1.64917 1.67568 1.70336 1.73216 1.76199

867 359 398 516 085

1.81252 1.84776 1.88523 1.92503 1.96725

953 547 335 509 237

1.98263 2.02813 2.07735 2.13070 2.18865

957 570 219 052 839

2.15651 2.21319 2.27537 2.34390 2.41984

565 470 643 472 165

70 72 74 76 78

1.41993 1.43365 1.44684 1.45927 1.47073

796 925 001 266 163

1.59590 1.61661 1.63693 1.65651 1.67495

624 644 134 218 873

1.79268 1.82402 1.85566 1.88713 1.91779

736 292 175 308 814

2.01192 2.05903 2.10843 2.15978 2.21243

798 582 282 295 977

2.25177 2.32070 2.39615 2.47892 2.56980

995 416 610 739 281

2.50455 2.59981 2.70806 2.83267 2.97856

008 973 762 258 895

80 82 84 86 88

1.48098 1.48977 1.49690 1.50215 1.50537

006 975 410 336 033

1.69181 1.70658 1.71876 1.72786 1.73350

489 456 033 543 464

1.94682 1.97316 1.99562 2.01290 2.02384

231 666 118 452 126

2.26527 2.31643 2.36313 2.40153 2.42718

326 897 736 358 003

2.66935 2.77736 2.89146 3.00370 3.09448

045 748 664 926 898

3.15338 3.36986 3.65185 4.05275 4.74271

525 803 597 817 727

90

1.50645 424

20 22 __ 24 26 28

~~

5 15 25 35 45 55 65 75 85

1.1h7AA - - - . . 575

[);-(

1.13589 1.14740 1.17069 1.20625 1.25446 1.31490 1.38443 1.45316 1.49977

803

1.73541 516

2.02758 942

31 544 244 811 660 980 567 225 359 412

i

2.43624 605

(-3P 1.22344 1.23727 1.26548 1.30915 1.36971 1.44840 1.54409 1.64683 1.72372

604 471 460 104 948 433 676 711 395

1.31101 1.32732 1.36083 1.41338 1.48788 1.58817 1.71762 1.87145 2.00498

537 612 467 702 472 233 935 396 776

1.39859 1.41751 1.45663 1.51870 1.60847 1.73347 1.90483 2.13389 2.38364

928 762 012 347 673 444 674 514 709

. . . ~

3.13130 1 3 3

[( -3171

1.48619 1.50780 1.55273 1.62477 1.73081 1.88296 2.10348 2.43657 2.94868

317 533 384 858 713 142 169 614 876

00

1.57379 1.59814 1.64899 1.73124 1.85407 2.03471 2.30878 2.76806 3.83174

213 200 522 518 468 531 680 315 200

616

ELLIPTIC INTEGRALS

Table 17.6

ELLIPTIC INTEGRAL OF THE SECOND KIND E(+-\a)

(l-sinz

~(p\a)=,f:

a

sin2 0 ) ' d s

a\v

0"

O0 2 4 6

0 0. 0 0 0

0.08726 0.08726 0.08726 0.08726 0.08726

646 633 592 525 432

0.17453 0.17453 0.17452 0.17452 0.17451

293 185 864 330 587

0.26179 0.26179 0.26178 0.26176 0.26174

939 579 503 715 224

0.34906 0.34905 0.34903 0.34899 0.34893

585 742 218 025 181

0.43633 0.43631 0.43626 0.43618 0.43607

231 608 745 665 403

0.52359 0.52357 0.52348 0.52335 0.52315

878 119 856 123 981

10 12 14 16 18

0 0 0 0 0

0.08726 0.08726 0.08725 0.08725 0.08725

313 168 999 806 590

0.17450 0.17449 0.17448 0.17446 0.17444

636 485 137 599 879

0.26171 0.26167 0.26162 0.26157 0.26151

041 182 664 510 743

0.34885 0.34876 0.34866 0.34853 0.34840

714 657 055 954 412

0.43593 0.43575 0.43555 0.43531 0.43505

011 552 106 765 633

0.52291 0.52261 0.52227 0.52187 0.52142

511 821 039 317 828

20 22 24 26 28

0 0 0 0 0

0.08725 0.08725 0.08724 0.08724 0.08724

352 094 816 521 208

0.17442 0.17440 0.17438 0.17436 0.17433

985 926 712 353 862

0.26145 0.26138 0.26131 0.26123 0.26114

391 485 056 141 778

0.34825 0.34809 0.34791 0.34773 0.34753

492 262 800 187 510

0.43476 0.43445 0.43411 0.43375 0.43337

831 488 749 767 709

0.52093 0.52040 0.51982 0.51921 0.51856

770 357 827 436 461

30 32 34 36 38

0 0 0 0 0

0.08723 0.08723 0.08723 0.08722 0.08722

881 540 187 824 453

0.17431 0.17428 0.17425 0.17422 0.17419

250 529 714 817 852

0.26106 0.26096 0.26087 0.26077 0.26067

005 867 405 666 697

0.34732 0.34711 0.34689 0.34666 0.34642

863 342 050 093 580

0.43297 0.43256 0.43212 0.43168 0.43122

749 075 880 368 748

0.51788 0.51716 0,51643 0.51566 0.51488

193 944 040 820 638

40 42 44 46 48

0 0 0 0 0

0.08722 0.08721 0. 08721 0.08720 0.08720

075 692 307 920 535

0.17416 0.17413 0.17410 0.17407 0.17404

835 779 700 613 531

0.26057 0.26047 0.26036 0.26026 0.26016

545 261 893 492 110

0.34618 0.34594 0.34569 0.34545 0.34520

625 343 850 266 710

0.43076 0.43029 0.42981 0.42933 0.42885

236 055 431 594 776

0.51408 0.51327 0.51246 0.51163 0.51081

862 866 037 767 454

50 52 54 56 58

0 0 0 0 0

0.08720 0.08719 0.08719 0.08719 0.08718

152 774 402 039 686

0.17401 0.17398 0.17395 0.17392 0.17389

472 449 477 571 745

G.26005 0.25995 0.25985 0.25975 0.25966

795 600 574 765 224

0.34496 0.34472 0.34448 0.34425 0.34402

302 162 409 159 529

0.42838 0.42791 0.42744 0.42699 0.42655

212 0.50999 501 134 0.50918 310 775 *0.50838 287 368 0.50759 831 138 0.50683 341

60 62 54 66 58

0 0 0 0 0

0.08718 0.08718 0.08717 0.08717 0.08717

345 017 704 408 130

0.17387 0.17384 0.17381 0.17379 0.17377

013 388 883 511 283

0.25956 0.25948 0.25939 0.25931 0.25924

996 126 660 640 104

0.34380 0.34359 0.34339 0.34320 0.34302

631 575 465 404 487

0.42612 0.42571 0.42531 0.42494 0.42459

308 097 712 358 224

0.50537 0.50469 0.50404 0.50343

811 523 700 686

70 72 74 76 78

0 0 0 0 0

0.08716 0.08716 0.08716 0. 08716 0. 08716

871 633 416 223 053

0.17375 0.17373 0.17371 0.17370 0.17368

210 302 568 018 659

0.25917 0.25910 0.25904 0.25899 0.25894

090 634 767 519 917

0.34285 0.34270 0.34256 0.34243 0.34233

805 443 478 984 022

0.42426 0.42396 0.42368 0.42344 0.42322

495 339 913 363 817

0.50286 0.50234 0.50186 0.50143 0.50106

804 359 633 886 351

80 82 84 86 88

0 0 0 0 0

0.08715 0.08715 0.08715 0. 08715 0.08715

909 789 695 628 588

0.17367 0.17366 0.17365 0.17365 0.17364

498 539 789 250 926

0.25890 0.25887 0.25885 0.25883 0.25882

983 737 195 370 271

0.34223 0.34215 0.34209 0.34205 0.34202

650 915 857 507 889

0.42304 0.42289 0.42277 0.42268 0.42263

389 175 258 700 547

0.50074 0.50047 0I50026 0.50011 0.50003

232 707 923 993 003

90

0

0.08715 574

8

5"

15"

10"

0.17364 818

20"

25"

30"

0.50609 207

0.25881 905

0.34202 014

0.42261 826

0.50000 000

0.26177 0.26160 0.26127 0.26082 0.26031 0.25980 0.25935 0.25902 0.25884

0.34901 0.34860 0.34782 0.34677 0.34557 0.34436 0.34329 0.34250 0.34207

0.43623 0.43543 0.43394 0.43190 0.42957 0.42721 0.42512 0.42356 0.42272

0.52342 0.52207 0.51952 0.51605 0.51204 0.50798 0.50436 0.50164 0.50018

[(-;)41

"-231 0.08726 0 0.08725 0 0.08724 0 0.08723 0 0.08721 0 0.08719 0 0.08717 0 0.08716 0 0.08715 85 0 Example 14.

5 15 25 35 45 55 65 75

562 905 671 006 113 220 554 317 659

0.17452 0.17447 0.17437 0.17424 0.17409 0.17394 0.17380 0.17370 0.17365

624 391 550 275 157 015 680 770 493

698 165 157 567 693 639 592 064 192

329 188 632 648 562 714 797 043 467

105 791 028 776 525 938 769 271 556

670 785 597 197 932 838 656 622 720

Compiled from K. Pearson, Tables of the complete and incomplete elliptic integrals,Canibridge Univ. Press, Cambridge, England, 1934 (with permission). Known errors have been corrected.

617

ELLIPTIC INTEGRALS

ELLIPTIC INTEGRAL OF THE SECOND KIND

E(q\a)

Table 17.6

E(a\u)= .fop_ (l-sin2. sin2 s)+c/s

35"

2a\,v O0 4 6 8

0.61086 oI6ioSZ 0.61069 0.61047 0,61018

10 12 14 16 18

50"

45"

40" 524

55"

60"

365 983 171

0.69813 0.69806 0.69788 0.69756 0.69713

170 905 136 935 427

0.78539 0.78531 0.78505 0.78461 0.78401

816 125 085 792 409

0.87266 0.87254 0.87220 0.87162 0.87081

463 883 183 487 998

0.95993 0.95978 0.95933 0.95859 0.95755

109 184 459 083 301

1.04719 1.04701 1.04644 1.04551 1.04421

755 051 996 764 646

0.60980 0.60933 0.60879 0.60817 0.60748

055 793 577 636 229

0.69657 0.69590 0.69511 0.69420 0.69318

784 226 023 492 999

0.78324 0.78230 0.78120 0.77994 0.77853

162 343 308 473 323

0.66979 0.86853 0.86707 0.86539 0.86350

001 a63 031 034 481

0.95622 0.95461 0.95271 0.95054 0.94810

460 005 478 522 878

1.04255 1,04052 1.03814 1.03542 1.03236

047 491 615 177 049

20 22 24 26 28

0.60671 0.60588 0.60498 OI60402 0.60300

652 229 319 308 616

0.69206 0.69084 0.68953 0.68812 0.68663

954 814 083 308 077

0.77697 0.77527 0.77343 0.77147 0.76939

402 316 735 387 059

0.86142 0.85914 0.85668 0.85405 0.85126

062 545 781 695 295

0.94541 0.94246 0.93928 0.93587 0.93225

386 984 709 699 186

1.02897 1.02526 1.02126 1.01696 1.01238

221 804 023 224 873

30 32 34 36 38

0.60193 0.60081 0.59966 0.59846 0.59723

687 994 035 332 431

0.68506 0.68341 0.68171 0.67994 0.67813

023 817 170 830 578

0.76719 0.76489 0.76250 0.76003 0.75749

599 908 947 726 309

0.84831 0.84522 0.84201 0.83868 0.83525

663 958 414 340 115

0.92842 0.92441 0.92022 0.91588 0.91140

504 083 452 234 150

1.00755 1.00247 0.99717 0.99167 0.98598

556 977 966 469 560

40 42 44 46 48

0.59597 0.59470 0.59341 0.59211 0.59081

897 312 278 406 324

0.67626 0.67439 0.67248 0.67056 0.66863

229 630 651 191 167

0.75486 0.75223 0.74954 0.74682 0.74409

809 383 234 605 773

0.83173 0.82814 0.82449 0.82080 0.81709

189 080 369 700 775

0.90660 0.90209 0.89731 0.89246 0.88758

017 742 325 858 513

0.98013 0.97414 0.96803 0.96184 0.95558

430 397 899 497 873

50 52 54 56 58

0.58951 0.58823 0.58696 0.58571 0.58450

664 065 171 622 056

0.66670 0.66479 0.66290 0.66104 0.65922

515 183 130 317 707

0.74137 0.73865 0.73597 0.73332 0.73074

047 766 286 979 229

0.81338 0.80968 0.80601 0.80239 0.79884

346 217 230 262 217

0.88268 0.87779 0.87293 0.86812 0.86340

551 305 184 660 261

0.94929 0.94300 0.93673 0.93051 0.92439

830 285 272 931 505

60 62 64 66 68

0.58332 0,58218 0.58109 0.58006 0.57908

103 382 497 032 549

0.65746 0.65575 0.65412 0.65257 0.65110

255 905 585 197 612

0.72822 0.72578 0.72345 0.72122 0.71911

416 915 085 260 737

0.79538 0.79202 0,78879 0.78571 0.78279

015 582 839 685 987

0.85878 0.85430 0.84997 0.84583 0.84191

561 169 709 811 082

0.91839 0.91254 0.90689 0.90146 0.89630

329 821 460 778 323

70 72 74 76 78

.~ -

0,57733 0.57657 0.57588 0.57528

__ 641 189 663 450

0.64973 0.64847 0.64731 0.64628 0.64537

667 154 812 328 322

0.71714 0.71532 0.71366 0.71216 0.71085

767 545 196 766 210

0.78006 0.77753 0.77521 0.77312 0.77129

562 157 434 952 143

0.83822 0.83479 0.83165 0.82682 0.82631

090 335 223 031 879

0.89143 0.88690 0.88273 0.87896 0.87563

642 237 530 810 185

80 82 84 86 88

0.57476 0.57434 0.57400 0.57376 0.57362

897 302 912 921 470

0.64459 0.64394 0.64344 0.64307 0.64286

347 879 316 973 075

0.70972 0.70879 0.70805 0.70753 0.70721

381 019 745 050 289

0.76971 0.76840 0.76737 0.76663 0.76619

298 544 830 912 339

0.82416 0.82238 0.82097 0.81996 0.81935

694 177 770 631 604

0.87275 0.87036 0.86847 0.86712 0.86629

520 381 970 068 990

90

0.57357 644

5 15 25 35 45 55 65 75 85

0.61059 0.60849 0.60451 0,59906 0.59276 0.58633 0.58057 0.57621 0.57387

210

734 557 051 618 408 563 051 910 732

"-$'"I

0.64278 761

0.69774 0.69467 0.68883 0.68083 0.67152 0.66196 0.65333 0.64676 0.64324

083 152 790 664 549 758 844 548 351

0.70710 678

[(-:PI 0.78485 0.78059 0.77247 0.76128 0.74818 0.73464 0.72232 0.71289 0.70776

586 337 109 304 650 525 215 304 799

0.76604 444

0.81915 204

[(-;I

~

4

1

0.87194 0.86625 0.85539 0.84036 0.62265 0.80419 0.78723 0.77414 0.76697

199 642 342 234 424 500 820 195 232

0.95899 0.95166 0.93760 0.91807 0.89489 0.87052 0.84788 0.83019 0.82042

51 964 385 971 186 714 066 276 625 232

0.86602 540

[(-p71 1.04603 1.03682 1.01914 0.99445 0.96495 0.93361 0.90415 0.86079 0.86773

012 664 662 152 146 692 063 972 361

618

ELLIPTIC INTEGRALS

ELLIPTIC INTEGRAL OF THE SECOND KIND E ( p\a) E(e\a)= (1--sin2 a sin2 8 ) 'tle

Table 17.6 a\v

I;

65"

70"

75"

80"

85"

90"

O0 2 4 6 8

1.13446 1.13423 1.13354 1.13240 1.13081

401 517 929 837 573

1.22173 1.22145 1.22063 1.21926 1.21735

048 628 443 717 820

1.30899 1.30867 1.30770 1.30609 1.30385

694 442 767 916 297

1.39626 1.39589 1.39477 1.39291 1.39031

340 024 165 030 062

1.48352 1.48310 1.48182 1.47970 1.47674

986 448 929 717 288

1.57079 1.57031 1.56888 1.56649 1.56316

633 792 372 679 223

10 12 14 16 18

1.12877 1.12629 1.12338 1.12004 1.11628

602 522 066 099 624

1.21491 1.21193 1.20844 1.20443 1.19992

274 748 065 195 262

1.30097 1.29747 1.29335 1.28863 1.28331

484 215 393 089 541

1.38697 1.38292 1.37815 1.37268 1.36651

886 302 292 017 823

1.47294 1.46831 1.46287 1.45662 1.44959

312 652 363 693 085

1.55888 1.55368 1.54755 1.54052 1.53259

720 089 458 157 729

20 22 24 26 28

1.11212 1.10757 1.10265 1.09736 1.09173

778 834 204 439 228

1.19492 1.18945 1.18352 1.17715 1.17036

542 465 618 743 745

1.27742 1.27096 1.26396 1.25643 1.24840

153 502 337 578 326

1.35968 1.35218 1.34405 1.33531 1.32596

233 961 903 146 967

1.44178 1.43321 1.42392 1.41391 1.40321

179 813 023 049 335

1.52379 1.51414 1.50366 1.49236 1.48029

921 692 214 871 266

30 32 34 36 38

1.08577 1.07950 1.07295 1.06614 1,05909

404 942 961 728 660

1.16317 1.15560 iii4768 1.13943 1.13087

686 796 469 273 946

1.23988 1.23091 iIZZi5i 1.21170 1.20152

858 635 705 870

1.31605 1.30560 1.29463 1.28318 1.27128

841 436 629 499 343

1.39185 1.37986 1.36727 1.35411 1.34041

532 503 328 306 965

1.46746 1.45390 1.43966 1.42476 1.40923

221 780 215 031 972

40 42 44 46 48

1.05183 1.04438 1.03677 1,02904 1.02122

322 435 875 677 034

1.12205 1.11298 1.10371 1.09426 1.08468

408 760 291 484 023

1.19101 1.18018 1.16909 1.15777 1.14625

036 648 366 077 899

1.25896 1.24627 1.23324 1.21991 1.20633

675 240 019 241 398

1.32623 1.31158 1.29652 1.28110 1.26535

066 614 865 340 837

1.39314 1.37650 1.35937 1.34180 1.32384

025 433 700 606 218

50 52 54 56 58

1.01333 1.00542 0.99751 0.98966 0.98190

305 010 835 632 414

1.07499 1.06525 1.05550 1,04578 1.03614

796 908 682 671 663

1.13460 1.12284 1.11104 1.09923 1.08748

200 604 010 604 883

1.19255 1.17861 1.16458 1.15051 1.13645

255 873 621 210 710

1.24934 1.23311 1.21672 1.20024 1.18373

449 580 971 724 339

1.30553 1.28695 ___ _ _.1.26814 1.24918 1.23012

909 314 653 162 722

60 62 64 66 68

0.97427 354 780 0.95958 158 0.95261 084 0.94595 256

1.02663 1.01731 1.00822 0.99942 0.99099

689 023 192 966 354

1.07585 1.06440 1.05318 1.04228 1.03176

669 132 814 653 998

1.12248 1.10866 1.09507 1.08178 1.06889

590 752 580 986 476

1.16725 1.15089 1.13472 1.11882 1.10330

747 364 145 658 172

1.21105 1.19204 1.17317 1.15454 1.13624

603 568 938 668 437

70 72 74 76 78

0.93965 0.93376 0.92833 0.92340 0.91901

447 462 088 024 802

0.98297 0.97544 0.96845 0.96208 0.95638

583 068 360 074 776

1.02171 1.01220 1.00333 0.99517 0.98783

634 781 091 606 670

1.05648 1.04465 1.03350 1.02317 1.01376

221 133 951 331 904

1.08824 1.07377 1.06000 1.04707 1.03513

773 505 556 504 640

1.11837 1.10106 1.08442 1.06860 1.05377

774 217 522 953 692

80 82 84 86 88

0.91522 0.91206 0.90956 0.90776 0.90667

691 588 905 445 305

0.95143 0.94729 0.94400 0.94162 0.94017

847 291 544 171 677

0.98140 0.97598 0.97165 0.96849. 0.96657

781 331 228 392 142

1.00543 0.99831 0.99255 0.98830 0.98568

295 000 019 025 915

1.02436 1.01495 .- . . 1.00715 1.00123 0.99748

393 896 - .. 650 026 332

440 _1.04011 _ .__

90

0.90630 779

0.93969 262

5 15 25 35 45 55 65 75

1.13303 1.12176 1.10005 1.06958 1.03292 0.99358 0.95606 0.92579 0.90857

1.22001 1.20649 1.18039 1.14359 1.09900 1.05063 1.00378 0.96518 0.94269

85

o;%d8i

[( 3 9 9 1

553 337 236 479 660 365 011 978 873

"-"'1

878 962 569 813 829 981 508 626 813

30s

0.96592 583

0.98480 775

~

[(-y][(-y]

1.30698 342

ii29io6 i 2 8

1.26026 1.21665 1.16345 1.10513 1.04769 0.99915 0.96992

405 853 846 448 389 744 212

- . .

~

1.02784 1.01723 1.00864 1.00258

362 692 796 409

0.99619 470

1.00000 000

1.48087 1.45984 1.41900 1.36076 1.28885 1,20849 1.12673 1.05342 1.00394

1.56780 1.54415 1.49811 1.43229 1.35064 1.25867 1.16382 1.07640 1.01266

384 990 286 208 906 656 373 632 027

907 050 493 097 388 963 796 511 351

619

ELLIPTIC INTEGRALS

Table 17.7

JACOBIAN ZETA FUNCTION Z(v\a)

a\v

0"

O0

0

2 4 6 8

8

10

K (a)Z(v\a) = K (a)E (.\a) - E (a)F (v\a) K(90")Z(v\a)=K(90")Z(?r~l)=K(90") tanh U = - for all 5" 10" 20" 15"

U

25"

30"

0 0 0

0.000000 0.000083 0.000332 0.000748 0.001331

0.000000 0.000164 0.000655 0.001474 0.002621

0.000000 0.000239 0.000957 0.002155 0.003832

0.000000 0.000308 0.001231 0.002770 0.004928

0.000000 0.000367 0.001467 0.003302 0.005875

0.000000 0.000414 0.001658 0.003734 0.006644

12 14 16 18

0 0 0 0 0

0.002080 0.002997 0.004082 0.005337 0.006761

0.004098 0.005905 0.008043 0.010516 0.013324

0.005992 0.008635 0.011765 0.015384 0.019496

0.007706 0.011107 0.015136 0.019796 0.025094

0.009188 0.013246 0.018055 0.023621 0.029951

0.010393 0.014987 0.020433 0.026740 0.033919

20 22 24 26 28

0 0 0 0 0

0.008357 0.010125 0.012067 0.014186 0.016483

0.016470 0.019958 0.023791 0.027972 0.032508

0.024105 0.029216 0.034834 0.040968 0.047624

0.031035 0.037627 0.044878 0.052799 0.061401

0.037055 0.044942 0.053626 0.063119 0.073438

0.041981 0.050941 0. 060814 0.071617 0.083373

30 32 34 36 38

0 0 0 0 0

0.018962 0.021625 0.024476 0.027520 0.030761

0.037403 0.042664 0.048298 0.054315 0.060725

0. 05481 1 0.062540 0.070823 0.079674 0.089108

0.070696 0.080700 0.091430 0.102905 0.115148

0.084599 0.096624 0.109534 0.123356 0.138120

0.096103 0.109834 0.124596 0.140421 0.157347

40 42 44 46 48

0 0 0 0

0.034205 0.037860 0.041734 0.045835 0.050177

0.067540 0.074714 0.082444 0.090569 0.099172

0.099145 0.109807 0.121118 0.133109 0.145813

0.128185 0.142046 0.156765 0.172383 0.188947

0.153860 0.170614 0.188428 0.207353 0.227450

0.175418 0.194683 0.215197 0.237025 0.260240

50 52 54 56 58

0 0 0 0

0.054771 0. 059634 0. 064786 0.070249 0.076052

0.108280 0.117925 0.128146 0.138989 0.150510

0.159273 0.173536 0.188661 0.204716 0.221785

0.206513 0.225145 0.244921 0.265933 0.288294

0.248789 0.271452 0.295538 0.321161 0.348462

0.284929 0.311193 0.339150 0.368940 0.400731

60 62 64 66 68

0 0 0 0 0

0.082227 0.088818 0.095876 0.103468 0.111676

0.162776 0.175872 0.189901 0.204994 0.221320

0.239971 0.259398 0.280221 0.302637 0.326895

0.312138 0.337632 0.364981 0.394446 0.426356

0.377610 0.408811 0.442321 0.478462 0.517644

0.434726 0.471170 0.510371 0.552710 0.598675

70 72 74 76 78

0 0 0 0

0.120612 0.130420 0.141301 0.153537 0.167542

0.239097 0.258615 0.280272 0.304631 0.332519

0.353322 0.382351 0.414575 0.450832 0.492356

0.461145 0.499384 0.541857 0.589673 0.644462

0.560402 0.607444 0.659739 0.718657 0.786214

0.648900 0.704225 0.765797 0.835238 0.914934

80 82 84 86 88

0 0 0' 0 0

0.183967 0.203902 0.229402 0.265091 0.325753

0.365230 0.404937 0.455734 0.526833 0.647691

0.541075 0.600229 0.675918 0.781873 0.962000

0.708771 0.786884 0.886859 1.026844 1.264856

0.865556 0.961976 1.085434 1.258352 1.552420

1.008608 1.122523 1.268462 1.472953 1.820811

90

m

00

5 15 25 35 45 55 65 75 85

0 0

0

0

0

0.000519 0. 004688 0.013105 0 0.025973 0 0.043755 0 0.067477 0 0.099601 0 0.147228 0 0.245478 0 Example 16.

m

0.001023 0.009238 0.025838 0.051258 0.086448 0.133487 0.197305 0.292070 0.487761

00

0.001496 0.013513 0.037836 0.075176 0.127026 0.196567 0.291216 0.432134 0.723644

0

0.001923 0.017387 0.048754 0.097073 0.164459 0.255266 0.379430 0.565011 0.949910

00

m

0.002292 0.020743 0.058271 0.116329 0.197748 0.308149 0.460039 0.688264 1.163313

0.002592 0.023479 0.066098 0.132373 0.225942 0.353807 0.531121 0.799407 1.360551

Compiled from P.F. Byrd and M.D. Friedman, Handbook of elliptic integrals for engineers and physicists, Springer-Verlag, Berlin, Germany, 1954 (with permission).

620

ELLIPTIC INTEGRALS

Table 17.7

a\q

35"

40"

45"

50"

55"

60"

O0 2 4 6 8

0.000000 0.000450 0.001800 0.004052 0.007212

0.000000 0.000471 0.001886 0.004248 0.007561

0.000000 0.000479 0.001916 0.004314 0.007681

0.000000 0.000471 0.001887 0.004250 0. 007567

0.000000 0.000450 0.001800 0.004056 0. 007224

0.000000 0.000415 0.001659 0.003739 0. 006660

10 12 14 16 18

0.011284 0.016276 0.022197 0.029060 0. 036876

0.011833 0.017073 0.023293 0.030505 0.038728

0.012023 0.017353 0.023683 0.031029 0.039411

0.011849 0.017106 0.023354 0.030610 0.038897

0.011313 0.016337 0.022312 0.029257 0.037194

0.010433 0.015070 0.020588 0.027006 0.034347

20 22 24 26 28

0.045662 0.055435 0.066216 0.078026 0.090893

0.047979 0.058279 0.069655 0.082132 0.095744

0.048850 0.059372 0.071005 0.083783 0.097742

0.048238 0.058663 0.070203 0.082895 0.096782

0.046150 0.056156 0. 067246 0.079461 0.092844

0.042639 0.051912 0.062203 0.073551 0.086003

30 32 34 36 38

0.104844 0.119914 0.136138 0.153557 0.172220

0.110525 0.126515 0.143758 0.162305 0.182211

0.112924 0.129375 0.147147 0.166300 0.186898

0.111909 0.128330 0.146103 0.165296 0.185983

0.107447 0.123327 0.140549 0.159186 0.179319

0.099613 0.114438 0.130548 0.148018 0.166934

40 42 44 46 48

0.192178 0.213492 0.236228 0.260466 0.286295

0.203541 0.226365 0.250764 0.276831 0.304671

0.209016 0.232738 0.258158 0.285383 0.314535

0.208248 0.232187 0.257907 0.285531 0.315196

0.201042 0.224459 0.249691 0.276871 0.306156

0.187395 0.209512 0.233413 0.259243 0.287169

50 52 54 56 58

0.313816 0.343151 0.374438 0.407844 0.443565

0.334405 0.366173 0.400138 0.436490 0.475457

0.345755 0.379203 0.415067 0.453565 0.494956

0.347064 0.381317 0.418166 0.457861 0.500691

0.337723 0.371776 0.408552 0.448328 0.491428

0.317383 0.350108 0.385601 0.424167 0.466161

60 62 64 66 68

0.481836 0.522947 0.567251 0.615191 0.667330

0.517310 0.562378 0.611064 0.663870 0.721434

0.539547 0.587709 0.639896 0.696670 0.758741

0.547003 0.597211 0.651822 0.711460 0.776910

0.538238 0.589220 0.644933 0.706068 0.773487

0.512007 0.562214 0.617399 0.678320 0.745922

70 72 74 76 78

0.724397 0.787359 0.857536 0.936789 1.027859

0.784577 0.854390 0.932355 1.020563 1.122089

0.827024 0.902728 0.987491 1.083621 1.194508

0.849178 0.929590 1.019938 1.122735 1.241670

0.848294 0.931931 1.026343 1.134246 1.259612

0.821411 0.906356 1.002860 1.113848 1.243568

80 82 84 86 88

1.135017 1.265447 1.432669 1.667113 2.066078

1.241721 1.387516 1.574623 1.837147 2.284127

1.325428 1.485245 1.690632 1.979107 2.470622

1.382470 1.554749 1.776579 2.088611 2.620801

1.408589 1.591484 1.827639 2.160541 2.729164

1.398577 1.589820 1.837791 2.188502 2.788909

90

W

W

m

5 15 25 35 45 55 65 75 85

0.002813 0.025510 0.071991 0.144695 0.248154 0.390865 0.590735 0.895883 1.538234

0.002815 0.025662 0.073210 0.149686 0.263028 0.428046 0.674774 1.078397 1.977347

0.002594 0.023683 0.067742 0.139108 0.246077 0.404479 0.647089 1.056317 1.995386

m

0.002948 0.026774 0.075754 0.152865 0.263583 0.418002 0.636916 0.975016 1.692810

0

0.002994 0.027228 0.077249 0.156547 0.271538 0.433972 0.667669 1.033955 1.820471

W

0.002949 0.026855 0.076403 0.155518 0.271473 0.437641 0.680968 1.069585 1.916972

ELLIPTIC INTEGRALS

JACOBIAN ZETA FUNCTION Z(v\a) K (a)Z(.\a)

=

k' (a)E (.\a>

-E

(a)F (P\ a)

R(90")Z(rp\a)=K(900)2(uil)=h-(900)tanh a\q

O0

65"

70"

75"

Table 17.7

U=-

80"

for all U

85"

90"

2 4 6 8

0.000000 0.000367 0.001468 0.003308 0.005893

0.000000 0.000308 0.001232 0.002776 0.004946

0.000000 0.000239 0.000958 0.002160 0.003849

0.000000 0.000164 0.000656 0.001477 0.002633

0.000000 0.000083 0.000333 0.000750 0.001337

0 0 0 0 0

10 12 14 16 18

0.009233 0.013341 0.018231 0.023922 0.030438

0.007751 0.011202 0.015312 0.020098 0.025581

0.006032 0.008718 0.011920 0.015649 0.019924

0.004127 0.005966 0.008158 0.010713 0.013642

0.002096 0.003030 0.004143 0.005442 0.006930

0 0 0 0 0

20 22 24 26 28

0. 037803 0.046047 0.055206 0.065319 0.076431

0.031783 0.038732 0.046459 0.055000 0.064397

0.024763 0.030188 0.036225 0.042905 0.050260

0.016959 0.020680 0.024823 0.029411 0.034466

0.008617 0.010509 0.012617 0.014952 0.017526

0 0 0 0 0

30 32 34 36 38

0.088594 0.101867 0.116315 0.132015 0.149053

0.074696 0.085951 0.098224 0.111585 0.126114

0.058332 0.067164 0. 076808 0.087324 0.098779

0.040018 0.046099 0.052747 0.060004 0.067920

0.020354 0.023454 0.026845 0.030550 0.034595

0 0 0 0 0

40 42 44 46 48

0.167527 0.187551 0.209254 0.232785 0.258315

0.141905 0.159064 0.177713 0.197996 0.220078

0.111254 0.124839 0.139641 0.155784 0.173414

0.076554 0.085973 0.096255 0.107493 0.119798

0.039011 0.043833 0.049104 0.054874 0.061201

0 0 0 0 0

50 52 54 56 58

0.286045 0.316206 0.349070 0.384960 0.424255

0.244154 0.270454 0.299246 0.330854 0.365664

0.192704 0.213858 0.237121 0.262789 0.291220

0.133299 0.148154 0.164550 0.182720 0.202947

0.068157 0.075826 0.084312 0.093745 0.104281

0 0 0 0 0

60 62 64 66 68

0.467411 0.514976 0.567621 0.626169 0.691653

0.404143 0.446860 0.494517 0.547987 0.608372

0.322854 0.358236 0.398048 0.443155 0.494668

0.225584 0.251076 0.279993 0.313069 0.351277

0.116121 0.129521 0.144812 0.162430 0.182965

0 0 0 0 0

70 72 74 76 78

0.765385 0.849072 0.944993 1.056298 1.187535

0.677086 0.755975 0.847508 0.955095 1.083634

0.554038 0.623195 0.704762 0.802400 0.921408

0.395917 0.448779 0.512376 0.590350 0.688163

0.207230 0.236382 0.272114 0.317015 0.375226

0 0 0 0 0

80 82 84 86 88

1.345674 1.542281 1.798909 2.163806 2.790834

1.240571 1.438150 1.698985 2.073357 2.721008

1.069839 1.260828 1.518315 1.894760 2.555104

0.814374 0.983236 1.220780 1.583040 2.241393

0.453784 0.565578 0.736684 1.028059 1.628299

0 0 0 0 0

90

m

m

m

m

5 15 25 35 45 55 65 75 85

0.002295 0.020975 0.060141 0.124003 0.220781 0.366615 0.596098 0.998480 1.962673

0.001926 0.017619 0.050625 0.104764 0.187640 0.314676 0.520463 0.899033 1.866624

0.001498 0.013718 0.039483 0.081953 0.147536 0.249634 0.419877 0.751288 1.686113

0.001025 0.009390 0.027060 0.056296 0.101748 0.173397 0.295957 0.549278 1.380465

m

0.000520 0.004769 0.013755 0.028657 0.051923 0.088901 0.153297 0.293208 0.860811

W

0 0 0 0 0 0 0 0 0

622

ELLIPTIC INTEGRALS

Table

9\90"-

a)1

30"

.

O0

n

0. 5 6. 0871 .- - -

2 4 6 8

0

10 12 14 16 18

0

0

0 0 0

0 0 0

0.087129 0.087050 0.086917 0.086732

0.173648 _ -_ - 0.173595 0.173437 0.173173 0.172804

0.258879 0: 258740 0.258504 0.258111 0.257562

0.342020 0; 341916 0.341604 0.341084 0.340359

0.422618 0; 422490 0.422104 0.421462 0.420566

0.500000 0.499848 0.499391 0.498633 0.497574

0.086495 0.086206 0.085866 0.085476 0.085037

0.172332 0.171757 0.171080 0.170303 0.169429

0.256858 0.256001 0.254994 0.253838 0.252536

0.339430 0.338299 0.336969 0.335445 0.333729

0.419419 0.418024 0.416385 0.414506 0.412394

0.496219 0.494572 0.492638 0.490424 0.487937

20 22 24 26 28

0 0 0 0

0.084549 0.084013 0.083432 0. 082806 0.082136

0.168458 0.167393 0.166236 0.164991 0.163661

0.251092 0.249509 0.247790 0.245941 0.243966

0.331827 0.329743 0.327483 0.325052 0.322458

0.410054 0.407492 0.404717 0.401736 0.398558

0.485184 482176 0.478920 0.475428 0.471710

30 32 34 36 38

0 0 0 0 0

0.081425 0.080674 0.079884 0.079058 0.078198

0.162247 0.160755 0.159187 0.157548 0.155842

0.241870 0.239657 0.237335 0.234908 0.232383

0.319707 0.316806 0.313764 0.310587 0.307286

0.395191 0.391645 0.387930 0.384057 0.380037

0.467777 0.463642 0.459316 0.454813 0.450147

40 42 44 46 48

0 0 0 0

0.077307 0.076385 0. 075436 0.074463 0.073469

0.154073 0.152246 0.150367 0.148439 0.146470

0.229767 0.227068 0.224292 0.221447 0.218543

0.303869 0.300346 0.296727 0.293022 0.289242

0.375880 0.371600 0.367209 0.362720 0.358145

0.445330 0.440378 0.435306 0.430127 0.424860

50 .~

52 54 56 58

0 0 0 0 0

0.072455 0.071426 0.070385 0.069336 0.068281

0.144464 0.142428 0.140370 0.138295 0.136211

0.215587 0.212589 0.209558 0.206506 0.203443

0.285399 0.281505 0.277573 0.273616 0.269648

0.353500 0.348799 0.344057 0.339290 0.334516

0.419519 0.414121 0.408685 0.403228 0.397769

60 62 64 66 68

0 0 0 0 0

0.067226 0.066175 0.065131 0.064100 0.063088

0.134126 0.132049 0.129989 0.127955 0.125958

0.200380 0.197331 0.194307 0.191324 0.188396

0.265684 0.261739 0.257832 0.253979 0.250200

0.329751 0.325015 0.320328 0.315710 0.311185

0.392328 0.386926 0.381586 0.376331 0.371186

70 72 74 76 78

0 0 0 0 0

0.0621 00 0.061143 0.060223 0.059348 0.058528

0.124009 0.122121 0.120307 0.118583 0.116967

0.185540 0.182774 0.180119 0.177596 0.175231

0.246517 0.242952 0.239531 0.236282 0.233238

0.306778 0.302515 0.298427 0.294547 0.290914

0.366180 0.361342 0.356706 0.352309 0.348194

0.115479 0.114143 OI 112988 0.112053 0.111392

0.173054 0.171099 _ _ -. 0.169410 0.168043 0.167078

0.230436 0.227922 0; 225750 0.223992 0.222751

0.287571 0.284573 0; 281983 0.279887 0.278408

0.344410 0.341017 0.338088 0.335718 0.334046 0.333333

0

0

80 -~

0

82 84 86 88

0

0 0 0

0.057773 0.057095 0; 016508 0.056034 0.055698

90

0

0.055556

0.111111

0.166667

0.222222

0.277778

(-512 [ 5 1

(-515 [ 5 1

(-517 [ 6 1

(-519 [ 6 1

(-4)l [ 6 1

0.086990 0.085677 0.083124 0.079476 0.074953 0.069861 0.064614 0.059779 0.056256

0.173318 0.170704 0.165625 0.158377 0.149408 0.139334 0.128968 0.119433 0.112490

0.258327 0.254434 0.246882 0.236134 0.222878 0.208034 0.192809 0.178839 0.168682

0.341370 0.336231 0. 326288 0.312192 0.294884 0.275597 0.255897 0.237883 0.224814

0.421815 0.415475 0.403252 0.386013 0.364976 0.341676 0.318009 0.296459 0.280867

5 15 25 35 45 55 65 75 85

0 0 0 0

0 0 0 0 0

o.

[(-64) '1

0.499050 0.491565 0.477203 0.457086 0; 432729 0.405958 0.378946 0.354475 0.336826

Compiled from C. Heuman, Tables of complete elliptic integrals, J. Math. Phys. 20, 127-206, 1941 (with perndssion).

623

ELLIPTIC INTEGRALS

HEUMAN'S LAMBDA FUNCTION Ao(p\a)

l a h l e 17.8

Ag(p\a)

= F(a'900-a) ---__ +: K (a)z((o\900-a) =;2 { K ( c E(p\9Oo ) - a) - [ K(a)-E: (a)]F(a\9Oo- a) I

K' (a)

35"

40"

45"

50"

55"

60"

0.573576 0.573402 0.572878 0.572009 0.570795

0.642788 0.642592 0.642006 0.641032 0.639674

0.707107 0.706891 0.706247 0.705177 0.703687

0.766044 0.765811 0.765113 0.763956 0.762347

0.819152 0.818903 0.818157 0.816922 0.815210

0.866025 0.865762 0.864975 0.863674 0.861876

10 12 14 16 18

0.569244 0.567360 0.565150 0.562623 0.559789

0.637940 0.635836 0.633373 0.630561 0.627412

0.701786 0.699484 0.696794 0.693729 0.690306

0.760298 0.757822 0.754937 0.751660 0.748011

0.813034 0.810416 0.807375 0.803935 0.800123

0.859602 0.856877 0.853731 0.850194 0.846297

20 22 24 26 28

0.556657 0.553238 0.549546 0.545591 0.541389

0.623939 0.620157 0.616080 0.611725 0.607107

0.686540 0.682450 0.678054 0.673372 0.668422

0.744012 0.739683 0.735049 . . - .. . 0.730130 0.724951

0.795963

0.786709 0.781667 0.776384

0.842073 0.837553 0.832766 0.827743 0.822510

30 32 34 36 38

0.536953 0.532297 0.527437 0: 5Z2386 0.517165

0.602244 . ..__ 0.597153 0.591851 0.586356 0.580687

0.663225 0.657801 0.652170 0.646351 0.640365

0.719533 0.713900 0.708073 0.702074 0.695923

0.770883 0.765190 0.759326 0.753314 0.747177

0.817093 0.811517 0.805804 0.799976 0.794052

40 42 44 46 48

0.511786 0.506266 0.500622 0.494873 0.489034

0.574862 0.568898 0.562815 0.556632 0.550366

0.634231 0.627970 0.621600 0.615142 0.608615

0.689642 0.683251 0.676769 OI 670217 0.663613

0.740932 0.734602 0; 728203 0.721756 0.715277

0.788051 0.781992 0.775891 0.769764 0.763627

5a 52 54 56 58

0.483125 0.477164 0.471170 0.465163 0.459163

0.544038 0.531275 0.524879 0.518502

0.602038 0.595432 0.588017 0.582212 0.575640

0.656976 0.650326 0.643682 0.637064 0.630491

0.708785 0.702298 0.695832 0.689405 0.683037

0.757496 0. 75.1385 0.745310 0.739286 0.733329

60 62 64 66 68

0.453192 0.447272 0.441428 0.435683 0.430065

0.512167 0.505895 0.499711 0.493642 0.487715

0.569122 0.562680 0.556339 0.550124 0.544062

0. 623985 0.617567 0.611258 0.605085 0.599072

0.676745 0.670549 0.664469 0.658528 0.652749

0.727455 0.721680 0.716024 0.710504 0.705142

70 72 74 76 78

0.424604 0.419332 0.414284 0.409500 0.405026

0.481959 0.476408 0.471098 0.466070 0.461371

0.538183 0.532519 0.527106 0.521985 0.517202

0.593247 0.587641 0.582290 0.577231 0.572511

0.647159 0.641784 0.636659 0.631818 0.627303

0.699961 0.694985 0.690244 0.685770 0.681601

80 82 84 86 88

0.400915 0.397229 0.394049 0.391477 0.389662

0.457055 0.453189 0.449853 0.447157 0.445255

0.512813 0.508883 0.505494 0.502754 0.500823

0.568181 0.564307 0.560967 0.558268 0.556366

0.623166 0.619464 0.616276 0.613700 0.611884

0.677782 0.674368 0.671427 0.669053 0.667379

90

0.388889

"-"'1

0.444444

[(-;)'I

0.500000

0.611111

0.666667

a\q

O0 2 4 6 8

5 15 25 35 45 55 65 75 85

0.572487 0.563926 0.547600 0.524935 0.497760 0.468167 0.438541 0.411857 0.392679

0: 537668

0.641567

0. 63201 0 0.613936 0.589127 0.559735 0.528076 0.496661 0.468546 0.448417

"-;)'I

0.705765 0.695307 0.675748 0.649283 0.618381 0.585512 0.553214 0.524506 0.504034

I';-[

0.555556

0.764592 0.753346 0.732623 0.705094 0.673501 0.640369 0.608153 0.579721 0.559529

OI i9I483

[(-2I' 0.817600 0.805703 0.784220 0.756337 0.724985 0.692612 0.661480 0.634200 0.614903

0.864388 0.852010 0.830282 0.802903 0.772830 0.742291 0.713246 0.687972 0.670162

ELLIPTIC LNTEGRALS

624 Table 17.8

65"

a\p

70"

75"

Ao('p\a)

80"

85"

90"

2 4 6 8

0.906308 0.906032 0.905210 0.903857 0.901997

0.939693 0.939407 0.938559 0.937172 0.935282

0.965926 0.965633 0.964769 0.963376 0.961512

0.984808 0.984511 0.983652 0.982315 0.980599

0.996195 0.995903 0.995130 0.994063 0.992833

1 1

10 12 14 16 18

0.899660 0.896881 0.893699 0.890152 0.886280

0.932934 0.930177 0.927061 0.923634 0.919940

0.959244 0.956638 0.953755 0.950646 0.947355

0.978597 0.976384 0.97401 6 0.971534 0.968969

0.991511 0.990135 0.988727 0.987299 0.985858

1 1 1

20 22 24 26 28

0.882119 0.877704 0.873068 0.868240 0.863249

0.916018 0. 91.1904 0.907630 0.903221 0.898703

0.943918 0.940364 0.936718 0.933000 0.929226

0.966343 0.963671 0.960968 0.958241 0.955500

0.984410 0.982958 0.981506 0.980054 0.978604

1 1

30 32 34 36 38

0.858117 0.852869 0.847523 0.842100 0.836615

0.894095 0.889416 0.884681 0.879904 0.875099

0.925409 0.921563 0.917695 0.913817 0.909935

0.952751 0.949998 0.947247 0.944502 0.941766

0.977159 0.975719 0.974286 0.972861 0.971445

1

40 42 44 46 48

0.831085 0.825524 0.819946 0.814365 0.808792

0.870277 0.865449 0.860625 0.855814 0.851026

0.906056 0.902188 0.898337 0.894508 0.890708

0.939042 0.936335 0.933647 0.930981 0.928341

0.970039 0.968644 0.967262 0.965894 0.964540

1

50 52 54 56 58

0.803241 0.797724 0.792252 0.786839 0.781496

0.846269 0.841553 0.836887 0.832280 0.827742

0.886942 0.883216 0.879537 0.875911 0.872345

0.925731 0.923152 0.920610 0.918108 0.915649

0.963204 0.961885 0.960586 0.959309 0.958055

60 62 64 66 68

0.776237 0.771077 0.766029 0.761110 0.756338

0.823283 0.818913 0.814645 0.810490 0.806464

0.868846 0.865421 0.862080 0.858831 0.855685

0.913240 0.910884 0.908588 0.906357 0.904198

0.956826 0.955626 0.954457 0.953321 0.952223

1 1

70 72 74 76 78

0.751731 0.747312 0.743104 0.739137 0.735442

0.802581 0.798860 0.795319 0.791983 0.788877

0.852654 0.849751 0.846990 0.844390 0.841 972

0.902119 0.900129 0.898237 0.896456 0.894800

0.951166 0.950154 0.949193 0.948288 0.947446

1 1

80 82 84 86 88

0.732059 0.729036 0.726434 0.724333 0.722852

0.786036 0.783497 0.781312 0.779549 0.778307

0.839759 0.837783 0.836083 0.834711 0.833745

0.893286 0.891933 0.890770 0.889831 0.889170

0.946677 0.945990 0.945400 0.944923 0.944587

1 1 1

90

0.722222

"-'1

0.777778

[(-;)91

0.833333

[(-;)71

0.888889

0.944444

1

0.904599 0.891969 0.870676 0.844820 0.817155 0.789537 0.763552 0.741089 0.725315

0.937930 0.925384 0.905441 0.882297 0.858217 0.834576 0.812552 0.793624 0.780373

0.964135 0.952226 0.934867 0.915757 0.896419 0.877717 0.860443 0.845669 0.835352

0.983037 0.972787 0.959607 0.945873 0.932311 0.919353 0.907464 0.897332 0.890270

0.994624 0.988015 0.980779 0.973573 0.966576 0.959944 0.953885 0.948733 0.945145

1 1 1 1 1

O0

'

HEUMAN'S LAMBDA FUNCTION

5 15 25 35 45 55 65 75 85

[(-;)51

[(-;I21

1

1 1

1 1

1 1

1 1

1 1 1

1 1 1 1

1 1

1 1 1

1 1 1

1 1

1 1

1

1 1

1 1

i

625

ELLIPTIC INTEGRALS

ELLIPTIC INTEGRAL OF THE THlHD KIND n (n;P\=)

n

O0

Table 17.9

0

15" 0.26180 0.26200 0.26254 0.26330 0.26406 0.26463 0.26484

0.52360 0.52513 0.52943 0.53562 0.54223 0.54736 0.54931

0.78540 0.79025 0.80437 0.82602 0.85122 0.87270 0.88137

60" 1.04720 1.05774 1.08955 1.14243 1.21260 1.28371 1.31696

75" 1.30900 1.32733 1.38457 1.48788 1.64918 1.87145 2. 02759

1.57080 1.59814 1.68575 1.85407 2.15651 2.76806

0.1

0 15 30 45 60 75 90

0 0 0 0 0 0 0

0.26239 0.26259 0.26314 0.26390 0.26467 0.26524 0.26545

0.52820 0.52975 0.53412 0.54041 0.54712 0.55234 0.55431

0.80013 0.80514 0.81972 0.84210 0.86817 0.89040 0.89939

1.07949 1.09058 1.12405 1.17980 1.25393 1.32926 1.36454

1.36560 1.38520 1.44649 1.55739 1.73121 1.97204 2.14201

1.65576 1.68536 1.78030 1.96326 2.29355 2.96601

0. 2 0.2 0.2 0.2 0.2 0. 2 0.2

0 15 30 45 60 75 90

0 0 0 0 0 0 0

0.26299 0.26319 0.26374 0.26450 0.26527 0.26585 0.26606

0.53294 0.53452 0.53896 0.54535 0.5521 7 0.55747 0.55948

0.81586 0.82104 0.83612 0.85928 0.88629 0.90934 0.91867

1.11534 1.12705 1.16241 1.22139 1.30003 1.38016 1.41777

1.43078 1.45187 1.51792 1.63775 1.82643 2.08942 2.27604

1.75620 1.78850 1.89229 2.09296 2.45715 3.20448

0. 0. 0. 0. 0. 0. 0.

3 3 3 3 3 3 3

0 15 30 45 60 75 90

0 0 0 0 0 0

0.26359 0.26379 0.26434 0.26511 0.26588 0.26646 0.26667

0.53784 0.53945 0.54396 0.55046 0.55739 0.56278 0.56483

0.83271 0.83808 0.85370 0.87771 0.90574 0.92969 0.93938

1.15551 1.16791 1.20543 1.26812 1.35193 1.43759 1.47789

1.50701 1.52988 1.60161 1.73217 1.93879 2.22876 2.43581

1.87746 1.91309 2. 02779 2.25038 2.65684 3.49853

0.4 0. 4 0. 4 0.4 0. 4 0. 4 0.4

0 15 30 45 60 75 90

0 0 0 0 0 0 0

0.26420 0.26440 0.26495 0.26572 0.26650 0.26708 0.26729

0.54291 0.54454 0.54912 0.55573 0.56278 0.56827 0.57035

0.85084 0.85641 0.87262 0.89756 0.92670 0.95162 0.96171

1.20098 1.21419 1.25419 1.32117 1.41098 1.50309 1.54653

1.59794 1.62298 1.70165 1.84537 2. 07413 2.39775 2.63052

2.02789 2.06774 2.19629 2.44683 2.90761 3.87214

0. 5 0. 5 0. 5 0. 5 0, 5 0. 5 0. 5

0

15 30 45 60 75 90

0 0 0 0 0 0 0

0.26481 0.26501 0.26557 0.26634 0.26712 0.26770 0.26792

0.54814 0.54980 0.55447 0.56119 0.56837 0.57394 0.57606

0.87042 0.87621 0. 89307 0.91902 0.94939 0.97538 0.98591

1.25310 1.26726 1.31017 1.38218 1.47906 1.57881 1.62599

1.70919 1.73695 1.82433 1.98464 2.24155 2.60846 2.87468

2.22144 2.26685 2.41367 2.70129 3.23477 4.36620 m

0 15 30 45 60 75 90

0 0 0 0 0 0 0

0.26543 0.26563 0.26619 0.26696 0.26775 0.26833 0.26855 [( 51

0.55357 0.55525 0.56000 0.56684 0.57414 0.57982

0.89167 0.89770 0.91527 0.94235 0.97406 1.00123 1.01225

1.31379 1.32907 1.37544 1.45347 1.55884 1.66780 1.71951

1.85002 1.88131 1.98005 2.1621 0 2.45623 2.88113 3.19278

2.48365 2.53677 2.70905 3.04862 3.68509 5.05734

0. 0. 0. 0. 0. 0. 0.

ff\

15 30 45 60 75 90

0.1 0. 1. 0.1 0. 1 0.1 0. 1

0. 6

0.6 0. 6 0. 6 0. 6 0. 6 0.6

9

O0

0 0 0 0 0 0 0

0 0 0 0 0

0

0

See Exampies 15-20.

-j)

30"

45"

90"

m

m

03

m

W

m

626

ELLIPTIC INTEGRALS

ELLIPTIC INTEGRAL OF THE THIKD KIND n(n; V\Q)

Table 17.9

n

a\

O0

0"

1.38587 1.40251 1.45309 1.53846 1.65425 1.77459 1.83192

75" 2.03720 2.07333 2.18765 2.39973 2.74586 3.25315 3.63042

90" 2.86787 2.93263 3.14339 3.56210 4.35751 6.11030

0 0 0 0 0 0 0

0.26605 0.26625 0.26681 0.26759 0.26838 0.26897 0.26918

0

0.26668 0.26688 0.26745 0.26823 0.26902 0.26961 0.26982

0.56501. 0.56676 0.57168 0.57877 0.58635 0.59225 0.59449

0.94034 0.94694 0.96618 0.99588 1.03076 1.06073 1.07290

1.. 47370 1.49205 1.54790 1.64250 1.77145 1.90629 1.97080

2.30538 2.34868 2.48618 2.74328 3.16844 3.80370 4.28518

3.51240 3.59733 3.87507 4.43274 5.51206 7.96669

0 0 0

0.26731 0.26752 0.26808 0.26887 0.26966 0.27025 0.27047

0.57106 0.57284 0.57785 0.58508 0.59281 0.59882 0.60110

0.96853 0.97547 0.99569 1.02695 1.06372 1.09535 1.10821

1.58459 1.60515 1.66788 1.77453 1.92081 2.07487 2.14899

2.74439 2.79990 2.97710 3.31210 3.87661 4.74432 5.42125

4.96729 5.09958 5.53551 6.42557 a. 20086 12.46407

0 0 0 0 0 0 0

0.26795 0.26816 0.26872 0.26951 0.27031 0.27090 0.27112

0.57735 0.57916 0.58428 0.59165 0.59953 0.60566 0.60799

1.00000 1.00731 1.02866 1.06170 1.10060 1.13414 1.14779

1.73205 1.75565 1.82781 1.95114 2.12160 2.30276 2.39053

3.73205 3.81655 4.08864 4.61280 5.52554 7.00372 8.22356

m m

15 30 45 60 75 90

0. 8 0.8 0. 8 0. 8 0. 8 0. 8 0. 8

0 15 30 45 60 75 90

0.9 0. 9 0.9 0. 9 0. 9 0. 9 0. 9

0 15 30 45 60 75 90

0 0

1. 0

0 15 30 45 60 75 90

1. 0 1. 0 1.0 1. 0 1. 0 1.0

45" 0.91487 0.92116 0.93952 0.96784 1.00104 1.02954 1.04110

30" 0.55918 0.56090 0.56573 0.57270 0.58014 0.58592 0.58812

0. 7 0. 7 0.7 0. 7 0. 7 0. 7 0. 7

0 0 0 0 0 0

0 0

15"

[(-;)51

60"

m

00

m

m

m 00

m m

18. Weierstrass Elliptic and Related

Functions THOMAS H. SOTJTHARD

Contents Mathematical Properties . . . . . . , , . . . . . . . . . . . . 18.1. Definitions, Symbolism, Restrictions and Conventions . . 18.2. Homogeneity Relations, Reduction Forinulas and Processes . 18.3. Special Values and Relations . . . . . . . . . . . . . 18.4. Addition and Multiplication Forrhulas , . . . . . . . . 18.5. Series Expansions. . . . . . . . . . . . . . . . . . . 18.6. Derivatives and Differential Equations . . . . . . . . . 18.7. Integrals . . . . . . . . . . . . . . . . . . . . . . 18.8. Conformal Mapping . . . . . . . . . . . . . . . . . 18.9. Relations with Complete Elliptic Integrals K and K’ and Their Parameter m and with Jacobi’s Elliptic Functions . 18.10. Relations with Theta Functions . . . . . . . . . . . . 18.11. Expressing any Elliptic Function in Termsof @ and @’ . . 18.12. CaseA=O. . . . . . . . . . . . . . . . . . . . . . 18.13. Equianharmonic Case (g2=0, ga=l). . . . . . . . . . . 18.14. Lemniscatic Case (g2= 1, ga=O) . . . . . . . . . . . . . 18.15. Pseudo-Lemniscatic Case (ga= - 1, g3=O) . . . , , . . NumericalMethods . . . . . . . . . . 18.16. Use and Extension of the Tables References

1

. .

,

.. .. .. . .

, ,

... . . . . . ... . .... . . . . . . . . . . . . , . . , . , . . .

pwe

629 629 631 633 635 635 640 64 1 642 649 6 50 651 651 652 658 662 663 663 670

University of California, National Bureau of Standards, and California State College

at Hayward. 627

628

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

Table 18.1. Table for Obtaining Periods for Invariants g2 and g,

. . . . . . . . . . . Discriminant (3 Sg25a)

G2=g2ga-2').

Non-Negative

+ ic

,

wg:16, w'g:'6

Js ln(&-3);

wg:'", W'g:''/i;

T2=

wg:"

,

.

,

,

....

Page

673

T2=3(.05)3.4, 7D

3.4 (.1)5(.2)10, 7D

I: w" 'gtl6, 7i14/i;zi =.I (- .Ol)O, 7D

Non-Positive Discriminant (-

a0

<J2

<3)

w'zg:l6I T 2 1 ll4/i; Ji '=O(-. w2g;l6, Wjg:"/i; w2g:'6,

Ja'=

01) -.2, 7D

-.05) -1, 7D

-.2(

+x

7 O'2 g a1/6 8 1n(3-z2);

32=-1(.2)3,7D

Table 18.2. Table for Obtaining @, Q' and

., .

Real Half-Period-Period

Ratio a ) .

Positive Discriminant (0
< 1, 0
Za

. .

,

,

{ ,

on Oz and Oy (Unit

.., .

, ,

,

..

674

@ ( z ) , 2 @ ' ( z ) , z ~ ( z ) , u = ~1.05, , 1.1, 1.2, 1.4, 2, 4 ~=0(.05)1,y=0(.05) 1.1, 1.2 (-2) U, 6--8D

Negative Discriminant (0< x

< 1, 0
@ ( z ) , 2 @ ' ( z ) , z ~ ( z ) , u = ~1.05, , 1.15, 1.3, 1.5, 2, 4 x=0(.05)1, y=0(.05)1 (.l)b(b>a/2), 7D

Table 18.3. Invariants and Values at Half-Periods (1 < U < - )

.. . .. . ., Real Half-Period) . . ~=1(.02)1.6(.05)i.3(.1) 4, a, 6--8D Non-Negative Discriminant 92, g3, e1=

. .

,

. . . .

Q (1), e3= Q ( U ' ) , v=t(l),T'/i=t(w')/i,

,

.

(Unit ,

, ,

&), u ( w ' ) / i ,

.

680

44

Non-Positive Discriminant .~1, ~

3 ei, ,

Ta=t(l),

d/i=t(w;)/i,

~ ( 1 u(4)/i, )~

44

The author gratefully acknowledges the assistance and encouragement of the personnel of Numerical Analysis Research, UCLA (especially Dr. C. B. Tompkins for generating the author's interest in the project, and Mrs. H. 0. Rosay for programming and computing, hand calculation and formula checking) and the personnel of the Computation Laboratory (especially R. Capuano, E. Godefroy, D. Liepman, B. Welter and R. Zucker for the preparation and checking of the tables and maps),

629

18. Weierstrass Elliptic and Related Functions

Mathematical Properties Invariants gt and ga

18.1. Definitions, Symbolism, Restrictions and Conventions

An elliptic function is a single-valued doubly periodic function of a single complex variable which is analytic except at poles and whose only singularitiea in the finite plane are poles. If w and U’ are a pair of (primitive) half-periods of such a function j ( z ) , then f(z+2Mo+2Nw’)=f(z), M and N being integers. Thus the study of any such function can be reduced to consideration of its behavior in a fundamental period parallelogram (FPP). An elliptic function has a finite number of poles (and the same number of zeros) in a FPP; the number of such poles (zeros) (an irreducible set) is the order of the function (poles and zeros are counted according to their multiplicity). All other poles (zeros) are called cong m n t to the irreducible set. The simplest (nontrivial) elliptic functions are of order two. One may choose the standard function of order two either a function with two simple poles (Jacobi’s choice) or one double pole (Weierstrass’ choice) in a FPP. Weierstrass’ @-Function. Let w , w’ denote a pair of complex numbers with $(w’/w)>O. Then @(z)=@(zIw, a’) is an elliptic function of order two with periods 2w1 20‘ and having a double pole at z=O, whose principal part is z - ~ ;@ ( Z ) - Z - ~ is analytic in a neighborhood of the origin and vanishes at z=O. Wekrstrass’ I-Function ((2) =c(zlw, U ’ ) satisfies the condition I’(z)=-@(z); further, f(z) has a simple pole at z=O whose principal part is z-*; { ( z ) - z - : vanishes at z=O and is analytic in a neighborhood of the origin. ~ ( z )is NOT an elliptic function, since it is not periodic. However, it is quaai-periodic (see “period” relations), so reduction to FPP is possible. Weierstrass’ a-Function a ( z )=u(zIw, U ’ ) satisfies the condition u ’ ( z ) / a ( z ) = t ( z ) ;further, u ( z ) is an entire function which vanishes at the origin. Like I , it is NOT an elliptic function, since it is not periodic. However, it is quasi-periodic (see (‘period” relations), so reduction to FPP is possible.

Let W=2Mw+2Nw‘, M and N being integers. Then

g2=602‘W-‘ and ga=1402’W-’

18.1.1

are the INVARIANTS, summation being over all pairs M, N except M=N=O. Alternate Symbolism Emphasizing Invariants

@ ( z > = @ ( z ; 91, gal @ ’ ( z > = @ ’ ( z ; gz, ga)

18.1.2 18.1.3 18.1.4 18.1.5

J.k>=t(z; 92, g*> a ( z > = ~ (gal~ gal ;

Fundamental Differential Equation, Discriminant and Related Quantities

18.1.6 18.1.7

@ ” ( 2 ) = 4 @‘a(z)-g, @ ( Z ) - g a =4(@(z>-el>(@(z>-ez>(@

(z>-ea)

18.1.8 A=

-27& =16 (e2 -e3)‘(e3-el)‘(el-e2)’

18.1.9

++

e:ea +e2e3>=2 (8 4 e’B> ga=4elelea=9(e:+e:+e:) e: + e l + ea= 0 e: 4 +et= &I8 4 e ~ - g 2 e , - g 8 = O ( i = l l 2 , 3)

g 2 = -4 (ele2+

18.1.10 18.1.11 18.1.12 18.1.13

+

Agreement about Values of Invariants (and Discriminant)

We shall consider, in this chapter, only real 91 and ga (this seems to cover most applications)hence A is real. We shall dichotomize most of what follows (either A>O or A
In this chapter, zl/” (n a positive integer) is used to denote the principal nth root of 2, as in chapter 3;B is used to denote the complex conjugate of z.

630

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

FPP’s, Symbols for Periods, etc.

A> 0

Y

A
I

iw

t

FIGURE 18.1

RECTANGLE

w1=w w2

=w

+w

RHOMBUS

w; =w

-w

wa=wf

REAL w’ PURE IMAG. lw’l hw, since g 3 h 0

REAL w l PURE IMAG.

w

~2

l w i l 2 w2, since g320 Fundamental Rectangles

Study of all four functions (@ , @ I t {, U) can be reduced to consideration of their values in a Fundamental Rectangle including the origin (see 18.2 on homogeneity relations, reduction formulas and processes). A>O

Fundamental Rectangle is tices 0, U ,

A
FPP, which has ver-

Fundamental Rectangle has vertices 0,

wa, w2+$,

and w‘ Y

FIQURE 18.2 There is a point on the right boundary of Fundamental Rectangle where @ =O.

Denote it by 26.

631

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

18.2. Homogeneity Relations, Reduction Formulas and Processes

The Case ga
Put t=i and obtain, e.g.,

Homogeneity Relations (Suppose t # O ) Note that Period Ratio is preserved.

18.2.1 18.2.2

@’(tzltw, t ~ ’ ) = t - ~ @ ’ ( Z I w , w’)

@ (tzltw, fw’) =t-’@

f(tzltw, tw’) =t-1S’(z(w,

18.2.4

u(tz)tw, tw’) =tu(z)w,

U‘)

18.2.5

gz(tw, tu’)=t-4gz(w,

U’)

18.2.6

g3(tw, t

18.2.7

e,(tw, tw’)=t-2et(w,

18.2.8 18.2.9

18.2.18

U’)

A(tw, tw’) =t-12A(w,

2, 3

18.2.10

~ ( t w ~, w ’ ) = Q ( w , a’)

(See 18.10)

18.2.11

m(to, t w ’ ) = m ( w , w ’ )

(See 18.9)

18.2.12

@ ’ ( t ~ ;t-’gS, t-6gg3)=t-a@’(~; ga, ga)

18.2.13

@ ( t z ; t-‘g*,

@ (2 ; ga, ga)

r ( t z ; t-’g2, t-*ga)=t-’{(z;

18.2.15

~ ( t zt ;- 9 2 , t - ” g a ) = t ~ ( z ; 91, g3)

@ (z+2Mw+2Nw’)

=@ ( z )

f(z+2Mw+2No’)=5(2) +2MTJ+2N?)’

(See 18.3)

18.2.14

@’(z+2Mw+2Nw’)= @ ’ ( z )

18.2.19

U’)

H t ( t ~t ,~ ’ ) = t - ~ H iU(’ ~) , ,i=1, 2, 3

t-”ga) =t-’

- @ (iz;g2, -gal

“Period” Relationa and Reduction to the FPP (M,N integers)

0’)

w ’ ) , i=1,

92, 93)

ga>O.

18.2.17

~ ’=t-’ga(w, )

@ (2;

Thus the cam g3<0 can be reduced to one where

( z ( w , U’)

18.2.3

18.2.16

gz, ga)

18.2.20 U(2

+2Mw +2Nw’)

= (-

1)MtN+MN U(Z)

exp [ (z+Mw+Nw’) (2Mv

+2Nv’)l 18.2.21

where q=f(w), q’=f(w’) ‘%onjugate” Values

fO=j(z),where j is @,

P,5, 6.

any one of the functions

632

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

Reduction from 1/r FPP to Fundamental Rectangle i n Case A
We need only be concerned with the case when 2 w ‘ - 2 is in triangle

z is in triangle A2 (therefore Ad. 18.2.34

@ ( 2 ) =@ ( 2 w ’ - 2 )

18.2.35

@ ’( 2 ) = -@ ’(20’ -2 )

18.2.36

((2)=29’-((2w’-2)

18.2.37

FIGURE 18.3

u(z)=u(2w’-z)

exp

[29’(2-w’)]

1

I

Reduction to Case where Real Half-Period is Unity

(preserving period ratio) A>O

A

a

(U2 =w

+w ’)

NOTE: New real half-period is

-+-=-0 w‘ @+U’ w w

w

-I

633

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

18.3. Special Values and Relations Values at Periods

@, @’, and

are infinite,

U

is zero at z=2wl, i = 1 , 2 , 3 and at 2w;(A
A>O

A
18.3.1

@ (wt)=et(i=1,2,3)

18.3.2

@ ’( U t ) =0 (i=1,2,3)

18.3.3

9t=t(w)(i=l,2,3)

18.3.4

91=9, rla=9+9’,

93=9’

Hi=2ef+ejel, ( i , j , k = l , 2,3; i#j, i # k , j # k )

18.3.5

=(et-e,)

18.3.6

(et-ek)=2e4+-=3ei--p g3 4et

g2

18.3.7

ef real

ea real and non-negative

18.3.8

el>O> e2>e3

(ea=O when g3=O) el=-cr+ij3,

(equality when g3=O)

e,=&

where a 2 0, /3>0 (equality when ga=O)

d = t ( 4=rl’--tl

18.3.9

q>O

18.3.10

< if ?‘/if0

18.3.11

Jw’J/wf1.91014 050 (approx.)

18.3.12

Hi>O, H3>0

18.3.13

H 2 = i a

18.3.14

U(.)

18.3.15

u(o‘)=ieq‘w‘/2/Ipala

18.3.16

~~(w~)=e~2~2/(--H~)

18.3.17

arg [

=eq012/M/2

$0

7r

U ( U ~ ) ] = ~ + ~

rli/i$O if lw61/w2f3.81915 447 (approx.)

634

WEIERSTRASS ELLIPTIC AND REUTED FWN’CTIONS

A>O

18.3.25

U( 0‘/2)=

A
ielt’o’18 21/4Hy8(2H3-3e3)lI8 =u(02+0;/2) exp [-rl’wzl

18.3.26

@ (wz/2)=e2-H2

@ (w’/2)=e3-H3

18.3.27

@’(U&) = -2H&W2-3edt

@ ’(0’/2) = -2iH3(2H3-3e3)*

18.3.28

f (w2/2)=+[v2-i(2H2 -3eJ *I

l(0’/2) =4[7’-1(2H3-3e3)41

18-3-29

d 4 2 ) = [ 4 p ( 2 H 2 - 3e2)1118

ltw iseir 14

ePzw218eir14

U( w’/2) =

[4H:( 2H3-3e3)]118

One-Third Period Relations

At z=20,/3(i=l, 2, 3) or 2~:/3,@”’=12@@’~; equivalently: 48 @

18.3.30

-2492 @2-48g3 @ -gi= 0

0 - 0

18.3.35

A
-exp [ 27’0’/9] ~ ( 2 4 3= ) [ @ ’( 2 0 ’ / 3 ) ] ~ / ~ e * r ~ / ~

Legendre’s Relation

18.3.37

-71 ’U=

710’

iri/2

(also valid for A
tjaO:-~;wz=ni

Relations Among the Hi

18.3.38

H:+z+E= 3gd4

18.3.39

W#+WT+EE=O

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

mHiE= -All6

18.3.40 18.3.41

16H:-12g2H:+A=O(i=l,

*

2, 3)

18.4. Addition and Multiplication Formulas Addition Formulasa(51#az)

18.4.1 18.4.2 18.4.3 18.4.4

421+22)~(21- 22) =-&2,)u2(22)[

@ (21) - @ ( 4 1

Duplication and Triplication Formulas

[Note that @"=6

@O(

z)-$)

@'2( 2)=4

p3(2)g2@ ( 2 ) -

g3 and @ " I ( z)= 12 @(2 ) @ ' ( z ) ]

18.4.5

@ '(22)=

18.4.6

-4 @ ''( 2)+12 @ ( 2 ) @ "( 2 ) @"( 4 @13(2>

18.4.7 18.4.8

18.5. Series Expansions Laurent Series

18.5.1 18.5.2

where

and k-2

3 ck=(2k+l)(k-3)

18.5.3

m-2

Cmck-m,

18.5.4 m

18.5.5 18.5.6

* Formulas for r and

U

are not true algebraic addition formulas.

kZ4

2)-

@"3(~)

635

1 ~

636

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

it being understood that am,.=O if either subscript is negative. (The radius of convergence of the above series for P - z - ~@, ’ + + z - ~ and p-z-’ smallest of 12~1,12w’( and 12wlf2w’I; series for U converges for all 2 . )

is equal to the

Values of Coefficients ck in Terms of ca and ca

$13

18.5.9

c4 =

18.5.10

c,=3c2ca/ll

18.5.11

ce= [2c:+3&]/39

18.5.12

C7 =2$Ca/33

(11c:+ 36~;)/7293

18.5.13

~ g =5 ~ 2

18.5.14

cQ=c3(29c; 1lc;)/27 17

+

18.5.15

c~O=

18.5.16

~11=

18.5.17

1 4 ~ 2(389~!+369~:)/3187041 ~3

(1149504+ 1 0 8 0 0 0 0 ~ ~166617~;)/891678645 ~~+

Cla=

18.5.18

~,,=10~~~3(297~;+530~!)/11685817

18.5.19

c14=

18.5.20

c15=

2 ~ (528770~;+7 2 164675~;~5+29896024) (306735)(215441)

+

c1e-

18.5.22

c17=

i

(5909761)(5132565) C~CS(

301719554+ 126138075~;~:+281517394) (920205)(6678671)

1541470~949003~~+30458088737 1155~;~!+122378650673.378~;~;+2348703* 887777~: 8 ~ (1342211013)(4695105713) 0

~

1

18.5.24 c19=

2&( 3365544215 4 +429852433 * 4544+ 85277434774) (91100295)(113537407)

NOTES: were computed and checked independently by D. H. Lehmer; these were double-checked by substituting g1=20 c2, gs=28 c3 in values given in [18.10]. 2. clrc18 were derived from values in [18.10] by the same substitution. These were checked (numerically) for particular values of ga, g8. 1.

+

4 ~(6292 3 18154 179865450~;~: 140513674) (179685)(38920531)

-ci( 58957855c;+ 1086511320cic2,+ 875341836~;)

18.5.21

18.5.23

(242~:+ 1455~:~$)/240669

c4-cIb

3. cIo is given incorrectly in [18.12] (factor 13 ie missing in denominator of third term of bracket); this value was computed independently. 4. No factors of any of the above integers with more than ten digits are known to the author. Tllisisnot necessarily true of smaller integers, which have, in many instances, been arranged for convenient use with a desk calculator.

I

Value of CoeflGcients a, ,, 1

I1

-2.3. 8

649

-107896773 1 I*

-2.3. 7

6-23

.m7.18018 6 1

6

D

f

6-59

-10'1886773 1 )

I I1

8 I9

-2.34

-2.34.23

-2.3- 6-7

-2-3- 6.7

.229.m

267

.181.1699

-41. W 7

.18019

.m

.4922497

I 8

I I*

I 1

6

1 11

-2.3.

2-34

-2.3.5.m

-2.3.

.9103

.m

.40670423

6

6 1

-2.3.

6.7

.59-179

412131 1 8 1

2.34

11

2-3.6.9103

4 .a1 1 4 1

I 4

2.3-23

2.3-5-31

8

6 8

-2.3.5491

.13.37.41

.83609

11

2-3.5.17

.IOQ -2.3

I

a: 2.3-23

16

2.3.6.63

2

-2.3

-3

219

1

0

]

.313.190387

.283.4648963 4 8

~176!&380144977 4 1

4 8

-2.3683

-2-3.6.603

-2.3-31

-2.3.5.7.11-29

-2.3.6.7d13

3911

.I66217

.315989M9

tl3.1129.9661

.I7606225081

I

a 4

2-3.637

-2-3-6-13

-167

a37

r2

4 1

-2.3-61.151

.653

I

2.3.311

3.23

-1

0

-2.3.5.7.19

4 6

3 8

11

1

1

6 1

I 8

-2.3.6-7

3.5.2oRo7

I

4 1

I T

1

-2.3.5.7-17

2.3.5.7

2(8'1260103

,2967-41189

.67.195651069

&%66647631901

6

T

3 6

6

1

*

-3.17

-2-3-7-13

-3

-2.3-5-7.193

1.7

-2509

~1578257

2742587

.u8811293Mo9

,13679.274973

89666603841079

1

3.7.23.37

4

:

-2.3.5.17.63

?-313.603

6

-3.7

7

6

4

I

4

.686973 3

1

-2-3

-2-3.1 1

3

3.107

I 1

I

4

-3.7-1123

3-11-37

-3.7.23.14387

-3.71

2.57981

40763

-176309760638

8

9

10

¶

.383.739.&539 11

+m

4 V a l m of a., I In unhictomd form for 4m+6R+1<35 sre given In IlS.261,p. 7; of (a. in factored form in I18.151, Vol. 4, p. 89 for 4m+6n+1<26. derL &&tors; prlmallty of Large factors was established with the aid of SWAC (National Bureau of Staudards Western Automatic Computer).

Additional value3 were computed and checked on

638

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

Reversed Series' for Large

I@ I

Reversed Series for Large I @ '1

18-5-29 Z = A ~ U + A ~ U ' + A T U ~ + A ~. U . .~ +

18.5.25

18.5.30

2u+c~u5+c3u7+-4 U g +F ull 3

where

U=( @'19-1efr13

18.5.31 1 ( 3 ~ 8 + 6 0 ( ~ ) u l l + ~ (~1~2 u~ l~~~ ++ ~ 7a~)u'~ 13 68 18.5.32

~~=21/3

+-

+5*" (aff7a:) 19

2

u19+3 (3a: 4

+1Oag)uZ1

As=-- a2 A: 5

A7=-

18.5.33

35a24y3 (9a:+4ffg)U* +92

18.5.34

Ag=O

18.5.35

All=8a2a3A:/l 1

18.5.36

AN=- 39 ( d 4 - 6 d )

7 +-200 (33~~+180~~a~+10~~)~~~

+-74a3 (lla;+lOa8) 12

-4a3A1 7

10a1

U27

+% ( 1 4 3 ~ y Z B + l 1 5 5 ~ ~ ~ ~ fu2' 210~~) 23 29

18.5.37

A16=-96ga3/175

18.5.38

A 1 7 = - M (4+124 51

18.5.39

where a2=g2/6,a3=g3/6 Reversed Series for Large

18.5.40

33w3 $23.5.7 (195cu~$456aiai+42af)u3'

$2O.37 11( 1105ai+ 16380a&g+ 10920a~4

+168~~8)

33cY;a3

28

(85&-280&$+56&) u3'

+2 (3234+6120a:d 4- 6300abf+336a:) +

u41

+

(1615ai+ 714OaBd 252044 +244) u~~

+ 0(~46)3' 18.5.26 18.5.27 18.5.28

{-'

18.5.42

A6= -6215

18.5.43

A7= -6317

18.5.44

A9=6:/7

18.5.45

All=36263/11

18.5.46

U=(

A13=-

18.5.47

17 (--86:+762,) 1001

A15= -416;63/91 A 1 7 =6m 2 (13496;-41166:)

18.5.49

AIg=-

=93/8

18.5.50

where 62=g2/12

@ -')4

18.5.51

where a2=g2/8 (113

..

Z=U+A~U'+A~U~+A~U~+.

18.5.41 where U =

18.5.48

In this and other series a choice of the value of the root has been made 80 that z will be in the Fundamental Rectangle (Figure 18.2). whenever the value of the given function is appropriate.

If I

263 (1154316:-2256861,) 323323

6a=g3/20

639

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

18.5.57

Other Series Involving @

where

Series near zo [ @(zO)=O1

U=

(z- w J 2

Other Series Involving @'

18.5.52

Series near zo

@ = @;U

[1-3czu4-4c3u~+- 13

114CzC3ulo 11

OC; U"-

18.5.58

- 10c2u-56c3u3+30c~u5$264~Z~3u7

(@ ,'- @ ;)=[

+5C;U4+33CZC3U6+

84~:--10~; 3

1363~i~3u'O 33

+

5~2(55~;-2316~:)~~~ 143

18.5.53

where u=(z-zo),@;=

164) +70c2(55c&23 143

u13]

+ @ ;[- 15czu4-28c3u6+30c~u8+1

1

4

~

@,'(zo)=i&

18.5.54 '2

U=

@:[v+av2+2a~+(++5a3

)

v4+:

( 3 ~ ~ 4 18.5.59

+1593@;2+70a3)@+2a2(2 pi4$79, @i2+ 21a3)v6 +( 93++@ { g:+20a3} @:4+15a2g3@~2+132a6 v7 '6

?

where

U=

(z-zo)

18.5.60

30): 2 A4+3(c2+b2)Aa

(z-zz,)=A-bA3-

+ l o b @ iA6-3[36ca-3@

:(

-3@

~ ~ $ 2b2)AB+$ 1

+1004-279@

h+4b3]A7 (285b2c2

i2b+132b4

18.5.61

where A = ( @ ,'-@ 6)/(-10cz)

18.5.62

and b =4g,/gp

Series near wi 18.5.63

+

@ '=2 (3ei- 5cz)a+4 (10c2ef 2 1c3)a3+6 (7c2e:

+2 lc3e

->

1oc; 3

f$

+ lOcie, 22c2,e:+ 92czc3e+1054

5c:)as+ 24 (6cae:

+24

(g

110 c2c3e:+T &ef

681 9150 +-171 4e?+c;c3ef+c&+ 13 11 143

13

+..., 18.5.64

where

a=(z-uf).

~

~

~

640

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

Other Series Involving f

Series neaT zo [ @

18.6. Derivatives and Differential Equations Ordinary 18.6.1

(a)=01

18.5.65

-T+2+2-3-U' CZU' C3U8

CiU'O

22 26 7cL U' 5c2u7 l C ZC 3U Q (loc:-84cg + L.l-

7

2

3

+

l3 C2(2316C;-55C;) + 1363CiC8 T + 429

18.5.66

where u=(z-zo),

18.5.67

To =f (20)

Series near

Ull

33

uq+

.. , *

f'(4 = - @ ( 4

=t(z)

18.6.2 18.6.3

19C$3Ula

(c2=g2/20, c3=g3/28) ~'(Z)/4Z>

@ 'yz) =4 @3( 2 ) -g2 @ ( 2 ) -g3 =4( @3--6c2 @ - 7c3) 18.6.4 @ " ( 2 ) =6 @ '(2) - 3g2=6 @ 2- 10~2 18.6.5 @ '''( 2 ) = 12 @@ ' 18.6.6 @ (4) ( Z ) =12( @@ '' @ '@ ')

+

=5!

18.6.7 @ (6) ( 2 ) = 12(@@

wt

c

'"+ 2 @ '@

18.6.8 @ (') ( z ) = 12(@@ (4)

g3-3C2@

'I+

--14~31 5

@ '' @ ')

=3*5!@'[@2-c*]

+3 @ '@ "'+ 3 @

+

"

@"

+@'If@

')

18.6.9 =7![ @'-4~2 @ -4 ~ @ 3 5~:/7] 18.6.10 @ ( " ( ~ ) = 4 . 7 ! @ ' [ @ ~ - - 2 ~ ~ @ - ~ ~ ] 18.6.11

@ '8'(Z)=g![@'-~C~@a-5c~@2 +(1O~~@+ll~a~~)/31

18.6.12

+= 607 cg3) -f a13

+= 68 1

18.5.69

dc3ei+-

@ ( " ( 2 ) =5*9!@ '[ @ '-33~2 @ '-2ca @ +24/3]

(z

18.6.13

c i e : +171 x 4e:

@ "O'(~)=11![@ '-6~2 @ *+7&@ (3&2C3@ 4-844- 1oCi)/33] 18.6.14 @(l1)(2)=6.1I! @ '[ @ '--4~2 @ '-3~a@ (774 @ 57c2c8)/33] 18.6.15 @ (12)( z ) = 13![ @ 7-7~2@ '-7~3 @ 'f354 @ a/3 4-2 1Ocg3@ "1 1+ (844- 35~:)@ /I 3- 1363Cl,c3/429] 18.6.16

+

9150 143

where a= (z-

wt)

Reversed Series for Small 101

18.5.70

@ I

18.5.71

where y2=92/48

18.5.72

Y3=ga/120

.

.

,

For reversion of Maclaurin series, see 3.6.25 and [18.18].

( z ) =7 * 13!@ '[ @

+

'+

+

+

'-5c2 @ ' - h a

@a

+ '+

54 @ ~ O C Z@ C ~/I 1 (124-54) /I 31 18.6.17 @ (14)(z)=15! [@ '-8~2 @ '-8~s @ 52~4@ '/3 328~2~3 @ 1 (444%-328~:) Q 2/39 - 4 8 8 4 ~@ ~ /33 cg(554-23lSc',)/429] 18.6.18 @ (l') (Z ) =8.15!,P'[ ,P'I- 6 ~@ 2 -5ca @ '4-26d @ '/3 1 2 3 ~@ ~ 8'/11+ (1114-824) @ /39-61&/33]

+

+

+

+

'

641

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

Partial Derivatives with Respect to Invariants

18.6.19

18.6.20

y”=[a@ (z)+b]y (LamB’s equation)see [18.8], 2.26 For other (more specialized) equations (of orders 1-3) involving @ ( z ) , see [18.8], nos. 1.49, 2.28, 2.72-3, 2.439-440, 3.9-12. For the use of @ ( z ) in solving differential equations of the form y””+A(z,y)=O, where A(z,y) is a polynomial in y of degree 2m, with coefficients which are analytic functions of z, see [18.7], p. 312ff.

18.6.21

18.7. Integrals Indefinite 18.7.1

J@’(z)dz=g 1 @ ’(z)+s 1 gaZ

+;z(9g3@+;g;)-;g2@’ 18.6.22

18.6.23

A

(formulas for higher powers may be derived by integration of formulas for @ ( ” ) ( z ) ) For J @ ” ( z ) d z , n any positive integer, see [18.15] vol. 4, pp. 108-9.

*=-+z

3 9 1 9 g Z d f g3&- 8 g ~ z ’ a - gear‘ ~ ag3 2

18.6.24

If @ ’(a) # O 18.7.3

=2zt(a)+ln a(z-a)-In

For J d z / [ @ (2)- @ (a)]”,( @ ’(a)#O) nanypositive integer, see [18.15], vol. 4, pp. 109-110.

Differential Equations

18.6.25 Equation y’3 =y“(9-a)’

a(z+a)

Solution a 27 @’ y=-+2 16

Defnite A>O

729

18.6.26

A
18.7.4

4-3d

g3

=- 27

18.7.5

18.6.27 18.7.6

where t is real and

18.7.7

s(t)=4t~-gd--gs

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

642

18.8 Conformal Mapping w=u+iv

A>O

A
w=@

maps the Fundamental Rectangle onto the half-plane v I 0 ; if (w’I=w(g3=O), the isosceles triangle 0wwz is mapped onto ~ 2 0 vIO. ,

maps the Fundamental Rectangle onto the half-plane v10; if (w;I=oz(g3=O), the isosceles OQW’ ‘5 mapped onto U 2 0 1

w=@’(Z> maps the hndamental onto the w-plane less quadrant 111;if Iw’l=w, triangle Owwz is mapped onto v 20, v 2 u .

w=@ ’ ( 2 ) maps the Fundamental Rectangle onto most of the w-plane less quadrant 111; if Iw:(=wz, the triangle Owzw’ is mapped onto vzo,

w=@

(2)

the

(2)

v10.

v2u.

(a=period ratio)

w=l(z) maps the Fundamental Rectangle onto the half-plane u>O. If a 51.9 (approx.), v50; otherwise the image extends into quadrant I. For very large a, the image has a large area in quadrant I. w = u ( z ) maps the Fundamental Rectangle onto quadrant I if a <1.9 (approx.), onto quadrants I and I1 if 1.9
a, arg[u(wz)]=-;

consequently the image winds

around the origin for large a. Other maps ’are described in [18.23] arts. 13.7 (square on circle), 13.11 (ring on plane with 2 slits in line) and in L18.241, p. 35 (double half equilateral triangle on half-plane) ,

w=l(z) maps the Fundamental Rectangle onto the half-plane u 2 0 . The image is mostly in quadrant IV for small a, entirely so for (approx.) 1.3 I a 1 3 . 8 . For very large a, the image has a large area in quadrant I. w=u(z) maps the Fundamental Rectangle onto quadrant I if a<3.8 (approx.), onto quadrants I and 11 if 3.8Ia<7.6 (approx.). For large a, arg

“2’1 2

[( U

w2+2

x-;

consequently the image

winds around the origin for large a. Other maps are described in [18.23] arts. 13.8 (equilateral triangle on half-plane) and 13.9 (isosceles triangle on half-plane).

Obtaining @ ’ from @ ’*

Fundamental Rectangle

Fundamental Rectangle

A >O

A < O

FU NDAME NTAL RECTANGLE

FUN DAM ENTAL

4/2: i0/2

RECTANGLE

tY

FIGURE 18.4 In region A %’(@’) > O i f y 1 . 4 a n d s I . 5 ; Y(@’)20elsewhere

In region A (1) If a21.05, use criterion for region A for A>O. (2) If 11a<1.05:

9(@’)20 if y 2 . 4 and s 5 . 4 , - ~ / 4 < a r g (@‘)<3~/4 if .4
643

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

In region B

In region B

Use the criterion for region B for A>O.

The sign (indeed, perhaps one or more significant digits) of @' is obtainable from the first term, -212, of the Lament series for @',

(Precisely similar criteria apply when the real half-period # 1) -3

-4

-2

w=l

A>O

Map: @(z)=u+iv 1 Near zero : @ ( 2 ) =-+Q 29

1

@ ( 2 ) =g+c222+e2

-

-5

w'=i

-6-

-

-5

wJ=1.4i

-61

-4

-5

U'

=2.0i

.

-3

-2

-I

0

I

2

3

4

-

-6-

FIGURE 18.5

.5

1.0

x

644

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

A
01=1

Map: @((z)=u+iu 1

Near zero: @(2)=?+e1

ui=1.5i

-.t

=2 .oi

lEpl 6.06

-6

I U;

lE,l 6.2

\

0 lEil 6 . 0 7

141 6 .03

-"

-6

FIGURE18.6

.5

I.o

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

A>O

w=l

Map: f(z)=u+iu 1

Near zero: f(z)=-+el 2

+

1 c2z3

f( 2 ) =--2 3

c2

3.5 -7"

3.5 -rU

lE1 I s , 0 0 7

1E21 s . 0 0 0 2

w'=1.4i

U'

=2.0i

FIGURE 18.7

645

646

WEIERSTRASS ELLIPTIC AND RELATED F"CTI0NB

A
1

WS=

Map: l(z)=u+iv 1

Near zero: l(z)=-+tl 2

v

3

1.0

1.5

2.0

2.5

3.0

I

I

I

1

I

I

3.5

Y

t

-3.0

U: = 1.5i

1% 11.007 let15.0009

1___---

->.'*'.

-3.5

FIGURE 18.8

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

A>O

w=l

Map: u(z) =u+iv

w'=i

w'= 1.4i .5

0

2.Oi

t"

1.0

-

"

1.8i

w'=2.0i 0

.5

1.0

U

FIQURE13.9

647

648

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

A
m=l

Map: u(z)=u+iv .5 i

h

.5

0

.4

.6

1.0

1.2

U

,

1.0

.5

w;=

2 .oi

FIGURE 18.10

649

WEIERSTRASS ELLIPTIC AND RELATED FUNCTION8

18.9. Relations with Complete Elliptic Integrals K and K’ and Their Parameter rn and with Jacobi’s Elliptic Functions (see chapter 16) (Here K ( m ) and K’(m)=K(l-m)

are complete elliptic integrals of the 1st kind; aee chapter 17.) A
A>O

18.9.1

18.9.2

e2=

(2m-l)P(m)

e2=

3w2

2(1-2m)Kym) 34

18.9.3

4(m2-m+1)K4(m)

18.9.4

18.9.5

18.9.6

gz=

g3=

4(16mZ-16m +1)K4( m ) gz= 3w:

3w‘

4( m - 2 ) ( 2 ~ ~ - 1)( m + l ) P ( m ) 27 we

A=

8(2m-l)( 32m2-32m-1)P( ga = 27 U:

16m2(m-l)zf12( m)

A=

W12

m)

( -256( m -m 2 ) F 2m) ,:a

18.9.7 18.9.8

@ ’( z)=

w =K(m) / (el

-ea) ‘1’

-2(e1-ea)~/’. cn(z*lm)dn( 2*lm)/sn3(z *I m)

where

18.9.13 18.9.14

@’(2)=

-4H~g/’sn(2’Im)dn(2’Im) [1-cn( 2’ Im)lZ

where q = c ( ~ ) = -K ( m ) [3E(m)+(m-2)K(

3w

q’= f(w’)=

lp’--]a?ri w

m)]

q 2 = t ( w ~ ) = [6E(m)+ ~

3wz I

.

[E(m)is a complete elliptic integral of the 2d kind (see chapter 17).]

(4m-5)K(m)]

650

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

18.10. Relations with Theta Functions (chapter 16)

The formal definitions of the four 9 functions are given by the series 16.27.1-16.27.4 which converge for all complex z and all p defined below. (Some authors use AZ, instead of z, as the independent variable.) These functions depend on z and on a parameter p, which is usually suppressed. Note that 9:(0)=s2(0)2113(O)94(O), where 9,(0)=tl),(O,

0 0 18.10.1

r=wf/w

18.10.2

=eirr =e - r K ’ t K

18.10.3

pis real and since ga>O(lw’12W), O
(v=az/2w)

j=1,2,3

18.10.8

U(

z)=-

2w exp a

18.10.9 12w2el=+[.2[6:(0)+S:(O)] 18.10.10 12w2e2=7r2[9:(0)-9:(0)] 18.10.11 12w2e3=-+[9:(0) +Sa‘(O)] A

18.10.12 (e2-es)t=-i(ea-ep)f=2w9:(0) 18.10.13 (el-ea)t= -i(ea-el)t=

lr

-8g(O) 2w lr

18.10.14 (el-e2)t=--i(ez-er)f=-9:(0) 2w 18.10.15 g2=i (&)l[O:(O) 18.10.16 ga=4eleZea 7?

18.10.17 A t = a 9 i 2 ( 0 )

+6:(0) +S!(O)]

p).

651

WEIERSTRASS ELLIPTIC AND RELATED FUNCTION&

18.10.20 18.10.21 18.10.22 18.10.23 Attainable Accuracy

A>O

A
Note: tJ2(0)=AefrI8, A>6;

Note: Sj(0)>O, j = 2 , 3 , 4

a&(o) >o;

-

(9.,(0)=&(0)

6,(0):

2 terms give at least 5s

2 terms give at least 3 s

j=2,3,4

3 terms give at least 11s

3 terms give at least 5s

4 terms give a t least 21s

4 terms give at least 10s

18.11 Expressing any Elliptic Function in Terms of @ and @'

If f(z) is any elliptic function and @ ( 2 ) has same periods, write

arw

+a[

18.11.1 f(4= +f( - 41 {f(4 -f( -4 1 { @ '(41 -'I @ (4. Since both brackets represent even elliptic functions, we ask how to express an even elliptic function g ( z ) (of order 2k) in terms of @ (2). Because of the evenness, an irreducible set of zeros can be denoted by at (i=ll 2, . . , , k) and the set of points congruent to - I t t (i=l, 2, . . . , k);correspondingly in connection with the poles we consider the points &bt, i= 1 , 2 , . . . , k. Then 18.11.2

a constant. If any at or bt is congruent to the origin, the corresponding factor is omitted from the product. Factors corresponding to multiple poles (zeros) are repeated according to the multiplicity. 18.12. Case A=O(c>O)

Subcase I

18.12.10

18.12.1 g2>0, ga
Hi=Ha=O, H 3 = 3 ~

18.12.3 @ ( z ; 12c2,-8~)=c+3c{sinh [(3c)+z]}-2

18.12.4 C ( Z ; 12c2,-88)=-~2+(3c)* coth [(3c)k~] 18.12.5

u(w)=O

18.12.11 18.12.12

u(w*)=O

18.12.13

@ ( w / 2 ) =c

18.12.14

@ (w/2)=0

18.12.15

c(w/2)=-

f

18.12.7 18.12.8

w=

)If = { ( U f )

= -cwf

18.12.19

CD

4 4 2 ) =0

18.12.16 sinh [ ( 3 ~ ) + z ] e - ~ ~ ~ / ~ 18.12.17 a, of = (12c)-+lri 18.12.18 q=f(w)=-m

u ( z ; 128,-88) = (3c)-4

18.12.6

m=l

q=l,

18.12.9

@ (w '/2) =-5c

r (w /2) =4 f

(--CO

+if/I. )

652

WEIERSTRASS ELLIPTIC AND RELATED FUNCTION&

18.12.20

U(

wf/2)=

era/geJZ

18.12.21

@ (wa/2)=c

18.12.22

@,'(wa/2)=0

- --cw'2

{( 4 2 ) =

18.12.23

Subcase I1

18.12.47

&/2)=-+i

18.12.48

U(

HI=3c, H,= H3=0

18.12.27 @ ( z ; 12c2,8c3)=-c+3c{sin [(3c)tz]} -a 18.12.28 (3c)t cot [(3c)tz]

18.12.29 u(z;

128, 82)= (3c)-* sin [(3c)~z]eezz~2

18.12.30

w=(12c)-hr, d = i w

18.12.31

q=f(w) =cw

18.12.32

qf = c ( w f ) =;a

18.12.33

2

~ 4 2= )0

18.12.49

ga=U, g3=0 (el =e2 =e3=0)

18.12.50

@ ( 2 ; 0, 0)=2-2

18.12.52 g2>0, g3>O: (el=2c, e2=e3=-c)

{ ( z ; 128, 82) =CZ+

CW

Subcase I11

18.12.51

18.12.25 18.12.26

(w2/2) =o

03

u(w,/2)=0

18.12.24

@

18.12.44

7r

18.12.53

((2;

0,O)=z-'

"(2;

0,O ) = r

-iwf=

a,

18.13. Equianharmonic Case (ga=O, g3=l)

If gz=O and g3>0, homogeneity relations allow us to reduce our considerations of @ to @ ( 2 ; 0, 1) ( p f ,{ and U are handled similarly). Thus @ ( z ; 0 , g 3 ) = g ~ 1 3 @ ( z g ~ l "1). ; 0 , The case ga=O, g a = l is called the EQUIANHARMONIC case.

t FPP; Reduction to Fundccmentd Triangle AI =AOwaZo is the Fundamental Triangle Let c denote eirI3 throughout 18.13. Q-1.5299

54037 05719 28749 13194 17231

'

m=O

4=0,

18.12.34

2w'

=o

18.12.35

U(.')

18.12.36

u(wJ=O

18.12.37

@ ( 4 2 ) =5c

18.12.38

@ '( 4 2 ) =-2w3

-I?

18.12.39 18.12.40 18.12.41

@ ( w f / 2 )=-c

18.12.42

@ f(wf/2)=0

18.12.43

c(0'/2) = +im

18.12.44

u(wf/2)=o

18.12.45

@ ( 4 2 ) = --c

WO

FIGURE18.11 8 This value was computed and checked by multiple precision on a desk calculator and is believed correat to 30s.

653

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

Reduction-for zz in Az: zl=eZo is in Al.

I

Special Values and Formulas

18.13.9 A=-27,

Hl=&(4-'/3)2,

H2 =&(4-19, 18.13.10 m = M 1 5 ' 2=- 67 18.13.4

&a)

=G(zJ

Reduction.for 23 in A3: zl=e-'(2w'--5)

is in AI

@ (Z3>=e-*@(zl> @ '(%I= @'(ZI)

18.13.5 18.13.6

18.13.11 18.13.12 18.13.13 18.13.14 where A=

18.13.7 f(z3) = -e-'f(zl) +2$, v'=f(o') 18.13.15 18.13.8 U(Za)=€U(zl) exp [(z3-o')(2rl')] Values at Half-periods

w2=

&(O)

7

Ji(4-119,

H3= q=ie

--5/2

=Aeir/8

S3(0)=Aef*/24

S,(O)=Ae+lz4 (w2/r)1/221/331/8=1.0086 67 ~ ( m ) 2 1 /3 r3(1/3) -31/4 47

18.13.16

lo Io ID

W B U 1

18.13.17 U2

18.13.18 w'

E wa

18.13.19 wal

Value€ f 18.13.20

!2w2/9 18.13.21 0213 I

18.13.22 4wd9

18.13.23 0212

18.13.24

2 ~ / 3

1

1

-43

18.13.25

849 7

Values at 2~2/9,h 1 / 9 and 802/9 from [18.141.

U

654

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

Values along (0, zo)

18.13.26 2012

1

-21/ae1

@‘

U

3i

18.13.28 10

Duplication Formulas

18.13.30 @’(22)=

where tan 4 = @ ’ ( x ) , 0<x<2w2 choose 4 in intervals

2 @ 6( 2 ) - 10p 3 (2 ) - 1 [05w13

@

x (i)j,P (g+y),

202

asd we must

x 4wa

@ (s+y)j

respectively.

Complex Multiplication

18.13.32

U(&)=-

@’(2)u4(2)

Trisection Formulas ( x real)

18.13.35

@(ez)=e-2@(2)

18.13.36

@’(ez)=-P’(z)

18.13.37

{(EZ)

18.13.38

= a-’t(z)

u(EZ)=tu(Z)

In the above, e denotes (as it does throughout The above equations are useful section 18.13), as follows, e.g.: If z is real, ez is on 00’ (Figure 18.11); if ez were purely imaginary, z would be on Ozo (Figure 18.11).

Conformal Maps -60°

Equianharmonic Case

Map: f(z)=u+iv

Near zero:

9 (z)=;i+el1

-1.2

1

.2

.4

.8

1.0

1.2

1.4

WEIERSTRASS ELLIPTIC AND RELATED FTJIVCTIONfi

655

.8

.5

@'(2) -2 Near zero: @ '(z)=-+el

z3 U

0

r(2)

1

Near zero: c(z)=-+el

.2

.4

-2.5 .6

-2.0 .8

-1.5 1.0

-1.0

-.5

0

1.2

-. 2

-.4

2

X

-.6

I

FIGURE 18.12

656

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONB-

Coefficients for Laurent Series for @, @' and f (c,=O for m#3k)

EXACT

k 1 2 3 4 5 6 7 8 9 10 11 12

APPROXIMATE

Cab

3. 5714 28571 42867

1/28 1/(13.28*)= 1110192

47409 21693 35758 8396I 79393 57961 79360 01329 68202 51463 4. 5541 38864 99184

13

Cab

.

x10-' 73312 40188X 10-8 55885 78983x10-7 61350 20301 X 10-10 00384 33643X io-18 36788 98x10-1' 41839 66x10-10 13026 87x10-fi 79236 40x10-U 63683X lO-" 05316X lO-*O 30391X 10-"

9.8116 1.8442 3.0400 k 6697 6.8662 9.7990 1.3685 1. 8800 2. 5497 3.4222

16954 88901 36650 9516i 18676 31742 06574 72610 66946 48599

6.0171

15776 98241 99591X 10-86

First 5 approximate values determined from exact values of c3k; subsequent values determined by using exact ratios c ~ ~ / c using ~ ~ - a~t ,least double precision rithmetic with a desk calculator. All approximate c's were checked with the lelieved correct to at least 21s; c3O-c80 are believed correct to 205.

Other Series Involving Reversed Series for Large

P PI

18.13.46 w-w2+d-;i

6

3 w4+- w' 7

18.13.39 z= (@-1)1/2

3u2 5us [I+-+-+-+-+7 26 38 U

7u4 6 3 d 40 248

18.13.47 where w =( @ -ea)/3ea

Other Series Involving @' 18.13.40 where U= @--"I8 and z is in the Fundamental Triangle (Figure 18.11) if @has an appropriate value.

Reversed Series for Large

I @ '1

18.13.48

Series near zo

18.13.41

z being in the Fundamental Triangle (Figure 18.11) if @' has an appropriate value.

18.13.42

Series near SO

18.13.43 where u=(z-zo) Series near w z

18.13.49

(@'-i)=z

[-2--iz+g

5

X2+%

3i P+O(X4)]

18.13.44 18.13.50 where x= ( z - ~ , , ) ~ 18.13.51 18.13.45 where U = ( Z - - ~ ) ~ , x=eau

2=&

132 $+T+o(CY4)]J

18.13.52 where a= (@ '-i)/(-4)

i

657

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

27*35*52*31 z26+28*38*5*9103 z31 + 25! 31!

Series near w2

18.13.53 @'=6ei(~-w2)

1+2v+3v2+?

24

-212*3Q*5*229*2683 37!

d

z37

25 24 285 +-?v4+v6+ve+o(v7) 7 91

-214*310*5*23*257*18049

1

z4a

43!

-215*312.5*59* 107895773

18.13.54 where v =e2 ( z -02) 18.13.55 (z-02)=(

o(Z 5 5 )

z4Q +

49! @'/64)

1-2w+9d-7

18.13.66

360 23

z=u+-

212058 + 3 3 0 w 4 - 2 2 6 8 w 5 + ~ we+ 0( W q ,

18.13.56 where w=

+

+

13.337~" 41d3 u7 23-3-5.7 27*32*52*11.13 210*34.53*11*17-19

@ "19

Other Series Involving Reversed Series for Large

r

Economized Polynomials (0 5 %11.53)

If(

6

18.13.67 z 2 @ ( z ) = C u,zeR+a(z)

18.13.57

0

I e(z)1<2X 10-7 y =t 4/20

18.13.58

~0=(-1)9.99999 96

~4=-(-9)2.20892

47

U1=(-2)3.57143

20

~,=(-10)1.74915 35

a2=(-5)9.80689

93

~,=-(-12)4.46863

aa= (-7)2.00835

02

Series near SO

18.13.59

18.13.60 where

6

U= (2- ~ 0 )

18.13.68 x3 @ '(2) =E 0 U,$"+

Series near w2

18.13.61 ( {-a2)= -e2(

3

18.13.62

U=

18.13.63

-ea

3

2

l+v+g v2+7 $+?

2-02)

el( z-

142) V*

12w2 26723 139w4 5 35 +5

w= ( f - d 2 / e 2 Series Involving U

18.13.65

uo= -2.00000

02)

1634208 d+0(w7)] --30192205 ,275 , , ,+

18.13.64

93

I< 4 x

c(2)

10-7

a4=-(-9)2.12719

00

66

al=(-1)1.42857

22

U5=(-10)6.53654

a2=(-4)9.81018

03

~~=-(-11)1.70510 78

aa=(-6)3.00511

93 6

18.13.69 zf(z) =

a$"

0

I&)

7

6=(-1)9.99999

98

67

+ (z) B

1<3 x 10-8 a4=(-10)6.12486

14

a1=-(-3)7.14285

86

~,=(-11)4.66919 85

a2=-(-6)8.91165

65 ~6=(-12)1.25014 65

a3=-(-8)1.44381

84

658

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

18.14. Lemniscatic Case (ga=l, ga=O) If g2>0 and ga=O, homogeneity relations allow us to reduce our consideration of @ to @ ( 2 ; 1, 0) (@', { and U are handled similarly). Thus @ ( 2 ; 92, O)=ga'@(Zga'; 1, 0). The g2=1, g3=0

is called the LEMNISCATIC case.

t

p (4=-F(Zd

18.14.1

@ '(Za>=iF(z1>

18.14.2

Y

-a4

S(s)= u(zJ =iS(zJ

18.14.3 18.14.4

r

=

t F P P ; Reduction to Fundamental Triangle A1=AOUUa is the Fundamental Triangle w=1.8540 74677 30137 1928 Reduction f o r 22 in Aa: Z 1 = G z is in A1

Special Values and Formulas

18.14.5 A= 1, H,=H3=2-+, H,=i/2,

w3):iw

m=sina 45O=4, q=e-* 18.14.6 &(O)=S,( O)= (UJZ/T)*; &( O)=( ~ U / T ) +

G~2.6220575542 92119 81046 48395 89891 11941 36827 54951 43162 is the Lemniscate constant [18.9]

,

This value was computed and checked by double precision methods on a desk calculator and is believed correct to 18s.

1

8

FIGURE 18.13

Values at Half-periods

Values along (0, U )

t

U

18.14.11 4 4

18.14.12

4 18.14.13 2w/3

18.14.14 3014

4-2

I

-+ T

8w

-=

-0

2&

er/l%(21/10) Ut

~

1

1

659

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

Values along (0, z o )

r

U

18.14.15

z0/4 18.14.16 2012 18.14.17 2 2013 18.14.18

3 2014 a=l+JZ

18.14.24

Duplication Formulas

18.14.19

@ (22) =[

+@'

@'(z)+tl"/I

(;)= @ '($1 T P@ +414 @ (2)

(2)-

3

@(4[4@'(4--11)

-[2@(2)-314@

18.14.20

<2)+3

-2@3/2((2) (See [18.13].)

[Use - on

O
+ on w < 2 < 2 w ]

Complex Multiplication

18.14.22

U( 2 2 ) =

-@ '( z)u4(2)

Bisection Formulas ( O < S < ~ W )

18.14.23

18.14.25

@(iz)=

- @' (2)

18.14.26

@ '(iz)=i @' '( 2)

18.14.27

l ( i z ) = --it(2)

18.14.28

a(iz)=iu(z)

The above equations could be used as follows, e.g.: If z were real, iz would be purely imaginary.

Conformal Maps Lemniscatic Case

Map: f(z)=u+iv @(z> Near zero: @(z)=-$-el1 x

1 z2 @(z)=2+,+e2,

Near zo: @ ( z ) =

14<1

-(Z-Z0)2+ea,

IEZI d ,0009

4

I z -zol <Jz @(z>=

-( 2- zoy 4

+szoyo (2-

+E4

660

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

X

-3

-2

0

-I

Near zo:@ ' ( z ) =

Near zero: @'(z)=-+el-2z3

2

I

4 2 -

20) +

2.0

I .5

X

I .o

.5

0

.5

I .o

FIGURE 18.14

I .5

2 .o

2.5

661

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS Coefficients for Laurent Series for @,

@ ', and r

(c, =0 for m odd) ~

EXACT

APPROXIMATE

czk

1/20 1/(3.20')= 1/1200 2/(3.13.203)= 1/156000 -, - - - 2/(32.13.17.20$= i1(31824x 105) = 1/(4964544X 10') 10/(3~~13a*17~20s) 4/(3.13*.17*29.20') = 1/(7998432X 10') 2453/ (3'. 11.13'. 17'.29.208) =958203 125/( 1262002599 X 10") 2.5.7.61/(33.13a.17'.29.37.209)= 833984375/( 18394643943X 10")

1 2 3

i

...

.8333 ,641025 .47134 ,31422 .2OI42 .12502 ,75927 ,45338

~

6 7 8 9

d

C ' k S - 93k-1

k = l , 2,

I @I

18.14.39 A=W3, v=Au4/6, and z is in the Fundamental Triangle (Figure 18.13) if @' has an appropriate value.

=a

Series near

18.144

w = @-'/8, and z is in the Fundamental Triangle (Figure 18.13) if @ has an appropriate value. 18.14.30

Series near SO

x=(z-20)2/2 x=-[w+,+,+-+

~3

@'

SO

+

[

(2- 20) -1 + 3 w -low2 7 + 3 35w3 0 (Wd)], w= ( 2- Z ~ ) ~ / ~ O

18.14.41 18.14.42

XI

18.14.31 2 @= -x+---+-+O(z9), 5 75 325

18.14.33

37651X10-la 59917X 10-l6 06092x10-1~

18.14.38

+ s + 4 0 0 + 4 6 4 + 1 2 8

18.14.32

32882X io-11

@' Reversed Series for Large I @ ' 1

429w7 195w8

x3 2x6

XlO-5 98944X 10-7 99296X 10-0

Other Series Iiivolving

18.14.29

3w6 231w'

..

...

Other Series Involving @ Reversed Series for Large

X10-3 . 641025 23831 07088 82554 04725 SSSSS 49183 45048 02941 19109 76468 43533 93461

Czk

u=4@'4

18.14.43

7 ~ 6 1 1 ~ 7 O(we)] 195 w=2@

Series near w

18.14.44

@ '=5

[1+x2+53

3

2

5 4 + 3 X6+G

1,

11

x8+mX'O+ 0(xl2)

Series near w

18.14.34

18.14.45

5=

(2 -u)

18.14.46 18.14.35

v= (2- uy/2

18.14.36

v=y

1-y+----+6y2 S$ 172y' 5 5 7 5

[

18.14.47 18.14.37

18.14.48

Other Series Involving 5 Reversed Series for Large If1 v Va 1368 1 3 4 9 ~ ' + ~ (

[

z =t-' 1--+---+5 7

1001

v= (4112

9163

.]'

v )

662

WEIERSTRASS ELLIPTIC AND RELATED FUKCTIONS Economized Polynomials (0 <x

Series near 50

18.14.49 1

(r-lo)=~(z-z0)

3[

31 7v 2v2 33 ---+---

6

39 v3 + 0 ( V 4 ) l J

18.14.57

2 @( z ) = C a , ~ ~ ~ + e ( ~ ) 0

(e(z)(<2x10-7

v= ( Z - Z ~ ) ~ / ~ O

18.14.50

(-1)9.99999

98

a4=(-8)4.81438

al=(-2)4.99999

62

a,=(-10)2.29729

21

U2=(-4)8.33352

77

a6=(-12)4.94511

45

uO=

Series near w

18.14.51

a3=

11213

211

5 1.86)

20

(-6)6.4D412 86

215

825 31200 9750

18.14.52

2 = (2-

6

U)

18.14.58

sc”@’(~)=Ca,z~~+~(~) 0

18.14.53

I€( 2)I< 4 x 10-7

x.w--+---+--w3 7w6 13w’ Y29w9 194w“ 3

30

63

4536

942883~’~ 891 + 3891888 +o(W15)

18.14.54 ~=-2({--71) Series Involving U

ao= -2.00000

a4= (-7)6.58947

00

~ ~ = ( - 1 ) 1 .0 0 0 0 0 02 a2=(-3)4.99995 38 a,=(-5)6.41145

52

~,=(-9)5.59262 49 ~,=(-11)5.54177 69

59

18.14.55 g=z----

z5 3 2 ~ 93 . 2 3 ~ ‘ ~ 3 . 1 0 7 ~ ’ ~33*7*23*3722’ 2.5! 22m9!+23.13!+24.17! + 25.21! 32*313*503~25 -34*7*685973~2e +0(~33) + 2e.25! 2’*29!

18.14.56 17.113 ~ ’ ~ 24-3*5 2’*3.7 213.34*7.11.13

z=u+-+-+

I9

U9

+

6

18.14.59

z l (2)=Conz4fi 0

e(2)

( € ( 4 < 3 x10-8 ~ =(-1 )9 .9 9 9 9 9 99 a1=-(-2)l.66666 74 a2= - (-4) 1.19036 70 a3=-(-7)5.86451

0,,=-(-9)2.57492 62 00 a,=-(-11)5.67008 a6= (- 13)9.70015 80

63

18.15. Pseudo-Lemniscatic Case (g2=-19

g3=0)

If g2<0 and g3=0, homogeneity relations allow us to reduce our consideration of @ to @ ( 2 ; -1, 0). Thus 18.15.1 @ ( z ; 92, O)=lgz11’2@(~1g211’4;-l, 0) [@’, l and U are handled similarly]. Because of its similarity to the lemniscatic case, we refer to the case g2=-1, g3=0 as the pseudo-lemniscatic case. It plays the same role (period ratio unity) for AO. &X (real half-period for lemniscatic case) =ij (the Lemniscate C o n s t a n t s e e 18.14.7)

wa=

2-U

FIGURE18.15

663

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

18.15.2

A=-1,

g2=--1,

g3=0

18.15.4

18.15.3

m=+,I J = ~ - * / ~

HI= -it&, H2=3, H3=i/&,

I

b2(0)=R21/4ei*18, b3(0)=Rei*I8, b4(0)=Re-i*/8,

18.15.5

where R =

Ju2JZ/r

Values at Half-Periods

18.15.6 18.15.7 18.15.8 18.15.9

WIW~

wz wf=w3 w2’

I

1 i/2

I

-i/2

~

Relations with Lemniscatic Values

18.15.10 @ ( z ; -1, O ) = i @ ( ~ e ‘ “ / ~ ; 1, 0) 18.15.11 @ ’ ( z ; -1, 0)=e3nf14@’(zeiT/4; 1, 0)

I

18.15.12 p(z; -1, 0)=eir/4p(zei*/4;1, 0) 18.15.13

u(z;

-1, 0)=e-t*/4u(zef*/4; 1, 0)

Numerical Methods 18.16. Use and Extension of the Tables

Example 1. Lemniscatic Case (a) Given z=z+iy in the Fundamental Triangle, find @(@’,(,U) more accurately than can be done with the maps. u-Use Maclaurin series throughout the Fundamental Triangle. Five terms give a t least six significant figures, six terms a t least ten. @, {-Use Laurent’s series directly “near” 0, (if lz1<1, four terms give a t least eight significant figures for @, nine for [; five terms a t least ten significant figures for @, eleven for {). Use Taylor’s series directly “near” zo. Elsewhere (unless approximately seven or eight significant figures are insufficient) use economized polynomials to obtain @ (x), @’(x) and/or {(z) as appropriate. To get @(iy), @’(;U) and/or {(iy), use Laurent’s series for “small” y, otherwise use economized polynomials to compute @ (y), @’(y) and/or p(y), then use complex multiplication to obtain @ (iy), @’(iy) andtor {(iy). Finally, use appropriate addition formula to get @ ( z ) and/or ( ( ~ 1 . @’-Use Laurent’s series directly “near” 0 (if Iz\< 1, four terms give a t least six significant figures, five terms a t least eight significant figures). Elsewhere, either use economized polynomials and addition formula as for @ and {, or get @’2=4@3- @ and extract appropriate square root (,(7@ ’20). (b) Given @(@’, {, U ) corresponding to a point in the Fundamental Triangle, conipute z more accurately than can be done with the maps, Only a few significant figures are obtainable from the use of any of the given (truncated) reversed series, except in a small neighborhood of the center of the series. For greater accuracy, use inverse interpolation procedures. Example 2. Equianharmonic Case (a) Given z=z+iy in the Fundamental Triangle, find @ (@’, {, U ) more accurately than can be done with the maps. u-Use Maclaurin series throughout the Fundamental Triangle. Four terms give a t least eleven significant figures, five terms a t least twenty one. @,{-Use Laurent’s series directly “near” 0 (if /PI< 1, four terms give a t least 10s for @, 11s for I ; five terms a t least 13s for @, 14s for p). Elsewhere (unless approximately seven or eight significant figures are insufficient) use economized polynomials to obtain @ (x),@’(z) andtor [(z), as appropriate. To get @(iy), @’(iy) and/or {(iy), use Laurent’s series. Then use appropriate addition formula to get @ (4 and/or t(z).

664

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

@’-Use Laurent’s series directly “near” 0 (if Izi< 1, four terms give a t least SS, five terms a t least 11s). Elsewhere, either proceed as for @ and {, or get @ ’ * = 4 P 3 - l and extract appropriate square root (g@ ’ 20). (b) Given @(@’,{,U) corresponding to a point in the Fundamental Triangle, compute z more accurately than can be done with the maps. Only a few significant figures are obtainable from the use of any of the given (truncated) reversed series, except in a small neighborhood of the center of the series. For greater accuracy, use inverse interpolation procedures. Example 3. Given period ratio a, find parameters m (of elliptic integrals and Jacobi’s functions of chapter 16) and q (of 29 functions). m-In both the cases A>O and A3, use the method of Example 6 in chapter 17. An alternative method is to use Table 18.3 to obtain the necessary entries, thence use m= (e2-e3)/(e,-e3) in case A>O, m=+-3e2/4H2 in case A
q-In both the cases A>O and AO, p=ie-ralz if AI. Determination of Values at Half-Periods, Invariants and Related Quantities from Given Periods (Table 18.3)

A>O

A
Given w and U ’ , form w ’ / i w and enter Table 18.3. Multiply the results obtained by the appropriate power of w (see footnotes of Table 18.3) to obtain value desired.

Given w2 and w2’, form wz’/iwz and enter Table 18.3. Multiply the results obtained by the appropriate power of w2 (see footnotes of Table 18.3) to obtain value desired.

Example 4.

Example 4.

Given w=10, w ’ = l l i , find er, gr, and A.

Given w z = l O , w 2 ’ = l l i , find et, g,, and A.

Here w’/iw=1.1, so that direct reading of Table 18.3 gives

Here w6/iw2= 1.1, so that direct reading of Table 18.3 gives el(l)= --.2166 2576$3.0842 589i

e1(1)=1.6843 041 ez(l)=-,2166 es(l)= -1.4676

258

e2(l)=.4332 5152= -2W(e1)

(=-el-e3)

783

e d l ) =&(1) g2(1)=-37.4874

g2(1)=10.0757 7364 g3(1) ~ 2 . 1 4 2 01000.

Multiplying by appropriate powers of obtain

912

gS(1) = 16.5668 099. U=

10 we

Multiplying by appropriate powers of w2= 10 we obtain

e1=.01684 3041 e2= - .00216 6258

el= --.00216 62576+.03084 2589&

e3= - .01467 6783

f?3=e1

g2= 1.0075 77364 x 10-3

g2=-3.7487

e2=.00433 25152

-

g3=2.1420 1000 x 10-’

whence

4912 x 10-3

g3=1.6566 8099 x lO-’

whence A=8.9902 3191 x 10-l’

A=-6,0092

019 x 10-*

665

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

Example 5. (A>O) Given w=10, w’=55i, find q , TJ’,u ( w ) , u ( w ’ ) and u(w2). Forming o’/iw=5.5 and entering Table 18.3 we obtain n=.82246704, d w ) =.96045 40. Using Legendre’s relation we find‘q’=go’--i/2 =2.9525 723i. Since interpolation for u(w’) and u ( w + w ’ ) is difficult, use is made of 18.3.15-18.3.17 together with 18.3.4 and 18.3.6. Values of g2,g3 and el can be read directly to eight significant figures and e3 to about five significant figures giving g2=8.1 174 243, g3=4.4508 759, el=1.6449 341, and e3 =-232247. Use of 18.3.6 ields H3=.00174 69 and H2=.00174 69i. Apptcation of 18.3.1518.3.17 yields u(w’)/i=.OO71177 and u(w2) = -.002016-.01055i. Multiplying the results obtained by the appropriate powers of w we obtain q = .08224 6704, q‘=.29527 723i, U(.) =9.6045 40, ~(0’)=.071177i and u(oz)= -.02016--.1055i.

Example 5. (A
Multiplying the results obtained by the appro6 q; priate powers of w2 gives ~ ~ ~ = . 0 0 1 526756, =--.00157 26664&, ~(~2)=1180.5028, =1190.152& and .(U’) =475.084+476.717i.

.(U:)

Determination of Periods from Given Invariants (Table 18.1.)

0 0 Given g2>0 and g3>0 such that A=g:-27g;>O (if g3=0, I w ’ ( = w ; see lemniscatic case), compute g2=g2g;2/3. From Table 18.1, determine ~ g and ~ ’ g ~ ~thence “ ‘ , w and U’.

-

Example 6. Given g2=10, g3=2, find w and w‘. With &= g~g,-~/~=6.2996 05249, from Table 18.1 1.1267 806 and ~ ‘ g ~ ~ / ’ = 1 . 2 3 2295i 4 whence U = 1.003847 and U‘ = 1.097970i.

A< 0

Given 92 and g3>0 such that A=$-27d
-

Example 6. Given -10, g3=2, find

ua

and wk. With

g~=g~g,-’~‘=--1011.5874 0105= -6.2996 053, from Table 18.1 02g31/6=1.5741 349 and w;g3’lB=1.7124 396i whence w2=1.40239 48 and 0:=1.52561 02i.

Example 7. Example 7. Given g2=7, 6, find w2 and U;. Given g2=8, g3=4, find w and U’. With &= g2g3-’ f=7/3.301 9 2725 =2.119974, from 52=9~93-~”=3.174802104, from Table 18.1 ~ g ~ ’ / ~ With = Table 18.1 uZg31i6=1.3423 442 and ~ ~ ‘ g ? / ~ = 3 . 1 4 4 1 1.2718 310 and 0 ’g ~ ~ /~ =1 .8 7425.7: 02 141i whencew2=.99579 976 and w;=2.33241 83i. whence w= 1.009453 and w‘= 1.484413i. Computation of @,

p t ,or f for Given z and Arbitrary g,,

ga

(or arbitrary periods from which g2 and ga can be computedin any case, periods must be known, at least approximately) First reduce the problem (if necessary) to computation for a point z in the Fundamental Rectangle by use of appropriate results from 18.2.

666

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

Method 1 (as accurate as desired) If both x and y are “sIna11,” (point in double-cross hatched region) use Laurent’s series in z directly. If either x or y is “large,” use Laurent’s series on Ox, then on Oy and finally use an addition formula. (For@’ an alternative is to get @, then compute the appropriate root of @’2=4@3-gz@-g3; see 18.8.) A>O

A
Method 2 (for @ or @ ’ only) Compute ei(i=1,2,3) (if only g2, g3 are given use Table 18.1 to get the periods, then get e t in Table 18.3; if periods are also given, use Table 18.3 directly). In any case, obtain m(=[e2-e3]/ [el-e3]), thence Jacobi’s functions sn(z*lm), cn(z*lm), dn(z*lm), from 16.4 and 16.21 and @ or @’ from 18.9.11-18.9.12.

Method 2 (for @ or @’ only) Compute e2 and H2 (if only g2,g3are given, use Table 18.1 to get the periods, then get ei in Table 18.3; if periods are also given use Table 18.3 directly). In any case, obtain m(=%-3e2/4Hz) thence Jacobi’s functions sn(z’lm), cn(z’Im), dn(z’lm), from 16.4 and 16.21 and @ or @’ from 18.9.11-18.9.12.

Method 3 (accuracy limited by Table 4.16 of and by the method of getting periods). Obtain periods, their ratio a, then q=e-ra from Table 4.16. Hence get S t ( 0 ) , i=2,3,4 from truncated series 18.10.21-.23. Compute appropriate 8 functions for z=x and for z=iy, whence get P(41 @’(d and/or T ( 4 , @ ( i Y ) l @’(id and/or {(iy), then use an addition formula (if either x or y is it is probably easier to use Laurent’s series).

Method 3 (accuracy limited as in the case A>O). Obtain periods, their ratio a, thence qz=e-.a/2 from Table 4.16. Then proceed as in the case A>O, using corresponding formulas.

Example 8. Given z=.O7+.li, g2=10, g3=2, find @. Using Laurent’s series directly with cz= .5

Example 8. Given z=.1$.03i, g2=-10, g3=2, find @ . Using Laurent’s series directly with Cz= - .5

c3= ,07142 85714 .O8333 33333 cS= .00974 02597 z - ~ = -22.97193 820-63.06022 25i +c&=.00255 OOO+ .00700 O O i +c3z4=.00001 214.00001 0% +c&=+ .OOOOO 024.OOOOO O l i C4=

Cgz.07142 85714 A0333 33333 ~ - ~ = 7 6 . 5 9 2 8938-50.50079 7 960i czz2= - .00455 000 - .00300 OOOi c3z4= +.OOOOO 334+ .OOOOO 780i c&= - .ooooo 002 .ooooo 01 li C4=

+

@ (~)=76.58833270-50.50379 169i.

@ ( z ) = -22.97450 010-63.05323 28i. Example 9. Given z=15+73&, g2=8, g3=4, find @. From Example 7, w=1.009453, a’= 1.484413i. From Table 18.3, e1=1.61803 37, e3=-.99999 96, whence m=.14589 79. From 18.2.18 with M = 7 and N=24, @((.867658+ 1.748176i)=@(l5+73i). Since z lies in Rz, by 18.2.31 @ ( 1 5 + 7 3 i ) = p ( . 8 6 7 6 5 8 + 1 . 2 2 0 6 5 i ) . From 16.4 with ~*=1.40390+1.97505i, sn(z*lm) =2.46550+1.96527i. Using 18.9.11, @(15+73i) - _ A7743 .067797i. -

+

Example 9. Given z=1.75+3.6il g2=7, g3=6, find @. From Example 7, w2=.99579 98, U;= 2.33241 83i. Using 18.2.18 with M = 1 , N=1, @ (1.75+3.6i) = @ (- .24159 9 6- 1 .O 6 4 8 3 6;) = @((.2415996+1.0648 36i). WithA
WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

A>O

Example 10. Given w=10, w’=2OiJ find ((9+19i) by use of theta functions, 18.10 and addition formulas. For the period ratio a=w‘lwi=2 with the aid of Table 4.16, q=e-2“=.00186 74427. Using the truncated approximations 18.10.2118.10.23 we compute the theta functions for argument zero. Using 16.27.1-16.27.4 we compute the theta functions for arguments v where z=z and z=iy. Then, with 18.10.5-18.10.7 together with 18.10.9 and 18.10.18 we obtain ((9)=.09889 5484, {(19i)=-.00120 0155i, @(9)=.01706 9647, @’(9)=--.00125 8460, @ (19i)=-.OOSSl 2615, @’(19i)=-.00003 757i. Using the addition formula 18.4.3, we obtain ((9+ 19i)=.O7439 49-.00046 8%.

667

A
Example 10. Given w2=5, wi=7i find @’(3+2i) by use of theta functions, 18.10 and addition formulas. With the use of Table 4.16 and 18.10.2, q=ie-.’* =.11090 12784i. The theta functions are computed for argument zero using 18.10.21-18.10.23 a n d t h e t h e t a functions for arguments v, and v2 corresponding to z=zl+z2 using 16.27.1-16.27.4. Using 18.10.518.10.6 together with 18 10.10, we find @ (3)= 773, @’ (3) = .10576 946, @ (2i) = -.24497 - .07474140, @ ’(24 =-.25576 007i. The addition formula 18.4.1 yields @(3+2i) =.01763 210 -.07769 187i, and 18.4.2 yields @’(3+2i)= -.00069 182 t.04771 305i.

Use of Table 18.2 in Computing @ , @’,

t for Special Period Ratios If the problem is reduced to computing @ , @ ’, { in the Fundamental Rectangle for the case when the real half-period is unity and pure imaginary half-period is ia,for certain values of a Table 18.2 may be used. Consider @ as an example. If / z \ is “small”, then use Laurent’s series directly for @ ( z ) [invariants for use in the series are given in Table 18.31. If z is “large” and y “sma11” use Table 18.2 to obtain z2@ (2) and $@’(LE) , thence @ (z) and @ ’(2); use Laurent’s series to obtain @ (iy) and @’(iy) ; finally, use addition formula 18.4.1. For z “small” and y “large”, reverse the procedure. For both z and y “large,” use Table 18.2 to obtain @ (z), @ ’(z) , @ (iy) and @ ’(iy) , thence use addition formula 18.4.1. Similar procedures apply to @‘ or [. For @‘, one can also first obtain @, then compute @ “ = 4 p 3 - g 2 @ -g3 and extract the appropriate square root (see 18.8 re choice of sign for @’). A>O

A< 0

Example 11. Compute @((.8+i)when a=1.2. Using Table 18.2 or Laurent’s series 18.5.1-4 with g2=9.15782 851 and g3=3.23761 717 from Table 18.3, @(.8)=1.92442 11, @’(.8)=-2.76522 05, @ (i)= - 1.40258 06 and @ ’(i)= - 1.19575 5%. Using the addition formula 18.4.1 @ (.8+i)= -.381433- ,149361i. Example 12. Compute {(.02+3i) for a=4. Using Table 18.2 or Laurent’s series 18.5.1-5 with g2=8.11742 426 g3=4.45087 587 from Table 18.3, (($02)=49.99999 89, @ (.02) =2500.00016, @ ’(.02) = -249999.98376, ( ( 3 i )=A9635 173i, @ (3i) = - ,82326 51 1, @’(3i) = - .00249 829i. Applying the addition formula 18.4.3, [(.02+3i)= .016465$.89635i.

Example 11. Compute @(,9+.1i) for a=1.05. Using Table 18.2 or Laurent’s series 18.5.14 with g2=-42.41653 54 and g3=9.92766 62 from Table 18.3, @ (.9) = ,34080 33, @’(.9) = -2.164801, @ (.li)=-99.97876, With the addition for@ ’(.li) = -2000.42553. mula 18.4.1 @ (.9 .li)= ,231859- .215149i. Example 12. Compute @’(.4+.9i) for a=2. Using Table 18.2 or Laurent’s series 18.5.14, with g2=4.54009 85, 1 ~ 3 ~ 8 . 3 8 5 394 7 from Table 18.3, @ (.4) =6.29407 07, @’(.4)= -30.99041, Q (.9i) = - 1.225548, Q,’(.Si)= -3.19127 03i. Using the addition formulas 18.4.1-2, @’(.4+ .9i) = 1.10519 76- .56489 OOi.

+

668

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

Computation of

U

for Given 5 and Arbitrary gz and g3

(or periods from which g2 and g3 can be computed-in any case, periods must be known, a t least approximately)

First reduce the problem (if necessary) to computation for a point z in the Fundamental Rectangle (see 18.2). After final reduction let z denote the point obtained. A>O A< 0 If gz>w2/2 or If 9 z > w / 2 or, Jz>w:/4, use duplication formula as in case A>O. Otherwise, use Maclaurin series for U 92>w'/2, use duplication formula directlv. U( Z ) = - @ '(212) U ' ( 2/2), . I

obtaining u(z/2) by use of Maclaurin series for U and @'(2/2) by method explained above. Otherwise, simply use Maclaurin series for U directly. An alternate method is to use theta functions 18.10 first computing p and tYi(0), i=!2,3,4, A>O

A
Example 13. Compute u(.4+1.3i) for g2=8, ga=4. From Example 7, 0=1.009453 and U' =1.484413i. Since Yz>w'/2, the Maclaurin series 18.5.6 is used to obtain u(z/2)=~(.2+.65;) =.19543 86f.64947 284 the Laurent series 18.5.4 to obtain @'(.2+.65i)=5.02253 80-3.56066 93i. The duplication formula 18.4.8 gives u(.4+ 1.3i) = .278080+ 1.27278%.

Example 13. Compute u(.8+.4i) for g2=7, ga=6. From Example 7, 02=.99579 976, wi =2.33241 83i. Since 9z>w2/2, the Maclaurin series 18.5.6 is used to obtain u(2/2)=u(.4+.2i) =.40038 019+.19962 017i, the Laurent series 18.5.4 to obtain @'(.4+.2i) = -3.70986 70 +22.218544&. The duplication formula 18.4.8 gives u(.8+.4i)=.81465 765-k.38819 473i.

Given U(@, @', f) corresponding to a point in the Fundamental Rectangle, as well as g2 and g3 or the equivalent, find z. Only a few significant figures are obtainable from the use of any of the given (truncated) reversed eeries, except in a small neighborhood of the center of the series. For greater accuracy, use inverse interpolation procedures. If t'he given function does not correspond to a value of z in the Fundamental Rectangle (see Conformal Maps) the problem can always be reduced to this case by the use of appropriate reduction formulas in 18.2. This process is relatively simple for @ ( z ) , more difficult for the other functions (e.g. if A>O and @=a+ib, where b>O, simply consider F=a--iband find z1 in R1 [Figure 18.11; then compute z2=B1+2w', the point in 23, corresponding to the given @).

I

A>O Given @=l-i,

Example 14. g2=10, g3=2, find z. Using the first three terms of the reversed series 18.5.25 zl= .727+ .423i. The Laurent series 18.5.1 gives

and

@ ( ~ l ) = @ (.727+.423i)=.825-.895i

@ (zJ= @ (.697+.393i)=.938-l1O38i.

Inverse interpolation gives Repeated applications of the above procedure yield 2= ,706231 .379893i. zjl)= .707+ .380i.

+

A
Example 14. Given @ = 1 +i, g2= - 10, g3=2, find z . From Example 6, w2=1.40239 48 and 0:=1.52561 02i. Since b>O, z exists in Rz and z is computed with Using 18.5.25 with m = 1.25, a3=.25, u = [ ( @ ) - ~ ]and ~ / ~the coefficients cn from Example 8

LE.

2u= 1.55377 3973 +.64359 42493i C~U'=

.08044 9281-.19422 174663

c~u'=-.01961 9359f.00812 660476

dug L=. l o l l 5 7160-,04190 06673i 3

669

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

A< 0

Stopping with the term in u7, zl=.81+.23i. Assuming Az= - .03-.0li, using 18.5.1, @ ( 3 1 .23i)=.91410 95-36824 37i, @ (.78+.22i)= 1.03191 60-.91795 22i; with inverse interpolation zjl) =. 7725 4-.2404i. Repeated applications of inverse interpolation yield z=. 772247 - ,239258i.

+

Given f=lO-l5i, g2=8, g3=4, find z. Using the reversed series 18.5.40 with Example 15.

yz=.

A6=-.13333

Given u = . 4 + . l i , 92=7, g3=6, Using the reversed series 18.5.70 with 14583, ya=.05

Example 15.

find

2.

333,

a=+.40000 000$.10000 OOOi

A7= -.02857 14286, U=

,03076 923076-k.04615 384615i

A , d = - .OOOOO 001402$ .OOOOO 006860i A7U7= Z=

-.OOOOO 000004 -.OOOOO 000003i ,03076 921670+.04615 391472i.

@=+.00011

783$.00032 6963

----.OOOOO Y3U7-

208+.00001 432i

5

7 3 y p --.OOOOO -14

093+.00000 126i

19yay3u11=55 .OOOOO 013$.00000 006i

~ = . 4 0 0 1 1469+.10034 260i Methods of Computation of @ (@‘, f or U ) for Given o and Given 82, g3 (or the equivalent), with the Use of Automatic Digital Computing Machinery

(a) Integration of Differential Equation @ and @’ may be generated for any z close enough to a “known point” z * ( @ ( z * ) and@’(z*)

being given) by integrating @ “ = 6 @“g2/2. A program to do this on SWAC, via a modification of the Hammer-Hollingsworth method (MTAC, July 1955, pp. 92-96) due to Dr. P. Henrici, exists at Numerical Analysis Research, UCLA (code number 00600, written by W. L. Wilson, Jr.). The program has been tested numerically in the equianharmonic case, using integration steps of various sizes. For example, if one starts with z*=a2, using an “integration step” (h,k), where h and k are respectively the horizontal and vertical components of a step, with (h,k) having one of the six ,values ( f2h0,0), ( f h o ,f k o ) , hi3=02/2OOO, ko= lw;1/2000, one can expect almost 8s in @ and 7s in @’ after 1000 steps, unless z is too near a pole. (b) Use of Series The process of reducing the computation problem to one in which z is in the Fundamental Rectangle can obviously be mechanized. Inside the Fundamental Rectangle the direct use of Laurent’s series is appropriate when the period

ratio a is not too large. However, if aZfi(A>O) or aL2v5(A<0), the series will diverge at the far corner of the Fundamental Rectangle, so that use may be made of an appropriate duplication formula, Alternatively, one may compute the functions on 02 and Oy, then use an addition formula. Even so, the series will diverge at z = b if a12(A>O) and at z = b / 2 if aL4(AO).

Use of Maps If the problem (of computing @, @’, or U for given z ) is reduced to the case where the real halfperiod is unity and imaginary half-period is one of those used in the maps in 18.8 inspection of the

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

670

appropriate figure will give the value of @ ( z ) [ { ( z ) or ~ ( z ) to ] 2-3s. If @ ’ is wanted instead, get @, use 18.6.3 to obtain @ and select sign ( 8 ) of @ ’ appropriately. (See Conformal Mapping (18.8) for choice of sign of square root of @‘2).

approximation to I zo/ by Graeff e’s process, we zo=w2 may use the fact that zo=w+iyo(A>O), +iyo(AO and

Computation of zo Given g2, g3 (or equivalent) Since

220 @ (2,)

[l Zy,/w>-

2 ?r

arccosh

f l (=.7297)].

yo/wz is a monotonic increasing function of a for A
=O, the Laurent’s series gives

o=1+c*u2tc~u~+c*u4+ .* . where u=zi. We may solve this equation [by Graeffe’s (root-squaring) process or otherwise] for its absolutely smallest root [having found an

Further data is available from Table 18.2 or from Conformal Maps defined by ,Cp (2).

References Texts and Articles

[18.14] E. S. Selmer, A simple trisection formula for the

[18.1] P. Appell and E. Lacour, Principes de la th6orie des

fonctions elliptiques et applications (GauthierVillars, Paris, France, 1897). [18.2] A. Erd6lyi et al., Higher transcrndental functions, vol. 2, ch. 13 (McGraw-Hill Book Co., Inc., New York, K.Y., 1953). (18.31 E. Graeser, Einfuhrung in die Theorie der elliptischen Funktionen und deren Anwendungen (R. Oldenbourg, Munich, Germany, 1950). [18.4] G. H. Halphen, Trait6 des fonctions elliptiques et de leurs applications, 1 (Gauthier-Villars, Paris, France, 1886). [18.5] €1. Hancock, Lectures on the theory of elliptic functions, vol. 1 (John Wiley & Sons, Inc., Iiew York, S.Y., 1910, reprinted, Dover Publications, Inc., New York, N.Y., 1958). 18.61 A. Hurwitz and R. Courant, Vorlesungen uber allgemeine Funktionentheorie und elliptische Funktionen, 3d ed. (Springer, Berlin, Germany,

[18.15]

[18.16] [18.17] [18.18] [18.19]

elliptic function of Weierstrass in the equianharmonic case, Norske Vid. Selsk. Forh. Trondheim 19, 29, 116-119 (1947). J. Tannrry and J. Molk, I!%ments de la th6orie des fonctions elliptiques, 4 vols. (Gauthier-Villars, Paris, France, 1893-1902). F. Tricomi, Elliptische Funktionen (Akademische Verlagsgesellschaft, Leipzig, Germany, 1948). F. Tricomi, Funzioni ellittiche, 2d ed. (Bologna, Italy, 1951). C. E. Van Orstrand, Reversion of power series, Phil. Mag. (6) 19, 366-376 (Jan.-June 1910). E. T . Wbittaker and G. R‘. Watson, A course of modern analysis, ch. 20, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952).

Guides, Collections of Formulas, etc.

1929) * 18.71 E. L. Ince, Ordinary differential equations (Dover Publications, Inc., New York, X.Y., 1944). 18.8) E. Kamke, Differentialgleichungen, Losungsmethoden und Losungen, vol. 1, 2d ed. (Akade-

mische Verlagsgesellsrhaft,

Leipzig, Germany,

1943). [18.9] D. H. Lehmer, The lemniscate constant, Math. Tables Aids Comp. 3, 550-551 (1948-49). [18.10] S. C. hfitra, On the expansion of the Wrierstrassian

and Jacobian clliptic functions in powers of the argument, Bull. Calcutta Math. Soc. 17, 159-172 (1926). [18.11] F. Oberhettinger and W. Maqnus, Anwendung der

elliptischen Funktionen in Physik und Technik (Springer, Berlin, Germany, 1949). [18.12] G. Prasad, An introduction to the theory of elliptic functions and higher transcendent& (Univ. of Calcutta, India, 1928). [18.131 U. Richard, Osservazioni sulla bisezione delle funzioni ellittiche di Weierstrass, Boll. Un. Mat. Ital. 3, 4, 395-397 (1949).

[18.20]P. F. Byrd and M. D . Friedman, Handbook of elliptic integrals for engineers and physicists, Appendix, sec. 1030 (Springer-Verlag, Berlin, Germany, 1954). [18.21] -4.Fletcher, Guide to tables of elliptic functions, Math. Tables Aids Comp. 3, 247-249 (1948-49). [18.22] S. Flugge, Handbuch der Physik, vol. 1, pp. 12CL 146 (Springer-Verlag, Berlin, Germany, 1956). [18.23J H . Kober, Dictionary of conformal representations (Dover Publications, Inc., New York, N.Y., 1052). [18.24] L. M. Milne-Thomson, Jacobian elliptic function

tables (Dover Publications, Inc., New York, N.Y., 1950). [18.25] I<. Weierstrass and H. A. Schwarz, Formeln und Lehrsatze ziim Gebrauche der rlliptischen Functionen. Nach Vorlesungen und Aufzeichnungen des Herrn K. Weierstrass bearbeitet und herausgegeben von H. A. Schwarz, 2 d ed. (Springer, Berlin, Germany, 1893).

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

671

Tables [18.28] T. H. Southard, Approximation and table of the Weierstrass @ function in the equianharrnonic (18.261 Chih-Bing Ling, Evaluation a t half-periods of case for real argument, Math. Tables Aids Comp. Weierstrass’ elliptic function with rectangular 1 primitive period-parallelogram, hlath. Comp. 143 11, 58, 99-100 (Apr. 1957). f ( u ) = @ ( u ) - - U2 69, 67-70 (19GO). Values of e , ( i = l , 2, 3) to 15D , to 7D , with modified central differences, for various period ratios in case A>O. U = 0(.1).8(.05) 1.55. [18.27] E. Jahnke and F. Emde, Tables of functions, 4th ed., pp. 100-106 (Dover Publications, Inc., New York, N.Y., 1945). Equianharmonic case, real 1 [18.29] D. A. Strayhorne, A study of an elliptic function (Thesis, Chicago, Ill., 1946). Air Documents a r g u m e n t , P ( u ) , P / (U), ~ ( u U) ,( u ) , u = o nivision T-2, AMC, Wright Field, Microfilm No. Ro-734F15000. @ ( z ; 37, -42), 4D, 4w -I mostly 4D. z= .04i(.04i) 1.36i. 3

(g)

i

673

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

Table 18.1 TABLE FOR OBTAINING PERIODS FOR INVARIANTS ("bq =gqg,t1

E2

w+14

W V 6

8 3

-

In (g2-3)

3. 00 3. 05 3.10 3.15 3.20

1.28254 1.27944 1.27637 1.27333 1.27031

98 73 43 03 49

1.52168 1.51892 1.51685 1.51505 1.51342

83 22 48 45 84

3.25 3.30 3.35 3.40

1.26732 1.26436 1.26143 1.25853

80 90 77 38

1.51193 1,51053 1.50923 1,50799

18 84 08 63

*g 3

3.4 3. 5 3.6 3. 7 3. 8 3.9

1.25853 1.25280 1.24718 1.24166 1.23624 1.23092

38 64 42 45 47 23

1.69503 1.64719 1.60789 1.57451 1.54548 1.51978

4. 0 4.1 4. 2 4.3 4.4

1.22569 1.22055 1.21551 1.21055 1.20568

47 95 44 69 50

1.49672 94 1.47581 86 1.45668 57 1,43905 10 1.42269 63

4. 5 4.6 4.7 4.8 4. 9

1.20089 1.19618 1.19156 1.18700 1.18253

62 86 83 18

1.40744 1.39316 1.37973 1.36706 1.35506

84 72 79 51 88

5. 0 5. 2 5.4 5.6 5. 8

1.17812 1.16953 1.16120 1.15314 1.14532

83 35 96 34 23

1.34368 1.32250 1.30316 1.28537 1.26889

10 70 60 08 69

6. 0 6. 2 6. 4 6.6 6.8

1.13773 1.13036 1.12321 1.11626 1.10950

46 91 55 38 49

1.25356 1.23923 1.22577 1.21310 1.20113

57 29 98 78 41

7.0 7.2 7.4 7. 6 7. 8

1.10293 1.09653 1.09030 1.08423 1.07831

00

11 03 04 46

1.18978 1.17901 1.16874 1.15895 1.14959

83 03 82 b7 65

8.0 8.2 8.4 8.6 8.8

1.07254 63 1.06691 95 1,06142 83 1.05606 74 1.05083 15

1.14063 1.13203 1.12377 1.11583 1.10817

29 51 59 09 84

9.0 9.2 9.4 9.6 9.8

1.04571 1.04071 1.03582 1.03104 1.02636 1.02178

1.10079 1.09367 1.08678 1,08012 1.07367 1.06742

87 40 83 69 66 51

10. 0

E2-I

0.10

00

58 56 65 44 52 54

%taf 1.81701 99 90 90 87 17

a

-0.00 -0.01 -0.02 -0.03 -0.04

2.62205 2.62025 2.61693 2.61258 2.60737

76 54 53 87 43

2.62205 2.62384 2.62710 2.63126 2.63611

76 98 11 10 20

- *

-0.05 -0.06 -0.07 -0.08 -0.09

2.60137 2.59464 2.58720 2.57909 2.57032

48 00 37 05 09

2.64151 2.64735 2.65355 2.66002 2.66669

34 75 47 55 74

-

-0.10 -0.11 -0.12 -0.13 -0.14

2.56091 2.55088 2.54025 2.52905 2.51729

33 61 86 23 09

2.67350 2.68037 2.68725 2.69409 2.70081

25 66 88 09 77

-0.15 -0.16 -0.17 -0.18 -0.19 -0.20

2.50500 2.49221 2.47895 2.46527 2.45118 2.43675

11 23 70 01 90 29

2.70738 2.71375 2.71986 2.72568 2.73117 2.73630

70 03 26 31 52 70

- 10 - 9 - 8 - 8 - 7 - 7 - 6 - 6 - 6 - 5

g2-'

w'gtg $/i 1.89818 1.89119 1.88476 1.87888 1.87354

bl 06 56 68 40

0. 05 0.04 0.03 0.02 0. 01

1.84028 47 1.84412 45 1.84756 35 1.85050 78 1.85280 73 1,85407 47

1.86873 53 1.86447 02 1,86077 37 1.85769 72 1.85534 90 1.85407 47

"-A)']

[(-A)']

~12= 0 . 2 0 4 1 24145

20 17 14 13 11

- 5

6

w2g3

49 74 94 71 14

1.82907 1.94863 2.04569 2.12452

88 05 84 34

-

5 4 3 3

-0.45 -0.50 -0.55 -0.60 -0.65

1.69080 1.68433 1.67705 1.66962 1.66240

53 20 44 98 65

2.24023 2.28267 2.31773 2.34701 2.37174

31 03 31 74 42

2 2 2 2 2

-0.70 -0.75 -0.80 -0.85 -0.90

1.65555 1.64914 1.64320 1.63771 1.63264

57 98 64 44 84

2.39284 2.41102 2.42683 2.44070 2;45294

34 56 68 05 88

-

1 1

-0.95 -1.00

1.62797 70 1.62366 67

2.46384 40 2.47359 62

-

E2 -1.0 -0.8 -0. 6 -0.4 -0.2

1.62366 1.60646 1.58820 1.56918 1.54967

67 93 63 06 81

3.03954 3.05518 3.06892 3.08070 3.09053

85 40 24 50 50

1.52995 1.51022 1.49067 1.47143 1.45262

40 67 44 75 13

3.09846 3.10458 3.10899 3.11182 3.11318

47 18 55 48 95

0. 8

11 13 14 17

1.2 1.4 1.6 1.8

1.43430 ii41652 1.39933 1.38273 1.36672

15 88 41 24 71

3.11320 3.11196 3.10955 3.10604 3.10147

22 36 78 84 38

20 25 33 50 100

2.0 2.2 2.4 2.6 2. 8 3. 0

1.35131 1.33647 1.32220 1.30847 1.29526 1.28254

24 63 24 11 10 98

3.09584 3.08910 3.08116 3.07175 3.06025 3.04337

00

0

-

1.62955 1.66926 1.68880 1.69574 1.69529

0. 0 0. 2 10

-100 - 50 - 33 - 25

-0.20 -0.25 -0.30 -0.35 -0.40

0. 4 0. 6

1.82207 1.82696 1.83165 1.83611

0. 00

A= 0

33 87 93 65 31 54

0.08 0. 07 0. 06

0. 09

@&E21 l

E2-

wii/i

&

E2

AND g3

Non-Positive Discriminant

Non-Negative Discriminant

' B

g2

1.0

=

74 35 37 10 67

0.40824 829

1

- 1 - 1

1 1

A= 0

674

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS TABLE FOR OBTAINING @, @' AND

Table 18.2

f

ON Ox AND Oy

(Positive Discriminant-Unit Real Half-Period) ZZ@(Z)

1.00

Z=3-\<1

1.4

1.2

1.1

1.05

4.0

2.0

34 41 41 77

1.00000 00 1.00000 32 1.00005 05 1.00025 59 1.00081 12

1.00000 1.00000 1.00004 1,00023 1.00074

00 29 59 31 02

1.00000 1.00000 1.00004 1.00021 1.00068

00 26 22 46 25

1.00000 1,00000 1.00004 1.00020 1.00066

00 25 08 75 02

1.00000 1.00000 1.00004 1.00020 1.00065

00 25 07 73 97

1.00212 1.00441 1.00821 1.01408 1.02269

32 61 33 14 65

1.00198 1.00414 1.00772 1.01326 1.02144

79 21 00 70 00

1.00181 1.00379 1.00709 1.01224 1.01985

79 79 99 31 94

1.00167 1.00351 1.00659 1.01140 1.01857

98 80 56 98 24

1.00162 1.00340 1.00640 1.01108 1,01807

64 97 03 69 36

1. 00162 1;00340 1.00639 1.01107 liO1806

51 71 57 93 19

54 92 23 99 15

1.03486 1.05152 1.07381 1.10307 1.14092

36 21 22 35

08

1.03302 1.04895 1.07036 1.09857 1.13524

47 81 11 95 09

1.03071 1.04572 1.06601 1.09291 1.12807

36 73 29 64 45

1.02883 1.04309 1.06246 1.08829 1.12222

08 40 70 58 46

1.02810 1.04207 1.06109 1.08650 1.11995

10 28 15 29 41

1.02808 1. 04204 1. 06105 1.08646 1,11990

38 87 91 07 05

0.85 0.90 0.95 1.00

1.19894 1.26229 1.34171 1.44091 1.56460 1.71879

38 01 37 81 22 62

1.18933 1.25071 1.32807 1.42515 1.54671 1.69885

40 86 28 17 40 59

1.18232 1.24227 1.31812 1.41364 1.53366 1.68430

81 98 18 80 04 41

1.17348 1.23162 1.30556 1.39912 1.51717 1.66592

94 95 03 31 65 77

1.16627 1.22292 1.29529 1.38725 1.50370 1.65090

18 96 60 23 31 68

1.16346 1.21955 1.29130 1.38264 1.49846 1.64507

98 14 97 14 94 17

1.16340 1.21947 1.29121 1.38253 1.49834 1.64493

37 17 57 27 59 41

0.00

1.00000

00

1.00000 00

37 91 91 57

1.00000 1.00005 1.00027 1.00086

31 03 42 14

1.00000

00

1.00000 1.00005 1.00029 1.00094

34 40 31 20

00

0.05 0.10 0.15 0.20

1.00000 1.00000 1.00005 1.00025 1.00080

1.00000 1.00004 1.00023 1.00072

29 57 54

1.00000 1,00000 1.00004 1.00021 1.00066

00 26 19 13 36

1.00000 1.00000 1.00004 1.00020 1.00063

00 25 05 39 99

1.00000 1.00000 1. 00004 1.00020 1.00063

00 25 04 37 94

0.25 0.30 0.35 0.40 0.45

1.00230 1.00479 1.00889 1.01520 1.02442

98 35 27 23 50

1.00210 1.00435 1.00804 1.01371 1.02194

86 37 93

1.00195 1.00403 1.00743 1. 01263 1.02016

05 04 81 dl 25

1.00176 1.00362 1.00667 1.01129 1.01792

15 91 40 28 92

1.00160 1.00330 1.00605 1. 01020 1.01612

81 38 50 38 33

1.00154 1.00317 1.00581 1,00978 1.01542

88 81 59 33 64

1.00154 1.00317 1.00581 1,00977 1.01540

75 52 03 34 99

0.00 0.05 0.10 0.15 0.20

1.00000 00 1.00000 37 1.00005 9 1 1.00029 91 1.00094 57

1.00000 1.00000 1.00005 1.00027 1.00086

0.25 0.30 0.35 0.40 0.45

1.00230 1.00479 1.00889 1.01520 1.02442

98 35 27 23 50

0.50 0.55 0.60 0.65 0.70

1.03738 1.05504 1.07855 1.10923 1.14872

0.75 0.80

00

4.0

14 08

n._.__ 517 _.__ 1.07738 _ 54 . _1.03345 -.._..04

05

0.55 0.60 0.65

1.05504 92 1.07855 23 1.10923 99

1.04901 44 1.06955 87 1.09614 60

1.03061 iio4466 1.06309 1.08675

34 92 37 16

1.02707 1 8 21 1.05504 64 1.07507 92

1.02421 ii03488 1.04856 1.06569

09 20 45 47

i.02310 ii03319 1.04606 1.06208

77 83 96 70

1.02308 i;om5 1,04601 1.06200 1;08148

17 85 09 18 16

0.75

1.19894 1;26229 1.34171 1.44091 1.56460

1.17264 1;22578 1.29157 1.37264 1.47224

1.15387 1;19980 1.25602 1.32443 1.40736

03 68 53 52 61

1.13065 1.16777 1.21233 1.26544 1.32835

1.11207 1.14221 1.17761 1.21873 1.26610

03 52 18 89 10

1.10494 1.13243 1.16435 1.20095 1.24247

84 76 46 66 14

1,10478 1.13220 1.16404 1.20053 1.24191

09 79 34 95 74

0180

0.85 0.90 0.95

38 01 37 81 22

1.71879 62

1.00 1. 05 1.10

(-314 [ 8 1 t=t/\tr

1.0

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2. 6 2.8 3. 0 3.2 3.4 3.6 3.8 4. 0

63 78 86 39 79

i;osszs

03 18 97 15 02

1.59449 89 1.74462 36

1.50769 66 1.62902 39 1.77589 10

1.40258 06

1.32024 1 7

1.28909 73

( - 3)3 [ 8 1

(-3)3 [ 7 1

(-3)1 [ 7 1

(-4P [ 6 1

(-416 [ 6 1

1.00

1.05

1.71879 62

1.59449 89

1.1 1.50769 66

1.2

1.4

1.40258 06 1.85616 29

1.32024 1 7 1.61789 95 2.09401 44

2.0 1.28909 1.52970 1.86127 2.28676 2,80921

1.28836 81

4.0 73 17 05 23 52

1.288368 1.527649 1.855916 2.273495 2.777516

3.43759 29

3.363868 4.028426 4 767658 5: 578809 6- 459856

If the real half-period + l , see 18.2 Homogeneity Relations. Interpolation with respect to i r will, in general, be difficult because of the non-uniform subintervals involved. Aitken's interpolation may be used in this case. As few as 3s may 7.409386 be obtained. For the computation of Ip, Q' or r at z= I . + ; ! / , an addition formula _ _ .. _. 10.660867 may be used (18.4and Examplec 11-12).

:;::,":::

11.877621 13.160574

675

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS TABLE FOR OBTAINING P, @ ' AND

f

ON Ox ARD Oy

Table 18.2

(Positive Discriminant-Unit Real Half-Period) :,U?'(:) z=r\a

1.05

1.2

1.4

2.0

00 32 17 07 79

-2.00000 -1.99999 -1.99989 -1.99948 -1.99836

00 -2.00000 00 37 -1.99999 43 89 -1,99990 80 63 -1.99953 1 0 70 -1.99850 41

-2.00000 -1.99999 -1.99991 -1.99956 -1.99861

00 -2.00000 47 -1.99999 53 -1.99991 73 -1.99958 55 -1.99865

-1.99572 -1i99107 -1.98332 -1.97121 -1.95319

57 69 00 06 16

-1.99598 -1i99158 -1.98420 -1.97260 -1.95525

17 17 07 99 47

33 67 95 35 77

-1.99656 -i;99273 -1.98621 -1.97581 -1.95998

50 38 31 22 33

-1.92339 -1i88593 -1.83488 -1.76619 -1.67451

01 -1.92730 50 83 -1.89106 43 99 -1.84127 27 53 -1.77376 97 43 -1.68307 45

-1.93016 -i;89480 -1.84594 -1.77931 -1.68934

21 -1.93377 03 97 -lie9954 33 09 -1.85184 82 45 -1.78633 89 72 -1.69729 96

-1.93671 -1i9034i -1.85668 -1.79209 -1i70382

95 -1.93786 73 -i;90492 71 -1.85856 80 -1.79433 60 -1i70636

93 -1.85861 37 95 -1.79439 25 76 -1i70642 75

-1.55271 -1.39118 -1.17683 -0.89169 -0.51095 0.00000

74 65 20 81 87 00

-1.56861 -1.40719 -1.19163 -0.90364 -0.51806

96 15 25 00 28

75 23 88 50 45

93 79 13 70 26

00

0.05 0.10 0.15 0.20

-2.00000 -1.99999 -1.99988 -1.99940 -1.99810

0.25 oi30 0.35 0.40 0.45

-1.99537 -1i99038 -1.98210 -1.96928 -1.95036

33

0.50 oi55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

0.00

1.1

-2.00000 26 -1.99999 1 8 -1.99989 1 6 -1.99945 75 -1.99825

1.00

23 95 90 3 :

(-212

-1.56189 -1.40041 -1.18536 -0.89858 -0.51505

13 70 53 18 33

0.00000

00

(-212

z/i=Yy\u 1.00 0.00 -2.00000 00

0.00000 00

-1.99630 -i;9922i -1.98530 -1.97437 -1.95785

-1.57715 -1.41579 -1.19959 -0.91006 -0.52188

61 -1.58416 29 -1.42286 24 -1.20613 69 -0.91535 70 -0.52503

0.00000 00

0.00000

00

4.0 00 -2.00000 00 49 -1.99999 49 81 -1.99991 82 14 -1.99958 17 86 -1.99865 97

-1.99666 ... 63 ._-1.99666 ... 88 ._ -1i99293 42 -i:99293 89 -1.98656 35 -1.98657 17 -1.97637 02 -1.97638 34 -1.96080 82 -1.96082 78

-1.58689 -1.42561 -1.20869 -0.91741 -0.52626 0.00000

53 -1.93789 23

ji -1;90495

00

Si

-1.58696 -1.42568 -1.20875 -0.91746 -0.52629

39 30 17 57 14

0.00000

00

(-212

1.05

1.1

1.4

1.2

4.0

2.0

0.05 0.10 0.15 0.20

-1.99999 -1.99988 -1.99940 -1.99810

26 18 16 75

-2.00000 -1.99999 -1.99989 -1.99945 -1.99828

00 -2.00000 32 -1.99999 2 1 -1.99989 48 -1.99949 08 -1.99840

00 37 95 33 62

-2.00000 -1.99999 -1.99990 -1.99954 -1.99856

00 43 89 15 33

-2.00000 -1.99999 -1.99991 -1.99959 -1.99873

00 49 94 59 99

-2.00000 -1.99999 -1.99991 -1.99959 -1.99874

49 95 62 11

0.25 0.30 0.35 0.40 0.45

-1.99537 -1.99038 -1.98210 -1.96928 -1.95036

33 23 95 90 13

-1.99581 -1.99133 -1.98398 -1.97268 -1.95619

31 -1.99613 14 82 -1.99202 89 06 -1.98533 03 69 -1.97513 44 80 -1.96039 48

-1.99652 -1.99289 -1.98701 -1.97818 -1.96561

94 -1.99685 19 -1.99697 25 -1.99359 12 -1.99386 63 -1.98837 91 -1.98890 68 -1.98065 01 -1.98159 82 -1.96982 60 -1.97144

66 12 48 94 57

-1.99697 -1.99386 -1.98891 -1.98162 -1.97148

95 76 71 18 38

0.50 0.55 0.60 0.65 0.70

-1.92339 -1.88593 -1.83488 -1.76619 -1.67451

01 83 99 53 43

-1.93299 -1.90123 -1.85861 -1.80221 -1.72827

84 75 50 44 05

-1.93989 -1.91218 -1.87553 -1.82780 -1.76629

10 -1.94845 17 25 -1.92574 23 39 -1.89643 1 6 48 -1.85930 08 64 -1.81290 09

-1.95533 -1.93661 -1.91313 -1.88437 -1.84984

26 23 16 77 78

74 35 74 96 68

-1.95803 -1.94088 -1.91967 -1.89418 -1.86425

95 17 77 46 71

0.75 0.80 0.85 0.90 0.95

-1.55271 -1.39118 -1.17683 -0.89169 -0.51095

74 65 20 81 87

-1.63184 -1.50639 -1.34312 -1.13018 -0.85145

71 22 50 63 23

-1.68753 -1.58698 -1.45865 -1.29452 -1.08387

62 80 26 95 84

-1.80902 -1.76134 -1.70615 -1.64263 -1.56972

61 -1.82937 52 -1.82985 21 96 -1.79034 89 -1.79102 80 96 -1.74698 46 -1.74793 96 75 -1.69950 14 -1.70082 95 20 -1.64818 82 -1.65001 75

1.00 1.05 1.10

0.00000 00

z/i=y\a 1.00 1.0 0.00000 00 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

-0.48485 79 0.00000 00

1.05 -0.48485

-0.81220 52 -0,45984 59 0.00000 00

1.1 79 -0.81220 52

-1.75545 -1.68471 -1.59780 -1.49093 -1.35912

41 79 32 18 08

-1.19575 58

1.2 -1.19575 58 0.00000 00

-2.00000 -1.99999 -1.99991 -1.99958 -1.99869

00 48 65 07 07

-1.95797 -1.94078 -1.91952 -1.89395 -1.86392

-1.48600 58 -1.59338 85

1.4

2.0

00

-1.59588 68

4.0

-1.48600 58 -1.59338 85 -1.59588 -0.99449 51 -1.34717 40 -1.35527 0. 00000 00 -1.07521 03 -1,09935 -0.78786 76 -0.85550 -0.46104 27 -0.64191

68 93 83 88 20

-0.46669 -0.33022 -0.22828 -0.15467 -0.10296

27 92 89 43 79

0.00000 00

-0. 06745 -0iO4346 -0. 02734 -0.01629 -0; 00795

48 22 75 07 66 0.00000 00

676

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS TABLE FOR OBTAINING P, P' AND

Table 18.2

(

ON Ox AND Oy

(Positive Discriminant-Unit Real Half-Period) 1.05

1.2

1.1

525 007 848 364 486 977

0,95059 0.93465 0.91473 0.89005' 0.85966 0.82239

446 128 876 936 076 820

0.95062 0.93468 0.91478 0.89010 0.85971 0.82246

155 503 003 902 964 703

146 973 617 588 222 906

0.94944 0.93322 0.91298 0.88795 0.85716 0.81947

0.99923 0.99840 0.99704 0,99494 0.99189

041 360 076 715 577

0.99929 0.99853 Oi99727 0.99534 0.99251

399 355 741 298 583

0.99934 0.99862 Oi99744 0.99563 0.99296

010 782 912 028 602

0.50 0.55 0.60 0.65 0.70

0.98762 0,98183 0.97419 0.96430 0.95174

541 783 386 782 028

0.98854 Oi98ji5 0,97599 0.96671 0.95486

726 105 894 478 674

0.98921 Oi98410 0.97731 0.96846 Oi95714

0.93598 0.91647 0.89251 0,86334 0,82800 0.78539

819 208 910 108 562 822

0.93995 0.92140 0,89855 0.87059 0.83659 0.79543

720 960 136 177 307 267

0,94284 0.92500 0,90294 0.87587 0,84284 0.80274

000

[ y 9 1

,.

1.05

1.2

1.1

905 476 299 725

0.99999 0.99998 0,99993 0.99978

0.99940 0.99877 0.99774 0.99618 0.99391

928 991 989 100 695

0.99945 0.99888 0.99794 0.99652 0.99448

935 517 811 557 077

725 820 291 464 810

0.99078 0.98659 Oi98113 0.97419 0.96553

600 385 295 175 386

0.95489 0.94200 0.92657 0.90827 0.88676

0.99999 0.99998 0.99992 0.99975

0.99929 0.99854 0.'99731 0.99541 0.99266

836 660 033 639 485

0.99934 0.99865 0,99750 0.99575 0.99322

758 014 544 586 092

541 783 386 782 028

0.98882 0.98364 o;9ike4 0.96808 0.95701

817 988 2% 373 320

0.98969 0,98495 0.97875 0,97080 0,96080

819 208 910 108 562

0.94322 0.92626 0.90559 0.88063 0.85068

518 102 833 688 069

0,94842 0.93328 0,91496 0.89299 0.86683

1.00000 000

0.99999 0.99998 0.99990 0.99971

0.25 0.30 0.35 0.40 0.45

0.99923 0.99840 0.99704 0,99494 0.99189

041 360 076 715 577

0. 50 0.55 0.60 0.65 0.70

0.98762 0.98183 0.97419 0,96430 0.95174

0.75 0.80 Oi85 0.90 0.95

0.93598 0.91647 Oi89251 0.86334 0.82800

1.00 1.05 1.10

0.78539 822

0.81491 420 0.77237 164

[( -;)91 1.05 0.81491 420

4.0

1.00000 000

887 200 891 234

000 876 031 029 483

2.0 000 912 601 935 752

000 895 322 516 226

1.00000 0,99999 0.99998 0,99990 0.99968

1.4 1.00000 0.99999 0.99998 0.99992 0.99977

1.00000 0.99999 0.99998 0,99991 0.99973

0.00 0.05 0.10 0.15 0.20

3.0 3. 2 3.4 3. 6 3.8 4.0

490 904 833 272 132

0.94648 0.92952 0.90847 0.88252 0.85073 0.81195

0.25 0.30 0.35 0.40 0.45

2.0 2. 2 2.4 2.6 2. 8

0,99101 0.98666 0.98083 0,97317 0.96326

503 321 299 177 790 283

0.99999 0.99998 0.99992 0.99975

1.2 1.4 1.6 1.8

867 012 605 633 002

0.99074 Oi98628 0.98030 0.97246 0.96233

895 319 481 030

1.00

0.99100 0,98666 0.98082 0.97315 0.96324

855 511 146 726 343

0.99999 0.99998 0.99991 0.99973

0.78539 822

340 174 531 106 582

0.99005 0;98530 0.97896 0.97066 0.96000

887 198 871 119

:#?=?/\a

364 045 954 099 438

683 521 096 489 079

0.99999 0.99998 0.99990 0.99971

1.0

0.99946 0.99888 0.99790 0.99640 0.99417

501 235 008 469 196

1.00000 000

1.00

321 957 793 831 016

0.99944 0.99884 Oi99784 0.99628 0.99399

000

z/i=Y\lL

0.99946 0.99887 Oi99790 0.99639 0.99417

799 617 478 122 179

1.00000

[(-,y]

000 915 644 115 148

0.99939 0.99874 Oi99766 0.99599 0.99353

1.00000 000

0.75 0.80 0.85 0.90 0.95 1.00

915 643 109 130

905 471 246 429

1.00000 0.99999 0.99998 0.99993 0.99978

000 876 031 029 483

4.0 1.00000 0.99999 0,99998 0.99993 0,99978

000 912 595 868 377

1.00000 0.99999 0.99998 0.99990 0.99968

'

2.0

1.4 1.00000 0.99999 0.99998 0.99992 0.99977

0.00 0.05 0.10 0.15 0.20

0.83587 315 0.79939 419 0.75655 714

.. 1.1

0.83587 315

915 649 181 537

1.00000 0,99999 0.99998 0,99993 0.99978

000 916 650 187 555

0.99947 0.99892 0.99802 0.99665 0.99469

871 586 472 871 855

0.99947 0.99892 0.99802 0.99666 0.99470

917 682 653 184 368

438 0.99166 445 357 0.98791 646 896 Oi98306 740 926 0.97694 003 710 0.96935 061

0.99200 0.98842 Oi98381 0.97799 0.97081

425 700 123 651 949

0.99201 0.98843 Oi98382 0.97802 0.97085

225 902 874 138 406

807 908 574 878 908

0.96211 Oi95172 0.93947 0.92521 Oi90878

557

0.96216 0.95178 0.93955 0.92532 0.90892

276 405 644 176 628

1.00000 000

0.96010 0.94902 0.93589 0.92051 0.90268

986 381 412 815 849

0.86166 128 0.88219 209

1.2

1.4

0,86166 128 0.88219 209 0.71573 454 0.76897 769 0.59293 450

061

230 144 307

0.89003 731

0.89022 154

2.0

4.0

0.89003 0.78909 0.64073 0.43846 tO.17708

731 0.89022 0,78956 505 496 0.64184 0,44095 099 802 t0.18250

15 60 73 77 43

-0.13652 -0.51809 -0.96348 -1.47349 -2.04858

01 61 97 03 16

-2.68905 -3.39508 -4.16677 -5. 00417 -5.90734 -6i87630

52 38 17 86 21 32

-0.14800 012

677

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS TABLE FOR OBTAINING @, @’ AND

(Negative Discriminant-Unit

f

Table 1 8 . 2

ON Ox AND Oy

Real Half-Period)

:?@ ( f )

1.00

z=z\\(,

0.00

1.05 00

1.15

1.3

1.5

2.0

4.0

0.05 0.10 0.15 0.20

1.00000 0.99998 0.99976 0.99880 0.99622

52 37 40 33

1.00000 0.99998 0.99978 0.99893 0.99663

00 68 83 08 32

1.00000 0.99998 0.99983 0.99918 0.99743

98 74 15 55

1.00000 0.99999 0,99990 0.99950 0.99845

38 10 43 77

1.00000 0.99999 0.99996 0.99980 0.99940

75 06 51 30

1.00000 1.00000 1.00002 1,00011 1,00038

00 14 30 83 24

1.00000 1.00000 1.00004 1.00020 1.00065

00 25 07 71 92

0.25 0.30 0.35 0.40 0.45

0.99079 0,98097 0.96495 0.94070 0.90617

63 82 11 57 03

0.99182 0.98317 0.96915 0.94811 0.91839

47 67 65 25 70

0.99381 0.98736 0.97703 0.96174 0.94051

16 11 14 61 05

0.99631 0.99255 0.98664 0.97810 0.96656

17 06 20 01 45

0.99860 0.99725 0.99525 0.99255 0.98928

26 51 02 94 71

1.00096 1.00205 1.00396 1.00705 1.01183

01 83 14 13 11

1.00162 1.00340 1.00639 1.01107 1.01805

38 46 11 17 02

0.50

0.85939 0; 79882 0.72356 0.63382 0.53123

83 11 52 07 69

0.87853 0; 82744 0.76469 0.69080 0.60756

56 45 39 48 14

0.91254 55

0.95189 16

0.83537 63 0.78725 05 0.73495 90

0.91429 23 0.89316 80 0.87276 38

0.98031 24 0.98063 64 Oi98521 20

1.01895 I: 02925 1.04381 1.06395 1;09136

1. 0 7 8 ~ hh

0; 93426 12

0.98573 01 0; 98244 30

42

0; 87744 80

01

32

1.06102 67 1.08641 83 ii11984 70

0.41930 0.30366 0.19233 0.09574 0.02666 0,00000

23

0.51830 0.42820 0.34438 0.27605 0.23446 0.23286

84 16 12 07 42 11

0.68155 0.63143 0.59046 0.56611 0.56753 0.60563

0.85577 0.84585 0.84771 0.86731 0.91197 0.99060

0.99643 1.01739 1.05201 1.10523 1.18314 1.29335

13 07 81 21 77 96

1.12815 1.17693 1.24098 1,32440 1.43234 1.57134

05 44 76 72 85 70

1.16333 1.21939 1.29112 1.38242 1.49822 1.64479

0; 55

0.60 0.65 0.70 0.75 0.80

0.85 0.90 0.95 1.00

33 10 08 27 00

(-315 [ 8 1

(-3)5 [ 8 1

00

50 16 32 51 12 48

(-3)5 [ 8 1 1.15

00

68 35 96 78 25 83

(-314 [ 8 1

00

(-3)4 [ 8 1

SS I: 04202 45

05

(-314 [ 8 1

76 20 16 38 24 64

(-314 [ 8 1

0. 00 0.05 0.10 0.15 0.20

1.00000 0.99998 0; 99976 0.99880 0.99622

00 52 37 40 33

1.05 .. 1.00000 0.99998 0; 99978 0.99892 0.99658

00 67 76 27 78

1.00000 0.99998 Oi99983 0.99916 0.99734

00 98 59 47 10

1.00000 0.99999 Oi99989 0,99948 0.99834

00 37 93 51 96

1.00000 0.99999 0.99995 0.99978 0.99931

00 75 93 96 61

1.00000 1.00000 1.00002 1.00011 1.00034

00 14 24 15 41

1.00000 1.00000 1.00004 1.00020 1.00063

00 32 04

0.25 0.30 0.35 0.40 0.45,

0.99079 0.98097 0.96495 0.94070 0.90617

63 82 11 57 03

0.99165 0.98266 0.96786 0.94525 0.91264

20 22 42 04 56

0.99345 16 0.98628 83 0.97433 43 0.95576.47 0.92846 67

0.99589 0.99132 0.98354 0.97122 0.95268

95 10 71 41 27

0.99827 0.99626 0.99275 0.98701 0.97806

12 60 81 30 19

1.00081 1.00162 1,00285 1.00459 1.00684

39 14 94 41 49

1.00154 1.00317 1.00580 1.00976 1.01539

61 22 47 35 36

0.50 0.55 0.60 0. 65 0. 70

0,85939 83

0,86784 46 0,80881 13

0,89009 57 0.83817 66 0.77024 24

0.92592 0.88861 0.83812 0.77163

17 10 71 28

0.96465 0.94522 0.91784 0.88019 0.82955

71 83 50 00 45

1.00955 1.01258 1.01563 1.01827 1.01983

92 51 95 41 61

1.02305 1.03311 1.04595 1.06191 1.08136

58 90 22 71 14

0.76286 31

1.01942 1.01585 1.00758 0.99269 0.96882 0.93312

61 25 28 39 29 29

1.10461 1.13197 1.16373 1,20012 1.24136 1.28763

36 83 23 24 39 91

,

~

1.3

0. 75

[(-;12]

[(-s3)2]

z/i=y\. 1.1 1.2 1.3 1.4 If the real half-period +l,see 18.2 Homogeneity Relations. 1.6

i:1.9 2.0

2.0

1.5

“-;)‘I

4.0

Interpolation with respect to ‘6 will, in general, be difficult because of the non-uniform subintervals involved. Aitken’s interpolation may be used in this case. As few as 3 s may be obtained. For the computation of @, @’ or f at z= r+;!/, an addition formula may be used (18.4 and Examples 11-12).

35 88

4.0 1.39585 1.52559 1.67719 1. 85056 2. 04521

80 80 97 87 26

2.26025 62

:3.01245 :;:z;iz:16 3.29069 52

678

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS TABLE FOR OBI'AININ(; 9,IP' AND t ON Ox AND Oy

Table 18.2

(Negative Discriminant-Unit ::@'(:)

c = ,T\

1.00

If

1.05

1.15

Real Half-Period)

2.0

1.5

1.3

4.0

0. 00 0. 05 0.10 0.15 0.20

-2.00000 -2.00002 -2.00047 -2.00239 -2. 00753

00 95 25 01 43

-2.00000 -2,00002 -2.00042 -2;00212 -2.00667

00 65 27 69 30

-2.00000 -2.00002 -2.00032 -2;0016l -2.00502

00 04 37 92 56

-2.00000 -2.00001 -2.00019 -2.00097 -2.00297

00 24 63 17 32

-2.00000 -2.00000 -2.00007 -2.00037 -2.00110

00 50 74 44 66

-2.00000 -1.99999 -1.99995 -1.99975 -1.99919

00 71 34 65 66

-2.00000 -1.99999 -1.99991 -1.99958 -1.99866

49 83 21 07

0.25 0.30 0.35 0.40 0.45

-2.01829 -2.03755 -2.06843 -2.11379 -2.17550

41 78 88 74 18

-2.01608 -2.03274 -2.05907 -2.09713 -2.14789

73 55 94 03 87

-2.01196 -2,02397 -2.04247 -2.06835 -2.10148

38 99 95 37 48

-2.00694 -2.01358 -2. 02334 -2.03614 -2.05106

49 73 71 78 10

-2.00246 -2.00448 -2.00696 -2.00922 -2.00992

05 84 68 15 37

-1.99793 -1.99544 -1.99095 -1.98338 -1.97120

23 16 74 63 64

-1.99667 -1.99294 -1.98657 -1.97639 -1.96084

11 36 99 65 72

0.50 0.55 0.60 0.65 0.70

-2.25339 -2.34395 -2.43881 -2.52318 -2.57463

16 53 27 49 40

-2.21047 -2.28098 -2.35140 -2.40840 -2.43241

72 85 73 49 27

-2.14013 -2.18023 -2.21466 -2.23248 -2.21839

46 97 43 50 89

-2.06592 -2.07692 -2.07815 -2.06116 -2.01460

49 41 03 83 73

-2.00685 -1.99665 -1.97452 -1.93392 -1.86620

64 49 31 01 81

-1.95234 -1.92399 -1.88246 -1.82286 -1.73878

05 70 83 83 53

-1.93791 -1.90499 -1.85865 -1.79444 -1,70648

93 42 81 54 76

0.75 0.80 0.85 0.90 0.95 1.00

-2.56240 -2.44770 -2.18496 -1.72414 -1.01321 0.00000

86 16 84 78

-2.39712 -2.26959 -2.01105 -1.57813 01 -0.92423 00 0.00000

18 69 50 99 16

-2.15233 -2.00933 -1.75959 -1.36864 -0.79716

79 39 77 82 03

-1.92378 -1.77031 -1.53168 -1.18057 -0.68374

08 11 32 88 39

-1.76023 -1.60178 -1.37288 -1.05066 -0,60580

25 75 13 42 78

-1.62181 -1.46089 -1.24141 -0.94387 -0.54202

13 21 08 76 52

84 81 20 44 04

00

0.00000

00

0.00000

00

0.00000

00

-1.58702 -1.42574 -1.20881 -0.91751 -0.52632 0.00000

(-213

0.00000 00

00

00

(-212

(-213

1.00

1.05

0.05 0.10 0.15 0.20

-2.00000 -2,00002 -2.00047 -2.00239 -2.00753

-2.00000 95 -2.00002 25 -2.00042 01 -2,00216 43 -2.00685

00 65 55 12 42

-2.00000 -2.00002 -2.00032 -2.00168 -2.00540

00 05 97 65 32

-2.00000 -2.00001 -2.00020 -2.00104 -2.00340

00 25 30 87 55

-2.00000 -2.00000 -2.00008 -2.00043 -2.00145

00 50 28 62 41

-2.00000 -1.99999 -1.99995 -1.99978 -1.99935

00 72 58 38 00

-2.00000 -1.99999 -1.99991 -1.99959 -1.99874

00 49 91 6t 22

0.25 0.30 0.35 0.40 0.45

-2.01829 -2.03755 -2.06843 -2.11379 -2.17550

41 78 88 74 18

-2.01677 -2.03479 -2.06420 -2.10841 -2.17036

67 40 40 06 66

-2.01340 -2.02825 -2.05319 -2. 09200 -2.14879

12 59 59 85 02

-2.00859 -2.01849 -2.03567 -2. 06346 -2.10597

22 50 60 12 25

-2.00378 -2.00844 -2.01691 -2.03134 -2,05462

54 -1.99851 -1.99718 87 -1.99536 51 -1.99323 43 -1.99120

75 99 97 08 21

-1.99698 -1.99387 -1.98892 -1.98164 -1.97152

24 40 95 41 19

0.50 -2.25339 16 -2.25173 01 -2.22747 67 -2.16805 0.55 -2.35170 68 -2.33108 42 -2.25504 0. 60 -2.46061 76 -2.37230 0.65 -2.52442 0. 70

61 79 39 19

-2.09057 -2.14403 -2.22089 -2.32798 -2.47283

56 61 13 29 02

-1.99006 -1.99107 -1.99605 -2.00760 -2.02919

63 16 96 83 12

-1.95810 -1.94097 -1.91982 -1.89440 -1.86458

18 97 80 95 73

-2.66308 69 -2.06534 -2.12187 -2.20596 -2.32643 -2.49375 -2.72008

90 04 83 60 12 43

-1.83032 -1.79170 -1.74889 -1.70215 -1.65184 -1.59838

90 68 39 68 57 35

5/1=?/\0

0.00

00

1.15

1.3

0.75 0.80 0. 85 0. 90 0.95 1.00

(-312 [

S

(-312 ]

[ S ]

(-3)3 [

S

]

(-314 [ 8 1

1.5

2.0

10

(-316 [ 8 1

4.0

(-3)7 [

S

]

[

(-416 6 1

qi=?/\n 1.1 1.2 1.3 1.4 1.5

-1.48398 -1.36337 -1.24144 -1.12345 -1.01509

95 47 17 13 75

1.6 1.7 1.8 1.9 2.0

-0.92286 -0.85472 -0i82134 -0.83783 -0.92645

21 55 27 54 86

4.0

679

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

TABLE FOR OBTAINING p, P AND

r

'rabic 18.2

ON O*.AND 0,.

(Negative Discriminant-Unit Real Half-Period) :I'(:)

1.5

2.0

4.0

00 21 31 65 15

1.00000 00 1,00000 08 1.00001 32 1;00006 60 1.00020 48

1.00000 00

1.00000 00

1;00027 40 1.00086 16

1.00000 1.00000 1.00003 1;00016 1.00052

0.99999 0.99999 0;99996 0.99987

95 24 10 51

0.99999 0.99998 Oi99993 0.99978

92 65 12 17

09 06 07 00 07

1.00208 1.00429 1.00787 1.01325 1.02090

94 54 32 74 50

1.00125 1.00256 1.00467 1.00779 1.01217

79 91 27 77 02

1.00048 1.00098 1.00175 1.00285 1.00433

81 15 16 61 47

0.99968 0.99934 0.99875 0.99781 0.99639

98 32 38 57 49

0.99946 0.99888 0.99791 0.99640 0,99417

41 13 11 37 86

1.04227 1.06102 1.08493 1.11454 1.15021

15 21 81 88 58

1.03127 1,04478 1.06180 1.08258 1.10724

19 39 26 64 76

1.01799 1.02543 1.03459 1,04547 1.05796

52 63 22 45

1.00619 1.00840 1.01087 1.01343 1.01581

68 79 54 17 69

0,99432 0.99139 0.98734 0.98186 0.97457

31 16 37 55 57

0.99102 0.98667 0.98085 0.97318 0.96328

12 79 06 91 27

1.19206 1.23993 1.29329 1.35118 1.41220 1.47443

86 78 24 37 03 48

1.13570 1.16765 1.20248 1.23929 1.27679 1.31332

79 25 62 22 52 66

1.07181 1.08659 1.10165 1.11613 1.12887 1.13842

59 33 80 35 36 65

1.01765 1.01845 1.01754 1.01408 1.00702 0.99506

94 41 58 73 76

0.96501 0.95262 0.93672 0.91653 0.89105 0.85912

30 09 94

0.95064 0,93471 0,91482 0.89015 0,85977 0.82253

87 88 13 86 85 59

92 65 19 57

1.05 .. ~

0.00

49 88 88 98

1.00000 1.00000 1.00007 1;00035 1.00112

00 44 06 70 60

1.00307 1.00636 1.01176 1.01999 1.03186

33 38 23 45 18

1.00274 1.00566 1.01043 1.01767 1.02805

1.04821 1.06990 1.09776 1.13248 1.17462

35 78 14 70 06

0.05 0.10 Oil5 0.20

1.00000 1.00000 1.00007 1;00039 1.00125

0.25 0.30 0.35 0.40 0.45 0.50 0.55

0.60 0.65 0.70

./;=?/\.

00

1.05

1.00 1.00000 1.00000 1.00007 1.00039 1,00125

00 49 88 88 98

1.00000 1.00000 1.00007 1.00035 1.00113

44 08 86

0.25 0.30 0.35

1.00307 1,00636 1.01176 1;01999 1.03186

33 38 23 45 18

1.00277 1.00576 1. 01069 1.01824 1.02921

0.45

[(-;I 51 1.15

00

0.00 0.05 0.10 0.15 0.20

0;40

1-15 1.00000 00 1.00000 34 1.00005 43

13

50

15

46 29

[( -;141

1.3

1.5

2.0

00

1.00000 00

1.00000

00

05

1.00000 1,00000 1.00003 1.00017 1.00054

21 34 04 31

1.00000 1.00001 1.00006 1.00022

08 35 91 22

0.99999 0.99999 0.99996 0.99988

95 25 24 28

4.n 1.00000 0.99999 0.99998 0.99993 0.99978

14 03 42 97 32

1.00134 1.00281 1.00529 1.00917 1.01496

04 53 28 72 03

1.00055 1.00117 1.00225 1.00396 1.00658

43 94 03 67 42

0.99971 0.99943 0.99897 0.99830 0.99739

90 06 41 68 10

0.99947 0.99892 0.99802 0.99666 0.99470

88

1.03581 72 1.05277 97 1.07515 67

1.02322 1.03466 1.05006 1.07029

84 71 29 97

1.01042 1.01588 1.02344 1.03369 1.04730

41 39 73 45 93

0.99619 0.99471 0.99295 0.99095 0.98878

89 80 77 58 64

0.99202 0.98845 0.98384 0.97804 0.97088

03 10 63 63 86

00

34 46 73

51

1.00000 1.00000 1.00005 1.00027 1,00088

55 38 02 62 31

1.00216 1.00451 1. 00841 1.01445 1.02333

1.04444 39 1.06483 58

1.3 ~~

00

96 78 83 50

0. 75 0. 80

0.85 0. 90 0. 95 1.00

z/i=y\a

.1.1.

4.0

1.2 1.3 1.4 1.5

0.84561 0,79003 0.72274 0.64295 0.55003

98 67 36 89 38

1.6 1.7 1. 8 1.9 2.0

0.44345 0,32282 0.18790 +0.03858 -0.12508

14 70 92 90 40

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS

680

INVARIANTS AND VALITES A T HALF-PERIODS

’lu111~* 18.3

(Non-Negative Discriminant-Unit Real Half-Period) ni q=.P(l) r<=P(w’) ?={(I)

(12

rr=w’li

1.00 1. 02 1. 04 1. 06 1.08

11.81704 11.37372 10.98419 10.64177 10,34065

384 107 347 794

0.00000 0.55318 1.03485 1.45484 1.82151

1.10 1.12 1.14 1.16 1.18

10.07577 9.84269 9.63754 9.45693 9.29789

364 185 049 072 413

2.14201 2.42241 2.66798 2.88320 3.07195

000 937 153

1.20 1.22 1.24 1. 26 1.28

9.15782 9.03445 8.92575 8.82999 8.74560

851 117 843 055 138

1.30 1. 32 1.34 1.36 1.38

8.67123 8.60568 8.54791 8.49698 8.45209

1.40 1.42 1.44 1.46 1.48

500

000

q ‘ i i = {(U’)/ i

1.71879 1.71005 1.70235 1.69556 1.68958

64 96 77 79 18

-1.71879 -1.66138 -1.60783 -1.55787 -1.51123

64 15 69 59 63

0.78539 0; 78979 0.79367 0.79708 0.80009

816 7I8 192 535 279

-0.78539 -0.76520 -0.74537 -0.72588 -0.70669

82 32 75 58 61

918

1.68430 1.67965 1.67554 1.67193 1.66874

41 08 80 04 05

-1.46767 -1.42698 -1.38894 -1.35338 -1.32011

83 19 48 12 96

0.80274 0.80507 0.80713 0.80895 0.81054

283 817 637 045 949

-0.68777 -0.66910 -0,65066 -0.63241 -0,61434

92 88 09 38 79

3.23761 3.38308 3.51088 3.62320 3.72197

717 317 223 977 756

1.66592 1.66344 1.66126 1.65933 1.65763

77 74 03 17 09

-1.28900 -1.25988 -1.23262 -1.20710 -1.18320

20 23 55 65 95

0.81195 0.81320 0.81429 0.81526 0.81611

906 168 717 299 453

-0.59644 -0.57869 -0.56106 -0.54356 -0.52616

54 03 78 50 97

169 628 374 890 746

3.80885 3.88529 3.95256 4.01178 4.06392

265 056 351 462 870

1.65613 1,65480 1.65364 1.65261 1,65170

11 86 22 37 67

-1.16082 -i;i3985 -1.12021 -1.10180 -1.08454

70 91 33 85

0.81686 0;8I752 0.81811 0.81862 0.81907

533 732 103 572 958

-0.50887 -0.49166 -0.47452 -0.45746 -0.44046

14 03 75 53 65

8.41252 8.37763 8.34687 8.31975 8.29583

263 305 283 228 997

4.10985 4.15029 4.18593 4.21732 4.24498

014 819 045 438 728

1.65090 1.65020 ii64957 1.64903 1.64854

68

92 06 68

-1.06837 -1.05321 -1.03899 -1.02566 -1.01316

47 20 58 55 45

0,81947 0,81983 0.82014 0.82041 0.82066

977 269 389 831 031

-0.42352 -0.40663 -0.38978 -0.37298 -0.35621

46 39 91 56 91

1.50 1.52 1.54 1. 56 1.58

8.27475 8.25616 8.23977 8.22531 8.21257

580 484 191 684 036

4.26936 4.29084 4.30978 4.32647 4.34119

502 965 602 752 120

1.64812 ii64774 1.64741 1.64711 1.64686

02 39 20 94 13

-1.00144 -0iwo44 -0.98012 -0.97045 -0.96137

04 37 84 19 37

0.82087 0;82106 0.82122 0.82137 0; 82150

370 i9i 787 423 329

-0.33948 58 -0.32278 22 -0.30610 4 -0.28945 55 -0.27282 11

1. 60 1.65 1. 70 1.75 1.80

8.20133 8.17870 8.16217 8.15011 8.14129

033 308 907 147 812

4.35416 4.38026 4.39931 4.41322 4.42337

210 291 441 294 818

1.64663 1.64617 1.64584 1.64559 1.64541

38 54 08 63 78

-0.95285 64 -0.93379 17 -0.91752 88 -0i90365 1 8 -0. 89180’8?

0.82161 0.82184 0.82201 0; 82213 0.82222

711 628 364 569 516

-0.25620 -0.21475 -0.17337 -0.13205 -0,09079

90 00 32 85 10

1. 85 1. 90 1.95 2.00 2. 05

8.13486 8.13016 8.12672 8.12421 8.12238

127 001 634 844 671

4.43079 4.43620 4.44016 4.44305 4.44516

368 896 375 205 152

1.64528 1.64519 1.64512 1.64507 1.64503

73 21 25 17 45

-0.88169 -0.87306 -0.86569 -0.85939 -0.85402

10

0.82229 0.82233 0.82237 0.82239 0.82241

038 800 281 820 676

-0.04955 -0.00835 tO.03283 0.07400 0.11515

91 41 07 01 80

2.10 2.15 2.20 2.25 2.30

8.12104 8.12007 8.11935 8.11883 8.11845

883 164 791 660 583

4.44670 4.44782 4.44864 4i44924 4.44968

219 746 934 963 808

1.64500 1.64498 1.64497 1.64496 1.64495

74 76 32 26 49

-0.84942 -0;84550 -0.84215 -0.83928 -0.83684

78 41 20 80 11

0.82243 0:8ZZ44 0.82244 0.82245 0.82245

032 022 745 274 659

0.15630 0.19745 0.23858 0.27972 0.32085

73 01 81 23 38

2.4 2. 5 2.6 2.7 2.8

8.11797 8.11771 8.11758 8.11750 8.11746

459 785 087 782 884

4.45024 4.45053 4.45069 4.45077 4.45082

222 785 555 969 457

1.64494 1.64494 ii64493 1.64493 1.64493

51 71 57 49

-0.83296 -0.83013 -0.82806 -0.82655 -0.82545

37 28 54 58 33

0.82246 0.82246 0.82246 0.82246 0.82246

146 406 546 619 659

0.40311 0.48536 0,56761 0.64986 0.73211

12 38 39 24 01

2.9 3. 0 3.1 3.2 3.3

8.11744 8.11743 8.11743 8.11742 8.11742

804 694 103 787 619

4.45084 4.45086 4.45086 4.45087 4.45087

852 130 811 174 368

1.64493 1;64493 1.64493 1.64493 1.64493

45 43 42 41 41

-0.82464 -0.82406 -0.82363 -0.82331 -0.82308

81 01 06 68 78

0.82246 0.82246 0.82246 0.82246 0.82246

680 691 698 701 702

0.81435 0,89660 0.97885 1.06109 1.14334

74 44 13 81 48

3.4 3.5 3.6 3. 7 3. 8

8.11742 8.11742 8.1’1742 8.11742 8.11742

529 481 455 441 434

4.45087 4.45087 4.45087 4.45087 4.45087

472 528 556 572 581

1.64493 1.64493 1.64493 1.64493 1.64493

41 41 41 41 41

-0.82292 04 -0.82279 82 -0.82270 89 -0.82264 37 -0.82259 61

0.82246 0.82246 0.82246 0.82246 0.82246

703 703 703 704 704

1.22559 1.30783 1.99008 1.47233 1.55457

16 83 50 17 84

3. 9 4. 0

8.11742 430 8.11742 426

4.45087 585 4.45087 587

1.64493 4 1 1.64493 41

-0.82256 13 -0.82253 59

0.82246 704 0.82246 704

m

8.11742 426

4.45087 590

1.64493 4 1

-0.82246 70

0.82246 704

A=O

992 699 521 890

000

13

00

31

76 52 37 82

1.63682 51 1,71907 18

[(‘p]

(-319 (-4)1 (-4)5 (-517 [ 8 1 [ 5 1 [ 6 1 [ 6 1Forn-1: g2=w4, q.,=O, rI=w2/2, ~ ~ = - w 2 1 2 , ~ rl’= ~I =4- ,T ’ ~ . For n = : 92-*4/12, q,=+1/216, c I = ~ ? / 6o,I = -TL 12, V = T ~ 12, ? ’ i t = ( w = 1.85407 4677 is the real half-period in the Lemniscatlc case 18.14.) For 4
-

(-317 [ 8 1

m.

681

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS INVARIANTS AND VALUES AT HALF-PERIODS

(Non-Negative Discriminant-Unit Real Half-Period) U ( . ’ ) i d1) W W 2 )

u=w‘ll

1. 00

1.02 1. 04 1. 06 1. 08

0.94989 oi95iI4 0.95224 0.95321 0. 95407

1.10 1.12 1.14 1.16 1.18

88

TnLlc 18.3

LJ c(w2)

92 98 54

0.949899 0.967481 0.984884 1.002097 1.019107

1.182951 1.170397 1.157316 1.143695 1.129522

1.182951 1.218650 1.253864 1.288619 1.322935

0.95482 0.95549 0.95608 0.95659 0.95705

97 47 10 79 36

1.035904 1.052476 1.068811 1.084899 1.100727

1.114782 1. 099457 1.083531 1.066989 1.049814

1.356827 1.390301 1.423362 1.456007 1.488231

1.20 1.22 1.24 1.26 1.28

0.95745 Oi95780 0.95812 0.95839 0.95864

55 98

22 77 07

1.116285 1.131562 1.146546 1.161227 1.175594

1. 031991 1.013507 0.994349 0.974506 0.953970

1.520022 1.551369 1.582254 1.612657 1.642557

1.30 1.32 1.34 1.36 1.38

0.95885 0.95904 0.95921 0.95935 0.95948

49 38 04 73 68

1.189636 1.203344 1.216707 1.229716 1.242361

0.932733 0.910790 0.888138 0.864776 0.840704

1.671930 1.700750 1.728989 1.756618 1.783607

1.40 1.42 1. 44 1. 46 1.48

0.95960 0,95970 0.95979 0.95986 oi95993

10 18 06 89 80

1.254633 1.266522 1.278021 1.289120 1.299811

0.815927 0.790449 0.764278 0.737425 0.709900

1.809925 1.835542 1.860425 1.884541 1.907860

1.50

1. 54 1. 56 1.58

0.95999 0.96005 0.96010 Oi96014 0.96017

90 27 01 19 87

1.310087 1.319941 1.329364 1.338351 1.346895

0.681719 0.652896 0.623452 0.593404 0.562777

1.930348 1.951974 1.972707 1.992515 2.011370

1.60 1.65 1.70 1.75 1.80

0. 96021 0,96027 0,96032 0.96035 Oi96038

13 67 45 94 49

1.354990 1.373224 1.388539 1.400869 1.410170

0.531593 0.451372 0.368286 0.282840 0.195588

1.85 1.90 1.95 2. 00 2. 05

0.96040 0.96041 0.96042 0.96043 0.96043

35 71 70 43 96

1.416408 1.419573 1.419665 1.416707 1.410733

2.10 2.15 2.20 2. 25 2.30

0.96044 0.96044 0.96044 0,96044 0.96045

35 63 84 99 10

2.4 2.5 2.6 2.7 2. 8

0.96045 0,96045 0.96045 0.96045 0.96045

2.9 3. 0 3.1 3.2 3. 3

1. 52

io

2. 029242 2; 069439

2.102914 2.129313 2.148344

0.107125 018074 -0.070918 -0.159199

2.159783 2.163478 2.159353 2.147412 2.127732

1.401800 1.389977 1.375349 1.358018 1.338098

-0.331019 -0.413290 -0,492330 -0.567579 -0.638522

2.100473 2; 065864 2. 024211 1.975882 1. 921308

24 31 35 37 38

1.291016 1.235264 1.172151 1.103091 1.029557

-0.765682 -0.870782 -0.951807 -1.007808 -1.038896

1.795415 1.650936 1.492779 1.326086 1.155967

0,96045 0.96045 0.96045 0.96045 0.96045

39 40 40 40 40

0.953025 0.874937 0.796655 0.719428 0.644360

-1. 046157 -1.031530 -0.997636 -0.947586 -0.884775

0.987255 0; 824296 0.670787 0.529666 0.403050

3.4 3.5 3.6 3.7 3. a

0.96045 0.96045 0.96045 0.96045 0.96045

40 40 40 40 40

0.572395 0.504299 0.440663 0.381903 0.328268

-0.812687 -0.734720 -0.654024 -0.573398 -0.495196

0.292246 0.197780 0.119493 0.056643 to. 008033

3.9 4. 0

0.96045 40 0.96045 40

0.279851 0.236623

-0.421291 -0.353075

-0.027857 -0. 052740

m

0.96045 40

0.000000

0.000000

0.000000

[(-;)91

[(-;)31

[‘-l’2]

A=O

[(3 -

to.

1 + w’, e 2 = @ (1+ U ’ ) = - (el +ej), v2 = ((1t w ’ ) = ? +?’. For n = l : ~(1) = f l / 8 2 1 ’ 4 / ~ , o(.‘) =iu(l), =,,&“lqe~d4/,. for^=-: ~(1)=2$~’?4 w, u ( w ’ ) = O , u ( u 2 ) = 0 . (~=1.854074677 is the real half-period in the Lemniscatic case 18.14.) To obtain the corresponding values of tabulated quantities when the real half-period w # l , multiply by w. w2=

Q

682

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS INVARIANTS AND VALUES AT HALF-PERIOD3

Table 18.3

.=.a,;

(Non-Positive Discriminant- -Unit Real Half-Period) 9PI = !Re,= ll#=<(W;)/i

92

1. 00 1.02 1. 04 1. 06 1.08

-47.26818 -45.35272 -43.40071 -41.42954 -39.45420

1.10 1.12 1.14 1.16 1.18

00 19 30 84 53

0.

noooo on

4:4i9ob 00

0.

ooobo ,000 3.43759 29

1.57079 1.53091 1.49282 1.45647 1.42184

63 63 30 87 01

-1.57079 -1.58005 -1.58905 -1.59772 -1.60600

63 81 67 52 53

99 74 96 18 81

-1.61384 -1.62120 -1.62804 -1.63434 -1.64006

68 68 93 46 85

8.23156 58 11.49257 28 14;25448 26

-0i04867 -0. 09452 -0.13769 -0.17834

810 083 202 547

3.36827 3.29802 3.22711 3.15578

69 68 39 40

.-37.48749 12 -35.54027 17 -33.62168 02 -31.73930 91 -29.89938 64

16.56680 18.47603 20:02550 21.25543 22.20294

99 08 17 82 45

-0.21662 -0.25266 -0i28660 -0,31854 -0.34862

576 894 315 915 086

3.08425 3.01273 2.94140 2.87040 2i79990

89 84 90 2"

1.38885 1.35748 1.32766 1.29935 1;27247

1.20 1.22 1.24 1.26 1.28

-28.10693 45 -26i36591 62 -24.67936 58 -23.04950 83 -21.47786 60

22.90208 23.38397 23.67693 23.80660 23.79610

34 82 85 45 09

-0.37692 -0.40356 -0.42863 -0.45222 -0.47442

571 512 481 513 139

2.73000 2.66084 2.59249 2.52505 2.45859

96 07 39 44 58

1.24699 1.22283 1.19995 1.17830 1.15780

24 82 95 09 77

-1.64520 -1.64973 -1.65364 -1.65693 -1.65959

18 00 28 36 88

1. 30 1.32 1.34 1.36 1.38

-19.96535 -18.51237 -17.11886 -15.78441 -14.50828

52 16 71 82 67

23.66620 -.. ...-.08 23.43548 95 23.12052 98 22.73602 29 22.29496 60

-0.49530 -0; 51494 -0.53342 -0.55081 -0.56715

414 941 897 058 817

2.39318 2.32886 2.26569 2.20369 2.14291

14 49 11 72 32

1.13842 1.12010 1.10279 1.08644 1.07100

65 52 31 09 10

-1.66163 -1.66305 -1.66384 -1.66403 -1.66361

82 38 99 31 13

1.40 1.42 1. 44 1.46 1.48

-13.28947 -12.12676 -11,01876 - 9.96396 - 8196012

27 19 70 40 32

21.80880 21.28756 20.74000 20.17372 19.59530

22 31 36 81 70

-0.58253 -0.59698 -0.61058 -0.62336 -0.63538

209 926 339 513 226

2.08336 2.02506 1.96802 1.91226 1. 85777

24 27 64 13 09

1.05642 1.04267 1.02970 1.01747 1.00593

75 61 43 14 83

-1.66259 -1,66099 -1.65881 -1.65608 -1.65280

42 26 85 44 40

1. 50 1. 52 1.54 1. 56 1.58

-

8.00739 7.10204 6.24304 5.42853 4.65668

71 36 63 20 53

19.01038 18.42378 17.83959 17.26123 16.69159

59 52 12 98 27

-0,64667 980 -0.65730 023 -0.66728 357 -0.67666 751 -0.68548 761

1.80455 1.75261 1.70192 1.65250 1.60432

50 00 94 41 26

0.99506 0.98482 0.97517 0.96608 0.95751

76 36 21 09 90

-1.64899 -1.64466 -1.63982 -1.63450 -1.62871

13 08 76 65 26

3.92570 2.26537 0.82241 + 0.42844 1.51045

12 64 58 48 44

16.13300 14.79653 13.56033 12.43388 11.41927

57 23 77 94 28

-0.69377 -0.71238 -0.72831 -0.74194 -0.75360

734 375 198 441 961

1.55737 1.44527 1.34049 1.24271 1.15159

16 36 21 21 40

0.94945 0.93130 0.91571 0.90232 0.89084

69 88 53 74 07

-1.62246 -1,60493 -1.58487 -1.56251 -1.53807

17 31 67 97 94

1.85 1.90 1. 95 2. 00 2. 05

2.44471 3.25015 3.94365 4.54009 5; 05259

18 81 25 85 79

10.51370 9.71138 9.00473 8.38537 7.84470

92 21 54 94 38

-0.76358 -0.77212 -0,77942 -0.78567 -0.79101

973 691 883 351 353

1.06678 0.98792 0.91466 0.84665 0.78355

48 73 65 46 46

0.88099 0.87254 0.86531 0.85912 0.85382

10 91 67 29 00

-1.51175 -1.48374 -1.45422 -1.42334 -1.39126

93 94 51 69 17

2.10 2.15 2.20 2.25 2.30

5.49261 5.87014 6.19388 6.47134 6.70905

57 76 05 49 42

7.37428 6.96611 6.61278 6.30752 6.04422

09 56 90 86 78

-0.79557 -0.79948 -0.80282 -0.80567 -0.80811

957 352 119 458 383

0.72504 67080 0.62056 0.57401 0.53092

25

06 95 40

0.84928 0; 84539 0.84207 0.83923 0.83679

11 69 37 09 93

-1.35810 -1.32398 -1.28903 -1.25332 -1.21695

23 93 05 31 43

2.4 2. 5 2.6 2.7 2. 8

7.08692 7.36377 7.56643 7.71470 7.82312

59 30 61 39 83

5.62231 5.31058 5.08099 4.91228 4.78851

14 54 59 49 39

-0.81198 -0.81480 -0.81687 -0.81837 -0.81948

137 718 167 985 158

0.45410 0.38831 0.33200 0.28383 0.24262

32 56 75 23 75

0.83294 0.83012 0.82805 0.82655 0.82545

16 09 92 25 16

-1.14253 28 -1.06629 03 -0.98863 87 -0.90990 09 -0,83032 82

2. 9 3. 0 3.1 3.2 3.3

7.90239 07 7.96032 11 8.00265 32 8.03358 32 8;05618 01

4.69782 4.63142 4.58284 4.54731 4.52134

05 26 25 53 25

-0.82028 -0.82087 -0.82130 -0.82161 -0,82184

636 422 361 725 634

0,20739 0.17726 0.15151 0.12949 0.11067

21 58 09 50 62

0.82464 0.82405 0,82363 0.82331 0.82308

72 96 03 67 77

-0.75011 -0.66941 -0.58833 -0.50697 -0.42540

58 39 87 92 32

3.4 3.5 3.6 3.7 3. 8

8.07268 8. 08474 8. 09355 8.09999 8,10469

4.50235 4.48848 4.47835 4.47094 4.46553

93 72 14 62 65

-0.82201 -0.82213 -0.82222 -0.82229 -0.82233

368 590 517 038 800

0.09459 0.08084 0.06909 0.05904 0.05046

10 29 25 97 65

0.82292 0.82279 0.82270 0.82264 0.82259

04 82 89 37 61

-0.34366 -0.26179 -0.17984 -0.09781 -0.01572

33 91 06 10 75

3.9 4. 0

8.10812 30 8.11063 05

4.46158 47 4.45869 80

-0.82237 279 -0.82239 820

0. 04313 08 0.03686 13

0.82256 13 0.82253 59

m

8.11742 43

4.45087 59

-0.82246 703

0.00000 00

0.82246 70

1. 60 1. 65 1. 70 1.75 1. 80

80 69 57 01 00

0:

17

Si

t0.06639 64 +0.14855 08

(-2)3 (-2P (-414 (-3)l (-4)3 F-$3] [ 9 1 [ 8 1 [ 7 1 [ 6 1 [ 6 1 For a = 1: 92 = -4~4,q? =0, & l =O, g e l = w 2 , q2 = ~ ! 2q,; / z = - 7 2. For a = - : g2=n* 12, gj=+/216, !Rel=-rrz/12,9e,=0, q2=+/12, s ; / l = - . ( w = 1.85407 4677 is the real half-period in the Lemniscatic case 18-14.) For 4
A=o

683

WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS INVARIANTS AND VALUES A T HALF-PERIODS

Table 18.3

(Non-Positive Discriminant-Unit Real Half-Period)

44) '

%(U')

SU(W')

1. 00 1.02 1. 04 1. 06 1. 08

1.18295 1.17091 1.15940 1.14841 1.13793

13 79 62 45 68

1.182951 1.219157 1.255842 1.292964 1.330480

0.474949 0.475654 0.476433 0.477275 0.478169

0.474949 0.483826 0.492792 0.501851 0.511006

1.10 1.12 1.14 1.16 1.18

1.12796 1.11848 1.10948 1.10094 1.09285

39 38 26 49 44

1.368342 1.406502 1.444910 1.483513 1.522257

0.479107 0.480078 0.481074 0.482085 0.483104

0.520259 0.529611 0.539064 0.548616 0.558268

1.20 1. 22 1. 24 1. 26 1.28

1.08519 1. 07794 1.07109 1. 06461 1. 05850

40 61 31 72 11

1.561089 1.599952 1.638790 1.677548 1.716167

0.484122 485132 0.486126 0.487098 0.488041

0.568019 0. 577866 0.587809 0.597843 0.607968

1.30 1.32 1.34 1.36 1.38

1.05272 1. 04727 1. 04214 1.03729 1.03272

75 97 12 63 96

1.754591 1.792765 1.830630 1.868133 1.905218

0.488949 0.489817 0.490639 0.491410 0.492126

0.618179 0.628474 0.638850 0.649302 0.659828

1. 40 1.42 1. 44 1.46 1.48

1.02842 1.02437 1.02055 1.01696 1.01357

64 26 48 00 57

1.941832 1.977922 2.013437 2.048327 2.082544

0.492783 0.493376 0.493902 0.494357 0.494739

0.670422 0.681082 0.691804 0.702582 0.713414

1. 50 1. 52 1. 54 1. 56 1. 58

1.01039 1.00739 1.00457 1.00191 0.99942

05 28 23 88 27

2.116040 2.148771 2.180693 2.211766 2.241950

0.495045 0.495272 0.495418 0.495480 0.495458

0.724295 0.735221 0.746189 0.757192 0.763229

1.60 1.65 1.70 1.75 1. 80

0.99707 0.99179 0.98727 0.98340 0.98008

51 98 79 36 56

2.271208 2.340071 2.402437 2.457895 2.506120

0.495348 0.494687 0.493456 0.491645 0.489246

0.779295 0.807059 0.834917 0.862812 0.890687

1. 85 1.90 1.95 2. 00 2. 05

0.97724 0.97481 0.97273 0.97095 0.96943

49 36 30 31 05

2.546866 2.579972 2.605345 2.622973 2.632902

0.486255 0.482673 0.478503 0.473748 0.468417

0.918490 0.946170 0.973680 1.000975 1.028011

2.10 2.15 2.20 2.25 2. 30

0.96812 0,96701 0.96606 0.96524 0.96455

82 46 23 80 19

2.635245 2.630169 2.617892 2.598678 2.572828

0.462516 0.456054 0.449041 0.441488 0.433405

1.054750 1. 081151 1.107179 1.132799 1.157978

2.4 2. 5 2.6 2. 7 2. 8

0.96344 0.96264 0.96205 0.96162 0.96130

79 13 18 12 65

2.502604 2.410244 2.299090 2.172666 2.034544

0.415693 0.395997 0.374417 0.351055 0.326022

1.206881 1.253647 1.298044 1.339858 1.378084

2.9 3.0 3.1 3. 2 3. 3

0.96107 0.96090 0.96078 0.96069 0.96063

67 89 62 67 12

1.888235 1.737097 1.584242 1.432486 1.284291

0.299435 0.271420 0.242114 0.211664 0.180224

1.414929 1.447812 1.477367 1.503441 1.525899

3.4 3. 5 3.6 3.7 3. 8

0.96058 34 0.96054 86 0.96052 3 1 0.96050 44 0.96049 08

1.141740 1.006520 0.879924 0.762869 0.655914

0.147962 0.115052 0.081678 0. 048028 +O. 014297

1. 544621 1.559512 1. 570495 1.577513 1.580552

3.9 4. 0

0.96048 09 0.96047 37

0.559298 0.472982

-0.019318 -0.052618

1.579595 1.574671

m

0.96045 40

0.000000

0.000000

0.000000

A=O

[(-;)33

n.

For (,=I:c ( l ) = e T 4 U, o ( u b ) = m ( l ) , ~ ( ~ ' ) = e " ~ ~ l ~ / ~ / 2 1 / 4 ~ . Fora=-: U ( I ) = ! P ~ ~o~( w~i ) =~OT, ,u ( w ' ) = O . (~=1.854074677 is the real half-period in the Lemniscatic case 18.14.) To obtain the corresponding values of tabulated quantities when the real half-period wz+1, multiply U by w2.

19. Parabolic Cylinder Functions J. C. P. MILLER

Contents

....................

Mathematical Properties 19.1. The Parabolic Cylinder Functions, Introductory . . . . . The Equation 2d2Y(~x2+a)y-0 dx 19.2 to 19.6. Power Series, Standard Solutions, Wronskian and Other Relations, Integral Representations, Recurrence Relations , , . . . . . . . . , . . . . , . . . . . . 19.7 to 19.11. Asymptotic Expansions . . . . . . . . . . . . . 19.12 to 19.15. Connections With Other Functions . . . . . . . The Equation d2Y a+(ax2--a)y=0

19.16 to 19.19. Power Series, Standard Solutions, Wronskian and 0ther Relations, Integral Representations , . . . . . . . 19.20 to 19.24. Asymptotic Expansions . . . . . . . . . . . . 19.25. Connections With Other Functions . , . . . . . . . . . 19.26. Zeros . . . . . , , . . . , . . . , , . . . . . . . . 19.27. Bessel Functions of Order ki as Parabolic Cylinder Functions . . . , . . . . . . . . . . . . . . . . .

&a,

......................

Numerical Methods 19.28. Use and Extension of the Tables References

Page

686 686

686 689 69 1

692 69 3 695 696 697

.

697 697

..........................

700

. . . . . . . . . . .

702

. . . .

Table 19.1. U(a,x) and V(a,x) (05x55) &~=0(.1)1(.5)5;~ = 0 ( . 1 ) 5 , 5 s Table 19.2. W(a, &x) (05x55). . . . &a=0(.1)1(1)5; x=0(.1)5, 4-5D or S Table 19.3. Auxiliary Functions .

,

,

,

,

.

,

.

,

. .

. . . . . . . . . .

712

,

. . . . . . . . . . . . .

720

The author acknowledges permission from H.M. Stationery Office to draw freely from [19.11] the material in the introduction, and the tabular values of W ( a , 5) for a=-5(1)5, Other tables of W ( a ,5) and the tables of U ( a ,2) and V ( a ,2) were prepared on +z=0(.1)5. EDSAC 2 at the University Mathematical Laboratory, Cambridge, England, using a program

prepared by Miss Joan Walsh for solution of general second order linear homogeneous differential equations with quadratic polynomial coefficients. The auxiliary tables were prepared a t the Computation Laboratory of the National Bureau of Standards.

1 The University Mathematical Laboratory, Cambridge, England. contract with t h e National Bureau of Standards.)

(Prepared under 085

,

19. Parabolic Cylinder Functions Mathematical Properties 19.1. The Parabolic Cylinder Functions

I

19.2.1

Introductory

These are solutions of the differential equation ~+(ux~+bx+c)y=O dx2

19.1.1

with two real and distinct standard forms

3- ($xz+u)y=O

19.1.2

dxZ

19.1.3

The functions 19.1.4 y(a, 2)

19.2.3

yz=xe-tz2M(&x+$, yb,

-2)

y(-a, ix)

y(-a,

are all solutions either of 19.1.2 or of 19.1.3 if any one is such a solution. Replacement of a by -iu and x by xeftr converts 19.1.2 into 19.1.3. If y(a, x) is a solution of 19.1.2, then 19.1.3 has solutions: 19.1.5 * y(-iu, xefl")

y(-iu, -xef'") y(ia, -xe-*'")

y(iu, xe-f'")

Both variable x and the parameter U may take on general complex values in this section and in many subsequent sections. Practical applications appear to be confined to real solutions of real equations; therefore attention is confined to such solutions, and, in general, formulas are given for the two equations 19.1.2 and 19.1.3 independently. The principal computational consequence of the remarks above is that reflection in the y-axis produces an independent solution in almost all cases (Hermite functions provide an exception), so that tables may be confined either to positive x or to a single solution of 19.1.2 or 19.1.3.

The Equation

q-(i xz+U) dx

4., +x2)

-ix)

y=o

19.2.4

=xet'*M( -+U+$,

8, -+x2)

these series being convergent for all values of (see chapter 13 for M(u, c, 2 ) ) . A1terna tively , 19.2.5

19.2.6

in which non-zero coefficients a, of x"/n! are connected by

19.2. Power Series in x

Even and odd solutions of 19.1.2 are given by 686

2

19.2.7

an+a=a

*

U,+-

1

4 n(n-1)

687

PARABOLIC CYLINDER FUNCTIONS

19.3. Standard Solutions

These have been chosen to have the asymptotic behavior exhibited in 19.8. The first is Whittaker's function [19.8, 19.91 in a more symmetrical no tation.

I

19.4*4

=U(% 2 ) +U(a, - 2 ) 19.4.5

19.3.1

U ( a ,2) =D-,-&)=cos?r(+++a).P1 -sin a (3++a). Y 19.3.2

V(a,z)=-

1

r (3-4

{sin 7r(+++a).Y1 +cos .rr(4++a).~7zj

in which

19.5. Integral Representations

A full treatment is given in [19.11] section 4. Representations are given here for U ( a , z ) only; others may be derived by use of the relations given in 19.4.

2@+t h a ( $ - + a )

V ( a ,0) =

r (3 -#a)

19.5.2 2@+3sina(a-$.z)

V'(a,0 )=-

r (fi -+a)

I n terms of the more familiar D,(z) of Whittaker,

U(a,z) =D-,-*(z)

19.3.7

I

where ci and 0 are the contours shown in Figures 19.1 and 19.2. When a++ is a positive integer these integrals become indeterminate; in this case

19.3.8 1

V ( U , 2) =7r

r(++a){ ~ i n . ~ r a . ~ - , -+D-,-*(-z) +(z)

}

19.4. Wronskian and Other Relations 19.4.1

t

S PLANE

W{U,V}=&i

PLANE

a

19.4.2 T V ( U , Z ) = ~ ( ~ + ~{) s ~ ~ T u . u ( u , z ) + u (-z) u,

}

19.4.3

r (++a)u(a, S) = a

secz a a { V(a,-2)

-sin na-V(a,z)}

FIGURE19.1 --r<arp a
FIGURE 19.2 --r
(z+t)>r

688

PARABOLIC CYLINDER FUNCTIONS

19.5.10

where 19.4.

e, c3

and

~4

are shown in Figures 19.3 and

The contour ll is such that ( t z z + v ) goes from m e - t s to m e t s while v=+z2 is not encircled; ( tz2-v)-fa-f has its principal value except possibly in the immediate neighborhood of the branch-point when encirclement is being avoided. Likewise 7, is such that ( t z 2 - v ) goes from me'" to e-'" while encirclement of v= - a zz is similarly avoided. The contours (11)and (ql) may be obtained from ll and 71 by use of the substitution

v =izzt. The expressions 19.5.7 and 19.5.8 become indeterminate when a=#, 5, I+, . . .; for these values I

FIGURE 19.3 -Is<arg a < f r

FIGURE 19.4 On On

ea

tT<arg a<+*

(4

--:s<arg s<-fr

19.5.11

19.5.7

Again 19.5.9 and 19.5.10 become indeterminate when a=+J 8, Q , . . .; for these values 19.5.12

Barnes-Type Integrals

where the contour separates the zerosof

r(S)from

19.6. Recurrence Relations 19.6.1 U'(U,x)+32U(aJz)+(a+$)U(a+l, z)=O 19.6.2 []'(a, z)-$zU(a, z)+U(a-l,~)=O 19.6.3

2U'(a, ~ ) + U ( a - l , ~)+(a+$)U(a+l,z)=O

19.6.4 zU(a,~ ) - U ( a - l ,

~)+(a+$)U(a+l, z)=O

These are also satisfied by r(i-a)V(a, z). 19.6.5

V'(a,z)-+zV(~, 2)- (a-$)V(a-I, s)=O

those of r ( a f 3 - 2 s ) .

Similarly

19.6.7 2 V'(a,2 ) - V(a+ 1,2)- (a-

$)V(a-1,2)=o

19.6.8

+ (a- 3) V(a-

zV(a,2)- V(a+ 1,2)

=o

1,2)

These are also satisfied by U(a,z>/I'($-a) 19.609 y;(a, z ) + $ ~ ~ i (%)=(a+3)yz(a+1,2) a,

689

PARABOLIC CYLINDER FUNCTIONS

19-6-11

y;(a, ~ ) + $ q z ( ax, ) = y i ( a f 1 , 2 )

19.6.12

y;(a,x ) - $ x ~ ~ ( uz ,) = Y ~ ( u - ~2) ,

sense of Watson [19.6], although valid for a wider range of /arg 21 in Poincard's sense; the second series is completely valid only for 2 real and positive.

Asymptotic Expansions

19.9. Expansions for a Large With z Moderate 19.7. Expressions in Terms of Airy Functions

When a is large and negative, write, for

OIx< X=24m[

(i) a positive When a>>x2, with p=&, then

t= (41a1)'T

19.7.1 T

= -($83) 1 1 -

O3=+s,

(is1)

arccos [-it-

.J1-szds=+

19.7.2 T=

+

where 19.9.3

($O*)t E -

.Js2-lds=+[dm-+

arccosh [ ([> l ) (a++

Then for x> 0, a+-

w

The upper sign gives the first function, and the lower sign the second function.

19.7.3

U ( a , X)- 2 - t - w

(+-+a)

(A)aAi(t) 5 -1

(ii) a negative When -a>>x2, with p = J q , then

19.7.4

r (+-a)

CO)

V(a,x ) - ~ ~ ' - + T(+-+a) ( r yt B i ( t ) 5 -1

19.9.4 U ( U 2) , +ir

(+-U)

V(U,

2)

Table 19.3 gives 7 as a function of 5 . See [19.5] for further developments. 19.8. Expansions for 2 Large and a Moderate

When s>>

la I

where 19.9.5

19.8.1

Further expansions of a similar type will be found in [19.11]. 19.10. Darwin's Expansions

+

(a-+) (a-% (a-$) (a-%) +

..,

2 *4x4

(i) a positive, zZ+4a large. Write 19.10.1

(x++

-1

These expansions form the basis for the choice of standard solutions in 19.3. The former is valid for complex 2, with larg z I < b , in the complete

8=4aOI(z/2.\Ta)=&

X = uJX+X Xdx=+sX+a In 24i X

=E4 d w a + a arcsinh 2fi

690

PARABOLIC CYLINDER FUNCTIONS

19.10.11

(see Table 19.3 for & ) , then

V ( a ,s)= 2

___ e0r sin

{&r++na+t)+v,)

(2r)fJr(+-a) where where

19.10.12

vT--&

da YlZ 4 2 In Y--+--. Y6

, ,

19.10.4 v(a, x)--*

vi-3-8 d d

y 3 ye+ . *

In X 4 - C ( - l ) S d 3 s / X 3 s

.

(x2+4a+-00)

6=1

(a>O, z2+4a++

a)

and dB8is given by 19.10.13. (ii) a negative, x2+4a large and positive. Write 19.10.5

In each case the coefficients d3rare given by 19.10.13

X=J22-4/al da=t x2-2a

(see Table 19.3 for &), then d,,==

8

x4- 1 8 6 ~ 2 ~80a2 4-

See [19.11] for d I 5 , . alternative form.

,

,,

d24,

and [19.5] for an

19.11. Modulus and Phase

where again

When a is negative and 12)<24a(, the functions U and V are oscillatory and it is sometimes convenient to write

19.10.8 v(a, z)--+

In X + C (-1)sd3s/x3s a-1

(a
a)

and daais given by 19.10.13. (iii) a large and negative and 2 moderate. Write

19.11.1

V ( a , x)+iI’($-a)V(a, z)=F(a, x)efxx(Q*2’

19.11.2

U’(a, z)+ir ($-a)V’(a, x)=-G(a, z)et+(’,”’

Then, when a>x2, 19.11.3

Y = J W

19.10.9

e= 4 lal6*(x/2Ji7) = i s , ’ ~ d x = + z ~ + lal arcsin X 2TVI

(see Table 19.3 for t9,=&r-tY3), 19.10.10

then

where v7, wf are given by 19.9.5 and p=d% Alternatively, with p = J m , and again -a>>x2, 19.11.4

691

PARABOLIC CYLINDER FUNCTIONS

19.12.2

U ( U~, ) = 2 - t ' ~ - t W - t ~-t($x2) ,

19.12.3

19.11.6

-$x6-l76x2 - . . (4P)

19.11.7

*-

.}

19.12.4 ~ ( ax ),= 2 - f - f a e - t z 2 ~ ( + a + t ,+,+x2)

(&a-+)r+px

=2 -$-+axe-

Again, when x2+4a is large and negative, with Y=44= then 19.11.8

+f , , i x z )

t z 2 (+a ~

Expressions for V(a,x ) may be obtained from these by use of 19.4.2. 19.13. Connection With Hermite Polynomials and Functions

When n is a non-negative integer where 0, v7 and vUt are given by 19.10.9 and 19.10.12. Another form is

19.13.2

19.11.9

F-

19.13.1

2 Jrt$-a) 3 5a 621 (2r)tJj7 (1+----,+--,+,+ 4 Y Y 32Y

*.

.)

V(n++,x ) = Jii$efZ2He*,(x) = 2 - + " e f z 2 ~ * , ( x / J Z ) in which H , ( x ) and He,(x) are Hermite polynomials (see chapter 22) while d" 19.13.3 He*,(x)=e-fz2 d j 19.13.4 H*,(x)=e-z2

while (c. and

x are connected by X

+:+647Y

-&r--- y3 ( 1

214a 14483

Pf40Y8+

. . .)

(x2+4a-+-

d" eZ2=(-i)nH,(ix) dx"

19.14. Connection With Probability Integrals and Dawson's Integral (see chapter 7 )

If, as in [19.10]

Connections With Other Functions

19.14.1

19.12. Connection With Confluent Hypergeometric Functions (see chapter 13)

19.14.2

19.12.1

(-i)"He,(ix)

This gives one elementary solution to 19.1.2 whenever 2 a is an odd integer, positive or negative.

19.11.11

9-x-

etz2=

~ h - ~ (=e-fzz x )

692

PARABOLIC CYLINDER FUNCTIONS

V (-1 , X) = (32);(Yi Yi)

+

19-15-15 V( -2 , X) =$ (32) ( 2 ~ 5 39*-3%)

v(+,x) =@eiz2

19.14.4

+

19.15.14

Correspondingly

19.15.16

and

V(-3, x>=g 5 ( 3 x ) f ( 5 ~ + 9 . ~ - 5 Y ~ - 4 ; )

19.14.5 V(-n-+,

x)=e-iz2

{l e - i t ' V ( - n

+3, tjat

U(-+, x ) = . J ~ ( + x ) K *

19.15.17

U(-%, X ) = - ( ~ X ) * ~ K +

19.15.18

19.15.19 U(-%, x)=@(~x)'(~K+-K;) Here V(-$, x) is closely related to Dawson's integral let2dt These relations give a second solution of 19.1.2 whenever 2a is an odd integer, and a second solution is iinobtainahle from U ( a , x) by reflection in the y-axis. 19.15. Explicit Formula in Terms of Bessel Functions When 2a Is an Integer Write 19.15.1

I-,-In= (2/7r) sinnr Kn

19.15.2

I-,+I,=cos n7r 4,

where the argument of all modified Bessel functions is ix2. Then

U (1 , X) =2~-*(3x)'(-Ki+Kt)

19.15.3

19.15.4 U(2, ~ ) = 2* gr-t(3~)$(2Ki-3K0+K~)

V(3,X)= ( 3 4 ( I t + l - t )

19.15.20 19.15.21

V(8, X) = (3~)~(21++21-4)

19.15.22 V(+,X) = (3~)~(51++5I-+-I;-I-t)

The Equation %+(: a22

22-a)

y=O

19.16. Power Series in x Even and odd solutions are given by 19.2.1 to 19.2.4 with --ia written for a and xeitr for x; the series involves complex quantities in which the imaginary part of the s u m vanishes identically. Alternatively, 19.16.1 X2

y1=l+a -+(a*-+) 2!

24

3+(a3--tu)

X6

6!

28

+(a4-11a2+q) --+(a5-25a3+Ya) 8! 19.16.2 23

n+. .. XIO

27

25

19.15.5 yz=x+a g+(a2-4) FJ+(a3---Jnga) fi U ( 3 , x ) = 2 . Q 5r-t(42)~(-5Kt+9K*-5K~+K;) +(a4-17a2+9) $ - ( u 5 - 3 5 a 3 + ~ )Xl1 19.15.6 V(1, s)=+(3s)"A-9i;>

.. z+.

19.15.7

V(2, ~ ) = 3 ( 3 ~ ) + ( 2 9 i - 3 A + 9 + )

19.15.8 V(3, ~ ) = 3 ( 3 ~ ) t ( 5 ~ - 9 ~ + 5 5 - - - ) U ( 0 ,x)=r-'(3x)+K~

19.15.9 19.15.10

U(-1, X ) = T - ~ ( ~ X ) ; ( K ~ + K + )

b(-

an+2=a.un-in(n- 1)an-z

19.17. Standard Solutions (see [19.4]) 2fi

+

2, X) =7r-* (ax)i (2Ki 3K3-K3) 19.17.2

19.15.12

+

U (-3 , 2) =r-+(3s)4(5K* 9Kt -5K*-K;) 19.15.13

19.16.3

(cosh ?ra)f 19.17.1 W(a,kx)= (Giyi T JZG~YZ)

19.15.11 .

in which non-zero coefficients a, of xn/n! are connected by

V(0,4=4(34+4

693

PARABOLIC CYLINDER FUNCTIONS

19.20.1 7= - (gs,)?

At x = O ,

19.17.5 19.20.2 7=

Complex Solutions

19.17.6

+(#W

E(u,~)=k-4W(a,z)+iktW(a, -2)

19.17.7 E*(u,z)=k-*W(a,z)-ik*W(a, -2) where

19.17.8 k =d

p -

1/k.=

e *a

41+e +e 2*a

*a

In terms of U(a,z) of 19.3,

19.17.9 E(a,z) =$Ze*ra+*f*+it@z U(ia,ze-fi*) with 42=arg

19.17.10

Table 19.3 gives T RS a function of f . See [19.5]for further developments.

r($+ia)

where the branch is defined by &=O and by continuity elsewhere. Also

when a=O

19.21. Expansions for z Large and a Moderate

19.17.1 1

&&(ia,

ze-ft;)=r($-ia) { ef*a-ff*U(-ia, %eti") +e-)*a+tc*U(-ia,-zef**)}

19.18. Wronskian and Other Relations 19.18.1

W { W(U, z),W(U, -T)} = 1

19.18.2

W { E ( az), , E*(u,z)}= -2i

19.18.3 J1+e2*"E(a, z)=e*aE*(a, z)+iE*(a,-z)

E*(a,z)=e-r'%l+f*)E(-a, iz)

19.18.4 19.18.5

d-E*(a,

z)=e-ff* J m ~ ( - u iz) ,

19.19. Integral Repr7sentations These are covered for 19.1.3 as well as for 19.1.2 in 19.5 (general complex argument).

Asymptotic Expansions 19.20. Expressions in Terms of Airy Functions When a is large and positive, write, for OIz<

z=2&

t= (4a)*7

w

&(a, 2)-----

U2

v42

,+us-+-3!23s6 4!24~8

1!2z2 2!2 z

08

* *

'

694

PARABOLIC CYLINDER FUNCTIONS

(see Table 19.3 for &), then 19.23.2

W(U,x ) = J23e0r cos (tn+e+v,)

19.23.3 W(a,- z ) = m e o r sin ( ~ n + e + v t ) 19.22. Expansions for a Large With x Moderate

(i) a positive When a>>x2, with p = J Z , then 19.22.1

W(a,x ) = W(a,0 ) exp ( -pz+vl)

19.22.2

W(a,-z)=W(a, 0 ) exp (px+v2)

where 19.23.4

...

In X-B+-d 4 2 X6 XI2

v,--$-

(x2-4a+

CO)

and d3, is given by 19.23.12.

where W(a,0 ) is given by 19.17.4, and

(ii) a positive, 4a-x2>>0

19.22.3

Write 19.23.5

=+I

Y= J ~ u - x ~ e = 4 ~ 9 , ( ~ / 2 & ) arcsin X 2&

Ydx=+xY+a

(a++

m)

The upper sign gives the first function, and the lower sign the second function.

(see Table 19.3 for tJ4=+7r-03), then 19.23.6

(ii) a negative When -u>>x2, with p=~'=-&, then

19.23.7

19.22.4

W(a,x ) =exp { -d+v(a, W(a,-2) =exp { €+(a,

2) } -2)

)

where

W ( a ,z)+iW(a, -x) =JzW(a, 0) exp {v,+i(px+t.rr+vi) j

19.23.8

v(a, x)--+

In Y+,+--,+--,+ d3 d6

Y

where W(a,0) is given by 19.17.4, and

Y

.. .

Y

).

(x2-4u+-

19.22.5

and d3, is again given by 19.23.12. (iii) a negative, x2-4a>>0 Write (U+-

CO)

Further expansions of a similar type wil be found in [19.3]. 19.23. Darwin's Expansions (i) a positive, x2-4a>>0 Write 19.23.1

2 m

=+xdx2+41al-

(see Table 19.3 for

fil)

24i X = + x d G - a arccosh 24i

19.23.11

X

a arcsinh 2

m

then

19.23.10 W(a,x )

+x =+xX-a In x-

In x+x

=+xX-U

W(a, -z)=&%e"r

cos ( b + e + u i ) sin

(&r+@+vi)

where U, and v, are again given by 19.23.4. each case the coefficients dat are given by

In

695

PARABOLIC CYLINDER FUNCTIONS

19.24.8

19.23.12

a"-;

1 (G-g.x) 23 19.24.9

d6=$x2+2u

1 7 ""=$(5760 153

d l z = s

49

31 12

19.24.10

- . . .)

~ ~ + 1 8 6 ~ ~ ~ + 8 0 ~ ~

See [19.11] for dls, ternative form.

. . ,, d24, and

[19.5] for an al-

19.24.11

19.24. Modulus and Phase

When a is positive, the function W(a,x) is and when x>24i; oscillatory when x<-2& when a is negative, the function is oscillatory for all x. In such cases it is sometimes convenient to write 19.24.1

k-+W(a,x)+ikiW(a,-x)=E(a, x)=Fefx

(x>O)

Again, when a>0, with X=d-, then

F-4evr X=t?r+O+v< where 8, v1 and vr are given by 19.23.4 and 19.23.9. Another form also when a>O, a?-4a-+o~ is 19.24.12

19.24.13

19.24.2 19.24.14

Then, when x2>>laI, while $ and x are connected by

19.24.3

10a2-3

30a3-47a + 4x6

19.24.15

19.24.4

X-ix2-U 19.24.5

G-

In x+&z+tn+-

4:

a 6a2-5 (1 -----

x2

4x4

4a2-3 4 ~ " 1 9 ~ + , 8x2 + 8x4

14a3-63a 4x6

-.

*

.,

.)

19.25. Connections With Other Functions Connection With Confluent Hypergeometric and Bessel Functions

19.25.1 f

W(a,fx)=2-*

d$ H(-+, I

7

&U,

tx')

19.24.6

$-$x2-a where +2

4a2+5 4a3+29a 8x2 + 8x4 is defined by 19.17.10.

In x+g2-+7r+-

+...

19.25.2

H(m, n, z)=e-fZIFl(m+l -in; 2m+2; 2ix)

When a>x2 19.24.7

F-&V(u,

where

19.25.3

O)evr

where v, is given by 19.22.5 with p=d-a.

Also

= c L Z M ( m +1-in, 2m+2,2ix)

19.25.4

W(0,f 2 ) = 2 - J ~ ? F i { J - t ( t s 2 ) f J i ( t x 2 ) }( 2 2 0 )

696

PARABOLIC CYLINDER FUNCTIONS

19.25.5

integer for W(a,x) or its derivative, or an even integer for U7(a,-2) or its derivative, the zeros hc, hc’have expansions

d & W ( 0 ,h 3 ) = -2-3xJ?ra:{ J*(tx’) J - * (4x2) } (z 20)

19.26. Zeros

Zeros of solutions U(a,5 ) , V(a,x) of 19.1.2 occur when a is negative. A single only for I J : I <24-a exceptionalzero is possible, for any a, in the general solution; neither U(a,x) nor V ( a ,x) has such a zero for x>O. Approximations may be obtained by reverting the series for II, (or x for zeros of derivatives) in 19.11, giving II, (or x) values that are multiples of +a, odd multiples for U(a,s), even multiples for V ( a ,2). Writing a=(?y-$a-I

4)

a

as an approximation to a zero of the function, or

p = (+

$a+

t )T

as an approximation to a zero of the derivative, we obtain for the corresponding zero c or c‘, with --a=p2 the expressions 19.26.1

c

a 2a3--3a 5 2 ~ ~ ” 2 4 0 a ~ + 3 1 5 ~ ~ -+f... 7680pQ P 48P5 +

19.26.2

C

’

p 2p3+3p 52p5+28Op3-285/3 O

P

-

+

~

48P

+

7680~’

f...

These expansions, however, are of little value in the neighborhood of the turning point x=2,/=7a. Here first approximations may be obtained by use of the formulas of 19.7. If a, (negative) is a zero of Ai(t), the corresponding zero c of U(a,x) is obtained approximately by solving 19.26.3

c=2JI;EIS

(a<<())

This may be done by inverse use of Table 19.3. For a zero of V ( a ,x), a, must be replaced by b,, a zero of Bi(t). For further developments see [ 19.51. Zeros of solutions W ( a , z ) , V7(a,-x) of 19.1.3 occur for Isl>2& when a is positive; the general solution may, however, have a single zero between -2& and +2&. If a is negative, zeros are unrestricted in range. Approximations may be obtained by reverting the series for \c, (or x ) in 19.24. With -a=p2, ( Y = ( ~ T - ; ) T , g = ( + r + t ) ~ , r>O being an odd

19.26.4

C=---

19.26.5

C’

P

2 d - 3 ~ ~52a5-240a3+315a 7680~~ 48P5 +

f...

p 2p3+ 3p 52p5+280p3--285p -+... 7680pQ P 48P5 -k ~

When x is large and a moderate, we may solve inversely the series 19.24.4 or 19.24.6 with a= 3 (rA-&T-+J, p=+ (TA+ + + J , r odd or even as above; the presence of the logarithm makes it inconvenient to revert formally, The expansions 19.26.4 and 19.26.5 fail when z is in the neighborhood of 2 m . When a is positive, a zero c of W(a,-x) is obtained approximately by solving 19.26.6

with the aid of Table 19.3. For a zero of W(a,z) we replace a, by b,. When a is negative we solve, again with the aid of Table 19.3, 19.26.7 (n-t)n

b , = + { [ & v + a r c s i n h I }=- 414

where n=1, 2 , 3! . . . for an approximate zero of W(a,-x), and n=+, 8, 4, . , . for an approximate zero of W(a,2). Further developments are given in [19.5]. Any of the approximations to zeros obtained above may readily be impcoved as follows: Let c be a zero of y, and c’ a zero of y’, where y is a solution of 19.26.8

y”-Iy-rO

Here I=a& $ 2 2 , I f = h 32, I“= & 3; the method is general and the following formulae may be used whenever I”’=O. Then if y, y’ are approximations to the zeros c , c’ and 19.26.9 u=y(r) /y’(r)

~ = y (7’) ’ /PY (7’)

with I = I ( y ) or I d ( ? ’ ) respectively, then

697

PARABOLIC CYLINDER FUNCTIONS

2t 19.27.1 J*i($z2)=- {W(O,-2) FW(0, X ) 1 4s

19.26.10

c

-

y -U

-8 Iu3 ++‘2I ’U4

-22 19.27.2 J**(+z2)=-xdax {W(O,x ) h W ( O , -2) }

-(;p++r2)&

19.26.11 y’ (c)

-

y’ (y) { 1-3 IU2+Q I’u3

Functions of other orders may be obtained by use of the recurrence relation 10.1.22, which here becomes 19.27.3 +x2JV+ 1 (+x2)-2 d V ( + x 2++x2JV) 1 (+x’) =O

Again 19.27.4

I - , ( + z 2 ) ) + I t ( $ ~ 2’ ) = Vz( 0 , X )

The process can be repeated, if necessary, using as many terms a t any stage as seems convenient. Note the relations, holding at zeros, 19.26,14

u’(a,C)=--/V(a,c)

19.26.15

V’(a,~’)=@/U(a,c’)

19-26-16

W’(a,C)=-l/W(a, -c) =-- 4 d - -U ( 0 , x )

19.26.17

W(a,c ’ ) = l /

{&W(u, -2))

=-l/W’(u,

x J T X dx

-4)

2- c’

As before, Bessel functions of other orders may be obtained by use of the recurrence relation 10.2.23, which here becomes

19.27. Bessel Functions of Order & $ , h $ as Parabolic Cylinder Functions

Most applications of these functions refer to cases where parabolic cylinder functions would be more appropriate. We have

2 I,( ~ $x2)-3x2 Iu- 1 ($z’)

19.27.8

$2’ Iu+l($x’) $

19.27.9

$x2KV+1 (3~’)- ~ v K ~ ( $ x ~ ) -($x2)=0 ~x~K~-~

=0

Numerical Methods 19.28. Use and Extension of the Tables

For V ( a , x), V(a,x ) and W ( a , x), interpolation x-wise may be carried out to 5-figure accuracy almost everywhere by using 5-point or 6-point Lagrangian interpolation. For (a1 5 1, comparable accuracy a-wise may be obtained with 5- or 6-point interpolation. For \a[> 1, V ( a ,x) and V ( a ,2) may be obtained by use of recurrence relations from two values, possibly obtained by interpolation, with la ( 5 1; such a procedure is not available for W(a, h x ) , I

la(> 1.

In cases where straightforward use of the a-wise recurrence relation results in loss of accuracy by cancellation of leading digits, it may be worth while to remark that greater accuracy is usually attainable by use of the recurrence relation in the

I

reverse direction, from arbitrary starting values (often 1 and 0) for two values of a somewhat beyond the last value desired. This is because the recurrence relation is a second order homogeneous linear difference equation, and has two independent solutions. Loss of accuracy by cancellation occurs when the solution desired is diminishing as a varies, while the conipanion solution is increasing. By reversing the direction of progress in a, the roles of the two solutions are interchanged, and the contribution of the desired solution now increases, while the unwanted solution diminishes to the point of negligibility. By starting sufficiently beyond the last value of a for which the function is desired, we can ensure that the unwanted solution is negligible but, because the starting values were arbitrary, we have an un-

698

PARABOLIC CYLINDER FUNCTIONS

relations 19.6.1-2. If less accuracy is needed they can be found by use of mean central differences of U(a,s), V ( a ,s) and also of W ( a ,s) with the formula

Backward Recurrence

Forward Recurrence I

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

I

(-6) 5. 2847* (-7) 9. 172* (-7) 1. 5527 (-8) 2. 5609 (-9) 4. 1885 (-10) 6. 2220 ( - 1 O ) t l . 2676 ( L l l ) - O . 1221 ( - 1 1 ) + 1 . 2654 ( - 1 2 ) - 5 . 6079 ( - 1 2 ) $ 3 . 2555

Final Values I

(12) (11) (10) (9) (9) (8) (7) (6) (0)

1. 59035 2. 76028 4. 67131 7. 72041 1.24785 1. 97488 3.06369 4. 66352 697082 102444 14789 2111 292 42 5 1+

(-6) (-7) (-7)

(-8)

(-Gj

(-10) (-10) (-11) -12j -13) -14) -15) -16)

5.2847** 9. 1724 1. 55227 2. 5655 4. i4SS 6. 5625 1.01806 1.5497 2.3164 3.404 4. 91 7. 01 9.7

using h=.1; this usually gives a 3- or 4-figure value of duldx. If greater accuracy is needed for d W ( a , x)/dx it may be obtained by evaluating d 2 W / d 9 with the help of the differential equation satisfied by W and integrating this second derivative numerically. This requires one accurate value of dW/dx to start off the integration; we describe two methods for obtaining this, both making use of the difference between two fairly widely separated values of W , for example, separated by 5 or 10 tabular intervals. (i) Write f,,fi, f:’ for W ( a , zo+rh) and its first two derivatives, then ji may be found from

O+

*From tables. +Starting values. **This value was used to obtain the constant multiplier L=(-6)5‘2847-(18)3.32298 for converting the prek* (12)1.59035vious column into this one.

The second column shows forward recurrence starting with values at a=3,4 from Table 19.1. Backward recurrence starts with values 0 and 1 at a=19 and 18, containing a multiple kU(a, 5 ) and a subsequently negligible multiple of the other solution r(+-a)V(a, 5 ) . Rounding errors convert kU(a, x ) into k*U(a,2) without a$ecting the values in the last column. The value of l / k * is identified from the known value of U ( 3 , 5 ) ,and used to obtain the final column by multiplying throughout by l / k * . The improvement in U(5,5) is evident by comparison with Table 19.1. Derivatives. These are not tabulated here. Since the functions U(a,s), V(a,s) and W ( a ,s) satisfy differential equations, values of derivatives are often required. For all these functions the equation is second order with first derivative absent, so that second derivatives may be readily obtained from function values by use of the differential equation. First derivatives can be obtained for U(a,s) and V ( a ,s) by applying the appropriate recurrence

(ii) Consider a solution y of the differential equation for “ ( a , s), namely y”= ( -tx2++a)y. If we are given values y and y’ a t a particular x=xo and write T,,=Hny(“)/n!,T-l=T-2=0, then we may compute T2,T3,T,,. . . in succession by use of the recurrence relation obtained from the differential equation,

-+HZTn- 21 These are computed, to a fixed number of decimals until they become negligible, thus giving

...

y(sfH)=~of~l+T2f~8+

This may be applied, with H=rh, h being the tabular interval, and r a small integer, say r=5, to the solutions y=yl, y=y2 having

y,(ro) = W ( a , To)

yl(s0) =W*’(a,

y2 (so) =0

!/:(so) =1

30)

in which W*’(a,zo) is an approximation to W’(a, so), not necessarily a good one; it may be

I

699

PARABOLIC CYLINDER FUNCTIONS

obtained from differences, for example. We thus obtain yl(xo&H) and y2((rortH). Now suppose WI(a,50) =W*’(a,xo) +X

yl(l)=W(-3, 1)=-.61113 to 5 decimals to about 3 decimals yI( 1) = - .745

then, for aIl x

W(a,4=y,(4

+Xy2(x)

the following values result, with H=.5,

and in particular

W a ,~ofH)=y,(~ort~+~Y2(~ofH)

l/1

The values of W(a,x o & H ) may be read from the tables and two independent estimates of X obtained, whence

W’(a,so)=W*l(a,zo) +X to a suitable accuracy. Example 2. Evaluate W’(-3, From Table 19.2 W(-3,

.5)=-.05857

W(-3,

0. 4 0. 5 0. 6 0. 7 0. 8

2’

1)=-.61113 1.5)=-.69502

+.. 58191 17937

-. -. -. -.

05857 18832 31226 42646

1. 0

-.

61113

1.98617

1. 1

-. -. -. -. -. -.

67522

2. 22991

71706 73488 72761 69502 63774

2.40932 2. 51513 2.53936 2.47601 2. 32137

0. 9

1. 2 1. 3 1. 4 1. 5 1. 6

6

FO. 07298 -0.22186

-.

52722

69

68

+ 131

T5

-

34081

- 1095 - 1032

29775 24374 17941

T7 Ts

.OOOO 677 - 26

X= .000203/.4323

-

+

24 2

+

5 4 5 y(1.5)--.695223 +.4323 ~ ( . 5 )--.058363 -.4371

+

=.000470

so W’(-3,

1)

- .745+X -- - .744530

At ~ = . 5 -.058363--.4371X - -.05857 X=.000207/.4371 =.000474 s o W‘(-3, 1) =-.745 +A = -,744526

Example 3. Evaluate the positive zero of U(-3, 2). We use 19.7.3 to obtain a first approximation, see 19.26.3. The appropriate zero of Ai(t) is at

-9129

1000

(10.38874)

whence T=

-L 100 { 24 1 (54149)-- 1440 l1 (-.02127)} - .0103887 - .0001918-

.0002272

- (2.338) X (12)-1= -,4461

Hence, from Table 19.3, f = .3990 and the approximate zero is s=2.\im(=1.382. We improve this by using 19.26.10, but take, for convenience, z=1.4 as an approximation, so that the value of U can be read directly from the tables. U‘ can be obtained as in the section following Example 1. We find

10001 {fa (2.29664)-- 240 (-.09260)}

= - .0744527

X- .695223 =- ,69502

279 3 1344-

+

+.4323X

+.5000

1=(4l~1)!~=-2.338

*W’(-3,1)

=- .0636450

1407 4

4 =Yl+xY2

.oooo At ~ = 1 . 5

5 decimals.

The fifth decimal in W”(-3,z) is only a guard figure which is hardly needed. Only the differences needed have been computed. Then =&(-.69502+.05857)--

+ .24827 2 + 5680 9

-

W ( -3,

1/2

.61113 .37250

Thus W’( -3, 1) = - .74453 which is correct to

.97503 1.34761

1. 68842

-

TZ TS Td

T6

(i) Using the first method V”(-3,

To Tl

1) using r = 5 .

W(-3,

2

Thus W’(-3, 1)=-.74453. This might have an error up to about 13 units in the last figure but is, in fact, correct to 5 decimals. (ii) Using the second method, with

U(-3,

1.4)=.02627

U’(-3,

1.4)=2.0637

Then 19.26.9 gives U=

and

u/u~=.oi2730

r-z.51

I’= .7

I”=.5

700

PARABOLIC CYLINDER FUNCTIONS C=

~ ’ ( c )=2.0637(1+.000203) =2.0641

1.4- .012730+,000002= 1.38727

which is correct to 5 decimals, while 19.26.11 gives

I

compared with the correct value 2.06416.

References Texu

[19.1]H. Buchholr, Die konfluente hypergeometrische Funktion (Springer-Verlag, Berlin, Germany, 1953). 119.21 C. G. Darwin, On Weber’s function, Quart. J. Mech. Appl. Math. 2, 311-320 (1949). [19.3] A. Erdklyi et al., Higher transcendental functions, vol. 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [19.4]J. C. P. Miller, On the choice of standard solution8 to Weber’s equation, Proc. Cambridge Philos. Soc. 48, 428435 (1952). [19.5]F.W. J. Olver, Uniform asymptotic expansions for Weber parabolic cylinder funcpions of large order, J. Research NBS 63B,2,131-169 (1959), RP63B2-14. [19.6]G. N. Wataon, A theory of asymptotic series, Philos. Trans. Roy. Soc. London, A 211, 279-313 (1911). [19.7] H.F. Weber, Ueber die Integration der partiellen Differential-gleichung: @u/bx*+@u/bv*+k*u=O, Math. Ann. 1, 1-36 (1869). 119.8) E. T. Whittaker, On the functions associated with the parabolic cylinder in harmonic analysis, Proc. London Math. Soc. 35, 417-427 (1903).

[19.9]E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed. (Cambridge Univ.. Press, Cambridge, England, 1952). Tablee

j19.101 British Association for the Advancement of Science, Mathematical Tables, vol. I, Circular and hyperbolic functions, exponential, sine and cosine integrals, factorial (gamma) and derived functions, integrals of probability integral, 1st ed. (British Association, London, England, 1931; Cambridge Univ. Press, Cambridge, England, 2d ed., 1946,3d ed., 1951). [19.11]National Physical Laboratory, Tables of Weber parabolic cylinder functions. Computed by Scientific Computing Service Ltd. Mathematical Introduction by J. c. P. Miller. (Her Majesty’s Stationery Office, London, England, 1955). [19.12]National Physical Laboratory Mathematical Tables, vol. 4, Tables of Weber parabolic cylinder functions and other functions for large arguments, by L. Fox (Her Majesty’s Stationery Office, London, England, 1960).

I

702

PARABOLIC C Y L I N D E R FUNCTIONS

Table 19.1

L7(-5.0, Z)

T

3.0522 3.6547 4.0753 4.2934 4.2988

I

3.3 3.4

!!:

0 01 01 0 ( 0)

3.5 3.6

f 01 5.3376 ( 0 5.4473

;3.9:

0 5.4614 01 5.3895 0 5.2427

4.4

0 5.0332 01 4.7733 01 4.4753 0 4.1508 0 3.8106

1 :::$ ; 1 4.5 4.6 4.7 4.8 4.9 5.0

1

0 01 0 0 0

3.2021 3.8377 4.3739 4.8038 5.1246

1

U(-4.5, Z)

0 01 0 0 ( 01

1

U ( - 4 . 0 , ~ ) U(-3.5, .E)

3.0000 2.9328 2.7341 2.4132 1.9846

0 ) 1.5204 01 1.1869 (-1 8.0608 -1 +3.9325 -1 -0.3518

-3.0982 -3.0617 -2.9073 -2.6435 -2.2824

0) -1.8583 01 -1.6076 0 -1.3029 -1 -9.5564 -1)-5.7791

[

1

1

0 2.5823 01 2.6273 01 2.6304 0 2.5957 ( 0) 2.5279

0 01 01 0 ( 0)

f 0) 3.7212

f 0) 2.4320 2.3134 2.1771 2.0282 0) 1.8714

1

0 01 01 0 0

1.7108 1.5502 1.3927 1.2408 1.0967

-1 9.5241 1-11 8.4694 7.4740 -1 6.5463 1-1 -11 5.6918

(-1) 4.4586

(-1) 2.1235

1 11

( 0) 3.6501

0 0

0 3.5331 01 3.3781 0) 3.1929

0 01 01 0 0

2.9854 2.7630 2.5323 2.2992 2.0689

1

C7(-2.O, z)

U(-1.5,

z)

-1 -6.4101 1-11-9.7121 -1 -0.9080 (-1 +1.9218 ( -11 4.7004

3.1620 3.4108 3.5883 3.6963 3.7388

0 0 01 0 ( 0)

[[

U ( - 3 . 0 , ~ ) U(-2.5, z)

1.8972 1.8543 1.7910 1.7109 1.6175

f 0) 1.5142 0) 1.4043 0 1.2906 01 1.1760 0 1.0626

i It

-1 5.2527 [-1 4.7497 -1 4.2658 1-1 3.8056 (-1) 3;3729

-1 3.1620 -1 2.8052 -1 2.4738 -1 2.1684 (-1)1;8896

(-1) 5.2617 -1j 4;6840 -1 4.1408 1-11 3.6358 -1 3.1709

(-1) 2.9700

I-111-6370 -1 1.4099 -1 1.2073 /-;/;:0;80 -2 8.7028

1 2.7473

1 1.4313

1-1 -11 2.8991

2.0226 -1 1.7190 -1 1.4517 1-11

1.0242 -2 8.5874 1-2 -11 7.1578

-2 7.3263 1-216.1328 1-2/5:1052 -2 4.2261 -2 3.4791

(-2)

(-2)

( - 2 ) 2.1262

(-3)9.6523

0 0 01 0 (-1)

[

1.2985 1.2306 1.1553 1.0749 9.9150

(-1) 9.0701 -1) 8.2306 -1) 7.4107 1-1) 6.6219 -1) 5.8733 -1) 5.1716

I-1, 4.5215 3.9256

-1 3.3849

1 8.4319

[Ill 7.7913 -1 7.1430 1-11 6.4987 (-1) 5.8688

-i

2;5987 -1 2.2595 1-11 1.9525 -1 1.6768

1111 2.3649 1111 1.2144

3.4641 3.1197 2.7843 2.4632 2.1608

( 0) 1.8800

(-1) 9.2276

For interpolation, see 19.28.

9.9802

4.6331

703

PARABOLIC CYLINDER FUNCTIONS

Table 19.1

0.0

0.1 0.2 0;3 0.4

-5.8311 -4.3898 -2.7299 -0.9344 +0.9074

-2 -2 -2 -2

0.0000 2.6397 5.1612 7.4519 9.4102

I-11

1.3071 1.5417 -1 1.7149 -1 1.8199 -1 1.8527 -1

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

1.2852

(-1) 5.2761

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 -2 -1 -1 -1

3.0 3.1 3.2 3.3 3.4

-1

3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4

2.8040 3.1270 3.4421 3.7545 4.0712 6.2301 7.4913 8.6933 9.8444 1.0959

4.5 4.6 4.7 4.8 4.9 5.0

9.7155 1.3693 1.7522 2.1187 2.4688

it

-1 4.4004 -1

-1 1.3540 -11 1.5202 -1 1.6819 -1 1.8422 -1 2.0048

-1 -1 -1 -1) 3.8093 -1 4.1462 -1 4.5368 -1 4.9967 -1 5.5449

ii (-1) 1.8370

(-1) 3.3533

(-1)

6.2047

-1 -1 -1 -1

0

0 7.5384 01 8.9563

6.6389 7.3192 8.1309 9.1078 1.0291 1.1734

( 0) 2.2757

( 0) 4.5254

( 0) 9.2067

( 1) 1.9107

704

PARABOLIC CYLINDER FUNCTIONS

Table 19.1 X

0.0 0.1 0.2 3.3 0.4

U ( - 1.0,5)

U ( - 0.9,Z)

(-1)7.7267

U ( - 0.8,~)

U ( - 0.7,~)

U ( -0.6,~)

U ( - 0.42)

-1) 8.7077

0.5 0.6 0.7 0.8 0.9

(-1)8.0270

(-1)8.4462

(-1)8.4842

( -1)8.5144

1.0 1.1 1.2 1.3 1.4

-1)8.4220 -1)8.2967

(-1)8.2808

(-1)7.9828

-1)6.4345

I!

I- \

I- \

-1 6.3372 1 5.9266

-1 6.9716 1 6.5948

1.5 1.6 1: 7 1.8 1.9

-1 -1 -1 -1 -1

-1)4.6264

2.0 2.1 2.2 2.3 2.4

-1)2.0961 1-11 1.8452 -1 1.6162

2.5 2.6 2.7 2.8 2.9

5.3826 4.9566 4.5424 4.1429 3.7603

-1 1.8987 1-11 1.6657 -1 1.4541 -1 1.2632 (-1)1.0920

(-1)1.1816

3.0 3.1 3.2 3.3 3.4

( -2)7.1436

(-2)6.3032

(-2)7.7900

3.5 3.6 3.7 3.8 3.9

( -2)4.1098

(-2)3.8747 (-2)2.1094

4.0 4.1 4.2 4.3 4.4

-3) 6.8002

4.5 4.6 4.7 4.8 4.9 5.0

U(-0.5,~)

I- 1

3 6.3297 -3 5.0418

I -3-3I 5.4320 4.3177

(-3)1.9305

(-3) 1.6401

-3)4.6914 (-3)4.3375

(-3)3.6919

(-3)3.1412

(-3)2.6716

(-3)2.2714

705

PARABOLIC CYLINDER FUNCTIONS

2

V(-I.O,X)

V(-0.9,~) -5.5730 -4.7818

0.0 0.1 0.2 0.3 0.4

-3.9477 -3.0785 -2.1823

1.0 1.1 1.2 1.3 1.4

-11-3.5487 -1 -4.3852

-1 -2.6839 -1 -1.7980 -1 -0.8980

V (- 0 . 4 , ~ )

V (- 0 . 5 , ~ )

(

[

I- 1 I

1 -1 (-1) -1) -1)

1.3217 2.1900 3.0449 3.8823 4.6988

(-1) 4.2420

-1 9.9812 011.0555

-1) 9.1046 -1 9.5974 0) 1.0295

0) 1.0211

1 ( 0) 1.8424

0

1.4936

i:?:;:

0 1.8355 ( 0) 1.9837

1 I: I 0

1.5902

: 78%!;

0 1.9700 0) 2.1371

( 0) 2.3321

3.5 3.6 3.7 3.8 3.9

I 0) 3.9868

0 1.6625 0) 1.7781

0) 1.5837 0) 0 01 0 0)

1.6912 1.8134 1.9535 2.1157 2.3045

0

1.7975

01 1.9338

1

0 1.9104 012.0631 0 2.2404

1

0 2.0311 012.2029 0 2.4032

( 0) 3.5319 ( 015.4531

I3 0 0

4.5323 5.1887 ( 0) 5.9818 ( 0) 6.9437

0

4.7686 01 5.4910

( 0) 8.7448

I

( 1) 4.0344

( 0) 8.8266

( 019.5693

( Oj9.9703

( 1j1.1276

1 1.1779 111.4002

( 111.3367

(

I 1 (

4.5 4.6 4.7 4.8 4.9 5.0

V (- 0.6,~)

-1) 8.4234

3.0 3.1 3.2 3.3 3.4

4.0 4.1 4.2 4.3 4.4

V (- 0.7,~)

(-1 5.8660

(-11 6.6605

1.5 1.6 1.7 1.8 1.9

2.5 2.6 2.7 2.8 2.9

V(-O.~,X)

-1)+0.0088

0.5 0.6 0.7 0.8 0.9

2.0 2.1 2.2 2.3 2.4

1

Table 19.1

1 1.6981 1 2.0559 1 2.5044 3.0694 11 3.7844 1

( 1) 4.6937

1

I 1

1 2.2417 1 2.7297

1 2.8928 3.5549 11 4.3944

( 1) 5.4639

1) 3.8676

1 5.5441 11 6.9051

( 1) 6.3641

( 1) 7.4168

( 1)8.6484

( 2)1.0090

706

PARABOLIC C Y L I N D E R F U N C T I O N S

Table 19.1

0.0 0.1 0.2 0.3 9.4

U ( - 0.2,Z)

U (- 0 . 3 , ~ )

5

1

U(O.2,x)

U(0.3,~)

1

-1 9.9016

-1 9.8431

-1 9.3856 -11 8.8661

I1

-1 7.5920 -1 7.0542

(-1)6.5307

1.0 1.1 1.2 1.3 1.4 5.0730 I -1-11 4.6492

-1)4.1927 -1 3.7912 1-11 3.4139

(-1)1.6064 ( -1)1.7179

1

-1 -1 -1 -2/

1.5020 1.3069 1.1317 9.7528

/ -11

-1 3.9191 -1 3.5288 3.1647 (-1 2.8266 (-1 2.5142

1

(-1)1.3136

I -2-11 1.1387 9.8278

(-1)1.5526 -1 1.3529 -1 1.1734 -1 1.0129 -2/ 8.7027

1

ii -1 -1 -1 -1 -1

( -1)2.2270

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

U(O.l,x)

0 1.1535 0 1.1887 0 1.2163 I 001 1.1105 1.0843 I 01 1.1161 I 0011.0412 01 1.1406 I 01 1.1581 1.0914 011.0421 0 1.1000

0.5 0.6 0.7 0.8 0.9

1.5 1.6 1.7 1.8 1.9

U@,4

U ( - 0.1,x)

5.9622 5.4457 4.9566 4.4953 4.0619

-i 1-d

-1 3.6565 -1 3.2790 2.9290

ii -1 -1 -1 -1 -1

2.0381 1.7913 1.5678 1.3665 1.1859

3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

-1 3.4055 -1 3.0417 2.7065

-1 1.8627 -1 1.6315 -1-1.4232 -1 1.2363 (-1)1.0695

iI -2 -2 -2 -2 -2

9.2134 7.9031 6.7502 5.7406 4.8608

(-2)1.4664 1.1951 I -3-21 9.6928

4.0 4.1 4.2 4.3 4.4

7.7613 I -3-31 6.2526 -3) 3.7034

-3)4.6596

4.5 4.6 4.7 4.8 4.9

( -3)1.7909

5.0

(-3)1.3929

-3)3.9954

(-3)1.1825

-3 3.4243 -3 2.7050 -3 2.1265 -3/1.6637 (-3)1.2952

-3)2.9336 -3 2.3127 -3 1.8145 -3/1.4168 (-3)1.1009

1

I 1

( -3)1.0035

(-4)8.5136

( -4)7.2201

(-3)2.5122 -3 1.9765 -3 1.5477 (-4)7.9449 (-4)6.1210

(-4)5.1875

707

PARABOLIC CYLINDER FUNCTIONS

Table 19.1 X

V (- 0 . 3 , ~ )

V (- 0 . 2 , ~ )

V( - 0.1, Z)

V(0,X)

V(O.1,x)

V(0.2,x)

V(0.3,~)

0.0 0.1 0.2 0.3 0.4

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

I

1.5 1.6 1.7 1.8 1.9 2.0 2.1 2;2 2.3 2.4

f 011.5220

f 0)1.5886

f 011.6563

f 011.7265

f 011.8005

f 011.8799

I 011.9665 I

2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 317 3.8 3.9

f 016.7730

f 0)7.5658

f 0)8.4638

f 019.4818

I 1)1.0637

( 1)1.1948

( 1)1.3438

( 2)1.1778

( 2)1.3756

( 2)1.6073

( 2)1.8791

( 2)2.1979

( 2)2.5720

( 2)3.0112

4.0 4.1 4.2 4.3 4.4 4.5 416 4.7 4.8 4.9 5.0

708

PARABOLIC CYLINDER FUNCTIONS

Table 19.1 X

U(0.4,~)

0.0 0.1 0.2 0.3 0.4

0)1.2579 0)1.1672

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

U(0.8,~)

U(0.7,x)

0)1.2292

(-1)1.9402

(-1)1.7749

0 1.2106

-1 7.3870 -1)6.6567

(-1)4.0318

(-1)3.0739

-1)2.8550

(-1)2.7238 (-1)2.4053

-1) 2.5204

-1) 1.6216

( -1)1.4798

-1) 2.6475

I 1

(-1)1.2278

(-1) 1.1165

(-2 4.2627 -2 3.5839 -2 3.0017

(-2)3.0068

(-2)2.6615

-2 4.7714 -2 4.0227 -2 3.3782 -2 2.8258 ( -2) 2.3543

/-2/2.0810 -2 2.5042

( -2) 2.5078 1-21 2.0830 -2 1.7228 -2 1.4189 ( -2) 1I1636

(-2)2.2142 1-211.8344 -2 1.5134 -2 1.2434 -2) 1.0172

(-2)1.9535

(-2)1.7224

I

-1)2.1487

( -2) 7.0055

( -2)3.8288

( -2)1.7351

1

( -1) 2.4514

-1 2.3288 -1 2.0419 -1'1.7 844 -111.5540 -1)1.3487

-2)5.9630

(-2 1.2468 (-2 1.0184

U(1.0,x)

1-11917063 -1 8.7549 -1 7.8843

-1)1.0088 -2) 8.6728

3.5 3.6 3.7 3.8 3.9

U(O.9,x)

01 1.1000

f -1)4.8280

-1) 3.3042

2.5 2.6 2.7 2.8 2.9

4.0 4.1 4.2 4.3 4.4

0 1.2533 i1 0 1.1564

(-1)5.3894 -1)4.8780 -1)4.4008

2.0 2.1 2.2 2.3 2.4

3.0 3.1 3.2 3.3 3.4

U(0.6,~)

(-1) 8.2327

0.5 0.6 0.7 0.8 0.9 1.0

U(0.5,~)

I 1

(-3)8.8881

-3 6.2937 -3 7.2238 I -3-31 8.2868 6.7217 I -31 5.8462 I -3 1 5.0820

-2 1.0887 -3 8.8715 -3 5.0391 -314.0432

-3) 5.0176 -3 4.0160 -31 3.1995 -3 2.5373 -3 2.0029

1

( -4)7.1610

(-4)9.8235

4.5 4.6 4.7 4.8 4.9

-4)5.7255

5.0

(-4)4.3948

-4 3.8680 -412.9611 (-4)3.7221

(-4)3.1512

(-4)2.6671

(-4)2.2566

(-4)1.9086

(-4)1.6138

709

PARABOLIC CYLINDER FUNCTIONS

Table 19.1 5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4

V(0.5,~)

V(0.4,~)

!I

-1 -1 -1 -1 -1

8.3485 8.3808 8.4468 8.5475 8.6844

V(0.6,x)

-1) 7.9788

V(0.7,~)

V(0.8,x)

-1) 6.4988 -1 6.5836 -1 6.7147 -1 6.8936 -1 7.1224

ii

-1) 5.4924

-1) 9.1481

I

( 011.2908

-1) 8.5344

( -1j 8.3040

I-1)9.3507

f -1) 9.2267

011.3870

il

1 1

I 1

0) 1.4003 0 1.5132 0 1.6433 0 1.7936 0 1.9674

0 2.0622 0 2.2705

0 2.1689 0 2.4030

012.5130 0 2.7961 0 3.1275

012.6757 0 2.9943 0 3.3676

11

0 1.4569 0 1.5992

1

( 0 2.2886 0 2.5514 0012.8578 3.2160 0 3.6363

1

0 2.4236 0 2.7182

0013.0620 3.4644 ( 0 3.9371

0 1.5420 0 1.7196

1 1 1 0 2.5760 0 2.9058

0)2.7481 0 3.1169

0 2.9424 0 3.3542

3.7428 3.2911 10 4 , 2 7 4 1

0 4.0548 3.5483 i4.6517

3.8368 0 4.4044 j5.0747

( 2)6.6688

( 2)7.8285

( 2)9.1938

4.2 4.3 4.4

1 1 1 1 1 47.0397 1 5.7065 1 8.7286

1186.5642 .1183 2 1.0091

1 7.5559 49.3682 2 1.1673

41.0817 1 8.7031 2 1.3511

5.0

( 2)3.5270

( 2)4.1331

( 2)4.8456

( 2)5.6833

4.0 4.1

-1) 871984 -1) 8.8253

0) 1.0245

( 0)1.0434

0)1.3779

V(l.O,x)

V(O.9,x)

1 3.8072 1 4.6493

1 4.3563 1 5.3341

1 4.9884 1 6.1242

1 5.7165 1 7.0364

(

710

PARABOLIC CYLINDER FUNCTIONS Table 19.1

-4) 5.3467

2.9

( - 2 ) 1.1116

( -4) 1.9134

( -3) 1.5460

( - 3 ) 5.8431

1-51 9.6913 -5 7.3727 -5 5.5875

1-51 -5 -5 -5 ( -5)

4.4015 3.3191 2.4937 1.8667 1.3920

( - 6 ) 6.0609

( -6) 2.6194

( -6) 2.2089

( -7) 9.1724

(-7) 3.7849

(-5) 2.3767

5.0

( -5) 6.9418

( -5) 2.9634

( -5) 1.2558

(-6) 5.2847

( -7) 1.5523

71 1

PARABOLIC CYLINDER FUNCTIONS

Table 19.1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

/-1/1:6ilE -1 0.7999

-1 2,4481 -1 3.3218

1-114.5665 -1 3;959i 1-118.38~4 -1 8;0788 1-116.9684 -1 5;856i 1-il4.8999 -1 23076 -1 5.2660 -1 6.0721

-1 4.2467 -1 5.2381 1-116.3li0

-1 7.0024 1-119.3~7 -1 8.0774

-1 7.4906 -1 8.7928

0 1.0764 0 1.2440

1.0 1.1 1.2 1.3 1.4

1 1 1

j 1

o011.2810 ii0483

-1 8.8948 -1 9.6332

-1 8.2911 -1 9.8651

-1 7.5647 0 1.0497

0 1.0617 011.3438 0 1.1873

0 1.1740 011.6644 0 1.3975

0 1.3802 012,2033 0 1.7600

0 1.5356 0 1.7683

0 1.9833 0 2.3652

0 2.7266 0 3.3501

0 4.2777 ( 015.2386

1 1.1028 111.3461 1 1.6454 1 2.0145 132.4708

1 1.8048 1 2.2284 1 2.7558 1 3.4139 14.2370

0 1.5652 0 1.9126

1

ooj2.6124 2;4477 ( 0oj2.6952 Z;iG 0 2.8954 0 3.3098

0 3.3715 0 4.2178

( 1 3.0195

( 1 5.1442 ( 1 6.4978

0 2.3376 0013.4955 2.8579

0)3.8551

1.5 1.6 1.7 1.8 1.9 0)6.7480 0 7.9725 01 9.4452 1) 1.1222 1) 1.3374

2.0 2.1 2.2 2.3 2.4

( 1)1.7910

1) 2.2765

1

1)2.1774 1 2.6535 113.2418 1)3.9709

1

1

( 1)5.2689

113.7699 1 4.7150 1 5.9076 d7.4155

1

1 8.2198 2 1.0415 11.3218

1) 9.3262

2 2.7325 2)1.2873

( 212.3822

( 213.4948 ( 2)4.4794

2.5 2.6 2.7 2.8 2.9 ( 3)2.0877

3.0 3.1 3.2 3.3 3.4

2)1.8032

3.5 3.6 3.7 3.8 3.9 1.5567 I 331 2.0173

4.0 4.1 4.2 4.3 4.4

ii

4 1.2465 4 1.6584 4 2.2145

3) 4.4934

4.5 4.6 4.7 4.8 4.9 5.0

( 4j4.9845 (

( 3)2.0666

( 3)4.6909

( 4)1.0746

( 4)2.4833

( 4)5.7864

( 5)1.3589

4)6.7384

( 5)3.2156

( 5)7.6639

712

PARABOLIC CYLINDER FUNCTIONS Table 19.2

W(- 5 . 0 , ~ )

W(-4.0,2)

W(-3.0,2)

0.0 0.1 0.2 0.3 0.4

0.47348 0.35697 0.22267 +0.07727 -0.07200

0.50102 0.39190 0.26715 +0.13172 -0.00899

0.53933 0.43901 0.32555 0.20231 +0.07298

0.60027 0.51126 0.41203 0.30453 0.19088

0.47348 0.56641 0.63113 0.66435 0.66434

0.50102 0.59017 0.65576 0.69515 0.70666

0.53933 0.62350 0.68900 0.73381 0.75649

0.60027 0.67730 0.74078 0.78939 0.82206

0.5 0.6 0.7 0.8 0.9

-0.21764 -0.35231 -0.46911 -0.56198 -0.62597

-0.14933 -0.28362 -0.40634 -0.51236 -0.59713

-0.05857 -0.18832 -0.31226 -0.42646 -0.52722

+0.07334 -0.04569 -0.16377 -0.27838 -0.38697

0.63099 0.56583 0.47199 0.35408 0.21799

0.68972 0.64485 0.57370 0.47898 0.36441

0.75622 0.73285 0.68690 0.61955 0.53268

0.83798 0.83665 0.81785 0.78173 0.72875

1.0 1.1 1.2 1.3 1.4

-0.65752 -0.65470 -0.61732 -0.54700 -0.44716

-0.65688 -0.68881 -0.69121 -0.66357 -0.60670

-0.61113 -0.67522 -0.71706 -0.73488 -0.72761

-0.48704 -0.57617 -0.65204 -0.71255 -0.75583

+0.07061 -0.08044 -0.22724 -0.36189 -0.47700

0.23458 +0.09483 -0.04897 -0.19063 -0.32388

0.42880 0.31103 0.18303 +0.04890 -0.08688

0.65975 0.57594 0.47890 0.37059 0.25333

1.5 1.6 1.7 1.8 1.9

-0.32290 -0.18077 -0.02851 +0.12535 0.27194

-0.52270 -0.41495 -0.28803 -0.14758 -0.00009

-0.69502 -0.63774 -0.55733 -0.45625 -0.33785

-0.78031 -0.78484 -0.76869 -0.73166 -0.67412

-0.56602 -0.62369 -0.64634 -0.63218 -0.58147

-0.44262 -0.54122 -0.61480 -0.65945 -0.67250

-0.21962 -0.34454 -0.45694 -0.55237 -0.62680

0.12978 +0.00294 -0.12397 -0.24749 -0.36405

2.0 2.1 2.2 2.3 2.4

0.40253 0.50907 0.58468 0.62416 0.62438

+0.14739 0.28751 0.41299 0.51702 0.59364

-0.20633 -0.06661 +0.07581 0.21503 0.34495

-0.59707 -0.50217 -0.39174 -0.26879 -0.13696

-0.49661 -0.38212 -0.24445 -0.09171 +0.06678

-0.65271 -0.60042 -0.51764 -0.40802 -0.27680

-0.67684 -0.69989 -0.69432 -0.65962 -0.59652

-0.47006 -0.56198 -0.63649 -0.69061 -0.72184

2.5 2.6 2.7 2.8 2.9

0,58460 0.50668 0.39507 0.25669 +0.10057

0.63810 0.64722 0.61968 0.55625 0.45985

0.45960 0.55333 0.62119 0.65920 0.66463

-0.00046 +0.13603 0.26749 0.38872 0.49459

0.22095 0.36067 0.47637 0.55973 0.60434

-0.13062 +0.02276 0.17482 0.31672 0.43980

-0.53704 -0.39454 -0.26363 -0.12008 +0.02936

-0.72830 -0.70889 -0.66340 -0.59265 -0.49853

3.0 3.1 3.2 3.3 3.4

-0.06260 -0.22123 -0.36354 -0.47850 -0.55672

0.33555 0.19042 +0.03320 -0.12614 -0,27701

0.63631 0.57472 0.48225 0.36312 0,22333

0.58021 0.64123 0.67411 0.67637 0.64681

0.60627 0.56451 0.48124 0.36184 0.21471

0.53615 0.59915 0.62397 0.60808 0.55155

0.17727 0.31588 0.43747 0.53481 0.60167

-0.38404 -0.25332 -0.11153 +0.03530 0.18042

3.5 3.6 3.7 3.8 3.9

-0.59128 -0.57849 -0.51836 -0.41490 -0.27601

-0.40886 -0.51196 -0.57820 -0.60177 -0.57982

+0.07050 -0.08654 -0.23816 -0.37452 -0.48622

0.58576 0.49519 0.37883 0.24205 +0.09180

+0.05079 -0.11714 -0.27544 -0.41066 -0.51073

0.45725 0.33088 0.18074 +0.01731 -0.14737

0.63325 0.62663 0.58111 0.49849 0.38313

0.31672 0.43701 0.53447 0.60305 0.63793

4.0 4.1 4.2 4.3 4.4

-0.11306 +0.05995 0.22741 0.37359 0.48406

-0.51295 -0.40534 -0.26474 -0.10210 +0.06923

-0.56500 -0.60443 -0.60059 -0.55252 -0.46263

-0.06370 -0.21535 -0.35365 -0.46937 -0.55413

-0.56615 -0.57098 -0.52367 -0.42750 -0.29056

-0.30058 -0.42985 -0.52406 -0.57448 -0.57571

0.24189 +0.08387 -0.08010 -0.23812 -0.37804

0.63597 0.59605 0.51937 0.40960 0.27290

4.5 4.6 4.7 4.8 4.9

0.54726 0.55583 0.50770 0.40664 0.26226

0.23443 0.37847 0.48758 0.55059 0.56028

-0.33674 -0.18393 -0.01604 +0.15314 0.30893

-0.60118 -0.60601 -0.56693 -0.48549 -0.36666

-0,12531 +0.05237 0.22465 0.37342 0.48233

-0.52643 -0.42982 -0.29363 -0.12977 +0.04660

-0.48847 -0.55975 -0.58492 -0.56059 -0.48753

+0.11769 -0.04573 -0.20576 -0.35036 -0.46788

5.0

0.08936 [(-;)71

0.51440 [(-;)71

0.43707 [(-;)6-j

-0.21874

0.53861 [(-;I71

0.21827

-0.37095

-0.54818

z

W(-2.0,2)

[(-;)q

W ( - 5 . 0 , - ~ ) W(-4.0,-~)

[(-;)“3

W ( - 3 . 0 , - ~ ) W(-2.0,-2)

[(-;PI [(-;)51

Values of W (a,z),for integral values of a are from National Physical Laboratory, Tables of Weber parabolic cylinder functions. Computed by Scientific Computing Service Ltd. Mathematical Introduction by J. C. P. Miller. Her Majesty’s Stationery Office, London, England, 1955 (with permission).

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714

PARABOLIC CYLINDER FUNCTIONS

Table 19.2

W (- 1.0,X) W (- 0.9,X)

W(- 0.8,~) W(- 0 . 7 , ~ ) W (- 0.6,~)

0.0 0.1 0.2 0.3 0.4

0.73148 0.65958 0.58108 0.49671 0.407 26

0.75416 0.68457 0.60881 0.52750 0.44133

0.77982 0.71267 0.63980 0.56175 0.47908

0.80879 0.74421 0.67441 0.59981 0.52089

0.84130 0.77940 0.71281 0.64187 0.56693

0.87718 0.81803 0.75477 0.68766 0.61 696

0.91553 0.85912 0.79925 0.73610 0.66984

0.5 0.6 0.7 0.8 0.9

0.31359 0.21659 0.11723 +0.01657 -0.08429

0.35102 0.25734 0.16111 +0.06324 -0.03529

0.39240 0.30233 0.20958 0.11490 +0.01912

0.43811 0.35200 0.26311 0.17206 +0.07954

0.48837 0.40658 0.32198 0.23506 0.14637

0.54293 0.46584 0.38601 0.30379 0.21956

0.60064 0.52866 0.45409 0.37715 0.29811

1.0 1.1 1.2 1.3 1.4

-0.18412 -0.28164 -0.37549 -0.46422 -0.54635

-0.13342 -0.23002 -0.32384 -0.41357 -0.49783

-0.07684 -0.17198 -0.26523 -0,35538 -0.44119

-0.01369 -0.10679 -0.19880 -0.28870 -0.37536

+0.05650 -0.03384 -0.12386 -0.21269 -0.29933

0.13380 +0.04704 -0.04009 -0.12687 -0.21246

0.21727 0.13503 +0.05185 -0.03172 -0.11502

1.5 1.6 1.7 1.8 1.9

-0.62034 -0.68464 -0.73771 - 0.77 808 -0.80439

-0.57517 -0.64409 -0.70310 -0.75070 -0.78547

-0.52130 -0.59431 -0.65875 -0.71317 -0.75611

-0.45753 -0.53393 -0.60317 -0.66382 -0.71446

-0.38270 -0.46162 -0.53480 -0.60091 -0.65854

-0.29594 -0.37627 -0.45231 -0.52280 -0.58645

-0.19728 -0.27764 -0.35510 -0.42857 -0.49684

2.0 2.1 2.2 2.3 2.4

-0.81541 -0.81014 -0.78787 -0.74822 -0.69124

-0.80610 -0.81144 -0.80054 -0.77279 -0.72790

-0.78618 -0.80212 -0.80282 -0.78741 -0.75531

-0.75365 -0.78003 -0.79238 -0.78960 -0.77089

-0.70628 - 0; 74273 -0.76654 -0.77649 -0.77153

-0.64186 -0.68765 -0.72243 -0.74486 -0.75373

-0.55864 -0.61261 -0.65738 -0.69156 -0.71385

2.5 2.6 2.7 2.8 2.9

-0.61743 -0.52785 -0.42412 -0.30847 -0.18374

-0.66601 -0.58777 -0.49436 -0.38753 -0.26968

-0.70633 -0.64071 -0.55918 -0.46303 -0.35416

-0.73570 -0.68391 -0.61582 -0.53224 -0.43455

-0.75086 -0.71398 -0.66079 -0.59164 -0.50739

-0.74799 -0.72686 -0.68984 -0.63684 -0.56821

-0.72301 -0.71801 -0.69802 -0.66256 -0.61149

3.0 3.1 3.2 3.3 3.4

-0.05335 +0.07873 0.20811 0.33 006 0.43974

-0.14378 -0.01339 + 0.11 741 0.24412 0.36198

-0.23506 -0.10884 +0.02083 0.14977 0.27340

-0.32474 -0.20540 -0.07973 +0.04850 0.17504

-0.40948 -0.29995 -0.18146 -0.05729 +0.06875

-0.48485 -0.38820 -0.28034 -0.16395 -0.04232

-0.54517 -0.46444 -0.37075 -0.26614 -0.15327

3.5 3.6 3.7 3.8 3.9

0.53233 0.60334 0.64885 0.66575 0.65207

0.46613 0.55184 0.61476 0.65118 0.65834

0.38695 0.48557 0.56460 0.61986 0.64786

0.29527 0.40440 0,49761 0.57035 0.61858

0.19236 0.30891 0.41 360 0.50168 0.56868

+0.08071 0.20083 0.31342 0.41373 0.49706

-0.03541 +0.08365 0.19963 0.30797 0.40397

4.0 4.1 4.2 4.3 4.4

0.60721 0.53214 0.42952 0.30382 0.16115

0.63466 0.58002 0.49593 0.38565 0.25422

0.64616 0.61356 0.55042 0.45874 0.34234

0.63904 0.62958 0.58939 0.51923 0.42158

0.61072 0.62476 0.60892 OI56270 0.48725

0.55906 0.59598 0.60496 0.58437 0.53398

0.48303 0.54088 0.57391 0.57944 0.55599

4.5 4.6 4.7 4.8 4.9

+O. 00918 -0.14329 -0.28674 -0.41153 -0,50861

+0.10831 -0.04397 -0.19348 -0.33057 -0.44572

0.20677 +OI05918 -0.09193 -0.23720 -0.36694

0.30072 0.16266 +0.01497 -0.13360 -0.27352

0.38544 0.2 6194 +0.12315 -0.02310 -0.16782

0.45522 0.35129 0.22716 +0.08947 -0.05374

0.50355 0.42375 0.31998 0.19740 +0.06277

5.0

-0.57025

-0.53023

-0.47182

-0.39516

-0.30146

-0.19341

-0.07580

X

W (- 0.5, X)

W(-0.4,~)

71 5

PARABOLIC CYLINDER FUNCTIONS

Table 19.2

x

W(-1.0,-2)

W (-0.5,

-2)

W ( - 0.4,-2)

0.0 0.1 0.2 0.3 0.4

0.73148 0.79607 0.85267 0.90067 0.93946

0.75416 0.81697 0.87241 0.91990 0.95892

0.77982 0.84073 0.89490 0.94182 0.98099

0.80879 0.86771 0.92053 0.96682 1.00612

0.84130 0.89814 0.94958 0.99522 1.03467

0.87718 0.93193 0.98201 1.02707 1.06677

0.91553 0.96827 1.01711 1.06178 1.10197

0.5 0.6 0.7 0.8 0.9

0.96849 0.98722 0.99521 0.99202 0.97734

0.98892 1.00940 1.01990 1.01997 1.00923

1.01192 1.03413 1.04713 1.05048 1.04374

1.03797 1.06191 1.07745 1.08414 1.08151

1.06749 1.09323 1.11143 1.1 2160 1.12325

1.10070 1.12843 1.14951 1.16343 1.16966

1.13729 1.16736 1.19170 1.20981 1.22114

1.0 1.1 1.2 1.3 1.4

0.95092 0.91262 0.86244 0.80055 0.72729

0.98738 0.95418 0.90952 0.85341 0.78603

1.02655 0.99859 0.95962 0.90954 0.84835

1.06912 1.04657 1.01355 0.96978 0.91515

1.11589 1.09904 1.07228 1.03523 0.98760

1.16769 1.15695 1.13693 1.10714 1.06714

1.22511 1.22112 1.20855 1.18680 1.15529

1.5 1.6 1.7 1.8 1.9

0.64322 0.54911 0.44603 0.33528 0.21849

0.70774 0.61912 0.52099 0.41443 0.30081

0.77623 0.69355 0.60091 0.49914 0.38936

0.84963 0.77341 0.68684 0.59053 0.48532

0.92923 0.86006 0.78025 0.69014 0.59032

1.01659 0.95525 0.88304 0.80004 0.70659

1.11351 1.06102 0.99750 0.92281 0.83697

2.0 2.1 2.2 2.3 2.4

+0.09757 -0.02528 -0.14758 -0.26660 -0.37941

0.18179 +0.05934 -0.06427 -0.18651 -0.30459

0.27298 0.15171 +0.02758 -0.09709 -0.21967

0.37236 0.25309 0.12930 +0.00305 -0.12323

0.48166 0.36531 0.24278 +0.11588 -0.01322

0.60326 0.49090 0.37070 0.24419 +0.11327

0.74025 0.63319 0.51665 0.39182 0.26028

2.5 2.6 2.7 2.8 2.9

-0.48297 -0.57415 -0.64990 -0.70733 -0.74387

-0.41552 -0.5 1623 -0.60356 -0.67449 -0.72615

-0.33731 -0.44698 -0.54551 -0.62975 -0.69663

-0.24685 -0.36487 -0.47416 -0.57149 -0.65363

-0.14203 -0.26774 -0.38730 -0.49748 -0.59492

-0,01983 -0.15248 - 0.281 78 -0.40451 -0.51729

+0.12398 -0.01472 -0.15309 -0.28802 -0.41615

3.0 3.1 3.2 313 3.4

-0.75737 -0.74633 -0.70996 -0164841 -0.56281

-0.75605 -0.76219 -0.74323 -0.69863 -0.62881

-0.74331 -0.76738 -0.76692 -0.74077 -0.68862

-0.71748 -0.76019 -0.77937 -0.77320 -0.74065

-0.67629 -0.73841 -0.77841 -0.79386 -0.78300

-0.61660 -0.69897 -0.761 08 -0.79994 -0.81309

-0.53384 -0.63739 -0.72310 -0.78743 -0.82721

3.5 3.6 3.7 3.8 3.9

-0.45542 -0.32961 -0.18992 -0.04191 +0.10799

-0.53525 -0.42059 -0.28860 -0.14423 +0.00657

-0.61114 -0.51016 -0.38867 -0.25086 -0.10208

-0.68160 -0159701 -0.48899 -0.36092 -0.21739

-0.74490 -0167961 -0.58833 -0.47349 -0.33883

-0.79874 -0.75603 -0.68515 -0.58750 -0.46582

-0.83985 -0182349 -0.77725 -0.70141 -0.59756

4.0 4.1 4.2 4.3 4.4

0.25266 0.38471 0.49679 0.58208 0.63477

0.15702 OI29976 0.42722 0.53205 0.60759

+0.05134 0.20225 0.34303 0.46597 0.56372

-0.06416 +0109203 0.24366 0.38285 0.50171

-0.18934 -0103124 +0.12831 0.28140 0.41981

-0.32421 -0.16811 -0.0 0420 +0.15987 0.31572

-0.46872 -0.31938 -0.15545 +O. 015 87 0.18634

4.5 4.6 4.7 4.8 4.9

0.65055 0.62708 0.56440 0,46513 0.33464

0.64841 0.65075 0.61301 0.53614 0.42379

0.62979 0.65910 0.64846 0.5 9705 0.50672

0.59285 0.64997 0.66833 0.64531 0.58085

0.53543 0.62083 0.66982 0.67800 0.64328

0.45473 0.56851 0.64950 0.69154 0.69050

0.34702 0.48877 0.60280 0.68125 0.71794

5.0

0.18091

0.28240

0.38215

0.47771

0.56635

0.64481

0.70889

716

PARABOLIC CYLINDER FUNCTIONS

Table 19.2 X

W(-0.3,~)

W ( - 0.2,5)

W(-0.1,s)

W(0,X)

w (0.1,x)

W(0.2,x)

w(0.3,x)

0.0 0.1 0.2 0.3 0.4

0.9 5 411 0.9 0030 0.84377 0.78461 0.72293

0.98880 0.9372 5 0.88381 0.82851 0.77137

1.01364 0.96381 0.91299 0.86116 0.80828

1.02277 0.97388 0.92496 0.87595 0.82673

1.01364 0.96480 0.91691 0.86984 0.82344

0.98880 0.93920 0.89145 0.84540 0.80084

0.95411 0.90311 0.85480 0.80896 0.76536

0.5 0.6 0.7 0.8 0.9

0.65878 0.59225 0.52341 0.45236 0.37924

0.71237 0.65150 0.58875 0.52410 0.45756

0.75426 0.69902 0.64245 0.58445 0.52493

0.77719 0.72716 0.67647 0.62496 0.57244

0.77753 0.73192 0.68637 0.64067 0.59459

0.75757 0.71533 0.67388 0.63296 0.59228

0.72375 0.68386 0.64540 0.60809 0.57163

1.0 1.1 1.2 1.3 1.4

0.30421 0.22751 0.14946 +0.07042 -0.00912

0.3891 8 0.31906 0.24734 0.17425 0.10007

0.46383 0.40111 0.33677 0.27090 0.20361

0.5 1877 0.46381 0.40744 0.34961 0.29032

0.54790 0.50038 0.45186 0.40217 0.35118

0,55160 0,51063 0.46915 0.42691 0.38374

0.53573 0.50010 0.46446 0.42854 0.39209

1.5 1.6 1.7 1.8 1.9

-0.08857 -0.16725 -0.24435 -0.31894 -0.38999

+0.02522 -0.04982 -0.12443 -0.19788 -0.26933

0.13514 +0.06577 -0.00407 -0.07387 -0.14299

0.22960 0.16760 0.10454 +0.04073 -0.02340

0.29883 0.24510 0.19006 0.13384 0.07667

0.33945 0.29393 OI24713 0.19904 0.14975

0.35491 0.31679 0.27761 0.23725 0.19569

2.0 2.1 2.2 2.3 2.4

-0.45633 -0.51674 -0.56989 -0.61444 -0.649 03

-0.33779 - 0.4021 9 -0.46135 -0.51400 -0.55882

-0.21066 -0.27600 -0.33802 -0.39560 -0.44755

-0.08731 -0.15034 -0.21170 -0.27048 -0.32569

+0.01891 -0.03902 -0.09655 -0.15300 -0.20756

0.09941 +0.04828 -0.00327 -0.05478 -0.10567

0.15296 0.10917 0.06450 +0.01926 -0.02617

2.5 2.6 2.7 2.8 2.9

-0.67233 -0.68311 -0.68033 -0.66313 -0.63097

-0.59448 -0.61966 -0.63315 -0.63385 -0.62088

-0.49261 -0.52947 -0.55686 -0.57356 -0.57846

-0.37619 -0.42082 -0.45833 -0.48749 -0.5071 0

-0.25934 -0.30731 -0.35040 -0.38745 -0.41729

-0.15523 -0.20267 -0.24709 -0.28749 -0.32283

-0.07129 -0.11 551 -0.15811 -0.19829 -0.23518

3.0 3.1 3.2 3.3 3.4

-0.58369 -0.52157 -0.44541 -0.35655 -0.25697

-0.59365 -0.55190 -0.49584 -0.42 613 -0.34402

-0.57063 -0.54943 -0.51451 -0.46594 -0.40427

-0.51607 -0.51344 -0.4985 1 -0.47084 -0.43039

-0.43878 -0.45085 -0.45256 -0.44315 -0.42215

-0.35203 -0.37401 -0.38777 -0.39239 -0.38713

-0.26783 -0.29526 -0.31648 -0.33055 -0.33663

3.5 3.6 3.7 3.8 3.9

-0.14924 -0.03654 +0.07742 0.18846 0.29213

-0.25134 -0.15050 -0.04453 +0.06302 0.16814

-0.33055 -0.24643 -0.15413 -0.05645 +0.04330

-0.37754 -0.31318 -0.23871 -0.15612 -0.06794

-0.38941 -0.34517 -0.29013 -0.22 549 -0.15299

-0.37148 -0.34523 -0.30852 -0.26190 -0.20639

-0.33401 -0.32218 -0.30091 -0.27027 -0.23072

4.0 4.1 4.2 4.3 4.4

0.38382 0.45904 0.51364 0.54413 0.54793

0.2 6651 0.35370 0.42535 0.47744 0.50658

0.14132 0.23354 0.31572 0.38368 0.43357

+0.02278 0.11257 0.19762 0.27395 0.33764

-0.07486 +0.00615 0.08689 0.16386 0.23342

-0.14349 -0.07518 -0.00389 +0.06754 0.13597

-0.18313 -0.12880 -0.06948 -0.00733 +0.05511

4.5 4.6 4.7 4.8 4.9

0.52370 0.47151 0.39312 0.29197 0.17327

0.51029 0.48726 0.43762 0.36308 0.26703

0.46212 0.46690 0.44663 0.40138 0.33274

0.38503 0.41300 0.41921 0.40237 0.36248

0.2 9194 0.33601 0.36270 0.36981 0.35608

0.19809 0.25059 0.29037 0.31476 OI32171

0.11504 0.16948 0.21549 0.25027 0.27144

5.0

0.04376

0.15455

0.24393

0.30095

0.32145

0.31009

0.27719

71 7

PARABOLIC CYLINDER FUNCTIONS

Table 19.2 X

W(-O.3,-Z)

W ( - 0 . 2 , - ~ ) W(-O.l,-z)

W(0,-4

W(0.1,-2)

W(0.2,--5)

"(0.3, -z)

0.0 0.1 0.2 0.3 0.4

0.95411 1.00506 1.05296 1.09759 1.13868

0.98880 1.03835 1.08581 1.13097 1.17362

1.01364 1.06245 1.11016 1.15665 1.20172

1.02277 1.07165 1.12050 1.16924 1.21771

1.01364 1.06348 1.11435 1.16622 1.21899

0.98880 1.04037 1.09399 1.14968 1.20741

0.95411 1.00797 1.06483 1.12477 1.18782

0.5 0.6 0.7 0.8 0.9

1.17589 1.20884 1.23706 1.2 6006 1.27725

1.21344 1.25007 1.28307 1.31193 1.33606

1.24510 1.28645 1.32534 1.36129 1.39368

1.26568 1.31285 1.35884 1.40315 1.44521

1.27248 1.32644 1.38053 1.43429 1.48719

1.26706 1.32845 1.39129 1.45520 1.51968

1.25396 1.32307 1.39494 1.46928 1.54567

1.0 1.1 1.2 1.3 1.4

1.28802 1.29171 1.28761 1.27501 1.25320

1.35480 1.36744 1.37321 1.37129 1.36083

1.42185 1.44504 1.46241 1.47304 1.47598

1.48433 1i51974 1.55054 1.57575 1.59429

1.53855 1.58760 1.63341 1.67498 1.71113

1.5 8412 1.64775 1.70967 1.76885 1.82408

1.62356 1.70224 1.78087 1.85841 1.93366

1.5 1.6 1.7 1.8 1.9

1.22150 1.17926 1.12596 1.06115 0.98458

1.34098 1.31091 1.26983 1.21705 1.15200

1.47020 1.45469 1.42841 1.39039 1.33973

1.60502 1.60672 1.59813 1.57800 1.54509

1.74059 1.76201 1.77390 1.77474 1.76299

1.87401 1.91713 1.95181 1.97628 1.98870

2.00522 2.07150 2.13072 2.18093 2.22000

2.0 2.1 2.2 2.3 2.4

0.89620 0.79618 0.68503 0.56357 0.43300

1.07426 0.98365 0.88026 0.76448 0.63710

1.27565 1.19757 1.10510 0.99819 0.87711

1.49825 1.43644 1.35882 1.26478 1.15405

1.73709 1.69557 1.63706 1.56041 1.46471

1.98714 1.96968 1.93446 1.87972 1.80390

2.24569 2.25565 2.24752 2.21894 2.16770

2.5 2.6 2.7 2.8 2.9

0.29492 0.15140 +0.00489 -0.14168 -0.28503

0.49932 0.35277 0.19959 +0.04242 -0.11563

0.74256 0.59571 0.43825 0.27241 +0.10100

1.02673 0.88342 0.72523 0.55388 0.37173

1.34942 1.21444 1.06021 0.88776 0.69887

1.70575 1.58440 1I43949 1.27129 1.08078

2.09177 1.98946 1.85956 1.7 014 0 1.51507

3.0 3.1 3.2 3.3 3.4

-0.42150 -0.54722 -0.65815 -0.75027 -0.81974

-0.27098 -0.41967 -0.55742 -0.67978 -0.78229

-0.07258 -0.24442 -0.41011 -0.56487 -0.70368

+0.18182 -0.01213 -0.20574 -0.39404 -0.57158

0.49606 0.28264 +0.06279 -0.15855 -0.37567

0.86979 0.64105 0.39827 + 0.14618 -0.10952

1.30151 1.06267 0.80159 0.52249 +0.23083

3.5 3.6 3.7 3.8 3.9

-0.86311 -0.87754 -0.86098 -0.81248 -0.73233

-0.86067 -0.91101 -0.93010 -0.9 1559 -0.86631

-0.82147 -0.91331 -0.97470 -1.00185 -0.99193

-0.73259 -0.67118 -0.98158 -1.05844 -1.0971 9

-0.58228 -0.77162 -0.93674 -1.07077 -1.16728

-0.36221 -0.60449 -0.82836 -1.02554 -1.18779

-0.06670 -0.36232 -0.64721 -0.91187 -1.14634

4.0 4.1 4.2 4T3 4.4

-0.62227 -0.48559 -0.32717 -0.15346 +0.02771

-0.78249 -0.66595 -0.52024 -0.35070 -0.16437

-0.94343 -0.85640 -0.73270 -0.57611 -0.39249

-1.09434 -1.04786 -0.95753 -0.82515 -0.65483

-1.22069 -1.22662 -1.18240 -1.08743 - 0.943 50

-1.30732 -1.37730 -1.39231 -1.34891 -1.24610

-1.34070 -1.48554 -1.57256 -1.59514 -1.5 49 01

4.5 4.6 4.7 4.8 4.9

0.20739 OI37594 0.52351 0.64069 0.71919

+ 0.0301 4

OI22299 0.40359 0.56113 0.68534

-0.18962 +0.02291 0.23414 0.43218 0.60494

-0.45301 -0.22843 +o. 00810 0.24408 0.46598

- 0.75 508 -0152942 -0.27649 -0.00874 +0.25940

-1.08573 -0T87285 -0.61582 -0.32626 -0.01876

-1.4 3285 -1.24877 -1.00271 -0.70462 -0.36835

5.0

0.75259

0.76721

0.74090

0.65996

0.51219

+ 0.2 8970

-0.01132

[i-p]

718

PARABOLIC CYLINDER FUNCTIONS

Table 19.2

W(0.4,~) W(0.5, x)

W(0.6,~)W(0.7, x)

W(0.8,~)

W(O.9,x)

W(1.0,x)

0.1 0.2 0.3 0.4

0.91553 0.86271 0.81331 0.76709 0.72376

0.87718 0.82232 0.77155 0.72456 0.68104

0.84130 0.78433 0.73205 0.68408 0.64007

0.80879 0.74973 0.69590 0.64687 0.60222

0.77982 0.71874 0.66339 0.61328 0.56794

0.75416 0.69116 0.63436 0.58321 0.53718

0.73148 0.66667 0.60852 0.55639 0.50970

0.5 0.6 3.7 0.8 0.9

0.68304 0.64462 0.60820 0.57347 0.54011

0.64064 0.60305 0.56793 0.53495 0.50380

0.59964 0.56244 0.52810 0.49629 0.46666

0.56155 0.52446 0.49058 0.45952 0.43095

0.52692 0.48979 0.45614 0.42558 0.39774

0.49578 0.45853 0.42499 0.39476 0.36745

0.46791 0.43051 0.39703 0.36704 0.3401 3

1.0 1.1 1.2 1.3 1.4

0.50782 0.47630 0.44523 0.41435 0.38338

0.47414 0.44567 0.41808 0.39108 0.36438

0.43889 0.41266 0.38765 0.36358 0.34015

0.40452 0.37992 0.35682 0.33494 0.31399

0.37228 0.34888 0.32720 0.30697 0.28790

0.34271 0.32020 0.29960 0.28063 0.26299

0.31594 0.29412 0.27435 0.25634 0.23981

1.5 1.6 1.7 1.8 1.9

0.35206 0.32018 0.28752 0.25395 0.21934

0.33771 0.31084 0.28354 0.25561 0.22689

0.31709 0.29416 0.27111 0.24773 0.22384

0.29370 0.27382 0.25410 0.23433 0.21430

0.26973 0.25219 0.23506 0.21812 0.20115

0.24643 0.23071 0.21559 0.20085 0.18629

0.22451 0.21019 0.19662 0.18361 0.1 7094

2.0 2.1 2.2 2.3 2.4

0.18363 0.14682 0.10899 0.07029 +0.03094

0.19726 0.16665 0.13504 0.10248 0.06908

0.19927 0.17390 0.14767 0.12054 0.09255

0.19384 0.17280 0.15107 0.12857 0.10528

0.18398 0.16644 0.14841 Oil2976 0.11045

0.17173 0.15700 0.14195 Oil2647 0.11045

0.15845 0.14595 0.13331 0.12038 0.10707

2.5 2.6 2.7 2.8 2.9

-0.00872 -0.04827 -0.08719 -0.12486 -0.16058

0.03504 +0.00063 -0.03378 -0.06773 -0.10069

0.06378 0.03440 +0.00466 -0.02513 -0,05457

0.08121 0.05645 0.03113 +0.00547 -0.02025

0.09043 0.06972 0.04840 0.02659 +0.00447

0.09385 0.07662 0.05879 0.04042 0.02163

0.09330 0.07900 0.06416 0.04879 0.03296

3.0 3.1 3.2 3.4

-0.19356 -0.22295 -0.24788 -0.26746 -0.28083

-0.13202 -0.16105 -0.18700 -0.20910 -0.22656

-0.08319 -0.1 1043 -0.13568 -0.15826 -0.17749

-0.04569 -0.07041 -0.09392 -0.11569 -0.13511

-0.01 769 -0.03960 -0.06087 -0.08106 -0.09969

+0.00259 -0.01649 -0.03531 -0.05355 -0.07080

0.01677 +0.00038 -0.01602 -0.03216 -0.04774

3.5 3.6 3.7 3.8 3.9

-0.28722 -0.2~8 -0.27664 -0.25895 -0.23299

-0.23861 -0.24455 -0.24381 -0.23596 -0.22079

-0.19265 -0.20307 -0.20814 -0.20735 -0.20033

-0.15158 -0.16446 -0117317 -0.1771 8 -0.17604

-0.11623 -0.13014 -0.14088 -0.14793 -0.15084

-0.08664 -0.10061 -0.11222 -0.12101 -0.12652

-0.06242 -0.07581 -0.08750 -0.09707 -0.10411

4.0 4.1 4.2 4.3 4.4

-0.19913 -0.15813 -0.11115 -0.05975 -0.00585

-0.19835 -0.16901 -0.13343 -0.09266 -0.04811

-0.18692 -0.16717 -0.14143 -0.11032 -0.07481

-0.16946 -0.15730 -0.13965 -0.11684 -0.08947

-0.14922 -0.14284 - 0.1 3162 -0.11566 -0.09531

-0.12836 -0.12624 -0.11996 -0.10948 -0.09494

-0.10824 -0.10912 -0.10653 -0.10030 -0.09046

4.5 4.6 4.7 4.8 4.9

+0.04828 0.10016 0.14714 0.18659 0.21607

-0.00149 +0.04518 0.08968 0.12967 0.16286

-0.03614 +0.00411 0.04416 0.08203 0.11567

-0.05843 -0.02485 +0.00985 0104406 0.07604

-0.07112 -0I04392 -0.01477 +0.01506 0.04414

-0.07669 -0105525 -0.03141 -0,00614 +0.01943

-0.07716 -0i06075 -0.04174 -0.02086 +0.00100

5.0

0.23350

0.18712

0.14307

0.10399

0.07092

0.04399

0.02281

x 0.0

3.3

"-:'"I

719

PARABOLIC CYLINDER FUNCTIONS

Table 19.2

W(O.4, - x )

W(0.5,- x )

W(0.6,-2)

W(0.7,-2)

W(0.8,-2)

W(O.9, -2)

W(l.O,-X)

0.0 0.1 0.2 0.3 0.4

0.91 5 53 0.97201 1.03235 1.09671 1.16520

0.87718 0.93642 1.00031 1.06911 1.14300

0.841 30 0.90331 0.97072 1.04386 1.12302

0.80879 0.87352 0.94433 1.02166 1.10591

0.77982 0.84714 0.92122 1.00258 1.09173

0.75416 0.82396 0.90115 0.98636 1.08022

0.73148 0.80361 0.88375 0.97265 1.07106

0.5 0.6 0.7 0.8 0.9

1.23789 1.31475 1.39567 1.48046 1.56879

1.22215 1.30664 1.39648 1.49158 1.59174

1.20846 1.30040 1.39896 1.50419 1.61602

1.19746 1.29663 1.40371 1.51888 1.642 2 5

1.18917 1I29538 1.41079 1.53574 1.67051

1.18338 1.29644 1.42000 1.55459 1.70068

1.17975 1.29949 1.43106 1.57519 1.73254

1.0 1.1

1.2 1.3 1.4

1.6602 1.7541 1.8497 1.9460 2.0418

1.6966 1.8057 1.9184 2.0337 2.1506

1.7343 1.8586 1.9884 2.1230 2.2613

1.7738 1.9133 2.0603 2.2144 2.3746

1.8153 1.9700 2.1345 2.3083 2.4909

1.8586 2.0286 2.2107 2.4048 2.6102

1.9037 2.0891 2.2891 2.5037 2.7327

1.5 1.6 1.7 1.8 1.9

2.1358 2.2263 2.3115 2.3891 2.4570

2.2677 2.3833 2.4956 2.6023 2.7009

2.4020 2.5437 2.6843 2.8216 2.9529

2.5397 2.7083 2.8785 3.0480 3.2141

2.6811 2.8777 3.0788 3.2823 3.4854

2.8264 3.0520 3.2856 3.5249 3.7674

2.9756 3.2316 3.4991 3.7762 4.0605

2.0 2.1 2.2 2.3 2.4

2.5125 2.5529 2.5754 2.5770 2.5548

2.7886 2.8623 2.9188 2.9546 2.9660

3.0752 3.1853 3.2793 3.3532 3.4030

3.3737 3.5231 3.6583 3.7748 3.8678

3.6849 3.8770 4.0573 4.2209 4.3624

4.0097 4.2479 4.4775 4.6931 4.8889

4.3487 4.6368 4.9201 5.1930 5.4490

2.5 2.6 2.7 2.8 2.9

2.5061 2.4283 2.3192 2.1772 2.0013

2.9496 2.9018 2.8196 2.7001 2.5413

3.4241 3.4124 3.3634 3.2734 3.1389

3.9321 3.9626 3.9538 3.9007 3.7984

4.4760 4.5555 4.5944 4.5863 4.5251

5.0582 5.1940 5.2887 5.3346 5.3240

5.6811 5.8811 6.0405 6.1502 6.2008

3.0 3.1 3.2 3.3 3.4

1.7914 1.5484 172746 0.9733 0.6496

2.3419 2.1015 178213 1.5038 1.1529

2.9573 2.7270 2.4478 2.1206 1.7487

3.6430 3.4312 371612 2.8324 2.4466

4.4050 4.2211 379697 3.6486 3.2576

5.2495 5.1041 4.8822 4.5794 4.1934

6.1832 6.0883 5.9081 5.6359 5.2669

X

3.5 3.6 3.7 3.8 3.9

+0.3098 -0.0381 -0.3848 -0.7198 -1.0317

0.7746 +0.3767 -0.0314 -0.4 385 - 0.831 9

1.3369 0.8923 +0.4244 -0.0553 -0.5332

2.0074 1.5210 0.9962 +0.4445 -0.1199

2.7987 2.2767 1.6994 1.0779 +0.4263

3.7241 3.1746 2.5511 1.8636 1.1259

4.7985 4.2315 3.5700 2.8225 2.0016

4.0 4.1 4.2 4.3 4.4

-1.3084 -1.5382 -1.7095 -1.8124 -1.8391

-1.1977 -1.5216 -1.7893 -1.9871 -2.1032

-0.9940 -1.4209 -1.7966 -2.1039 -2.3268

-0.6804 -1.2184 -1.7136 -2.1453 -2.4930

-0.2378 -0.8941 -1.5199 -2.0907 -2.5817

+0.3558 -0.4249 -1.1915 -1.9160 -2.5692

1.1251 +0.2152 -0.701 3 -1.5936 -2.4280

4.5 4.6 4.7 4.8 4.9

-1.7844 -1.6469 -1.4292 -1.1387 -0.7876

-2.1283 -2.0567 -1.8870 -1.6231 -1.2742

-2.4513 -2.4668 -2.3670 -2.1513 -1.8252

-2.7376 -2.8632 -2.8579 -2.7153 -2.4359

-2.9685 -3I2291 -3.3452 -3.3040 -3.0995

-3.1213 -3.5437 -3.8110 -3.9027 -3.8054

-3.1692 -3.7818 -4.2 32 6 -4.4924 -4.5392

5.0

-0.3927

-0.8557

-1.4010

-2.0281

-2.7346

-3.5149

-4.3599

720

PARABOLIC CYLINDER FUNCTIONS

AUXILIARY FUNCTIONS

Table 19.3

The functions 9 , , 02, 93 of 19.10 and 19.23 are needed in Darwin’s expansion and also the function of 19.7 and 19.20. I 0.0 0; 1 0.2 0.3 0.4

0.00000 0; 05008 0.10066 0.15222 0.20521

0.39270 0; 34278 0.29337 0.24498 0.19817

-0.70270 -0; 64181 -0.57855 -0.51304 -0.44540

0.5 0.6 0.7 0.8 0.9

0.26006 0.31713 0.37678 0.43929 0.50492

0.15355 0.11182 0.07387 0.04088 0.01468

-0.37574 -0.30415 -0.23071 -0.15549 -0.07857

E

91

93

7

5

91

92

5.0

5.5506 _

5.1 5.2 5.3 5.4

6.9519 .. _ 7.2093 7.4716 7.7388 8.0109

5.7981 6.0507 6.3084 6.5712

4.2291 4.3511 4.4738 4.5972

5.5 5.6 5.7 5.8 5.9

8. 2880 8.5700 8.8569 9.1487 9.4454

6.8391 7.1120 7.3901 7.6732 7.9614

4.7213 4.8461 4.9716 5.0977 5.2246

._.

__ __

7

4.1079 ..--. .

7

0.57390 0.64640 0.72261 0.80265 0.88666

0.00000 0.01513 0.04341 0.08086 0.12617

0.00000 0.08015 0.16185 0.24502 0.32964

E 6.0 6.1 6.2 6.3 6.4

92

1.0 1.1 1.2 1.3 1.4

9.7471 10.0537 10.3652 10.6817 11.0031

8.2546 8.5530 8.8564 9.1649 9.4784

5.3521 5.4803 5.6092 5.7387 5.8688

1.5 1.6 1.7 1.8 1.9

0.97473 1.06696 1.16344 1,26422 1.36937

0.17866 0.23786 0.30347 0.37527 0.45309

0.41566 0.50304 0.59175 0.68175 0.77300

6.5 6.6 6.7 6.8 6.9

11.3295 11,6608 11.9970 12.3382 12.6843

9.7970 10.1207 10.4494 10.7832 11.1220

5.9996 6.1310 6.2631 6.3958 6.5290

2.0 2.1 2.2 2.3 2.4

1.47894 1.59299 1.71155 1.83466 1.96236

0.53679 0.62626 0.72142 0.82220 0.92853

0.86549 0.95917 1.05403 1.15004 1.24716

7.0 7.1 7.2 7.3 7.4

13.0354 13.3914 13.7524 14.1183 14.4892

11.4659 11.8148 12.1688 12.5278 12.8919

6.6629 6.7974 6.9325 7.0682 7.2045

2.5 2.6 2.7 2.8 2.9

2.09467 2,23163 2.37325 2.51956 2.67058

1.04036 1.15764 1.28034 1.40843 1.54187

1.34539 1.44470 1.54506 1.64646 1.74888

7.5 7.6 7.7 7.8 7.9

14.8651 15.2459 15.6316 16.0223 16.4180

13.2610 13.6352 14.0144 14.3987 14.7880

7.3414 7.4789 7.6169 7.7555 7.8947

3.0 3.1 3.2 3.3 3.4

2.82632 2.98681 3.15205 3.32207 3.49688

1.68063 1.82470 1.97406 2.12867 2.28853

1.85229 1.95669 2.06206 2.16837 2.27562

8.0 8.1 8.2 8.3 8.4

16.8186 17.2242 17.6348 18.0503 18.4708

15.1823 15.5817 15.9861 16.3956 16.8101

8.0344 8.1747 8.3155 8.4569 8.5989

3.5 3.6 3.7 3.8 3.9

3.67648 3.86089 4.05011 4.24416 4.44305

2.45363 2.62394 2.79946 2.98017 3.16606

2.38378 2.49285 2.60281 2.71365 2.82536

8.5 8.6 8.7 8.8 8.9

18.8962 19.3266 19.7620 20.2024 20.6477

17.2296 17.6542 18.0838 18.5184 18.9581

8.7413 8.8844 9.0279 9.1720 9.3166

4.0 4.1 4.2 4.3 4.4

4.64678 4.85537 5.06880 5.28711 5.51028

3.35712 3.55335 3.75474 3.96127 4.17295

2.93791 3.05131 3.16554 3.28058 3.39643

9.0 9.1 9.2 9.3 9.4

21.0980 21.5532 22.0135 22.4787 22.9488

19.4028 19.8525 20.3073 20.7671 21.2319

9.4617 9.6074 9.7535 9.9002 10.0474

4.5 4.6 4.7 4.8 4.9

5.73833 5.97126 6.20908 6.45178 6.69938

4.38976 4.61169 4.83875 5.07093 5.30822

3.51308 3.63051 3.74872 3.86770 3.98743

9.5 9.6 9.7 9.8 9.9

23.4240 23.9041 24.3892 24.8792 25.3742

21.7017 22.1766 22.6565 23.1414 23.6314

10.1951 10.3433 10.4920 10.6411 10.7908

5.0

6.95188

5.55062

4.10792

10.0

25.8742

24.1264

10.9410

91

92

7

91

( 4)6 (-4)7 (-411 (-4)6 (-3)2 (-4)3 L-31 [ 3 I [ 3 I [ 4 1 [ 6 1 E 3 1 When interpolating for o2 and 93 for c near unity, it is better to interpolate for I and then use

20. Mathieu Functions GERTRUDE BLANCH l

Contents Mathematical Properties . . . . , . . . . . . . . . . . . . . 20.1. Mathieu’s Equation . . . . . . . . . . . . . . . . 20.2. Determination of Characteristic Values . . . . . . . . 20.3. Floquet’s Theorem and Its Consequences . , . . , . . 20.4. Other Solutions of Mathieu’s Equation . . . . . . . . 20.5. Properties of Orthogonality and Normalization . . . . . 20.6. Solutions of Mathieu’s Modified Equation for Integral Y . 20.7. Representations by Integrals and Some Integral Equations 20.8. Other Properties . . . . . , . , . . . . , . . . . . 20.9. Asymptotic Representations . . , . . . . , , . . . . 20.10. Comparative Notations . . . . . . . . . . . . . . . References . . . . , . , , . , . . . , . . . . , . . . . . .

.

.

. . . . .

. .

. . .

Table 20.1. Characteristic Values, Joining Factors, Some Critical Values ( O < q < c o ) . . . . . . . . . . . . . . . . . . . . .

Page

722 722 722 72 7 730 732 732 735 73 8 740 744 74 5 748

Even Solutions

Odd Solutions

q=0(5)25, a,+2q-(4rS2)

8D or S

&, br+zq-(4~--2) h

q-*=. 16 (- .04)0,

8D

r=O, 1, 2, 5, 10, 15 Table 20.2. Coefficients A , and B ,

. . .

,

., . . .. . .. . . .

q=5, 25; T = O , I, 2, 5, 10, 15,

1

750

9D

Aeronautical Research Laboratories, Wright-Patterson Air Force Base, Ohio.

72 1

20. Mathieu Functions Mathematical Properties 20.1. Mathieu’s Equation Canonical Form of the Differential Equation ~

20.1.1 Mathieu’s Modified DifferentialEquation

%-(a-2q du2

20.1.2

(v=iu, y=f)

cosh 2u)f=O

Relation Between Mathieu’s Equation and the Wave Equation for the Elliptic Cylinder

The wave equation in Cartesian coordinates is

A solution W is obtainable by separation of variables in elliptical coordinates. Thus, let

x= p

p

cosh U cos v; y=

p

sinh

U

sin U; z= z;

a positive constant; 20.1.3 becomes

20.1.4

D2W+

* bz2

2 ZU’ b2W p2 (cosh 2u-cos 2v) ( & + ~ ) + k 2 w = o

where a is a constant. The above are equivalent to 20.1.1 and 20.1.2. The constants c and a are often referred to as separation constants,due to the role they play in 20.1.5 and 20.1.6. For some physically important solutions, the function g must be periodic, of period A or 2 ~ . It can be shown that there exists a countably infinite set of characteristic values a,(q) which yield even periodic solutions of 20.1.1; there is another countably infinite sequence of characteristic values b,(q) which yield odd periodic solutions of 20.1.1. It is known that there exist pei’odic solutions of period k r , where k is any positive integer. In what follows, however, the term characteridc value will be reserved for a value associated with solutions of period A or 2~ only. These characteristic values are of basic importance to the general theory of the differential equation for arbitrary parameters a and q. An Algebraic Form of Mathieu’s Equation

20.1.7

Assuming a solution of the form

w =co(z)f(u)9(v)

Relation to Spheroidal Wave Equation

and substituting the above into 20.1.4 one obtains, after dividing through by W ,

*

20.1.8

Thus, Mathieu’s equation is a special case of c=a+2q.

20.1.8, with E = - * ,

where .~ 2 * G= p2 (cosh 2u-cos

Since z, that

U ,v

20.2. Determination of Characteristic Values

2v)

are independent variables, it follows

20.2.1

20.1.5

U,

v

d2f 1 (k2-c) * a=- du2f -+ p z cosh 2 u 2 722

m-0

COS

mz+ B, sin mz)

20.2.2 m

m=-2

20.1.6

y=C (Am

where B, can be taken as zero. If the above is substituted into 20.1.1 one obtains

where c is a constant. Again, from the fact that G=c and that are independent variables, one sets

A solution of 20.1.1 with v replaced by z, having period A or 2 r is of the form

+

[(a-~2)Am-~(Am-2+Am+~)l cm m z m

m=-1

[(a-m2)Bm-q(Bm-2+Bm+2)] sin mz=O A-m,B-,=O

‘see page U.

m>o

MATHIEU FUNCTIONS

Equation 20.2.2 can be reduced to one of four simpler types, given in 20.2.3 nnd 20.2.4 below

eo

y l = C Bzm+,sin (2m+p)z1

20.2.4

m=O

p=O or 1

If p=Ol the solution is of period a ; if p = l , the solution is of period 2a.

20.2.18

723 Vz=Go4, for odd solutions of period

A,

along with 20.2.15 These three-term recurrence relations among the coefficients indicate that every Gmcan be developed into two types of continued fractions. Thus 20.2.15 is equivalent to 20.2.19

Recurrence Relations Among the Coefficients

Even solutions of period a: 20.2.5

a A o-qA, =0

20.2.6

(a-4)Az- q(2Ao+A4) =0

20.2.7

(a-mmZ)Am-q(Am~z+Am+z) =O

(m1 3 )

where

a = d =0; cpo= 2 , if Gm+,=A2r/A2s-2

Even solutions of period 2n:

C P ~ = ~ = ( P O = O , if Gm+2=B2a/B2s-2

(a-l)Al-p(Al+A3)=0,

20.2.8 ‘

po=d=I, if Gm+2=A2a+i/Aa,-i CP= ~ d=+o = 1 if Gm+a=Bzs+lt&-t CPI=--~;

along with 20.2.7 for m r 3 .

1

Odd solutions of period a:

(a-4) Bz- qB4=0

20.2.9

*

20.2.10 (U-m*)B,-q(Bm-,+Bm+Z)=O

(m 2 3 )

Odd solutions of period 2a:

+

20.2.11

(a- l)Bl q(Bl-B3)=0, along with 20.2.10 for m 2 3 . Gem=Am/An-z, Gom=Bm/Bm-z;

Gm=Gem or Gom when the same operations apply to both, and no ambiguity is likely to arise. Further let V m = (a-m2)/q.

20.2.13

Equations 20.2.5-20.2.7 are equivalent to 20.2.14 20.2.15

a=a,, associated with even periodic solutions

a =b,, associated with odd periodic solutions If 20.2.19 is suitably combined with 20.2.13-20.2.18

Let 20.2.12

The four choices of the parameters ql, 'pal d correspond to the four types of solutions 20.2.320.2.4. Hereafter, it will be convenient to separate the Characteristic values a into two major subsets:

2

Ge2= V,; Ge4= Vz--Ge2 Gm=I/(Vm-Gm+z) (m>3),

for even solutions of period Similarly

A.

20.2.16 V,-1=Ge3; for even solutions of period 2a, along with 20.2.15 20.2.17 Vl+l=Go,, for odd solutions of period 27r, along with 20.2.15 *See page U.

there result four types of continued fractions, the roots of which yield the required characteristic values 20.2.21

2 vo--vz-

1 1 --. . . = O v4-

vg-

Roots: azr

20.2 22

1 -1 1 v,-1-- Vav,- v,- . . . = O 1 1 1 20.2.23 Vs---. . . = O v,v,v4-

Roots:

~2,+l

Roots: b27

20.2.24

V,+l--

1 v3-

1

1

--. , . = O

vs- v,-

Roots:

bi,+l

If a is a root of 20.2.21-20.2.24, then the corresponding solution exists and is an entire function of z, for general complex values of p. If p is real, then the Sturmian theory of second order linear differential equations yields the

724

MATHIEU FUNCTIONS

following: (a) For a fixed real p, characteristic values a, and b, are real and distinct, if p#O; aoO and ar(a>, br(d 8 ~ proach r2 &s p approaches zero. (b) A solution of 20.1.1 associated with a, or b, has T zeros in the interval 0 Iz < r , (p real). (c) The form of 20.2.21 and 20.2.23 shows that if azr is a root of 20.2.21 and p is different from zero, then azr cannot be a root of 20.2.23; similarly, no root of 20.2.22 can be a root of 20.2.24 if q # O . It may be shown from other considerations that for a given point (a, p) there can be at most one periodic solution of period 7r or 27r if p#O. This no longer holds for solutions of period ST, 8 1 3 ; for these all solutions are periodic, if one is.

I

Power Series for Characteristic Value8

20.2.25 -

ao(n),-Pz+--2

7q4 29qe 68687pS 128 2304'18874368'

'

al(-q) =I- p--+----p2 a" p4 1lqS +- 49qe 8 64 1536 36864 589824 bi(q) .-.

+...

---55p7

9437184 35389440

-

289p6 bz(p)=4--+- pz 5q4 12 13824 79626240 21391pS '458647 142400

+...

5q2 763q4 1002401$ az(p)=4+--12 13824' 79626240

- 1669068401$

Or

br

458647142400' aa(-p)=9+-

b4a)

* * '

--+-+-

P2 a" 13q4 5qb 16 64 20480 16384

1961qs 609p' -23592960+104857600 +...

br (a) =16+---p2

317q' 10049q'I 30 864000+2721600000

+...

433q4 5701qe 30 864000 2721600000

p2 ad (n) =16+-+

+...

--

as(- p) =25+-+--- nz

bdd

-24

L

\

FIGURE 20.1. Characteristic Values U,, b,

~=0,1(1)5

I

11q4 !$ 48 774144 147456

37q6 '891813888

+. .

p" 187q4 5861633$ =36+@+43904000-929~5987200000

+...

p* 187q' 6743617q' =36+@+43904000+92935987200000

+...

*

For r 1 7 , and IpI not too large, aI is approximately equal to br, and the following approximation may be used 20.2.26

(9r4+589+29) pa +M(fl-l)yfl-4) (9-9)'

*

MATHIEU FTJNCTIONS

The above expansion is not limited to integral values of r, and it is a very good approximation for r of the form n++ where n is an integer. I n case of integral values of T = n , the series holds only up to terms not involving r2-n2 in the denominator. Subsequent terms must be derived specially (as shown by Mathieu). Mulholland and Goldstein [20.38] have computed characteristic values for purely imaginary q and found that a. and a2have a common real value for 1q1 in the neighborhood of 1.468; Bouwkamp [20.5] has computed this number as qo= fi 1.46876852 to 8 decimals. For values of +>+pot a. and az are conjugate complex numbers. From equation 20.2.25 it follows that the radius of convergence for the series defining a. is no greater than (qol. It is shown in [20.36], section 2.25 that the radius of convergence for azn(q),n > 2 is greater than 3, Furthermore ar--br=0(qr/rr-1),

725

20.2.28

-cos [ (r-2)z-p(n/2)]

.

4(r-1)

-

32

with p=O for ce,(z, q), p = l for se,(z, q), r>3.

T+co.

Power Series in q for the Periodic Functions (for sufficiently small Iq I)

20.2.27

cos 22+q2

(T-&) 42 COS

)+ .]

cos 62 11 cos 22 128

cel(z, p)=cos

2--

sel(z, q)=sin

2-P

* *

P cos 32 8

8

sin 32

1

-'[ ce&,

-+----sin 72 sin 5 2 sin 32 sin z 9216

p)=cos 2z--q

1152

COS42 (--12

3072

512]+"*

FIGURE 20.2. Even Periodic Mathieu Functions, Orders 0-6 q= 1.

)+...

1 cos 62 19 COS 22 4)+"(=288

726

MATHIEU FUNCTIONS

L 4 t

FIGURE20.3. Odd Periodic Mathieu Functions, Orders 1-5

-8

q= 1. . I O L

FIGCRE20.5. Odd Periodic Mathieu Functions, Orders 1-5 q= 10.

For coefficients associated with above functions 20.2.29 A!(O)=2-*; Ai(O)=B;(O)= 1, T>O

Aj+2",[(-l)"!

Q8/48(T+S)!

s!]

Br+2a

c:+ .

*

.

rs>O, C ;=A ; or B; (T-el)!

-c+.. . p 8

A;-2sor B;+= S!(T-l)! 4s

Asymptotic Expansion for Characteristic Values, q>> 1

Let w=2r+l, p=w4cp, cp real. Then 20.2.30 a -br+1--2q+2w&----

w2+ 1 8

where d 1 =5 -1.0

FIQURE 20.4.

Even Periodic Alathieu Functions, Orders 0-5 q= 10.

+-w34 2+-w9

33 410 405 dz=-+-+ w w3 2

z7d?

727

MATHIEU FUNCTIONS

proportional to F, (- z ) ; the second, independent solution of 20.3.1 then has the form

63 1260 2943 486 d3-w2 - - + - + Tw+ 4- p

d

527 15617 ~ =

69001 41607 ~ + ~

OD

+

p)+C ~20.3.6+ y2=Zcer(z, ~

&k+p

k=O

sin (2k+p)z,

associated with ce,(z, q) - a r ~ 2 4 r + 6 ~ ~ ~ r + ,~ e -p+4 ~ / r ! 20.3.7 (given in [20.36]without proof.)

20.2.31

br+l

20.3. Floquet's Theorem and Its Consequences

Since the coefficients of Mathieu's equation 20.3.1

y"+ (a-2q cos 2z)y=O

are periodic functions of z, it follows from the known theory relating to such equations that there exists a solution of the form 20.3.2

F,(zj= e f v * P ( z ) ,

where v depends on a and q, and P ( z ) is a periodic function, of the same period as that of the coefficients in 20.3.1, namely T . (Floquet's theorem; see [20.16]or [20.22]for its more general form.) The constant v is called the characteristic exponent. Similarly 20.3.3

FY(- z) = e - f

-z )

satisfies 20.3.1 whenever 20.3.2 does. Both Fy(z) and F , ( - z ) have the property

0

Y2=Zser(z, p ) + C f i k + p k=O

COS

(2k+p)z,

associated with se,(z, q)

The coefficients d2,+, andjik+p depend on the corresponding coefficients A , and B,, respectively, of 20.2, as well as on a and p. See [20.30],section (7.50)-(7.51) and [20.58],section V, for details. If v is not an integer, then the Floquet solutions F,(z) and FY(-z) are linearly independent. It is clear that 20.3.2 can be written in the form 20.3.8

5

~ ~ (= 2 )

czkei(v+2k)z.

kr-ar

From 20.3.8 it follows that if v is a proper fraction mllmz, then every solution of 20.3.1 is periodic, and of period at most 2 m 2 . This agrees with results already noted in 20.2; i.e., both independent solutions are periodic, if one is, provided the period is different from T and 2 ~ . Method of Generating the Characteristic Exponent

Define two linearly independent solutions of 20.3.1, for fixed a , q by

20.3.4

y1(0)= 1 ; y; (0) =0.

y ( z + k r ) = C k y ( z ) , y = F , ( z ) or F,(-z), C=efv"for F,(z),C=e+" for F J - z )

20.3.9 y,(O) =0; yi(0) = 1 .

Then it can be shown that Solutions having the property 20.3.4 will hereafler be termed Floquet solutions. Whenever Fy(z) 20.3.10 cos Tv-y,(T)=o and F y ( - z ) are linearly independent, the general solution of 20.3.1 can be put into the form 20.3.11 COS T V - 1 - 2 ~ ; Y2 @=o

6)

20.3.5

y=AF,(z) +BFv(-z)

If A B Z O , the above solution will not be a Floquet solution. I t will be seen later, from the mgthod for determining v when a and q are given, that there is some ambiguity in the definition of V ; namely, v can be replaced by v+2k, where k is an arbitrary integer. This is as it should be, since the addition of the factor exp (2ikz) in 20.3.2 still leaves a periodic function of period ?r for the coefficient of exp ivz. It turns out that when a belongs to the set of characteristic values a, and b, of 20.2, then v is zero or an integer. It is convenient to associate Y=T with a,(p), and v = - r with b r ( p ) ; see [20.36]. In the special case when v is an integer, F,(z) is

Thus v may be obtained from a knowledge of y1( T ) or from a knowledge of both y;

(') and y2 (d

*

For nuIr,erical purposes 20.3.11 may be more desirable because of the shorter range of integration, and hence the lesser accumulation of roundoff errors. Either v, - v , or f v + 2 k (k an arbitrary integer) can be taken as the solution of 20.3.11. Once v has been fixed, the coefficients of 20.3.8 can be determined, except for an arbitrary multiplier which is independent of z. The characteristic exponent can also be computed from a continued fraction, in a manner analogous t o developments in 20.2, if a sufficiently close first approximation to v is available. For

728

MATHIEU FUNCTIONS a

systematic tabulation, this method is considerably faster than the method of numerical integration. Thus, when 20.3.8 is substituted into 20.3.1, there result the following recurrence relations: 20.3.12

VznCzn= c2 n-2

4

+

c2 n + ~

------ -.

I

where 20.3.13

Vzn=[~-(2n+~)2]/q, --CO <%<-CO. When v is complex, the coefficients Vznmay also be complex. As in 20.2, it is possible to generate the ratios Gm=c,/c,-z and H - m= c- ,-&-

from the continued fractions 20.3.14

G,=-

1

1

vm- v,+,- . . . 1

H-,= V-,-2-

1

m20 m20.

V-,-4... From the form of 20.3.13 and the known properties of continued fractions it is assured that for sufficiently large values of ImJ both IGml and IH-ml converge. Once values of G, and H - m are available for some sufficiently large value of m, then the finite number of ratios Gm-2, Gm-4, . . ., Go can be computed in turn, if they exist. Similarly for H-,+z, . . ., Ho.It is easy to show that v is the correct characteristic exponent, appropriate for the point (a, q), if and only if HoGo=l. An iteration technique can be used to improve the value of Y , by the method suggested in t20.31. One coefficient c j can be assigned arbitrarily; the rest are then completely determined. After all the cI become available, a multiplier (depending on q but not on z) can be found to satisfy a prescribed normalization. It is well known that continued fractions can be converted to determinantal form. Equation 20.3.14 can in fact be written as a determinant with an infinite number of rows-a special case of Hill's determinant. See [20.19], [20.36], [20.15], or [20.30] for details. Although the determinant has actually been used in computations where high-speed computers were available, the direct use of the continued fraction seems much less laborious.

FIQURE20.6. Characteristic Ezponent-First Two Stable Regions y=eipzP(z) where P ( z ) i s a periodic function of period T . Dejnition of v; I n j r s t stable region, 0 Iv 51, I n second stable region, 1 5 v 5 2. (Constructed from tabular values supplied by T.Tamir, Brooklyn Polytechnic Institute) a

Special Cases (a, q Real)

Corresponding to q=O, yl=cos &z, y,=sin 4Zz; the Floquet solutions are exp (iaz) and exp (-iaz). As a, q vary continuously in the q-a plane, Y describes curves; Y is real when (q, a), q 2 0 lies in the region between a,(q) and b,+,(q) and

a

FIGURE20.7. Characteristic Ezponent i n First Unstable Region. Differential equation: y"+ ( a - 2 9 cos 22)y=O. The Floquet solution y=ei*zP(z), wh.ere P ( z ) i s a periodic funclion of period T . I n the j r s t unstable region, v=ia; p i s given f o r a 2 - 5 . (Constructed at NBS.)

72 9

MATHIEU FUNCTIONB

FIGURE 20.8

t

-& 2.0

1.8

1.6

I.4

1.2

2q _-

FIQURE 20.9

0

I .o

t

.8

.6

.4

Charts of the Characteristic Exponent. (From 8.J. Zaroodny, An elementary review of the Mathieu-Hill equation of real variable based on numerical solutions, Ballistic Research Laboratory Memo. Rept. 878, Aberdeen Proving Ground, Md., 1955, with permission.)

-s=eirr--constant; in unstable regions

- - - - v =constant; in stable regions - . - . - Lines of constant values of -q.

730

MATHIEU FUNCTIONS Expansions for Small q (120.361 chapter 2)

If v, p are fixed: 20.3.15

a=v2+-

2(v2-1) p2

-P4 +3 2 ((5v2+7) ~~-1)~(~*-4)

(9v4f58v2+29)pe +64(v2- l)’(v2-4) (v2-9)

+ . . . (v#1,2,3).

For the coefficients czj of 20.3.8 20.3.16

(v2+4v+7)? 128(v+l)3(v+2)(v-1)

---

--P c / “-4(v+l)

t

c4/co=q2/32(v+l) (vf2)

+.

+ ,

.. (v#l,2)

*

.

c ~ ~ / c ~ = ( - (v+1)/228~!r ~ ) ~ ~ T (v+~+i)+ . . . 20.3.17 ~ ” ( z ) = c[etvz~

a

.

.

(v not an integer)

For small values of a 20.3.18 FIGURE20.10. Chart of the Characteristic Exponent.

an2 a2r4 24

(1--+-+2

cos v7r=

(From S. J. Zaroodng, An elementarq review of the Mathieu-Hill equation of real variable based on numerical solutions, Ballistic Research Laboratory Memo. Rept. 878, Aberdeen Proving Ground, Md., 1955, with permission)

. ..

-@ 4 [I+a (I-$)+

-s=eim--constant; ----

- .- .-

i n unstable regions v=constant; i n stable regions Lines of constant values of - q .

,

.

.]

(&%+ . . . )+ . . .

+a4

20.4. Other Solutions of Mathieu’s Equation

all solutions of 20.1.1 for real z are therefore bounded (stable) ; v is complex in regions between b , and a,; in these regions every solution becomes infinite at least once; hence these regions are termed “unstable regions”. The characteristic curves a,, b, separate the regions of stability. For negative q, the stable regions are between b 2 r + l and b 2 r + 2 , a2, and az,+l;the unstable regions are between a2,+1 and b 2 r + i , a?, and b 2 r . In some problems solutions are required for real values of z only. In such cases a knowledge of the characteristic exponent v and the periodic function P(z) is sufficient for the evaluation of the required functions. For complex valuw of z, however, the series defining P(z) converges slowly. Other solutions will be determined in the next section; they all have the remarkable property that they depend on the same coefficients c, developed in connection with Floquet’s theorem (except for an arbitrary normalization factor).

Following Erdklyi [20.14], [20.15], define 20.4.1

pk(z)=[e‘* cos (z-b)/cos (a+b)]”Jk(f)

where 20.4.2

(~+b)]?, and J k c f ) is the Bessel function of order k; b is a fixed, arbitrary complex number. By using the recurrence relations for Bessel functions the following may be verified: f=2[9

COS ( 2 - b )

COS

20.4.3

--

fdz2 lpk

2q(cos 2z)pkfP(pk-2+pk+2)

+f%k=O.

It follows that a formal solution of 20.1.1 is given by 20.4.4

Y=

5

n--m

Czntan+r

731

MATHIEU FUNCTIONS

where j satisfies 20.4.2. An examination of the ratios $2n+v/$2n+vshows that

where the coefficients cZnare those associated with Floquet’s solution. In the above, Y may be complex. Except for the special case when v is an integer, the following holds : (P-Zn+u

(02n+u-2--

-

(P-z~+~+z

~2n+u

-4n2 P [COS (z-b)I2

(n+w)

m

1

will be a solution provided \cos (z-b)j>l;

]cos (z+b)J>l.

The above two conditions are necessary even when Y is an integer. Once b is fixed, the regions in which the solutions converge can be readily established. Following [20.36] let 20.4.7

=zp (2);

J,(2)

It follows that 20.4.4 converges absolutely and uniformly in every closed region where

If z is replaced by --iz in 20.4.5 and 20.4.6 solutions of 20.1.2 are obtained. Thus

\cos (z-b)]>d,>l.

20.4.8

There are two such disjoint regions:

yjj)(z)=

(1) J(z-b)>da>O; (Ices (z--b)I>d,>l) (11) Y ( z - b ) -dzd,> 1)

20.4.9

If v is an integer 20.4.4 converges for all vaIues of

y$)(z)=

Various representations are found by specializing b. 2.

(Icos zl>l, larg 2& cos z ( < T )

5

n=-m

c2,Zii)+,(2& sinh z )

The relation between vjJ)(z) and y P ( z ) can be determined from the asymptotic properties of the Bessel functions for large values of argument. It can be shown that

5 c2n(-1)n~2n+v(2hcos z )

20.4.6

I

20.4.10 y:’) (z>/yi’)(2) = [F,(0

U

m

If b=-j y=

Z)

(Isinh zl>l, j=1, 2, 3, 4)

20.4.5 na-m

5 ~ 2 ~ ( - 1 ) ~ 2 $ ; ’ + ~ ( 2cosh &

n=-m

(Icosh Zl>l)

<

If b=o, y=e*rv/2

Y,(r)=zp (2);

H p (2)=zp (2); H f ’ (2)=2:) (2)

~ 2 ~ J ~ ~ + ~sin( 2z )i &

/F,

(;)I

et v r / 2

(9’Z>O).

When v is not an integer, the above solutions do not vanish identically. See 20.6 for integral (Isin z / > I , larg 2& sin z l ~ a ) values of v. If b-ai, y reduces to a multiple of the solution Solutions Involving Products of Bessel Functions 20.3.8. The fact that 20.3.8, 20.4.5, and 20.4.6 20.4.11 are special cases of 20.4.4 explains why it is that 1 these apparently dissimilar expansions involve y ~ (2) ” =C z n (- 1)n ~ i ’ vi + s (fief.) Jn - 8 (Ge-is) Cz.3 n = - m the same set of coefficients czn. (j=1, 2, 3, 4) Since 20.4.4 results from the recurrence propersatisfies 20.1.1, where Zi’)( U ) is defined in 20.4.7, ties of Bessel functions, J&‘) can be replaced by the coefficients cznbelong to the Floquet solution, Hi’)cf),j=1, 2, where Hi’) is the Hankel function, and s is an arbitrary integer, czsf 0. The solution at least formally. Thus let converges over the entire complex z-plane if qZ 0. Written with z replaced by -iz, one obtains solutions of 20.1.2. 2

na-m

5

732

MATHIEU FUNCTIONS

I

20.4.12

20.5.4

2Ai+Ai+.

. . =A:+A:+. . .

= @ + E + .. . =@-I-@+ . . . =l.

20.5.5

provided czs #O. If c2,=0, the coefficient of l / c Z s in 20.4.11 vanishes identically. For details see [20.43],[20.15], [20.36]. If s is chosen so that Iczs(is the largest coefficient of the set Ic2,1, then rapid convergence of 20.4.12

is obtained, when 93"z>O. Even then one must be on guard against the possible loss of significant figures in the process of summing the series, especially so when q is large, and ( z ( small. (If jz1, then the phase of the logarithmic terms occurring in 20.4.12 must be defined, to make the functions single-valued.) 20.5. Properties of Orthogonality and Normalization

If a ( v + 2 p , q ) , a ( v + 2 s , q) are simple roots of 20.3.10 then 20.5.1

I'

F Y + 2 P ( ~ ) F Y + 2 , ( - ~ )ifd ~ p#s. =0,

Define

n#O

'

For integral values of U , the functions ce,(z, q) and se,(z, q) form a complete oIthogona1 set for the interval 0 5 2 5 2 ~ .Each of the four systems ce2,(z),~ e ~ , + ~ se2,(z), ( z ) , ~ s e Z r f l (is z ) complete in the smaller interval G 5 z 5 + T , and each of the systems cer(z), se,(z) is complete in O l z l ~ . If q is not real, there exist multiple roots of 20.3.10; for such special values of a(q), the integrals in 20.5.3 vanish, and the normalization is therefore impossible. In applications, the particular normalization adopted is of little importance, except possibly for obtaining quantitative relations between solutions of various types. For this reason the normalization of F,(z), for arbitrary complex values of a, p, will not be specified here. It is worth noting, however, that solutions ace&,

d,

Bser(z,

defined so that

cev(z, q ) , se,(z, p) are thus even and odd functions of z, respectively, for all U (when not identically

zero). If Y is an integer, then ce,(z, q ) , se,(z, q) are either Floquet solutions or identically zero. The solutions ce,(z, q) are associated with a,; se,(z, q) are associated with b,; r an integer. Normalization for Integral Values of U and Real q

are always possible. This normalization has in fact been used in [ 2 0 . 5 9 ] , and also in I20.581, where the most extensive tabular material is available. The tabulated entries in [20.58] supply the conversion factors A= l / a , B= l / B , along with the coefficients. Thus conversion from one normalization to another is rather easy. In a similar vein, no general normalization will be imposed on the functions defined in 20.4.8. 20.6. Solutions of Mathieu's Modified Equation 20.1.2 for Integral U (Radial Solutions)

Solutions of the first kind

For integral valuea of

Y

the summation in

20.3.8 reduces to the simpler forms 20.2.3-20.2.4; on account of 20.5.3, the coefficients A,,, and B,

(for all orders r ) have the property

i

733

MATHIEU FUNCTIONS m

20.6.2

Se2r+p(z, P)=-hez,+,,(iZ, q) =E

(q) sinh ( 2 k + p ) z , associated with b,

k=O

writing A:;:; (q)=AZk+, for brevity; similarly for BzkCp; p=O, 1 ,

20.6.5

-se'r(o' -

coth z

PBF

2

2 k B z J z k ( 2 f i sinh z)

See [20.30] for still other forms. Solutions of the second kind, as well as solutions of the third and fourth kind (analogous to Hankel functions) are obtainable from 20.4.12.

+

m

20.6.7

Mci? (2, q) =E (- 1 ) '+'A%(a)[Jn-a(q)Zi'ia(u2) Jk+*(udWa(wJ I/eaAi: k=O where e0=2, e a = l , for s=1, 2 ,

=zo

. . .; s arbitrary, associated with az,

m

20.6.8

20.6.9

MCJp+l(Z, q)

(-1)'+kAAar+l zk+l

(P) [Jk-a(ul) Zi'ia+l(G) +J'+a+l

(uJZica(~J I/Ai:Sf

-

associated with

aZr+l

Msi?(z, q)=C( - l ) k + f ' B ~ ; ( q ) [ J n ~ , ( u l ) Z ~ ~ ~ , ( ~ z ) - J k + ,associated ( u 1 ) Z ~ ~with , ( ~bzr z)]/B~~, k-1

m

20-6-10

M&)+ i (2, P) =E (-1>"+'B%$ Kq)[Jk-Jui)zi'L + 1 ( ~ 2 ) - J t + r + i k-0

(U,) Zi'L

(UZ)I/B:CS:

associated with

bgr+l

where

ul=fie-', %=fie', PdSPp, Ai:$Pp#O, p=O, 1 . See 20.4.7 for definition of Z$)(z). Solutions 20.6.7-20.6.10 converge for all values of z, when p#O. If j = 2 , 3, 4 the logarithmic terms entering into the Bessel functions Y,,,(%)must be defined, to make the functions single-valued. This can be accomplished as follows: Define (as in [20.58]) 20.6.11

In (fie')=ln

See [20.15] and [20.36],section 2.75 for derivation.

(fi)+z

734

MATHIEU FUNCTIONS

Other Expressions for the Radial Functions (Valid Over More Limited Regions) m

20.6.12

Mc$(z, q)=[cezr(O, q)]-l

(-1)”+‘A~;(q)Zi$)(2fi cosh z )

k =O

m

Mci?+l(Z, a) =[CeZr+i (0, a) I-’

C (k =O

l)”+‘A”z~:(q)zi$’+i(2fi cosh

2)

m

20.6.13

Msil,)(z,q)=[seil(O, q)]-l tanh z

(-1)”+‘2kB,2;(p)Z~f(2fi cosh z )

/:=1

m

M ~ i l , ) + ~q)=[se6r+l(0, (z, a)]-’ tanh z C ( - 1 ) ” ‘ ( 2 k + l ) B ~ ; S : ( q ) Z i $ ) + 1 (cosh 2 ~ z) k =O

Valid for %z>o, /cosh z1>1; if j=1, valid for all z. They agree with 20.6.7-20.6.10 if the Bessel functions Ym(2q+cosh z ) are made single-valued in a suitable way. For example, let 2 Y d u )=; On 4Jm(4 ++(U) With u=2q+ cosh z, define -l r <arg $ ( l + e - 2 z ) )7T~ 20.6.14 In (2qf cosh z)=ln 2q++z+ln ~(l+e-*’) 2(If q is not positive, the phase of In 2qt must also be specified, although this specification will not affect continuity with respect to z. If Y,(u) is defined from some other expression, the definition must be compatible with 20.6.14.) where

$(U)

is single-valued for all finite values of

U.

25

- l-I2 1 /6

FIGURE 20.13. Radial Mathieu Function of the Second Kind. FIGURE 20.11. Radial Mathieu Function of the First Kind. (From J . C . Wiltse and M. J. King, Values of the hlathieu functions, The Johns Hopkins Univ. Radiation Laboratory Tech. Rept. AF-53, 1958, with permission)

(From J. C . Riltse and hl. J. King, Values of the Mathieu functions, The Johns IIopkins Univ. Radiation Laboratory Tech. Rept. AF-53, 1958, with permission) &M:;lz,-

q)

t 1

1.4-

\

‘,

I

\

1.2-

1

1.0I\

\

FIGURE 20.12. Derivative of thc Radial Mathieu Function of the First Kind.

FIGURE 20.14. Radial Mathieu Function ojth,e Third Kind.

(From J. C. Wiltse and M. J. King, Derivatives, zeros, and other data pertaining to Mathieu functions, The Johns Hopkins Univ. Radiation Laboratory Tech. Rept. AF-57, 1958, with permission)

(From J. C. Wiltse and hf. J . King, Values of the Mathieu functions, The Johns Hopkins Univ. Radiation Laboratory Tech. Rept. AF-53, 1958, with permission)

735

MATHIEU FUNCTIONS

If j= 1, Mc$+, and Ms$:)+,,p=O, 1 are solutions of the first kind, proportional to Cezr+,and S e n r + p , respectively. Thus 20.6.15

20.6.19 (-

I

2,

n) = -Mc$*)( 2 , q ) -2je, ,MCj" ( 2 , $7)

20.6.20

M S y ( - z , p)=MSy'(z,

q)",,,Ms',"(z,

a)

where 20.6.21

Sezr(2,

a)=

Senr+l(Z,

a)=

se;, (0, q)se;r (- 1)rq,:,

6'

q)

See [20.58]. In particular the above equations can be used to extend solutions of 20.6.12-20.6.13 when 9 z < O . For although the latter converge for 9 z < O , provided only [cosh zj>l, they do not represent the same functions as 20.6.9-20.6.10.

MsF(2,a)

seL+l(O,a) sezr+l q)Ms::il(2,q) ( -1)rfiB:r+ 1

6

20.7. Representations by Integrals and Some Integral Equations

The Mathieu-Hankel functions are

Let

20.6.16

q ) + i M y ' ( z , q) M y ( 2 , q>=Ml"(z,q)-iM!2)(2,a) M $ j ) =Mc J" or Ms If) . From 20.6.7-20.6.11 and the known properties of Bessel functions one obtains MY'(2, q ) - - M ! " ( z ,

'c

be defined for U in a domain U and let the contour C belong to the region T of the complex t-plane, with t=yo as the starting point of the contour and t=yl as its end-point. The kernel K(u, t ) and the function V(t) satisfy 20.7.3 and the hypotheses in 20.7.2.

~

20.6.17

M a ( ? + p ( ~ + i q) n~, = (-1)"P[Mi?+v(Z, !?)+2niM8+,(2,n)l MA?+P ( 2 i n r , a) = (- 1)"P[Mi?+,(z,n) - 2nM,'?+,(z,n) I MI?+p(z+inT, q) = (-l)"P[M1?+,(z, !2)+2nMi?+p(z, q>l where M= Mc or Ms throughout any of the above equations.

20.7.2

+

and

U

[%&"K] L dt

xa=Msy' ( 2 , q) -Msy'

q) ; (- 2 , q ) 2,

Since X , is an even solution it must be proportional to McJ';(z, q) ; for 20.1.2 admits of only bne even solution (aside from an arbitrary constant factor). Similarly, X 2 is proportional t o MsJ1)(z, q). The proportionality factors can be found by considering values of the functions at z=O. Define, therefore,

Yo

O ; d2V -+(a-2*

cos 2t)V=0.

dt2

If K satisfies 20.7.4

+

dV in U; V and - are continuous in t. at

20.7.3

Consider

Xl =.McJ2' ( 2 , q) McJ2'(-

K(u,t ) and its first two partial derivatives U and t are continuous for t on C

with respect to

Other Properties of Characteristic Functions, q Real (Aseociated With a, and b,)

20.6.18

G(u)=$ K(u, t)V(t)dt

20.7.1

1

b2K b2K bu2 at

-+T+~~ 2~-C (Oc S2t)K=O os~

then G(u) is a solution of Mathieu's modified equation 20.1.2. If K(u, t ) satisfies 20.7.5

b2K b2K COS 2 ~ - C O S2t)K=O &U2 at2

then G(u) is a solution of Mathieu's equation 20.1.1, with U replacing 8.

736

MATHIEU FUNCTIONS

Kl(z,t)=Z:j)(u)[M(z,

t)]-Y12,

(9z>O)

I

where u=&q(cosh

20.7.7 20.7.8

I

I where F,(t) . . is the Floquet solution 20.3.8.

Kernels G(s, t ) and C ( 5 , t )

20.7.6

~Z+COS

2t)

M ( z , t)=cosh (z+it)/cosh (z-it)

To make

single-valued, define

20.7.9

The path C is chosen so that G(z, t, a ) exists, and 20.7.2, 20.7.3 are satisfied. Then it may be verified that &(2, t , a ) , considered as a function of z and t, satisfies 20.7.4; also, considered as a function of a and t , K3 satisfies 20.7.5. Consequently G(z, q, a )= Y(z,q)y(a, q ) , where Y and y satisfy 20.1.2 and 20.1.1, respectively. Choice of Path C. Three paths will be defined: 20.7.16

cosh (z+iIr)=ef* cosh z cosh (z-in) =e-‘* cosh z M ( z , 0) =1

Path C3:from -dl+ia

--dl<arg [&{cosh ( z + i a > f l } l < 7 r - d l

[M(z,? ~ ) ] - i ” = e - ~ ~ * 0) M(z,

Let

20.7.10 G ( z ,a)=-

l

A

l

K~(u, t)Fp(t)dt,

to d2-iw, d l , d2 real

-d2<arg [&{cosh (9?~>0)

(2-ia)

f1 }1<7r-d2

20.7.17

Path C4: from d2-iw to 2s+iw -dl

where FY(2)is defined in 20.3.8. I t may be verified that KIFvsatisfies 20.7.3, K satisfies 20.7.2 and 20.7.4. Hence B is a solution of 20.1.2 (with z replacing U). It can be shown that Kl may be replaced by the more general function

(same dl, d2 as in 20.7.16) 20.7.18

20.7.11

K2(z,t ) = Z ( ~ ) + 2 s ( ~ ) [ M t)]-*+, ( z , s any integer. See 20.4.7 for definition of Z(!)+2s(~). From the known expansions for 2(!)+,28(u) when 9 z is large and positive it may be verified that

where Mi(z, q) is also given by 20.4.12. 20.7.19 Path C,: from -dl+iw

to 2 s - d l + i a

20.7.12

M y ( 2 , q) =

w z > o , .@(Y+3)>0) where Mbj)( z , q) is given by 20.4.12, s=O, 1 , . . ., czs#O, and Fv(t) is the Floquet solution, 20.3.8. Kernel K&, t , a)

20.7.13

&(Z,

t, a)=earfiw

where 20.7.14 w=cosh z cos a cos t+sinh z sin a sin t

See [20.36], section 2.68. If U is an integer the paths can be simplified; for in that case FY(t)is periodic and the integrals exist when the path is taken from 0 to 21r. Still further simplifications are possible, if z is also real. The following are among the more important integral representations for the periodic functions ce,(z, q ) , se,(z, q) and for the associated radial solutions. Let r=2s+p, p=O or 1 20.7.20

737

MATHIEU FUNCTIONS r12

20.7.21 20.7.22

ser(z, q ) = p r l i 2 sin (2hcos z cos t + p

where

2 ?r

4

“2

pr=- ce2,

G,

q)

cezs(O, q)/&(q)

/&

-2 (q) ;~ = O P ~ ce;,+l = ~

z

COS

t]ce,(t, q)dt

sin z sin t ser(t, q)dt

6,

q)

/ f i ~ ~ + l ( qif)

p = l , for functions cer(z, q )

4 ur=- ce2,+l(0,q)/A;’++’(q), if p = l ; associated with functions cer(z, q)

if p = O ;

a

se;,(O,q )/fiB?(q),if p=O;

2

if p = l ; ,ur=- “sel,+l(O, q)/&P+’(q),

associated with ser(z, q)

Integrals Involving Bessel Function Kernels

Let 20.7.25

COS

ser(z, q ) = c r l l zsinh ( 2 4 sin z sin t ) [ ( l - p ) cos z cos t+p]se,(t, q)dt

20.7.23 20.7.24

cosh ( 2 4 sin z sin t ) [ ( I - p ) + p

cer(z, q)=c.S,

u=d2q(cosh 2z+cos 2 t ) , (B’cosh 22>1; if j=1, valid also when z=O)

20.7.26 c

Ms:;’(z, q)=

20.7.27

(-l)‘+%q sinh 22

Z$j)(u)sin 2t Selr(t, q)dt

“B;‘

U2

In the above the j-convention of 20.4.7 applies and the functions Mc, Ms are defined in 20.5.120.5.4.

(These solutions are normalized so that they approach the corresponding Bessel-Hankel functions as Wz+ 00 .) Other Integral#for Mc$”(5, q ) and Msj”(5, q )

20.7.28 20.7.29

20.7.31

Mc!”(z,

q)=r,l’

Ms,’f)+,(z,q)=s

[ ( 1 - p ) + p cosh z cos t ] cos ( 2 4 sinh z sin t)cer(t, q)dt

s$:lTb‘,;P l’sinh

z sin t cos (2& cosh z

COS t ) s e z r + l ( t , q)dt

738

MATHIEU FUNCTIONS

Further with w=cosh z cos a cos t+sinh z sin a sin t cer(a,q ) M c ? ) ( z , a>=

20.7.34 20.7.35

Serb,

(- 1 ) ('1 - r e 2 i 43 wce 2r (t , a>d t

a > M s P ( z , a>= (-')'(-')' 21r l r e 2 i GWse, ( t , q>dt.

The above can be differentiated with respect to a, and we obtain 20.7.36 20.7.37

ce:(a, q ) M c l l ) ( z ,q)= se:(a, q)MsS1)(z,a>=

er(t, a>dt

lr

(-1)a+p(')-p+lfi lr

2*

. bw e2'4Fwzs

4 ,!?)at

Integrals With Infinite Limits

r=2s+p

In 20.7.38-20.7.41 below, z and 20.7.38

Q

are positive.

McJ1)(z,q ) = y r l m sin

(2 J? cosh z cosh t + p

6,

q ) /TA:, if p=O

yr=2ceZs

20.7.39

Ms',')(z,q ) = y r L m sinh z sinh y r = -4sei,

20.7.40

(;,

p)

/filrEs,if

t

p=o

if p = l ~ ~ = 2 c e i ~ + ~ /fi TA:~+', 2 f i cosh z cosh t - p E)] Ms:"(t, q)dt 2 y r = -4se28+1

(;,

q)

/ l r ~ s + l ,

if p = l

Mcy"(z, q ) = y r J m cos ( 2 4 cosh z cosh 0

y I = -2ce2,(3n,

20.7.41

MsF'(z, q ) = y r

q)/?rAF,if p=O

som

7 1 = 2 ~ e ; l s + 1 ( 3p)/?rfiA?+', n, if p = l

sin ( 2 4 5 cosh z cosh t + p

sinh z sinh t Ms:"(t, q)dt

y r = - 4 s e ; , ( 3 ~ , q)/fi T B ~ ~ if 'p=O ,

Additional forms in [20.30],[20.36],[20.15]. 20.8. Other Properties Relations Between Solutions for Parameters p and -p

Replacing z by 3 7 - 2 in 20.1.1 one obtains 20.8.1 ,,I+ ( a + 2 q cos 2z)y=O Hence if u(z) is a solution of 20.1.1 then U(+"-z) satisfies 20.8.1. It can be shown that

20.8.2 a(-v, q)=a(v, -q)=a(v, q), v

not an integer

c ; , ( - q ) = ~ ( - l ) ~ c z Y , ( a ) , v not an integer (c2, defined in 20.3.8) and p depending on the normalization ;

Fy(z, -p)=pe-f~rI*FV

739

MATHIEU FUNCTIONS OD

Ko~r+l(z,q) =C A2t+1[Ik-s (ul)Kk+,+l (us) k-0

+

It+s+l(ul)Kk-r

(u~>l/A~a+l

where Im(z),Km(z)are the modified Bessel functions, ul, uz are defined below 20.6.10. Superscripts are omitted, c,=2, if s=O, c , = l if s # O . Then for functions of first kind: 20.8.10

Mc2 (2, -a> = ( - 1)‘Iezr (2, q) Msj?(z, -a)

= (-

l)‘IOa,(Z,

M~a’f)+ 1(z, -a> = (- 1)‘ikr+l (2, q>

Mdf)+ 1(z, -a)

=

- 1)‘iI02r

+1(z,

I

q)

I

i

For the Mathieu-Hankel function of first kind:

i

20.8.11

Mc$ ( z , (2,

Mc$+ ( z ,

-q) = (-a)

2

-

1 )?+li B Ke2,(2, q)

= (-

2

!

I

I

Kosr(Z , q)

!

2

i

-q) = ( - 1)‘+ l -* Ke2,+1(z, q)

M~j8,)+ 1(Z, -a)

=(

-

i

2

- Koar+l(z,

B

I

For M:’)(z, -a), j = 2 , 4, one may use the definitions

M:a)=-i(Mia)-Mp));Mr=Mc, or Ms,

i

also

M?’(z, -q)=2M:”(z, -q)-M:*’(z, -q) M=Mc or Ms; for real z, q, M?(z, -q)

I

are in general complex if j =2,4.

i

I

Zeros of the Functions tor Real Valuea of q.

See [20.36], section 2.8 for further results. Zeros of ce.(s, q ) and se,(s, q ) , Mc!”(s, q), MS!”(%q). In O I z c . ~ , c e , ( z ,q) and ser(z, q) have T real zeros. There are complex zeros if q>O. If ~ ~ = z o + iisy ~any zero of ce,(z, q), ser(z,q) in

are also zeros, k an integer. *See page

n.

*

740

MATHIEU FUNCTIONS 7r

7r

In the strip --
Ince [20.56] tabulates the first ‘hon-trivial”

)

zero (i.e. different from 0, I,7~ for cer(z), ser(z), 2 r = 2 ( 1 ) 5 and for seB(z) to within OlO-‘, for q = O ( l ) 10(2)40. He also gives the “turning” points (zeros of the derivative) and also expansions for them for small p. Wiltse and King [20.61,2]tabulate the first two (non-trivial) zeros of M c p ) ( z , q) and Ms!’)(z, q) and of their derivatives r=O, 1, 2 for 6 or 7 values of q between .25 and 10. The graphs reproduced here indicate their location. Between two real zeros of Mcp)(z,q ) , MsP)(z,q) there is a zero of McF2)(z,a), Msy)(z,q ) , respectively. No tabulation of such zeros exists yet. Available tables are described in the References. The most comprehensive tabulation of the characteristic values a,, br (in a somewhat different notation) and of the coefficients proportional to A , and B , as defined in 20.5.4 and 20.5.5 can be found in [20.58]. In addition, the table contains certain important “joining factors”, with the aid of which i t is possible to obtain values of McF’)(z, q ) and Ms?(z, q).as well as their derivatives, a t z=O. Values of the functions cer(z, q ) and ser(x, q ) for orders up to five or six can be found in [20.56]. Tabulations of less extensive character, but important in some aspects, are outlined in the other references cited. In this chapter only representative values of the various functions are given, along with several graphs.

The functions f0,,, g o . r , j e , r g, e , r are tabulated in [20.58] for q 5 2 5 . 20.9.

Asymptotic Representations

The representations given below are applicable to the characterislie solutions, for real values of q, unless otherwise noted. The Floquet exponent v is defined below, as in [20.36] to be as follows: In solutions associated with a , : V = T In solutions associated with b,: v = - T .

For the functions defined in 20.6.7-20.6.10:

N

q cosh z---Dm e f (2 J 7 3 5 m=o [ -4ifi(cosh T + ~ ” ‘ ( c ozs- u~ ) +

Z-U)]”

where D-l=D-z=O; D o = l , and the coefficients Dm are obtainable from the following recurrence formula : 20.9.2

Special Values for Arguments 0 and 1 2

20.8.12

d-1=d-2=O; d o = l , and

741 20.9.4

20.9.9

-

Fo(z) 1++2p-a]drn+(m-i)

1 '2048q

1)(1- U ~ ) ~ , - ~ = O .

In the above -2a<arg

-

4 cosh z < a /cosh z - - ~ [ > [ ~ & 1 1%'z>O, ,

is otherwise arbitrary. If u2=1, 20.9.2 and 20.9.4 become three-term recurrence relations. Formulas 20.9.1 and 20.9.3 are valid for arbitrary a, p, provided v is also known; they give multiples of 20.4.12, normalized so as to approach the corresponding Hankel functions HL1)(fiez), H?(fie'), as z+-. See [20.36], section 2.63. The formula is especially useful if [cosh zlis large and p is not too large; thus if U = -1, the absolute ratio of two successive terms in the expansion is essentially but

8& cosh2 z

[16q(l-~~)+8i~urn]d,_~

+4q(2m-3)(2m-

U

I(<

)/(cosh z + l )

2+-+2 :4

.

If a, p, z, v are real, the real and imaginary components of McJ3'(z,p) are Mcjl)(z,p) and Mcj2)( z , a), respectively; similarly for the components of M s : ~ ) (p). z , If the parameters are complex

W

w4+86w2+105 -w4+22w2+57 cosh4 z cosh2 z

[

1

- (w5+14w3+33w) +16384q3I2 cosh2 z (2w6+124w3+ 1122w) 3w5+290w3+ 1627w cosh4 z 4cosh6 z

[

I+...

20.9.10

F1(z)

sinh z cosh2 z [2u2+3+L 32fi 512q (w3+3w+

r y p

COSh2

4w3+44w)

12 cosh2 z

12139w2+10395) + (w6+ 505w4+12 cosh4 z -}I+.

.

.

See [20.18] for details and an added term in q-512; a correction to the latter is noted in [20.58]. The expansions 20.9.7 are especially useful when p is large and z is bounded away from zero. The order of magnitude of Mc,"(O,p) cannot be obtained from the expansion. The expansion can also be used, with some success, for z=iz, when p is large, 1 if lcos zI>>O; they fail at z=z a. Thus, if p, z are real, one obtains 20.9.11

20.9.6

i

M c ~ ' ( za)=-, ~ ) - M c ~ ) a)] (z, 2 [MC;"(Z,

Replacing c by s in the above will yield corresponding relations among Ms? ( z , q). Formulas in which the parameter a does not enter explicitly :

+W2[Po(z)+ P l ( X ) 20.9.12

ser+l(z,a) -s4+1(0, q)Tr+l W'i[f'o(z) -Pi(z)I

-Wz[Po(z) + P l ( Z )

Goldstein's Expansions

20.9.7

MC ;3) (2, q) iMS !":1 ( 2 , a) = [Fo( z ) -iFl( z)]ei+/~*qt(cosh z)*

20.9.13

+=2&

sinh z--

1 ( 2 r + l ) arctan sinh z, 2

Bz>O, q>>l, w = 2 r + l

11

In the above, Po(z) and Pl(z) are obtainable from Fo(z), Fl(z) in 20.9.9-20.9.10 by replacing cosh z with cos z and sinh z with sin z. Thus Po(z)=Fo(k) ; Pl(Z)= -iF1(?k) :

where 20.9.8

11

742

MATHIEU F'UNCTIONB

10.9.20

20.9.14

r4+2Ta-1211.2-122r-84 2048q

f 2=2TS+5r4-416fl-

6291.2- 1162r- 476

It should be noted that 20.9.15 is also valid as an approximation for ser+](z, q), but 20.9.16 may give dightly better results. See [20.4.]

Expansionr in Terms of Parabolic Cylinder Functions

(Good for angles close to ir, for large values of q, especially when lcos 21<2*/q+.) Due to Sips

Explicit Expansions for Orders 0, 1, to Terms in q-O'* (Q L w e )

[20.44-20.461. cer(z, q)-Cr[Zo(a) +Zi(a)I

20.9.15

-%

+

See 20.9.23-20.9.24 for expressions relating to cer(O, q) and se:(O, q). When lcos z l > d w / q * , 20.9.11-20.9.12 are useful. The approximations become poorer as r increases.

20.9.16

ser+1(5, q)-Sr[&(a)-Z1(a)]

Let Dk=Dk(a)=(- 1)'efm

a=2$

sin z,

For r=O:

20.9.21

Zo-Do--+- D4

64fi

COS 5.

1

169

(--+-) Dd Ds 8 512

*

a' d k

- e+*.

dak

20.9.17

For r=l :

20.9.22

20.9.18

+r(r-1)

(--73-27~+10) 64

Dr-2--

()

45 r 4 6 Dr-e+

.

*

-1 Formula8 Involving cer(O,k)and ser(O,q )

20.9.19

20.9.23

+r4+213+263r2+262r+1O8 2048q +-+ f]=6TS+ 'See page n.

15r4+1280?+19M?+

fl

..

1778r+572

.I-%

MATHIEU FUNCTIONS

743

For higher orders, these ratios are increasingly more difficult to obtain. One method of estimating values at the origin is to evaluate both 20.9.11 and 20.9.15 for some x where both expansions are satisfactory, and so to use 20.9.11 as a means to solve for ce,(O, a ) ; similarly for sei(0, q). Other asymptotic expansions, valid over various regions of the complex z-plane, for real values of a, p, have been given by Langer [20.25]. I t is not always easy, however, to determine the linear combinations of Langer’s solutions which coincide with those defined here.

20.10. Comparative Notations Thb Volume

Stratton-Mom. [m. 591 etc.

[m.151 Batemsn Manusrrlpt A

0

h

Y

s

e

or

or 4

4

r),

a.

Bee Note 1.

B: 1

tKv*dz,vbtheStandari

Bee Note 1.

Solution of 20.1.1. Floquet's S o I u t l m s ~ 3 . 8 . ~ ~ Chsraetedstic Exponent.---Normaliurtlons of Floquet's Bolutlons.

4

3

Wutlons of Mndified Eqm. tlon 20.1.2.

3 El

3

Bee Note 2.

Joinlno Factors__._. -. ~. .....

Bee Note a.

Game as this volume

NOTE:1. The conversion factors A' and B' are tabulated in t20.581 along with the coefficients. 2. The multipliers p , and 8 - are defined in [20.30], Appendix 1, section 3, equations 3, 4, 5, 6. 3. See [20.591,sections (5.3) and (5.5). In eq. (316) of (5.5), the tkst term should have a minus sign. ~

*See page U.

MATHIEU FUNCTIONS

745

References Texts

[20.1]W. G. Bickley, The tabulation of Mathieu functions, Math. Tables Aids Comp. 1, 409-419 (1945). [20.2]W. G. Bickley and N. W. McLachlan, Mathieu functions of integral order and their tabulation, Math. Tables Aids Comp. 2, 1-11 (1946). [20.3]G. Blanch, On the computation of Mathieu functions, J. Math. Phys. 25, 1-20 (1946). [20.4]G. Blanch, The asymptotic expansions for the odd periodic Mathieu functions, Trans. Amer. Math. SOC.97,2, 357-366 (1960). [20.5]C. J. Bouwkamp, A note on Mathieu functions, Kon. Nederl. Akad. Wetensch. Proc. 51, 891-893

(1948). [20.6]C. J. Bouwkamp, On spheroidal wave functions of order zero. J. Math. Phys. 26, 79-92 (1947). [20.7]M. R. Campbell, Sur les solutions de pBriode 2 ST de 1’6quation de Mathieu associee, C.R. Acad. Sci., Paris, 223, 123-125 (1946). [20.8]M. R. Campbell, Sur une catBgorie remarquable de solutions de 1’6quation de Mathieu associBe, C.R. Acad. Sci., Paris, 226, 2114-2116 (1948). [20.9] T. M. Cherry, Uniform asymptotic formulae for functions with transition points, Trans. Amer. Math. Soc. 68, 224-257 (1950). [20.10]S. C. Dhar, Mathieu functions (Calcutta Univ. Press, Calcutta, India, 1928). (20.111 J. Dougall, The solution of Mathieu’s differential equation, Proc. Edinburgh Math. Soc. 34, 176-

196 (1916). [20.12]J. Dougall, On the solutions of Mathieu’s differential equation, and their asymptotic expansions, Proc. Edinburgh Math. Soc. 41, 26-48 (1923). [20.131 A. ErdBlyi, u b e r die Integration der Mathieuschen Differentialgleichung durch Laplacesche Integrale, Math. Z. 41, 653-664 (1936). [20.14]A. ErdBlyi, On certain expansions of the solutions of Mathieu’s differential equation, Proc. Cambridge Philos. Soc. 38, 28-33 (1942). [20.15]A. Erd6lyi et al., Higher transcendental functions, vol. 3 (McGraw-Hill Book Co.,Inc., New York, N.Y., 1955). 120.161 G. Floquet, Sur les Bquations diffbrentielles linhaires A coefficients periodiques, Ann. h o l e Norm. Sup. 12,47 (1883). [20.17]S. Goldstein, The second solution of Mathieu’s differential equation, Proc. Cambridge Philos. SOC.24, 223-230 (1928). [20.18]S. Goldstein, Mathieu functions, Trans. Cambridge Philos. Soc. 23, 303-336 (1927). [20.19]G. W. Hill, On the path of motion of the lunar perigee, Acts Math. 8, 1 (1886). [20.20]E. Hille, On the zeros of the Mathieu functions, Proc. London Math. Soc. 23, 185-237 (1924). (20.211E. L. Ince, A proof of the impossibility of the coexistence of two Mathieu functions, Proc. Cambridge Philos. Soc. 21, 117-120 (1922).

[20.22]E. L. Ince, Ordinary differential equations (Longmans, Green & Co., 1927, reprinted by Dover Publications, Inc., New York, N.Y., 1944). [20.23]H. Jeffreys, On the modified Mathieu’s equation, Proc. London Math. Soc. 23, 449-454 (1924). [20.24]V. D. Kupradze, Fundamental problems in t h e mathematical theory of diffraction (1935). Translated from the Russian by Curtis D. Benster, NBS Repor1 200 (Oct. 1952). [20.25] R. E. Langer, The solutions of the Mathieu equation with a complex variable and at least one parameter large, Trans. Amer. Math. Soc. 36,

637-695 (1934). [20.26]S. Lubkin and J. J. Stoker, Stability of columns and strings under periodically varying forces, Quart. Appl. Math. 1, 215-236 (1943). [20.27]E. Mathieu, MBmoire sur le mouvement vibratoire d’une membrane de forme elliptique, J. Math. Pures Appl. 13, 137-203 (1868). [20.28]N. W. McLachlan, Mathieu functions and their classification, J. Math. Phys. 25, 209-240 (1946). [20.29]N. W. McLachlanJMathieu functions of fractional order, J. Math. Phys. 26, 29-41 (1947). [20.30]N. W. McLachlan, Theory and application of Mathieu functions (Clarendon Press, Oxford, England, 1947). [20.31]N. W. McLachlan, Application of Mathieu’s equation t o stability of non-linear oscillator, Math. Gaz. 35, 105-107 (1951). [20.32]J. Meixner, Uber das asymptotische Verhalten von Funktionen, die durch Reihen nach Zylinderfunktionen dargestellt werden konnen, Math. Nachr. 3, 9-13, Reihenentwicklungen von Produkten zweier Mathieuschen Funktionen nach Produkten von Zylinder und Exponentialfunktionen, 14-19 (1949). Integralbeziehungen zwischen l20.331 J. Meixner, Mathieuschen Funktionen, Math. Nachr. 5,

371-378 (1951). [20.34]J. Meixner, Reihenentwicklungen vom Siegerschen Typus fur die Spharoid Funktionen, Arch. Math. Oberwolfach 1, 432-440 (1949). [20.35]J. Meixner, Asymptotische Entwicklung der Eigenwerte und Eigenfunktionen der Differentialgleichungen der Spharoidfunktionen und der Mathieuschen Funktionen, Z. Angew. Math. Mech. 28, 304-310 (1948). [20.36]J. Meixner and F. W. Schafke, Mathieusche Funktionen und Spharoidfunktionen (SpringerVerlag, Berlin, Germany, 1954). “20.371 P. M. Morse and P. J. Rubinstein, The diffraction of waves by ribbons and by slits, Phys. Rev. 54,

895-898 (1938). f20.381 H . P. Mulholland and S. Goldstein, The characteristic numbers of t h e Mathieu equation with purely imaginary parameters, Phil. Mag. 8,

834-840 (1929). [20.39]L. Onsager, Solutions of the Mathieu equation of period 4r and certain related functions (Yale Univ. Dissertation, New Haven, Conn., 1935).

746

MATHIEU FUNCTIONS

[20.40] F. W. Schiifke, uber die Stabilitatskarte der. Mathieuschen Differentialgleichung, Math. Nachr. 4, 175-183 (1950). [20.41] F. W. Schiifke, Das Additions theorem der Mathieuschen Funktionen, Math. Z. 58,436-447 (1953). [20.42] F. W. Schiifke, Eine Methode aur Berechnung des charakteristischen Exponenten einer Hillsohen Differentialgleichung, 2. Angew. Math. Mech. 33, 279-280 (1953). [20.43] B. Sieger, Die Beugung einer ebenen elektrischen Welle an einem Schirm von elliptischem Querschnitt, Ann. Physik. 4, 27, 626-664 (1908) [20.44] R. Sips, Representation asymptotique des fonctions de Mathieu et des fonctions d’onde sphbroidales, Trans. Amer. Math. Soc. 66, 93-134 (1949). [20.45] R. Sips, Reprbsentation asymptotique des fonctions de Mathieu et des fonctions spheroidales 11, Trans. Amer. Math. Soc. SO, 2, 340-368 (1959). [20.46] R. Sips, Recherches sur les fonctions de Mathieu, Bull. Soc. Roy. Sci. Liege 22, 341-355, 374-387, 444-455, 530-540 (1953); 23, 37-47, 90-103 (1954). [20.47] M. J. 0. Strutt, Die Hillsche Differentialgleichung im komplexen Gebiet, Nieuw. Arch. Wisk. 18 31-55 (1935). [20.48] M. J. 0. Strutt, LamBsche, Mathieusche und verwandte Funktionen in Physik und Technik, Ergeb. Math. Grenageb. 1, 199-323 (1932). [20.49] M. J. 0. Strutt, On Hill’s problems with complex parameters and a real periodic function, Proc. Roy. Soc. Edinburgh Sect. A 62, 278-296 (1948). 120.501 E. T. Whittaker, On functions associated with elliptic cylinders in harmonic analysis, Proc. Intl. Congr. Math. Cambr. 1, 366 (1912). [20.51] E. T. Whittaker, On the general solution o f Mathieu’s equation, Proc. Edinburgh Math. SOC. 32, 75-80 (1914). [20.52] E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952). Tablee

[20.53] G. Blanch and I. Rhodes, Table of characteristic values of Mathieu’s equation for large values of the parameter, J. Washington Acad. Sci. 45, 6, 166-196 (1955). Be&) =u,(q)+2q-2(2r+l) Bo,(t)=b,(q) +2q-2(2r-l) at=1 / 2 , / 5 r=0(1) 15, O < t < . l , with 62, a‘*; 8D (about); interpola ble [20.54] J. G. Brainerd, H. J. Gray, and R. Merwin, S o h tion of the Mathieu equation, Am. Inst. Elec. Engrs. 67 (1948). Characteristic exponent over a wide range. p, M fo; ~ = 1 ( 1 ) 1 0 ;k = . l ( . l ) l , 5D; g(t), h(t) for t=0(.1)3.1, r, 5D;e=l(l)lO, k = , l ( . l ) l , where g(t), h(l) are solutions of y”+c(l+k COS t ) y = O , with g(O)-=h‘(O)=l, g’(O)=h(O)=O, cos 2rp=2g(r)h’(*)-l1, M = 1- g(*)g‘ (d/ h ( r )h ‘ ( 4 P. [20.55] J. G. Brainerd and C. N. Weygandt, Solutions of Mathieu’s equation, Phil. Mag. 30, 458-477 (1940).

.

[20.56] E. L. Ince, Tables of the elliptic cylinder functions, Proc. Roy. Soc. Edinburgh 52, 355-423; Zeros and turning points, 52, 424-433 (1932). Characteristic values a ~ ,UI, . . ., us, bl, bz, . . ., be, and coefficients for 0=0(1)10(2)20(4)40; 7D. Also ce,(z, O), ser(z, 8 ) , 8=O(l)lO, z=Oo(l0)9Oo; 5D, corresponding to characteristic values in the tables. U,= ber-24; b,= bo,-2q; O=q. [20.57] E. T. Kirkpatrick, Tables of values of the modified Mathieu function, Math. Comp. 14, 70 (1960). Ce,fu, q), r=0(1)5, Se,(u, q), r=1(1)6; U = l ( . l ) l ; q=1(1)20. [20.58] National Bureau of Standards, Tables relating to Mathieu functions (Columbia Univ. Press, New York, N.Y., 1951). Characteristic values be&), bo,(s) for 0 <s 100, along with P*,interpolable to 8D; coefficients Der(s), Do&) and conversion factors for ce,(q), se,(q), same range, without differences but interpolable t o 9D with Lagrangian formulas of order 7. “Joining factors” drgi,r, drg0,,, srfg,,, sffo,.along with 6z*; interpolable to 8s. [20.59] J. A. Stratton, P. M. More, L. J. Chu and R. A. Hutner, Elliptic cylinder and spheroidal wave functions (John Wiley & Sons, Inc., New York, N.Y., 1941). Theory and tables for bo, bl, b2, bat b,, b;, b;, b;, b;, and coefficients for Se,(s, x ) and So& z) for c=0(.2)4.4 and .5(1)4.5; mostly 5 s ; ~-2q4, br=u,+2q, b;=br+2q. [20.60] T. Tamir, Characteristic exponents of Mathieu * equations, Math. Comp. 16, 77 (1962). The Floquet exponent Y , of the first three stable regions; namely r=O, 1, 2; q=.1(.1)2.5; u=r ( . l ) r + l , 5D. [20.61] J. C. Wiltse and M. J. King, Values of the Mathieu functions, The Johns Hopkins Univ. Radiation Laboratory Technical Report AF-53, Baltimore, Md. (1958). (Notation of [20.58] used: cen(u,q)/A, sen(u, q)/B for 12 values of q b e t w e s .25 and 10 and from 8 to 14 values of U; d s / 2 Mc!(u, q), d a M$(u, q), j=1, 2 for 6 to 8 values of q between .25 and-10 and about 20 values of U, Y=O, I, 2; d=/2 Mc?)(-~uI, q), d a MS:*)(-JuI,q), r=O, 1, 2 for about 9 values of U and q, 2 to 4 D in all. [20.62] J. C. Wiltse and M. J. King, Derivatives, zeros, and other data pertaining t o Mathieu functions, The Johns Hopkins Univ. Radiation Laboratory Technical Report AF-57, Baltimore, Md. (1958). [20.63] S. J. Zaroodny, An elementary review of the Mathieu-Hill equation of real variable based on numerical solutions, Ballistic Research Laboratory Memorandum Report 878, Aberdeen Proving Ground, Md. (1955). Chart of the characteristic exponent. See also [20.18]. It contains, among other tabulations, values of a,, b, and coefficients for ce,(z, q ) , ac,(x, q ) , q=40(20)100(50)200; 5 0 , r 52.

<

*See page

11,

748

MATHIEU FUNCTIONS

Table 20.1

CHARACTERISTIC VALUES, JOINING FACTORS, SOME CRITICAL VALUES

EVEN SOLUTIONS r

q

“r

0

0 5 15 20 25

-

0.00000 5.80004 13.93697 22.51303 31.31339 40.25677

000 602 996 776 007 955

0 5 10 15 20 25

4.00000 7.44910 7.71736 5.07798 t 1.15428 - 3.52216

974 985 320 288 473

0

100.00000 100.12636 100.50677 101.14520 102.04891 103.23020

-1) 7.07106 -2) 4.48001 -3) 7.62651 -3 1.93250 -4) 6.03743 -4) 2.15863

781 817 757 832 829 018

(-1) 7.07106 1.33484 1.46866 1.55010 1.60989 1.65751

1.00000 7.35294 2.45888 7.87928 2.86489 1.15128

000 308 349 278 431 663

(-1 -7.24488 (-1 -9.26759 -1.01996 -1.07529 -1.11627

15 26 O0 62 32 90

000 922 002 345 602 480

1.00000 1.02599 1.05381 1.08410 1.11778 1.15623

000 503 599 631 863 992

-1.00000

00

000 754 240 513 142 969

1.00000 000

5 10 15 20 25

ar 1.00000 + 1.85818 2.39914 - 8.10110 14.49130 21.31489

5

0 5 10 15 20 25

25.00000 25.54997 27.10376 31.95782 36.64498 40.05019

000 175 873 125 973 099

1.00000 1.12480 1.25801 1.19343 (-1 9.36575 (-1)6.10694

000 725 994 223 531 310

15

0 5 10 15 20 25

225.00000 225.05581 225.22335 225.50295 225.89515 226.40072

000 248 698 624 341 004

1.00000 1.01129 1.02287 1.03479 1.04708 1.05980

000 373 828 365 434 044

10

2

10

5 10 15 20 25

r 1

9 0

-

000

-

i

-1 -1 -2 -2) -2)

( -1) 7.97884 1.97009 2.40237 2.68433 2.90011 3.07743

78 87 05 82 09 03

56 00 95 53 25 91

l-l*ooooo

.

-3 5.05181 376 -31 1.91105 1 5 1

(Ip

0.16 0.12 0.08 0.04 0.00

1

0 -0.25532 -0.25393 -0.25257 -0.25126 -0.25000 For ge,

994 098 851 918 000

-1.30027 -1.28658 -1.27371 -1.26154 -1.25000

2 212 972 191 161 000

, and f,, see 20.8.12.

-3.45639 -3.39777 -3.34441 -3.29538 -3.25000

4.29953 32 1.11858 69

-1 -9.05710 78 -11 -8.82691 92

.

ce:(3.,

4)

-1.00000 -3.46904 -4.85043 -5.76420 -6.49056 -7.10674

00

21 83 64 58 15

-5.00000 -5.39248 -5.32127 -5.11914 -5.77867 -7.05988

00 61 65 99 52 45

1

(49) “ge, r (9) 1.59576 9 1 111.35943 7.26039 1 1.91348 1 2.42144 1 2.89856

.

+2q- ( 4 ~ + 2 ) 4 5

483 782 938 745 000

2.54647 9 1 1.02263 46

49 84 51 01 94

Compiled from National Bureau of Standards, Tables relating to Mathieu functions, Press, New York, N.Y., 1951 (with permission). q -i\r

.

-17.84809 -16.92019 -16.25305 -15.70968 -15.25000

551 225 645 373 000

10 -76.04295 -76.84607 -63.58155 -58.63500 -55.25000

314 855 264 546 000

-

ilumbia Univ.

15

80.93485 -141.64507 -162.30500 -132.08298 -120.25000



048 841 052 271 000

39 69 156 625 03

~

< q > =nearest integer to q. Compiled from G. Blanch and I. Rhodes, Table of characteristic values of Mathieu’s equation for large values of the parameter, Jour. Wash. Acad. Sci., 45, 6, 1955 (with permission).

749

MATHIEU FUNCTIONS

CHARACTERISTIC VALUES, JOINING FACTORS, SOME CRITICAL VALUES

Table 20.1

ODD SOLUTIONS r 2

4 0

20

0 5 10 15 20 25

r

4

1

0

15

000 922 946 229 286 004

( 1)1.00000 00

br 1.00000 5.79008 13.93655 22.51300 31.31338 40.25677

000 060 248 350 617 898

1.00000 00

25.00000 25.51081 26.76642 27.96788 28.46822 28.06276

000 605 636 060 133 590

+

-

5 10 15 20 25 5

100.00000 100.12636 100.50676 101.14517 102.04839 103.22568

-

25 10

045 824 680 325 062

+

5 10 15

-

0 5 10 15 20 25

000

sei(tn, 4 ) -2.00000 00 -3.64051 79 -4.86342 21 -5.76557 38 -6.49075 22 -7.10677 19

se;(O, 4) 2.00000 00 -1) 7.33166 22

br 4.00000 2.09946 2.38215 8.09934 14.49106 21.31486

9.73417 9.44040 9.11575 8.75554 8.35267

Se;(O,

32 54 13 51 84

11

1 -1.00000 00

1 -1.02396 46 1 -1.04539 48 1 -1.06429 00 1)-1.08057 24 ( 1)-1.09413 54

1.00000 1.33743 1.46875 1.55011 1.60989 1.65751

00 39 57 51 16 04

1.00000 9.06077 -11 8.46038 -1 8.37949 -11 8.63543 -1) 8.99268

00 93 43 34 12 33

-3 5.07788 49 -31 2.04435 94

00 00 68 65 69 62

-1

0

5 10 15 20 25

226.40072 004

3.91049 85

11 1.51800 43 111 1.56344 50 11) 1.62453 03

23)2.30433 72 23) 2.31909 77

se,(ta, 9 )

9)

5.00000 4.33957 3i40722 2.41166 1.56889 (-1) 9.64071

6.38307 - -.... . 65 --

( 1) 1.40643 73

1.59576 2.27041 2,63262 2.88561 3.08411 3.24945

91 76 99 87 21 50

-1.00000 00

19 3.73437 81 191 3.78055 49

(-1)-9.46708 70

19)4.06462 83

2.54647 91

hr+2q-(4r-2) J; -0.25532 -0.25393 -0.25257 -0.25126 -0.25000

For

go,

5

2

1 0.16 0.12 0.08 0.04 0.00

994 098 851 918 000

-1.30027 -1.28658 -1.27371 -1.26154 -1.25000

164 971 191 161 000

-11.53046 -11.12574 -10.78895 -10.50135 -10.25000

10 855 983 146 748 000

-51.32546 -56.10964 -51.15347 -47.72149 -45.25000

and j , , see 20.8.12.

< q > =nearest integer to q.

875 961 975 533 000

-

15

55.93485 -108.31442 -132.59692 -114.76358 -105.25000

112 060 424 461 000

39 69 156 625 m

MATHIEU FUNCTIONB COEFFICIENTS 8, AND R,,

Table 20.2

Anl

9=5 0

ni\r

2 +0.43873 +0.65364 -0.42657 +0.07588 -0.00674 +0.00036 -0.00001

7166 0260 8935 5673 1769 4942 3376 +O.OOOOO 0355 -0.00000 0007

10 +o.ooooo 1679

ni\,i

3619 2987 4807 5121 5640 6780 2962 9166 +o.ooooo 4226 -0.00000 0071 +o.ooooo 0001

3 5 7 9 11 13 15 17 19 21 23 25

1

+0.00003 +0.00064 +0.01078 +0.13767 +0.98395 -0.11280 +0.00589 -0.00018

1

5

+O. 76246 3686 +0.07768 5798 -0.63159 6319 ~0.139684806 -0.01491 5596 +0.00094 4842 -0.00003 9702 +o.ooooo 1189 -0.00000 0027 +o.ooooo 0001

~0.303751030 +0;92772 8396 -0.20170 6148 +0.01827 4579 -0.00095 9038 +0.00003 3457 -0.00000 0839 +O.OOOOO 0016

15 0.00000

0000

+O.OOOOO +O.OOOOO

0106 4227 8749 1393 2014 4092 7946 6409 6394 1092 0014

+o.ooooo 0002 +0.00014 +0.00428 +0.08895 +0.99297 -0.07786 +0.00286 -0.00006 +O.OOOOO

-0.00000

q=25 2

10

2 4 6 8 10

-0.69199 +0.36554 -0.13057 +0.03274 -0.00598 12 +0.00082 14 -0.00008 16 +O.OOOOO 18 -0.00000 20 +o.ooooo 22 -0.00000 24 26

9610 4890 5523 5863 3606 3792 7961 7466 0514 0029 0001

-0.04661 4551 -0.64770 5862 +0.55239 9372 -0.22557 4897 +0.05685 2843 -0.00984 6277 +0;00124 8919 -0.00012 1205 +o.ooooo 9296 -0.00000 0578 +o.ooooo 0030 -0.00000 0001

28

1

ni\,r

0 +0.42974 1038 +0.33086 5777 +O. 00502 6361

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

+0.02075 4891 +0;07232 7761 +0.23161 1726 +0.55052 4391 +0.63227 5658 -0.46882 9197 +0.13228 7155 -0.02206 0893 +0.00252 2374 -0.00021 3672 +0.00001 4078 -0.00000 0746 +O.OOOOO 0032 -0.00000 0001

5

t0.39125 2265 -0.74048 2467 +0.50665 3803 -0.19814 2336 ~0.05064 0536 -0.00910 8920 +0.00121 2864 -0.00012 4121 +0.00001 0053 -0,00000 0660 +O.OOOOO 0036 -0.00000 0002

+0.65659 0398 +0.36900 8820 -0,19827 8625 -0.48837 4067 +0.37311 2810 -0.12278 1866 +0.02445 3933 -0.00335 1335 +0.00033 9214 -0.00002 6552 +o.ooooo 1661 -0.00000 0085 +O;OOOOO 0004

15 +o.ooooo 4658

+0.00003 7337 +0.00032 0026 +0.00254 0806 +0;01770 9603 +0.10045 8755 +0.40582 7402 +0.83133 2650 -0.35924 8831 +0.06821 6074 -0.00802 4550 +0.00066 6432 -0.00004 1930 +O.OOOOO 2090 -0.00000 0085

+o;ooooo

0005

Bm

q=5 ni\r 2 4 6 8 10 12 14 16 18

2 +0.93342 -0.35480 t0.05296 -0.00429 +0.00021 -0.00000

94'42 3915 3730 5885 9797 7752 +b.ooooo 0200 -0.00000 0004

20 22

10 +0.00003 3444 +0.00064 2976 +0.01078 4807 t0.13767 5120 +0.98395 5640 -0.11280 6780 +0.00589 2962 -0.00018 9166 +O.OOOOO 4227 -0.00000 0070 +o.ooooo 0001

1

ni\r

1 3 5 7 9 11 13 15 17 19 21 23 25

+0.94001 -0.33654 +0.05547 -0.00508 +0.00029 -0.00001

9024 1963 7529 9553 3879 1602 +O.OOOOO 0332 -0.00000 0007

5 +0.05038 +0.29736 +0.93156 -0.20219 +0.01830 -0.00096 +0.00003 -0.00000 +O.OOOOO

2462 5513 6997 3638 5721 0277 3493 0842 0017

16 -_ 0.00000 0000 +O.OOOOO 0002 e0.00000 0106 +O.OOOOO 4227 +0.00014 8749 +0.00428 1392 +0;08895 2014 +0.99297 4092 -0.07786 7946 +0;00286 6409 -0.00006 6394 +O.OOOOO 1093 -0,00000 0013

q=25 ni\r

2 4 6 8 10 12 14 16 18 20 22 24 26

For

2 +0.65743 9912 -0.66571 9990 ~0.33621 0033 -0.10507 3258 +0.02236 2380 -0.00344 2304 r0.00040 0182 -0;00003 6315 +O.OOOOO 2640 -0.00000 0157

+o;ooooo 0008 A,

10 +0.01800 3596 +0;07145 6762 +0.23131 0990 +0.55054 4783 +0;63250 8750 -0.46893 3949 +0.13230 9765 -0;02206 3990 +0.00252 2676 -0.00021 3694 +o;ooooi 4079 -0.00000 0746 +O.OOOOO 0033

and B, see 20.2.3-20.2.11

1

9 11 13 15 17 19 21 23 25 27 29 31

+0.81398 3846 -0.52931 0219 +0.22890 0813 -0.06818 2972 +0.01453 0886 -0.00229 5765 +0.00027 7422 -0.00002 6336 +O.OOOOO 2009 -0.00000 0126 +O.OOOOO 0007

5 +0.30117 4196 ~0.627198468 +0.17707 1306 -0.60550 5349 +0.33003 2984 -0.09333 5984 +0.01694 2545 -0.00217 7430 +0.00021 0135 -0.OWO1 5851 +O.OOOOO 0962 -0.00000 0048 +o.ooooo 0002

15 +O.OOOOO

+0.00003 +0.00032 +0.00254 +0.01770 +0.10045 +0.40582 +0.83133 -0.35924 +0.06821 -0.011802

+0.00066 -0.00004 +O.OOOOO

-0.00000 tO.00000

3717 7227 0013 0804 9603 8755 7403 2650 8830 6074 4551 6432 1930 2090 0086 0003

Compiled from National Bureau of Standards, Tables relating to Mathieu functions, Columbia Univ. Press, New York, h.Y., 1951 (with permission).

21. Spheroidal Wave Functions ARNOLDN. LOWAN'

Contents Mathematical Properties, , , . . . . . . . . . . . . . . . . . 21.1. Definition of Elliptical Coordinates . . . . . . . . . . 21.2. Definition of Prolate Spheroidal Coordinates . . . . . . 21.3. Definition of Oblate Spheroidal Coordinates. . . . . . . 21.4. Laplacian in Spheroidal Coordinates . . . . . . . . . . 21.5. Wave Equation in Prolate and Oblate Spheroidal Coordinates 21.6. Differential Equations for Radial and Angular Spheroidal Wave Functions . . . . . . . . . . . . . . . . . . 21.7. Prolate Angular Functions . . . . . . . . . . . . . . 21.8. Oblate Angular Functions . . . . . . . . . . . . . . 21.9. Radial Spheroidal Wave Functions . . . . . . . . . . 21.10. Joining Factors for Prolate Spheroidal Wave Functions . 21.11. Notation . . . . . . . . . . . . . . . . . . . . . . References

,

,

,

,

.

,

., ., .

Table 21.1. Eigenvalues-Prolate m=0(1)2, n=m(l)m+4 ~ ~ = 0 ( 1 ) 1 6~-'=.25(--.01)0, ,

. . . . . . . . . . . . . . and Oblate . . . . . . . . . . . ,

752 752 752 752 752 752 1

753 753 756 756 757 758

1

,1

759 760

4-6D

Table 21.2. Angular Functions-Prolate m=0(1)2, %=m(1)3, 7 =0(.1)1 e=oo(ioo)900, C= 1(1)5, 2-4D

and Oblate

. . . . .

Table 21.4. Oblate Radial Functions-First m=O, 1, n = m ( l ) m + 2 ; m, n = 2 ,$=O, .75, c=.2, .5, .8, 1(.5)2.5, 5s Table 21.5. Prolate Joining Factors-First m=O, 1, n=m(l)m+2; m, n = 2 c=1(1)5, 4s

766

768

. . . .

769

. . . . . . . . . .

769

and Second Kinds

Kind

I

. ..

Table 21.3. Prolate Radial Functions-First and Second Kinds m=0(1)2, n=m(1)3 ,$=1.005, 1.02, 1.044, 1.077, c=1(1)5, 4s

1 Yeshiva University. Standards.) (Deceased.)

Page

(Prepared under contract with the National Bureau of 751

21. Spheroidal Wave Functions Mathematical Properties 21.1. Definition of Elliptical Coordinates 21.1.1

Z=T

rl and r2 are the distances to the foci of a family of confocal ellipses and hyperbolas; 2f is the distance between foci. a=jI, b = j m ,

21.1.2

e=-f a

cos 4; x=r sin 4; 0 1 4 5 2 ~

where [,7and 4 are oblate spheroidal coordinates. Relations Between Carteeian gnd Oblate Spheroidal Coordinates

21.3.2 x = j t q sin 4; y = j J ( I ~ - I > ( I - ~ ~ )z=jIq ; cos Q,

a=semi-major axis; b=semi-minor axis; e=eccentricity.

21.4. Laplacian in Spheroidal Coordinates 21.4.1

Equation of Family of Confocal Ellipses

Equation of Family of Confocal Hyperbolas

Relations Between Cartesian and Elliptical Coordinates

21.2. Definition of Prolate Spheroidal Coordinates

Metric doefficients for Prolate Spheroidal Coordinates

If the system of confocal ellipses and hyberbolas referred to in 21.1.3 and 21.1.4 revolves around

21.42

the major axis, then x2 ra -+---=pi €2 €2-1

21.2.1

y = r cos 4;

22 -.---

72

Z=T

ra 1-72

-$

Metric Coefficientu for Oblate Spheroidal Coordinates

sin 4; 0 1 4 1 2 ~

21.4.3

where f , 7 and 4 are prolate spheroidal coordinates. f

Relations Between Cartesian and Prolate Spheroidal Coordinates

21.2.2

21.5. Wave Equation in Prolate and Oblate Spheroidal Coordinates Wave Equation in Prolate Spheroidal Coordinates

21.5.1 21.3. Definition of Oblate Spheroidal Coordinates If the system of confocal ellipses and hyperbolas referred to in 21.1.3 and 21.1.4 revolves around the minor axis, then 762

(c=k *See page 11.

jk)

753

SPHEROIDAL WAVE FUNCTION&

Wave Equation in Oblate Spheroidal Coordinates

21.5.2

(21.6.3 may be obtained from 21.6.1 by the transformations [+&it, c 4 T i c ; 21.6.4 may be obtainedfrom21.6.2 by the transformation c+Fic.) 21.7. Prolate Angular Functions 21.7.1 7) =

S%C,

21.5.2 may be obtained from 21.5.1

transforms,tions

[+*it,

21.7.2

21.6. Differential Equations for Radial and

SZi(c, 7>=

Angular Prolate Spheroidal Wave Functions @=Rmn(c, E)fJmn(C,7)

cos

(7)

=Prolate angular function of the first kind

by the

c+i=ic.

If in 21.5.1 we put

5 dY ( 4 E+,

r=O, 1

m4

then the “radial solution” Rmn(c,E ) and the “angular solution” Smn(c,7) satisfy the differential equations

=Prolate angular function of the second kind ( e ( q ) and @(q) are associated Legendre functions of the first and second kinds respectively. However, for - 12Z< 1, = ( z ) =(1-Zz)m’2dmPn( z ) / dzm(see 8.6.6). The summation is extended over even values or odd values of r.)

Recurrence Relations Between the Coefficients

21.6.1

21.6.2

2’ dY”’c)QE+,(tl)

r=--oo

ffk=

(2m+k+2) (2m+k+l)c2 (2 m+2k+3) (2 m+2k+5)

fik=(m+k)(m+k+l) 2(m+k)(m+k+l)-2m2-1

-k (2m+2k-l)(2m+2k+3) where the separation constants (or eigenvalues) A,, are to be determined so that Rmn(c,E) and Smn(c, 7) are finite at E=&1 and 7 = & 1 respectively. (21.6.1 and 21.6.2 are identical. Radial and angular prolate spheroidal functions satisfy the same differential equation over different ranges of the variable.)

‘”(2m+2k-3)

k(k-l)c2 (2m +2k- 1)

Transcendental Equation for A,,”

Differential Equations for Radial and Angular Oblate Spheroidal Functions

21.6.3

21.6.4

(The choice of r in 21.7.4 is arbitrary.)

c2

754

SPHEROIDAL WAVE FUNCTIONS Power Serieo Expansion for A.,

21.7.5

+

zo=n (n 1)

Id=

- (n- m+ 1) (n- m +2) (n+ m+ 1) (n+ m $2)

+

(n- m- 1) (n- m) (n+ m -1) (n+m)

2(2n+ 1) (2n+3)3(2n+5)

+

2(2n-3) (2%- 1)'(2n+ 1)

+ +

le= (4m2- 1) (n- m 1) (n- m +2) (n m 1) (n+ m +2) (2n-l)(2n+l)(2n+3)'(2n+5)(%+7)

[

Z8=2(4ma-1)2A+A=

1

-

m) (n+ m I) (n+ m) (2n-5)(2n-3)(21t-1)6(%+l)(2n+3)

1

1

C+a D

+

+

+

+ +

(p- m- 1) (n- m) (n m -1)(n m) (n- m 1) (n- m +2) (n m 1) (n+ m +2) (2n- 5 ) (2n -3) (2.11-1)'I(271 1) (% 3) 2(%- 1)'(271 1) (2n 3) 'I(% 5 ) (2n 4-7) a

B= (n-

C=

1 16

- (n- m -I) (n-

' + + + + + m -3) (n- m -2) (n- m -1) (n- m ) (n+ m -3) (n+ m -2) (n+ m -1) (n+ m)

+

(2?a-7) (271-5) a (%-3) a (2n -1) (2n 1)

+1)'(n- m +2)a(n+m +1) (n+m +2) -(n- m -1)"n- m) "n+ m-1)a(n+m) a (2n+ 1) (271+3) (2n+ 5 ) (271-3) (2n-1) (2n +1) m - 1) (n- m)(n- m +1)(n- m +2) (n+ m -1) (n+m) (n+ m+ 1) (n+m +2) (2n- 3) (2n -1) (2n +1)'(2n+ 3) (2nf5)

(n-m

D= (n-

a

a

a

a

'I

Asymptotic Expaneion for A,

1

($+5)--64~

a

Refinement of Approximate Valuee of

21.7.6

hn(C)=cq+m2-g

'I

A-n

If G1iis an approximation to bnobtained either from 21.7.5 or 21.7.6 then

(f+ll-32m2)

1 --1024~' [5(q4+26qa+21) -384ma($+l)]

(33qb-k 1594@+5621q)

--ma (376+167q)+g

q]

128

1 -1 [(63q6+4940q4f43327 $+22470) c4 2562

-512 ma (115q4+1310$+735)+~

(q2+1)]

+...

(527q7+61529q6+1043961$ f2241599d-32.1024 ma +29895 1q)

(573fkf+127550@

+*512 (355 $+1505q) -%]+

0(c-")

q=2(n-m)+l

(2m+r)(2m+r-l)aa w=(2m+2r-1)(2m+2r+l)

d, d,-2

(r 2 2 )

r(r- 1 (2m +r) (2m +T- 11 ~ 4 $ 2 ~ - 1)2(2m.+2r+ 1) (2m f2r-3)

755

SPHEROIDAL WAVE FUNCTIONS

Evaluation of Coefficients

21.7.15 I--

Step 1 . Calculate NT's from

21.7.8 (n-m) even

21.?.16

-2"-m

(- 1) (n+r m+ 1)~ ! + ("-:-') + ! l(n-m) ) odd n-m-1

2

do and Step 2. Calculate ratios dl from dar &+I

21.7.9

$=($)($). . . vy)

(The normalization scheme 21.7.13 and 21.7.14 is also used in [21.10].)

("8> ym)

21.7.10

&=(5) 4p+i d3 & , . . A P + l and the formula for NY in 21.7.7. The coefficients d?" are determined to within the arbitrary factor do for r even and dl for r odd. The choice of these factors depends on the normalization scheme adopted. Normalization of Angular Functions Meixner- Schiifke Scheme

where the D,(z)'s are the parabolic cylinder functions (see chapter 19).

Stratton-More-Chu-Little-Corbat6Scheme

21.7.12

E' (r+2m)! r=o,1 r!

dr=-

and the Hr(x)are the Hermite polynomials (see chapter 22). (For tables of h i r & see [21.4].)

(n+m)!

(n-m)!

(This normalization has the effect that Smn(c,q)+ W O ) as O-+1.>

Epandon of Sln(c,

r))

in Powers of 7

21.7.18 Flammsr Scheme [21.4]

21.7.13

Smn(c,

T)=(l-+)m'a

+ +

2'

r=O, 1 PT(C)O'

+ +

(r 1) (r 2)p7;:;2(c) -[r(r+2m+ 1) m (m 1)

-X , n ( ~ ) ] p y " ( ~ ) -c*p??:-"2(c)=O (n-m) even

21.7.14 9l-Sll-1

(n-m) odd The above lead to the following conditions for

dY

(The derivation of the transcendental equation for A, is similar to the derivation of 21.7.4 from 21.7.3.)

756

SPHEROIDAL WAVE FUNCTIONS

gn = -2-%~[33q'+ 114q2+37 -2m2(23q2+25)

+ 13m4]

p;"" = -2-"[63

a'+

340q4+239q2+ 14

- 10mz(10q4+23q2+3) +m4(39qZ- 18) -2me] (n-m) even

gn=v(v+m)a;'+

(U+

q = n + l for (n-m)

c:: =

l)(v+m+l)a:l

even; q=n for (n-m) odd

(For the definition of aFr see 21.8.3.)

. rPk (2m+2r+1)! (2r+l)! (-r)k(m+-T+i)F:+, 2"k!(m+k)IC 1 OD

(n-m)

Asymptotic Expansion for Oblate Angular Functions

odd

. . . (.+k+l)

(4k=4a+1)(a+2)

21.8.3 ~ , , ~ - i c ,q)-(l--*)m/2

+ (-l)n-me-c(l+v)~(m) U + # [241+?)lj

Prolate Angular Functions-Second Kind

Expansion 21.7.2 ultimately leads to

2 AY {e-""-~'~lm,![2c(l-q)1

E--#

(The d:"'s are the coefficients in 21.7.1.)

where the L!m)(z)are Laguerre polynomials (see chapter 22) and

21.7.21

(Expressions of ayfrare given in [21.4].) (The coefficients dll.. are the same as in 21.7.1; the coefficients dz; are tabulated in [21.4].)

21.9. Radial Spheroidal Wave Functions 21.9.1

21.8. Oblate Angular Functions Power Series Expansion for Eigenvalues m

L n = k C=O

21.8.1

(-l)kL~2k

8-

where the Zk'S are the same as in 21.7.5. Asymptotic Expansion for Eigenvalues [21.4]

21.8.2 X,n=-cZ+2c(2v+m+

1)-2v(v+m+

1)

-(m+ 1) + A m n 1

v = ~(n-m) for (n-m) even;

v=2 1 (n-m-1) m

A -

mn-z

&"'c-~

for (n -m) odd

(Jn+*(z)and Y,++(z) are Bessel functions, order n+3, of the first and second kind respectively

(see chapter lO).) 21.9.2

R:~(c,t )=R:i(c, [) + i R g i ( ~[),

21.9.3

R$,(c, [)=R$,!(c,4) - ~ R $ ; ( c ,()

Asymptotic Behavior of

21.9.4

*See page

and R$i(cl €1

1

R $ ~ ( c , [ ) - - COS [c[-$(n+l)r] et-)-

21.9.5

Rgi(clE )

R:~(c,€)11.

c€

1

- sin [c[-$(n+l)r]

C€

SPHEROIDAL WAVE FUNCTIONS

21.10. Joining Factors for Prolate Spheroidal Wave Functions 21.10.1

(The expression for joining factors appropriate to the oblate case may be obtained from the above formulas by the transformation c + - k . )

757

21.11. Notation Notation for Prolate Spheroidal Wave Functions I

Ang. coord.

Stratton, Morse, Chu, Little and CorbaM

9

Flammer and this chapter

9

Chu and Stratton

9

Meixner and Schafke

9

Morse and Feshbach

*=cos 6

Page

€

IndeRad. coord. pendent variable

€

Ih

€

lk-

Ang. wave function

Rad. wave function

Eigenvalue

Xmn(c) R%c,

U

Normalization of angular functions

Remarks

Smn(c,O ) = e ( O ) (n-m) ever S&, 0 )=R' (0) (n- m) odc

A mt

2 =Flammer's n Aml=Xmn

9

IL

1=Flammer's aim=Xmn-C2

Notation for

n

we Functions

Stratton, Morse, Chu, Little and Corbatb

9

1 =Flammer's n

Flammer and this chapter

9

Smn( - ic,O) =e ( 0 )

Chu and Stratton

9

S!!,)(-ic, 0) = e + 1 ( 0 ) (1 even) 1=Flammer's n - m A'$)'( -ic, 0)=e k z ( 0 ) ( I odd) B I m = - Xm, n-m

Meixner and Schafke

9

Aml=Xmn

(n-m) even Smn(-icJ O ) = P f ( O ) (n- m)odd

w-9) -- 2 (n+m)! 2n+1 ( n - m ) ! -

~

Morse and Feshbach

Le'itner and Spence The notation

?=cos 6

1=Flammer's n Amz=Xmn

1)

1=Flammer's n aim=Xmn+*

I

this chapter closely follows the notation in [21.4].

SPHEROIDAL WAVE FUNCTIONS

759

References [21.1] M. Abramowite, Asymptotic expansion of spheroidal wave functions, J. Math. Phys. 28, 195-199 (1949). [21.2] G. Blanch, On the computation of Mathieu functions, J. Math. Phys. 25, 1-20 (1946). [21.3] C. J. Bouwkamp, Theoretische en numerieke

behandeling van de buiging door en ronde opening, Diss. Groningen, Groningen-Batavia, (1941). [21.4] C. Flammer, Spheroidal wave functions (Stanford Univ. Press, Stanford, Calif., 1957). [21.5] A. Leitner and R. D. Spence, The oblate spheroidal wave functions, J. Franklin Inst. 249, 299-321 (1950). [21.6] J. Meixner and F. W. Schafke, Mathieusche

Funktionen und Sphliroidfunktionen (SpringerVerlag, Berlin, Gottingen, Heidelberg, Germany, 1954).

[21.7] P. M. Morse and H. Feshbach, Methods of

theoretical physics (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [21.8] L. Page, The electrical oscillations of a prolate spheroid, Phys. Rev. 65, 98-117 (1944). [21.9] J. A. Stratton, P. M. Morse, L. J. Chu and R. A. Hutner, Elliptic cylinder and spheroidal wave functions (John Wiley & Sons, Inc., New York, N.Y., 1941). [21.10] J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and F. J. Corbitt6, Spheroidal wave functions (John Wiley & Sons, Inc., New York, N.Y., 1956).

760

SPHEROIDAL WAVE FUNCTIONS

EIGENVALUES-PROLATE

Table 21.1

AND OBLATE

PROLATE L n ( c ) -m(m f 1)

*

Xh(C)

0 0.000000 0.319000 0.611314 0.879933 1.127734

2.000000 2.593 084 3.172127 3.736869 4.287128

2 6.0 000 00 6.5 33471 7.084258 7.649317 8.225713

3 12.000000 12.514462 13.035830 13.564354 14.100203

4 20.000000 20.5 08274 21.020137 21.535636 22.054829

1.357356 1.571 155 1.771183 1.959206 2.136732

4.822809 5.343903 5,850492 6.342739 6.820888

8.810735 9.401958 9.997251 10.594773 11.192938

14.643458 15.194110 15.752059 16.31 7122 16.889030

22.577779 23.104553 23.635223 24.169860 24.708534

10 11 12 13 14

2.305040 2.465217 2.61 8185 2.764731 2.905523

7.285254 7,736212 8.174189 8.599648 9,013085

11.790394 12.385986 12.978730 13.567791 14.152458

17.467444 18.051962 18.642128 19.237446 19.837389

25.251 312 25,798254 26.349411 26.904827 27.46453 0

15 16

3.041137 3.172067

9.41 501 0 9.805943

14.732130 15.306299

20.441413 21.048960

28.028539 28.596854 [(-;)51

C2\n

0

I 2 3 4

1

[(-;PI

[(-:)21

[( - 2 9 1

C- '[Aon(c)]

c - 1\n

0.25 0.24 0.23 0.22 0.21

0 0.79301 6 0.802442 0.811763 0.820971 0.830059

2.451485 2.477117 2.503218 2.529593 2.556036

2 3.826574 3.858771 3.895890 3.937869 3.984499

3 5,26224 5.25133 5.25040 5.26046 5.28251

4 7.14921 7.05054 6.96237 6.88638 6.82460

0.20 0.19 0.18 0.17 0.16

0.83 9025 0.847869 0.856592 0.8652 00 0.873698

2.582340 2.608310 2.633778 2.658616 2.682743

4.035382 4.089903 4.147207 4.206229 4.265772

5.31747 5.3 6610 5.42883 5.50551 5.59516

6.77941 6.75360 6.75030 6.77286 6.82451

0.1 5 0.14 0.13 0.12 0.11

0.882095 0.890399 0.898617 0.906758 0.914827

2.706127 2.728784 2.750762 2.772133 2.792971

4.324653 4.381 878 4.436798 4.489168 4.539096

5.69566 5.80359 5.91452 6.023 83 6.12 806

6.90779 7.02356 7.16962 7.33916 7.52035

0.10 0.09 0.08 0.07 0.06

0.922830 0.930772 0.938657 0.946487 0.954267

2.813346 2.833316 2.852927 2.872213 2.891203

4.586895 4.632927 4.677506 4.720863 4.763160

6.22 577 6.31730 6.40385 6.48655 6.5661 8

7.69932 7.86638 8.01951 8.16148 8.29538

0.05 0.04 0.03 0.02 0.01 0.00

0.961998 Oi969683 0.977324 0.984923 0.992481 1.00 0000

2.909920 2.928382 2.946608 2.964611 2.982404 3.000000 [( -;)91

4.804519 4.845033 4.884779 4.923820 4.962212 5.000000

6.643 26 6.71 812 6.79104 6.86221 6.93182

8.42315 8.54 594 8.66452 8.77945 8.891 16 9.00000 [( -;)41

*See page 11.

1

761

SPHEROIDAL WAVE FUNCTIONS

EIGENVALUES-PROLATE

AND OBLATE

Table 21.1

OBLATE Am"(

-ic) - m(m i o n ( ic)

-

+ 1)

*

0 1 2 3 4

0 0.000000 -0.348602 -0.729391 -1.144328 -1.594493

1 2.000000 1.393206 0.773097 +0.140119 -0.505243

6.000000 5.486800 4.996484 4.531027 4.091509

3 12.000000 11.492120 10.990438 10.494512 10.003863

4 20.000000 19.495276 18.994079 18.496395 18.002228

5 6 7 8 9

-2.079934 -2.599668 -3.1 51 841 -3.733981 -4.343292

-1.162477 -1.831050 -2.510421 -3.200049 -3.899400

3.677958 3.289357 2.923796 2.578730 2.251269

9.51 7982 9.036338 8.558395 8.083615 7.611465

17.511597 17.024540 16.541110 16.061382 15.585448

10 11 12 13 14

-4.976895 -5.632021 -6.306116 -6.996903 -7.702385

-4.607952 -5.325200 -6.05 0659 -6.783867 -7.524384

1.938419 1.637277 1.345136 1.059541 0.778305

7.141 427 6.6730 01 6.205705 5.739084 5.272706

15.1 13424 14.645441 14.1 81652 13.722230 13.267364

15 16

-8.420841 -9.150793

-8.271 79 5 -9.02571 0

0.499495 0.221407

4.8061 65 4.339082

12.817261 12.3721 44

c2\,n

~

4

1

"-;)'I

2

~ C-'[Xon(

3

1

[(-:)*I

"-:)7

-ic)]

0.25 0.24 0.23 0.22 0.21

-0.571924 -0.585248 -0.599067 -0.613 349 -0.628058

1 -0.564106 -0.579552 -0.595037 -0.610591 -0.626242

+O. 013837 -0.009136 -0.031481 -0.053477 -0.075480

3 0.2711 92 0.213225 0.157464 0.103825 0.052196

4 0.77325 0.67822 0.58772 0.50191 0.42099

0.20 0.19 0.18 0.17 0.16

-0.643161 -0.658625 -0.674418 -0.690515 -0.706891

-0.642016 -0.657938 -0.674031 -0.690310 - 0.7 06792

-0.097943 -0.121 428 -0.146603 -0.174201 -0.204894

+O. 002437 -0.045635 -0.092251 -0.137692 -0.182301

0.34521 0,27490 0.21043 0.15215 0.10020

0.15 0.14 0.13 0.12 0.11

-0.72353 0 -0.740416 -0.757541 -0.774896 -0.792476

-0,723486 -0.740399 -0.757535 -0.774894 -0.792476

-0.2391 09 -0.276886 -0.317881 -0.361548 -0.407352

-0.226469 -0.2 70627 -0.31 5206 -0.360594 -0.407081

0.05428 +0.01332 0.02 476 -0.06337 -0.10723

0.10 0.09 0.08 0.07 0.06

-0.81 0279 -0.828301 -0.846539 -0.864992 -0.883657

-0.810279 -0.828301 -0.846539 -0.864992 -0.883657

-0.454896 -0.503937 -0.554337 -0.606021 -0.658931

-0.454839 -0.503928 -0.554337 -0.606021 -0.658931

-0,16065 -0.22419 -0.2951 3 -0.37117 -0.45125

0.05 0.04 0.03 0.02 0.01 0.00

-0.902532 -0.921 616 -0.940906 -0.960402 -0.980100 -1.0 00000

-0.902532 -0.921616 -0.940906 -0.960402 -0.980100 -1.000000

-0.713025 -0.768262 -0.824608 -0.882031 -0.940503 -1.000000 [( -;)41

-0.713025 -0.768262 -0.824608 -0.882031 -0.940503 -1.000000

-0.53495 -0.62200 -0.71218 -0.80533 -0.90131 -1.000 00

c - l\n

0

'See puge 11.

2

-

I":-

762

SPHEROIDAL WAVE FUNCTIONS

Table 21.1

EIGENVALUES-PROLATE

AND OBLATE

PROLATE Am.(C)--m(?n+l) X,n(c) - 2

2

* * 4

4

1 0.000000 0.195548 0.382655 0.5 6 19 75 0.734111

4.000000 4.424699 4.841 7 18 5.251162 5.653149

3 10.000000 10.467915 10.937881 11.409266 11.881493

18.000000 18.481696 18.965685 19.451871 19.940143

5 28.000000 28.488065 28.977891 29.469456 29.962738

5 6 7 8 9

0.899615 1.058995 1.21 2711 1.361183 1.504795

6.047807 6.435272 6.815691 7.1 89213 7.555998

12.354034 12.826413 13.2 98196 13.768997 14.238466

20.430382 2 0.922458 21.416235 21.911569 22.40 8312

30.457716 30.954363 31.452653 31.952557 32.454044

10 11 12 13 14

1.643895 1.778798 1.909792 2.037141 2.1 61081

7.916206 8.2 70004 8.617558 8.959038 9.294612

14.706292 15.172199 15.635940 16.097297 16.556078

22.906311 23.405410 23,905451 24.406277 24.907729

32.957080 33.461629 33.967652 34.475109 34.983956

15 16

2.281832 2.399593

9.624450 9.948719 [(-;v]

17.012115 17.465260

25.409649 25.911881

35.494147 36.005634

c2\

n

0 1 2

3

"-:"I

[( -,4)41

[(-:)31 C'[Xln(c)

-21

[(-:PI

*

1

2

3

4

0.25 0.24 0.23 0.22 0.21

0.599898 0.613295 0.627023 0.641073 0.655431

2.487179 2.491544 2.497852 2.506130 2.51 63 83

4.36631 5 4.338520 4.315609 4.297923 4.285792

6.47797 6.38296 6.29522 6.21556 6.14494

5 9.00140 8.80891 8.62445 8.44916 8.28436

0.20 0.19 0.18 0.17 0.16

0.670084 0.685014 0.7 002 04 0.715632 0.731281

2.528591 2.542705 2.558644 2.576296 2.595516

4.279522 4.279366 4.285495 4.297965 4.316672

6.08438 6.03498 5.99788 5.97420 5.96496

8.1 3 163 7.99282 7.87010 7.76598 7.68328

0.15 0.14 0.13 0.12 0.11

0.747129 0.763159 0.779353 0.795696 0.812174

2.616135 2.637968 2.660829 2.684536 2.708934

4.341320 4.371397 4.406191 4.444844 4.486445

5.97090 5.99230 6.02874 6.07889 6.1 4051

7.62508 7.59446 7.59407 7.62539 7.68773

0.10 0.09 0.08 0.07 0.06

0.828776 0.845493 0.86231 6 0.879237 0.896251

2.733891 2.759305 2.785099 2.811212 2.837600

4.530151 4.575277 4.621329 4.667984 4.71 5031

6.21063 6.28624 6.36482 6.44473 6.52505

7.77728 7.88714 8.00897 8.1 3 579 8.26355

0.05 0.04 0.03 0.02 0.01 0.00

0.913352 0.930535 0.947796 0.965129 0.982531 1.000000

2.864224 2.891056 2.91 8069 2.945243 2.972 5 5 8 3.000000

4.762333 4.809790 4.857332 4.904906 4.952472 5.000000

6.60532 6,68528 6.76480 6.84378 6.92219 7.00000

8.39048 8.51592 8.63963 8.76153 8.88164

c-l\n

i

[( - p 4 1 *See page 11.

763

SPHEROIDAL WAVE FUNCTIONS

EIGENVALUES-PROLATE

Table 21.1

AND OBLATE

OBLATE

- 7n(mt 1) XI .c- ic) - 2

Xm"(-ic)

*

*

0 1 2 3 4

1 0.00 0000 -0.204695 -0.419293 -0.644596 -0.881446

2 4.000000 3.567527 3.127202 2.678958 2.222747

3 10.0 00000 9.534818 9.0731 04 8.615640 8.1 63245

18.000000 17.520683 17.043817 16.569461 16.097655

5 28.000000 27.51 3713 27.029223 26.546548 26.065 706

5 6 7 8 9

-1.130712 -1.3 93280 -1.670028 -1.961809 -2.2 69420

1.758534 1.286300 0.806045 +O. 317782 -0.1 78458

7.716768 7.277072 6.845015 6.421425 6.007074

15.628426 15.161 786 14.697727 14.236229 13.777252

25.586715 25.109592 24.634357 24.161031 23.689634

10 11 12 13 14

-2.593577 -2.934882 -3.293803 -3.670646 -4.065548

-0.68263 0 -1.194673 -1.714511 -2.242055 -2.777205

5.602649 5.208724 4.825732 4.453947 4.093464

13.320743 12.866634 12.414640 11.965266 11.517803

23.220190 22.752726 22.287271 21.823856 21.362516

15 16

-4.4 78470 -4.909200

-3.319848 -3.869861 [(-$)l]

3.744202 3.405903

11.072331 10.628718

20.903290 20.446222

c2 \n

"-:"I

4

~ c-"[X1,(-ic)

-21

3

1

~ 2 3 1

*

0.25 0.24 0.23 0.22 0.21

.1 -0.306825 -0.318148 -0.330984 -0.345469 -0.361702

2 -0.241866 -0.266693 -0.291340 -0.31 5894 -0.34 0450

3 0.21286 0.17062 0.1 3125 0.09476 0.06107

4 0.66429 0.57759 0.49460 0.41533 0.33974

5 1.2778 1.1420 1.0120 0.8879 0.7697

0.20 0.19 0.18 0.17 0.16

-0.3 79735 -0.399564 -0.4211 25 -0.444308 -0.468974

-0.365113 -0.389998 -0.415222 -0.440907 -0.4671 66

0.03001 +O.O 0127 -0,02563 -0.05142 -0,07710

0.26779 0.19942 0.13449 0.07282 +0.01411

0.6575 0.5515 0.4520 0.3591 0.2735

0.15 0.14 0.13 0.12 0.11

-0.494976 -0.522180 -0.550474 -0.579775 -0.610027

-0.4941 04 -0.521805 -0.550335 -0.579732 -0.61 0016

-0.1 0406 -0.13412 -0.16924 -0.21076 -0.25868

-0.04205 -0.09625 -0.14929 -0.20210 -0.25572

0.1958 0.1271 0.0680 +0.0183 -0.0250

0.10 0.09 0.08 0.07 0.06

-0.641193 -0,673251 -0.706186 -0.739985 -0.774638

-0.641191 -0.673251 -0.706186 -0.739985 -0.774638

-0,31185 -0.36901 -0.42934 -0.49242 -0.55807

-0.31111 -0.36888 -0.4293 2 -0.49242 -0.55807

-0.0685 -0.1219 -0.1907 -0.2714 -0.3598

0.05 0.04 0.03 0.02 0.01 0.00

-0.810135 -0.846468 -0.883628 -0.921608 -0.960401 -1.000000

-0.810135 -0.846468 -0.883628 -0.921608 -0.960401 -1.000000

-0.62616 -0.69657 -0.76923 -0.84406 -0.92100 -1.00000

-0.62616 -0.69657 -0.76923 -0.84406 -0.92100 -1.0 0000

-0.4542 -0.5540 -0.6588 -0.7682 -0.8820 -1.0000

c-l\n

*See page XI.

"-:)21

764

SPHEROIDAL WAVE FUNCTIONS

Table 21.1

EIGENVALUES-PROLATE

AND OBLATE

PROLATE

Ln.(C)-m((m+l)

*

-6

*

bn(C)

0 1 2 3 4

2 0.000000 0.140948 0.2 78219 0.41 2 006 0.542495

3 6.000000 6.331101 6.657791 6.980147 7.2982 50

4 14.0000 00 14.402353 14.804100 15.205077 15.605133

5 24.000000 24.436145 24.872744 25.309731 25.747043

6 36.000000 36.454889 36.910449 37.366657 37.823486

5 6 7 8 9

0.669857 0.794252 0.91 5832 1.034738 1.151100

7.61 21 79 7.922016 8.227840 8.5 29734 8.827778

16.004126 16.401931 16.798429 17.193516 17.587093

26.184612 26.6 22373 27.060261 27.498208 27.936151

38.280913 38.738910 39.197451 39.656510 40.116059

10 11 12 13 14

1.265042 1.376681. 1.486122 1.593469 1.698816

9.122052 9.412636 9.699610 9.983 052 10.263039

17.979073 18.3 69377 18,757932 19.144675 19.529549

28.374023 28.811761 29.249302 29.686584 30.123544

40.576070 41.036514 41.497364 41.958589 42.420160

15 16

1.802252 1.903860

10.539650 10.812958

19.91 2501 20.293486

30.5601 25 30.996267

42.882048 43.344222

cz\n

c-'[bn(c) -61 c-'\n

*

0.25 0.24 0.23 0.22 0.21

2 0.475965 0.489447 0.503526 0.518220 0.533551

3 2.703239 2.683149 2.665356 2.650003 2.637236

4 5.073371 4.994116 4.919290 4.849313 4.784640

5 7.74906 7.58138 7.41971 7.26479 7.11743

0.20 0.19 0.18 0.17 0.16

0.549534 0.566185 0.583513 0.601526 0.620224

2.627196 21620017 2.615819 2.614701 2.616735

4.725757 41673177 4.627427 4.589031 4.558480

6.97858 6184931 6.73081 6.62442 6.53155

9.5023 9.2649 9.0409 818323 8.6417

0.15 0.14 0.13 0.12 0.11

0.639604 0.659659 0.680376 0.701737 0.723722

2.621954 2.630349 2.641862 2.656384 2.673764

4.536196 4.522485 4.517479 4,521086 4.532956

6.45371 6.39236 6.34878 6.32389 6.31794

8.4718 8.3260 8.2078 8.1208 8.0678

0.10 0.09 0.08 0.07 0.06

0.746308 0.769471 0.793186 0.817429 0.842175

2.693817 2.716339 2.741120 2.767960 2,796673

4.552484 4.578871 4.611219 4.648642 4.690346

6.33030 6.35935 6.40263 6.45738 6.52096

8.0507 8.0688 8;1184 8.1932 8.2864

0.05 0.04 0.03 0.02 0.01 0.00

0.867402 0.893087 0.919209 0.945747 0.972684 1.000000

2.827089 2.859059 2.892449 2,927138 2.963019 3.000000

4.735658 4.784022 4.834980 4.888160 4.943252 5.000000

6.591 27 6.66670 6.74607 6.82 849 6.91 330

8.3919 8.5057 8.6249 8.7477 8.8730

*See page 11.

6 10.8360 10.5536 10.2781 10.0103 9.7512

765

SPHEROIDAL WAVE FUNCTIONS

EIGENVALUES-PROLATE OBLATE

hmm(-i c )

-m(m+ 1)

b r n ( - i ~-6 ) c2\n

AND OBLATE

Table 21.1

* *

4

0 1 2 3 4

2 0.000000 -0.14483 7 -0.293786 -0.447086 -0.604989

3 6.000000 5.664409 5.324253 4.979458 4.629951

14.000000 13.597220 13.194206 12.791168 12.388328

5 24.000000 23.564371 23.129322 22.694912 22.2 61201

6 36.0 00000 35.545806 35.092330 34.639597 34.187627

5 6 7 8 9

-0.767764 -0.935698 -1.1 09090 -1.288259 -1,473539

4.275662 3.916525 3.552475 3.183450 2.809393

11.985928 11.584224 11.183489 10.784014 10.3 86106

21.828245 21.396098 20.964812 20.534436 20.105013

33.736444 33.286069 32.836522 32.387826 31.940000

10 11 12 13 14

-1.665278 -1.863838 -2.069595 -2.2 82933 -2.504245

2.430250 2.045970 1.656508 1.261822 0.861875

9.990084 9.596286 9.205059 8.816762 8.431761

19.676587 19.249195 18.822869 18.397640 17.973532

31.493066 31.047043 30.601952 30.157814 29.714648

15 16

-2.733927 -2.972375 (-y]

0.456635 0.046076

8.050424 7.673121

17.550565 17.128753

29.272476 28.831317

[

[1(-:)71

~ c - ~ [Xi.(

5

-i ~-)61

1 l':-C

*

c - 1\n 0.25 0.24 0.23 0.22 0.21

2 -C.185773 -0.190754 -0.196680 -0.203790 -0,212386

3 +0.002879 -0.030028 -0.062228 -0.093813 -0.124893

4 0.47957 0.41280 0.34933 0.28933 0.23297

5 1.07054 0.95365 0.84167 0.73461 0.632 51

6 1.8019 1.6261 1,4577 1.2965 1.1428

0.20 0.19 0.18 0.17 0.16

-0.222841 -0.235596 -0.251126 -0,269873 -0.292149

-0.155607 -0.1 86120 -0.216631 -0.247375 -0.278624

0.1 8049 0.13215 0.08816 0.04864 +O. 01342

0.53537 0.44322 0.35607 0.2 7389 0.19662

0.9964 0.8574 0.7260 0.6022 0.4863

0.15 0.14 0.13 0.12 0.11

-0.318047 -0.347414 -0.379928 -0.41 5213 -0.452947

-0.310677 -0.343847 -0.378432 -0.414688 -0.452800

-0.01 813 -0.04727 -0.07609 -0.10778 -0.14643

0.12409 ~0.05600 -0.00822 -0.06954 -0.12937

0.3785 0.2795 0.1901 0.1120 +0.0470

0.10 0.09 0.08 0.07 0.06

-0,492902 -0.534942 -0.578991 -0.625006 -0.672956

-0.492871 -0.534937 -0.578991 -0.625006 -0.672956

-0.19508 -0.25333 -0.31876 -0.38955 -0.46494

-0.18959 -0.2521 7 -0.31861 -0.38955 -0,46494

-0,0051 -0.0517 -0.1076 -0.1844 -0.2768

0.05 0.04 0.03 0.02 0.01 0.00

-0.722813 -0.774556 -0.828164 -0.883618 -0.940902 -1.000000

-0.722813 -0.774556 -0.828164 -0.883618 0.94 0902 -1.000000

-0.54456 -0.62821 -0.71571 -0.80691 -0.901 71 -1.00 000

-0.54456 -0.62821 -0.71571 -0.80691 -0.90171 -1.00000

-0.3791 -0.4895 -0.6073 -0.7319 -0.8629 -1.0000

+See psge 11.

-

766

SPHEROIDAL WAVE FUNCTIONS

Table 21.2

ANGULAR FUNCTIONS-PROLATE

AND OBLATE

PROLATE

Smn(c,cos e) m

n

c\e

0"

10"

20"

30"

40"

0

0

1 2 3 4 5

0.8481 0.5315 0.2675 0.1194 0.0502

50"

60"

0.8525 0.5431 0.2815 0.1312 0.0585

0.8651 0.5772 0.3242 0.1689 0.0861

0.8847 0.6320 0.3967 0.2379 0.1419

70"

80"

90"

0.9091 0.7032 0.4980 0.3442 0.2380

1 2 3 4 5

0.9046 0.6681 0.4034 0.2042 0.0916

0.8936 0.6665 0.4099 0.2138 0.1001

0.8602 0.6598 0.4273 0.2415 0.1262

0.8035 0.6429 0.4489 0.2833 0.1703

0.7225 0.6081 0.4630 0.3294 0.2279

0.9815 0.9355 0.8805 0.8271 0.7776 0.3381 0.3270 0.3110 0.2929 0.2752

1.000 1.000 1.000 1.000 1.0 00

1

0.9606 0.8654 0.7571 0.6589 0.5742 0.4878 0.4540 0.4068 0.3566 0.3104

0.9952 0.9831 0.9682 0.9530 0.9383

0

0.9354 0.7842 0.6226 0.4885 0.3839 0.6169 0.5472 0.4543 0.3618 0.2840

0.1731 0.1717 0.1695 0.1669 0.1643

0

2

1 2 3 4 5

1.022 1.064 1.041 0.8730 0.6018

0.9795 1.030 1.023 0.8768 0.6233

0.8553 0.9271 0.9640 0.8787 0.6792

0.6621 0.7579 0.8497 0.8513 0.7407

0.4198 0.5296 0.6660 0.7549 0.7537

0.1556 0.2602 0.4104 0.5553 0.6494

-0.0988 -0.0192 +0.1061 0.2512 0.3844

-0.3105 -0.2668 -0.1938 -0.0998 +0.0008

-0.4509 -0.4385 -0.4171 -0.3879 -0.3542

0

3

1 2 3 4 5

0.9892 0.9590 0.9090 0.8197 0.6650

0.9042 0.8864 0.8546 0.7877 0.6560

0.6692 0.6816 0.6957 0.6868 0.6183

0.3400 0.3840 0.4485 0.5087 0.5245

-0.0045 +0.0560 0.1501 0.2591 0.3482

-0.2816 -0.2261 -0.1364 -0.0215 + 0.0 971

-0.4259 -0.3907 -0.3319 -0.2514 -0.1575

-0.4085 -0.3949 -0.3714 -0.3376 -0.2952

-0.2467 -0.2447 -0.2412 -0.2361 -0.2293

0.1578 0.1194 0.0776 0.0449 0.0239

0.3134 0.2437 0.1654 0.1018 0.0588

0.4643 0.3757 0.2724 0.1832 0.1179

0.6067 0.5149 0.4030 0.2994 0.2162

0.7355 0.6562 0.5546 0.4537 0.3650

0.8450 0.7892 0.7144 0.6353 0.5602

0.9290 0.9000 0.8597 0.8150 0.7698

0.9819 0.9740 0.9627 0.9497 0.9361 0.5119 0.5088 0.5039 0.4979 0.4911

1

1

1 2 3 4 5

1

2

1 2 3 4 5

0 0 0 0

0.4788 0.3896 0.2780 0.1762 0.1011

0.9054 0.7509 0.5538 0.3683 0.2254

1.232 1.052 0.8148 0.5813 0.3896

1.417 1.253 1.030 0.7968 0.5906

1.435 1.316 1.149 0.9643 0.7879

1.276 1.212 1.118 1.008 0.8957

0.9562 0.9335 0.8992 0.8575 0.8127

0

-0.50 00 -0.5000 -0.5000 -0.5000 -0.5000 0

0 0 0

0

1,000 1.000 1.000 1.000 1.000 0 0 0 0 0

1

3

1 2 3 4 5

0 0 0 0 0

0.9928 0.9559 0.8745 0.7393 0.5662

1.745 1.710 1.611 1.418 1.146

2.075 2.092 2.063 1.934 1.691

1.903 1.998 2.097 2.128 2.047

1.280 1.432 1.640 1.841 1.975

0.3775 0.5298 0.7606 1.032 1.299

-0.5521 -0.4541 -0.2972 -0.0951 +0.1319

-1.244 -1.214 -1.174 -1.097 -1.017

-1.500 -1.500 -1.500 -1.500 -1.500

2

2

1 2 3 4 5

0 0

0 0 0

0.0844 0.0690 0.0500 0.0328 C.0198

0.3295 0.2744 0.2051 0.1405 0.0898

0.7111 0.6092 0.4773 0.3487 0.2414

1.189 1.054 0.8738 0.6876 0.5212

1.710 1.572 li380 1.171 0.9701

2.211 2.101 1.944 1.764 1.580

2.627 2.566 2.475 2.367 2.251

2.903 2.886 2.859 2.827 2.791

3.000 3.000 3.000 3.000 3.000

1 2 3 4 5

0 0 0 0 0

0.4222 0.3597 0.2765 0.1934 0.1244

1.570 1.358 1.070 0.7758 0.5226

3.116 2.755 2.255 1.723 1.243

4.596 4.175 3.576 2.909 2.269

5.530 5.170 4.641 4.025 3.395

5.548 5.327 4.994 4.588 4.150

4.501 4.417 4.286 4.122 3.936

2.522 2.510 2.491 2.466 2.437

2

3

0 0 0 0 0

From C. Flammer, Spheroidal wave functions. Stanford Univ. Press, Stanford, Calif., 1957 (with permission).

767

SPHEROIDAL WAVE FUNCTIONS

ANGULAR FUNCTIONS-PROLATE OBLATE Smn(-%

m 0

n 0

AND OBLATE

Table 21.2

11)

c\tl

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1

1.000 1.000 1,000 1,000 1.000

1.002 1.008 1.022 1.047 1.083

1.007 1.032 1.089 1.191 1.341

1.016 1.073 1.205 1.449 1.835

1.028 1.132 1.377 1.854 2.648

1.044 1.210 1.617 2.452 3.952

1.064 1.310 1.940 3.319 6,000

1.088 1.434 2.366 4.557 9.211

0.1001 0.1004 0.1011 0.1016 0.1032

0.2009 0.2034 0.2079 0.2150 0.2252

0.3027 0.3114 0.3273 0.3526 0.3884

0.4065 0.4274 0.4664 0.5298 0.6252

0.5128 0.5542 0.6338 0.7681 0.9804

0.6222 0.6952 0.8398 1.096 1.525

0.7353 0.8539 1.098 1.552 2.369

0.8530 1.035 1.425 2.195 3.684

0.9760 iI241 1.842 3.105 5.741

1.105 ~ _ . _ 1.484 2.378 4.396 8.970

2

3 4 5

0.8 1.115 1.585 2.923 6.323 14.23

0.9

1.0

1.147 1.767 3.648 8.837 22.11

1.183 1.986 4.589 12.42 34.48

0

2

1 2 3 4 5

-0.5000 -0.5000 -0.5000 -0.5000 -0.5000

-0.4863 -0.4897 -0.4943 -0.4994 -0.5061

-0.4450 -0.4585 -0.4766 -0.4966 -0.5234

-0.3757 -0.4052 -0.4448 -0.4891 -0.5495

-0.2779 -0.3277 -0.3952 -0.4716 -0.5780

-0.1507 -0.2231 -0.3223 -0.4356 -0.5977

+0.0070 -0.0872 -0.2183 -0.3681 -0.5869

0.1965 +0.0849 -0.0721 -0.2485 -0.5067

0.4197 0.2999 +0.1311 -0.0458 -0.2880

0.6784 0.5660 0.3845 0.2868 0.1892

0.9749 0.8930 0,7958 0.8201 1.132

0

3

1 2

0 0

4 5

0 0

-0.1477 -0.1480 -0.1486 -0.1495 -0.1504

-0.2810 -0.2839 -0.2885 -0.2949 -0.3033

-0.3855 -0.3947 -0.4097 -0.4306 -0.4589

-0.4466 -0.4668 -0.4998 -0.5415 -0.6123

-0.4491 -0.4839 -0.5421 -0.6270 -0.7489

-0.3768 -0.4275 -0.5140 -0.6432 -0.8356

-0.2130 -0.2757 -0.3841 -0.5540 -0.8080

+0.0600 -0.0015 -0.1091 -0.2765 -0.5447

0.4613 0.4274 Oi3711 0.2912 0.1715

1.011 1.051 lil38 1.327 1.723

0.9961 0.9994 1.006 1;020 1.041

0.9838 0.9973 1.025 1.079 1.174

0.9628 0.9923 1.055 1.178 1.406

0.9316 0.9827 1.093 1.319 1.776

0.8884 0.9652 1.135 1.498 2.242

0.8299 0.9340 1.172 1.708 2.878

0.7506 0.8802 1.188 1.920 3,642

0.6402 0.7864 1.149 2.067 4.400

0.4731 0.6118 0.9724 1.950 4.651

0 0 0 0 0

0.2987 0.2985 0.3005 0.3022 0.2990

0.5897 0.5950 0.6043 0.6213 0.6400

0.8643 0.8815 0.9140 0.9640 1.040

1.113 1.153 1.228 1.349 1.537

1.322 1.398 1.541 1.780 2.165

1.478 1.600 1.837 2.250 2.947

1.554 1.730 2.082 2.723 3.868

1.508 1.734 -- 2.200 3.092 4.786

1.247 1.487 2;ooo 3.033 5.138

0 0

0 0 0

3

1

1

1 2 3 4 5

0

1.000 1.000 1.000 1.000 1.000

5

0

~

1

3

1 2 3 4 5

-1,500 -1.500 -1.500 -1.500 -1.500

-1.421 -1.431 -1.447 -1.467 -1.486

-1.189 -1.228 -1.289 -1.364 -1.442

-0.8136 -0.8941 -1.024 -1.184 -1.353

-0.3165 -0.4427 -0.6502 -0.9148 -1.198

0.2710 +0.1060 -0.1738 -0.5415 -0.9435

0.9015 0.7174 +0.3916 -0.0538 -0.5506

1.501 1.329 1.006 0.5403 0.0161

1.946 1.826 1.572 1.177 0.7471

1.988 1.951 1.834 1.619 1.439

0 0 0 0 0

2

2

1 2 3 4 5

3.000 3.000 3.000 3.000 3.000

2.972 2.979 2.992 3.013 3.052

2.889 2.915 2.965 3.052 3.211

2.748 2.805 2.915 3.111 3.469

2.549 2.644 2.830 3.170 3.813

2.291 2.425 2.693 3.200 4.202

1.970 2.138 2.481 3.157 4.564

1.585 1.770 2.161 2.966 4.746

1.131 1.305 1.687 2.512 4.460

0.6041 0.7234 0.9944 1;615 3.188

0 0

1.486 1.488 1.494 1.498 1.509

2.886 2.906 2.943 2.996 3.073

4.115 4.180 4.295 4.475 4.738

5.086 5.226 5.482 5.891 6.515

5.704 5.954 6.413 7.166 8.347

5.877 6.251 6.951 8.132 10.07

5.503 5.982 6,904 8.515 11.28

4.477 4.990 6.008 7.857 11.21

2.683 3.077 3.879 5.408 8.354

0 0 0 0 0

0

0 0

i

768

SPHEROIDAL WAVE FUNCTIONS

PROLATE RADIAL FUNCTIONS-FIRST

Table 21.3

n 0

c\t

0

0

1

1 2 3 4 5

m

0

2

0

3

1

1.005

-1)3.153 -1 5.289 -1 6.064 -1 5.892 -1 5.381

/i

1.020

1.044

-1 3.190 -1 3.249 1-11 5.298 1-11 5.308 -1 5.960 -1 5.786 -1 5 612 -1 5 162 -11 4:888 -11 4:125

[

[

1.077

/ [

-1 3.328 -11 5.311 -1 5.529 -1 4.542 -11 3.137

AND SECOND KINDS

1.005

1.020

/

;:?!I 0 -1.133

1

[

-1 -6 741 -1) -4:293

[

1%;

-1 -7.365 -1 -3.528 -11-1.390

1.044

1

I?:67?

-1 -4.987 -1 -11534 -21 3.87

1.077

0 2 920 1-11 19:216 -1 -3.207

):1

-!:;94

1-2)4.470

4 1-112.107 3 -2 9.956 5

1

1

1

2

1

3

2

2

2

3

-1 3.298

1 (-4)6.612

-4 3.845 -4 8.736 1-91 2.896 1-31 6.525 -3 8.889 -2 1.974 -2) 11862 -21 47048 -2 3.150 -2 6.657

From C. Flammer, Spheroidal wave functions. Stanford Univ. Press, Stanford, Calif., 1957 (with permission).

769

SPHEROIDAL WAVE FUNCTIONS

OBLATE RADIAL FUNCTIONS-FIRST

AND SECOND KINDS

REi(-ic, it)

Rzi(-ic, it) nL

n

c\l

0

0

0.2 0. 5 0.8 1.0 1.5 2. 0 2.5

0

1

0.2 0.5 0.8 1.0 1.5 2. 0 2.5

0

2

0.2 0. 5 0. 0 1.0 1.5 2.0 2.5

1

1

0. 2 0.5 0. 0 1. 0 1.5 2.0 2.5

1

2

0.2 0.5 0. 8 1. 0 1.5 2. 0 2.5

1

3

0.2 0. 5 0. 8 1.0 1.5 2.0 2.5

2

2

0.2 0. 5 0. 0 1. 0 1.5 2.0 2.5

0

-2 -1 -1 -1 -1 -1 -1

0.75

0

0.75

6.6454 1.6336 2.5333 3.0762 4.1700 4.8229 5.0170 0 0 0 0 0 0 0

-1) -3.3594

PROLATE JOINING FACTORS-FIRST C

Table 21.4

11

KIND

Table 21.5

K$~(c)

p 13

(1)

K22

From C. Flammer, Spheroidal wave functions. Stanford Univ. Press, Stanford, Calif., 1957 (with permission),

.

22 Orthogonal Polynomials URS W. HOCHSTRASSER

Contents Mathematical Properties . . . . . . . . . . . . . . . . . . . . 22.1. Definition of Orthogonal Polynomials . . . . . . . . . . 22.2. Orthogonality Relations . . . . . . . . . . . . . . . . 22.3. Explicit Expressions . . . . . . . . . . . . . . . . . . 22.4. Special Values . . . . . . . . . . . . . . . . . . . . . 22.5. Interrelations . . . . . . . . . . . . . . . . . . . . . 22.6. Differential Equations . . . . . . . . . . . . . . . . . 22.7. Recurrence Relations . . . . . . . . . . . . . . . . . 22.8. Differential Relations . . . . . . . . . . . . . . . . . 22.9. Generating Functions . . . . . . . . . . . . . . . . . 22.10. Integral Representations . . . . . . . . . . . . . . . 22.11. Rodrigues' Formula . . . . . . . . . . . . . . . . . . 22.12. Sum Formulas . . . . . . . . . . . . . . . . . . . . 22.13. Integrals Involving Orthogonal Polynomials . . . . . . . 22.14. Inequalities . . . . . . . . . . . . . . . . . . . . . 22.15. Limit Relations . . . . . . . . . . . . . . . . . . . 22.16. Zeros . . . . . . . . . . . . . . . . . . . . . . . . 22.17. Orthogonal Polynomials of a Discrete Variable . . . . . . NumericalMethods . . . . . . . . . . . . . . . . . . . . . . 22.18. Use and Extension of the Tables . . . . . . . . . . . . 22.19. Least Square Approximations . . . . . . . . . . . . . 22.20. Economization of Series . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . Table 22.1. Coefficients for the Jacobi Polynomials Pg*fl)(z) . . . . .

Page

773 773 774 775 777 777 781 782 783 783 784 785 785 785 786 787 787 788 788 788 790 791 792 793

n=0(1)6

Table 22.2. Coefficients for the Ultraspherical Polynomials CP)(2) and for znin Terms of Cg)(z). . . . . . . . . . . . . . . . . . . .

794

n=0(1)6

Table 22.3. Coefficients for the Chebyshev Polynomials T,(z) and for z n i n Terms of T,,,(z) . . . . . . . . . . . . . . . . . . . . .

795

n=0(1)12

Table 22.4. Values of the Chebyshev Polynomials T,,(z) . . . . . . . n=0(1)12. z=.2(.2)1, 10D Table 22.5. Coefficients for the Chebyshev Polynomials U ,(z) and for z"in Terms of U,,,(z) . . . . . . . . . . . . . . . . . . . . n=0(1)12

.

Table 22.6. Values of the Chebyshev Polynomials U&) n=0(1)12. z=.2(.2)1, 10D

795

796

. . . . . .

796

.

1 Guest Worker. National Bureau of Standards. from The American University (Preaently. Atomic Energy Commission. Switzerland.) 77 1

772

ORTHOGONAL POLYNOMIALS

Table 22.7. Coefficients for the Chebyshev Polynomials C,(z) and for z" in Terms of C,(z) .....................

Page

797

n=0(1)12

Table 22.8. Coefficients for the Chebyshev Polynomials S,(z) P i n T e r m s of S,(z) . . . . . . . . . . . . . . . . . .

and for

...

797

n=0(1)12

Table 22.9. Coefficients for the Legendre Polynomials P,(z) and for z" in Terms of P,(z) . . . . . . . . . . . . . . . . . . . . . .

798

n=0(1)12

Table 22.10. Coefficients for the Laguerre Polynomials L,(z) and for z" in Terms of L,(z) . . . . . . . . . . . . . . . . . . . . . .

799

n=0(1)12

Table 22.11. Values of the Laguerre Polynomials L,(z) . . . . . . . n=0(1)12, 2=.5, 1, 3, 5, 10, Exact or lOD Table 22.12. Coefficients for the Hermite Polynomials El&) and for zn in Terms of H,(z) . . . . . . . . . . . . . . . . . . . . . .

800

801

n=0(1)12

Table 22.13. Values of the Hermite Polynomials HJx) n=0(1)12, z=.5, 1, 3, 5 , 10, Exact or 11s

. . . . . . .

802

22. Orthogonal Polynomials Mathematical Properties where g(z) is a polynomial in x independent of n.

22.1, Definition of Orthogonal Polynomials

A system of polynomialsf,(z), degree [fn(z)]=n, is called orthogonal on the interval u < z < b , with respect to the weight function w(x), if

{g}

The system

consists again of orthogonal

polynomials.

22.1.1

l

W(z>jn(z>jm(z>dz=O (n#m;n,m=O,1,2,.

. .)

The weight function w(z)[w(z) 201 determines the systemfn(z) up to a constant factor in each polynomial. The specification of these factors is referred to as standardization. For suitably standardized orthogonal polynomials we set 22.1.2

1

w(z)f?,(z)dz=hn,fn(x)=knz"+k~sn-l+. . . (n=O,1,2,

* *

.)

These polynomials satisfy a number of relationships of the same general form. The most important ones are: Differential Equation

22.1.3

g,(z))f::+gl(s>f:+a~n=o

where g&), gr(x)are independent of n and un a constant depending only on n. Recurrence Relation

where 22.1.5

-I

Rodriguea' Formula

FIGURE 22.1. Jacobi Polynomials P!ei e)(x>, a=1.5, 8=-.5,

n=1(1)5. 773

22.2. Orthogonality Relations Standardization

. m e of Polynomial 22.2.1

lacobi

22.2.2

lacobi

22.2.3

Ultraspherical (Gegenbauer)

1

-1

0 -1

(I - 2 ) q

k,=

-q>

1

(1 -2')-t =( n

+

1

( I -2')-t

Tn(l)= 1

Chebyshev of the first kind

-1

22.2.5

Chebyshev of the second kind

-1

1

(1-2')t

Un(1)=nfl

22.2.6

Chebyshev of the first kind

-2

2

( -$)-t

C,(2)=2

22.2.7

Chebyshev of the second kind

-2

2

(1

22.2.8

Shifted Chebyshev of the first kind

1

(2- 2') -t

-$)#

Sn(2)= n + 1

2T

7

a=O

5X

nfo

a-

n=O

a2

223.9

Shifted Chebyshev of the second kind

22.2.10

Legendre (Spherical)

22.2.11

Shifted Legendre

*See page

11.

0

-1

0

1

(x-

1

1

1

1

2') t

T,*(l)= 1

G'r(l)=n+l

Pn(l)= 1

nfo n=O

4%

8aa%

0

8>-1 - 1 , q>o

a>-+

:7

2 C?)(I)=-, n C?)(l)=l

22.2.4

Remarks

z>-l,

+Z)b

1 1

h.

5

n#O

a-

n=O

* -

*

8

2

2n+l 1 2n+ 1

22.2. Orthogonality Relations-Gntinued Generalized Jiaguerre

0

L.(z)

Iapuerre

0

ff,(z)

Hermite

-m

Hermite

-m

L:'

22.2.12 22.2.13 22.2.14

I

1

(2)

*See page 11.

a>-l

I

22.3. Explicit Expressions

f n k ) =dn

N

cngr(z) m=O

-__

Reinarks 22.3.1

u>-l,@>-l

22.3.2

a>-1,@>--1

22.3.3

1

p-q>-l,

a > - 5 '1 a f O

22.3.4 22.3.5

1

22.3.6

n 2

22.3.7

1

22.3.8

q>o

2" n

nfO

n#O, Cho'(l)=l

2"

1 -

2"

22.3.9

1

2"

22.3.10

n!

( 2 4 n-*-

2"

22.3.11

n!

Zn-Zm

1

n!

a>-I

see 22.11

776

ORTHOGONAL POLYNOMIALS 2xplicit Expressions Involving TrigonometricFunctions

I

n

fn(coa

12.3.12

c:)

(COS

~ ~ 3 . 1 3 P,(COS

e)=C U,,, m=O

00s (n-2rn)e

r ( a+m) r (a+ n -m) m!(n- m)![r(U)12

e)

1 2m 2n-2m G ( m > ( n-m

e)

C

a#O

1

IXl

X

FIQURE 22.3. Jacobi Polynomials P!O.p)(z), a=1.5, @=-.8(.2)0, n=5.

FIQURE 22.4. Gegenbaw (Ultrasphehal) Polynomials C~u)(~), a=.5, n=2(1)5.

ORTHOGONAL POLYNOMIALS

777

22.4. S icial Values

.io(z) -

fn(1)

(?)

22.4.1

*

1

(0,n = 2 m + 1 22.4.2

(- 1)" C t ' ( 2 )

("+:--':

1

2a2

1

22

(-l)m, n=2m 0, n = 2 m + l

1

2

(- l ) m , n = % m 0, n = 2 m + l

1

22

1

t

22.4.7

1

-x+a+l

22.4.8

1

22

2, n#O

22.4.3

[o, n = 2 m + 22.4.4

1

22.4.5

n+l

22.4.6

1

I I

1

10, n = 2 m + 1

CA"'(

X)

22.5. Interrelations

t

I ci"

Interrelations Between Orthogonal Polynomials of the Same Family

Jacobi Polynomials

22.5.1

22.5.2

(see [22.21]).

22.5.3

Fn(p,q,z)=(-l)"n!

r (P) P $ J - ~ ~ ~ - 1 ) ( 2 x - l ) F(a+n>

(see [22.13]). Ultraspherical Polynomials

22.5.4

1

C~o)(z)=lim - Cp)(z) a-ma

Chebyshev Polynomiab

FIQURE 22.5. Gegenbaw (Ultraephericd) Polynomials Cia)(z), a=.2(.2)1, n=5. 'See page U.

778

ORTHOOONAL POLYNOMIALS

PionO) (5) =Pn (3)

32.5.24 22.5.9

Un(5)=S,(25)=U::

Ultraspherical Polynomiah

22.5.25

22.5.11

22.5.28

Chebyshev Polynomials

Generalized Laguerre Polynomials

22.5.16 22.5.17

Q)

(z)=L,(z)

d" z m [Ln+m(s)I

Limn,(~)=(-l)"

Hermite Polynomials

22.5.18

He ,,(5)=2-"'* H,

(see [22.20]).

22.5.19

(3

H , ( 5 )=2"We, (zJz>

(see [22.13], [22.20]). Interrelations Between Orthogonal Polynomials of Ditrerent Families Jacobi Polynomials

22.5.20

22.5.21

FIQURE 22.6.

Chebyshev Polynomials T,(x), n= 1(1)5.

*See page 11.

779

ORTHOGONAL POLYNOMIALS U&)

I

t

Legendre Polynomials

22.5.35

P.(s)=PAo*o)(z)

22.5.36

P,(z)=cy)(z)

22.5.37

d"

[ P n ( ~ ) ] = 1 *. 3. . (2m-1)C$!+,+)(z)

(mI n )

Generalized Laguerre Polynomials

L i - 1 1 4 ) ( ~ ) HZ,(@) =s

22.5.38

t::

FIGURE 22.7. Chebyshev Polynomials Un(x),

LA1/a)(z)=nhit!&

22.5.39

H2n+l(&5)

n= 1(1)5. Hermite Polynomials

22.5.33 22.5.M 22.5.34

U,(2) =Cil)(5)

22.5.41

Ham(%)= (-l)m2"m!L~-1'a'(~2)

HZ,+~(S)= (-1)"2"+'m!~L"'~' m

(z2>

Orthogonal Polynomials as Hypergeometric Functions (see chapter 15) f n ( z ) = W a , b; C; g(z))

For each of the listed polynomials there are numerous other representations in terms of hypergeometric functions. d

b

a

22.5.42

-n

n+a+a+l

22.5.43

-n

-n--a

22.5.44

-n

-n-8

22.5.45

-n

-n--a

22.5.%

-n

n+2-a

22.5.47

-n

n

22.5.48

-n

n+2

22.5.49

-n

n+l

22.5.50

-n

-n

22.5.51

--n

I-fl -

22.5.52

-n

n+3

22.5.53

-n

n+%

2

*See page 11.

2

*

780

ORTHOGONAL POLYNOMIALS

Orthogonal Po!ynomials as Confluent Hypergeometric Functions (see chapter 13)

Orthogonal Polynomials as Parabolic Cylinder Functions (see chapter 19)

22.5.55 22.5.56

Orthogonal Polynomials as Legendre Functions (see chapter 8)

H , ( 2 )= 2 W Hz,(~)=(--1)"7 ('m)! M ( - m ,

i,

22.5.57

* Hzm+,(4=(-1)"

22.5.60

( ' m + 1 ) ! 2 % ~ (- m, m!

x2)

c p ( x )=

i,

x2)

Ln (XI

3

-2

t

I FIQURE 22.9. Lagwrre Polynomials Ln(X), n=2(1)5.

FIGURE 22.8. Legendre Polynomials P,(x), n=2(1)5. *See page n.

FIGURE 22.10. Hermite Polynomiale n=2(1)5.

H n(x),

7

ORTHOGOXAL POLYNWIALS

22.6. Differential h u a t i o n s

Q Z ( 4 Y' '

+

QI(2)

Y'

+ g o ( 4 Y =o

Y 22.6.1

1- 2 2

8-

22.6.2

1-22

a-p+

22.6.3

1

0

22.6.4

1

0

a-

(aS8-t 2)x (a+B-

n ( n + a + 8 + 1)

2)x

+ (n+-)' 22.6.5

1-2'

- (2a+ 1 )

n(nt2a)

22.6.6

1-21

(20-3)z

(n+ 1 ) (n 2a - 1 )

22.6.7

1

0

22.6.8

1

0

22.6.9

1-21

-2

nz

22.6.10

1

0

n1

22.6.11

1-21

- 3x

nz- 1

22.6.12

1-21

- 32

22.6.13

1-21

-2l.

22.6.14

1

0

22.6.15

I:

a+1-2

22.6.16

1:

2+

22.6.17

1

0

2 n + ~ + l 1-az 22 42'

22.6.18

1

0

4n t 2a

22.6.19

1

- 2z

2n

22.6.20

1

0

2n+

22.6.21

1

-z

n

*See pnge I I

+

It

1 ~

1 4

+ 2-z*+- 1 -4x24a2

1-21

782

ORTHOGONAL POLYNOMIALS

22.7. Recurrence Relations Recurrence Relations With Respect to the Degree n alnfn+l(x)

=

= (aan+ asnr)f n(z) -arnfn-l(x)

aa n

aa n 22.7.1

2(n+l)(nSa+8+1) (2n+ a + 8 )

22.7.2

(2n+ P - 2)4(2n+ P - 1 )

22.7.3

n+1

22.7.4

1

22.7.5

1

22.7.6

1

22.7.7

1

22.7.8

1

22.7.9

1

22.7.10

n+l

22.7.11

n+l

22.7.12

n+l

1

ad

4n+2

n

-1

n+a

22.7.13

2

2n

22.7.14

1

n

Miscellaneous Recurrence Relations Jacobi Polynomials

22.7.15

(l--s)Pp+"~'(Z) = (n+a+l)P,'~~8'(5)-(n+l)P',";81'(2)

22.7.16

(l+z)Pp+4+1)(5)

(n+;+;+l)

= (n+B+Z)P?fl)(4

22.7.17

+(n+l>P2ml9

Chebyshev Polynomials

22.7.24

+

(1-,)Pp+l*fl) (5) (1+z)Pp*fl+" (5)=2Pp'fl) (5)

22.7.18 (2n+ a+

+a +B)Pp*B'

B)Pp-1*8) (5)= (n

(5)

- (n+B)PZ!'(s) 22.7.19 (2n+ a

22.7.20

+B)Pp

8-1) (2) = (n+

a

+g)PpJ)

(5)

+(n+a)P2?(5)

P ~ @ - " (-Pp-'*b'(~) Z) =P!.,a)

(2)

*Bee page

11.

22.7.31 22.7.29

Llp

L?+’)(z)=;1 [ (n+a+l)L?’h)

+ +

1

+

= ; [ (5-n)L f ) (2) (a n)Lfi$(z)]

1) (2)

- (n+l)Llp!l (z)]

22.7.32 1

22.7.30

i

fn

Bo

+8 ) (1- x * )

+B)zl

+

22.8.1

(2n+

22.8.2

1-22

-nz

n+2a-l

22.8.3

1-22

-nz

n

22.8.4

1-21

-na

n+l

22.8.5

1-21

-nx

n

22.8.6

2

n

-(n+a)

22.8.7

I

0

2n

22.8.8

1

0

n

U

n[a- B- (2n+

a

2(n a)(n+ i3)

22.9. Generating Functions

Remarks

+

22.9.1

R-’ (1 - z

22.9.2

R - ~ ( ~ - z I +R)+-”

22.9.3

1

22.9.4

R)-a(

1

+ r + R) -8

R-h --In Ra

22.9.5

X=COS n

a2.9.6

L

22.9.7

$(;)

22.9.8

-1

e

(?+1)

n

1-3 In R’

1

1-21 R’

-1<2<1

22.9.10

R-’

-

22.9.11

B1

-1<2<1

22.9.9

do= I

-l
14<1 (l-~z+R)-”*

14<1 ‘See page

n.

784

ORTHOGONAL POLYNOMIALS

22.9. Generating Functions-Continued

R 4-222 + 2' Remarks

an

--

R-1

22.9.12

1

22.9.13

J

22.9.14

1

22.9.15

1

22.9.16

r(n+ a

22.9.17

1 nl

1

(1 - I)-=-I exp 1

(2)

+ 1) elaa-al

(-.l)n

22.9.18

(an)! (-1)"

22.9.19

(2n+ l)!

22.10. Integral Representations go(4

fn(z)=,,i

Contour Integral Representations

sc

[gl(zJ z)pg& z)dt where C is a closed contour taken around z=a in the positive sense

a

Bo(4

22.10.1 22.10.2

-

(1 Z ) ' (

1

1

-

+

1

(1- 2)'(1+

ZIP

G

& 1 outside C

3

Both zeros of 1-%t+ aloulside C,

D

Both zeros of 1- &z E) outside C

2-2

2)

(1-22?+

zf)-az-l

Remarks

a>O 1-21

- &2+

22.10.3

I( 1

22.10.4

1

22.10.5

1

22.10.6

5

22.10.7

e'z-'

22.10.8

1

22.10.9

n!

1

1

I( 1- 8 2

29

+ IS)

-1 (1-222+2')-1~* 2

1 *

O

+ Both zeros of 1- 2 z z + zf outgide C

0

Both zeros of 1-&E+ z* outside C

z

2-2

e-a

2

Zero outside C

0

E= - 2

2--T

E

Miscellaneous Integral Repreeentationr

0

outside C

785

ORTHOGONAL POLYNOMIALS

22.10.12 Pn(cos e)=;

'6'

(cos e+i sin e cos +)"d4

lrn ? !lrn fi

22.10.14 L ~ ) ( z ) =ez2-9 n! 22.10.15

Hn(s)=ez2

e-'t"+f Ja(2&)dt

e-"tn cos (2zt-i x ) dt

22.11. Rodrigues' Formula

The polynomials given in the following table are the only orthogonal polynomials which satisfy this formula.

22.11.1

Pf' U ) (2)

(1 -X)"(l

22.11.2

c y (2)

(1-22).-1

1-21

22.11.3

T.(z)

(1 - 2 y t

1-22

22.11.4

U.(z)

(1 -2*y

1-21

22.11.5 22.11.6 22.11.7 22.11.8

Pn(z)

Lp)( 2 )

I $i!$)

22.12. Sum Formulas Christoffel-Darboux Formula

22.12.1

Miscellaneous Sum Formulas (Onlv a Limited Selection

22.13.2

22.12.2 n-1

22.12.3 22.12.4 22.12.5 22.12.6 22.12.7 22.12.8

*See page

11.

+r)b

1-22

1

1-21

e-*P

2

p 2 e-.2/2

1

1

786

ORTHOGONAL

OLYNOMIALS

22.14. Inequalities

22.13.8

22.14.1

22.13.9

d maximum point nearest to (x>-2) 22.13.10

22.13.11

IC F (4II

("+;-'> IC?)(z')I

~

(a>O)

(-l

x'=O if n=2m; z'=maximum point nearest zero if n='2m 1

+

22.14.5

22.13.18

t

22.14.2

a-a

a+8+1

ldql sn2

(- 15 25 1)

787

ORTHOGONAL POLYNOMIALS

22.14.15 22.14.16

(H2,(z)I I ez2l22"m!

22.15.2

n+a

(2m+2)! (m+l>!

IHz,+l(z)lI s e z z / 2

( 2 20)

k= 1.086435

22.14.17

lim

22.15. Limit Relations

22.15.4 lim

[$ L p ) (;)]=~-~"Ja(2&"

[w (&)I=$ H2n+l

2 sin z

n+ m

22.15.5

22.15.1 22.15.6

lim PF,6) (1 -$)=@)(,)

8-1=

lim 1 C'?) ajm

an/z

(%)=1 H,(z) n!

For asymptotic expansions, see [22.5] and [22.17].

z 2 ) m t h zero of fn(z)(z,'")
2:,,,+l(o<e!w)<ep<

,

. .
. . . <ep
j , , , mth positive zero of the Bessel function J&) o<j,,<j,z< Relation

22.16.1 22.16.2 22.16.3 22.16.4 22.16.5

22.16.6

22.16.7

22.16.8

For error estimates see t22.61.

...

788

ORTHOGONAL POLYNOMIALS

C w * ( z t ) is finite. The constant factor which is

22.17. Orthogonal Polynomials of a Discrete Variable

1

still free in each polynomial when only the orthogonality condition is given is defined here by the explicit representation (which corresponds to the Rodrigues' formula)

In this section some polynomialsf,(z) are listed which are orthogonal with respect to the scalar product 22.17 1

(fa, fm)

=?w*

(5t)fn

f,(z)=-

22.17.2

(z t)fm (zt )

1

raw*(x)

A"[w*(z)g(s,n ) ]

where g(s,n)=g(s)g(s-1) . . . g(z-n+l) g ( t ) is a polynomial in s independent of n.

The st are the integers in the interval a<si
and

Name

b

a

-

Remarks

tn

Chebyshev

0

N-1

1

Krawtchouk

0

N

p=qN-=

Charlier

0

m

(-

Meixner

0

m

C"

Hahn

0

m

n!

(3

l/n!

(3

qnz!

(- l)nn!

Pl q>O; p+q=l

(x-n)!

l)nda.n!

Z!

a>O

(2-n)! Z!

b>Ol O
(x-n)! x!r(b+x) (z - n) !r(b z- n)

+

For a more complete list of the properties of these polynomials see [22.5] and [22.17].

Numerical Methods 22.18. Use and Extension of the Tables

Evaluation of an orthogonal polynomial f o r which the coefiients are given numerically. Example 1. Evaluate Le(1.5) and its first and second derivative using Table 22.10 and the Horner scheme. 1

-36

450 1. 5

2=1.5 1 1.5

-34.5 1. 5

1 1.5

-33.0 1. 5

1

-31.5

- 2400

5400

- 4320

720

-51. 75

597.375

-2703. 9375

4044. 09375

-413. 859375

398. 25

- 1802. 625

2696.0625

-275. 90625

306. 140625

-49. 5

523. 125

348. 75

- 1279. 500

-47, 25

452. 250

301. 50

-827. 250

- 1919. 25 776.8125 - 1240. 875

- 464.0625

1165. 21875

306. 140625 720 =. 42519 53

L6=

889. 3125 t

889. 3125 720 = 1.23515 625

Le=-

[ - 46406251 720 = 1I . 28906 25

L;'=2

789

ORTHOGONAL POLYNOMIALS

Evaluation of an orthogonal polynomial using the explicit representation when the coeficients are not given numerically. If an isolated value of the orthogonal polynomial fn(z)is to be computed, use the proper explicit expression rewritten in the form f&) =dn(4ao(x) and generate ao(z)recursively, where bm a m - l ( z ) ~ l - - - - f ( ~ ) a m ( z )(m=n, n - I , . . ., 2 , 1 , a , ( z ) = l ) . Cm

The o!,,(x), b,, c , , f ( z ) for the polynomials of this chapter are listed in the following table: bm

(n- m+ 1)(a+ P+ n+

m am bm Cm

8 1 18

136

4

2m(a+ m)

2(n-m+ l ) ( a + n + m - 1)

m(2m-1)

2(n-m+l)(a+n+m)

m(2m+ 1)

2(n- m+ 1)(n+ m- 1)

m(2m- 1)

2 (n - m+ 1)(n+ m)

m@m+ 1)

2(n-m+1)(n+m)

m(2m- 1)

2(n--++)(n+m+l)

m(2m+l)

(n- m+ 1)(2n+2m- 1)

m(2m- 1)

(n-m+1)(2n+2m+1)

m@m+ 1)

n-m+l

m(u+m)

2(n-m+ 1)

m(2m- 1)

2(n-m+ 1)

m(2m+ 1)

7

6

5

4

3

2

1

1. 132353 34 105

1. 366667 48 78

1. 841026 60 55

3.008392 70 36

6.849651 78 21

26.44156 84 10

223. 1091 88 3

n Ci*)(2.5)

2 3. 65625

3 13. 08594

4

-50. 87648

5 207. 0649

6 867.7516

0

6545.533 90 0

790

ORTHOGONAL POLYNOMIALS Change of Interval of Orthogonality

In some applications it is more convenient to use polynomials orthogonal on the interval [O, 11. One can obtain the new polynomials from the ones given in this chapter by the substitution x=2Z- 1. The coefficients of the new polynomial can be computed from the old by the following recursive scheme, provided the standardization is not changed. If n f n ( ~ ) = C

m=O

n

amxm, f : ( ~ ) = f n ( 2 ~ - I ) = C n:xm m ==O

then the a: are given recursively by the am through the relations

* .

m=n-1, n-2, . . ., 9 ; 3=0, 1 , 2, u & - ~ ) = u , /m=O, ~, 1 , 2, . . R. a ~ ’ ) = 2 4 z a , , j = 01,, 2, . . ., n and aLm:’=a:; m=O, I , 2, . m

. . *,n

.)

Example 4.

. ., n.

Given T5(x)=5x-20x3+16~5,find Tg(s).

\m

\

1

0

2.5=aj-’)

0

a

-1 16 32 64 128 256 512=a;

- 16 - 64 - 192 - 512 - 1280=a;

4

-4 56 304 1120= a;

-l=a;

1 50=a;

-48

-400=~;

Hence, T,*(x)=512x6-1200x4+1120x3-400x2+50x-l. D a Continuous Interval

22.19. Least Square Approximations

Problem: Given a function f(s) (analytically or in form of a table) in a domain D (which may be a continuous interval or a set of discrete points).2 Approximate j ( z ) b y a polynomial Fn(z) of given degree n such that a weighted sum of the squares of the errors in D is least. Solution: Let w(x)>O be the weight function chosen according to the relative importance of the errors in different parts of D. Let fm(x) be orthogonal polynomials in D relative to w(x), i.e. (fm, fn)=O for m # n , where

Example 5.

Find a least square polynomial of

degree 5 f o r J ( z ) = L , in the interval 2 5 x 1 5 , 1 +x

using the weight function 1

w(x) = J(z-~)(~-z) which stresses the importance of the errors a t the ends of the interval. Reduction to interval [-1, 11, t=-

(fl

s) =

12

if D is a continuous irterval w(zm>j(zm)g(zm> if D is a set of N discrete points x m.

From 22.2, f m ( t )= Tm(t)and

Tm(t)dt

*f(z) has t o be square integrable, see e.g. [22.17]. *See page 11.

22-7 3

( m ZO)

791

ORTHOGONAL POLYNOMIAL&

Evaluating the integrals numerically we get - -1 . 2 3 5 7 0 3 - . 0 8 0 8 8 O T ~ ( ~22-7 )+.013876T~ l+x

( 722-7 )-.002380T3

(3) 22 7

+.000408T, ( 722-7 )-.000070T,

(T) 22 7

D a Set of Diecrete Points

If zm=m(m=O, 1, 2, . . ., N ) and w(x)=l, use the Chebyshev polynomials in the discrete range 22.17. It is convenient to introduce here a slightly different standardization such that nfm

j n ( x ) =m-0k ( - 1 ) m ( t ) ( (fn, f n ) =

2

10 12 14

16 18

0

.3162 .2887 .2673 .2500 . 2357

4

5

2. 5

3. 5

f(X)

f(x)

.271580

+

+

1 -2 0 2 -1

- 117

-

f*(3

fn)

G)

1

fi(Z)

i

.fa

--11;

fo(D

(f j n ) (jni

fa@)

1 11; - 11;

1 1 1 1 1

1 2 3

1.3579 an=-

-m )!N!

fl(23

fo (2)

z=- 2-10 2

---

(2

(N+n+l)!(N--n)! (2n+l)(N!)2

-

f(4

)

z!(N-m)!

fa(3 10

,09985

.01525

,0031

.039940

.0043571

.000310

+

.27 158 .03994(3.5- .252) .0043571(23.5-3.5~$. 1259) ,0003l(266 -59.83332 +4.3752- .10417$) .59447- .043658~+.00190092a- .000032292$ 22.20. Economization of Series

Problem: Given f(x)= -1 I s 2 1 and R>O.

n

m -0

Then, since 1 T,,,(s)151(- 1S s 5 1)

a d m in the interval N

F i n d y ( z ) = C b,,,zmwithN m =O

as small as possible, such that ( j ( z )-j(s)l
within the desired accuracy if

J ( x ) is evaluated most conveniently by using the recurrence relation (see 22.7).

792

ORTHOGONAL POLYNOMIALS

Example 7. Economize f(z) = 1 +z/2 + 9 / 3 $214 4- x4/54-216 with R= .05.

so

References Texts [22.1] Bibliography on orthogonal polynomials, Bull. of the National Research Council No. 103, Washington, D.C. (1940). [22.2] P. L. Chebyshev, Sur l’interpolation. Oeuvres, vol. 2, pp. 59-68. i22.3) R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, ch. 7 (Interscience Publishers, Xew York, X.Y., 1953). [22.4] G. Doetsch, Die in der Statistik seltener Ereignisse auftretenden Charlierschen Polynome und eine damit eusammenhangende Differentialdifferenzengleichung, Math. Ann. 109, 257-266 (1934). [22.5] A. ErdBlyi et al., Higher transcendental functions, vol. 2, ch. 10 (McGrrtw-Hill Book Co., Inc., New York, N.Y., 1953). [22.6] L. Gatteschi, Limitazione degli errori nelle formule asintotiche per le funzioni speciali, Rend. Sem. Mat. Univ. Torina 16, 83-94 (1956-57). [22.7] T. L. Geronimus, Teoria ortogonalnikh mnogochlenov (Moscow, U.S.S.R., 1950). [22.8] W. Hahn, Vber Orthogonalpolynome, die qDifferenzengleichungen genugen, Math. Nachr. 2, 4-34 (1949). [22.9] St. Kaczmare and H . Steinhaus, Theorie der Orthogonalreihen, ch. 4 (Chelsea, Publishing Co., New York, N.Y., 1951). [22.10] M. Krawtchouk, Sur une g6nBralisation des polynomes d’Hermite, C.R. Acad. Sci. Paris 187, 620-622 (1929). [22.11] C. Lanceos, Trigonometric inteipolation of empirical and analytical functions, J. Math. Phys. 17, 123-199 (1938). [22.12] C. Lanczos, Applied analysis (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1956). [22.13] W. Magnus and F. Oberhettinger, Formeln und Satze fiir die speziellen Funktionen der mathematischen Physik, ch. 5, 2d ed. (SpringerVerlag, Berlin, Germany, 1948).

[22.14] J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. London Math. Soc. 9, 6-13 (1934). [22.15] G. Sansone, Orthogonal functions, Pure and Applied Mathematics, vol. I X (Interscience Publishers, New York, N.Y., 1959). (22.161 J. Shohat, Thborie g6n6rale des polynomes orthogonaux de Tchebichef, MBm. Soc. Math. 66 (Gauthier-Villars, Paris, France, 1934). [22.17] G. Szego, Orthogonal polynomials, Amer. Math. Soc. Colloquium Publications 23, rev. ed. (1959). [22.18] F. G. Tricomi, Vorlesungen uber Orthogonalreihen, chs. 4, 5, 6 (Springer-Verlag, Berlin, Germany, 1955). Tables [22.19] British Association for the Advancement of Science, Legendre Polynomials, Mathematical Tables, Part vol. A (Cambridge Univ. Press, Cambridge, England, 1946). P,(z), z=0(.01)6, n=1(1)12, 7-8D. [22.20] N. R. Jorgensen, Undersogelser over frekvensflader og korrelation (Busck, Copenhagen, Denmark, 1916). He&), z=0(.01)4, n=1(1)6, exact. [22.21] L. N. Karmazina, Tablitsy polinomov Jacobi (Izdat. Akad. Nauk SSSR., Moscow, U.S.S.R., 1954). G , ( p , q, z), ~ = 0 ( . 0 1 ) 1 , q=.1(.1)1, p=1.1(.1)3, 7t=1(1)5, 7D. [22.22] National Bureau of Standards, Tables of Chebyshev polynomials S,(z) and C,,(z), Applied Math. Series 9 (U.S. Government Printing Office, Washington, D.C., 1952). 2=0(.001)2, n=2(1)12, 12D; Coefficients for Tn(z), U,,@), C,(z), S,(z) for n=0(1)12. [22.23] J. B. Russel, A table of Hermite functions, J. Math. Phys. 12, 291-297 (1933). e-z212Hn(z), z =O( .04)1 (. 1) 4(.2)7( .5)8, 7t =O( 1) 11, 5D. [22.24] N. Wiener, Extrapolation, interpolation and smoothing of stationary time series (John Wiley & Sons, Inc., New York, N.Y., 1949). L,(z), 7t= 0(1)5, Z= 0(.01).1 (. 1) 18(.2)20( .5)21(1)26(2)30, 3-5D.

n

Coefficients for the Jacobi Polynomials P$p)

c,(z-

(2)=U;' m=O

1)"

Table 22.1

794

ORTHOGONAL POLYNOMIALS

3 1

1%

1

6

t t

W 0

n

7 3 . W2

*

n

V

U

2

*

n

t t

W

n 3

V

n M

rg

rl

I (?

i;s

t

t

Y

k

Y

U m n

0

h rl

t

V

h

V

3

t

m

v

U

I!

a

U

V

rl

" U

M

H

-

rl

k 3

I

795

ORTHOGONAL POLYNOMIALS

Table 22.3 Coefficients for the Chebyshev Polynomials T,(z) and for z" in terms of T,(z)

Chebyshev Polynomials T,(z) 0.2

n\z

0

i

2 3 4 5

6 7 8 9 10 11 12

+ -. 1.00000 00000 -

.-

io.20000 -0.92000 -0.56800 t-0.69280 $0.84512 -0.35475 -0.98702 -0.04005 $0.97099 $0.42845 -0.79961 -0.74830

- . ..

ooooo

00000 00000 00000 00000 20000 08000 63200 82720 56288 60205 20370

0.4

+ 1.00000 00000 ~. ... . .

+o.40000 -0.68000 -0.94400 -0.07520 $0.88384 $0.78227 -0.25802 -0.98868 -0.53292 $0.56234 +0.98280 $0.22389

-

.

ooooo

00000 00000 00000 00000 20000 24000 99200 95360 62912 65690 89640

0.6

+ 1.00000 00000 +o:6oooo 00000

-0.28000 -0.93600 -0.84320 -0.07584 $0.75219 $0.97847 $0.42197 -0.47210 -0.98849 -0.71409 4-0.13158

00000 00000 00000 00000 20000 04000 24800 34240 65888 24826 56097

Table 22.4 0.8

+ 1.00000 00000

+o~sO000 00000

+0.28000 -0.35200 -0.84320 -0.99712 -0.75219 -0.20638 $0.42197 $0.88154 $0.98849 $0.70005 +0.13158

00000 00000 00000 00000

20000 72000 24800 31680 65888 13741 56097

1.0 1 1 1 1 1 1

1 1 1 1 1 1 1

796

ORTHOGONAL POLYNOMIALS

Table 22.5 Coefficients fordtheChebyshev Polynomials U,(%) and for

ua

I

I

-I

in terms of V,,,(Z)

4 11

-1 128 1 I

I

-32

----- -500 UII

I

I

-12

I I

280

1792

-2304

1024 1

11

- 1792

U11

- 5370

1120

----

Table 22.6

Chebyshev Polynomials U,(%)

n\z 0 1 2 3 4 5

0.2 +1.00000 +0.40000 -0.84000 -0.73600 +0.54580 +0.95424

0.4

00OOo OOO00

+1.00000 00000

OOOOO

-0.38000 -1.08800 -0.51040 +0.67908

ooOo0 00000 00000

+0.80000 OOo00

00OOO

00OOO 00000 OOOOO

7 8 9 10

11

-0.16390 40000 ._.. .... -1.01980 is000 -0.24401 00400 4-0.92219 49440 +0.61289 40176 -0.07703 70970

-0.92323 -0.90222 +0.20145 1.06338

12

-0.88370 94504

+0.04925 46703

0

~

+1.05414 40000

+o;iS383 SWO

+

U10

58400 38720 67424 92059

0.0

+l.ooOoO 00000 $1.20000 00OOO

0.8

-1.24040 00000 -0.82308 00000

+l.OooOo o0000 +1.m 00o00 +1.66OOO OOO00 +0.89600 o0000 -0.12840 OoooO -1.09824 00000

+0.25798 40000

-1.63078 40000

$0.44000 00000 -0.67200 ooOo0

+i;i33i6 OsoOo

+1.10192 $0.18905 -0.87506 - 1.23913

89800 39520 42176 10131

-0.61189

29981

-i;siioi

eujoo

-0.78083 +0.25207 +1.19015 1.05217

90400

19360 41370 40842

+ + 1.45332 53571

1.0 1 2 3 4 5 6

7 8 9 10 11 12 13

797

ORTHOGOKAL POLYNOMIALS

*See page XI

cs(2) =zS -62( +922-

2

zS=lOCu+ 15G+ 6C4+ Cs

Coefficients for the Chebyshev Polynomials Sn(z) and for z" in terms of &(z)

&(Z) =zS'See page

n.

5244- W- 1

zS= 5so

+9&+

5s4

+ss

Table 22.8

798

ORTHOGONAL POLYNOMIALS

K

0

H

z "H

-

k

k

c

H

"H

0

H

k

k

ta

k

k

-

B

k k

H

Coe5cients for the Laguerre Polynomials L,,(z) and for 2" in terms of Lm(z) 9l

L.(x) =a;1

93

d

d,L,(z)

x"=

CA?

m-0

a.

m=O

d

2'

2

Table 22.10

382880

6

36Z&XlO(

389153001

479001800

~~

-4

1

______ 2

2

61

61

-4 -181

1

2 9 1-1

-18

-6

-1200

-96 1

2

4

25

-36

7350

-882

176400

10800

-im

450

-176400

-1IUw)

tj00

-I

1o5840

10800

1200

144

18

-35280

-4320

-600

-96

-4320 1

- 1 W

35280

120 49

-1

-5040

-322x30 1128wIo

-27.57W 2822u)o

-2257920 1128860

--322560

-32851)20 13083680 -30481920

45722880 -45722880 30181920

-13063680

-100

M8856ooo

-15jeirgSeo

--a161410560

L&)

=m 1 [a9-3~+4w-240+54w-43202+7201

1 1 m13280

--2813600

614718120

-75271680

34=720&-4320L1+

183350

--6050 -2oou)(1

10800La- 14400&+ 10soOIy-4320L~+720L~

800

ORTHOGONAL POLYNOMIALS

Laguerre Polynomials L,(z)

Table 22.11 n\z

0.5

0

+ 1.00000 00000

1.0

2 3 4 5

4-0.50000 4-0.12500 -0.14583 -0.33072 -0.44557

00000 00000 33333 91667 29167

6 7 8 9 10 11

-0.50414 -0.51833 -0.49836 -0.45291 -0.38937 -0.31390

49653 92237 29984 95204 44141 72988

12

-0.23164 96389

1

+ 1.00000.00000 0.00000 -0.50000 -0.66666 -0.62500 -0.46666

00000 00000 66667 00000 66667

-0.25694 -0.04047 +0.15399 +0.30974 +0.41894 $0.48013

44444 51905 30556 42681 59325 41791

$0.49621 22235

3.0

5.0

10.0

+-2.00000 1.00000 00000 00000

+-4.00000 1.00000 00000 00000

-0.50000 1.00000 1.37500 +0.85000

++

00000 00000 00000 00000

+3.50000 +2.66666 1.29166 -3.16666

-

00000 66667 66667 66667

+-9.00000 1.00000 00000 00000

-45.66666 66667 4- 11.00000 00000 4-34.33333 33333

-0.01250 -0.74642 -1.10870 -1.06116 -0.70002 -0.18079

00000 85714 53571 07143 23214 95130

-2.09027 +0.32539 t2.23573 +2.69174 1.75627 +0.10754

77778 68254 90873 38272 61795 36909

-3.44444 -30.90476 16.30158 -f 14.79188 +27.98412 t 14.53695

i-0.34035 46063

- 1.44860

42948

+

-t31.00000 00000

-

44444 19048 73016 71252 69841 68703

-9.90374 64593

Coefficients for the Hermite Polynomials H,,(z) and for Z" in terms of H,,,(x)

11

bm

2

16 2

Ho HI H2

-4 I 12 ,

H4

1-

I

Hi1

1-

6o 121

- 160

120

-1680

3360 - 13440

30240

-80640

1 1 1

-665280

1I

1 ;:

1-1

1680

1 1

I

13440

32

1

-1344

I

1 1

I

180

48384

1

a9

1

I

-3584

128 1

1

H~(z) =W 4802'4- 7209 - 120

56

I

I

* 30240

j

1

I 506880 1 26

I I

1512 2520

1 256

I

90 512

1 1024 1

I -56320 28

I 332640

I

I

4096 665280

55440

3960

I

1-1

I

1 *=[12OHo+ 180Hz+30H4+ He] 64

1

H2 H3 H4

Hs

110880

He

H7 5940

H8 Ho -

132

- 135168 lp14096 ZQ

I I

__

__

110

2048

Ha HI

1 1 I

72

1520640 27

2048

75600

-23040

-7096320

a9

I

1-1 1995840 I 277200 I 25200 I I 831600

3360

-9216

I -1774080 I 24

12 1 1

161280

13305600

-7983360

64

I

I

1024

15120

I

- 403200

2217600

2'

-480

I

302400

I

32

- 48

- 30240

HIZ 665280

I 6I

1

-12

I I

He

1

720

I 1680

H8

4

- 120

H7

Hio

1

-2

H3

__

2

12

Table 22.12

I

His Hit

1

ORTHOGONAL POLYNOMIALS

Hermite Polynomials Hn(x)

Table 22.13 nb

0.5

2 3 4 5

++11.ooooo .00000 - 1.00000 - 5.00000 + 1.ooooo (1)+4.10000

6 7 8 9 10 11

(1)+3.10000 (2) - 4.61000 (2) -8.95000 (3) +6.48100 (4) +2.25910 (5) - 1.07029

12

(5) -6,04031

0 1

1.o

++2.00000 1.oO000

3.0

5.0

++6.00000 1.00000 00 00

(2)3.98000 00000 (3j7.88000 00000 (5)1.55212 00000 (6)3.04120 00000 (7)5.92718 80000 (9)1.14894 32000 (iom.21490 57680 (iij4.24598 06240 (12)8.09327 82098 (14) 1.53373 60295 (15)2.88941 99383

00 00 00 00

00 00

(4j - i.07200

00 --

(3) 4-8.22400 (5) $2.30848

(4) +l.41360 (4) $3.90240 14) +3.62400 - ----(5) -4.06944 (6) -3.09398 (7) - 1.04250

00 40 24

(5)7.17880 00000 (6)6.21160 00000 m5.20656 80000 (8j4.2ii7i 20000 (9)3.27552 97600 (10)2.43298 73600

(5) $2.80768

(6)+5.51750 40

(11)1.71237 08128

+

(2) 1.84000 (2) $4.64000 (3) - 1.64800

\-, 1

1.ooooo OOOOO

1.00000 00000 (1)l.OOOOO OOOOO (1)9.80000 00000 (2j9.40000 00000 (3)8.81200 00000 (4)8.06000 00000

(1) +3.40000 (2j +i.8oooo (2) +8.76000 (3) +3.81600

+2.00000 -4.00000 (1) -2.00000 (0) -8.00000

10.0 ( 1)2.00000 00000

23. Bernoulli and Euler PolynomialsRiemann Zeta Function EMILIE V. HAYNSWORTR' AND KARLGOLDBERQ~

Contents Mathematical Properties . . . . . . . . . . . . . . . . . . . . 23.1. Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula.. . . . . . . . . . . . . . . . . . . . . 23.2. Riemann Zeta Function and Other Sums of Reciprocal Powers.. . , . . . . . . .. . . . . . . , . . . References . , . . . . , . . . . . . . . . . . . . . . . . . .

.

,

804

807

808 809

... .. . ..

810

. ... . . . . . . ..

811

Table 23.2. Bernoulli and Euler Numbers , . . . . B, and E,, n=O, 1, 2(2)60, Exact and B, to 10s

.

804

.. .

Table 23.1. Coefficients of the Bernoulli and Euler Polynomials B,(z) and E,(%),n=0(1)15

Table 23.3. Sums of Reciprocal Powers

Page

n=1(1)42

Table 23.4. Sums of Positive Powers m

k -1

., ., . ., .

,

..

. ..

813

. . .

,

.. . . ..

818

,

k",n=1(1)10, m=1(1)100

Table 23.5. z"/n!, z=2(1)9, n=1(1)50,

10s

, ,

The authors acknowledge the aasistance of Ruth E. Capuano in the preparation and checking of the tables.

1 National Bureau of Standards. ZNational Bureau of Standards.

(Presently, Auburn University.)

23. Bernoulli and Euler Polynomials-Riemann

Zeta

Function Mathematical Properties 23.1. Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula Generating Functions

tezt

23.1.1 -et-1

-2B n ( 5 ) at" n = ~

n=O, 1, .

Bn=B,(O)

23.1.2

1 1 23.1.3 Bo= 1, B,=-s, B2=-, 6

Bd=--

..

1 30

n=O, 1 , . . .

E,=2"En Eo=l, E2=-1, E4=5

(For occurrence of B, and E, in series expansions of circular functions, see chapter 4.) Sums of Powers

23.1 5 Bk (Z)=nBn - 1 (2)

B,(Z+~)-B,(Z)=~S"-'

23.1.6

.. 1, . . .

.. 1,. . .

R=I, 2 , .

E:n(s)=nEn-1(2)

n=1,2,

n=O,

En(~+l)+En(~)=22"

n=O,

n=o, 1, *

*

..

n=O, 1 , . . .

23.1.8

B, (1-S ) =(- 1)"B, (z)

23.1.9

(-l)"Bn(-2)=Bn(2)+nzn-'

.. 1 , .. .

n=O, I , . n=o,

En(1-~)=(-1)"En(~)

n=O, 1 , . . .

(- l)"+'E,(-2)=En (z) -22"

n=O, 1 , . . .

23.1.10

n=O, 1 , . . . m=1,2,.

m=1, 3 , . . .

.. 2 n+l

E, (mz)=-- mn

5'(-1)k~,+' (z+k) k=O

n=O, 1 , . . . m=2,4,. 804

..

805

BERNOULLI AND EULER POLYNOMIALS, RIEMANN ZETA FUNCTIOPU'

Integrals

23.1.11 23.1.12

B,(t)dt=

(2)-B n + 1 (a)

n+l

l1

m !n!

B,(t)~,,,(t)dt=(-l)"-' -B m + n (m+n)! m , n = l , 2, . .

.

I

m,n=O, 1 , . , .

(The polynomials are orthogonal for m + n odd.) Inequalities

23.1.13

IB2nl>lBzn(~)I n=1, 2,

. . .,

1>~>0

23.1.14

n=1,2,.

.

.)

$>x>O

23.1.15

n=1,2,.

..

Fourier Expansions

23.1.16

B,(s)=-2

,n ! (2.rr)

z

= cos (2nkx-47rn)

kn n>l, 1 2 x 2 0

n>o, 1 2 x 2 0

n=l,l>z>O

n=O, 1>x>O

23.1.17

(-1)*2(2n-l)! B2n-1(4=

(27r)2"-1

= sin 2k7rz

zkz"-l n>l, 1 2 i 2 0 n=l,l>z>O

Ezn-1(4=

(- 1)"4(2n-l)!

n=1,2,.

.

.)

12220

23.1.18

n=1,2,..

9 )

n>O, 1 2 x 2 0 n=o, l>x>O

12x20 Special Values

23.1.19

BZ,+'=O

n=1,2,.

..

23-1-20 B, (0) = (- 13"B,(1) =B,

23-1-21 B, (4)=- (1-21-n) B,

Ezn+i=O

n=O,l,.

..

n=1,2,.

,

E m =-E, n=O,l,.

..

n=O,l,.

..

(1) = -2(n+l)-'(2n+l-1)

E, (3) =2 -"E,

B,+'

n=O, 1,

.

23.1.22

23.1.23

B, (+>= (- 1 )"B,(3) = -2-n(l--21-") Ba--n4-"Ea..1 n=1,2,.

&a

- 1(3) =-&a - 1 (3) =- (2n)- ( 1-3 1-2n) (2'"- 1 )B,,

..

n=1,2,.

..

B2,(+>=B2,(#) =-2-1(1-31-2a

)BZa

n=o, 1 , .

* *

23-1-24 Bz, (6) =Bzn(#) =2- 1( 1 -21-2n >(1-3'-'")Bzn

n=O,l,.

..

Symbolic Operations

+

23.1.25

p(B(z) 1 ) -p(B(z)) =P'(z)

23.1.26

B,(z+h)=(B(~)+h)" n=O, 1,

...

Herep($) denotes a polynomial in z and after expanding we set {B(z)}"=B,(z)and {E(z)}"=E,(z). Equivalent to this is

Relations Between the Polynomials

23.1.27 23.1.31

23.1.28

n=2,3,

...

23.1.29 A

Euler-Maclaurin Formulas

Let F(z) have its first 2n derivatives continuous on an interval (a, b ) . Divide the interval into m equal parts and let h=(b-a)/m. Then for some 8, l>8>0, depending on F(2a)(z)on (a, b ) , we have

Let B,(z) =Ba(z- [z]). The Euler Summation Formula is 23.1.32

23.1.30 m

hk-I

F(a+kh)=kl F ( t ) d t + ~ ~ { F ( b ) + F ( a ) ]

k- 0

-E P! h2 +-(2n)! BznE' F('"'(a+kh+8h) k-O

s' 0

&,(u-t)

{5' F(P)(a+kh+th) k=O

p52n11 I: 020

BERNOULLI AND EULER POLYNOMIALS, RIEMANN ZETA FUNCTION

23.2. Riemann Zeta Function and Other Sums of Reciprocal Powers 23.2.1

~(s)= k-*C

as>1

k= 1

23.2.2

f'(0) = - 3 In 2~

23.2.13 23.2.14

m

=II (l-p-')-l

Ws>l

P

(product over all primes p )

807

[(-2n) =O

..

n=l,2,.

-BZ" 2n

.

23-2-15 5 (1-2n) =

n=1,2,.

(242" 23-2-16 f ( 2 n ) = 7 IBZn( 2(2n).

n=1,2, * . .

23.2.17 23.2.3

sZ1, n=1,2,

*

. , ,,

..

n=2,3,.

..

,$?ts>-2n S u m s of Reciprocal Powers

23.2.4

The sums referred to are

23.2.5

23.2.18

where

23.2.19

r(n)

=gk-" k=l

Y,,=

-(In k)"

lim

k

m+ m

m

(In m)"+l} n+l

=2?rS'-'

23.2.7

1 =-$ r(s)

k-1

23.2.8

-

eE-1

(2k

X (n)= I

xa-l

-dx

23.2.20 m

sin ( 3 ~ s r) (1-8) f ( 1-8)

-

n=1,2,. . .

q ( n ) = C (- l)'+lk-"= (1-21qn) f(n)

as>o 23.2.6

n=1,2,.

as>1

1 (1-2 -a ) r (S)

23.2.9

+1)- =(1-2 -") f (n)

n=2,3,

...

23.2.21

B(n)

=5(- 111:(2k+1)

n=1,2,.

--(L

k-0

.,

These sums can be calculated from the Bernoulli and Euler polynomials by means of the last two formulas for special values of the zetrt function (note that q(l)=ln 2), and

23.2.10

product over all zeros p of f(s) with 9 ' p > O . The contour C in the fourth formula starts at infinity on the positive real axis, circles the origin once in the positive direction excluding the points f 2 n h for n=1, 2, . . ., and returns to the starting point. Therefore f(s) is regular for all values of s except for a simple pole at s = l with residue 1.

23.2.23

n=1,2,.

B(2) is known as Catalnn's constarit. Some other special values are 23.2.24

Special Values

23.2.11

..

f(0) = - 3 23.2.25

{(2)=1+-+3+ 1 1 22 3

..-

1 1 {(4)=1+7+7+ 2 3

. , . =a

=7rz

6 7r4

808

BERNOULLI AND EULER POLYNOMIALS, R I E M A " ZETA FUNCTION

23.2.26

1 1 7(2)= 1- -+ 7 22 3

23.2.27

1 7(4)=1--+724

23.2.28

1 1 X(2)=1+-+-3+ 32 5

1

3

7 2 _-12 ...-

23.2.29

71r4 _-720 . . .-

23.2.30

. . . =-f8

23.2.31

1 X(4)=1+7+3

1

54 + . . . =z

1 1 ,8(1)=1--+-3 5 ,9(3)=1--+~1 1 33 5

lr4

?r . . . -_ 4

-_??

' * *

-32

References Texts

Tables

[23.1] G. Boole, The calculus of finite differences, 3d ed. (Hafner Publishing Co., New York, N.Y., 1932). [23.2] W. E. Briggs and S. Chowla, The power series coefficients of {(s), Amer. Math. Vonthly 62, 323-325 (1955). [23.3] T. Fort, Finite differences (Clarendon Press, Oxford, England, 1948). [23.4] C. Jordan, Calculus of finite differences, 2d ed. (Chelsea Publishing Co., New York, X.Y., 1960). (23.51 K. Knopp, Theory and application of infinite series (Blackie and Son, Ltd., London, England, 1951). [23.6] L. M. Milne-Thomson, Calculus of finite differences (Macmillan and Co., Ltd., London, England, 1951). [23.7] N. E. Xorlund, Vorlesungen uber Differenzenrechnung (Edwards Bros., Ann Arbor, Mich., 1945). [23.8] C. H. Richardson, An introduction to the calculus of finite differences (D. Van Nostrand Co., Inc., New York, N.Y., 1954). [23.9] J. F. Steffensen, Interpolation (Chelsea Publishing Co., New York, N.Y., 1950). [23.10] E. C. Titchmarsh, The zeta-function of Riemann (Cambridge Univ. Press, Cambridge, England, 1930). [23.11] A. D . Wheelon, A short table of summable series, Report No. SM-14642, Douglas Aircraft Co., Ino., Santa Monica, Calif. (1953).

[23.12] G . Blanch and R. Siegel, Table of modified Bernoulli polynomials, J. Research NBS 44, 103-107 (1950) RP2060. [23.13] H. T. Davis, Tables of the higher mathematical functions, vol. I1 (Principia Press, Bloomington, Ind., 1935). [23.14] R. Hensman, Tables of the generalized Riemann Zeta function, Report KO.T2111, Telecommunications Research Establishment, Ministry of Supply, Great Malvern, Worcestershire, England (1948). {(s, a), s=-10(.1)0, a=0(.1)2, 5D; ( ~ - l ) r ( ~U ),, ~ = 0 ( . 1 ) 1 ,~ = 0 ( . 1 ) 2 , 5D. [23.15] D. H. Lehmer, On the maxima and minima of Bernoulli polynomials, Amer. Math. Monthly 47, 533-538 (1940). [23.16] E. 0. Powell, A table of the generalized Riemann Zeta function in a particular case, Quart. J. Mech. Appl. Math. 5, 116-123 (1952). f(3, a ) , a= 1 (.01)2(.02)5(.05)10, 10D.

809

ZETA FUNCTION

BERNOULLI AND EULER POLYNOMIALS, R I E M A "

COEFFICIENTS bk OF THE BERNOULLI POLYNOMIALS B, (x)= $ bkxk

Table 23.1

k=O

n\k

o

0

1

1

-12

-

1

-1

0

6

0

1

-+ +

0

1 42

0

-+

0

0

1

-+

1

-&

5

0

8

- -30

1

0

9

0

-1

10

5

0

11

0

5

0

12

-m

0

5

66

-2

1 6

15

0

8

9

3

-+

0

-3

0

0

0

2

-3

0

-+

-%

11

12

13

1 3 - 2 2

1

14

15

3 0

- 2

-+

1

-U 3

-4

0

6

5

0

-7

0

0

11

0

- 11

- -33

O 9

0

0

15. 2

1

5

0

- f30i

0

-3

O

22

0

2

-m 10

0

O

- 455

b

-9

0

9

o y -3.P o 0

- 2m

1

_ 29_

1

l+ - 5

-4

1

0

1

1

-1

2

3

4

5

6

7

b

2

O

l1

-?

0

0

1 -6

0-1001

y

0

9 1 - 7 6

30

0y2

k=O

0

1

2i-11

0

COEFFICIENTS ek OF THE EULER POLYNOMIALS E, (x)= n\k

10

1

-3

0

2730

14

7

1

f

. 3

6

O

6

1

10

13

5

4

1

T

3 4

3

1

+

2

2

0 - - a

0

2

1

25-15. 2

2

1

ekxk

8

10

9

11

12 13

14

15

1

2 2

0

3

1

0

0

1

0

0

3

0

4 5

-1

1

-5

4

-+

1 -2

1

-3-

1

b

0

-3

0

5

0

-3

1

7

17 8

0

-y

0

5

0

-3

1

8

0

17

0

- 28

0

14

0

-4

21

0

9

-5

0

10

0

0 691

l1

0 0

O

12

0

2073

13

- 5461

0

14

0

- 38227

15

929569

0

2

16

9

-155

7-

-63

-9 0 -

255 4

0

0

30

0

-5

1

- 231

0

-165 4

0

-5

0

1683

7

- 396

0

55

0

22165 2

0

7293

0

1287 2

0

lp

62881

0

0

7293

0

155155

0

109395

0

30!3

7

8

0943215 7 4 573405

-126

1

0

-3410

2 0

0

0 2 8 0 5

1

_ -29

-31031 0

7-

-1001

0

1 1

-6

o - + 91 0

1 0 -7

9

0-

1

15 2

1

810

BERNOULLI AND EULER POLYNOMIALS, RIEMANN ZETA FUNCTION

BERNOULLI AND EULER NUMBERS B,=NID

Table 23.2

N

D

0

1

1

1 2 4 6 8

-1 1 -1 1 -1

2 6 30 42 30

10 12 14 16 18

5 -691 7 -3617 43867

66 2730 6 51 0 798

74611 54513 64091 53103 61029

330 138 2730 6 870

58412 76005 93210 41217 76878 58367 30534 77373 39138 41559

14322 510 6 19 19190 6

n

20 22 24 26 28

-1 8 -2363 85 -2 37494

30

861 -770 257 -26315 27155 2 92999

32 34 36 38 40 42 44 46 48

-2 15 -278 5964 -560 94033

61082 20097 33269 51111 68997

71849 64391 57930 59391 81768

64491 80708 10242 21632 62491

22051 02691 35023 77961 27547

13530 1806 690 282 46410

05241 35489 84862 26753 41862

07964 95734 42141 68541 04677

82124 79249 81238 57396 59940

77525 91853 12691 63229 36021

66 1590 798 870 354

60 -121 52331 40483 75557 20403 04994 07982 02460 41491

567 86730

50 52 54 56 58

49 -80116 29 14996 -2479 39292 84483 61334

50572 57181 36348 93132 88800

B7l (

0) 1.0000 00000

( - 21 -3.3333 33333

( 34)-2.1399 94926

1 -1 5 61 1385

-

- 50521

10

12 14 16 18

27 -1993 1 93915 -240 48796

20 22 24 26 28

37037 11882 37525

30 32 34 36 38 40 42 44 46 48 50

52 54 56 58 60

-44 17751 -80 72329 41222 06033 -234 89580 52704

1 48511 50718 -1036 46227 33519 7 94757 94225 97592 -6667 53751 66855 44977 60 96278 64556 85421 58691 -60532 85248 650 61624 86684 -7 54665 99390 08739 9420 32189 64202 41204 -126 22019 25180 62187 19903

18862 60884 09806 20228 40923

18963 77158 14325 623% 72874

14383 70634 65889 90583 89255

02765 60981 12145 75441

15438 93249 93915 79539 92358 87898 95177 02122 31082 52017

02310 28943 06216 34707 82857

45536 66647 82474 96712 61989

82821 89665 53281 59045 47741

11498 61211 70360 43502 68574

00178 93979 80405 84747 28768

77156 57304 10088 73748 43153

78140 58266 74518 59763 07061 95192 19752 41076 97653 90444

84425 10201 73805 84661 35185

78511 08082 73674 22720 48234

16490 29834 42122 93888 10611

88103 83644 40024 52599 91825

49822 23676 71169 64600 59406

51468 15121 53855 76565

98586 45581

93949 05945 99649 20041

181089 11496 57923 04965 45807 74165 21586 88733 48734 92363 14106 00809 54542 31325

From H.T. Davis, Tables of the higher mathematical functions, vol. 11. Principia Press, Bloomington, Ind., 1935 (with permission).

BEWOULLI AND EULER POLYNOMIALS, R I E M A "

SUMS OF RECIPROCAL POWERS

ck - n k= W

n

811

ZETA FUNCTION

Table 23.3

c (-1)k-lk-n k=l 0)

c(n)=

v(n)=

1

1 2 3 4 5 6

1.64493 1.20205 1.08232 1.03692 1.01734

40668 69031 32337 77551 30619

48226 59594 11138 43369 84449

43647 28540 19152 92633 13971

0.69314 0.82246 0.90154 0.94703 0.97211 0.98555

71805 70334 26773 28294 97704 10912

59945 24113 69695 97245 46909 97435

30942 21824 71405 91758 30594 10410

7 8 9 10 11 12

1.00834 1.00407 1.00200 1.00099 1.00049 1.00024

92773 73561 83928 45751 41886 60865

81922 97944 26082 27818 04119 53308

82684 33938 21442 08534 46456 04830

0.99259 0.99623 0.99809 0.99903 0.99951 0.99975

38199 30018 42975 95075 71434 76851

22830 52647 41605 98271 98060 43858

28267 89923 33077 56564 75414 19085

13 14 15 16 17 18

1.00012 1.00006 1.00003 1.00001 1,00000 1.00000

27133 12481 05882 52822 76371 38172

47578 35058 36307 59408 97637 93264

48915 70483 02049 65187 89976 99984

0.99987 0.99993 0.99996 0.99998 0.99999 0.9f3999

85427 91703 95512 47642 23782 61878

63265 45979 13099 14906 92041 69610

11549 71817 23808 10644 01198 11348

19 20 21 22 23 24

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

19082 09539 04769 02384 01192 00596

12716 62033 32986 50502 19925 08189

55394 87280 78781 72773 96531 05126

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

80935 90466 95232 97616 98808 99403

08171 11581 58215 13230 01318 98892

67511 52212 54282 82255 43950 39463

25 26 27 28 29 30

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00298 00149 00074 00037 00018 00009

03503 01554 50711 25334 62659 31327

51465 82837 78984 02479 72351 43242

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99701 99850 99925 99962 99981 99990

98856 99231 49550 74753 37369 68682

96283 99657 48496 40011 41811 28145

31 32 33 34 35 36

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00004 00002 00001 00000 00000 00000

65662 32831 16415 58207 29103 14551

90650 18337 50173 72088 85044 92189

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99995 99997 99998 99999 99999 99999

34340 67169 83584 41792 70896 85448

33145 89595 85805 39905 18953 09143

37 38 39 40 41 42

1,00000 1.00000 1.00000 1.00000 1.00000 1.00000

00000 00000 00000 00000 00000 00000

07275 03637 01818 00909 00454 00227

95984 97955 98965 49478 74738 37368

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99999 99999 99999 99999 99999 99999

92724 96362 98181 99090 99545 99772

04461 02193 01084 50538 25268 62633

Q)

For n > 4 2 , f(n+l) = [ 1t {(n)] 7 (n+ 1)=f [1t v @ ) ] From H. T. Davis, Tables of the higher mathematical functions, vol. 11. Principia Press, Bloomington, Ind., 1935 (with permission).

812

BERNOULLI AND EULER POLYNOMIALS, R I E M A "

SUMS OF RECIPROCAL POWERS

Table 23.3 n

ZETA FUNCTION

x(n)=

5 (2k+l)

-?I

k=O

P(n)=

5 (-l)k(2k+l)-"

k=O

1 2 3 4 5 6

1.23370 1.05179 1.01467 1.00452 1.00144

05501 97902 80316 37627 70766

36169 64644 04192 95139 40942

82735 99972 05455 61613 12191

0.78539 0.91596 0.96894 0.98894 0.99615 0.99868

81633 55941 61462 45517 78280 52222

97448 77219 59369 41105 77088 18438

310 015 380 336 064 135

7 8 9 10 11 12

1.00047 1.00015 1.00005 1.00001 1.00000 1.00000

15486 51790 13451 70413 56660 18858

52376 25296 83843 63044 51090 48583

55476 11930 77259 82549 10935 11958

0.99955 0.99984 0.99994 0.99998 0.99999 0.99999

45078 99902 96841 31640 43749 81223

90539 46829 87220 26196 73823 50587

909 657 090 877 699 882

13 14 15 16 17 18

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

06280 02092 00697 00232 00077 00025

55421 40519 24703 37157 44839 81437

80232 21150 12929 37916 45587 55666

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

93735 97910 99303 99767 99922 99974

83771 87248 40842 75950 57782 19086

841 735 624 903 104 745

19 20 21 22 23 24

1.00000 1.00000 1.00000 1.00000 1.00000 1,00000

00008 00002 00000 00000 00000 00000

60444 86807 95601 31866 10622 03540

11452 69746 16531 77514 20241 72294

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99991 99997 99999 99999 99999 99999

39660 13213 04403 68134 89377 96459

745 274 029 064 965 311

25 26 27 28 29 30

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00000 00000 00000 00000 00000 00000

01180 00393 00131 00043 00014 00004

23874 41247 13740 71245 57081 85694

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99999 99999 99999 99999 99999 99999

98819 99606 99868 99956 99985 99995

768 589 863 288 429 143

31 32 33 34 35 36

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00000 00000 00000 00000 00000 00000

00001 00000 00000 00000 00000 00000

61898 53966 17989 05996 01999 00666

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99999 99999 99999 99999 99999 99999

99998 99999 99999 99999 99999 99999

381 460 820 940 980 993

37 38 39 40 41 42

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00000 00000 00000 00000 00000 00000

00000 00000 00000 00000 00000 00000

00222 00074 00025 00008 00003 00001

0.99999 99999 99999 998 0.99999 99999 99999 999

Q)

BERNOULLI A X D EULER POLYNOMIALS, R I E M A ”

813

ZETA FUNCTION

Table 23.4

SUMS OF POSITIVE POWERS g k n k=l

m\n 1 2 3 4 5

1 1 3 6 10 15

1 5 14 30 55

3 1 9 36 100 225

4 1 17 98 354 979

5 1 33 276 1300 4425

6 1 65 794 4890 20515

6 7 8 9 10

21 28 36 45 55

91 140 204 285 385

441 784 1296 2025 3025

2275 467 6 8772 15333 25333

12201 29008 61776 1 20825 2 20825

67171 1 84820 4 46964 9 78405 19 78405

11 12 13 14 15

66 78 91 105 120

506 650 819 1015 1240

4356 6084 8281 11025 14400

39974 60710 89271 1 27687 1 78312

16 17 18 19 20

136 153 171 190 210

1496 1785 2109 2470 2870

18496 23409 29241 36100 44100

2 3 4 5 7

21 22 23 24 25

231 253 276 300 325

3311 3795 4324 4900 5525

26 27 28 29 30

351 378 406 435 465

31 32 33 34 35

2

3 6 10 15 22

81876 30708 02001 39825 99200

37 67 115 190 304

49966 35950 62759 92295 82920

43848 27369 32345 62666 22666

33 47 66 91 123

47776 67633 57201 33300 33300

472 713 1054 1524 2164

60136 97705 09929 55810 55810

53361 64009 76176 90000 1 05625

9 17147 11 51403 1 4 31244 17 63020 2 1 53645

164 215 280 359 457

17401 71033 07376 70000 35625

3022 4156 5636 7547 9988

21931 01835 37724 40700 81325

6201 6930 7714 8555 9455

1 23201 1 42884 1 64836 1 89225 2 16225

26 31 37 44 52

10621 42062 56718 63999 73999

576 719 891 1096 1339

17001 65908 76276 87425 87425

13077 16952 21771 27719 35009

97101 17590 07894 31215 31215

496 528 561 595 630

10416 11440 12529 13685 14910

2 46016 2 78784 3 14721 3 54025 3 96900

61 72 84 97 112

97520 46096 32017 68353 68978

1626 1961 2353 2807 3332

16576 71008 06401 41825 63700

43884 54621 67536 82984 1 01367

34896 76720 44689 49105 14730

36 37 38 39 40

666 703 741 780 82 0

16206 17575 19019 20540 22140

4 4 5 6 6

43556 94209 49081 08400 72400

129 148 169 192 217

48594 22755 07891 21332 81332

3937 4630 5423 6325 7349

29876 73833 09001 33200 33200

1 23134 1 48792 1 78901 2 14089 2 55049

97066 23475 59859 03620 03620

41 42 43 44 45

861 903 946 990 1035

23821 25585 27434 29370 31395

7 8 8 9 10

41321 15409 94916 80100 71225

246 277 311 348 389

07093 18789 37590 85686 86311

8507 9814 11284 12934 14779

89401 80633 89076 05300 33425

3 3 4 4 5

07861 39605 02654 16510 82135

46 47 48 49 50

1081 1128 1176 1225 1275

33511 35720 38024 40425 42925

11 68561 1 2 72384 1 3 82976 15 00625 1 6 25625

02550 57440 20654 93217 76254

16838 96401 6 70997 79031 7 78789 94360 19132 41408 9 01095 84824 21680 45376 1 0 39508 72025 24505 20625 11 95758 72025 27630 20625 From H. T. Davis, Tables of the higher mathematical functions, vol. 11. Principia Press, Bloomington,

Ind., 1935 (with permission).

434 483 536 594 656

63767 43448 51864 16665 66665

814

BERNOULLI AND EULER POLYNOMIALS, RIEMA"

Table 23.4

SUMS OF POSITIVE POWERS

ZETA FUNCTION

gk*

k=l

51 52 53 54 55

1 1326 1378 1431 1485 1540

2 45526 48230 51039 53955 56980

17 18 20 22 23

3 58276 98884 47761 05225 71600

724 797 876 961 1052

4 31866 43482 33963 37019 87644

31080 34882 39064 43656 48688

45876 49908 45401 10425 94800

13 15 17 20 23

6 71721 69427 91071 39020 15826

59826 69490 30619 41915 82540

56 57 58 59 60

1596 1653 1711 1770 1830

60116 63365 66729 70210 73810

25 27 29 31 33

47216 32409 27521 32900 48900

1151 1256 1369 1491 1620

22140 78141 94637 11998 71998

54196 60213 66776 73925 81701

26576 18633 75401 99700 99700

26 29 33 37 42

24236 67201 47888 69693 36253

61996 09245 01789 35430 35430

61 62 63 64 65

1891 1953 2016 2080 2145

77531 81375 85344 89440 93665

35 38 40 43 46

75881 14209 64256 26400 01025

1759 1906 2064 2232 2410

17839 94175 47136 24352 74977

90147 99309 1 09233 1 19971 1 31573

96001 28833 65376 07200 97825

47 53 59 66 73

51457 19459 44694 31889 86078

09791 45375 47584 24320 14945

66 67 68 69 70

2211 2278 2346 2415 2485

1 1 1 1

98021 02510 07134 11895 16795

48 51 55 58 61

88521 89284 03716 32225 75225

2600 2802 3015 3242 3482

49713 00834 82210 49331 59331

1 1 1 1 2

44097 57598 72137 87778 04585

30401 55508 89076 20425 20425

82 91 101 111 123

12617 17201 05876 85057 61547

64961 47130 29754 92835 92835

71 72 73 74 75

2556 2628 2701 2775 2850

1 1 1 1 1

21836 27020 32349 37825 43450

65 69 72 77 81

33136 06384 95401 00625 22500

3736 4005 4289 4589 4905

71012 44868 43109 29685 70310

2 2 2 2 3

22627 41976 62707 84897 08627

49776 67408 39001 45625 92500

136 150 165 181 199

42550 35691 49033 91098 70883

76756 46260 72549 62725 78350

76 77 78 79 80

2926 3003 3081 3160 3240

1 1 1 1 1

49226 55155 61239 67480 73880

85 90 94 99 104

61476 18009 92561 85600 97600

5239 5590 5961 6350 6760

32486 85527 00583 50664 10664

3 3 3 4 4

33983 61051 89922 20693 53461

17876 02033 76401 32800 32800

218 239 262 286 312

97883 82106 34102 64977 86417

06926 87015 87719 43240 43240

81 82 83 84 85

3321 3403 3486 3570 3655

1 1 1 2 2

80441 87165 94054 01110 08335

110 115 121 127 133

29041 80409 52196 44900 59025

7190 7642 8117 8615 9137

57385 69561 27882 15018 15643

4 5 5 6 6

88329 25403 64793 06614 50985

17201 15633 56276 75700 28825

341 371 404 439 477

10712 50779 20183 33163 04658

79721 51145 24514 56130 71755

86 87 88 89 90

3741 3828 3916 4005 4095

2 2 2 2 2

15731 23300 31044 38965 47065

139 146 153 160 167

95081 53584 35056 40025 69025

9684 10257 10856 11484 12140

16459 06220 75756 17997 27997

6 98027 99001 7 47870 08208 8 00643 27376 8 56483 86825 9 15532 86825

517 560 607 657 710

50331 86593 30633 00446 14856

06891 07900 94684 85645 85645

91 92 93 94 95

4186 4278 4371 4465 4560

2 2 2 2 2

55346 63810 72459 81295 90320

175 183 191 199 207

22596 01284 05641 36225 93600

12826 13542 14290 15071 15885

02958 42254 47455 22351 72976

9 10 11 11 12

77936 43844 13413 86803 64181

08276 23508 07201 47425 56800

766 827 892 961 1034

93549 57099 27001 25699 76617

37686 39030 22479 03535 94160

96 97 98 99 100

4656 4753 4851 4950 5050

2 3 3 3 3

99536 08945 18549 28350 38350

216 225 235 245 255

78336 91009 32201 02500 02500

16735 17620 18542 19503 20503

07632 36913 73729 33330 33330

13 14 15 16 17

45718 31592 21984 17083 17083

83776 24033 32001 32500 32500

1113 1196 1284 1379 1479

04195 33915 92339 07141 07141

83856 88785 69649 19050 19050

m b

5

815

BERNOULLI AND EULER POLYNOMIALS, RIEMANN ZETA FUNCTION

SUMS OF POSITIVE POWERS

1 2 3 4 5

Table 23.4

9

8

7 1 129 2316 18700 96825

m\n

Sh

k=l

1 513 20196 2 82340 22 35465

1 257 6818 72354 4 62979

6 7 8 9 10

3 12 32 80 180

76761 00304 97456 80425 80425

21 79 246 677 1677

42595 07396 84612 31333 31333

123 526 1868 5743 15743

13161 66768 84496 04985 04985

11 12 13 14 15

375 733 1361 2415 4124

67596 99404 47921 61425 20800

3820 8120 16278 31035 56664

90214 71910 02631 91687 82312

39322 90920 1 96965 4 03575 7 88009

52676 33028 32401 79185 38560

16 17 18 19 20

6808 10911 17034 25972 38772

56256 94929 14961 86700 86700

1 2 4 7

99614 69372 79571 49407 05407

49608 07049 67625 30666 30666

14 26 46 78 129

75204 61082 44675 71552 91552

15296 91793 82161 79940 79940

21 22 23 24 25

56783 81727 1 15775 1 61640 2 22675

75241 33129 58576 30000 45625

10 16 24 35 50

83635 32394 15504 16257 42136

90027 63563 48844 63020 53645

209 330 510 774 1155

34353 07045 18572 36647 83620

26521 44313 05776 46000 11625

26 27 28 29 30

3 4 5 7 9

02993 07597 42526 15025 33725

55801 09004 37516 13825 13825

71 99 137 187 252

30407 54702 32722 35186 96186

18221 54702 53038 65999 65999

1698 2461 3519 4969 6938

78656 34631 19191 90651 20651

90601 75588 28996 04865 04865

31 32 33 34 35

12 15 19 25 31

08851 52448 78633 03866 47259

27936 66304 09281 59425 56300

338 448 588 767 992

25097 20213 84299 42238 60992

03440 31216 49457 54353 44978

9582 13100 17741 23813 31695

16872 60593 75437 45365 01751

65536 54368 56321 22785 94660

36 37 38 39 40

39 48 60 73 90

30901 80219 24375 96685 35085

20396 97529 80121 86800 86800

1274 1625 2060 2595 3251

72091 96886 74807 94900 30900

52434 06355 44851 05332 05332

41851 54847 71368 92241 1 18456

01318 18716 79729 63340 03340

63076 58153 21001 79760 79760

41 42 43 44 45

109 132 160 191 229

82628 88021 06208 98986 35680

60681 93929 05036 14700 67825

4049 5018 6186 7591 9273

80152 06672 88675 70911 22165

34453 30869 08470 33686 24311

1 1 2 3 3

51194 91861 42120 03932 79600

22684 36523 62642 81037 87463

73721 23193 60036 69540 47665

46 47 48 49 50

272 323 382 450 528

93857 60088 30771 13002 25502

25041 45504 87776 60625 60625

11277 13659 16477 19800 23706

98287 11154 03958 33264 58264

56247 18008 47064 16665 16665

4 5 7 8 10

71819 83732 18993 81834 77147

89090 93821 48427 84406 34406

16721 19488 14176 24625 24625

816

BERNOULLI A I D EULER POLYNOMIALS, R I E M A "

SUMS OF POSITIVE POWERS

Table 23.4

ZETA FUNCTION

-5.b

h=l

51 52 53 54 55

617 720 838 972 1124

99609 80326 27437 16689 41042

38476 41004 80841 90825 25200

28283 33629 39855 47085 55458

8 37709 34995 31899 51512 90891

87066 18522 29883 69019 59644

13 15 19 23 27

10563 88554 18530 08961 69498

9 86137 44973 80891 40014 05854

15076 50788 52921 66265 50640

56 57 58 59 60

1297 1492 1713 1962 2242

11990 60965 40807 27322 20922

74736 67929 35481 20300 20300

65130 76273 89079 1 03762 1 20559

64007 55578 86395 90771 06771

33660 45661 63677 67998 67998

33 39 46 55 65

11115 46261 89027 55326 63096

00335 19889 07286 65472 25472

95536 79593 24521 79460 79460

61 62 63 64 65

2556 2908 3302 3742 4232

48350 64496 54303 34768 57047

56321 62529 01696 12800 03425

1 39729 1 61563 1 86379 2 14526 2 46391

79901 80957 38760 88527 36656

65279 50175 17696 28352 18977

77 90 106 124 145

32510 86219 49600 51040 22232

86401 51863 93432 78527 06906

13601 77153 30976 12960 03585

66 67 68 69 70

4778 5384 6056 6801 7624

08654 15770 45658 09190 63490

04481 09804 28236 80825 80825

2 3 3 4 4

82395 23002 68718 20098 77746

42718 19494 51890 35635 36635

88673 45314 98690 27331 27331

168 196 227 262 303

98500 19153 27863 73072 08433

07044 51006 53971 32327 02327

03521 98468 28036 04265 04265

71 72 73 74 75

8534 9537 10641 11857 13191

14692 20822 94807 07610 91497

39216 43504 62601 35625 07500

5 6 6 7 8

42321 14542 95188 85107 85220

71947 13310 14229 61631 53135

73092 81828 75909 79685 70310

348 400 459 526 601

93283 93152 80311 34352 42821

09511 87653 54736 62487 25280

53296 82288 50201 29625 26500

76 77 78 79 80

14656 16261 18017 19938 22035

43442 28675 84364 23453 38653

79276 46129 01041 87200 87200

9 11 12 14 15

96524 20097 57109 08819 76592

01010 63925 07632 95731 11731

25286 72967 56103 62664 62664

686 781 888 1007 1142

01885 17055 03947 89106 10879

63746 08237 17370 77196 57196

04676 76113 60721 79040 79040

81 82 83 84 85

24323 26815 29529 32480 35686

06578 92048 52558 42905 19993

42161 98929 88556 44300 72425

17 19 21 24 27

61894 66308 91537 39413 11903

13620 22206 44528 33638 86142

14505 69481 08522 91018 81643

1292 1459 1646 1854 2086

20343 82298 76323 97898 59593

10166 14263 66939 52248 15080

78161 86193 26596 56260 59385

86 87 88 89 90

39165 42938 47024 51447 56230

47815 02610 78207 91556 88456

94121 81904 18896 14425 14425

30 33 36 40 45

11121 39333 98967 92626 23094

78853 46007 98488 86545 07545

47499 84620 39916 41997 41997

2343 2629 2945 3296 3683

92334 46750 94588 30228 72277

88197 30627 48916 85991 75991

23001 52528 18576 03785 03785

91 92 93 94 95

61398 66976 72993 79478 86462

49475 96076 96947 74541 11837

50156 73804 34561 53825 63200

49 55 60 66 73

93346 06565 66147 75716 39136

60306 47620 28587 22441 65570

93518 69134 19535 30351 20976

4111 4583 5104 5677 6307

65257 81394 22502 21982 46923

77288 10154 40039 62325 59571

92196 48868 36161 52865 62240

96 97 98 99

93976 1 02056 1 10737 1 20058 1 30058

59315 42160 67693 33041 33041

74016 52129 76801 67500 67500

80 88 96 106 116

60526 44269 95032 17777 17777

23468 59412 61670 31113 31113

59312 36273 54129 33330 33330

7000 7760 8593 9507 10507

00323 23429 98205 49930 49930

17816 04362 25663 00499 00499

42496 07713 57601 98500 98500

7

m\n

100

BERNOULLI AND EULER POLYNOMIALS, R I E M A ”

SUMS OF POSITIVE POWERS g k n

Table 23.4

k=l

m\n I

51 52 53 54 55

613 757 932 1143 1396

10 38941 75112 94452 34603 83199 38258 66451 30907 95967 52098

62626 19650 32699 53275 93900

40451 15700 57524 41925 41925

56 57 58 59 60

1700 2062 2493 3004 3608

26516 29849 10270 21945 88121

43060 57628 26623 59629 59629

08076 99325 05149 46550 46550

10 1 1025 60074 11 08650 108 74275

1 2 3 4 5 6 7 8 9 10

713 3538 14275 49143 1 49143

817

ZETA FUNCTION

m\n

11 12 13 14 15

4 10 24 52 110

08517 27691 06276 98822 65326

66526 30750 22599 77575 68200

61 62 63 64 65

4322 5161 6146 7299 8645

22412 52349 45378 37528 64962

76258 34941 53759 99828 44456

29151 69375 60224 07200 97825

16 17 18 19 20

220 422 779 1392 2416

60442 20381 25054 35716 35716

95976 96425 23049 80850 80850

66 67 68 69 70

10213 12036 14150 16596 19421

98650 82430 74713 94119 69368

53564 99082 00654 07202 07202

93601 55050 65674 25475 25475

21 22 23 24 25

4084 6740 10882 17223 26760

34526 33754 98866 32676 06992

59051 50475 64124 29500 70125

71 72 73 74 75

22676 26420 30718 35642 41273

93723 84347 46930 45970 81117

17301 43545 40581 14140 23612

06676 94100 51749 29125 94750

26 27 28 29 30

40876 61465 91085 1 33156 1 92205

77949 89270 56937 29270 29270

23501 18150 13574 13775 13775

76 77 78 79 80

47702 55029 63365 72833 83570

70010 38057 15640 43249 85073

47012 72874 85236 11504 11504

36126 36775 36199 83400 83400

31 32 33 34 35

2 3 5 7 10

74168 86758 39916 46353 22208

12139 11208 01061 78601 52136

94576 37200 01649 61425 77050

81 82 83 84 85

1 1 1 1

95728 09473 24989 42479 62166

51619 31932 36051 48338 92382

02074 38034 10093 76074 16796

12201 70825 24274 16050 81675

36 37 38 39 40

13 18 24 33 43

87824 68682 96503 10544 59120

36537 80261 98741 59593 59593

40026 57875 46099 37700 37700

86 87 88 89 90

1 2 2 2 3

84297 09139 36989 68171 03039

08171 42312 52073 24066 08467

04827 96263 05665 05327 05327

52651 21500 33724 17325 17325

41 42 43 44 45

57 74 95 122 156

01386 09406 70554 90290 95353

52694 33911 57044 66428 55588

90101 67925 52174 70350 85975

91 92 93 94 95

3 3 4 4 5

41980 85419 33817 a7679 47552

69648 54190 77262 28403 97795

23434 47066 26359 21259 59638

62726 76550 94799 64975 55600

46 47 48 49 50

199 251 316 396 494

37428 97341 89847 69074 34699

30416 52774 73860 36836 36836

62551 92600 37624 49625 49625

96 97 98 99 100

6 6 7 8 9

14036 87778 69485 59924 59924

24155 65424 93493 14243 14243

51139 46067 33614 42419 42419

60176 86225 75249 24250 24250

818

BERNOULLI AND EULER POLYNOMIALS, RIEMANN ZETA FUNCTION

Table 23.5 nb

n!

2

3

5

4

I- 1

0 1.0125 00000 1 4.3392 85714

-

0 1.2232 47480 115.0968 64499

4 2.0527 69883 5i4.8300 46785

I- 1

-10 2.0473 98767 11 2.7919 07410

26 27

1-191 1.6640 28884 -20 1.2326 13988

-11) 1.1167 10881

I- 1

1 3 5.6630 23524 -14 8.8484 74256

I 1 41 42

1-381 6.5735 64027 -39 3.1302 68584

For x = 1, see Table 6.3.

-25 4.0348 -26 3.2715 2.5827 1.9867 1.4900

74405 19788 78780 52907 64681

1.0902 7.7877 5.4333 3.7045 2.4697

91230 94498 44998 53408 02272

1.6106 1.0280 6.4255 3.9340 2.3604

75395 90678 66735 20450 12270

1.2694 1.3724 1.4446 1.4816 1.4816

76128 06625 38553 80567 80567

I 1

-17 3.9118 75343 -18 5.2863 1 8 0 3 1

BERNOULLI AKD EULER POLYNOMIALS, R I E M A "

xn/n! 6

n\x

1 (

5

0)6.0000 1.8000 3.6000 5.4000 6.4800

00000 00000 00000 00000 00000

( ( ( ( (

7 0 7.0000 00000 112.4500 00000 1 2 2 1.4005 83333

[

1)4.9536 20388 2.8896 11893

I

( (

6 7 8 9

10

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

819

ZETA FUNCTION

Table 23.5 8 0) 8.0000 00000 1

9 (

0 9.0000 114. 0500 2) 1.2150 2 2.7337 214.9207

00000 00000 00000 50000 50000

24. Combinatorial Analysis K. GOLDBERO,~ M. NEW MAN,^ E. HAYNSWORTH~

Contents Mathematical Properties. . . . . . . . . . . . . 24.1. Basic Numbers . . . . . . . . . . . . . 24.1.1 Binomial Coefficients . . . . . . . 24.1.2 Multinomial Coefficients . . . . . . 24.1.3 Stirling Numbers of the First Kind . 24.1.4 Stirling Numbers of the Second Kind 24.2. Partitions . . . . . . . . . . . . . . . 24.2.1 Unrestricted Partitions . . . . . . . 24.2.2 Partitions Into Distinct Parts . . . . 24.3. Number Theoretic Functions . . . . . . . 24.3.1 The Mobius Function . . . . . . . 24.3.2 The Euler Function . . . . . . . . 24.3.3 Divisor Functions . . . . . . . . . 24.3.4 Primitive Roots . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . Table 24.1. Binomial Coefficients

.

.

.

.

.

.

.

. . , . . a

.

a

s

.

.

.

.

a

.

. . . . .

.

.

.

. . . .

. . . . .

. . . .

.

.

.

.

.

.

. . . . .

. . . .

.

.

.

. . . . . .

.

.

.

.

a

. . . . .

.

. . . . .

. . . . .

.

.

. . . . .

.

. . . .

. . . .

.

. . . . . . . * . ..

a

*

*

PBge

822 822 822 823 824 824 825 82 5 825 826 826 826 827 827 827 828

n150, mS25

Table 24.2. Multinomials (Including a List of Partitions)

. . . .

I

83 1

I

833

n510

Table 24.3. Stirling Numbers of the First Kind SArn)..

.. .. .

n525

. .. ... .

835

. .

836

. ... .... ... .....

840

....................

844

... . .....

864

. ... . ............. .. ...

870

Table 24.4. Stirling Numbers of the Second Kind S;;'") n525

Table 24.5. Number of Partitions and Partitions Into Distinct Parts Ph),

n5500

Table 24.6. Arithmetic Functions

$44, uo(n),U l h ) , n 5 1000 Table 24.7. Factorizations n<10000

Table 24.8. Primitive Roots, Factorization of p - 1 n<10000

Table 24.9. Primes p_<106

1.

* National Bureau of Standards.

* National

Bureau of Standards. (Presently, Auburn University.) 82 1

24. Combinatorial Analysis Mathematical Properties In cnch sub-section of tliis clinptcr we use a fi\etl formfit wliicli eliiplinsiscs t tic usc t l t i t l inet Iiotls of c\tciiding tlic iic~c.ortil)iiii~.il~~ t n1)lcs. Tlir foriiint follows tliis for111 : I . Definition4

A . ('oni1)inatorial B. Genernting functions C. Closed form

n specinl n n t l easily recogniznhle syrnhol, nnd

sytiihol tnust 1)c easy to write. R r have n sc8r;pt capital 3 without any certainty t l i n t we liave settled tliis quest ion prrriinriently . \.Ye fcel tlint t h e subscript-superscrir,t notation crnplinsixes tlic gcncrnting friiictioris (which :we pbwers of niiitiinlly itiversc. functions) froin wliich most of the iriiportnnt relations flow.

yet

tlint

settlctl

011

11. Relations

24.1. Basic Numbers

A. Recurrences B. Checks in computing C. Basic use in numerical nnalysis

24.1.1

Binomial Coefficients I. Definitions

111. Asymptotic and Special Values

In genernl the notations uscd are standard. This includes the difference operator A tlcfincd on functions of x by AJ(z) = f ( r i - 1) -.f(x), A"+!f(rr) =A(Anf(x)), the Kronecker delta a,, the Rieniann zetn functioii { ( s ) and the greatest common divisor symbol ( m ,n). The range of tliesunimands for a sumination sign without limits is esplained to the riglit of the formula. The notations whicli are not standard arc those for the multinomials whicli nre arbitrary sliorthand for use in this chapter, and those for the Stirling numbers whicli have never been standardized. A short table of various notations for these numbers follows :

A.

(i) is tlie number of

Fzrsl Kind

SA*) s;m'

n=O,l,. .

8; S:

W'

* *

C. Closed form

11. Relations

A . Recurrences

0::

n>m>l s(n, m)

(-l)n-mSl(n-l,n-rn)

S(n-m+l,

n)

S(n, m)

&(m, n-m)

=(:)+( m--1 )'--l)+

...

n> m +("im)

,S"

B. Checks u (n , m )

We feel that a capital S is natural for Stirling numbers of the first kind; i t is infrequendy used for other notation in this contest. B u t once it is used we have difficulty finding a suitable symbol for Stirling numbers of the second kind. The numbers are sufficiently important to warrant 822

n2m

-n(n-1) . . . ( n - m + l )m!

(24.91 Milne-Thomson

[24.15] Riordan [24.1] Carlitz} [24.3) Gould Miksa (Unpublished tables) (24.171 Gupta

I

,4<1

Second Kind

yy1 E:;

m

ohjects froill i i collection of n cliqtinct objects wit Iiout rcgnrcl to order. 13. Generating fuiictioiis

Notations for the Stirling Numbers

Reference This chapter [24.2] Fort 24.71 Jordan [24.10] hloser and Wyman

\ynys of clloosing

'Set' page 11.

r

r>

)i

$1

823

COMBINATORIAL ANALYSIS

where

f:

m = kC= O mkpk

n = k=O C nkpk,

(;)f'x-m'

m=0

m

OD

p h , , nk 20

n-k-1

I

C. Numerical analysis

f (x-s)

8-k

k=O

s
111. Special Values

(3 (3 =

=1

2"(2n-l)(2n-3) n!

. . . 3.1

Multinomial Coefficients

24.1.2

I. Definitions

A. (n;n1,.n2, . . ., n,) is the number of ways of putting n=nl+nz+. . . +nm different objects into m different boxes with nk in the k-th box, k = 1 , 2 , . . ., m. (n; al, az, . . ., an)* is the number of permutations.of n=a1+2a2+ . . . +na, symbols composed of ak cycles of length k for k=1, 2 , . . ., n. (n; a ] , az, . . ., an)' is the number of ways of partitioning a set of n=al+2az+. . . +na, different objects into ak subsets containing k objects for k=1, 2 , . . ., n.

B. Generating functions (q+xz+

. . . +xm)"=z(n;n,, n2, . .

(29

tk)l=m!Z,z

(g2

m

tk)

t"

., n,)x;l@

. . . x:m

summed over nl+n2+.

z(n;a,, a2, . . .,an)*z:lz;a. , . zn:

summed over a1+2az+ and al+az+

t"

=m! n-m

7 z(n; al,az, , . ., an)'z;lz;s , . .x:n

. . +n,=n

. . . +na,=n

. . . +an=m

72.

C. Closed forms

nl+n2+. . .+nm=n a1+2q+ . . .+na,=n a1+2a2+. . . +nan=n

(n;nl,n2,. . ., n m ) =n!/nl!n2! . . . n,! (n;a ] ,a2, . . .,a,) *=n!/l"la1!2"2az! . . . nGa,! . . . (n!)".a.! (n;al,Q . ~ ,, . ., an)'=n!/(l!)'lal!(2!)"~a2! 11. Relations

A. Recurrence (n+m;nl+1,n2+1, . . .,n,+l)=C

rmn

B. Checks

*

Z(n;nl,Q,. .,nm)= m! $:m) I

m

k-1

all ni 3 1

2(n;al,a2,. . ., an)*=(-l)n-mSkm' Z(n;al,az,. . .,a,)'= BAm)

(n+m-l; nl+l,

..

1

.,nk-l+l,nk,nk+l+l, .

summed over nl+nZ+ . . . +n,,,=n

summed over al+2az+

. . . +na,=n

C. Numerical analysis (Fah di Bruno's formula)

dn

-j(g(z))=

dz"

Cn f ( " ) ( g ( z ) ) Wa;] , . . ., an)' { g'(z)

m=O

summed over a1+2az+ 'See page U.

a2,

. . . +na,=n

. . ,71m+l)

and

)"I

{ g"(z)

al+q+.

}"z

. . . { g("'(z) )"*

. . +an=m.

and al+az+.

. . +an=m

COMBINATORIAL ANALYSIS

824 0

P I

2

P 2

PI

Pn-1

summed over a 1 + 2 6 + sum equal n!Zzlzz . . . z, 24.1.3

*.. ...

1

0

... ...

Pn-2

...

0

. ..

n-1

...

P I

. . . +mn=n;

e.g. if P k = 2 j - l x ~ for k = l , 2, . . ., n then the determinant and the latter sum denoting then-th elementary symmetric function of zl, xz, . . ., xr.

Stirling Numbers of the First Kind

-

I SArn)I

I. Definitions

111. Asymptotic8 and Special Values

(n- 1) !(y+ln n)m-l/(m -1) ! for m=o(ln n)

A. (- l)n-mSim)is the number of permutations of n symbols which have exactly m cycles. B. Generating functions x(x-1).

n

. . ( x - n + l ) = Cm-0

Sim)xm

C. Closed form (see closed form for

@hm))

82'=1

11. Relations

A. Recurrences S(m) -S~m,m-I)-nS~m) n+l-

24.1.4 Stirling Numbers of the Second Kind I. Definitions

A. B;&%sthe number of ways of partitioning a set of n elements into m non-empty subsets. B. Generating functions B. Checks n

sp=0 m-1

n

xn=C

m-0

@Lm)x(x-1)

(eZ-l)m=m!

. . . (z--m+l)

m

BAm'2 2"

n-m

Ixl<m-l

C. Numerical analysis

C. Closed form if convergent.

825

COMBINATORIAL ANALYSIS

B. Generating function Ixl
B. Checks n (-l)n-mrn!

m-0

p - 2 4

giap)=l

where

( ( s ) ) = x - [ z ] - 3 if z is not an integer if x is an integer =O 11. Relations

A. Recurrence

C. Numerical analysis Amf(x)=m!

sim)

m

n-m

-f(")(x) n!

if convergent

B. Check 111. Asymptotics and Special Values

-

lim m-*Ig$,m)= n+

m1-1

111. Asymptotics

man

SATmwS

1

P(4- -

for n=o(m*)

*m.\l;s

4 n a e

24.2.2 Partitions Into Distinct Parts I. Definitions

A. q(n) is the number of decompositions of n into distinct integer summands without regard to order. E.g., 5=1+4=2+3 so that q(5)=3. B. Generating function m

24.2. Partitions 24.2.1 Unrestricted Partitions

n =O

a

m

q(n)xn=nn= l (l+zn)= nni l (1-xZn-l)-l

I4<1

C. Closed form

I. Definitions

A. p (n ) is the number of decompositions of n into integer summands without regard to order. E.~.,5=1$4=2+~=1+1+3=1+2+2=1+1+ 1+2=1+1+1+1+1 so that p(5)=7.

where &(x) is the Bessel function of order 0 and A2k-l(n)was defined in part I.C. of the previous subsection.

826

COMBINATORIAL ANALYSIB 11. Relations

A. Recurrences

and if

=O otherwise

If(mnz)I =Euo(n)If(nz) 1 converges.

m-1 n-1

n-1

The cyclotomic polynomial of order n is II (f- l ) d n / d ) dln

B. Check O
111. Aeymptotice

(-l)kq(n-(3P&k))=1 if n=-

P-T

2

=O otherwise. 111. Asymptotics

24.3. Number Theoretic Functions

24.3.2 The Euler Totient Function

24.3.1 T h e Mobius Function

I. Definitions

I. Definitions

A. p(n) is the number of integers not exceeding and relatively prime to n. B. Generating functions

A. p(n)=l

if n=l = (- l ) k if n is the product of

k distinct primes =O if n is divisible by a square >1. B. Qenerating functions

2 P(nM-a=l/t(s)

0-1

9s>l

C. Closed form 11. Relations

A. Recurreme

over distinct primes p dividing n.

r(mn)=p(m)p(n) if ( m , n)=1

=O

if ( m , n)>l

B. Check

11. Relations

A. Recurrence (P(mn)=(P(m)(PP(n)

p(d)=L

C. Numerical analysis g(n) f(d) for all n if and only if

(m,n)=l

B. Checks

=F n

f ( n ) = Z rc(d)g(n/d)for all n g(n)=11 f(d) for all n if and only if dln f(n) =11 g(n/d)r(')for all n

a*(")= 1(mod n)

dln

g(z)

=gf(z/n) for all z>O if and only if

111. Asymptotice

n-1

nek-1

(a, n)=1

COM3INATORIAL ANALYSIS

24.3.3

Divisor Functions

l nu,(m,=$+O

I. Definitions

n2 m-1

'

A. uk(n)is the sum of the k-th powers of the divisors of n . Often uo(n)is denoted by d(n), and u*(n>by d n ) . B. Generating functions

2 uk(n)n-*=5.(s)5.(s-k)

n-1

9s>k+l

C. Closed form

24.3.4

(k)

827

Primitive Roots I. Definitions

The integers not exceeding and relatively prime to a fixed integer n form a group; the group is cyclic if and only if n = 2 , 4 or n is of the form p k or 2p' where p is an odd prime. Then g is a primitive root of n if it generates that group; i.e., if g, g2, . . ., g V ( " ) are distinct modulo n . There are cp(cp(n)) primitive roots of n . 11. Relations

A. Recurrences. If g is a primitive root of a prime p and gp-' $ l(mod p2) then g is a primitive root of p k for all k. Jf g P - l = 1(mod p 2 ) then g+p is a primitive root of p' for all k. If g is a primitive root of p' then either g or g+pk, whichever is odd, is a primitive root of 2pk. B. Checks. If g is a primitive root of n then g' is a primitive root of n if and only if (k, cp(n))= 1, and each primitive root of n is of this form.

11. Relations

A. Recurrences (mn)=uk(m)uk(n)

(my n)= 1

References Texts

124.11 L. Carlitz, Note on Norlunds polynomial Bt), Proc. Amer. Math. SOC.11, 452-455 (1960). [24.2]T. Fort, Finite differences (Clarendon Press, Oxford, England, 1948). (24.31 H . W. Gould, Stirling number representation problems, Proc. Amer. Math. Soc. 11, 447-451 (1960). [24.4]G. H.Hardy, Ramanujan (Chelsea Publishing Co., New York, N.Y., 1959). [24.5] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed. (Clarendon Press, Oxford, England, 1960). [24.6]L. K. Hua, On the number of partitions of a number into unequal parts, Trans. Amer. Math. Soc.

51, 194-201 (1942). [24.7]C. Jordan, Calculus of finite differences, 2d ed. (Chelsea Publishing Co., New York, N.Y.,

1960). [24.8]K. Knopp, Theory and application of infinite series (Blackie and Son, Ltd., London, England,

1951). [24.9]L. M. Milne-Thomson, The calculus of finite differences (Macmillan and Co., Ltd., London, England, 1951). [24.101 L. Moser and M. Wyman, Stirling numbers of the second kind, Duke Math. J. 25, 29-43 (1958). [24.111 L. Moser and M. Wyman, Asymptotic development of the Stirling numbers of the first kind, J. London Math. Soc. 33, 133-146 (1958). P4.121 H. H. Ostmann, Additive Zahlentheorie, vol. I (Springer-Verlag, Berlin, Germany, 1956).

[24.13]H . Rademacher, On the partition function, Proc. London Math. Soc. 43, 241-254 (1937). [24.14]H. Rademacher and A. Whiteman, Theorems on Dedekind sums, Amer. J. Math. 63, 377-407 (1941). [24.15]J. Riordan, An introduction to combinatorial analysis (John Wiley & Sons, Inc., New York, N.Y., 1958). [24.16]J. V. Uspensky and M. A. Heaslet, Elementary number theory (McGraw-HB1 Book Co., Inc., New York, N.Y., 1939). Tables

[24.17]British Association for the Advancement of Science, Mathematical Tables, vol. VIII, Number-divisor tables (Cambridge Univ. Press, Cambridge, England, 1940). n1lO'. [24.18]H.Gupta, Tables of distributions, Res. Bull. East Panjab Univ. 13-44 (1950);750 (1951). [24.19]H. Gupta, A table of partitions, Proc. London Math. Soc. 39, 142-149 (1935) and 11. 42, 546-549 (1937). p ( n ) , n=1(1)300; p ( n ) , n=301

(1)600. [24.20]G. KavBn, Factor tables (Macmillan and Co., Ltd., London, England, 1937). n 1256,000. [24.21]D. N. Lehmer, List of prime numbers from 1 to 10,006,721,Carnegie Institution of Washington, Publication No. 165, Washington, D.C. (1914). [24.22]Royal Society Mathematical Tables, vol. 3, Table of binomial coefficients (Cambridge Univ. Press, Cambridge, England, 1954). (:).for r s i n 5 100. [24.23]G. N. Watson, Two tables of partitions, Proc. London Math. Soc. 42, 550-556 (1937).

828

COMBINATORIAL ANALYSIS

BINOMIAL COEFFICIENTS

Table 24.1

(i)

n\m

o

1

2

3

4

5

1 2 3 4 5

1 1 1 1 1

1 2 3 4 5

1 3 6 10

1 4 10

1 5

1

6 7 8 9 10

1 1 1 1 1

6 7 8 9 10

15 21 28 36 45

20 35 56 84 120

15 35 70 126 210

6 21 56 126 252

1 7 28 84 210

1 8 36 120

1 9 45

11 12 13 14 15

1 1 1 1 1

11 12 13 14 15

55 66 78 91 105

165 220 286 364 455

330 495 715 1001 1365

462 792 1287 2002 3003

462 924 1716 3003 5005

330 792 1716 3432 6435

165 495 1287 3003 6435

16 17 18 19 20

1 1 1 1 1

16 17 18 19 20

120 136 153 171 190

560 680 816 969 1140

1820 2380 3060 3876 4845

4368 6188 8568 11628 15504

8008 12376 18564 27132 38760

11440 19448 31824 50388 77520

12870 24310 43758 75582 1 25970

21 22 23 24 25

1 1 1 1 1

21 22 23 24 25

210 231 253 276 300

1330 1540 1771 2024 2300

5985 7315 8855 10626 12650

20349 26334 33649 42504 53130

54264 74613 1 00947 1 34596 1 77100

26 27 28 29 30

1 1 1 1 1

26 27 28 29 30

325 351 378 406 435

2600 2925 3276 3654 4060

14950 17550 20475 23751 27405

65780 80730 98280 1 18755 1 42506

2 2 3 4 5

31 32 33 34 35

1 1 1 1 1

31 32 33 34 35

465 496 528 561 595

4495 4960 5456 5984 6545

31465 35960 40920 46376 52360

1 2 2 2 3

69911 01376 37336 78256 24632

36 37 38 39 40

1 1 1 1 1

36 37 38 39 40

630 666 703 741 780

7140 7770 8436 9139 9880

58905 66045 73815 82251 91390

3 4 5 5 6

41 42 43 44 45

1 1 1 1 1

41 42 43 44 45

820 861 903 946 990

10660 11480 12341 13244 14190

101270 111930 123410 135751 148995

7 8 9 10 12

a

6

1 1 2 3 4

16280 70544 45157 46104 80700

2 3 4 7 10

03490 19770 90314 35471 81575

30230 96010 76740 75020 93775

6 8 11 15 20

57800 88030 84040 60780 35800

15 22 31 42 58

62275 20075 08105 92145 52925

7 9 11 13 16

36281 06192 07568 44904 23160

26 33 42 53 67

29575 65856 72048 79616 24520

78 105 138 181 235

88725 18300 84156 56204 35820

76992 35897 01942 75757 58008

19 23 27 32 38

47792 24784 60681 62623 38380

83 102 126 153 186

47680 95472 20256 80937 43560

302 386 489 615 769

60340 08020 03492 23748 04685

49398 50668 62598 86008 21759

44 52 60 70 81

96388 45786 96454 59052 45060

224 269 322 383 453

81940 78328 24114 20568 79620

955 1180 1450 1772 2155

48245 30185 08513 32627 53195

46 1 46 1035 15180 163185 13 70754 93 66819 535 24680 2609 32815 47 1 47 1081 16215 178365 15 33939 107 37573 628 91499 3144 57495 48 1 48 1128 17296 194580 17 12304 122 71512 736 29072 3773 48994 49 1 49 1176 18424 211876 19 06884 139 83816 859 00584 4509 78066 50 1 50 1225 19600 230300 21 18760 158 90700 998 84400 5368 78650 From Royal Society Mathematical Tables, vol. 3, Table of binomial coefficients. Cambridge Univ. Press, Cambridge, England, 1954 (with permission).

829

COMBINATORIAL ANALYSIS

BINOMIAL COEFFICIENTS 9

10

9 10

1 10

1

11 12 13 14 15

55 220 715 2002 5005

16 17 18 19 20

11440 24310 48620 92378 1 67960

(i)

Table 24.1

11

12

13

11 66 286 1001 3003

1 12 78 364 1365

1 13 91 455

1 14 105

8008 19448 43758 92378 1 84756

4368 12376 31824 75582 1 67960

1820 6188 18564 50388 1 25970

560 2380 8568 27132 77520

21 22 23 24 25

2 4 8 13 20

93930 97420 17190 07504 42975

3 6 11 19 32

52716 46646 44066 61256 68760

3 7 13 24 44

52716 05432 52078 96144 57400

2 6 13 27 52

93930 46646 52078 04156 00300

2 4 11 24 52

03490 97420 44066 96144 00300

26 27 28 29 30

31 46 69 100 143

24550 86825 06900 15005 07150

53 84 131 200 300

11735 36285 23110 30010 45015

77 130 214 345 546

26160 37895 74180 97290 27300

96 173 304 518 864

57700 83860 21755 95935 93225

104 200 374 678 1197

00600 58300 42160 63915 59850

31 32 33 34 35

201 280 385 524 706

60075 48800 67100 51256 07460

443 645 925 1311 1835

52165 12240 61040 28140 79396

846 1290 1935 2860 4172

72315 24480 36720 97760 25900

1411 2257 3548 5483 8344

20525 92840 17320 54040 51800

2062 3473 5731 9279 14763

53075 73600 66440 83760 37800

36 37 38 39 40

941 1244 1630 2119 2734

43280 03620 11640 15132 38880

2541 3483 4727 6357 8476

86856 30136 33756 45396 60528

6008 8549 12033 16760 23118

05296 92152 22288 56044 01440

12516 18524 27074 39107 55868

77700 82996 75148 97436 53480

23107 35624 54149 81224 1 20332

89600 67300 50296 25444 22880

41 42 43 44 45

3503 4458 5639 7089 8861

43565 91810 21995 30508 63135

11210 14714 19173 24812 31901

99408 42973 34783 56778 87286

31594 42805 57520 76693 1 01505

61968 61376 04349 39132 95910

1 1 2 2

78986 10581 53386 10906 87600

54920 16888 78264 82613 21745

1 2 3 5 7

76200 55187 65768 19155 30062

76360 31280 48168 26432 09045

46 47 48 49 50

11017 13626 16771 20544 25054

16330 49145 06640 55634 33700

40763 51780 65407 82178 1 02722

50421 66751 15896 22536 78170

1 1 2 2 3

33407 74171 25952 91359 73537

83196 33617 00368 16264 38800

3 5 6 9 12

89106 22514 96685 22637 13996

17655 00851 34468 34836 51100

10 14 19 26 35

17662 06768 29282 25967 48605

30790 48445 49296 83764 18600

830

COMBINATORIAL ANALYSIS

BINOMIAL COEFFICIENTS

Table 24.1

14 15

14 1 15

16 17 18 19 20

120 680 3060 11628 38760

n?m

15

16

1 18 171 1140

1 19 190

1 20

54264 1 70544 4 90314 1 3 07504 32 68760

20349 74613 2 45157 7 35471 20 42975

5985 26334 1 00947 3 46104 10 81575

1330 7315 33649 1 34596 4 80700

21 0 1540 8855 42504 1 77100

1

26 27 28 29 30

96 200 401 775

57700 58300 16600 58760

77 173 374 775 1551

26160 83860 42160 58760 17520

2651 82525 4714 35600

3005 5657 10371 is559 32479

40195 22720 58320 43160

55679 93641 1 54712 2 51408 4 02253

02560 99760 86560 40660 45056

1 2 3 6

73078 28757 22399 77112 28521

72110 74670 74430 60990 01650

8188 _ - _ _ 09200 ~

13919 75640 23199 59400 97200 86800 54100 04396 29840

19

17 153 969 4845

16280 19770 17190 61256 57400

37962 61070 96695 1 50845 2 32069

18

16 136 816 3876 15504

*

1 3 8 19 44

36 37 38 39 40

17

1

21 22 23 24 25

31 32 33 34 35

(z)

11735 37895 21755 63915 22675

31 84 214 518 1197

24550 36285 74180 95935 59850

15 46 131 345 864

62275 86825 23110 97290 93225

6 22 69 200 546

57800 20075 06900 30010 27300

3005 40195 6010 80390 11668 03110 22019 6i430 40599 28950

2651 5657 11668 23336 45375

82525 22720 03110 06220 67650

2062 4714 10371 22039 45375

53075 35600 58320 61430 67650

1411 3473 8188 18559 40599

20525 73600 09200 67520 28950

85974 1 59053 2 87811 5 10211 8 87323

96600 68710 43380 17810 78800

90751 1 76726 3 35780 6 23591 11 33802

35300 31900 00610 43990 61800

85974 76726 53452 89232 12824

96600 31900 63800 64410 08400

53 130 304 678 1454

67520

41 42 43 44 45

3 5 7 11 16

52401 28602 83789 49558 68713

52720 29080 60360 08528 34960

6 9 15 22 34

74896 27616 56696 17056 25584

10 16 26 41 64

30774 65097 51821 67148 66264

46706 21602 49218 05914 22970

15 25 42 68 110

15844 46619 11716 63537 30686

80450 27156 48758 97976 03890

20 35 60 102 171

21126 36971 83590 95306 58844

40600 21050 48206 96964 94940

46 47 48 49 50

23 34 48 67 93

98775 16437 23206 52488 70456

44005 74795 23240 72536 56300

51 17387 60544

99 150 225 334 492

14938 32326 48489 81089 36896

48554 09098 13647 92991 95575

174 274 424 649 984

96950 11888 44214 92703 73793

26860 75414 84512 98159 91150

281 456 730 1155 1805

89530 86481 98370 42584 35288

98830 25690 01104 85616 83775

nm \

20

20

1

21 22 23 24 25

34322 86724 15326 99116 48674

21

22

21 231 1771 10626 53130

1 22 253 2024 12650

23 276 2300

24 300

1 25

26 27 28 29 30

2 8 31 100 300

30230 88030 08105 15005 45015

65780 2 96010 11 84040 42 92145 143 07150

14950 80730 3 76740 1 5 60780 58 52925

2600 17550 98280 4 75020 20 35800

325 2925 20475 1 18755 5 93775

31 32 33 34 35

846 2257 5731 .. 13919 32479

72315 92840 66440 -. . .. 75640 43160

36 37 38 39 40

73078 1 59053 3 35780 6 89232 1 3 78465

72110 68710 00610 64410 28820

443 1290 3548 9279 23199

52165 24480 17320 83760 59400

1 2 6 13

55679 28757 87811 23591 12824

02560 74670 43380 43990 08400

23

1 3 6 13

24 46626 70200

415 697 1154 1885 3040

42466 31997 18478 16848 59433

71960 70790 96480 97584 83200 25

24

1

201 645 1935 5483 14763

1 1

26 351 3276 23751 1 42506

60075 12240 36720 54040 37800

78 280 925 2860 8344

88725 48800 61040 97760 51800

26 105 385 1311 4172

29575 18300 67100 28140 25900

7 33 138 524 1835

36281 65856 84156 51256 79396

37962 97200 93641 99760 2 22399 74430 5 io2ii i 7 8 i o 11 33802 61800

23107 61070 1 54712 3 77112 8 87323

89600 86800 86560 60990 78800

12516 35624 96695 2 51408 6 28521

77700 67300 54100 40660 01650

6008 18524 54149 1 50845 4 02253

05296 82996 50296 04396 45056

41 42 43 44 45

26 51 96 176 316

91289 37916 05669 10393 98708

37220 07420 18220 50070 30126

26 53 105 201 377

91289 82578 20494 26164 36557

37220 74440 81860 00080 50150

24 51 105 210 411

46626 37916 20494 40989 67153

70200 07420 81860 63720 63800

20 44 96 201 411

21126 67753 05669 26164 67153

40600 10800 18220 00080 63800

15 35 80 176 377

15844 36971 04724 10393 36557

80450 21050 31850 50070 50150

46 47 48 49 '50

560 976 1673 2827 4712

82330 24796 56794 75273 92122

07146 79106 49896 46376 43960

694 1255 2231 3904 6732

35265 17595 42392 99187 74460

80276 87422 66528 16424 62800

789 1483 2738 4969 8874

03711 38976 56572 48965 98152

13950 94226 81648 48176 64600

823 1612 3095 5834 10804

34307 38018 76995 33568 32533

27600 41550 35776 17424 66600

789 1612 3224 6320 12154

03711 38018 76036 53032 86600

13950 694 41550 1483 83100 3095 18876 6320 36300 12641

316 98708 30126 35265 38976 76995 53032 06064

80276 94226 35776 18876 37752

I

831

COMBINATORIAL ANALYSIS

Table 24.2

Multinomials and Partitions

. . ., nan, n=a1+2az+ . . . +m,, m=al+h+ . . . +an Ml= (n; nl, %, . . ,, n,) =n!/(1!)01(2!)a2 . . . (n!)”* M2= (n; al, az, . . ., U*)*=n!/lala1!2”2az! . . . nonan! Ma=(n; al, az, . . ., ~,)’=n!/(l!)~ia~!(2!)”2~~2! , . . (n!>”*a,! r=1a1,

n 1

m 1

2

1 2

7r

1

2 12

202,

Mi

Ma

Ma 1 1 2

1 1 1

1

n 8

m 1 2

1 1

3 3

4

1 2 3

3 192

1 2

4 1, 3 22 12, 2

3 4 5

1 2 3

4 5 6

1 2

3 4 5 6 7

1 2 3

4

1 3 6

2 3 1

1 3 1

1 4 6 12 24

6 8 3 6 1

1 4 3 6 1

5 L4 2, 3 12, 3 1,2a 18, 2 1’

1 5 10 20 30 60 120

24 30 20 20 15 10 1

1 5 10 10 15 10 1

6 1, 5 21 4 32 12, 4 192, 3 28 13, 3 l2,22 14, 2

1 6 15 20 30 60 90 120 180 360 720

120 144 90 40 90 120 15 40 45 15 1

1 6 15 10 15 60 15 20 45 15 1

1 7 21 35 42 105 140 210 210 420 630 840 1260 2520 5040

720 840 504 420 504 630 280 210 210 420 105 70 105 21 1

1 7 21 35 21 105 70 105 35 210 105 35 105 21 1

18

14

16

7 1, 6 295 3,4 18, 5 1, 294 1, 32 22, 3 18, 4 1 2 , 2, 3 1, 28 14,3 18, 21 16, 2 17

4

5 6

7 8 9

1 2

3

4

5

6 7 8 9

7r

8 1, 7 296 3, 5 42 l’, 6 1,2, 5 1,3,4 22, 4 2,3a 18, 5 12, 2, 4 12, 32 1 22, 3 25 14,4 18, 2, 3 1 2 , 28 l’, 3 1 4 , 22 16, 2 l8

9 1, 8 2, 7 3, 6 4, 5 12,7 1, 2 , 6 1,3, 5 1,42 22, 5 2,3,4 38 13, 6 1 2 , 2, 5 1’,3,4 1, 22, 4 1, 2, 32 28, 3 14,5 13, 2, 4 18, 32 12,2a,3 i,24 l’, 4 14,2 , 3 18, 28 16, 3 1 6 , 23 17,2 19

Mi

Ma

M3

1 8 28 56 70 56 168 280 420 560 336 840 1120 1680 2520 1680 3360 5040 6720 10080 20160 40320

5040 5760 3360 2688 1260 3360 4032 3360 1260 1120 1344 2520 1120 1680 105 420 1120 420 112 210 28 1

8 28 56 35 28 168 280 210 280 56 420 280 840 105 70 560 420 56 210 28 1

1 9 36 84 126 72 252 504 630 756 1260 1680 504 1512 2520 3780 5040 7560 3024 7560 10080 15120 22680 15120 30240 45360 60480 90720 181440 362880

40320 45360 25920 20160 18144 25920 30240 24192 11340 9072 15120 2240 10080 18144 15120 11340 10080 2520 3024 7560 3360 7560 945 756 2520 1260 168 378 36 1

1 9 36 84 126 36 252 504 315 378 1260 280 84 756 1260 1890 2520 1260 126 1260 840 3780 945 126 1260 1260 84 378 36 1

1

I

832

COMBINATORIAL ANALYSIS

Multinomials and Partitions

Table 24.2

n

m

U

10

1 2

10 1, 9 2, 8 3, 7 4, 6 52 12, 8 1, 2, 7 1, 3, 6 1,4, 5 22, 6 2, 3, 5 2, 42 32, 4 13,7 12;2, 6 12, 3, 5 1 2 , 42 1, 22, 5 1, 2 , 3 , 4 1, 33

3

4

'See page 11.

Mi 1 10 45 120 210 252 90 360 840 1260 1260 2520 3150 4200 720 2520 5040 6300 7560 12600 16800

M2

M3

1 362880 10 403200 45 226800 120 172800 210 151200 126 72576 45 226800 360 259200 840 201600 181440 1260 630 75600 *120960 2520 1575 56700 50400 2100 120 86400 151200 1260 120960 2520 1575 56700 3780 90720 151200 12600 22400 2800

n

m

10 5

6

7 8

9 10

n

Mi

M2

M3

23,4 18900 18900 3150 22, 32 25200 25200 6300 5040 25200 210 14,6 15120 60480 2520 13,2, 5 50400 4200 25200 13,3,4 12, 22, 4 *37800 *56700 9450 50400 50400 12600 12, 2, 32 75600 25200 12600 I , 23,3 113400 945 945 25 16, 5 30240 6048 252 75600 18900 3150 14,2,4 100800 8400 2100 14, 3 2 13, 22, 3 151200 25200 12600 226800 4725 4725 12~24 151200 16, 4 1260 210 302400 5040 2520 1 6 , 2, 3 453600 3150 3150 14,23 604800 240 120 17, 3 907230 630 630 1 6 , 22 1814400 45 45 18, 2 1 1 110 3628800

833

COMBINATORIAL ANALYSIS

STIRLING NUMBERS OF THE FIRST KIND Sim)

11 12 13 14 15

-120 720 -5 040 40320 -3 62880

274 -1764 13068 -1 09584 10 26576 -106 1205 - 14864 1 98027 -28 34656

8 71782 91200 -130 2092 - 35568 6 40237 -121 64510

16 17 18 19 20 21 22 23 24 25

24

1 -3 11 -5 0

-6

6 7 8 9 10

76743 27898 74280 37057 04088

68000 88000 96000 28000 32000

2432 90200 81766 40000

- 51090 94217 17094 40000

1 11 24000 72777 76076 80000 -41 965 -258 52016 73888 49766 40000 6204 48401 73323 94393 60000 -23427

91630 42823 55905 80585 68176

01600 93600 79200 21600 38400

94803 81078 77933 65249 39871

67616 01702 54547 30662 85664

00000 40000 20000 40000 00000

-8752 86244 48476 38966 87216

35 -225 1624 _._ - 13132 1 18124 -11 72700 127 -1509 19315 -2 65967 39 21567 -616 10299 -1 82160 34 01224 -668 60973

-1 0

1

6 7 8 9 10

85 -_ -735 6769 - 67284 7 23680

-1 5 175 -1960 22449 -2 69325

53576 17976 59552 17056 97824

58176 14720 22448 37120 24446 24640 95938 22720 03411 53280

5

4 5

6

1

-84 1052 - 14140 2 03137 -31 09892

11 12 13 14 15

34 -459 6572 - 99577 15 97216

09500 58076 14888 53096 60400

-270 4836 .- - - 90929 17 95071 -371 38478

505 69957 03824

16

-.-. . .--

17

-R707 7 7, A R R 75904 ._,-.

18 19 20

1 58331 39757 27488 -30 32125 40077 19424 610 11607 57404 91776

- I

21 22 2 23 -65 24 1573 25 -39365

1 -6

28640 43840 42880 59040 47360

433 -7073 1 22340 -22 37698 431 56514

4

n\m

3

2

1 1 -1 2

n\m 1 2 3 4 5

Table 24.3

12870 84093 48684 75898 61409

93124 31590 85270 28594 13866

51509 18114 30686 15107 31181

88800 68800 97600 32800 31200

-1 42 -1050 26775

8037 81664 80722 05310 03356

81182 97952 86535 75591 42796

68133 60092 - - - .99058 22809 73452 26450 06970 71471 74529 03823

From unpublished tables of Francis L. Miksa, with permission.

1 -2 1 322 -4536 63273 -9 133 -2060 33361 -5 66633

16930 95730 06836 03756 05680 45600

-33424 - .- .

44112 21504 28000 51776 76096 42912 -20 84576 507 62624 -13237

100 -1886 3690i -7 55152 161 42973 -3599 83637 21687 79532 14091

97951 38169 37691 53430 57918

02055 39535 70150 18786 66760

96721 15670 26492 75920 65301

07080 58880 34384 63024 18960

79476 95448 06827 28501 58577

07200 02976 41568 98976 60000

834

COMBINATORIAL ANALYSIS Table 2.1.3

STIRLINC NUMBERS OF THE FIRST KIND S',") 1

n

1 -2 8 546 -9450

-3 6 870

c

n\m 7 8 9 10

11 12 13 14 15

1 -26 449 -7909 1 44093

-27 537 - 11022 2 35312 -52 26090

28032 45234 84661 50405 33625

10680 77960 84200 49984 12720

21 22 23 24 25

1206 64780 37803 58339 73354 7 20308 21644 09246 -185 88776 35505 19497 4969 10165 05554 96448

73360 47760 53696 76576 36800

- 28939

n\ rn

I

11 12 13 14

- 55770

5 -114 2487 - 55792 12 95363

29553 83528 97936 47048 43896

-311 33364 31613 7744 65431 01695 -1 99321 97822 i k i 53 04713 71552 54458 -1459 01905 52766 26492

90640 76800 37360 12976 88000

14 74473 -373 12275

15

-10 276 -7707 2 20984 -65 08376

21 22 23 24 25

-2 57 -1471 38192

9280 30571 79248 07534 20555

95740 59840 94833 08923 02195

14229 01910 40110 45497 17966

98655 92750 12973 94337 81468

11450 35346 61068 ii3G 50000

n\ rn 13 14 15

-2 83 -2996 1 02469 -34 22525

16 17 18 19 20

n\m

21 _22_ 23 13 24 -640 25 29088

17 1 -153 13566 -9 20550 533 27921 67171 05903 66798

7 49463 -166 69653 3684 11615

- 82076

18 -430 10241 -2 50385 63 -1634 43714 -12 04749 342 18695

03081 98069 22964 26016 95940

59531 81053 77407 87554

28000 77553 01929 32658 67550

20992 72465 95944 17376 71489

94896 83456 12832 32496 92880

11

12

1 -6 6 2717 - 91091 27 49747

1 -7 8 3731 -1 43325

-785 21850 -6 02026 166 15733 -4628 06477

58480 31420 93980 86473 51910

30753 60053 23088 71136 52782

40395 59745 49736 04996 37300

50105 50868 11859 85742 52146

48 -1569 48532 -14 75607 446 52267

99622 52432 22764 03732 5i38i

- 13558 51828 99530 4 -129 4070 -1 30770

15482 00665 38405 92873

38514 98183 70075 67558

30525 31295 69521 73500

16

15

14

13 1 -91 5005

-105

1

18400 94022 50806 37272 11900

6580 23680 96582 89282 33630

-120 8500 -4 68180 223 23822 -9739 41900

1131 02769 95381 - 37310 09998 02531 12 36304 58470 86207 -413 35671 43013 14056 13990 94520 02391 06865

17 18 19 20

1 -37 1103 - 33081 10 14945

1320

- 32670

57423 26634 36473 53775

46311 69012 18452 16815 69899

-5 5 1925

16 17 18 19 20

1 -45

- 18150

3 -69 1350 - 26814

10

10

21 22 23 24 25

1

57773 37558 90231 43153 22928

16 17 18 19 20

9

1

-3 138 -5497 2 06929 -75 2718 97125 34 70180 -1246 20006

-

61111 86118 04609 64487 90702

84500 69881 39913 04206 15000

4 -159 6238 -2 40604 92 44691

in

19

20

1 -171 16815

1 -190

1

27946 -12 56850 20615 67686 797 21796 -16 89765 45460 47198 1168 96626 -22 57942 1684 24 12764 43496 - 72346 69596 36096 67135 -1219 12249 80000 41 49085 13800 -1 12768

01717 97183 24164 60386 13761

-210 25025 40315 23871 -29 42500 2388

1 -136

ioaii

-6 62796 349 16946

- 16722

52896 60911 25118 63218

80820 68850 03430 00831 64500

23

24

25

1 1 -231 -253 1 30107 1 35926 -276 32776 10495 -37 95000 42550 3 0 0

1

71630 88730 21941 44556 73550

21

7 -325 13727 -5 70058

22

835

COMBINATORIAL ANALYSIS

ll\ttll

1 2 3 4 5

1 1 1 1 1

STII{LIM; \ L ' ~ ~ B E HOSF T I I E SECOND K I N D

ap)

2

s

:I

1

3 7 15

1 6 25

1 10

I

6 7 8 9 10

1 1 1 1 1

31 63 127 255 511

90 301 966 3025 9330

11 12 13 14 15

1 1 1 1 1

1023 2047 4095 8191 16383

28501 86526 2 61625 7 88970 23 75101

16 17 18 19 20

1 1 1 1 1

32767 65535 1 31071 2 62143 5 24287

21 22 23 24 25

1 1 1 1 1

11'

10 20 41 83 167

71 214 644 1934 5806

48575 97151 94303 88607 77215

1 6 25 103 423

45750 11501 32530 91745 55950

1717 6943 27988 1 12596 4 52321

98901 37290 06985 66950 15901

18 72 291 1168 4677

15090 77786 63425 10566 12897

m 1 28 462 5880

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

41686 57825 39010 48101 06446

17423 43625 52280 79450 1 56863 35501 4 70632 00806 14 11979 91025

7 8 9 10

2 19 149 1114

6 57 493 4087

63987 27396 15424 29280 41333

32818 57081 74624 29246 35540

82604 04786 83400 34839 45652

1 18 209 2166

8231 09572 14948 60276 23799 67440 4 38264 19991 17305 31 67746 38518 04540 227 83248 29987 16310

1 9 82 690

1

65 350 1701 7770 34105

15 140 1050 6951 42525 2 13 75 400 2107

70050 23825 74750 34501 38810

46730 79400 08501 75035 66920

1 13 93 634 4206

79487 23652 21312 36373 93273

10961 90550 56527 51651 2 89580 95545 1 4 75892 84710 74 92060 90500

27349 1 75057 11 06872 69 30816 430 60788

26558 49898 51039 01779 95384

379 1913 9641 48500 2 43668

12625 78219 68881 07834 49741

9

1 36 750

1 45

16 17 18 19 20

289 5120 83910 12 94132 190 08424

21 22 23 24 25

2682 68516 89001 36628 25008 70286 4 86425 13089 51100 63 10016 56957 75560 802 35590 44384 62660

)I

ttt

1

16 17 18 19 20

120 7820 3 67200 139 16778 4523 29200

21 22 23 24 25

tl

1 30874 34 56159 847 94044 19582 02422 4 29939 46553

62580 43200 29331 47080 47200

3251 0239 1195 1828 2372

10153 90799 50199 21583 11183

47084 91620 00400 20505 68583

1 12 120 1167

12327 24196 32006 62257 92145.

24764 33035 88117 43260 10929

20

llt

20

1

21 22 23 24 25

210 23485 1 8 59550 1169 72779 62201 94750

30420 33560 27734 45907 07848

1 136 9996 5 27136 223 50954 44464 74004 27264 65555 95960

65204 33920 96900 9 72500 108 73005 1203

7118 83514 59340 25408 16339

71322 37993 12973 17849 21753

1 105 2 84 2435 63025

6020 49900 08778 77530 24580

14 93040 51652 91758 10216 03608

04500 81331 07115 41000 96000

28924 1258 39160 54638 6 10686 60380 120 2249 40128 6 88883 114 48507

49092 68618 25603 60579 33437

18331 68481 41390 22000 44260

I? 1 153 12597 7 41285 349 52799 14041 42047 4 99169 88803 161 09499 36915 4806 33313 93110

21

22

1 231 28336 24 54606 1685 19505

1 253 33902 32 00450

10 533 23648 9 24849 327 56785

23

1 276 40250 . ..

91275 77954 13460 31500 87500

II

1 65620 49 10178

From unpublished tables of Francis L. Miksa, with permission. 'See page U.

47 72970 33785 591 75849 64655

91 4550

57118 22324 28866 51300 33391 30178 51137 83930 58260 74680

1937 54990 27583 34150

13

16

8099 60465 23611 71824 51616

1

3 71121 63803

27 620 12563 2 34669 41 10166

2 76 2067 52665

10

8207 84250 95288 22303 1 0 61753 95755 114 46146 26805 1201 12826 44725

64053 95028 65010 03480 62679

1 .i

15

62804 45225 83405 84530 02430

55 1705 39325 7 52752 126 62650

21417 04159 90360 97510 09326

683 10882 1 67216 24 93020 362 26262

56794 53393 98579 60360 70000

1155 22275 3 59502 51 35130 671 28490

1 78 3367 1 06470

36908 60978 04908 17791 29486

2658 16330 99896 6 09023 37 02641

11880 59027 99612 12320 27840

2 18 170 1517

1 66 2431 66066 14 79478

68401 12055 84100 95250 10751

8

12 11 12 13 14 15

1 21 266 2646 22827

329

6862 1 36209 25 95811

18

19

1 171 15675

1 190

23435 74629 85369 25445 94925

13 797 38807 16 62189

24

1 300

19285 89850 81779 39170 69675

2.5

1

*

COMBINATORIAL ANALYSIS A ( VRER O F I'AK'L'l'flO\S A V I ) l'AR'fl'1'lOAS 1\'1'0 1)IS'llhC'l PARTS !I

d,O voo

0

II

II

d/O

/ ~ O l )

dl1)

2 2 2 3 3

04226 39943 81589 29931 86155

3658 4097 4582 5120 5718

100 101 102 103 104

1905 2144 2412 2712 3048

69292 81126 65379 48950 01365

4 4 5 5 6

44793 83330 25016 70078 18784

150 151 152 153 154

4 4 4 5 6

08532 50606 96862 47703 03566

35313 24582 88421 36324 73280

194 207 222 238 255

06016 92120 72512 53318 40982

1 2 3 5

1 2 2

50 51 52 53 54

9

7 11 15 22 30

3 4 5 6 8

55 56 57 58 59

4 5 6 7 8

51276 26823 14154 15220 31820

6378 7108 7917 8808 9792

105 106 107 108 109

3423 3842 4311 4835 5419

25709 76336 49389 02844 46240

6 7 7 8 9

71418 28260 89640 55906 27406

155 156 157 158 159

6 7 8 8 9

64931 32322 06309 87517 76627

82097 43759 64769 78802 28555

273 292 ' 313 335 358

42421 64960 16314 04746 39008

10 11 12 13 14

42 56 77 101 135

10 12 15 18 22

60 61 62 63 64

9 11 13 15 17

66467 21505 00156 05499 41630

10880 12076 13394 14848 16444

110 111 112 113 114

6071 6799 7610 8513 9520

63746 03203 02156 76628 50665

10 10 11 12 13

04544 87744 77438 74118 78304

160 161 162 163 164

10 11 i2 14 15

74381 81590 99139 27989 69194

59466 68427 04637 95930 75295

383 409 438 468 500

28320 82540 12110 28032 42056

15 16 17 18 19

176 231 297 385 4YO

27 32 38 46 54

65 66 67 68 69

20 23 26 30 35

12558 23520 79689 87735 54345

18200 20132 22250 24576 27130

115 116 117 118 119

10641 11889 13277 14820 16536

44451 08248 10076 74143 68665

14 16 17 18 20

90528 11388 41521 81578 32290

165 166 167 168 169

17 18 20 22 25

23898 93348 78904 82047 04389

00255 22579 20102 32751 25115

534 571 610 651 695

66624 14844 00704 39008 45358

20 21 22 23 24

627 792 1002 1255 1575

64 76 89 104 122

70 71 72 73 74

40 46 53 61 70

87968 97205 92783 85689 89500

29927 32992 36352 40026 44046

120 121 122 123 124

18443 20561 22913 25523 28419

49560 48051 20912 38241 40500

21 23 25 27 29

94432 68800 56284 57826 74400

170 171 172 173 174

27 30 33 36 39

47686 13848 04954 23268 71250

17130 02048 99613 59895 74750

742 792 845 901 962

36384 29676 43782 98446 14550

25 26 27 28 29

1958 2436 3010 3718 4565

142 165 192 222 256

75 76 77 78 79

81 92 106 121 138

18264 89091 19863 32164 48650

48446 53250 58499 64234 70488

125 126 127 128 129

31631 35192 39138 43510 48352

27352 22692 64295 78600 71870

32 34 37 40 43

07086 57027 25410 13544 22816

175 i76 177 178 i79

43 47 52 57 62

51576 67158 21158 17016 58467

97830 1 0 2 6 57290 1 0 9 4 31195 1 1 6 6 05655 1 2 4 3 53120 1 3 2 5

14114 20549 58616 54422 35702

30 31 32 33 34

5604 6842 8349 10143 12310

296 340 390 448 512

80 81 82 83 84

157 180 205 233 265

96476 04327 06255 38469 43660

77312 84756 92864 101698 111322

130 131 132 133 134

53713 59645 66208 73466 81490

15400 39504 30889 29512 40695

46 50 53 58 62

54670 10688 92550 02008 40974

180 181 182 183 184

68 74 81 89 98

49573 94744 98769 66848 04628

90936 1412 11781 1 5 0 4 08323 1 6 0 2 17527 1 7 0 7 80430 1818

31780 73568 93888 27424 10744

35 36 37 38 39

14883 17977 21637 26015 31185

585 668 760 864 982

85 86 87 88 89

301 342 388 441 499

67357 62962 87673 08109 95925

121792 133184 145578 159046 173682

90358 36076 135 136 1 00155 81680 137 1 10976 45016 1 3 8 1 22923 41831 139 1 36109 49895

67 72 77 83 89

11480 15644 55776 34326 53856

185 186 187 188 189

107 117 128 139 152

18237 14326 00110 83417 72735

74337 92373 42268 45571 99625

82642 84096 58315 51098 10816

40 41 42 43 44

37338 44583 53174 63261 75175

1113 1260 1426 1610 1816

90 91 92 93 94

566 641 725 820 926

34173 12359 33807 10177 69720

189586 206848 225585 245920 267968

1 4 0 1 50658 141 1 66706 142 1 84402 143 2 03909 144 2 25406

78135 89208 93320 82757 54445

96 103 110 118 127

17150 27156 86968 99934 69602

190 191 192 193 194

166 182 198 216 236

77274 07011 72768 86271 60227

04093 2642 88462 00652 2811 38048 56363 2990 i b 6 0 8 05469 3179 04256 41845 3381 04630

45 46 47 18 49

89134 105558 124754 147273 173525

2048 2304 2590 2910 3264

95 96 97 98 99

1046 1181 1332 i50i 1692

51419 14304 30930 98136 29875

291874 317788 345856 376256 409174

145 2 49088 58009 146 2 75170 52599 147 3 03886 71978 1 4 8 3 35494 1 9 4 9 7 149 3 70273 55200

136 146 157 168 181

99699 94244 57502 93952 08418

195 196 i97 198 199

258 281 306 334 364

08402 45709 88298 53659 60724

12973 87591 78530 83698 32125

50

204226

3658

100

1905 69292

444793

150

194 06016

200

397 29990 29388

1 2 3 4 5 6 7 8

1 1

4 08532 35313

1935 2060 2193 2334 2484

3594 3820 4060 4315 4584

44904 75868 72422 13602 82688

4870 67746

values of /)Or) from H. Gupta, A table of partitions, Proc. London Math. Soc. 39, 142-149, 1935 and 11. 12,546-549, 1937 (with permission),

837

COMBINATORIAL ANALYSIS

Table 24.5

NUMBER OF PARTITIONS .4ND PARTITIONS INTO DISTINCT PARTS

s(n)

n

PO1 1

n

4(n)

200 201 202 203 204

397 432 471 513 559

29990 83636 45668 42052 00883

29388 58647 86083 87973 17495

4870 5173 5494 5834 6195

67746 61670 62336 73184 03296

250 251 252 253 254

23079 24929 26923 29072 31389

35543 14511 27012 69579 19913

64681 68559 52579 16112 06665

85192 89949 94961 1 00243 1 05807

80128 26602 58208 00890 47264

205 206 207 208 209

608 662 720 784 852

52538 29877 68417 06562 85813

59260 08040 06490 26137 02375

6576 6980 7408 7862 8341

67584 87424 90786 12446 94700

255 256 257 258 259

33885 36574 39472 42593 45954

42642 95668 36766 30844 57504

48680 70782 55357 09356 48675

1 1 1 1 1

11669 17844 24348 31199 38413

59338 71548 95064 20928 23582

21 0 211 212 213 214

927 1008 1096 1191 1295

51025 50658 37072 66812 00959

75355 85767 05259 36278 25895

8849 9387 9956 10558 11195

87529 48852 45336 52590 55488

260 261 262 263 264

49574 53471 57667 62183 67044

19347 50629 26749 74165 81230

60846 08609 47168 09615 60170

1 1 1 1 1

46009 54008 62428 71293 80624

65705 01856 82560 59744 90974

215 216 217 218 219

1407 1528 1660 1802 1957

05456 51512 15981 81825 38561

99287 48481 07914 16671 61145

11869 12582 13336 14133 14977

49056 38720 40710 83026 05768

265 266 267 268 269

72276 77905 83961 90476 97483

09536 06295 17303 01083 43699

90372 62167 66814 16360 44625

1 2 2 2 2

90446 00783 11660 23106 35150

44146 03620 75136 91192 17984

220 221 222 223 224

2124 2306 2502 2715 2945

82790 18711 58737 24089 45499

09367 73849 60111 25615 41750

15868 16811 17807 18860 19973

61606 16852 51883 61684 57056

270 271 272 273 274

1 1 1 1 1

05019 13123 21837 31205 41274

74899 85039 43498 18008 95651

31117 38606 44333 16215 73450

2 2 2 2 3

47820 61149 75170 89917 05427

61070 71540 53882 72486 58738

225 226 227 228 229

3194 3464 3756 4071 4413

63906 31263 11335 80636 29348

96157 22519 82570 27362 84255

21149 22392 23705 25091 26556

65120 29960 13986 98528 84608

275 276 277 278 279

1 1 1 1 2

52098 63729 76227 89656 04082

04928 39693 84330 41035 58525

51175 37171 57269 91584 75075

3 3 3 3 3

21738 38889 56923 75883 95815

19904 46600 20960 26642 57440

230 231 232 233 234

4782 5182 5613 6080 6585

62397 00518 81486 61354 15859

45920 38712 70947 38329 70275

28103 29737 31462 33284 35207

94454 72212 84870 23936 06304

280 281 282 283 284

2 2 2 2 2

19578 36221 54095 73287 93892

63116 91453 25900 31835 97939

82516 37711 45698 47535 29555.

4 4 4 4 5

16768 38791 61938 86265 11828

26624 78240 97032 19094 44672

235 236 237 238 239

7130 7719 8356 9043 9786

41855 58926 11039 68396 29337

14919 63512 25871 68817 03585

37236 39379 41639 44025 46543

75326 02688 89458 67324 00706

285 286 287 288 289

3 3 3 3 4

16013 39758 65243 92592 21938

78671 40119 08360 21614 85285

48997 86773 71053 89422 87095

5 5 5 6 6

38689 66911 96562 27710 60430

49522 97084 52987 98024 42088

240 241 242 243 244

10588 11454 12388 13397 14486

22467 08845 84430 82593 76924

22733 53038 77259 44888 96445

49198 52000 54955 58073 61360

87992 62976 97248 01632 27874

290 291 292 293 294

4 4 5 5 6

53425 87203 23437 62299 03976

31269 80564 10697 26919 38820

00886 72084 53672 50605 95515

6 7 7 8 8

94797 30892 68798 08604 50401

40554 09120 39744 19136 45750

245 246 247 248 249

15661 16929 18297 19772 21363

84125 67223 38898 65166 69198

27946 91554 54026 81672 20625

64826 68481 72335 76397 80679

71322 72604 19619 50522 55712

295 296 297 298 299

6 6 7 8 8

48667 96585 47956 03024 62049

41270 01441 50785 83849 62754

79088 95831 10584 43040 65025

8 9 9 10 10

94286 40360 88727 39499 92791

47940 04868 65938 71456 76298

250

23079 35543 64681

85192 80128

300

9 25308 29367 23602

11 48724 72064

838

COMBINATORIAL ANALYSIS

Table 24.5

NUMBER OF PARTITIONS AND PARTITIONS INTO DISTINCT PARTS

?I

3 00 301 302 3 03 304

P(l1)

9 25308 9 93097 10 65733 11 43554 12 26921

29367 23924 12325 20778 80192

23602 03501 48839 22104 29465

d.1

9(.) 11 48724 72064 12 07425 10607 12 69025 30816 1 3 33663 83848 14 01485 59930

n 350 3 51 352 353 354

279 298 318 340 363

36332 33006 55597 12281 11751

84837 30627 37883 00485 20481

02152 58076 29084 77428 10005

126 132 139 145 152

91829 93477 22769 80938 69267

24648 19190 71520 18816 15868

72642 47292 25601 07743 93899

18618 17536 42890 43642 64242

355 356 357 358 359

387 413 441 471 502

63253 76618 62298 31406 95756

29190 09333 19293 42683 65060

29223 42362 58437 98780 00020

159 167 175 183 192

89096 41824 28907 51867 12289

56578 09148 55072 38752 32216

84259 79022 78394 82593 91846

79304 32212 72390 94656 82870

360 361 362 363 3 64

536 572 610 651 695

67907 61205 89840 68887 14371

03106 88980 37518 99972 34589

91121 37559 84101 06959 46040

201 210 __ 220 230 241

11827 52205 35221 62751 36750

04478 02772 50410 50210 01278

29 22603 40224

365 366 367 368 369

958 72869 79123 38045

252 264 276 289 302

59255 32392 58376 39517 78222

33946 51488 86784 78822 57408

77000 38466 65347 60483 26834

44480 77248 41118 21048 76992

305 306 307 308 309

17 41418 01331 47295

14 15 16 17 17

310 311 312 313 314

18 20 21 23 24

67148 01742 45809 00000 65010

82996 67625 60373 66554 61508

00364 76945 52891 87337 30490

18 19 20 21 22

315 316 317 318 319

26 28 30 32 34

41580 30502 32618 48829 80095

76335 03409 19898 33514 48694

66326 96003 42964 66654 40830

320 321 322 323 324

37 39 42 45 49

27440 91956 74807 77235 00564

57767 55269 80359 85435 36352

48077 99991 54696 78028 37875

325 326 327 328 329

52 56 60 64 68

46204 15660 10534 32537 83488

42288 21128 98396 46091 59460

28641 74289 66544 14550 73850

330 331 332 333 334

73 78 84 90 96

65328 80125 30081 17543 45011

78618 53026 56362 49805 01922

50339 66615 25119 49623 02760

49 51 54 57 59

335 336 337 338 339

103 110 117 126 134

15146 30786 94949 10851 81918

63217 04252 15461 78337 06233

340 341 342 343 344

144 154 164 175 188

11793 04359 63747 94355 00864

345 346 347 348 349

200 214 229 244 261

88255 61829 2i228 90453 57890

350

d?l)

370 371 372 373 374

1022 1089 1161 1238 1319

14122 65764 53783 05779 51059

83673 44243 48499 41191 97274

45362 99782 62850 25085 73500

316 331 346 362 379

375 376 377 378 379

1406 1498 1596 1701 1812

20744 47874 67527 16942 35649

65614 35905 44907 79758 97394

84054 81081 56791 13525 72950

796 61401 30794

28759 15040 85792 13990 87918

380 381 382 383 3 84

1930 2056 2190 2332 2484

65607 51347 40133 82119 30529

23504 53366 24237 a5438 42654

65812 33805 65131 92336 18180

496 518 542 566 592

35325 92772 13972 96355 01520

62 87095 13216 65 91563 14788

69 i03i2 43iO

72 43991 92576 75 93279 10200

385 386 387 388 389

2645 41834 Ziib 75950 2998 96444 3192 70751 3398 70404

65278 73795 91657 98104 70522

73832 76030 61044 22753 92980

79 83 87 91 95

23110 64940 06890 94270 92704

390 391 392 393 394

3617 3850 4098 4361 4640

62876 97432 687iZ 74553 73511

83159 86299 iii50 82406 44125

100 58620 35461 105 30799 17632 iio 4 i i 9 2 ko9ii 115 66794 79970 121 16645 56454

395 396 397 398 399

279 36332 84837 02152

126 91829 24648

400

30 32 33 35 37

67552 19458 78644 45449 20225

58118 99993 53293 18543 96286

58881 41536 42016 61123 99699

32574 41664 88192 47722 12608

71276 53843 03453 10617 71312

06887 63701

32179 42792

02629 61523 20259 83119 54565

40968 80864 26436 27092 72864

619 39246 14094

77364 52194 a4335 32826 13581 60275

38676 46674 56265 07622 46996

04423 29186 94791 84114 23515

772 807 843 881 920

37784 10844 35537 18291 65799

71936 79444 42947 29614 74150

961 85031 43424

1604 Giii 32444

1049 67982 04736 1096 47115 85280 1145 28826 89344 6727 09005 17410 41926

1196 21634 00706

'

839

COMBINATORIAL ANALYSIS

NUMBER OF PARTITIONS AND PARTITIONS INTO DISTINCT PARTS n

PO!)

/I

Table 24.5 901)

400 401 402 403 404

6727 7154 7608 8091 8603

09005 64022 80284 20027 55175

17410 41926 26539 42321 33398 79269 64844 65581 93486 55060 1422 86674 81438

450 451 452 453 454

405 406 407 408 409

9147 9725 10339 10990 11682

67906 51251 09726 60006 31627

88591 37420 71239 37759 71923

17602 21729 47241 26994 17780

1485 1551 1619 1691 1765

75420 34186 74236 07292 45549

52794 29884 54282 29128 15430

455 456 457 458 459

1 1 2 2 2

79855 90581 01933 13948 26665

91645 04044 37928 90703 62143

39582 26519 51146 27330 58313

67598 31034 88629 69132 45565

410 411 412 413 414

12416 13196 14023 14902

67740 25896 78888 15629

31511 69254 35188 03099

90382 35702 47344 48968

1843 1923 2008 2096 2187

01696 88934 20999 12178 77334

07104 65516 30208 16576 80960

460 461 462 463 464

2 2 2 2 3

40123 54365 69435 85381 02253

65561 39575 60521 55524 16287

39251 85741 29549 19619 25766

92081 14888 91233 20640 99975 15506 48874 75476 94471 86287 36605

2283 2382 2486 2594 2707

31930 92048 74417 96435 76199

70488 69148 20078 42056 52640

465 466 467 468 469

415 416 417 418 419

9893 10307 10739 11188 11656

14440 93957 65687 96810 57102

61528 13070 10144 43072 54336

12143 12649 13176 13724 14295

19032 57862 51755 81881 32530

12544 22432 08648 00782 93376

420 421 422 423 424

22755 24167 25664 27253 28938

29021 05302 64021 16454 03725

65800 14413 38377 62304 70847

25259 63961 14846 21739 98150

2825 2947 3075 3208 3347

32529 84998 53960 60580 26867

77152 62528 09352 00384 45954

470 471 472 473 474

425 426 427 428 429

30724 32620 34629 36760 39020

98514 06861 70071 66724 14800

70950 74102 39035 18315 02372

51099 32189 75934 27309 59665

3491 3642 3i99 3962 4132

75707 60097 30895 45254

iiiii 07136

60256 14146 86891 79000

475 476 477 478 479

5 5 6 6 7

58055 23018 37870 67318 66282

99675 52926 28583 07232 38005

27271 28385 29543 30747 31998

99448 57585 43443 28468 90573

23232 65430 69603 94368 73738

430 431 432 433 434

41415 43955 46647 49501 52527

73920 47717 86328 89040 07072

71023 05181 42292 94051 91082

58378 16534 67991 50715 40605

4310 4495 4687 4888 5096

24877 03113 51640 01685 85706

85006 72460 62334 40672 20480

480 481 482 483 484

7 51666 00419 49931 7 95317 79841 47582 8 41457 02874 28236 a 90222 78495 19280 9 41761 78911 49976

25591 32180 49455 88294 98055

33300 34652 36059 37521 39040

14373 91433 20520 07873 67468

57056 03468 80640 43946 62530

435 436 437 438 439

55733 59131 62733 66549 70593

46514 71430 07137 43656 39364

46362 91696 60430 69662 65621

86656 18645 79215 97367 35510

5314 5540 5776 6022 6278

37439 91949 85678 56498 43769

57460 44512 02880 45546 39520

485 486 487 488 489

9 10 11 11 12

96228 53787 14608 78875 46778

80660 07886 77893 49115 71600

85734 24553 64264 57358 12729

11012 46513 84248 02646 19665

40620 42261 43968 45741 47585

21308 99712 41621 94910 16717

45496 45764 12802 51264 64998

440 441 442 443 444

74878 79418 84227 89322 94720

24841 06934 73040 95632 37025

94708 86233 64434 02240 77294 99781 13536 45667 78934 71820

6544 6822 7111 7411 7725

88391 32867 21361 99762 15750

85792 92200 67457 56080 89318

490 491 492 493 494

13 13 14 15 16

18520 94313 74382 58964 48308

40161 50322 57204 37499 54706

22702 44478 03639 49778 61724

33223 16939 53132 06173 38760

49500 51491 53560 55709 57943

73777 42772 16694 75216 45082

62304 84172 36938 10170 47040

00437 06493 12906 19698 26891

54417 05190 52519 71278 54269

17528 52391 91961 27202 09814

8051 8390 8743 9111 9494

18865 60575 94352 75744 62459

81728 94564 40798 62854 05984

495 496 497 498 499

17 18 19 20 21

42678 42351 47619 58791 76192

27774 03350 31798 47204 51543

77609 31598 76580 28849 92874

81187 91466 64007 01563 61625

445 446 447 448 449

1 1 1 1 1

47604 18581 03354 05954 18000

450 1 34508 18800 15729 23840 9893 14440 61528

24180 69iii 9SSG

25171 11509 01902 26201 03821 12696 66337 99397 34350 71304 10369

12186 20478 76365 20389 79823

500 23 00165 03257 43239 95027

840

COMBINATORIAL ANALYSIS

ARITHMETIC FUNCTIONS

Table 24.6

4 . 1

00

UI

1 2 3 4 5

1 1 2 2 4

1 2 2 3 2

1 3 4 7 6

51 52 53 54 55

32 24 52 18 40

6 7 8 9

2 6 4 6 4

4 2 4 3 4

12 8 15 13 18

56 57 58 59 60

24 8 120 36 4 80 28 4 90 58 2 60 16 12 168

106 52 4 162 107 106 2 108 108 36 12 280 109 108 2 110 110 40 a 216

11 10 12 4 13 12 14 6 15 8

2 6 2 4 4

12 28 14 24 24

61 62 63 64 65

60 30 36 32 48

2 62 4 96 6 104 7 127 4 84

111 72 112 48 113 112 114 36 115 88

16 8 17 16 18 6 19 18 20 8

5 2 6 2 6

31 18 39 20 42

66 67 68 69 70

20 66 32 44 24

8 144 2 68 6 126 4 96 8 144

116 117 118 119 120

21 22 23 24 25

12 10 22 8 20

4 4 2 8 3

32 36 24 60 31

71 72 73 74 75

70 2 72 24 12 195 72 2 74 36 4 114 40 6 124

121 110 122 60 123 80 124 60 125 100

26 27 28 29 30

12 18 12 28 8

4 4 6 2 8

42 40 56 30 72

76 77 78 79 80

36 6 140 60 4 96 24 8 168 78 2 80 32 10 186

31 32 33 34 35

30 16 20 16 24

2 6 4 4 4

32 63 48 54 48

81 82 83 84 85

36 37 38 39 40

12 36 18 24 16

9 2 4 4 8

91 38 60 56 90

41 42 43 44 45

40 12 42 20 24

2 8 2 6 6

46 47 48 49 50

10

4 72 6 98 2 54 8 120 4 72

101 100 102 32 103 102 104 48 105 48

2 8 2 8 8

102 216 104 210 192

151 150 152 72 153 96 154 60 155 120

2 8 6 8 4

152 300 234 288 192

156 48 12 392 157 156 2 158 158 78 4 240 159 104 4 216 160 64 2 378

201 202 203 204 205

132 4 272 100 4 306 168 4 240 64 12 504 160 4 252

206 102 4 312 207 132 6 312 208 96 10 434 209 180 4 240 210 48 16 576 2 6 4 4 4

212 378 288 324 264

166 82 4 252 167 166 2 168 168 4a 16 480 169 156 3 183 170 64 8 324

216 72 16 217 180 4 218 108 4 219 144 4 220 80 12

600 256 330 296 504

171 108 172 84 173 172 174 56 175 120

4 0 2 8 4

152 248 114 240 144

161 132 162 54 163 162 164 80 165 80

56 6 72 6 58 4 96 4 32 16

210 182 180 144 360 133 186 168 224 156

4 0 2 6 8

192 363 164 294 288

~

211 212 213 214 215

210 104 140 106 168

6 6 2 8 6

260 308 174 360 248

221 192 4 252 222 72 8 456 223 222 2 224 224 96 12 504 225 120 9 403

126 36 12 312 127 126 2 128 128 64 8 255 129 84 4 176 130 48 8 252

176 80 10 177 116 4 178 88 4 179 178 2 180 48 18

372 240 270 180 546

226 112 4 342 227 226 2 228 228 72 12 560 229 228 2 230 230 88 8 432

54 5 121 40 4 126 82 2 84 24 12 224 64 4 108

131 130 2 132 132 40 12 336 133 108 4 160 134 66 4 204 135 72 8 240

181 180 182 72 183 120 184 88 185 144

2 8 4 8 4

182 336 248 360 228

231 232 233 234 235

86 87 88 89 90

42 4 56 4 40 8 88 2 24 12

132 120 180 90 234

136 64 8 137 136 2 138 44 8 139 138 2 140 48 12

186 60 187 160 188 92 I85 ioi 190 72

8 384 4 216 6 336

42 96 44 84 7a

91 92 93 94 95

72 44 60 46 72

112 168 128 144 120

141 92 4 192 142 70 4 216 143 120 4 168 144 48 15 403 145 112 4 180

191 190 2 192 192 64 14 508 193 192 2 194 194 96 4 294 195 96 8 336

22 4 72 48 46 7 16 16 124 42 3 57 20 6 93

96 97 98 99 100

146 72 4 147 84 6 148 72 6 149 148 2 150 40 12

196 84 9 399 197 196 2 198 198 -60 12 468 199 198 2 200 200 80 12 465

4 6 4 4 4

32 12 252 96 2 98 42 6 171 60 6 156 40 9 217

3 4 4 6 4

270 138 288 140 336

222 228 266 150 372

8 lio

8 360

120 8 384 112 8 450 232 2 234 72 12 546 184 4 288

236 116 6 237 156 4 238 96 8 239 238 2 240 64 20

420 320 432 240 744

241 242 243 244 245

240 110 162 120 168

2 6 6 6 6

242 399 364 434 342

246 247 248 249 250

80 216 120 164 100

8 4 8 4 8

504 280 480 336 468

From British Association for the Advancement of Science, Mathematical Tables, vol. VIII, Number-divisor tables. Cambridge Univ. Press, Cambridge, England, 1940 (with permission).

,

I

841

COMBINATORIAL ANALYSIS

ARITHMETIC FUNCTIONS

251 252 253 254 255

250 2 252 72 ia 728 220 4 2aa 126 4 384 128 8 432

Table 24.6

)(. 400 224 300 226 2aa

n 301 302 303 304 305

r(n)

n

r(n)

no

UI

252 4 352 150 4 456 200 4 408 144 10 620 240 4 372

351 352 353 354 355

8 Go 12 352 2 116 a 280 4

560 756 354 720 432

401 402 403 404 405

400 132 360 200 216

2

402 a16 448 714 726

n 451 452 453 454 455

8 720 4 456 6 1080 2 410 a 756

456 457 458 459 460

144 16 1200 456 2 458 228 4 690 2aa 8 720 176 12 iooa

no

01

216

a

4 6 0

256 128 9 257 256 2 258 a4 a 259 216 4 260 96 12

511 258 528 304 588

306 307 308 309 310

96 12 702 306 2 308 120 12 672 204 4 416 120 a 576

356 357 358 359 360

176 6 630 192 8 576 178 4 540 358 2 360 96 24 1170

406 407 408 409 410

168 360 128 408 160

272 4 204 6 348 4 i32 12 328 4

-0

9

4 6 4 4 8

504 798 608 684 672

261 262 263 264 265

168 130 262

6 4 2 ao 16 208 4

390 396 264 720 324

311 312 313 314 315

310 2 312 96 16 840 312 2 314 156 4 474 144 12 624

361 362 363 364 365

342 3 180 4 220 6 144 12 288 4

3 81 546 532 784 444

411 412 413 414 415

552 728 480 936 504

461 462 463 464 465

460 2 462 120 16 1152 462 2 464 224 10 930 240 a 768

266 267 268 269 270

108 a 176 4 132 6 268 2 72 16

480 360 476 270 720

316 317 318 319 320

156 6 316 2 104 8 280 4 128 14

560 318 648 360 762

366 367 368 369 370

120 a 366 2 176 10 240 6 144 a

744 368 744 546 684

416 192 12 882 417 276 4 560 418 i a o a 720 419 418 2 420 420 96 24 1344

466 467 468 469 470

232 4 702 466 2 468 144 ia 1274 396 4 544 184 a a64

271 272 273 274 275

270 2 272 128 10 558 144 a 448 136 4 414 200 6 372

321 212 4 432 322 132 8 576 323 288 4 360 324 ioa 15 847 325 240 6 434

371 372 373 374 375

312 4 120 12 372 2 160 a 200 a

432 a96 3 74 648 624

421 422 423 424 425

420 210 276 208 320

2 4 6 8 6

422 636 624 558

471 472 473 474 475

312 232 420 156 360

376 377 378 379 380

184 a 336 4 ioa 16 378 2 144 12

720 420 960 380 840

426 427 428 429 430

140 360 212 240 168

4 6

a

a64 496 756 672 792

476 477 478 479 480

192 12 1008 312 6 702 238 4 720 478 2 480 128 24 1512

512 576 3a4 1020 576

431 432 433 434 435

430 2 432 144 20 1240 432 2 434 iao 8 768 224 a 720

481 4a2 483 484 485

432 240 264 220 384

162 12 1092 486 2 4aa 240 a 930 324 4 656 168 12 1026

276 88 12 277 276 2 278 138 4 279 180 6 280 96 16

672 278 420 416 720

326 327 328 329 330

162 216 160 276

ao

4 492 4 440 a 630 4 384 16 a64

a a

aio

4

a 4 a

6

4 4

632 900 528 960 620

281 282 283 284 285

280 92 282 140 144

2 8 2 6

282 576 284 504 a 480

331 332 333 334 335

330 164 216 166 264

332 5aa 494 504 408

381 382 383 384 385

252 4 190 4 382 2 128 16 240 a

286 287 288 289 290

120 a 504 240 4 336 96 18 819 272 3 307 112 a 540

336 337 338 339 340

96 20 992 336 2 338 156 6 549 224 4 456 128 12 756

386 387 388 389 390

192 4 582 252 6 572 192 6 686 388 2 390 96 16 ,1008

436 216

6

770

437

396

4

480

i 8aa 439 438 2 440 440 160 16 ioao

486 487 4aa 489 490

291 292 293 294 295

192 144 292

392 578 294 684 360

341 342 343 344 345

300

4 384 12 780 294 4 400 168 a 660 176 a 576

391 392 393 394 395

352 4 168 12 260 4 196 4 312 4

432 a55 528 594 480

441 442 443 444 445

252 9 741 192 a 756 442 2 444 144 12 1064 352 4 540

491 492 493 494 495

490 2 492 160 12 1176 448 4 540 216 a a40 240 12 936

296 297 298 299 300

144 I80 148 264

8 570

ia

1092 398

446 447 448 449 450

222 4 672 296 4 600 192 14 1016 448 2 450 120 i a 1209

496 497 498 499 500

240 10 992 420 4 576 164 a 1008 498 2 500 200 12 1092

4 6 2 a4 12 232 4

80

a iao

4 450 4 336 18 868

2 6 6 4 4

ioa

346 172 4 522 347 346 2 348 348 iiZ 12 840 349 348 2 350 350 120 12 744

396 120 397 396 398

198

399 2i6

2 4

i

400 160 15

600 -__

640 961

4ia i 4 4

a

9 4

532 726 768 931 5aa

COMBINATORIAL ANALYSIS

ARITHMETIC FUNCTIONS

'Table 24.6 n v(.> UI, QI 672 4 501 332 756 4 502 250 2 504 503 502 504 144 24 1560 612 4 505 400

n v(n> an Q l 600 4 551 504 552 176 16 1440 640 4 553 468 4 834 554 276 912 8 555 288

n 601 602 603 604 605

~ ( n )an 600 252 396 300 440

2 8 6 6 6

dn)

QO

602 1056 884 1064 798

651 652 653 654 655

360 324 652 216 520

8 6 2 8 4

1024 1148 654 1320 792

n ~ ( n ) an QI 701 700 2 702 702 216 16 1680 703 648 4 760 704 320 14 1524 705 368 8 1152

QI

QI

506 507 508 509 510

220 312 252 508 128

8 864 6 732 6 896 2 510 1 6 1296

556 557 558 559 560

276 556 180 504 192

6 2 12 4 20

980 558 1248 616 1488

606 607 608 609 610

200 606 288 336 240

8 1224 2 ,608 1 2 1260 8 960 8 1116

656 657 658 659 660

320 432 276 658 160

10 6 8 2 24

1302 962 1152 660 2016

706 707 708 709 710

511 512 513 514 515

432 256 324 256 408

4 10 8 4 4

592 1023 800 774 624

561 562 563 564 565

320 280 562 184 448

8 4 2 12 4

864 846 564 1344 684

611 612 613 614 615

552 192 612 306 320

672 4 1 8 1638 2 614 4 924 8 1008

6 6 1 660 662 330 663 384 664 328 665 432

2 4 8 8 8

662 996 1008 1260 960

7 1 1 468 712 352 713 660 714 192 715 480

6 1040 8 1350 4 768 1 6 1728 8 1008

516 517 518 519 520

168 460 216 344 192

12 4

1232 576 8 912 696 4 1 6 1260

566 567 568 569 570

282 324 280 568 144

4 10 8 2 16

852 968 1080 570 1440

616 617 618 619 620

240 616 204 618 240

16 2

8

666 667 668 669 670

216 616 332 444 264

12 4 6 4 8

1482 720 1176 896 1224

716 717 718 719 720

356 476 358 718 192

6 4 4 2 30

1060

2 12

1440 618 1248 620 1344

521 522 523 524 525

520 168 522 260 240

2 12 2 6 12

522 1170 524 924 992

571 572 573 5 74 575

570 240 380 240 440

2 12 4 8 6

572 1176 768 1008 744

6 2 1 396 622 310 623 528 624 192 625 500

8 4 4 20 5

960 936 720 1736 781

6 7 1 600 672 192 673 672 674 336 675 360

4 24 2 4 12

744 2016 674 1014 1240

721 722 723 724 725

612 342 480 360 560

4 6 4 6 6

832 1143 968 1274 930

526 527 528 529 530

262 480 160 506 208

4 4 20 3 8

792 576 1488 553 972

576 577 578 579 580

192 576 272 384 224

21 2 6 4 12

1651 578 921 776 1260

626 627 628 629 630

312 360 312 576 144

4 8 6 4 24

942 960 1106 684 1872

676 677 678 679 680

312 676 224 576 256

9 2 8 4 16

1281 678 1368 784 1620

726 727 728 729 730

220 726 288 486 288

12 2 16 7 8

1596 728 1680 1093 1332

531 348 532 216 533 480 534 176 535 424

6 12 4 8 4

780 1120 588 1080 648

58 1 582 5 83 5 84 585

492 192 520 288 288

4 8 4 8 12

672 1176 648 1110 1092

631 632 633 634 635

630 312 420 316 504

2 8 4 4 4

632 1200 848 954 768

681 682 683 684 685

452 300 682 216 544

4 8 2 18 4

912 1152 684 1820 828

731 732 733 734 735

672 240 732 366 336

4 12 2 4 12

792 1736 734 1104 1368

536 537 538 539 540

264 356 268 420 144

8 4 4 6 24

1020 720 810 684 1680

586 587 588 589 590

292 4 586 2 168 18 540 4 232 8

882 588 1596 640 1080

636 637 638 639 640

208 504 280 420 256

12 6 8 6 16

1512 798 1080 936 1530

686 687 688 689 690

294 456 336 624 176

8 4 10 4 16

1200 920 1364 756 1728

736 737 738 739 740

352 660 240 738 288

12 4 12 2 12

1512 816 1638 740 1596

541

24 4 12 4

-542

641 640 642 212 643 642 644 264 645 336

2 8 2 12 8

642 a296 644 1344 1056

692 693 694 695

344

2 16 8

792 1178 594 1440 864

690

816 728 1134 660

591 392 592 288 593 592 594 180 595 384

691

543 544 545

540 270 360 256 432

360 346 552

2 6 12 4 4

692 1218 1248 1044 840

741 742 743 744 745

432 312 742 240 592

8 8 2 16 4

1120 1296 744 1920 900

546 547 548 549 550

144 546 272 360 200

16 2 6 6 12

1344 548 966 806 1116

596 597 598 599 600

296 396 264 598 160

6 4 8 2 24

1050 800 1008 600 1860

646 647 648 649 650

8 2 20 4 12

1080 648 1815 720 1302

696 697 698 699 700

224 640 348 464 240

1 6 1800 4 756 4 1050 4 936 18 1736

746 747 748 749 750

372 492 320 636 200

4 1122 6 1092 12 1512 4 864 1 6 1872

542

4 10

288 646 216 580 240

352 600 232 708 280

4 4 12 2 8

1062 816 1680 710 1296

1260 960 720 2418

COMBINATORIAL ANALYSIS

ARITHMETIC FUNCTIONS n

~ ( n )Qn

751 752 753 754 755

750 2 368 10 500 4 336 8 600 4

756 757 758 759 760

n

801 802 803 804 805

528 6 1170 400 4 1206 720 4 888 264 12 1904 528 8 1152

851 852 853 854 855

V(") Qo Cl 792 4 912 280 12 2016 852 2 854 360 8 1488 432 12 1560

901 902 903 904 905

216 24 2240 756 2 758 378 4 1140 440 8 1152 288 16 1800

806 807 808 809 810

360 8 536 4 400 8 808 2 216 20

1344 1080 1530 810 2178

856 857 858 859 860

424 8 1620 856 2 858 240 16 2016 858 2 860 336 12 1848

761 762 763 764 765

760 2 762 252 8 1536 648 4 880 380 6 1344 384 12 1404

811 812 813 814 815

810 2 812 336 12 1680 540 4 1088 360 8 1368 648 4 984

861 862 863 864 865

766 767 768 769 770

382 4 1152 696 4 840 256 18 2044 768 2 770 240 16 1728

816 817 818 819 820

256 20 2232 756 4 880 408 4 1230 432 12 1456 320 12 1764

771 772 773 774 775

512 . 4 1032 384 6 1358 772 2 774 252 12 1716 600 6 992

821 822 823 824 825

776 777 778 779 780

384 8 432 8 388 4 720 4 192 24

1470 1216 1170 840 2352

781 782 783 784 785

QI

4qn)

Qfl

Ql

n

v(n)

Ql

n

972 1512 1408 1710 1092

951 952 953 954 955

632 4 1272 384 16 2160 952 2 954 312 12 2106 760 4 1152

906 907 908 909 910

300 8 906 2 452 6 600 6 288 16

1824 908 1596 1326 2016

956 957 958 959 960

478 ..-

4.

6 1680 8 1440 1 - 4. 4. 0816 4 1104 256 28 3048

480 8 1344 430 4 1296 862 2 864 288 24 2520 688 4 1044

911 912 913 914 915

910 2 912 288 20 2480 820 4 1008 456 4 1374 480 8 1488

961 962 963 964 965

930 432 636 480 768

3 8 6 6 4

866 867 868 869 870

432 4 1302 544 6 1228 360 12 1792 780 4 960 224 16 2160

916 917 918 919 920

456 6 1610 780 4 1056 288 16 2160 918 2 920 352 16 2160

966 967 968 969 970

264 16 2304 966 2 968 440 12 1995 576 8 1440 384 8 1764

820 2 822 272 8 1656 822 2 824 408 8 1560 400 12 1488

871 872 873 874 875

792 432 576 396 600

952 1650 1274 1440 1248

921 922 923 924 925

612 4 1232 460 4 1386 840 4 1008 240 24 2688 720 6 1178

971 972 973 974 975

970 2 972 324 18 2548 828 4 1120 486 4 1464 480 12 1736

826 827 828 829 830

348 8 1440 826 2 828 264 18 2184 828 2 830 328 8 1512

876 877 878 879 880

288 12 2072 876 2 878 438 4 1320 584 4 1176 320 20 2232

926 927 928 929 930

462 4 1392 612 6 1352 448 12 1890 928 2 930 240 16 2304

976 977 978 979 980

480 10 1922 976 2 978 324 8 1968 880 4 1080 336 18 2394

700 4 864 352 8 1296 504 8 1200 336 15 1767 624 4 948

831 832 833 834 835

552 4 1112 384 14 1778 672 6 1026 276 8 1680 664 4 1008

881 882 883 884 885

880 2 882 252 18 2223 882 2 884 384 12 1764 464 8 1440

931 932 933 934 935

756 464 620 466 640

1140 1638 1248 1404 1296

981 982 983 984 985

648 6 1430 490 4 1476 982 2 984 320 16 2520 784 4 1188

786 787 788 789 790

260 786 392 524 312

1584 788 1386 1056 1440

836 837 838 839 840

360 12 1680 540 8 1280 418 4 1260 838 2 840 192 32 2880

886 887 888 889 890

442 4 1332 886 2 888 288 16 2280 756 4 1024 352 8 1620

936 937 938 939 940

288 24 2730 936 2 938 396 8 1632 624 4 1256 368 12 2016

986 987 988 989 990

448 8 1620 552 8 1536 432 12 1960 924 4 1056 240 24 2808

791 792 793 794 795

672 4 912 240 24 2340 720 4 868 396 4 1194 416 8 1296

841 842 843 844 845

812 420 560 420 624

871 1266 4 1128 6 1484 6 1098

891 892 893 894 895

540 10 1452 444 6 1568 828 4 960 296 8 1800 712 4 1080

941 942 943 944 945

940 2 942 312 8 1896 880 4 1008 464 10 1860 432 16 1920

991 992 993 994 995

990

796 797 798 799 800

396 6 1400 796 2 798 216 16 1920 736 4 864 320 18 1953

846 847 848 849 850

276 12 1872 660 6 1064 416 10 1674 564 4 1136 320 12 1674

896 897 898 899 900

384 16 2040 528 8 1344 448 4 1350 840 4 960 240 27 2821

946 947 948 949 950

420 8 1584 946 2 948 312 12 2240 a64 4 1036 360 12 1860

996 997 998 999 1000

8 2 6 4 8

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3

4

4 8 6 8 8

832 400 504 448 720

Qo

4 8 8 8 4

752 488 008 ~ . 260 912

6 6 4 4 8

476 560

993 1596 1404 1694 1164

7

997

12 660 4 420 8 792 4

20ib 1328 1728 1200

480

328 12 2352 996 2 998 498 4 I500 648 8 i5i0 400 16 2340

!

2.5 3-5 2-3-5 2'.5 2.8 3-3.5 2.5.7 2'*5 2.32.5 21.5' 2.5.11 2'.3-5 2-5-13 3-5-7 2-3.8 25-5 2-5-17 3.32.5 2.5.19 28.5' 2.3-5-7 3 - 5 . 11 2.5.23 24-3-5 2-5' 2'-5-13 2-3'-5 2"5.7 2-5.29 3.3.5' 2-5.31 26-5 2-3.5.11 3-5-17 2-9.7 28-32-5 2.5-37 22-5.19 2.3-5-13

P.8 2.5-41 3.3-5.7 2.5-43 2'-5.11 2-32-5' 3.5-23 2.5-47 25-3-5 2.5-7'

I I1 3.7 31 41 3.17 61 71 34 7-13 101 3-37 11' 131 3.47 151 7.23 32-19 181 191 3.67 211 13.17 3.7.11 241 251 32-29 271 281 3-97 7-43 31 1 3.107 331 11-31 3'. 13 19 7.53 3-127 17-23 401 3.137 421 431 32-7' 11.41 461 3.157 13.37 491

2 21.3 2.11 25 2-3-7 3-13 2-31 28-32 2-41 3-23 2-3-17 24-7 2.61 3-3-11 2-71 2'019 2-34 3.43 2-7-13 Y.3 2-101 3.53 2-3-31 28.29 2-11' 3.32.7 2.131 2'-17 2-3-47 3.73 2-151 2'.3-13 2-7-23 3-83 2-3'-19 25.11 2.181 P.3-31 2.191 2a-p 2-3-67 3-103 2-211 2'-3' 2-13-17 22-113 2.3-7.11 2'.59 2-241 21-3-41

*

3 13 23 3-11 43 53 3-7 73 83 3-31 103 113 3.41 7-19 11-13 3.17 163 173 3-61 193 7.29 3-71 223 233 35 11-23 263 3-7-13 283 293 3-101 313 17.19 32-37 7' 353 3.11' 373 383 3-131 13.31 7.59 32-47 433 443 3.151 463 11.43 3-7-23 17-29

3 2-7 2a.3 2.17 3-11 2.3' 26 2-37 2'.3*7 2-47 2'*13 2-3-19 3-31 2.67 2'-3' 2-7-11 22-41 2-3.29 2'-23 2-97 3.3-17 2-107 25-7 2.3'.13 3.61 2.127 2'-3-11 2-137 3-71 2.3-'P 24.19 2-157 3-34 2.167 2'-43 p.59 '-7-13 2.1 1-17 27.3 2.197 21-101 2-32-23 2'.53 2-7-31 3.3-37 2.227 24-29 2-3-79 3.11' 2-13-19

5 3.5 52 5.7 32-5 5-11 5.13 3.52 5-17 5-19 3.5.7 5-23 5s 38.5 5-29 5-31 3-5-11 52.7 5.37 3-5-13 5-41 5.43 32.52 5.47 5.72 3-5-17 5.53 52-11 3.5-19 5.59 5.61 32.5-7 52.13 5.67 3.5-23 5.71 5-73 3.5' 5-7-11 5.79 34-5 5.83 52.17 3-5.29 5.89 5.7.13 3.5.31 52-19 5.97 3'-5-11

2.3 2' 2-13 22-32 2-23 2'-7 2-3-11 22-19 2-43 25.3 2-53 22-29 2.32.7 2'-17 2.73 3-3-13 2.83 24.11 2-3.31 22-7' 2.103 2a.3' 2-113 22-59 2-3-41 28 2-1-19 22-3-23 2.11.13 2'-37 2.3'.17 2'-79 2.163 24.3.7 2-173 3.89 2.3-61 2'-47 2-193 3.32-11 2.7.29 25-13 2-3-71 2'.109 2-223 2'-3-19 2-233 22-7-17 2-35 2'-31

7 17 3'

37 47 3-19 67 7-11 3-29 97 107 3'-13 I27 137 3.7' 157 167 3.59 11.17 197 32-23 7.31 227 3.79 13-19 257 3.89 277 7.41 3'- 1 1 307 317 3-109 337 347 3-7.17 367 13.29 32-43 397 11-37 3.139 7-61 19.23 3.149 457 467 32.53 487 7.71

23 2.32 22-7 2.19 24.3 2.29 2'-17 2.3.13 23.11 2.72 2'.3' 2.59 27 2.3-23 22.37 2.79 23-3-7 2.89 22.47 2.32.11 2'.13 2.109 22.3-19 2.7-17 2'.31 2-3.43 22-67 2-139 25-32 2-149 3.7-11 2-3-53 2'.41 2.13' 22.3-29 2.179 2'-23 2.3'-7 22-97 2-199 28.3-17 2-11-19 22.107 2-3-73 26.7 2.229 2'-3'-13 2.239 2'.61 2.3.83

KI

32 19 29 3-13 72 59 3.23 79 89 32.11 109 7.17 3.43 139 149 3-53 132 179 33.7 199 11-19 3-73 229 239 3.83 7.37 269 3'.31 172 13-23 3.103 11-29 7.47 3.113 349 359 3'.41 379 389 3.7.19 409 419 3-11-13 439 449 3'- 17 7-67 479 3.163 499

O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

P -

.

8

2. !a E.

8m % r 5

rp

g

50 51 52 53 54

2'. 5' 2-3.5.17 2"5- 13 2.5-53 22-38-5

3.167 7.73

55 56 _. 57 58 59

2.52-11 2'.5-7 2-3.5-19 22-5-29 2-5.59

19.29 3.11.17

60 61 62 63 64

2"3-5' 2.5-61 22.5.31 2.32-5-7 27-5

65 66 67 68 69 70 71 72 .~ 73 74

242.13 22.3.5711 2.5.67 2'-5.17 2.3-5.23 F.52-7 2.5-71 24.32.5 2-5.73 2'-5.37

75 76 77 78 79

2.3-5s 23-5-19 2.5.7.11 2'-3.5.13 2.5.79

80

84

25.52 2.3'-5 2'.5.41 2.5-83 23.3-5.7

85 86 87 88 89

2.5'.17 2'.5.43 2-3-5.29 24.5.11 2.5.89

23.37 3-7.41 13-67

90 91 92 93 94

22.32.52 2-5-7-13 2'-5.23 2.3.5.31 2'.5-47

17-53

95 96 97 98 99

2.5'-19 26.3.5 2.5.97 22.5.72 2.32.5.1 1

~~

81

82 83

521

33.59 541

571

7-83 3-197 601

13-47 3'-23 631 641

3-7-31 661 _ _ ~

11-61 3-227 691 701

32-79 7.103 17.43 3.13.19 751 761

3-257 11-71 7.113 32.89 811 821

3-277 2g2

881

3'. 11 911

3-307 72-19 941

3.317 3 l2 971 3'.109 991

2.251 29 2-32-29 P-7-19 2-271 23-3-23 2.281 22.11-13 2-3-97 2'-37 2-7-43 22-3'- 17 2.311 23-79 2-3-107 22.163 2-331 25.3.7 2-11.31 22-173 2.38.13 2'-89 2.192 22.3.61 2-7.53

503

3'-19 523

13-41 3.181 7-79 563

3-191 11-53 593

3'-67 613

7-89 3.21 1 643 653

3-13-17 673 683

3'.7.11 19.37 23.31 3-241 733 743

2'.3'-7 2-257 22.131 2.3.89 2'-17

5.101 5.103 3-52.7 5-107 5.109

2-11-23 2'.3.43 2.263 23-67 2.3.7-13

3.13' 11-47 17.31 3-179

2-277 22-3-47 2-7.41 2a.73 2.33-11

3.5.37 5.113 52.23 3'.5.13 5.7.17

22.139 2.283 26.32 2.293 22.149

557

2'-151 2-307 24.3.13 2.317 F-7.23

5.11' 3.5-41 5-127 3.5-43

2-3.101 23.7-11 2.313 2'.3.53 2-17.19

2.3-109 23-83 2.337 22.3'-19 2.347 26-11 2.3-7.17 22.181 2.367 2'-3-31

5.131 5.7.19 31.52 5.137 5.139 3.5.47 5.11.13 5'.29 3.5.72 5.149

2'-41 2.3'.37 22.132 2.73 23-3-29 2.353 22.179 2.3-1l2 25.23 2.373

32.73 23-29

2.13.29 22.191 2.3'-43 2'.7' 2.397

5.151 32.5.17 5'.31 5.157 3.5.53

22.33-7 2-383 23.97 2-3.131 22.199

757

2'-3.67 2.11.37 23.103 2-3.139 22.211

5.7.23 5-163 3.52.11 5.167 5-13'

2-13.31 2'.3.17 2.7.59 22.11.19 2-32-47

32.5.19 5.173 53.7 3-5.59 5-179

Z3- 107 2.433 22.3+73 2.443 27.7

54

2'.47 2-3-127 22.193 2-17.23 2'.3?.11

3.251 7.109

2-401 2'-7.29 2.3.137 2c.13 2-421

11.73 3.271

2'-3.71 2-431 23.109 2.32.72 2'.223

853 863

19-47

2.7.61 25-33 2.19.23 22.13.17 2.3.149

2.11.41 2'-3. 19 2.461 22.233 2-3.157

3-7.43 11.83 13.71 3.311 23-41

23.113 2-457 22-3.7-11 2.467 2'.59

5.181 3.5.61 52-37 5.11-17 33.5.7

2.3.151 22.229 2.463 28-32-13 2.1 1-43

23.7-17 2.13-37 22.35 2.491 25.31

953

2.3'.53 22-241 2.487 23.3.41 2-7-71

5.191 5.193 3.52.13 5.197 5.199

2'.239 2-3-7.23 2'.61 2-17.29 22.3.83

773

3'.29 13-61

823

7'-17 3-281

32.97 883

3'-107 7.139 983

3-331

547

34.7 577 587

3-199 607 617

3.11.19 7'. 13 647

677

3.229 17.41 7.101 3.239 727

11.67 3'.83 13.59 3.7.37 787 797

3.269 19.43 827

33.31 7.11' 857

3.172 877 887

3-13.23

2'.127 2.7.37 2'.3-11 2.269 2'.137 2.3'.31 23.5 1 2.17' 22.3.72 2.13.23 25-19 2.3.103 22.157 2.11.29 23.34 2.7.47 P-167 2.3.113 2'-43 2.349 2Y3-59 2.359 23.7.1 3 2.32.4 1 2'. 11-17 2-379 28.3 2.389 22-197 2-3.7.19 23.101 2.409 22.32.23 2-419 2'.53 2-3.11.13 2'-7.31 2.439 23-3.37 2.449

937 947

22.227 2.33.17 25.29 2-7-67 22-3-79

3-11.29

2.479

967 977

22.13.19 2-3.163

907

7.131 3'.103

3-7.47 997

23.112

2.4'39

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3-173 2327'.11 3'.61

50 51 0 52 53 54

13-43

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569

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3.7.29 619

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3I223 7-97 13.53 3.233 709 719 36

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61 62 63 64 65 66 67 d

69 68

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70 71 72 73 74

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19.41 3.263 17.47

77 78 79

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80 81 82 83 84

3 2 - 7. I

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829 839

3.283 859 11 -79

85 86 87 88 89

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90 91 92 93

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100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149

2'.5' 2-5-101 3-3.5.17 2-5-103 24-5-13 2.3.P-7 22-5-53 2-5-107 2'-3'-5 2.5-109 22.51.11 2-3-5-37 26-5-7 2.5-113 22-3.5-19 2-53.23 28-5-29 2-3'.5.13 3-5-59 2.5.7.17 3.3.5s 2-5-11' 2'.5.61 2-3.5.41 2'.5.31 2.5' 22.32-5-7 2.5-127 2 8 -5 2-3-5-43 3-51-13 2.5.131 2'.3.5-11 2.5.7-19 P.5.67 2-3'6' 2'-5-17 2.5-137 22-3.5-23 2-5-139 2'-53.7 2-3.5-47 3-5-71 2-5.11.13 26-3'.5 2-53.29 22-5-73 2-3.5.7' 2'-5.37 2-5.149

9-11-13 3-337 1021 1031 3-347 1051 1061 32-7-17 23-47 1091

3.367 11-101 19.59 3-13-29 7-163 1151 38.43 1171 1181 3-397 1201 7-173 3-11-37 1231 17-73 32.139 13.97 31-41 3-7-61 1291 1301 3-19-23 1321 118 3'-149 7-193 1361 3.457 1381 13-107 3.467 17-83 72.29 38-53 11-131 1451 3.487 1471 1481 3-7-71

2-3-167 22-11-23 2-7-73 28-3-43 2-521 22.263 2-32-59 3-67 2-541 22-3-7-13 2.19-29 2'-139 2-3-11.17 2'-283 2.571 2l-3' 2-7-83 P-293 2.3-197 2'-149 2-601 22.3-101 2-13-47 3-7-11 2.3'-23 P.313 2-631 2"3-53 2-641 21-17-19 2.3.7.31 25.41 2-661 2'-32-37 2-11-61 28.132 2-3.227 2'. 73 2-691 2'*3-29 2.701 P.353 2-3-79 2'-179 2-7.103 P-3-11' 2-17-43 3.23 2.3-13-19 22-373

17-59 1013 3-11-31 1033 7-149 3'.13 1063 29-37 3-19' 1093 1103 3.7-53 1123 11-103 32-127 1153 1163 3.17-23 7-13' 1193 3-401 1213 1223 32- 137 11.113 7-179 3-421 19-67 1283 3.431 1303 13-101 38-72 31.43 17-79 3-11-41 29.47 1373 3.461 7.199 23-61 3'- 157 1423 1433 3-13-37 1453 7-11-19 3-491 1483 1493

22-251 2-3-13' 2'0

2-11-47 3-32-29 2-17-31 2a-7-19 2-3-179 3.271 2-547 2'-3.23 2.557 22.281 2-3'-7 2'- 11.13 2.577 22-3.97 2.587 2'.37 2-3-199 P-7.43 2.607 23-32-17 2.617 22.311 2.3-11.19 24-79 2-7'.13 22.3.107 2.647 23.163 2.3'-73 P-331 2-23-29 P-3-7 2-677 3-11-31 2.3-229 25.173 2-17-41 22-3'-13 2-7-101 2'49 2-3.239 3-19 2.727 2'.3-61 2-11-67 22-7-53 2.32.83

3.5-67 5-7-29 52-41 32-5.23 5-11-19 5-211 3.5.71 58-43 5.7.31 3.5.73 5.13-17 5-223 32.53 5.227 5-229 3.5-7-11 5-233 52.47 3.5-79 5-239 5.241 35-5 52.72 5.13-19 3-5.83 5-251 5.11-23 3-52.17 5-257 5.7.37 3'-5.29 5.263 58.53 3.5-89 5-269 5.271 3-5-7-13 5%11 5-277 3'.5-31 5-281 5.283 3-52-19 5-7.41 5-175 3.5-97 5-293 58.59 33-5-11 5-13-23

2-503 28.127 2.3'019 3-7-37 2-523 26-3-11 2-13-41 22-269 2-3.181 2"137 2.7-79 22-32.31 2-563 2'.71 2.3.191 2s.172 2.11.53 2"3*72 2-593 22.13-23 2-3'-67 28-19 2.613 22-3-103 2-7-89 2'-157 2.3.21 1 22-11.29 2.643 2'-3' 2-653 22-7.47 2-3-13.17 2'-167 2-673 22-3-113 2-683 25-43 2.3'-7-11 22-349 2.19-37 2'.3-59 2-23.31 22.359 2-3-241 3.7.13 2-733 3-3-41 2.743 2'- 11-17

19-53 32-113 13-79 17-61 3.349 7-151 11-97 3.359 I087 1097 38-41 1117 72-23 3-379 31.37 13-89 3.389 11.107 1187 3'-7-19 17-71 1217 3.409 1237 29-43 3.419 7.181 1277 32-11.13 1297 1307 3.439 1327 7.191 3-449 23.59 1367 3'.17 19-73 11.127 3-7-67 13-109 1427 3-479 1447 31.47 3'.163 7-211 1487 3-499

24.32-7 2.509 22-257 2-3.173 2*.131 2.232 22-3-89 2-7'-11 2"17 2-3'.61 2'-277 2-13-43 2'-3-47 2-569 22.7-41 2.3-193 24.73 2-19.31 22-33-11 2.599 23.151 2.3-7-29 2'-307 2.619 2'03.13 2-17-37 2'-317 2.3'-71 2'.7-23 2-11-59 22.3.109 2.659 2'.83 2-3-223 2'.337 2-7-97 28-32-19 2-13-53 2'-347 2-3.233 27.11 2.709 3-3.7.17 2-719 2'-181 2-38 2'.367 2-739 2'-3.31 2-7-107

1009 1019 3.7' 1039 1049 3.353 I069 13.83 32. 112 7-157 1109 3.373 1129 17.67 3-383 19-61 7-167 3'-131 29.41 11.109 3-13-31 23.53 1229 3.7-59 1249 1259 3"47 1279 1289 3.433 7-11-17 1319 3-443 13-103 19-71 3'.151 372 7.197 3-463 1399 1409 3-11.43 1429 1439 3'-7.23 1459 13:113 3-17-29 1489 1499

100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149

2.761 21-383 2-3-257

31.167 17-89 1523 3-7-13 1543

2"47 2-757 21-3-127 2-13-59 2l.193

5.7-43 3-5-101 55-61 5.307 3.5.103

2.3-251 22.379 2-7-109 2'-3 2-773

11-137 37-41 3.509 29.53 1-13-17

21.13-29 2-3-1 1-23 20-191 2.769 21.32-43

3.503 71-31 11.139 34-19 1549

150 151 152 153 154

3e.11.47 7-223 1571 3-17-31 37-43

2'-97 2-11-71 3.3-131 2-7-113 20.199

1553 3.521 112-13 1583 3a.59

2.3-7.37 21-17-23 2-787 24-32-11 2-797

5-311 5-313 32.52.7 5-317 5-11-29

2'-389 2.38.29 2a-197 2-13-61 P.3-7-19

32.173 1567 19-83 3.23' 1597

2-19-41 25.72 2-3-263 21.397 2.17-47

1559 3-523 1579 7.227 3-13.41

155 156 157 158 159

20.55 2.5.7.23 22-34-5 2-5-163 20-5.41

1601 32.179 1621 7.233 3-547

2-32-89 3-13-31 2.811 26-3-17 2-821

7-229 1613 3-541 23-71 31.53

22.401 2.3-269 23-7-29 2-19-43 3.3-137

3-5-107 5-17-19 9-13 3-5-109 5-7.47

2-11 -73 24.101 2-3-271 2'.409 2.823

1607 3-72.11 1627 1637 3'-61

23-3-67 2-809 22. 11-37 2-32-7.13 2 ' . 103

1609 1619 38.181 11-149 17.97

160 161 162 163 164

165 166 167 168 169

2-3-55.11 21-5-83 2.5.167 24.3.5-7 2-5-13'

13.127 11-151 3-557 411 19.89

21-7-59 2.3.277 2*11- 19 2-29 21-32-47

3-19-29 1663 7-239 32-1 1.17 1693

2-827 2l.13 2.3'-31 21.421 2-7-11'

5-331 3.5-37 52.67 5-337 3-5-113

25-32-23 2-72-17 22-419 2-3-281 25-53

1657 1667 3.13.43 7-241 1697

2-829 22-3-139 2.839 23.211 2.3.283

3.7-79 1669 23.73 3-563 1699

165 166 167 168 169

170 171 172 173 174

21-55-17 23-5-19 23.5-43 2-5.173 21-3.5-29

35.7 29-59 1721 3-577 1741

2-23-31 24-107 2.3.7-41 21-433 2-13.67

13.131 3-571 1723 1733 3-7-83

2s-3-71 2.857 P.431 2.3-17' 24.109

5-11 -31 5-73 3.9-23 5.347 5.349

2.853 22-3.11-13 2-863 23.7.31 2-32-97

3-569 17.101 11-157 3'-193 1747

3-7-61 2-859 26-33 2.11.79 2'*19-23

1709 32-191 7.13-19 37.47 3.11 -53

170 171 172 173 174

175 176 177 178 179

2-8-7 2'-5-11 2-3.5.59 22.5.89 2.5-179

17.103 3.587 7-11 -23 13.137 32-199

20.3.73 2.881 22-443 2.34-11 28.7

1753 41.43 32-197 1783 11.163

2-877 22.32-P 2-887 2.223 2.3.13-23

33.5-13 5.353 52-71 3-5-7.17 5.359

22-439 2-883 2'.3.37 2-19-47 22-449

7.251 3-19-31 1777 1787 3-599

2-3.293 23-13-17 2.7-127 3.3.149 2-29-31

1759 29.61 3-593 1789 7.257

175 176 177 178 179

180 181 182 183 184

23.32-52 2-5.181 21-5-7-13 2.3.5-61 2'.5-23

1801 1811 3.607 1831 7-263

2-17-53 2'.3.151 2-911 2'-229 2-3-307

3.601 71-37 1823 3.13-47 19-97

22.11-41 2-907 25.3-19 2-7.131 22-461

5-19' 3-5-11' 52-73 5.367 32.5.41

2-3-7043 23-227 2.11.83 22-33.17 2.13-71

13-139 23-79 32-7-29 11-167 1847

2'.113 2-32.101 3.457 2-919 23.3.7.11

33-67 17.107 31-59 3-613 432

180 181 182 183 184

185 186 187 188 189

2.52.37 21-3.5.31 2.5-11-17 23.5.47 2-38.5.7

3.617 1861 1871 32-11-19 31-61

22.463 2.72.19 24-32-13 2-941 22.11.43

17.109 3'-23 1873 7.269 3.631

2-32-103 23-233 2-937 22.3-157 2.947

5-7-53 5-373 3-54 5-13.29 5.379

28.29 2-3.311 31-7-67 2-23-41 23.3.79

3-619 1867 1877 3.17-37 7.271

2-929 22-467 2.3-313 25-59 2.13.73

11.13' 3.7-89 1879 1889 32.211

185 186 187 188 189

190 191 192 193 194

3-51-19 2.5-191 27.3-5 2-5-193 21-5-97

1901 3.72-13 17.113 1931 3-647

2-3-317 2'.239 2-31' 22-3-7-23 2.971

11-173 1913 3.641 1933 29.67

24.7.17 2-3-11.29 3-13-37 2-967 2a. 36

3-5-127 5-383 52-7-11 32.5-43 5-389

2-953 22-479 2.32-107 2.7.139

1907 33.71 41-47 13-149 3.11-59

23.83 19-101 3-643 7-277 1949

190 191 192 193 194

195 196 197 198 199

2.3-55-13 23.5-79 2.5-197 22-32-5. 1I 2.5-199

1951 37-53 33-73 7-283 11.181

25-61 2.32-109 3.17-29 2-991 21.3.83

39-7-31 13-151 1973 3.661 1993

2.977 22.491 2.3-7-47 28-31 2-997

5.17.23 3.5.131 52-79 5.397 3-5.719

3-3-163 2-983 3-13-19 2.3.331 3.499

19.103 7-281 3-659 1987 1997

150 151 152 153 154

22-34? 205.151 2'.5.19 2-3~.5.17 21-5-7. 11

19-79 1511 3-13 1531 23-67

155 156 157 158 159

2.52.31 23.3-5.13 2.5-157 21-5-79 2-3-5053

160 161 162 163 164

2.751 21-3-7

24.112

22-32-53 2-7-137 23-241 2-3-17-19 22-487 2.11 -89 2'.3.41 2-23.43 22.7-71 2.33.37

3-653 11-179 1979 32-13.17 1999

-

N

0

1

2

3

4

5

6

7

8

9

N

m3c3 00 8OS m 00 A

rn

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P-52 2.3.5.67 22.5-101 2-5.7.29 3-3.5.17 2.52-41 22.5.103 2.32-5.23 2'.5.13 2.5.11-19 22-3.52.7 2.5-211 23-5-53 2.3.5.71 2'-5.107 2-52-43 2'.3'.5 2-5.7.31 23-5.109 2.3-5.73 3.52.11 2.5.13.17 3-3.5.37 2.5-223 2'-5-7 2.32-53 22.5-113 2.5.227 23.3.5.19 2.5.229 22.52-23 2.3.5.7.11 2'.5.29 2-5-233 2"-32-5.13 2.5-47 3-5-59 2-3.5.79 3.5.7.17 2.5.239 2'.3-5 2.5.241 22-5-11' 2-3'.5 3-5-61 2.51.72 22.3-5-41 2.5-13.19 2'-5-31 2.3-5.83

3.23.29 2011 43.47 3-677 13-157 7.293 32.229 19.109 2081 3-17-41 11-191 2111 3.7-101 2131 2141 3'-239 2161 13.167 3.727 7-313 31.71 3.11-67 2221 23.97 33-83 2251 7-17-19 3-757 2281 29.79 3.13-59 2311 11.211 39.7.37 2341 2351 3.787 2371 2381 3.797 7' 241 1 32.269 11.13.17 2441 3-19.43 23-107 7.353 3-827 47.53

2-7-11-13 22-503 2.3-337 24.121 2.1021 22.33.19 2-1031 2'-7.37 2.3.347 22-523 2.1051 2O-3-11 2.1061 22-13-41 2-3'.7.17 23.269 2.23-47 3.3-181 2-1091 24. 137 2.3.367 22.7-79 2.11.101 23-32-31 2.19.59 22.563 2.3.13.29 25.71 2.7.163 22.3-191 2-1151 3.172 2.3"43 22-11-53 2.1171 24-3-72 2.1181 22-593 2-3-397 23.13-23 2.1201 22s-32.67 2.7.173 27-19 2-3-11.37 22-613 2.1231 3-3.103 2.17-73 21-7-89

2003 3-11-61 7-172 19.107 3'.227 2053 2063 3.691 2083 7-13-23 3-701 2113 11-193 33.79 2143 2153 3-7-103 41.53 37-59 3-17-43 2203 2213 P.13.19 7-11-29 2243 3-751 31-73 2273 3-761 2293 72-47 32-257 23-101 2333 3-11.71 13-181 17-139 3.7-113 2383 2393 33.89 19-127 2423 3.811 7-349 11.223 3-821 2473 13-191 32.217

22.3.167 2-19.53 23.11-23 2-3-113 22.7.73 2-13-79 24.3-43 2.17.61 22-521 2-3.349 3.263 2-7-151 Z2-3'-59 2.11-97 25.67 2-3.359 22.541 2.1087 23.3.7.13 2-1097 3.19-29 2-33*41 24.139 2-1117 22-3-11-17 2.7'-23 23-283 2.3.379 2'.571 2-31-37 28-32

2-13-89 23.7.83 2.3.389 23-293 2.11-107 22-3-197 2.1187 24.149 2.32-7-19 22-601 2-17-71 23-3.101 2-1217 23-13-47 2.3-409 2'.7-11 2-1237 2-33-23 2-29-43

5.401 5-13.31 34.52 5.11.37 5-409 3.5-137 5.7.59 52-83 3.5-139 5-419 5-421 32-5-47 53.17 5-7.61 3-5.11-13 5-431 5.433 3-52.29 5.19-23 5.439 32.5.72 5.443 52-89 3.5.149 5.449 5-11-41 3.5-151 52.7.13 5.457 33.5-17 5.461 5-463 3.52-31 5-467 5.7.67 3.5.157 5-11-43 51-19 32.5-53 5.479 5.13-37 3-5-7-23 8-97 5.487 3.5.163 5.491 5.17-29 32-52.11 5.7-71 5.499

2.17.59 25-32-7 2-1013 22.509 2.3.1 1-31 23-257 2.1033 2'.3.173 2-7.149 2'.131 2-3'-13 22.232 2.1063 2'-3-89 2.29-37 2'-7'. 11 2-3-199 Z7-17 2.1093 2'.32.6 1 2-1103 23.277 2.3.7.53 22-13-43 2.1123 24.3.47 2-11-103 22.569 2-32-127 23.7-41 2.1153 22-3.193 2.1163 25-73 2.3-17-23 22-19.31 2.7-13' 23.33.11 2.1193 22.599 2-3.401 2'.151 2-1213 22.3.7-29 2-1223 23-307 2-3'-137 22.619 2.11-113 2O.3-13

32-223 2017 2027 3.7-97 23.89 112-17 3.13.53 31-67 2087 32-233 72.43 29-73 3-709 2137 19.113 3-719 11.197 7.311 37 133 2207 3.739 17-131 2237 3.7.107 37.61 2267 3-11-23 2287 2297 3.769 7.331 13-179 3.19.41 2347 2357 32-263 2377 7-11-31 3-17-47 29-83 2417 3.809 2437 2447 33.7.13 2467 2477 3-829 11-227

23.251 2.1009 22-3-132 2-1019 2" 2-3-73 2'.11.47 2.1039 23-32. 29 2-1049 22.17.31 2-3-353 2'-7-19 2.1069 2'-3.179 2.13.83 23.271 2-32-11' 22.547 2.7-157 25-3.23 2.1109 22.557 2.3-373 23.281 2-1129 22-3'-7 2-17-67 2'-11.13 2.3-383 22.577 2.19.61 23-3.97 2-722-587 167 2.32-131 26.37 2-29-41 2'.3.199 2-11-109 23.7.43 2.3-13-31 22.607 2-23-53 2'-32-17 2-1229 22.617 2.3.7.59 23.311 2.1249

72.41 3-673 2029 2039 3-683 29.71 2069 33.7.11 2089 2099 3.19.37 13.163 2129 3.23-31 7-307 17.127 32.24 1 2179 11.199 3-733 472 7.317 3.743 2239 13.173 3'.251 2269 43.53 3.7-109 112-19 2309 3-773 17.137 2339 3'.29 7-337 23-103 3.13.61 2389 2399 3.11.73 41-59 7.347 32.271 31.79 2459 3-823 37.67 19-131 3.72.17

200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249

g

2.7-179 22.17-37 ~. _ _ 2-3-421 2'.317 2-19.67

23-109 3-839 7-19 43.59 3'.283

22-3.11.19 2-1259 25-79 2*3',47 22.72-13

13.193 11-229 32.281 2539 2549

250 251 8 252 253 254

5.7-73 3'.5-19 52.103 5.11.47 3-5.173

22-32-71 2.i283 2'.7-23 2-3.431 22-11.59

2557

17-151 3-859 13-199 72.53

2.1279 23-3-107 2.1289 2'.647 2-3-433

3.853 7.367 2579 3.863 23.113

255 256 257 258 259

22.3.7-31 2.1307 26-41 2-3.439 22-661

5-521 5.523 3.53.7 5-17-31 5-23'

2.1303 2'.3. 109 2.13.101 2'.659 2-33-72

3.11-79 2617 .37-71 3'-293 2647

2'.163 2-7.11.17 22-32.73 2-1319 23.331

2609 33.97 11.239 7.13.29 3.883

260 261 262 263 264

7-379 2663 35-11 2683 2693

2-1327 2a-32-37 2-7-191 22-11-61 2-3.449

32.5.59 5.13.41 5'.107 3.5-179 5-7'-11

25.83 __ 2-31.43 22-3-223 2.17.79 23.337

2657 3-7-127 2677 2687 3-29-31

2-3-443 22.23.29 2.13-103 27-3.7 2.19.71

2659 17.157 3.19-47 2689 2699

265 266 267

3.17.53 2713 7-389 3-911 13.211

24.138 - -_ 2-23.59 22-3-227 2.1367 23.73

5.541 3.5-181 52. 109 5-547 3'-5.61

2.3-11-41 2'.7-97 2-29.47 2'-32.19 2.1373

2707 11-13.19 3'.101 7.1723 41-67

2'.677 2.3'.151 23.11.31 2.37' 22.3.229

32.7-43 2719 2729 3.11-83 2749

2753 3'-307 47.59 112-23 3-7'.19

2-34-17 - -. 2'-691 2.19.73 25.3-29 2-11-127

5.19.29 5.7.79 3.52.37 5.557 5.13.43

22.13.53 2.3.461 23.347 2.7.199 22-3-233

3.919 2767 2777 3.929 2797

2.7.197 2'. 173 2-3.463 2'.17.41 2-1399

31-89 3.13.71 7.397 2789 3'-311

2.3-467 22-19-37 2-17.83 2'.3-59 2-72-29

2803 29-97 3-941 2833 2843

22.701 2.3.7.67 23.353 2-13-109 2*-3'.79

3.5-11-17 5.563 8.113 34-5.7 5.569

2-23.61 28.11 2.3'.157 22.709 2.1423

7-401 3'.313 11.257 2837 3.13.73

23.33.13 2.1409 2'.7.101 2-3.11.43 25.89

532 2819 3.23.41 17.167 7-11-37

280 281 282 283 284

2-11-131 22-3-241

3'-317 7-409 132-17 3-31' 11-263

2.1427 2'-179 2-3-479 2'.7.103 2-1447

5.571 3.5.191 53-23 5.577 3-5-193 .

2'-3.7-17 2.1433 22.719 2-3-13-37 2'.181

2857 47.61 3-7.137 2887 2897

2.1429 2'.3.239 2-1439 23.1~2 2.32-7-23

3.953 19.151 2879 33.107 13-223

285 286 287 288 289

2909 3-7.139 29.101 2939 3.983

290 ~. 291 292 293 294

250 251 252 253 254

22.54 2-5-251 2'-32-5-7 2-5-11-23 22.5-127

41-61 3'-31 2521 2531 3-7-1 1 2

2-3'-139 2'-157 2-13-97 22-3-211 2-31-41

2503 7.359 . __. 3-29 17.149 2543

255 256 257 258 259

2-3.5'.17 20-5 2-5.257 2'-3.5.43 2-5.7.37

2551 Ei97 3.857 29-89 2591

2'-11-29 2-3-7-61 22-643 2-1291 2'.3'

3-23.37 11-233 31-83 3'-7.41 2593

2.1277 22-641 2-3'-11.13 2'- 17-19 2.1 297

260 261 262 263 264

2'.52-13 2.38.5-29 22-5-131 2-5.263 2'-3.5- 11

32.17a 7.373 . 2621 3-877 19.139

2.1301 22.653 2-3-19-23 2'*7-47 2-1321

19-137 3-13-67 43-61 2633 3.881

2.52.53 22-5-7-19 2.3.5-89 2'.5.67 2-5-269

11-241 3.887 _ ___ 267 1 7-383 32-13-23

22.3-13-17 2.1_ -1' 24-167 2-3'-149 22-673

265 ~

.

266 267 268 269

_

~

3-5.167 5-503 52.101 3-5-13' 5-509

~

270

22.38. .5t

272 273 274

2.51271 2'.5.17 2-3-5-7.13 22-5-137

37-73 2711 3.907 2731 2741

275 276 277 278 279

2.58.11 2'.3-5.23 2.5.277 2'.5.139 2-3'.5.31

3-7.131 ~-~ 11.251 17.163 3"103 2791

280 281 282 283 284

24.52.7 2.5.281 2.3.5.47 2.5-283 2'.5-71

2801 3-937 7.13.31 19-149 3.947

285 286 287 288 289

2.3.FiZ-19

290 1 29 ~. 292 293 294

22.fi2.29 2.315.97 23.5-73 2.5.293 22.3-5.72

3.987 - _._ 41.71 23.127 3.977 17-173

2.1451 25.7-13 2-3.487 22-733 2-1471

2903 3.971 37.79 7.419 33-109

23.3-11 2 2-31.47 22-17-43 2.3'.163 27-23

5-7.83 5-11.53 31-52.13 5.587 5.19.31

2.1453 22-30 2.7.11.19 23.367 2.3.491

32.17.19 2917 2927 3.11.89 7.421

22.727 2.1459 2'.3-61 2.13.113 22.11.67

295 296 297 298 299

2.52-59 2'.5-37 2.33.5.11 22.5.149 2.5-13.23

13.227 32-7-47 2971 11-271 3.997

23.32-41 2-1481 2'.743 2.3.7.71 2'- 11-17

2953 2963 3-991 19-157 41-73

2.7.211 2'.3.13.19 2.1487 23.373 2.3.499

3.5-197 5.593 52.7.17 3-5-199 5.599

22.739 2-1483 25.3.31 2.1493 22-7-107

2957 3.23.43 13.229 29.103 34-37

2.3.17.29 23-7.53 2-1489 22.3243 2.1499

5ii

~

~

2.7.193 ~. ~ . 2'-3.1 13 2-1361 22.683 2-3.457

23.313 2-3.419 - ~~. 22.631 2-7-181 2'.3.53

26.43 2-1381 22.3'-7.11 2-13.107 2'.349

_2851 _ _-

2861 32-11-29 43-67 72-59

_

~

1 1.289 ~ . ~~

2969 32-331 72-61 2999

.

c;l

o"4 00 N

0

1

2

3

4

5

6

7

8

9

N S E (P

300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 236 337 338 339 340 341 342 343 344 345 346 347 348 349

2a.3.5' 2.5.7-43 22-5.151 2.3.5.101 2'-5-19 2-51-61 22.3'-5-17 2-5-307 2'*5.7.11 2.3-5.103 22-9-31 2-5-311 2'-3.5.13 2-5-313 4-5-157 2.3-52-7 2'-5-79 2-5.317 22.3.5.53 2-5-11.29 27.9 2-3.5-107 2'-5-7-23 2-5-17-19 2'-3'-5 2-5a-13 22-5.163 2-3.5.109 2'-5.41 2-5.7.47 22-3.52.11 2.5-331 2'-5.83 2-3'-5-37 22.5-167 2-52.67 25-3.5-7 2-5.337 22-5.13' 2-3.5-113 2'-52-17 2.5-11.31 22-3'.5-19 2-5-72 2'.5-43 2-3-52.23 4-5-173 2-5-347 2'-3.5-29 2.5-349

3001 3011 3-19-53 7-433

3041 3'.113

3061 37.83 3.13.79 11.281 7.443 3-17-61

3121 31-101 3-349 23-137 29.109 3-7-151

3181 3191 3-11-97 13'-19

3221 35-359 7.463

3251 3.1087

3271 17-193 3-1097

3301 7-11-43 3'-41

3331 13-257 3-1117

3361 3371 3-7"23

3391 19.179 32-379 11.311 47-73 3-31.37 7-17-29

3461 3-13-89 59

3491

2-19-79 4.3-251 2.1511 2l.379 2-32-13' 22-7-109 2.1531 2'0.3 2.23-67 22.773 2.3-11-47 2'.389 2-7-223 22-3l-29 2.1571 2'-197 2-3.17-31 4-13.61 2-37-43 28-3-7-19 2.1601 22.11.73 2-3'-179 25.101 2-1621 22-3-271 2.7-233 2a-409 2-3-547 22-823 2-13-127 2'-3'.23 2-11.151 22-7'*17 2-3-557 21.419 2.411 22-3-281 2-19-89 28-53 2-35-7 22.853 2-29.59 2"3-11-13 2.1721 22.863 2-3-577 Y.7-31 2.1741 9-32-97

3.7-11-13 23-131

3023 32.337 17.179 43-71 3-1021 7.439

3083 3.1031 29-107 11.283 32-347 13.241 7-449 3.1051

3163 19.167 3.1061 31-103

3203 38-7-17 11-293 53-61 3-23-47

3253 13.251 3-1091 72-67 37-89 3'-367

3313 3323 3-11.101

3343 7-479 3-19-59

3373 17.199 3'*13-29 41.83

3413 3-7-163

3433 11.313 3-1151

3463 23.151 34-43 7-499

22.751 2.11-137 24-3-3-7 2-37-41 22.761 2-3-509 2'.383 2-29-53 22-3.257 2-7.13-17 25-97 2.3'-173 22.11-71 2.1567 25-3.131 2-19.83 22.7-113 2-3-23' 2'.199 2.1597 22-3'-89 2.1607 2"13.31 2-3.7'-11 22.811 2.1627 28.3-17 2-1637 21-821 2.33-61 23.7.59 2-1657 22-3.277 2-1667 2'-11.19 2-3-13.43 P.292 2.7-241 2'-3'-47 2.1697 22.23-37 2-3-569 25-107 2-17.101 2'.3.7.41 2.11.157 2'-433 2-3'.193 22-13-67 2-1747

5-601 3'-5.65 52.112 5-607 3-5.7-29 5-13-47 5-613 3-52.41 5.617 5-619 3a-5-23 5.749 55 3.5-11-19 5-17-37 5.631 3-5-211 52-127 5-7'-13 38-5-71 5-641 5.643 3-52.43 5.647 5-11-59 3-5-7-31 5-653 52.131 32-5-73 5-659 5.661 3-5-13.17 52-7-19 5.23.29 3-5-223 5.11.61 5-673 3a-53 5-677 5-7-97 3.5.227 5.683 5'*137 3-5.229 5.13.53 5-691 3'-5-7-11 52-139 5-17-41 3.5-233

2.32.167 2'.13-29 2-17-89 22-3-11-23 2.1523 2'.191 2-3.7-73 22.769 2.1543 2'.3'-43 2-1553 22-19-41 2-3.521 26-72 2.112-13 22.3-263 2-1583 23*397 2-P.59 22-17.47 2-7.229 2'-3-67 2.1613 2'4309 2-3.541 2"11-37 2-23-71 2'.3'-7.13 2.31-53 25.103 2-3-19-29 22-829 2-1663 23-3.139 2-7-239 2'.839 2.32.11-17 24-211 2-1693 22-3-283 2-13-131 2'-7-61 2-3-571 21-859 2.1723 27-38 2.1733 2*.11*79 2-3-7433 2'-19*23

31-97 7-431 3-1009

3037 11-277 3-1019

3067 17.181 3'.78 19-163 13-239 3.1039 53-59

3137 3.1049 7.11.41

3167 32-353

28-47 2.3-503 2'-757 2-7r-31 2'-3.127 2.1 1.139 2'-13.59 2.3'.19 2'.193 2.1549 2'.3.7*37 2.1559 23.17-23 2.3-523 22-787 2-1579 25-32-11 2-7-227 2'-797

3187 23-139 3-1069

3217 7.461 3-13-83 17.191

3257 35-11' 29.113 19-173 3.7.157

3307 31-107 3-1109 47-71

3347 32.373 7-13-37 11.307 3-1129 43-79

3407 3.17-67 23-149 7.491 31.383

3457 3467 3-19.61 11.317 13-269

2.3-13-41 23.401 2.1609 2'.3.269 2-1619 2'.7-29 2*3'-181 22.19-43 2.1 1.149 2'.3.137 2.17-97 2'*827 2.3-7.79 28-13 2-1669 22.3"31 2-23.73 2-3.421 2.3-563 22-7.112 2-1699 2'-3-71 2.1709 22.857 2-32.191 2'.431 2-7-13-19 22-3-1 7' 2-37-47 2'.109 2.3.11-53

3-17-59

3019 13-233 3.1013

3049 7-19-23 3'-11-31

3079 3089 3-1033

3109 3119 3.7.149 43-73 47.67 3'013

3169 11.172 3-1063 7-457

3209 3.29.37

3229 41.79 3'.19

3259 7-467 11.13.23 3.1093

3299 3.1103

3319 3329 32-7.53 17.197

3359 3.1123 31.109

3389 3.11.103 7.487 13-263 3'.127 19-181

3449 3.1153

3469 P.71 3.1163

3499

300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 33 1 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 g

8

350 351 352 353 354

22.53-7 2.33-5.13 28-5.11 2-5.353 22.3-5-59

32-389 3511 7-503 3-11-107 3541

2-17-103 2"439 2-3.587 22-883 2.7.11.23

31.113 3.1171 13-271 3533 3.1181

24.3.73 2.7.251 22-881 2.3.19.31 2'.443

5.701 5-19-37 3-52-47 5.7-101 5.709

2.1753 22-3.293 2-41-43 24.13.17 2.3'-197

3.7-167 3517 3527 33.131 3547

22.877 2-1759 23.32.72 2.29.61 2'.887

1 12.29 3'-17.23 3529 3539 3-7.13'

350 351 8 352 353 354

355 356 357 358 359

2.9.71 2"5-89 2-3-5.7-17 2'-5.179 2-5-359

53.67 3-1187 3571 3581 3'-7.19

2'.3-37 2-13-137 22-19-47 2.3'.199 21.449

11.17-19 7-509 32.397 3583 3593

2.1777 22.3'.11 2-1787 29-7 2.3.599

32-5.79 5.23.31 5'. 11.13 3.5-239 5.719

22.7-127 2.1783 2'.3.149 2-11.163 22-29.31

3557 3-29.41 72-73 17.211 3.11.109

2.3-593 2'.223 2.1789 2'.3.13.23 2-7.257

3559 43.83 3.1193 37.97 59.61

355 356 357 358 359

360 361 362 363 364

2'.32.53 2.5-19 22.5-181 2.3-5.11' 28.5.7.13

13.277 23-157 3-17-71 3631 11.331

2.1801 21-3-7-43 2.1811 2'-227 2-3.607

3-1201 3613 3623 3.7-173 3643

22.17-53 2-13.139 23.3.151 2-23.79 22-911

5.7.103 3.5-241 53-29 5.727 38-5

2.3.601 25.113 2*7'.37 22.32.101 2.1823

3607 3617 3'.13-31 3637 7.521

2'.11.41 2-33.67 22.907 2-17.107 26.3.19

8'.401 7.11-47 19.191 3.1213 41-89

360 361 362 363 364

365 366 367 368 369

2.9.73 22-3.5-61 2-5.367 25.5.23 2-32.5-41

3.1217 7.523 3671 3'-409 3691

22.11.83 2-1831 2'-3'.17 2-7-263 2'.13.71

13-281 32.1 1.37 3673 29-127 3-1231

2-3'-7-29 2'-229 2-11-167 22.3.307 2.1847

5.17-43 5.733 3.52.72 5-11.67 5-739

23.457 2.3.13.47 22.919 2.19.97 2'.3.7.11

3.23-53 19-193 3677 3.1229 3697

2.31.59 2'.7.131 2.3.613 2'.461 2.432

3659 3-1223 13-283 7-17.31 33.137

365 366 367 368 369

370 371 372 373 374

2'052.37 2.5-7-53 28.3.5.31 2-5.373 22.5-11-17

3701 3-1237 612 7-13-41 3.29-43

2.30617 27-29 2.1861 22-3-311 2-1871

7-23' 47.79 3-17-73 3733 19-197

2'.463 2.3-619 2'-7'- 19 2.1867 25.32.13

3.5.13.19 5.743 52.149 3'.5.83 5.7-107

2-17.109 2'.929 2.3'.23 23.467 2-1873

11-337 32.7.59 3727 37-101 3-1249

22.32.103 2.1 1.132 2'.233 2.3.7.89 22.937

3709 3719 3.11.113 3739 23.163

370 371 372 373 374

375 376 377 378 379

2.3.5' 2'-5-47 2-5.13.29 22.33.5.7 2-5.379

112-31 3761 32.419 19-199 17-223

2"7-67 2-32-11-19 22.23-41 2.31-61 2'.3-79

3a-139 53-71 7'.11 3-13.97 3793

2-1877 22.941 2.3-17.37 2a-11-43 2.7-271

5.751 3-5-251 52.151 5.757 3-5.11-23

22.3.313 2.7.269 28-59 2.3-631 22-13.73

13-17' 3767 3.1259 7.541 3797

2.1879 2'-3.157 2.1889 2'.947 2-32.211

3-7.179 3769 3779 32-421 29-131

375 376 377 378 379

380 381 382 383 384

2a-53-19 2-3-5.127 22.5-191 2.5-383 28.3-5

3-7-181 37-103 3821 3-1277 23-167

2-1901 2'-953 2.3-72-13 28-479 2-17-113

3803 3-31-41 3823 3833 32-7-61

22.3-317 2.1907 2'-239 2-3'-71 22.312

5-761 5-7.109 32.52-17 5.13.59 5.769

2.1 1.173 2'-32.53 2.1913 22.7.137 2.3.641

34.47 11-347 43.89 3-1279 3847

25.7.17 2-23-83 22.3.11-29 2.19.101 23.13.37

13.293 3-19.67 7.547 11.349 3.1283

380 381 382 383 384

385 386 387 388 389

2.53.7-11 22-5-193 2-32-5-43 2'-5.97 2.5.389

3851 3'- 11-13 72-79 3881 3.1297

22.32.107 2-1931 25.112 2-3-647 22.7.139

3853 3863 3-1291 11-353 17-229

2.41-47 2'-3-7.23 2.13-149 2'.971 2.3-11-59

3.5-257 5.773 53.31 3.5.7-37 5-19-41

24.241 2-1933 22.3.17.19 2.29-67 23.487

7.19.29 3-1289 3877 132.23 32.433

2.3.643 2'.967 2.7.277 24-35 2.1949

17-227 53.73 32.431 3889 7.557

385 386 387 388 389

390 391 392 393 394

22-3-9-13 2-5.17-23 24.5.72 2-3-5-131 3-5-197

47-83 391 1 3-1307 3931 7.563

2.1951 2'-3-163 2.37-53 22.983 2-3"73

3-1301 7-13-43 3923 32-19-23 3943

2l'-61 2.19-103 'La.32.109 2.7-281 2'-17-29

5.11.71 3"5-29 52.157 5-787 3-5.263

2.32.7.31 22-11449 2.13-151 25.3-41 2.1973

3907 3917 3-7-11.17 31.127 3947

22.977 2.3.653 2'.491 2-11.179 22.3-7.47

3-1303 3919 3929 3-13.101 11-359

390 391 392 393 394

395 396 397 398 399

2-53-79 2'. 32.5.11 2.5.397 22-5-199 2.3.5.7-19

32-439 17-233 11.19 3-1327 13-307

2'.13-19 2.7-283 22-3-331 2-11-181 2'-499

59.67 3.1321 29.137 7-569 3.11'

2-3.659 22.991 2.1987 2'-3-83 2.1997

5.7-113 5.13.61 3.52.53 5.797 5-17-41

22.23-43 2-3.661 28-7-71 2.1993 2'-3"37

3.1319 3967 41.97 32.443 7.571

2.1979 2'.31 2.32-13-17 22.997 2.1999

37.107 34.72 23-173 3989 3-31.43

395 396 397 W N 00 398 w rp Q, 3 9 9 g j C'L

400 40 1 402 403 404 405 406 407 408 409 410 41 1 412 413 414 415 416 417 418 419 420 42 1 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 44 1 442 443 444 445 446 447 448 449

25.53 2.5-401 22-3-5.67 2.5-13-31 2'-5-101 2.3'.52 22-5-7-29 2.5-11.37 2'.3.5.17 2.5.409 22.52-41 2.3.5.137 23-5.103 2.5-7.59 22.32.5.23 2.52433 26.5.13 2.3.5-139 22-5.11.19 2.5-419 2"3.52.7 2.5.421 22.5-211 2.32-5.47 2'.5-53 2.58.17 22.3.5.71 2.5.7.61 2'.5-107 2.3.5.11-13 22-52-43 2.5.431 25.3'.5 2-5.433 22.5.7-31 2-3.52-29 28.5-109 2.5.19-23 2'.3.5.73 2-5.439 24.52.11 2.32-5.72 22.5.13.17 2.5-443 2'.3-5-37 2.52-89 22-5-223 2-3-5.149 27.5-7 2.5.449

4001 3-7-191 4021 29.139 32-449 4051 31-131 3-23.59 7.11.53 4091 3-1367 4111 13.317 35-17 41.101 7.593 3.19.73 43-97 37.113 3-11.127 4201 4211 32-7.67 4231 4241 3-13-109 4261 4271 3.1427 7.613 11-17-23 32.479 29-149 61-71 3.1447 19.229 72*89 3-31-47 13.337 4391 3'.163 11.401 4421 3.7.211 4441 4451 3.1487 17-263 4481 32.499

2-3-23-29 22-17-59 2.2011 2'-3'-7 2.43.47 22.1013 2.3.677 23-509 2-13.157 2'.3-11.31 2-7.293 2'-257 2.32-229 3-1033 2-19.109 2a-3-173 2-2081 22-7-149 2.3-17.41 25.131 2-11-191 22-3'- 13 2-2111 28.232 2.3.7-101 22-1063 2.2131 2'-3-89 2.2141 22-29-37 2.32.239 2J.72.11 2-2161 22-3-19 2.13-167 26-17 2.3.727 22.1093 2.7-313 28-32-61 2-31-71 2'-1103 2.3.11.67 2'-277 2-2221 22-3.7-53 2-23-97 2'.13-43 2-3'43 22.1123

4003 4013 3'-149 37.109 13.311 3.7-193 17.239 4073 3-1361 4093 11-373 32.457 7-19-31 41 33 3.1381 4153 23-181 3-13-107 47-89 7.599 3'.467 11.383 41-103 3-17.83 4243 4253 3-72-29 4273 4283 34.53 13.331 19-227 3.1 1.131 7.619 43-101 3.1451 4363 4373 3'-487 23-191 7-17-37 3.1471 4423 11.13.31 3.1481 61.73 4463 3'-7-71 4483 4493

22.7-11-13 2-3'-223 2'.503 2-2017 22.3.337 2.2027 25-127 2-3.7-97 22-1021 2.23-89 2'.3'.19 2.112.17 22.1031 2-3-13-53 2'-7-37 2.31.67 22-3.347 2.2087 2'.523 2.32.233 22-1051 2.72-43 27.3.11 2.29.73 2'- 1061 2.3-709 23-13.41 2-2137 22.3'.7-17 2.19-113 2'.269 2.3-719 22-23.47 2.11.197 2'-3-181 2.7.311 22.1091 2.37 25.137 2-13' 2'.3.367 2.2207 23.7.79 2 - 3.739 22.11.101 2.17.131 2'.3'.31 2.2237 2'- 19-59 2.3.7.107

32.5-89 5.11.73 52.7-23 3.5-269 5-809 5.811 3-5.271 52.163 5.19-43 32-5-7.13 5.821 5-823 3.53-11 5-827 5.829 3-5.277 5.72.17 52.167 3'.5.31 5.839 5-29 3.5.281 52.13' 5.7-11' 3-5-283 5.23-37 5.853 32.52.19 5.857 5-859 3.5.7.41 5.863 5l.173 3.5.172 5.11-79 5.13.67 32-5.97 54.7 5.877 3-5-293 5.881 5.883 3.52.59 5.887 5.7.127 3'.5.11 5-19-47 52-179 3.5.13.23 5.29.31

2.2003 2'.251 2-3.11-61 2'.1009 2.7.17' 2'.3.132 2.19.107 2'-10 19 2.32-227 2'2 2-2053 22.3.73 2.2063 23-11.47 2.3.691 22.1039 2.2083 2'. 32.29 2-7.13.23 22-1049 2-3.701 23.17.31 2.2113 2'.3.353 2-11-193 25-7.19 2.33.79 22.1069 2.2143 2'.3.179 2-2153 2 2 - 13-83 2.3.7-103 2'.271 2-41-53 22.32.112 2.37-59 2'.547 2.3.17.43 22.7.157 2.2203 26-3.23 2.2213 22.1109 2-3'-13-19 2'.557 2.7.11-29 22.3.373 2.2243 24.281

4007 3.13.103 4027 11-367 3.19.71 4057 7'.83 3'.151 61.67 17-241 3.372 23-179 4127 3-7-197 11.13.29 4157 32.463 4177 53.79 3.1399 7-601 4217 3-1409 19-223 31.137 3'. 11.43 17.251 7.13.47 3.1429 4297 59-73 3.1439 4327 4337 3'.7.23 4357 11.397 3-1459 41.107 4397 3.13-113 7.631 19-233 32-17.29 4447 4457 3-1489 112-37 7.641 3-1499

23.3.167 2-72-41 22.19.53 2.3.673 2'-11-23 2-2029 2'.32.113 2.2039 23.7.73 2-3.683 22.13.79 2.29.71 25.3.43 2-2069 2'-17.61 2.3'-7.11 2'.521 2.2089 2'.3.349 2.2099 2'.263 2-3.19-37 2'.7.15 1 2.13.163 2'.32.59 2.2129 22.11.97 2.3.23-31 28.67 2.7.307 22.3.359 2.17-127 23.541 2.32.241 22.1087 2.2179 2'.3.7.13 2.1 1.199 22.1097 2.3.733 2'.19-29 2.472 22.33.4 2.7.317 1 25-139 2.3.743 22.1117 2.2239 23.3.11.17 2.13-173

19-211 4019 3.17.79 7.577 4049 32.1 1-41 13-313 4079 3-29.47 4099 7.587 3.1373 4129 4139 3'.461 4159 11-379 3.7-199 59-71 13.17.19 3.23-61 4219 4229 3'.157 7.607 4259 3.1423 11.389 4289 3.1433 31.139 7.617 3'.13.37 4339 4349 3.1453 17.257 29.151 3.7.11.19 53.83 4409 32.491 43.103 23.193 3.1483 73.13 41.109 3.1493 672 1 1.409

400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 $ 449 8

450 451 452 453 454

22.32.5' 2-5-11.41 2'-5-113 2-3.5.151 22.5-227

7-643 13.347 3-11-137 23.197 19-239

2.2251 25-3-47 2.7-17-19 21.11-103 2-3-757

3-19-79 4513 4523 3.1511 7-11-59

23-563 2-37-61 21-3-13-29 2-2267 26-71

5.17-53 3-5-7.43 5a-181 5-907 3'-5-101

2-3.751 2'-1129 2-31-73 2'-3'.7 2-2273

4507 4517 3'.503 13.349 4547

21.72-23 2.3?-251 2'-283 2.2269 2'.3.379

33-167 4519 7.647 3-17-89 4549

453 454

455 456 457 458 459

2-52.7-13 2-3.5-19 2-5.457 21-5-229 2-3"5-17

3-37-41 4561 7-653 3'-509 4591

3"569 2-2281 2'.3'- 127 2-29-79 2'*7-41

29.157 3'-132 17.269 4583 3-1531

2-3'-11-23 21.7-163 2.2287 2'*3-191 2-2297

5.91 1 5.11-83 3.52-61 5.7-131 5.919

2'-17-67 2.3.76 1 25-11.13 2.2293 2'-3-383

3.7'-31 4567 23.199 3.1 1.139 4597

2.43.53 23.571 2.3.7.109 2'.31-37 2.1 12.19

47.97 3-1523 19-241 13-353 32.7.73

455 456 457 458 459

460 461 462 463 464

2W-23 2-5-461 22-3-5-7-11 2.5-463 2'-5-29

43-107 3-29-53 4621 11.421 3-7-13-17

2.3.13-59 P.1153 2-2311 2'-3.193 2.11.211

4603 7-659 3-23-67 41-113 4643

3.1151 2-3-769 2'. 17' 2-7-331 22-33-43

3-5.307 5-13-71 58-37 3'-5.103 5.929

2-72-47 2'-577 2-32-257 2'.19.61 2.23.101

17-271 35-19 7-661 4637 3.1549

29-32 2.2309 22.13.89 2-3.773 23-7.83

11-419 31-149 3.1543 4639 4649

460 461 462 463 464

465 466 467 468 469

2-3-9-31 9.5.233 2-5-467 2a-32*5-13 2-5-7-67

4651 59.79 3"173 31-151 4691

21.1163 2.32-7.37 28-73 2-2341 22-3.17-23

3"11-47 4663 4673 3-7.223 13-19'

2-13-179 2'. 11.53 2.3.19.41 21.1171 2-2347

5-7'-19 3.5.311 5'. 11-17 5.937 3-5-313

24.3-97 2.2333 22-7-167 2.3-11.71 23-587

4657 13-359 3-1559 43.109 7-11-61

2.17.137 2'.3-389 2.2339 2'.293 2-3'.29

3.1553 7.23.29 4679 32.521 37.127

465 466 467 468 469

470 471 472 473 474

21-55-47 2.3-5-157 24-5-59 2.5-11-43 3-3-5079

3-1567 7-673 4721 3-19.83 11.431

2-2351 2'-19*31 2-3-787 2'.7-13' 2.2371

4703 3-1571 4723 4733 32-17-31

25.3.7 2.2357 22-1181 2-32-263 2'-593

5.941 5.23.41 35-52.7 5-947 5.13.73

2-13-181 2'-3'.131 2.17-139 2'-37 2.3.7.113

32.523 53.89 29-163 3,1579 47-101

2'. 11.107 2-7-337 F.3-197 2.23.103 22.1187

17.277 3.1 12-13 4729 7-677 3-1583

470 471 472 473 474

475 476 477 478 479

2-5'.19 2*-5-7-17 2-3'-5-53 22-5-239 2.50479

4751 32.23' 13.367 7-683 3-1597

2'.3'.11 2-2381 2'.1193 2-3-797 2"599

72.97 11-433 3.37.43 4783 4793

2-2377 21.3-397 2.7-1 1-31 2'-13-23 2.3.17-47

3.5.317 5.953 5'.191 3.5-11-29 5.7.137

21-29.41 2-2383 23.3.199 2.2393 22.11.109

67.71 3-7.227 17-281 4787 3'-13.41

2.3-13-61 25.149 2.2389 21.32.7-19 2-2399

4759 19.251 34-59 4789 4799

475 476 477 478 479

480 481 482 483 484

2'-3-52 2.5.13-37 22.5.241 2-3-5.7.23 2*.5-112

4801 17-283 3-1607 4831 47-103

2*74 2"3-401 2.2411 2"151 2-3'-269

3.1601 4813 7-13-53 35.179 29.167

22-1201 2-29-83 23.32-67 2.2417 21-7.173

5.31' 3'.5.107 5'.193 5.967 3.5.17.19

2-33.89 2'-7.43 2-19-127 2'.3.13-31 2.2423

11-19.23 4817 3.1609 7-691 37.131

23-601 2-3-11-73 22.17.71 2.41.59 2'.3.101

3.7.229 61-79 11-439 3-1613 13.373

480 481 482 483 484

485 486 487 488 489

2-5'-97 21.35.5 2.5.487 2'-5.61 2.3-5-163

32-72.11 4861 4871 3-1627 67.73

22.1213 2-11-13.17 25.3-7-29 2-2441 21.1223

23-211 3.1621 11-443 19-257 3-7-233

2-3-809 28-19 2.2437 3 - 3 - 11-37 2.2447

5-971 5.7.139 3.P.13 5.977 5-11-89

2'-607 2.3-811 P.23-53 2.7-349 25.32.17

3-1619 31-157 4877 3'-181 59.83

2.7.347 2'-1217 2.3'.271 23-13.47 2.31-79

43-113 3'-541 7-17-41 4889 3.23-71

485 486 487 488 489

490 49 1 492 493 494

21.55-P 2-5-491 2'-3-5-41 2-5-17-29 3.5.13.19

13'029 3-1637 7-19.37 4931 3'-61

2-3.19.43 2'0307 2.23-107 21-3'-137 2-7-353

4903 175 32.547 4933 4943

2'*613 2-3'-7-13 21-1231 2.2467 2'-3-103

3'.5-109 5.983 51-197 3-5.7-47 5-23-43

2-11.223 2'-1229 2-3.821 2'.617 2-2473

7-701 3.1 1-149 13.379 4937 3.17.97

2'-3.409 2.2459 2O.7-11 2-3-823 2'-1237

4909 4919 3.31.53 11.449 '12.101

490 491 492 493 494

495 496 497 498 499

2-3'S.ll P-5.31 2.5.7-71 21-3.5.83 2-5-499

4951 11'-41 3-1657 17-293 7-23-31

2'-619 2.3-827 21.11-113 2.47* 53 2'-3-13

3-13-127 7-709 4973 3-11-151 4993

2-2477 21-17-73 2.3-829 2'07.89 2-11.227

5.991 3-5.331 55-199 5.997 33-5.37

21.3-7-59 2-13-191 2'-311 2.3'-277 21.1249

4957 4967 32.7.79 4987 19-263

2.37-67 23.38.23 2.19.131 22-29-43 2.3.72.17

3'-19.29 4969 13-383 3.1663 4999

495 496

N

0

1

2

3

4

5

6

7

8

9

g2 E *

N a& (P

500 501 502 503 504

23-54 2-3.5-167 22-5-251 2-5-503 24-32*5.7

3-1667 5011 5021 32-13.43 71,

2.41-61 22-7-179 2-34.31 2-17-37 2-2521

5003 32-557 5023 7-719 3.412

22.32-139 2.23-109 25. 157 2-3-839 3.13-97

5-7-11.13 5.17-59 3-52-67 5.19.53 5.1009

2.2503 23-3.11-19 2-7-359 3.1259 2-3-292

3-1669 29.173 11.457 3-23.73 72. 103

24.313 2-13-193 3-3-419 2.11-229 23-631

5009 3-7.239 47.107 5039 3'.11-17

500 501 502 503 504

505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549

2-5a-101 22-5-11-23 2.3.5.13' 21-5.127 2.5.509 3.3-52.17 2.5-7-73 21O.5 2-38-5-19 22.5.257 2-P-103 23.3-5.43 2-5.11.47 22-5-7.37 2-3.5-173 24-52-13 2-5-521 22.3'.5.29 2.5.523 21-5-131 2.3.5a.7 22-5.263 2-5.17-31 2'.3-5-11 2.5-232 3.52-53 2-32-5-59 23-5.7.19 2-5.13-41 22-3-5-89 2-52.107 24.5.67 2-3-5.179 22.5-269 2-5-72.11 21-33-52 2.5.541 22.5.271 2.3-5-181 26.5-17 2.52.109 22-3-5.7-13 2.5.547 23-5.137 2.32.5-61

5051 3.7.241 11.461 5081 3.1697 5101 19.269 3'-569 7.733 53.97 3-17.101 13-397 5171 3-11-157 29.179 7.743 3.193 23.227 5231 3-1747 59-89 5261 3-7-251 5281 11-13-37 32.19.31 47-113 17.313 3.1777 72.109 5351 3-1787 41.131 5381 32.599 11.491 7.773 3-13.139 5431 5441 3-23-79 43-127 5471 3'-7-29 172.19

22-3.421 2-2531 2'.317 2-3-7.11' 22-19-67 2-2551 21.32.71 2-13-197 22.1283 2-3.857 25-7-23 2-29-89 22-3-431 2-2591 23-11.59 2-32-172 22-1303 2-7-373 24.3-109 2-2621 22-13.101 2.3.877 3.659 2-19.139 3.33.72 2-11.241 28.83 2.3-887 22.31-43 2-2671 23-3-223 2.7-383 3.17-79 2-32-13.23 24.337 2-37.73 22.3-11-41 2-2711 23.7-97 2.3-907 22.29-47 2-2731 26.32-19 2.2741 22.1373

31-163 61-83 3-19-89 13-17-23 11.463 36-7 5113 47-109 3-29.59 37-139 5153 3-1721 7-739 71.73 32.577 112-43 13.401 3-1741 5233 72-107 3-17-103 19-277 5273 32.587 67-79 5303 3-7-11-23 5323 5333 3-13-137 53-101 31-173 33.199 7-769 5393 3-1801 5413 11-17-29 3.1811 5443 7.19.41 32.607 13-421 5483 3-1831

2-7.192 23.3.211 2.43-59 3.31-41 2.32.283 2'-11-29 2-2557 22-3.7.61 2.17-151 23-643 2.3-859 22.1291 2.13.199 26-34 2.72.53 22.1301 2.3-11-79 23-653 2-2617 22-3.19-23 2.37-71 24-7.47 2.32.293 22.1321 2-2647 23-3.13.17 2.2657 22-I 13 2.3.7.127 25-167 2.2677 22.32.149 2-2687 23-673 2-3-29-31 22-7-193 2.2707 2'-3-113 2.11.13.19 22.1361 2,33.101 23.683 2.7- 17-23 22-3-457 2.41-67

3-5-337 5.1013 52-7.29 32.5.113 5.1019 5.1021 3-5-11.31 53.41 5.13-79 3-5-73 5.1031 5-1033 32-52.23 5-17-61 5-1039 3-5.347 5.7.149 911.19 3.5.349 5-1049 5.1051 3'-5-13 52.211 5.7-151 3.5.353 5.1061 5. i063 3.52.71 5.11.97 5-1069 32.5.7-17 5-29-37 53.43 3-5-359 5.13443 5-23-47 3-5-192 52.7.31 5-1087 32.5-112 5.1091 5.1093 3.52.73 5-1097 5-7-157

2"79 2-17.149 3.33.47 2-2543 23-72-13 2.3-23-37 3-1279 2.1 1.233 2'.3.107 2-31.83 22.1289 2.32-7-41 23-647 2-2593 22.3.433 2.19-137 2'-163 2-3-13.67 22.7.11-17 2-43-61 3.32-73 2.2633 22-1319 2-3-881 2'.331 2.7.379 22.3-443 2-2663 23-23.29 2.35.11 22.13-103 2.2683 28.3.7 2.2693 22.19-71 2.3.17.53 23-677 2.2713 22.32-151 2.7,389 2'. 11.31 2.3-911 3.372 2-13.211 3-3.229

13-389 32.563 5077 5087 3.1699 5107 7.17.43 3.1709 11-467 5147 33.191 5167 31-167 3-7.13-19 5197 41-127 3.37.47 5227 5237 3 2 - 11.53 7.751 23.229 3.1759 17.311 5297 3.29.61 13-409 7.761 32-593 5347 11-487 3.1789 19.283 5387 3.7-257 5407 5417 3'-67 5437 13.419 3-17-107 7-11-71 5477 3-31.59 23-239

2.32-281 22-7-181 2-2539 25.3-53 2-2549 22.1277 2.3.853 23.641 2.7.367 22-32-11-13 2.2579 24.17-19 2.3.863 22-1297 2-23.113 23-3.7-31 2-2609 22.1307 2.33.97 2'- 41 2-11-239 22-3.439 2.7.13.29 23-661 2.3.883 22.1327 2.2659 24.32-37 2.17.157 22-7.191 2-3.19-47 3-11-61 2.2689 22.3.449 2.2699 25-1 3 2 2-3'-7.43 22-23-59 2.2719 23.3.227 2.2729 22.1367 2.3.11.83 2'.73 2-2749

5059 37.137 3-1693 7.727 5099 3.13.131 5119 23-223 32.571 19.271 7-11-67 3-1723 5179 5189 3.1733 5209 17-307 32.7-83 132.31 29-181 3-1753 11*479 5279 3.41.43 7.757 5309 33-197 732 19-281 3-1783 23.233 7-13-59 3.11-163 17-317 5399 32.601 5419 61.89 3-72.37 5449 53. i03 3-1823 5479 11.499 32.13-47

505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 . ~ . 546 547 548 549 g

550 g 551 g 552 553 554

550 551 552 553 554

22-5'*11 2-5-19-29 2'-3-5.23 2-5.7-79 22-5-277

5501 3-11-167 5521 5531 3-1847

2-3-7-131 28-13.53 _ ~- ._ 2-11.251 3-3-461 2-17-163

5503 37-149 3.7-263 11-503 23.241

2'-43 2.3-919 2'.1381 2.2767 2'.3'-7-1 1

3-5.367 5.1103 P.13.17 3'-5-41 5.1109

2.2753 22-7-197 2.3'.307 25.173 2.47.59

5507 3'.613 5527 72.113 3.43'

22-3'.17 2.31-89 2'-691 2.3.13-71 22-19-73

7-787 5519 3-19-97 29-191 31-179

555 556 557 558 559

2.3-52-37 2"5-139 2-5-557 22.32.5.31 2.5.13-43

7-13-61 67-83 9.619 5581 5591

2'-347 2-38-103 2'-7-199 2.2791 2'*3.233

3'-617 5563 5573 3-1861 7-17.47

2.2777 22-13-107 2-3-929 2'.349 2.2797

5.11.101 3-5.7-53 5'-223 5.1117 3-5.373

2'-3-463 2.1 P.23 2'017-41 2-3.7'-19 2'. 1399

5557 19-293 3-1 1 . 1 3 2 37-151 29-193

2.7-397 26-3.29 2.2789 22-11.127 2.3'.311

3-17-109 5569 7.797 35.23 11.509

555 556 557 558 559

560 561 562 563 564

25.52.7 2-3-5.11-17 2'-5.281 2.5-563 2a.3.5.47

3-1867 31.181 7-11-73 3-1877 5641

2.2801 29.1I 2.7-13.31

13.431 3-1871 5623 43.131 3'- 11.19

2'.3-467 2-7.401 _ . ~.~ 2'019.37 2-3'-313 22-17-83

5-19-59 5.1123 32-54 5-72-23 5.1129

2.2803 2'3.13 2-29.97 2'. 1409 2.3.941

3'-7-89 41-137 ~ . 17-331 3-1879 5647

2'-701 2.532 2'.3.7-67 2.2819 2'.353

71.79 3-1873 13-433 5639 3-7-269

560 . ~ 561 562 563 564

565 566 567 568 569

2-5'-113 22-5-283 2.3'.5-7 2'.5-71 2.5.569

5651 32-17-37 53-107 13-19-23 3-7.251

22-32-157 2-19.149 2a-709 2-3-947 2'.1423

5653 7-809 3.3 1-61 5683 5693

2.1 1-257 25.3.59 2.3.13-73

3-5.13-29 - . 5.11-103 52.227 3.5-379 5.17-67

2'.7-101 2-2833 22.3.11.43 2.2843 26.89

5657 3-1889 7.811 112.47 3'.211

2.3-23-41 22.-13-109 2-17.167 23-32.79 2-7-11.37

5659 5669 3'-631 5689 41.139

565 566 567 568 569

3.1901 29.197 59.97 3'-72.13 5743

23-23.31 2.2857 22-33-53 2-47.61 2'.359

5.7-163 32.5-127 5'-229 5.31.37 3-5-383

2.32-317 2'.1429 2.7.409 2'.3-239 2-13'. 17

13.439 5717 35383 5737 7.821

2'.1427 2.3-9.53 _ - .__ 25.179 2.19-151 2'-3.479

3-11-173 7-19.43 . ~. __ 17-337 3-1913 5749

570 571 572 574

2.3.7.137 2"-11-131 2.2887 2'-3-241 2-2897

5.1151

2'-1439 2-3.312 24-19 2.1 1.263 22.32-7-23

3-19.101 73.79 53-109 3'.643 11.17.31

22.1447 2-13.223

13.443 32.641 5779 7-827 3.1933

578 579

5807 3.7.277 5827 13-449 3.1949

2'-3.112 2.2909 22.31.47 2-3.7.139 2j. 17-43

37.157 11-23' 3-29-67 5839 5849

580 581 582 583 584

~~

~~

.

570 571 572 573 574

22-3.52-19 2.5.571 2'-5. 11.13 2.3-5.191 2'.5-7-41

5701 5711 3-1907 11.521 5741

2-2851 2'.3-7.17 2.2861 2'-1433 2.32.11-29

575 576 577 578 579

2.5'-23 2'.3'-5 2.5.577 2'.5.17' 2.3-5.193

3'.71 7.823 29.199 3.41.47 5791

23-71 - . -9_

2-43-67 2'-3-13-37 2-7'-59 25.181

11-523 3-17.113 23-251 5783 3.1931

580 581 582 583 584

23-52.29 2.5.7.83 2'-3.5.97 2-5.11-53 2'.5.73

5801 3.13.149 5821 7'-17 32-11.59

2.3.967 22-1453 2.41.71 2'-3' 2-23.127

7.829 5813 3'-647 19-307 5843

22.1451 2-3'.17-19 26.7-13 2-2917 2'-3487

5'.233 3-5-389 5.7.167

2.2903 23.727 2-3.971 22-1459 2-37-79

585 586 587 588 589

2-3'.5'-13 2'-5.293 2.5.58'7 2'.3.5-72 2.5.19-31

5851 5861 3-19.103 5881 43-137

2'.7-11-19 2.3.977 2'.367 2-17.173 22-3.49 1

3.1951 11-13-41 7-839 3.37.53 71.83

2.2927 23-733 2.3.1 1-89 2'.1471 2.7.421

5-1171 3-5.17.23 53-47 5-11-107 32-5-131

25-3.61 2.7-419 2'- 13.113 2-33.109 23-11-67

5857 5867 3'-653 7-29 5897

2.29.101 22.32.163 2-293928-23 2.3.983

3'.7.31 5869 5879 3.13.151 17-347

585 586 587 588 589

590 59 1 592 593 594

2'.52-59 2-3.5-197 25.5.37 2-5-593 2"-33-5.1 1

3-7-281 23.257 31.191 3'.659 13.457

2.13.227 2'.739 2.3'.7.47 2'.1483 2.2971

5903 34.73 5923 17.349 3.7-283

24-32-41 2-2957 22.1481 2-3.23.43 2'-743

5.1181 5-7.13' 3.52-79 5.1187 5.29.41

2.2953 22.3-17-29 2.2963 2'.7-53 2-3.991

3.1 1.179 61-97 5927 3-1979 19.313

2'-7-211 2.11.269 2'-3- 13-19 2.2969 22-1487

19-311 3.1973 72.112 5939 32.661

590 591 592 593 594

595 596 597 598 599

2.52.7.17 23.5.149 2.3.5.199 2'.5.13-23 2-5.599

11.541 3.1987 7.853 5981 3-1997

26.3-31 2.11-271 2'.1493 2.3.997 2'.7.107

5953 67.89 3-11-181 31.193 13.461

2-13-229 22-3-7.71 2.29.103 25-11.17 2.3'.37

3.5-397 - . 5.1193 5'-239 32.5.7.19 5.11.109

2'*1489 2.19.157 23.32.83 2.41.73 22.1499

7.23-37 3'. 13.17 43.139 5987 3.1999

2.32-331 2'.373 2.7'-61 2a.3-499 2-2999

59.101 47-127 3.1993 53-113 '7.857

595 596

~

~~

~~

5-13-89 5.19.61 33.5-43

5.1163

~~

~~~

.

~

0

2 E;I gJ

z m

2

856 Table 24.7 6000

COMBINATORIAL ANALYSIS

Factoriza tions 6499

2.3251 2'- 11-37 2-3.1087 2'.23.71 2.3271

7-929 3-13.167 11-593 47.139 3'-727

23.3-271 2.3257 2'.7.233 2.33-112 2'-409

5.1301 5-1303 3'.5'-29 5.1307 5.7.11-17

2.3253 Z2.3'.181 2.13.251 2-'-19.43 2.3-1091

33.241 73.19 61.107 3-2179

6553 6563

3.133

23.32.7.13 2.17-193 22-31-53 2-3.1097 2".103

3-7-313 29.227 19.347

2.29.113 22-3.547 2.19.173 Z3-823 2.3.7-157

3.5.19-23 5.13.101 52-263 3.5.439 5.1319

Z3.3.5'.11 2.5.661 22-5-331 2-3.5.13-17 2'- 5.83

7.23.41 11.601 3.2207 19.349 29.229

2.3301 22.3-19-29 2.7.1 1.43 Z3.829 2.3*-41

3.31.71 17.389 37.1 79 32-11-67 7.13.73

22.13.127 2.3307 25-32.23 2.31.107 22.11.151

665 666 667 668 669

2.52.7.19 22.32.5.37 2.5.23.29 Z3.5-167 2-3.5.223

32.739

2'-1663 2-3331 2'.3-139 2.13.257 22.7-239

6653

670 671 672 673 674

22.5'.67 2-5.11.61 26.3.5.7 2.5.6 73 2'.5.337

675 676 677 678 679

2.33.53 23-5-132 2.5-677 2'-3.5.113 2.5.7-97

3.37-61

680 681 682 683 684

2' -5217 2.3.5.227 22-5.11.31 2-5-683 23.32.5.19

3.2267 72.139 19.359 3'-11-23

685 686 687 688 689

2.5'.137 22.5.73 2.3.5.229 25-5.43 2.5.13.53

13.17.31 3-2287

690 691 692 693 694

22.3.52.23 2.5.691 Z3.5.173 2.3'-5.7.11 22.5-347

67.103

695 696 697 698 699

2.5'+139 2'.3.5.29 2.5.17-41 2'.5.349 2.3.5.233

3.7.331

650 651 652 653 654

22.53-1 3 2-3.5.7-31 2'-5.163 2.5-653 2'.3.5-109

3-11-197 17.383

655 656 657 658 659

2.5'.131 25.5.41 2.3'.5-73 22-5.7.47 2.5.659

6551 6571 6581

660 661 662 663 664

6521

3-7.311 31.211 3*

6661

7-953 3-17.131 6691 6701

3.2237 11.13.47 53.127 3'-7.107 43.157 6761 6781 6791

684 1

6871

7.983 3.2297 6911

3'.769 29.239 11-631 6961 6971

3.13.179 6991

2.3-1117 23.839 2.3361 Z2.3'-11.17 2.3371 25.211 2.3.72.23 22-1693 2.3391 2Y3.283 2.19.179 2'. 13.131 2.32-379 2'.7-61 2.11.311

3.2221 6673

41.163 3.23.97 6703

7'.137 3'.83 6733

11.613 3.2251 6763

13.521 3.7-17-19 6793 6803

32.757 6823 6833

3.2281

23.283 3.41-53

6547

2'.1627 2.3259 2l.3.17 2-7.467 2'. 1637

2'. 11.149 2.7'.67 2'.3-137 2.37.89 Z2-17.97

79-83 3-11.199 6577 7.941 32-733

2.3-1093 23-821 2.1 1.13.23 22.33.61 2.3299

7-937

5-1321 33.5.72 53.53 5.1327 3.5-443

2.3'.367 Z3-827 2.3313 Z2.3.7.79 2-3323

6607

13-509 3.472 172.23

2'.7.59 2.3-1103 2'. 1657 2.3319 23.3-277

2-3-1109 23.72-17 2.47.71 22.3.557 2-3347

5.113 5-31.43 3-5'-89 5.7.191 5.13.103

2'. 13 2.3.11.101 2'.1669 2.3343 23.33.31

3.7.317 59.113 11.607 32-743 37.181

2.3329 2'. 1667 2.32.7.53 25.11.19 2.17-197

2'.4 19 2.3'.373 22.412 2.7-13-37 Z3-3.281

3'.5.149 5.17-79 5'.269 3.5.449 5.19-71

2.7.479 22-23.73 2.3-19.59 2'.421 2.3373

19-353 3.2239 7.31' 3.13.173

22.3-13-43 2.3359 23.292 2.3.1123 2 2 - 7.2 4 1

2.11.307 22-19.89 2-3.1129 2'.53 2.43-79

5.7.193 3.5.11.41 52.271 5.23.59 32-5-151

Z2.3.563 2.17-199 23.7.11 2 2.T.13-29 2'.1€99

29.233 67.101 33.251 11.617 7.971

2.31-109 2'.32.47 2.3389 Z2.1697 2.3.11.103

Z2.35.7 2.3407 23-853 2.3.17-67 2'.29.59

5.1361 5.29.47 3-5'.7.13 5.1367 5.372

2.41.83 25.3.71 2.3413 22.1709 2.3-7-163

3-2269 17.401 3.43.53 41.167

23.23.37 2-7.487 2'.3.569 2.13-263 Z6. 107

3-5.457 5.1373 5'-11 3'-5-17 5.7.197

23.857 2.3433 22-32-191 2.11.313 2'.431

3'.7.109 13.232 71-97 3.112.19

22.3.571 2.47-73 23.859 2.3.31-37 2'.1723

3.29.79 61.113

2.23.149 24.3.11-13 2-7.491 2'.1721 2.3'.383

2-7.17-29 28.33 2.3461 2'.1733 2.3.13-89

3'.13.59 31-223 7.23-43 3.2311 53.131

Z3.863 2.3457 Z2.3.577 2.3467 25-7-31

5.1381 3.5.461 5'.277 5.19.73 3.5-463

2.3.1 151 22.7-13-19 2.3463 Z3.3.17' 2.23-151

Z3.1 1.79 2.59' 2'.3.7-83 2.3491 2'-19-23

17.409 3-11.211 19.367

2.3.19-61 2'.1741 2-11.317 23.32.97 2.13.269

5.13.107 5.7.199 32.52-31 5.1 1.127 5-1399

22.37.47 2.3'-43 Z6.109 2.7.499 2'-3.11.53

7.11.89 6863 6883

6983

33.7.37

6637

6737

6827

6857

6907 6917

3.2309 7.991 6947

32.773 6967 6977

3.17.137 6997

6529

13.503 3.37-59 6569

3'-17.43 11-599 6599

3.2203 6619

7.947 3.2213 61-109 6659

33.13.19 6679 6689

3-7.11-29 6709 6719

3-2243 23.293 17.397 3'-751 7-967 6779

3.31.73 13.523 11.619 3-2273 6829

7.977 32.761

2.33.127 52.17.101 2.19. I81 23-3-7.41 2.3449

3-2293 832

2'.11-157 2.3.1153 2'.433 2-3469 22.32-193

3-72.47 11.17.37 132-41 33.257

2.7'-71 23.13.67 2-3-1163 22.1747 2.3499

3-23-101 7.997 29.241 3-2333

1g3 6869 6899

6949 6959

650 651 g 652 653 654 655 656 657 658 659 660 66 1 662 663 664 665 666 667 668 669 670 671 672 673 674 675 616 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694

N

0

1

2

3

4

5

6

7

8

9

2'-5'.7 2.5.701 22.33.5.13 2-5.19.37 27.5.11 2-3-52-47 22.5.353 2-5-7.101 2'.3.5-59 2-5.709 22.52.7 1 2-3'.5.79 2'-5.89 2-5.23.31 22.3.5.7.17 2.52-11.13 23.5-179 2.3.5.239 2'-5-359 2.5.719 2'-3'.52 2.5.7.103 22.5.192 2.3-5-241 2'-5-181 2-5'.29 22.3.5.11' 2-5-727 2'.5.7.13 2.36.5 22.52-73 2-5.17-43 23-3.5.61 2.5.733 22.5-367 2.3.5'.7' 26.5.23 2.5.11-67 22.32.5.41 2.5-739 23-52.37 2-3.5.13-19 22-5-7.53 2-5.743 2'.3-5.31 2-52-149 2'.5.373 2.3'-5-83 23-5-11.17 2-5-7.107

7001

2-32-389 2'. 1753 2-3511 Z3-3.293 2-7.503 22.41.43 2-3-11-107 25-13-17 2-3541 22-32-197 2-53.67 25-7-127 2-3.1187 2'- 1783 2-3571 2'.3.149 2.3581 22-11.163 2.3'-7-19 2'.29-31 2.13.277 2'.3.601 2-23-157 26-113 2.3-17-71 22-72.37 2-3631 23-32.101 2-11.331 22.1823 2-3-1217 2'.457 2.7.523 2'.3.13:47 2.3671 23-919 2-3'.409 Z2-19-97 2-3691 2'.3-7-11 2-3701 22-17.109 2.3-1237 2'-929 2.61' 22-3'-23 2-7-13-41 2'.467 2.3-29-43 22.1873

47-149

2'-17.103 2.3.7-167 2'-439 2.3517 2'.3.587 2.3527 2'.883 2.3'-131 22-7-11-23 2.3547 26-3-37 2.3557 22.13.137 2-3.29-41 23-19.47 2.7'-73 2'.3'-199 2.17-211 2'.449 2-3.11-109 22.1801 2-3607 2'.3.7.43 2.3617 22.1811 2.32-13-31 25-227 2-3637 22.3.607 2-7-521 23.11-83 2-3.23-53 22-1831 2-19.193 2'.3'-1 7 2-3677 22-7.263 2-3-1229 23-13-71 2-3697 22-3-617 2-11-337 28-29 2.35-7-59 2'- 1861 2.3727 2'-3-311 2.37-101 22-1871 2-3.1249

3.5-467 5.23.61 52.281 3.5-7-67 5-1409 5-17-83 3'.5-157 52-283 5.13-109 3-5011-43 5-7'-29 5-1423 3.5'-19 5.1427 5.1429 33-5-53 5.1433 52.7-41 3.5.479 5.1439 5.11-131 3.5-13.37 9-17' 5-1447 3'.5-7-23 5.1451 5-1453 3-52.97 5-31-47 5-1459 3-5.487 5.7.11-19 52.293 32.5.163 5-13-113 5.1471 3-5-491 55.59 5-7.211 3-5.17-29 5.1481 5.1483 3'-52. 11 5-1487 5-1489 3.5.7-71 5.1493 52.13.23 3-5.499 5-1499

2.31.1 13 23-877 2.3-1171 22.1759 2-13-271 24.32.72 2.3533 22-29-61 2-3-1181 23-887 2-11-17-19 P-3-593 2-7-509 25-223 2.3'.397 2'-1789 2.3583 23.3.13.23 2-3593 22-7.257 2.3-1201 2'-11-41 2.3613 22.3'.67 2.3623 23.907 2.3-7-173 22-17-107 2.3643 27.3-19 2.13.281 22.31.59 2-32-11.37 2'.7-131 2-3673 22.3.613 2.29-127 2'.461 2-3-1231 22.432 2.7-232 2a-32-103 2-47-79 22-11-13' 2-3-17.73 25-233 2.3733 3-3-7439 2-19-197 2'-937

72.11.13 3.2339

2'-3-73 2-1P.29 22-7.251 2.3'-17-23 23-881 2.3529 22-3.19.31 2.3539 2'-443 2.3-7-132 22.1777 2-3559 2'-3'-11 2-43-83 22.1787 2.3-1193 210.7 2-37-97 22-3.599 2-59-61 2'.17-53 2'.13.139 2.32.401

43-163

22 % 00

NoE

N I

700 701 702 703 704 705 706 707 708 709 710 71 1 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749

32-19.41 7-17-59 79.89 3-2347 11-641 23.307 3.2357 73-97 7-1013 3'.263 13-547 7121

3.2377 37.193 7151

3.7-11-31 71.101 43-167 32.17-47 19-379 7211

3.29.83 7-1033 13-557 3-2417 53-137 11.661 3'.809 23-317 7'.149 3.2437 7321 7331

3-2447 7351

17.433 3'.7-13 112-61 19.389 3.2467 7411

41-181 3.2477 7.1063 7451

3'.829 31-241 7481

3-11.227

7013

3-2341 13-541 7043

3.2351 7-1009 11-643 32-787 41-173 7103

3.2371 17-419 7-1019 3.2381 23.31 1 13.19-29 32.797 11.653 7193

3.74 7213

31.233 3-2411 7243 7253

3'-269 7.1039 7283

3-11-13-17 67.109 71.103 3-2441 7333

7.1049 3'. 19.43 37-199 73-101 3-23-107 7393

11.673 3-7.353 13-571 7433

37.827 29-257 17.439 3-47-53 7-1069 59.127

7027

31-227 35.29 7057

37.191 3.7-337 19.373 47-151 3-23-103 11-647 7127

32*13-61 7-1021 17-421 3-2389 7177 7187

3.2399 7207

7-1031 3'-11-73 7237 7247

3-41-59 132-43 19.383 3-7.347 7297 7307

33.271 17.431 11-23-29 3.31.79 7-1051 53.139 3-2459 83.89 13-569 3'-823 7417

7-1061 3-37-67 11-677 7457

3-19-131 7477 7487 3 2 - 72.17

2.7.1 1.47 2'-3.151 2.19-191 2'.23.79 2-3-1213 23.911 2-41-89 22.32.7.29 2.3659 25.229 2.3-1223 22.11.167 2.13-283 23-3-307 2-7.17.31 22.1847 2.33-137 24-463 2-3709 22-3.619 2-3719 23-72-19 2-3-11-113 2'.1867 2.3739 2'-32- 13 2.23.163

7019

3'- 11-71 7039

7.19-53 3-13-181 7069 7079

3-17.139 31.229 7109

32-7-113 7129

112.59 3-2383 7159

67-107 3.2393 7-13-79 23-313 3'-89 7219 7229

3.19-127 11-659 7-17.61 3.2423 29.251 37.197 3'-811 7309

13.563 3-7.349 41.179 7349

3-11-223 7369

47-157 3'.821 7'-151 31-239 3.2473 17-19.23 43.173 3-13-191 7459

7-11-97 3'-277 7489 7499

700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 ?22

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7 $

8*

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725 726 m 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 2 749 g

b $

g cc

B

750 2 751 8 752 753 754

750 751 752 753 754

2"3.5' 2.5.751 25.5.47 2-3-5-251 2'-5- 13.29

13.577 ~- ... 7-29-37 3-23-109 17.443 7541

2.1 lt.31 23-3.313 2-3761 2'-7-269 2.3*-419

3-41.61 11-683 7523 35-31 19.397

2'-7.67 2.13-17' 2'-3'-1 1-19 2-3767 23.23-41

5-19-79 3'-5*167 9-7.43 5.11.137 3-5-503

2-33.139 2'-1879 2.53.71 2'.3.157 2-73.1I

7507 7517 3.13-193 7537 7547

2'. 1877 2-3.7.179 23.941 2.3769 22-3.17-37

3.2503 73-103 7529 3-7.359 7549

755 756 757

759

2-5'. 15 1 23-33.5-7 2.5.757 22.5.379 2.3-5-11-23

3"839 7561 67-113 3-7-19' 7591

2'-59 2-19.199 2'-3-631 2.17-223 23-13-73

7-13-83 3.2521 7573 7583 3-2531

2-3-1259 F.31.61 2.7-541 2'.3.79 2-3797

5.1511 5.17.89 3-5'.101 5.37-41 5.7'.31

2'-1889 2-3.13-97 23.947 2.3793 22.32.211

3-11-229 7-23-47 7577 33.281 71.107

2.3779 2'.11-43 2.3'.42 1 22.7-271 2.29.131

7559 32-292 11.13.53 7589 3.17-149

755 756 757 758 759

760 76 1 762 763 764

2'.52*19 2-5.761 P-3.5.127 2-5.7.109 23.5-191

11-691 3-43.59 7621 13.587 33-283

2.3-7-181 2.37-103 2'.3'-53 2.3821

7603 23-331 3'-7.112 17-449 7643

2"1901 2*3'.47 Z3.953 2.11 -347 2"3.7'- 13

3'.5.13' 5-1523 53.61 3.5.509 5.11.139

2.3803 26-7.17 2.3.31.41 22.23.83 2.3823

7607 3-2539 29-263 7-1091 3.2549

23-3.317 2-13.293 2"1907 2.3-19.67 25-239

7-1087 19.401 3.2543 7639 7649

760 761 762 763 764

765 766 767 768 769

2-3'@- 17 P.5-383 2.5.13.59 2'-3-5 2.5-769

7-1093 47.163 3.2557 7681 7691

2'- 1913 2-3-1277 23-7-137 2.23.167 2'-3-641

3.2551 79.97 7673 3.13.197 7'. 157

2-43-89 2'.479 2.3-1279 22-17.113 2.3847

5.1531 3-5.7.73 5'-307 5-29-53 3'-5.19

23.3.11-29 2.3833 22-19.101 2.3'-7-61 2'.13.37

13-19.31 11.17-41 3'.853 7687 43.179

2-7.547 2=.33.71 2-11-349 23.312 2-3-1283

32.23.37 7669 7.1097 3.11.233 7699

765 766 767 . .. 768 769

770 771 772 773 774

2'.5'.7-11 2-3.5.257 23-5-193 2.5.773 2'.3'-5.43

3-17.151 11.701 7-1103 32.859 7741

2.3851 25.241 2-33-11-13 2'.1933 2-7'.79

7703 3'357 7723 11.19-37 3-2939

23.35.107 2.7.19-29 2"193 1 2.3.1289 28.112

5-23.67 5.1543 3.5'-103 5-7-13-17 5.1549

2.3853 2'- 3.643 2.3863 S.967 2.3.129 1

3.7.367 7717 7727 3.2579 61-127

22.41-47 2.17.227 2'-3-7.23 2.53-73 2'.13.149

13-593 3-31-83 59.131 71.109 33.7.41

770 771 772 773 774

775

23.337 3-13.199 19-409 31-251 3-72-53

2'-3.17- 19 2.3881 2'-29-67 2-3.1267 2'-487

7753 7-1109 3-2591 43-181 7793

2.3877 2'.3-647 2.13'.23 23-7-139 2.3'- 433

3.5.1 1.47 5.1553 5'-311 3'.5.173 5-1559

22-7.277 2.11-353 25.35 2.17.229 2'-1949

7757 3'.863 7-11-101 13.599 3-23-113

2.32-431 23.971 2.3889 22.3-11.59 2.7.557

7759 17-457 3.2593 7789 11.709

775

777 778 779

2.53-31 2'.5.97 2-3.5.7.37 2*.5.389 2-5-19-41

778 779

780 781 782 783 784

23.3-5l.13 2-5-11-71 2'.5.17.23 2-33.5.29 25.5.7'

29.269 73-107 3'*11.79 41-191 7841

2-47-83 2f-32-7.31 2.391 i 23-11.89 2.3-1307

33-172 13-601 7823 3.7.373 11-23-31

2"1951 2-3907 2'-3-163 2-3917 2'.37-53

5-7-223 3.5.521 5'.313 5-1567 3-5.523

2.3.1301 23.977 2-7.13.43 2'-3.653 2.3923

37.21 1 7817 17-461 7.19.59

27-61 2.3-1303 2'- 19.103 2-3919 23.3'.109

3.19-137 7.1117 7829 3'.13-67 47.167

780 781 782 783 784

785 786 787 788 789

2-5'-157 2'-3-5-131 2-5.787 23-5-197 2.3.5.263

3-2617 7.1123 17-463 3.37-71 13-607

2'.13-151 2-3931 26-3-41 2.7-563 22.1973

7853 3-2621 7873 7883 3'.877

2-3-7-11-17 2-31-127 22-33.73 2.3947

5-1571 5.1 12.13 32.53.7 5.19-83 5.1579

2'-491 2.3'- 19-23 22.11.179 2.3943 23-3.7.47

34-97 7867 7877 3-11.239 53.149

2.3929 22.7.281 2-3-13.101 2'.17-29 2.11.359

29.271 3.43.61 7879 7'-23 3.2633

785 786 787 788 789

790 79 1 792 793 794

b-5'-79 2-5-7-113

7901 33.293 89' 7-11.103 3-2647

2-3'-439 23.23*43 2-17.233 2'-3-661 2-11-19'

7.1129 41-193 3.19.139 7933 132.47

25.13.19 2.3.1319 22.7.283 2-3967 23.3.331

3-5-17.31 5.1583 5'.317 3-5-23' 5.7.227

2.59.67 22-1979 2.3.1321 28.31 2.29-137

7907 3-7.13.29 7927 7937 3'.883

2"3.659 2-37.107 23.991 2.3'. 7' 2'.1987

11-719 7919 3'.881 17.467 7949

790 791 792 793

795 796 797 798 799

2-3-5'.53 23-5-199 2-5-797 2'.3.5.7-19 2.5-17.47

7951 19-419 3-2657 23.347 61.131

24-7-71 2-3-1327 2'- 1993 2.13-307 23-3a-37

3-11-241 7963 7-17.67 3'.887 7993

2.41.97

5.37-43 33.5-59 52.11-29 5.1597 3.5-13-41

2'-3'.13*17 2.7.569 23.997 2-3-113 22.1999

73.109 3 1.257 3-2659 7'.163 11.727

2.23-173 2'.3.83 2.3989 22-1997 2.3.31 -43

3.7-379 13-613 79.101 3.2663 19.421

758

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810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837

838 839 840 841 842 843 844 845 846 847 848 849

26-53 2.3'.5.89 2'-5-401 2-5-11-73 2a.3-5-67 2-5'.7-23 22.5-13-31 2-3-5-269 2'.5.101 2.5-809 22.34.52 2.5.811 23.5.7.29 2-3-5-271 22.5.1 1.37 2.5'. 163 25-3.5.17 2.5.19.43 2"5.409 2.32-5-7-13 2'-5'.41 2.5.821 2'.3-5.137 2.5.823 2'.5.103 2-3-53-11 2'-5*7-59 2-5-827 23.32-5-23 2.5.829 2'.5'-83 2-3.5-277 2'.5-13 2.5-7'- 17 2'.3.5.139 2.5'.167 23.5.1 1-19 2-33-5.31 2'.5-419 2-5.839 2'.3.5'-7 2.5.29' 2'-5.421 2.3.5.281 23.5.21 1 2-5'-13' 2'-3'.5-47 2-5.7.11' 25.5.53 2.3.5-283

3'-7-127 8011 13.617 3.2677 11.17.43 83-97 3.2687 7-1153 8081 3"-29-31 8101 8111 3.2707 47.173 7-1163 3.11-13-19 8161 8171 3'.101 8191 59-139 3.7-17-23 8221 8231 3-41-67 37-223 11.751 3'.919 7'-13' 8291 3.2767 8311 53.157 3.2777 19.439 7.1 193 3'.929 11.761 17'.29 3.2797 31.271 13.647 3-7.401 8431 23-367 33.313 8461 43-197 3.11-257 7-1213

2.4001 2'-2003 2.3-7-191 25.251 2.4021 2'-3-11-61 2.29.139 23.1009 2.3'-449 2'.7*17' 2-4051 24.3-13' 2.31.131 2'- 19-107 2-3-23-59 23*1019 2.7-11-53 2'-3'.227 2-4091 213 2-3.1367 2l.2053 2-4111 23-3-73 2.13.317 2'-2063 2-35.17 2'-11-47 2.41.101 2'-3-691 2.7.593 2'. 1039 2-3.19.73 2"2083 2-43-97 25-32.29 2.37-113 22*7-13*23 2.3-11.127 23-1049 2.4201 2f-3-701 2-4211 2'-17-31 2.3'-7-67 2f-2113 2.4231 23-3-353 2-4241 F.11.193

53-151 3-2671 71-113 29-277 3-7.383 8053 11.733 33-13-23 59.137 8093 3-37-73 7-19-61 8123 3.2711 17-479 31-263 3'-907 11.743 72-167 3.2731 13.631 43-191 3.2741 8233 8243 3"7.131 8263 8273 3.11.251 8293 19'-23 3.17-163 7-29.41 13.641 3'.103 8353 8363 3.2791 83-101 7-11.109 3-2801 47-179 8423 31.937 8443 79-107 3-7.13-31 37-229 17.499 3-19-149

22-3-23-29 2.4007 23-17.59 2.3-13-103 22.2011 2.4027 27.32.7 2.1 1.367 2'.43.47 2-3.19.71 23.1013 2-4057 2'.3-677 2-7'*83 2'-509 2.3j.151 2"13.157 2.61-67 2'-3-11-31 2.17.241 2'.7.293 2.3.37' 25.257 2.23.179 22-32-229 2-4127 23-1033 2.3-7.197 2'.19.109 2.11.13-29 2'.3.173 2.4157 22.2081 2-32.463 23-7.149 2.4177 2'.3.17.41 2.53-79 26.131 2-3.1399 2'-11.191 2.7.601 23-34.13 2.4217 22-2111 2-3-1409 2'.232 2-19-223 2'-3*7*101 2-31-137

5-1601 5.7.229 3-52-107 5.1607 5.1609 3'.5.179 5.1613 52.17-19 3-5.7'.11 5-1619 5.1621 3.5.541 5'.13 5.1627 3'.5- 181 5.7-233 5-23-71 3~5~.109 5.1637 5.11-149 3-5-547 5-31.53 52.7-47 33.5-61 5.17-97 5.13.127 3-5.19-29 5'-331 5.1657 3.5.7.79 5.11-151 5.1663 32.52.37 5.1667 5.1669 3-5.557 5.7.239 53.67 3-5.13-43 5-23.73 5-41' 32.5.11*17 52.337 5-7-241 3.5.563 5.19-89 5.1693 3.5'-113 5-1697 5.1699

2-4003 2'.3.167 2-4013 2'.7'-41 2.33.149 23-19-53 2-37.109 2'-3-673 2-13-311 25-11.23 2-3.7.193 P.2029 2.17-239 23.32.113 2-4073 2=.2039 2.3-1361 2'.7.73 2-4093 2'.3.683 2.11.373 23.13-79 2.3'-457 2'.29.7 1 2-7-19-31 26-3-43 2-4133 2'-2069 2.3.1381 23.17.61 2-4153 22-33.7-1 1 2.23-181 2'-521 2.3.1 3-107 2'.2089 2-47-89 23.3-349 2.7.599 2'.2099 2.3'.467 25-263 2-11-383 22.3.19-37 2.41.103 23.7-151 2.3- 17-83 2'.13-163 2-4243 2'.3'-59

3-17-157 8017 23-349 3'. 19-47 13-619 7-1151 3.2689 41-197 8087 3-2699 112.67 8117 33.7.43 79.103 8147 3.2719 8167 13-17.37 3.2729 7-1171 29-283 3'. 11.83 19.433 8237 3.2749 23.359 7-1181 3-31.89 8287 8297 3'.13.71 8317 11.757 3-7.397 17-491 61-137 3-2789 8377 8387 33.311 7-1201 19-443 3-53' 11-13.59 8447 3-2819 8467 7'- 173 3'.23.41 29-293

23.7.11-13 2-19.211 2'.3'-223 2.4019 2'-503 2.3.17.79 2'.2017 2-7.577 23-3-337 2-4049 2'.2027 2.32.11.41 26.127 2-13.313 2"3.7-97 2.4079 23.1021 2.3.29.47 2'.23.89 2-4099 2'.33.19 2.7.587 22.112.17 2.3.1 373 23.1031 2.4129 2'.3-13.53 2.4139 25.7.37 2-3'.461 2*.31-67 2.4159 23.3.347 2.1 1-379 22.2087 2.3.7-199 2'.523 2.59.71 22.32.233 2.13-17.19 23.1051 2-3-23-61 22-72-43 2.4219 28-3-11 2.4229 22.29.73 2-33.I57 23.1061 2.7.607

8009 36.11 7-31-37 8039 3.2683 8059 8069 3-2693 8089 7-13.89 3'-17.53 23.353 11.739 3.2713 29.281 41.199 3-7.389 8179 19.431 3'-9 11 8209 8219 3.13-211 7.1 1.107 73-113 3.2753 8269 17.487 33.307 43.193 7-1187 3.47.59 8329 3 1.269 3.1 12.23 13.643 8369 32.72.19 8389 37.227 3.2803 8419 8429 3.29.97 7-17-71 11-769 3"941 61-139 13.653 3.2833

800 801 802 803 804 805 806 807 808 809 810 81 1 812 813 814 815 816 817 818 819 820 821

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850 851 852 853 854

2'.53.17 26-23-37 2a.3-5-11 2-5-853 22.5.7.61

3-2837 8521 19-449 3'.13-73

2.3.13.109 26.7.19 . ~. 2.4261 2.3a-79 2-4271

11.773 8513 39.947 7.23-53 8513

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35.5-7 5.13-131 5'-11-31 3-5.569 5.1709

2-4253 22.2129 2-3-72.29 23-11.97 2.4273

47.181 3.17.167 8527 8537 3.7-11-37

855 856 857 858 859

2-3'*5*-19 2'6. 107 2.5357 2'.3-5-11.13 2-5-859

17.503 7.1223 3.2857 8581 112.71

23-1069 2-3-1427 $-2143 2-7.613 2'-3-179

3-2851 8563 8573 3.2861 13-661

2-7-13-47 2'-2141 2-3.1429 23-29-37 2-4297

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22-3-23.31 2-4283 27.67 2.3'-53 2'-7.307

43.199 13-659 32.953 31.277 8597

2.11.389 23*32.7.17 2-4289 22.19.113 2.3.1433

33.317 11.19.41 23.373 3.7.409 8599

855 856 857 858 859

860 861 862 863 864

23-52-43 __ 2-3-5-7-41 2'. 5.431 2.5.863 2"3'-5

3-47-61 79.109 . _-.37.233 3'.7.137 864 I

2.1 1.17.23 2'-2153 2*3'.479 23-13-83 2.29.149

7-1229 33.11.29 8623 89.97 3.43.67

2'.3'-239 2-59-73 2'-7*-11 2-3-1439 2'-2 16 1

5-1721 5.1723 3-FY.23 5.11.157 5-7.13.19

2.13.331 2'-3-359 2-19.227 2'-17-127 2.3.11.131

3-19.151 7.1231 8627 3-2879 8647

25.269 2-31-139 2'.3.719 2-7.617 23.23.47

8609 3-13'-17 8629 53.163 32.312

860 86 1 862 863 864

2.F-173

2-3.5.17' 28-5.7-31 2.5.11-79

41-211 3.2887 13-23-29 8681 3-2897

2'-3.7.103 2.61.71 26-271 2-3-1447 2'-41-53

17-509 8663 3 -72.59 19-457 8693

2-4327 2'-3-19' 2-4337 2'-13-167 2.33.7-23

3 -5.577 5-1733 52.347 3'-5.193 5.37-47

2'-541 2.7-619 2'.3'.241 2.43-101 23.1087

11.787 3'. 107 8677 7.17-73 3-13.223

2.3'-13.37 2l.11.197 2.4339 2'-3.181 2.4349

7-1237 8669 3-11.263 8689 8699

865 866 867 868 869

2'.3*5'.29 2-5-13-67 2'.5-109 2.3'.5.97 F-5-19-23

7-11.113 31-281 33-17-19 8731 874 1

2.19-229 23-3'*11' 2.P-89 2'-37.59 2-3.31.47

3'-967 8713 11.13-61 3-41-71 7.1249

2Q.17 2-4357 2'-3-727 2-11.397 23-1093

5-1741 3.5.7.83 52-349 5-1747 3-5.11-53

2-3.1451 2'.2179 2-4363 25.3.7-13 2.4373

8707 23.379 3.2909 8737 8747

2'.7-311 2.3-1453 23.1091 2.17.257 2'.37

3.2903 8719 7-29-43 32.971 13-673

870 871 872 873 874

2*5'-7 28.3.5-73

3-2917 8761 7x.179 3-2927 59-149

2'447 2-13.337 2'-3.17-43 2.4391 2'-7-157

8753 3-23-127 31.283 8783 32.977

2-3-1459 2'-7-313 2-41-107 2'-3'.61 2.4397

5-17-103 5.1753 33-52-13 5-7.251 5.1759

21.1 1.199 2.3l.487 23.1097 2.23.191 2'.3.733

3'.7.139 11.797 67.131 3.29.101 19.463

2-29.151 Z'e.137 2.3.7.11.19 22.133 2.53.83

19.461 3-37.79 8779 11.17-47 3.7-419

879

13-677 3'. 11-89 8821 8831 3.7.421

2.33-163 2l.2203 2.11.401 27.3.23 2-4421

8803 7.1259 3-17.173 11l.73 37.239

2'-3 1 -71 2.3-13-113 23*1103 2-7.631 2'-3-11*67

3.5.587 5.41 -43 52.353 3.5-19-31 5.29.61

2.7.17.37 2'-19.29 2-3.1471 2l.47' 2.4423

8807 3.2939 7.13.97 8837 3'-983

23.3.367 2.4409 2.3'-491 2'.2207 2'.7.79

23.383 8819 3'.109 8839 8849

880 881 882 883 884

53-167

2'-2213 2.3-7-211 23.1109 2-4441 2'-3'-13.19

3.13-227 8863 19-467 31-7-47 8893

2-19.233 25-277 2-3*-17-29 21.2221 2.4447

5.7.1 1-23 3'.5-197 P.71 5-1777 3-5-593

23.33-41 2.11-13-31 2l.7.317

17.521 8867 3.1 1-269 8887 7-31.41

2-43.103 2'-3-739 2.23-193 23.11.101 2.3-1483

3-2953 72.181 13.683 3-2963 1 1-809

885 886 887 888 889

2.4451

25-3-7-53 2-4457 2'-23.97 2-3-1489 2'. 13-43

5-13.137 5.1783 3*5'.7-17 5.1787 5-1789

2.61-73 2'.3-743 2.4463 Z3.1117 2.3'.7.71

3.2969 37-241 79-113 33-331 23-389

2'- 17.13 1 2.73-13 2'.3'-3 1 2.41.109 Z2.2237

59.151 3'.991 8929 7.1277 3.19.157

890 89 1 892 893 894

2.11l.37 2'-3'.83 2-7-641 23-1123 2-3-1499

3'.5.199 5.11-163 52.359 3 -5.599 5.7.257

2'.2239 2.4483 2'.3.11-17 2.4493 22-13-173

2.3-1493 23-19*59 2.672 22.3.7.107 2-11-409

17l.31 8969 3.41-73 89.101 8999

865 866 867 868 869 870

__

871 -

872 873 874 875 876 877 878 879

2z.5-4.13 - -__

880 ~.~ 2 W -1 1 2.5.881 881 22.32.5.7' 882 245.883 883 2'45-13.17 884

885 886 887 888

889 890

__

89 -1

892 893 894 895 _..

896 897 898 899

8501 _ _ .~

~

2-3-5l.59 2'-5.443 - _ ___ 2-5.887 2'-3*5-37 2-5.7-127

3-2957 83.107 17-523

2'-5'.89 2.3'.5*11 2'.5-223 2-5-19.47 2'-36149

3'-23.43 7-19-67 11-811 3-13.229 8941

2.3-1487 2'.7-11-29 2-17.263

29.307 3-2971 8923 8933 3-11-271

2.5'- 179 28-5-7 2-3.5-13.23 2'.5-449 2.5.29-31

8951 3.29.103 8971 7.1283 35.37

25.3.373 2.4481 22.2243 2-3'*499 2'-281

7-1279 8963 32.997 13-691 17-23'

8861 __.-

24.557 __.

13'-53 3.72.61 - . .~ 47-191 11.19-43 3.2999

2'.3.709 2.4259 2'-13-41 2-3.1423 2'.2137

67-127 7-1217 3-2843 8539 83.103

850 851 8 852 $53 854

c1

0

2 52 2

5.

4

c3

U

cc

E

2

895 896 89700, lx 898 CO* Q, 899gj F

E

=e 00

N 900 901 902 903 904 905 906 907 908 909 910 91 I 912 913 914 915 916 917 918 919 920 92 1 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949

0

1

2

3

4

5

23-3'.55 2.5.17.53 2'-5-11-41 2.3-5-7.43 24-5-113 2.5'. 181 22-3-5.151 2-5.907 23.5-227 2.32-5-101 2'.5'.7-13 2-5-911 25.3-5-19 2-5-11.83 2'.5.457 2.3.5'-61 23-5-229 2.5.7-131 2l-3a.5-17 2.5.919 2'-5'-23 2-3.5.307 2'.5-461 2.5-13-71 23.3.5.7.11 2-8-37 2'.5-463 2-3'.5.103 2'.5-29 2-5.929 2'.3.5"31 2-5-7'.19 23.5.233 2-3-5.311 2'. 5.467 2-5'-1 1-17 2'*3'.5.13 2-5-937 2'.5.7.67 2.3-5-313 2a-5'-47 2-5-941 2'-3-5-157 2-5-23-41 2'.5-59 2-3'.5'-7 2'-5-11.43 2-5.947 23-3.5-79 2.5.13.73

9001 9011 3-31.97 11-821 9041 3-7.431 13-17-41 47-193 3'.1009 9091 19.479 3-3037 7-1303 23.397 3.1 1.277 9151 9161 3'-1019 9181 7.13-101 3.3067 61,151 9221 3.17.181 9241 11-29' 38-73 73.127 9281 3-19.163 71-131 9311 3.13-239 7-31-43 9341 3'-1039 11-23-37 9371 3.53.59 9391 7-17.79 3-3137 9421 9431 3l.1049 13.727 9461 3.7-11.41 19-499 9491

2-7.643 2'-3-751 2.13-347 2'-1129 2.3-11-137 2'-31-73 2.23.197 2'.3'.7 2-19.239 2'-2273 2.3-37.41 2'-17-67 2-4561 2'.3.761 2-7-653 28.11.13 2.3'-509 2'-2293 2.4591 2'-3-383 2.43-107 2'.7'*47 2.3-29-53 2'.577 2-4621 2'.3'-257 2-11-421 23.19*61 2.3.7.13-17 2'.23- 101 2-4651 2'-3*97 2-59-79 21.2333 2.3'0173 2'-7-167 2-31-151 2'-3.11.71 2-4691 2'.587 2-3-1567 22-13.181 2-7.673 2**3'-131 2.4721 21-17-139 2.3-19-83 2'.37 2-11-431 2'-3-7-113

3.3001 9013 7-1289 3-3011 9043 1 1-823 3'.19-53 43.211 31-293 3-7.433 9103 13-701 3.3041 9133 41.223 3'.113 7'-11.17 9173 3-3061 29.317 9203 3-37.83 23-401 7-1319 3'- 13-79 19-487 59-157 3.1 1-281 9283 9293 3.7-443 67-139 9323 3'. 17-61 9343 47-199 3-3121 7.13-103 1 1-853 3.31-101 9403 9413 33.349 9433 7-19-71 3-23-137 9463 9473 3-29-109 11.863

22.2251 2-4507 28-3-47 2-4517 2'.7-17-19 2-3'.503 23.11.103 2-13.349 2'03-757 2.4547 2'-569 2.3.72-31 2'-2281 2.4567 23-31127 2-23-199 2'-29.79 2-3.11-139 25-7-41 2.4597 2'.3-13.59 2-17.271 2'-1153 2-35-19 2'.2311 2-7-661 2'.3-193 2-4637 22.11.21I 2.3.1549 23.1163 2.4657 2'-3'-7-37 2.13-359 27-73 2-3.1559 2'.2341 2-43-109 2'-3-17-23 2.7.11-61 2"2351 2.3"523 2'-19.31 2-53-89 2'-3-787 2-29-163 23-7-13' 2.3-1579 2'-2371 2-47-101

5-1801 3.5-601 52.19' 5-13-139 33.5.67 5-1811 5-72-37 3.-5'.11' 5-23-79 5-17-107 3.5.607 5.1823 53.73 3'05-7-29 5-31.59 5-1831 3-5.13-47 5'*367 5-11.167 3-5.613 5.7-263 5-19.97 3'-5'.41 5.1847 5.43' 3-5.617 5-17.109 52.7.53 3.5.619 5-11 -13' 5.1861 3'.5.23 52-373 5-1867 3-5.7.89 5-1871 5.1873 3-55 5.1877 5. I879 3*-5.11-19 5-7-269 5'-13-29 3-5-17.37 5.1889 5-31-61 3-5.631 52.379 5-7-271 3*-5.211

6 2-3-19-79 23.72.23 2.4513 2'-32.251 2-4523 25.283 2.3-151I 22.2269 2.7-11.59 23-3.379 2.29.157 2'4-53 2-33-132 2'.571 2-17.269 2'*3-7*109 2-4583 23.31-37 2.3-1531 22-1-12-19 2.4603 210.32 2.7.659 2'.2309 2.3.23.67 23-13.89 2.41-113 2=-3.773 2-4643 24-7.83 2-3'.11-47 22-17.137 2.4663 23-3-389 2.4673 22.2339 2-3.7.223 2"293 2-13-19' 2'.3'*29 2-4703 23.11.107 2.3-1571 2'-7-337 2.4723 2'-3-197 2-4733 2'-23-103 2.3'--17-31 23.1187

7

8

9

N

9007 71-127 3'- 17-59 7-1291 83-109 3.3019 9067 29-313 3-13-233 11-827 7.1301 3'- 1013 9127 9137 3.3049 9157 89.103 3-7.19-23 9187 17-541 3'-1 1.31 13.709 9227 3.3079 7-1321 9257 3.3089 9277 37.251 1033 41-227 7-113 3.3109 9337 13-719 3.3119 17-19-29 9377 3*-7-149 9397 23-409 3.43-73 11-857 9437 3-47-67 7'.193 9467 36-13 53-179 9497

2'.563 2-33-167 2'-37-61 2.4519 23-3.13-29 2.7-647 22-2267 2-3.17-89 27-71 2-4549 2'-3'-11-23 2.47-97 23-7.163 2.3.1523 2'-2287 2-19-241 2'.3-191 2.13.353 2=-2297 2.3'-7-73 23-1151 2.11-419 2'.3.769 2-31.149 25-172 2-3-1543 2'.7.331 2.4639

3'-7.11-13 29-311 9029 3-23-131 9049 9059 3.3023 7-1297 61-149 33.337 9109 11.829 3.17-179 13-19-37 7.1307 3-43-71 53-173 67-137 3'- 1021 9199 9209 3.7.439 11.839 9239 3-3083 47-197 13-23-31 3'- 1031

900 901 902 903 904 905 906 907 908 909 910 91 1 912 913 914 915

23.33.43 2-4649 2'-13-179 2-3-1553 2'. 1 1.53 2-7.23.29 22.3.19.41 2.4679 2.3'-521 23-1171

7-1327 17-547 3-29-107 9319 19-491 3.1 1-283 9349 7'.191 33.347 83-113 41-229 3.13-241 97' 9419 3-7.449 9439 11 2359 3'- 1051 17-557 9479 3-3163 7-23-59

2'.2347 2.37.127 26.3;72 2.17-277 2'.2357 2.3.112-13 23.1181 2-4729 2'.3"-263 2.7.677 2'-593 2.3-1583

80 a oz

E3

t4

z

$

O

3 52

918 919 920 921 922 923 924 925 926 m 927 928 929 930 93 1 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 g

2 $ 8 0

s. g cc

E

3.3167

950 951 952 953 954

22.53-19 2.3.5.317 24.5.7.17 2-5-953 22-32.5.53

33.353 7.29.47

955 956 957 958 959

2.9-191 23.5.239 2.3.5.11-29 22.5.479 2.5-7-137

3-3187 17.563 11-13.67 3.23.139

960 961 962 963 964

27-3-52 2-5-312 2'.5.13.37 2.32-5.107 2'.5.241

7-1373 32.1069

965 966 967 968 969

2.52.193 22.3.5-7.23 2.5.967 2'.5-112 2.3.5.17-19

970 971 972 973 974

22.52.97 2-5-971 23.35.5 2-5-7.139 22.5.487

975 976 977 978 979

2.3-53.13 25.5-61 2.5-977 22-3.5-163 2.5.11.89

72-199 43.227 3.3257

980 981 982 983 984

23.9.72 2-32-5-109 22-5.491 2-5.983 24.3.5-41

3 4 - 112

985 986 987 988 989

2.9-197 3.5-17.29 2-3-5.7.47 23.5.13-19 2-5.23-43

990 99 1 992 993 994

22.32.52.11 2-5-991 26.5.31 2-3.5-331 22-5.7-71

995 996 997 998 999

2.52.199 23.3.5.83 2.5.997 3-5-499 2-33-5-37

9511 9521

9551

9601 9631

31-311 3.3217 9661

19.509 3.7-461 114381 89-109 32.13.83 9721

37-263 3.17.191

9781 9791 9811

7.23.61 3-29-113 13.757 9851

3-19.173 9871

41.241 32-7.157 9901

11.17-53 3.3307 9931 9941

3-31-107 7-1423 132-59 32.1 109 97.103

5-1901 5.11-173 3-52.127 5.1907 5-23-83

2.75-97 3.3.13.61 2-11.433 26-149 2-3-37-43

3-3169 31-307 7-1361 3-11.172

3-3181

25-33-11 2.67.71 22.2381 2.3-7-227 2'-1193

2'.3-199 2.7.683 22-2393 2.3.1597 2". 11.109

41-233 73-131 3-3191 7-372 53.181

2.17.281 22.3-797 2-4787 2'.599 2-Y.13.41

3.5-7a-13 5-1913 52.383 3'.5-71 5.19-101

2'-2389 2.4783 23.32.7-19 2.4793 23-2399

19-503 32.1063 61-157

2.4801 3.3'.89 2.17-283 25-7.43 2.3.1607

32.11-97

9643

3.74 2.11-19.23 23.3.401 2.4817 22.2411

5-17.113 3.5.641 53.7.11 5.41-47 3-5.643

2.3.1601 2'.601 2-4813 3.3. 11-73 2-7-13.53

3.19-127 2.4831 23-3.13.31 2.47.103 22-2423

7'-197 3-3221 17-569 23.421 33-359

2-3-1609 28.151 2.7-691 22.32-269 2-37>131

5.1931 5-1933 32.52.43 5.13.149 5.7.277

2.32.72.11 2'.607 2-4861 3.3.811 2.4871

31.313 11.883 3-7-463 9733 9743

23.1213 2.3-1619 3-11.13.17 2.31.157 24.3-7-29

23-23.53 2-3-1627 3-7.349 2-67-73 26.32.17

3.3251 13.751 29.337 32-1087 7.1399

2-132.29 22-11-223 2-3-i637 23-1229 2-7-19-37

9803

2-4751 23.29.41 2.32.232 22-2383 2-13.367

13-17-43 32-7-151 89.107 9533

9613 9623

3.132.19

3-3271 11.19-47 9833

3.17.193

22-3-821 2-4931 24-617 2.34-61 22.2473

59.167 7.1409 32.1097

2.4951 23-3.7.59 2-112.41 22-13.191 2.3-1657

3-3301 23.431

25.311 2.17.293 22.32.277 2.7.23.31 23-1249

9883

13.761

9923

3.7.11-43 61-163 37.269 35.41 9973

67-149 3.3331

22.2377 2-4759 23.3.397 2.19.251 2'.7-11.31

37-257 3.19-167 13.733 3'- 1061

950 a 951 952 953 954

2-3'.59 25-13.23 2.4789 22.3.17.47 2.4799

112.79 7-1367 3-31-103 43.223 29-331

955 956 957 958 959

13-739 59.163 3-3209 23.419 11-877

2.3.7.229 23.1201

3.3203

960 961 962 963 964

23-17-71 2.33.179 22-41.59 2-29.167 25.3.101

32-29.37 7.1381

2.11-439 22.2417 2.3.1613 23.7-173 2.13-373

13.743 3.11.293

3.5-647 5-29-67 9.389 3-5-11.59 5.1949

2.23.21 1 22.7.347 2-3-1621 23.1217 2.11.443

17.571 3-41.79 71.137 7.13.107 33.192

22-3-809 2.43-113 2.32.54 29-191

7-19-73 32.23.47

22.2437

9739 9749

2-4877 22.2441 2.33: 181 23-1223 2-59.83

5.1951 32-5.7-31 9-17-23 5-19.103 3.5-653

22.32.271 2-19-257 2'.13.47 2-3-7-233 22.31-79

11-887

2-7-17-41 23.3-11-37 2-4889 22-2447 2.3-23.71

22-3.19-43 2.7-701 25-307 2.3-11-149 23.23-107

5.37-53 5.13.151 3-9-131 5-7-281 5-11-179

2.4903 23.3-409 2-173 22.2459 2-32-547

2-13.379 23.32.137 2-4937 22.7.3 53 2.3.17.97

33.5-73 5-1973 53.79 3-5.659 5-1979

27-7.11 2.4933 3-3.823 2.4943 23.1237

3-11-13.23 7.17.83

2'-619 2.4957 22.3-827 2.4967 23-11.113

5.7-283 3-5.661 52-397 5-1987 3 2 - 5.13-17

2.3.13.127 22.37-67 2-7-709 24.33.23 2-4973

47-211 32.1103 19-523 73.29

2.32-7.79 22.47. 53 2-4987 2'-3-13 2.19-263

5.11-181 5-1993 3-9.7-19 5-1997 5-1999

22-19-131 2.3.11.151 23-29-43 2-4993 3-3-72.17

9547

9587

3.7.457

9677

3-3229 9697

9767

3.3259 9787

97-101 3.7-467 9817

31.317 32.1093 43-229 9857 9887

3-3299 9907

3-3319 9967

11.907 3.3329 13-769

22-29.83 2.61.79 24.32.67

24-613 2-4909 3-33.7.13 2.4919 23.1231 2.3.31.53 22.2467 2.1 1.449 25.3.103 2-72-101

9539

9619 9629

3'-7-17 9649

9679 9689

3.53.61 9719

3.3253

965 966 967 968 969 970 971 972 973

cl

3 m

$. k'

9769

975 976

7.11.127 3.13.251 41.239

:;; 8 979

17.577 32.1091 9829

980 981 982 983 984

9839

3.72-67 9859

71.139 11-29-31 3.37.89 19-521

22.2477 2-32.19-29 23.17.73 2.4969 22.3.829

33.367 7.13.109

2-13.383 2'-7-89 2.3.1663 22.11-227 2-4999

23-433 3.3323 17.587 7-1427 31-11.101

9929

3.3313 9949

@

985 986 987 988 989 990 991 992 993 994 995 996 997

2

I+

m

5

g

* 8

864

COMBINATORIAL ANALYSIS

Table 24.8 Primitive Roots, Factorization of p-1 g, G denote the least positive and least negative (respectively) primitive roots of p .

P

-

P-1

3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79

83 89 97 1111 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 24 1 251 257 263 269 271 277 281 283 293 307 31 1 313 317 331 337 347 349 353

whether 10, -10 both or neither are primitive roots. -

2 22 2.3 2.5 2'.3 24 2.32 2.11 22.7 2.3.5 22.32 23.5 2.3.7

2.5.7 23.32 2.3.13 2.41 23.11 2s.3 22.yjs 2.3.17 2.53 22.33 2'.7 2.3'.7 2.5.13 23. I 7 2.3.23 22.37 2.3.52 22.3.13 2.3' 2.83 22.43 2.89 22.32.5 2.5.19 26.3 22.72 2.32.11 2.3.5.7 2.3.37 2.113 22.3.19 23.29 2.7.17 2'.3.5 2.53 28 2.131 2'.67 2.33.5 22.3.23 23.5.7 2.3.47 22.73 2.3'.17 2.5.31 23,3.13 22.79 2.3.5.1 1 24.3.7 2.173 2'*3.29 25.11

B

-G

_-2 1 2 2 3 2 2 3 2 2 3 3 2 4 2 5 2 2 3 7 2 2 6 6 3 9 5 2 2 2 2 3 2 2 2 4 7 2 5 5 2 3 2 3 3 3 5 5 2 2 5 2 2 3 6 6 3 3 9 3 2 3 3 3 2 4 2 2 6 5 5 5 4 2 2 5 2 2 2 3 2 2 2 19 5 5 2 2 2 3 4 2 3 9 2 3 6 6 3 3 7 2 7 7 6 3 3 3 5 2 2 2 6 2 5 5 3 3 3 6 2 2 5 7 2 17 10 10 2 2 5 3 10 10 2 3 2 2

a

3

E

- 10 -_-10

____

-___ f 10

10 10 f 10 - 10 -___

____

- 10

10 -___ 10 f 10 - 10 - 10

____

-___ - 10

P 359 367 373 379 383 389 397 40 1 409 419 421 431 433 439 443 449 457 46 1 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 63 1 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 76 1 769 773 787 797 809

811

P- 1 2.179 2.3.61 2'.3.31 2.3q.7 2.191 22.97 22.32.11 24.52 23.3.17 2.11.19 22.3.5.7 2.5.43 24.33 2.3.73

2.3.7.11 2.233 2.239 2.35 2.5.72 2.3.83 2,251 22.127 23.5.13 2.32.29 22.33.5 2.3.7.13 22.139 2.281 23.71 2.3.5.19 25.32 2.293 2'.37 2.13.23 23.3.52 2.3.101 22.32.17 23.79 11 2.3.103 2.32.5.7 27.5 2.3.107 2.17.19 22.163 2.7.47 22.3.5.11 26.3.7 2'. 13' 2.11.31 2.3.5.23 22.52.7 22.3.59 2.359 2.3.11' 22.3.61 2.32.41 2.7.53 2.3.53 22.33.7 23.5.19 28.3 22.193 2.3.13 1 22.199 23.101 2.34.5

Q

denotes

-

-

B

-G

B

-G

P

P- 1

- -

7 6 2 2 5 2 5 3 21 2 2 7 5 15 2 3 13 2 3 2 13 3 2 7 5 2 3 2 2 2 2 2 3 3 5 2 3 7

2 2 2 4 2 2 5 3 21 3 2 5 5 5 3 3 13 2 2 3 2 2 4 5 2 2 3 4 2 4 2 3 3 5 5 3

82 1 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297

22.5.41 2.3.137 2.7.59 22.32.23 2.419 22.3.71 23.107 2.3.1 1.13 2.431 2'*3*73 24.5.11 2.32.7' 2.443 2.3.151 2.5.7.13 2.3'. 17 25.29 23.32.13 22.5.47 2.11.43 23.7.17 2.3.7.23 2.5.97 24.61 2,491 2.32.5.11 21.3.83

2 2 2 3 2 3 2 2 11 2 2 2 3 b 2 4 2 6 2 2 3 3 2 4 2 5 2 4 17 3 7 5 3 3 5 5 2 2 2 3 3 3 2 5 6 3 3 3 2 5 2 6 7 7 11 11 3 3 2 3 10 10 14 2 5 5 2 3 3 3 6 7 2 2 2 3 6 6 2 3 4 2 5 5 3 3 5 3 2 2 2 2 4 2 11 11 2 17 5 5 5 3 2 4 7 7 2 3 3 3 11 11 2 2 3 3 2 5 2 2 2 3 2 2

- -

7

3 2 3 2 3 3 11 5 2 2 2 5 2 5 3 2 2 11

5 6 3 5 3 2 6 I1 2 2 2 3 3

3

2 7 2 2 3 4 9 3 7 2 2 3 2 5 2 10 6 2 2 2

7 6 6 2 2 2

6

11 2 4 2 3

5

2.3a.59 22.3.89 2.3.181 2.5.109 22.3.7.13 25.137 2.19929 22.277 22.32.31 2.3.11.17

2.32.5913 22.5.59 2.593 22.3101 25.10 2.13.47 22.307 2.3.5.41 22.3.103 25.3.13 2.17.37 2'. 11.29 2.32.71 2.641 23.7.23 2.3.5.43 24-34

7

2 2 3 2 6 2 LO

7

3 2 2 3 6 4

10

865

COMBINATORIAL ANALYSIS

Table 24.8

Primitive Roots, Factorization of p-1 g,

G denote the least positive and least negative (respectively) primitive roots of p . e denotes whether 10, -10 both or neither are primitive roots. - - - P-1

P 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759 1777 1783 1787 1789 1801 1811 1823

22.52.13 2.3.7.31 2.653 2.659 23.3.5.11 2.3.13.17 24.5.17 2.683 22~7~ 22.3.5.23 2.3.233 2'. 11 2.32.79 2.23.31 21.3.7.17 23.179 2.719 2.3.241

~

2.743 24.3.31 22.373 2.7.107 2.5.15 1 2.761 2.32.5.17 2.3.257 22.3'. 43 24.97 2.19.41 2.38.29 2.5.157 2.3.263 2.7.113 22.3.7.19 26.52 2.11.73 23-3.67 22.13.31 2.809 22*3'.5 2.3.271 22.409 5.5.23 253.277 2.72.17 22.3.139 22.32.47 25.53 2.3.283 22.7.61 23.5.43 2.3.7.41 22.433 22.3.5.29

2'.3.37 2.3'*11 2.19.47 22.3.149 29.32.9 2.5.181 2.911

B

G

2 6 2 13 13 3

2 2 3 2 13 9 3 2 2 2 5

P- 1

- -

3 5 2

2

13 3 3 2 6 3 7 3 2 2

3 9

3

6 3 2 2 3 2 6 3 5 6 3 3 4 2 2 5 14 14 2 2 2 3 11 2 2 3 4 2 2 5 2 2 3 3 2 19 2 3 2 3 3 5 2 5 11 11 3 3 2 5 7 7 3 3 2 3 2 2 6 3 2 2 11 11 3 2 2 3 2 2 2 2 3 3 3 6 3 3 3 3 3 6 2 2 2 2 2 4 7 7 2 6 5 5 2 ia 2 3 6 6 11 11 6 3 e 2

1831 1847 1861 1867 1871 1873 10 _ _ _ _ 1877 1'879 10 _ _ _ _ 1889 1901 f10 1907 -10 1913 1931 ____ 1933 -10 1949 f10 1951 *lO 1973 -10 1979 10 1987 ____ 1993 -_-1997 ____ 1999 -10 _ _ _ _ 2003 2011 ____ 2017 10 2027 ____ _ _ _ _ 2029 2039 ____ 2053 -10 2063 -10 2069 10 2081 10 2083 f10 2087 f10 2089 -10 2099 10 2111 10 2113 10 2129 10 2131 2137 2141 10 2143 2153 2161 10 2179 f10 2203 2207 2213 2221 10 2237 -10 . _ _ _ _2239 2243 .____ 2251 f10 2267 .____ 2269 f10 2273 .____ 2281 .____ . _ _ _2287 _ 2293 f10 _ _ _ _ _ 2297 _ _ _ _ _ 2309 2311 -10 2333 f10 2339 10 2341 -10 2347 k10 2351 .___ _ 2357 10 2371 10

f10 10 -10 -10

____

____

_____

2.3.5.61 2.13.71 22.3.5.31 2.3.311 2.5.1 1.17 2'. 32.13 22.7.67 2.3.313 25.59 22.5'. 19 2.953 23.239 2.5.193 22.3.7.23 22,487 2.3.52.13 22.17.29 2.23.43 2.3.331 2Js3.83 22.499 2.33.37 2.7.11.13 2.3.5.67 25.32.7 2.1013 22.3.132 2.1019 22 38.19 2.1031 22.11.47 25.5.13 2.3.347 2.7.149 23.32.29 2.1049 2.5.211 26.3.11 24.7.19 2.3.5.71 23.3.89 22.5.107 2.32.7.17 23.269 2'.33.5 2.32.112 2.3.367 2.1103 22.7.79 22.3.5.37 22.13.43 2.3.373 2.19.59 2.32.53 2.1 i.103 2?.34.7 25.71 23.3.5.19 2.32.127 22.3.191 23.7.41 22.577 2.3.5.7.1 1 2'. 11.53 2.7.167 22.32.5.13 2.3.17.23 2.52.47 22.19.31 2.3.5.79

B

-G

3 5 2 2 14 10 2 6 3 2 2 3 2 5 2 3 2 2 2 5 2 3 5 3 5 2 2 7 2 5 2 3 2 5 7 2 7 5 3 2 10 2 3 3 23 7 5 5 2 2 2 3 2 7 2 2 3 7 19 2 5 2 3 2 2 7 3 13 2 2

9 2 2 4 2 10 2 2 3 2 3 3 3 5 2 2 2 3 4 5 2 5 3 5 5 3 2 2 2 2 2 3 4 2 7 3 2 5 3 4 10 2 9 3 23 5 7 2 2 2 2 2 3 5 3 2 3 7 7 2 5 2 2 2 3

-

P-1

-

7

6 3 2 4

____

I

2377 2381 2383 k 110 0 2389 -10 2393 -10 2399 f10 2411 ____ . _ _ _ 2417 _ _ _ _ 2423 2437 2441 -10 2447 k10 _ _ _ _ 2459 _ _ _ _ 2467 2473 f10 2477 2503 ____ 2521 10 2531 ____ _ _ _ - 2539 _ _ _ _ 2543 2549 -10 2551 -10 _ _ _ _ 2557 2579 f10 2591 -10 2593 f10 2609 -10 2617 ____ 2621 10 2633 f10 2647 ____ 2657 -10 _ _ _ _ 2659 2663 ____ 2671 10 2677 -10 2683 f10 _ _ _ _ 2687 _ _ _ _ 2689 2693 f10 2699 f10 2707 10 2711 f10 2713 ____ 2719 10 2729 -10 2731 10 _ _ _ _ 2741 2749 f10 2753 ____ 2767 -10 2777 -10 2789 10 2791 -10 2797 f10 2801 f10 2803 ____ 2819 2833 ____ 2837 f10 2843 f10 _ _ _ _ 2851 2857 ____ 2861 10 2879 f10 2887 -10 2897 -10 ____ 2903 2909 10

____

____

-2.3.397 22.3.199 23.13.23 2.11.109 2.5.241 2'*151 2.7.173 22.3.7.29 23.5.61 2.1223 2.1229 2.32.137 23.3.103 22.619 2.32.139 23.32.5.7 2.5.1 1.23 2.333 47 2.31.41 22.72. 13 2.3.52.17 22.32.71 2.1289 2.5.7.37 25.3' 24. 163 23.3.109 22.5.131 28.7.47 2.33.72 25.83 2.3.443 2.113

5.673 2.19.71 2.3.11.41 2.5.271 23.3.113 2.32.151 23.11.31 2.3.5.7.13 22.5.137 22.3.229 26.43 2% 46 1 28.347 22.17.41 2.32.5.31 22.3.233 24.52.7 C3Y467 2.1409 24.3.59 22.709 2.72.29 2:3; 51-19 23.3.7.17 22.5.11.13 2.1439 2.3.13.37 2'*181 2.1451 22.727

B

-G

5 3 5 2 3 11 6 3 5 2 6 5 2 2 5 2 3 17 2 2 5 2 6 2 2 7 7 3 5 2

5

-

3 3 3

2 5 7 2 2 5 19 2 2 2 7 5 3 3 3 2 6 3 3 3 2 6 2 3 2 2 5 2 2 2 11 2 7 5

3 5 2

3

13 2 3 2 3 3 2 2 6 2 3 4 5 2 2 17 3 4 2 2 2 2 3 2 7 3 5 2 3 2 3 4 2 5 2 4 3 19 2 3 4 2 5 2 3 5 2 6 3 9 3 2 7 2 3

4

3 5 2 4 4 11 2 2 2 3 2 2

866

COMBINATORIAL ANALYSIS

Primitive Roots, Factorization of p-1 Table 24.8 g, G denote the least positive and least negative (respectively) prjmitive roots of p .

whether 10, -10 both or neither are primitive roots. - - -

P -

P- 1

2917 2927 2939 2953 2957 2963 2969 2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 3089 3109 31 19 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221 3229 3251 3253 3257 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331 3343 3347 3359 3361 3371 3373 3389 3391 3407 3413 3433 3449 3457 3461 3462 3467 346E 3491 349s 3511 3515

22.36 2.7.11.19 2.13.113 23.31.41 22.739 2.1481 28.7.53 2.33.5.1 1 29 1499 23.3.53 2.5.7.43 2.3.503 2,1511 22.3.11*23 25.5.19

2.3'.19 2.23.67 2'. 193 22.3.7.37 2,1559 24*3,5*13 28.7' 2.3.17.31 -21583 2'. 32.11 28.3.5.53 2.38.59 2.5.11.29 2.1601 23,401 2'.3.67 22.5.7.23 22.3.269 243.13 22.3.271 2'. 11.37 2.32.181 2.3.5.109 2.17.97 22.3.52.11 2.3.19.29 2'.32:23 2.3.7.79 2.11.151 28.13 2.31.5.37 2.3.557 2.7.239 2.23.73 25.3.5.7 2.8.337 22.3.28 1 22.7.112 2.395.113

2.ii.iii

2'.853 23.3.11.13 23.431 2'. 33 22.5.173 2.3.577 2.1733 22.3.172 2.5.349 2.3.11.53 2.33.5.13 22.3.293

Q

-G

5 5 2 13 2 2 3 10 17 14 2 2 5 2 3

5 2 3 13 2 3

I

- - 1-

3

5 2 14 3 4 2 2 3 11 11 6 6 4 2 2 6 2 3 3 3 6 6 2 7 7 7 3 3 6 3 2 5 7 7 7 7 4 2 11 5 2 3 3 3 5 5 10 10 6 6 6 3 2 2 3 3 5 3 3 5 2 3 6 6 4 2 10 10 2 6 3 2 3 3 5 3 5 11 3 2 2 11 22 22 3 2 5 5 3 3 5 3 2 5 2 2 5 5 3 3 7 7 2 2 9 3 2 3 2 2 2 3 2 4 7 2 2 2

_--10_- -_ -_ _ _ 10 __

3539 3541 3547 3557 3559 -10 3571 ____ 3581 10 3583 10 3593 10 3607 3613 .____ 3617 . _ _ _ _ 3623 .____ 3631 -10 3637 -10 3643 -10 3659 . _ _ _ _ 3671 3673 -10 3677 . _ - _ _ 3691 3697 f10 -10 3701 10 3709 .____ 3719 .____ 3727 . - - - - 3733 ----3739 -10 3761 3767 3769 3779 f10 3793 10 3797 _ _ _ _ _ 3803 3821 f10 10 3823 -10 3833 10 3847 3851 f10 -10 3853 3863 f10 _ _ _ - - 3877 -10 3881 - _ _ - _ 3889 10 3907 10 3911 -10 3917 -10 3919 -__-3923 10 3929 -__-3931 3943 -110 -10 3947 10 3967 ----3989 4001 310 _ _ _ - - 4003 --_-4007 4013 f10 10 4019 -10 4021 4027 f10 _ _ _ - _ 4049 _ _ _ _ _ 4051 4057 -10 _ _ _ _ _ 4073

__ __ __ __ __ * - - - -

1

Q

-G

5 17 2 2 7 2 2 3 2 2 3 3 5 2 3 5 15 2 2 2 13 5 2 2 5 2 2 7 3 2 7 3 5 7 2 5 2 2 3 3 3 5 2 2

2 10 17 2 ____ 3 10 7 -___ 4 -10 2 -___ 2 -10 10 4 2 f10 2 ____ 3 -110 11 10 2 3 f10 2 10 10 -10 2 4 -10 3 10 2 5 f10 2 4 5 2 f10 2 f10 2 -10 2 10 2 -___ 5 -___ 3 ---10 2 7 ---10 3 5 ---2 -___ 3 -10 3 f10 9 ____ 3 f l O 10 2 4 --__ 2 10 2 2 13 11 - - _ _ 4 -10 2 -10 2 2 ____ 3 -10 3 4 -_-_ 10 9 3 -10 10 2 2 f10 3 -___ 4 -___ 10 2 2 10 4 2 -___ 6 -10 3 ____ 1'0 5 5 flO 3 flO

2.41.43 23.32.72 22,883 2.29.61 22.3.5.59 2.3a.igi 21.7.127 2.3.593 2.3.5.7.17 22-5.179 2.32.199 23.449 2.3.601 22.3.7.43 25.113 2.1811 2.3.5.112 22.32.101 2.3.607 2.31359 2.5.367 239 33.17 22.919 2.3"5*41 2.432 22*5'.37 22.32.103 2.11.132 2.3'.23 22.3.311 293.7.89 2'.5.47 2.79269 23.3.157 2.1889 2'. 3.79 22.13.73 2.1901 22.5.191 2.3.72. 13 23.479 293,641 2.52.7.11 22.32.107 2,1931

2.5.17.23 22.11.89 2.3.653 2.37.53 23.491 2.3.5.131 2.33.73 2.1973 2.3.661 21.997 25.53 2.3.23.29 2.2003 22.17.59 2.72.41 22.3.5.67 2.3.1 1.61 2'*11.23 2.3'*5' 23.3.132 23.509

5

2 13 11 2 13 2 3 2 3 2 3 2 6 2 3 2 5 2 2 2 3 3 10 5 3

____

____

____ ____ __ __ __ __ ____

____ ____ ____ ____ ____

____

4079 4091 4093 4099 4111 4127 4129 4133 4139 4153 4157 4159 4177 4201 4211 4217 4219 4229 4231 4241 4243 4253 4259 4261 4271 4273 4283 4289 4297 4327 4337 4339 4349 4357 4363 4373 4391 4397 4409 4421 4423 4441 4447 4451 4457 4463 4481 4483 4493 4507 4513 4517 4519 4523 4547 4549 4561 4567 4583 4591 4597 4603 4621 4637 4639 4643 4649 4651 4657 4663

s

denotes - -

P-1

-

Q

-G

2,2039 2.5.409 21.3.11.31 2.3.683 2.3.5.137 2.2063 26.3.43 22.1033 2.2069 22.3.173 22.1039 2.38.7.11 24.32.29 2'. 3.52.7 2.5.42 1 28.17.31

11 2 2 2 12 5 13 2 2 5 2 3 5 11 6 3 2 2 3 3 2 2 2 2 7 5 2 3 5

2 3 2 4 2 2 13 2 3 5 2 2 5 11 3 3 4 2 2 3 4 2 3 2 3 5 3 3 5 2 3 5 2 2 4 2 2 2 3 3 7 21 2 3 3 2 3 4 2 4 7 2 9 3 3 6

2.2141 20.67 28.3.179 2.3.7.103 24,271 2.32.241 22.1087 2'.3a.1la 2.3.727 22.1093 2.59439 21.7.167 28.1929 21.5.13.17 2.3.11.67 28.3.5.37 2.32.13.19 2.52.89 28.557 2.23.97 27.5.7 2.38.83 2'.1123 2.3.751 25.3.47 22.1129 2.32.2 51 2.7.17.1 9 2.2273 22.3-379 24.3.5.19 2.3.761 2.29.79 2.33.5.17 22.3.383 293.13.59 22.3.5.7.11 22.19.61 2.3.773 2.11.211 23.7.83 2.3.54.31 24.3.97 2.32.7.37

3 3

10 2 2 2 2 14 2 3 3 3 21 3 2 3 5 3 2 2 2 7 2 3 5 2 6 11 3 5 11 5 2 2 2 3 5 3 3 15 3

-

11

7 2 2 5 4 2 2 2 3 3 5 15 9

e

867

COMBINATORIAL ANALYSIS

Primitive Roots, Factorization of p-1 g, G denote the least positive and least negative (respectively) primitive roots of p .

whether 10, -10 both or neither are primitive roots. - -

4673 4679 4691 4703 4721 4723 4729 4733 4751 4759 4783 4787 4789 4793 4799 4801 4813 4817 4831 4861 4871 4877 4889 4903 4909 4919 4931 4933 4937 4943 4951 4957 4967 4969 4973 4987 4993 4999 5003 5009 5011 5021 5023 5039 5051 5059 5077 5081 5087 5099 5101 5107 5113 5119 5147 5153 5167 5171 5179 5189 5197 5209 5227 5231 5233 5237 5261 5273 5279 5281

P-1 26.73 2,2339 2.5.7.67 2.2351 2'. 5.59 2.3.787 23.3.197 22.7.132 2.9.19 3.3.13.61 2.3.797 2.2393 2'.3'.7.19 23.599 2.2399 26.3.52 22.3.401 24.7.43 2.3.5.7.23 2'. 35.5 2y5.487 22.23.53 23.13.47 2.3.19.43 22.3.409 2.2459 2.5.17.29 22.32-137 23.617 2.7.353 2-32.52.11 22.3.7.59 2.13.191 28933.23 2'. 11 * 113 2.31.277 27.3.13 2-3.72.17 2.41.61 24.313 2.3.5.167 22.5.251 2.34.31 2.11.229 2.8.101 2 3 - 5.127

2.2543 2.2549 22.3.5'. 17 2.3.23.37 28.32.71 - - .2.3.853 2.31.83 28.7.23 2.32.7.41 2.5.11.47

2.3.13.67 2.5.523 24.3.109 22.7.11.17 22.5.263 23.659 2-7.13.29 26.3.5.11

Q

-G

- 3 3 2 11 2 3 2 5 6 6 4 2 17 17 5 5 3 19 5 3 2 6 2 3 2 2 3 3 2 7 7 7 2 2 3 3 2 3 11 11 11 3 2 2 3 3 2 3 6 6 2 13 6 3 2 2 3 3 2 7 2 6 2 2 5 2 11 11 2 2 4 2 5 5 3 9 2 3 3 3 4 2 3 3 2 3 2 11 3 2 4 2 2 2 3 3 5 2 2 3 6 6 2 4 19 19 2 3 2 3 5 5 6 11 4 2 4 2 2 2 7 7 17 17 2 4 2 7 10 10 3 3 2 2 3 3 7 3 7 7

d

P

-

P- 1

5297 5303 5309 5323 5333 5347 5351 5381 5387 5393 5399 5407 5413 5417 5419 5431 5437 5441 5443 5449 5471 5477 5479 5483 5501 5503 5507 5519 5521 5527 5531 5557 5563 5569 5573 5581 5591 5623 5639 5641 5647 5651 5653 5657 5659 5668 5683 5689 5693 5701 571 1 5717 5737 5741 5743 5749 5779 5783 5791 5801 5807 5813 5821 5827 5839 5843 5849 5851 5857 5861

2'.331 2.113241 22.1327 2.3.887 22.31.43 2.35.11 2.52.107 22.5.269 2.2693 2'.337 2.2699 2.3.17.53 22.3.11.41 28.677 2.32.7.43 2.3.5.181 23.32.151 28.5.17 2.3.907 23.3.227 2.5.547 22.372 2.3.11.83 2.2741 22.53.11 2.3.7.131 2.2753 2.31.89 2'*3*5.23 2.32.307 2.597.79 22.3.463 2.38.103 28.3.29 22.7.199 22.32.5.31 2.5.13.43 2.3.937 2.2819 28.3.5.47 2.3.941 2.53.113 22.32.157 28.7.101 2.3.23.41 22.13.109 2.3.947 28.32.79 22,1423 22.33.19 2.5.571 22.1429 28.3.239 22.5.7.41 2.3'. 11.29 22.3,479 2.33.107 2.72.59 2.3.5.193 23.52.29 2,2903 22-1453 21.3.5.97 2.3.971 2.3.7.139 2.23.127 28.17.43 2.32.8.13 26.3.61 P.5.293

Q

-G

3 5 2 5 2 3 11 3 2 3 7 3 5 3 3 3 5 3 2 7 7 2 3 2 2 3 2 13 11 5 10 2 2 13 2 6 11 5 7 14 3 2 5 3 2 3 2 11 2 2 19 2 5 2 10 2 2 7 6 3 5 2 6 2 6 2 3 2

3 2 2 10 2 6 2 3 3 3 2 2 5 3 5 2 8 3 4 7 3 2 2 3 2 9 3 2 11 2 5 2 4 13 2 6 2 2 2 14 2 3 5 3 4 3 4 11 2 2 3 2 5 2 2 2 4 2 2 3 2 2 6 4 2 4 3 4 7 3

- -

7 3

P- 1

-

Q

-G

2.7.419 22.32.163 2.2939 28.3.5.72 23.11.67 2.3.227 2-32.7.47 2.2963 2.2969 28.3.31 22.5.13.23 2.41.73 2.3.7.11.13 2.5.601 22.1 1,137 22.3.503 - - _.. 2.3.19.53 2.3023 22.17.89 2.32.337 28.3.11123 2.3.1013 2a.761 2.3.5.7.29 22.8.61 26.191 28.32.5.17 2.5.613 22.3.7.73 2.37.83 2.3.52.41 2.3.13.79 22.1543 22.1549 2.3.1033 2.7.443 2.33.5.23 23.3.7.37 22.59311 22.31.173 2.1.347 24.1723 2.31.101 22.1567 2.3.6.11.19 21.3.523 2.7.449 - . -.. 2.47.67 21.32.52.7 2.5.631 229 1579 229.109 28.7.113 26.31.11 2.3.7.151 2'*397 2.11.17' 23.3.5.53 2.3.1061 22.38.59 2.3.1063 22.1597 21.3.13-41 22.3.6.107 2.38.7.17 24.13.31 2.3.51.43 21.3.71.11 28.809 24.34.5

5 2 11 31 3 5 2 5 2 7 3 2 3 2 2 5 5 5 2 2 10 17 3 7 2 3 7 2 5 5 3 3 2 2 3 2 2 5 3 2 5 8 5 2 11 2 7 2 10 7 2 2 3 10 3 3 13 19 3 2 2 2 2 6 3 3 3 2 3 7

3 2 2 31 3 2 4 2 3 7 3 3 9 4 2 5 6 2 2 4 10 7 3 11 2 3 7 3 5 2 7 6 2 2 2 3 4 5 3 2 2 3 2 2 17 2 2 3 10 2 2 3 3 10 2 3 2 19 2 2 4 2 2 6 6 3 6 2 3

P 5867 5869 5879 5881 5897 5903 5923 5927 5939 5953 5981 5987 6007 6011 6029 6037 6043 6047 6053 6067 6073 6079 6089 6091 6101 6113 6121 6131 6133 6143 6151 6163 6173 6197 6199 6203 6211 6217 6221 6229 6247 6257 6263 6269 6271 6277 6287 6299 6301 6311 6317 6323 6329 6337 6343 6353 6359 6361 6367 6373 6379 6389 6397 6421 6427 6449 6451 6469 6473 6481

denotes - -

- -

P -

Table 24.8 a

~

-

7

€

868

COMBINATORIAL A N A L Y S I S

Primitive Roots, Factorization of p-1 Table 24.8 g, G denote the least positive and least negative (respectively) primitive roots of p . denotes whether 10, -10 both or neither are primitive roots. - - - P 649 1 6521 6529 6547 6551 6553 6563 6569 6571 6577 6581 6599 6607 6619 6637 6653 6659 6661 6673 6679 6689 6691 6701 6703 6709 6719 6733 6737 6761 6763 6779 6781 6791 6793 6803 6823 6827 6829 6833 6841 6857 6863 6869 687 1 6883 6899 6907 6911 6917 6947 6949 6959 6961 6!7!i7 6971 6977 6983 6991 6997 7001 7013 7019 7027 7039 7043 7057 7069 7079 7103 7109

P- 1 2.5.11.59 28.5.163 27.3.17 2.3.1091 2.5'. 131 2'.3'. 7.13 2.17.193 23.821 2.3l.5.73 2'. 3,137 22.5.7.47 2.3299 2.3l.367 2.3.1 103 22.3.7.79 22. 1663 2.3329 22*3"5*37 2'. 3.139 2.32.7.53 2'*11.19 2.3.5.223 22.5'- 67 2.3.1117 22.3.13.43 2.3359 3'. 3'. 11.17 24.421 23.5*132 2.3.72.23 2133892'.3*5*113 2.5.7.97 2'*3.283 2.19.179 2.32.379 2,3413 2l.3.569 24.7.61 28*32*5*19 28,857 2.47.73 2'. 17.101 2.3.5.229 2-3.31.37 2,3449 2.3.1151 295.691 2'*7*13.19 2.23.151 22.31.193 2.7'.71 24.3*5.29 2.3'.43 2.5.17.41 2'. 109 2,3491 2.3-5.233 2'.3*11*53 23.5a*7 22.1753 2.1 12.29 2.3.1171 2*32.17*23 2.7.503 24.32.72 22.3.19.31 2.3539 2953.67 2'*1777

B

2 6 7 2 17 10 5 3 3 5 14 13 3 2 2 2 2 6 5 7 3 2 2 5 2 11 2 3 3 2 2 2 7 10 2 3 2 2 3 22 3 6 2 3 2 2 2 7 2 2 2 7 13 5 2 3 5 6 5 3 2 2 2 3 2 5 2 7 5 2

I

-G

- -

-I-

3 _---6 _____ 7 _____ 4 2 -10 11 10 10 3 7 5 _____ 14 _ ~~~. ____ 2 -10 2 4 10 2 _____ 2 _____ 10 3 6 5 5 3 4 I0 1 1 2 f10 2 10 2 f10 2 -10 2 3 +lO 3 4 3 10 2 3 --_-10 f 1 0 3 -10 2 3 2 *to 3 f10 22 11 3 f10 2 10 2 f10 9 -10 4 -10 3 10 /I 4 ----2 -10 2 _____ 3 -10 2 *lO 3 -10 13 _ _ _ _ _ 11 13 10 4 10 3 f10 2 10 2 -10 5 3 _____ 2 3 10 4 _____ 2 -____ 4 5 f10 2 f10 2 -10 2 I0 2 f10

1

_____

~

_____

_____

1

I

_____

_____

_____

P-1 7121 7127 7129 7151 7159

7211 7213 7219 7229 7237 7243 7247 7253

7321 7331 7333 7349 7351 7369 7393 7411 7417 7433 7451 7457 7450 7477 748i 7487 7489 7499 7507 7517 7523 7529 7537 7541 7547 7549 7559 7561 7.573 7577 7583 7589 7591 7603 7607 7621 7639 7649 7669 7673 7681 7687 7691 7699 7703 7717 7723 7727

2'.5.89 2.7.509 2'.3'.11 2.51.1 1.13 2.3.1193 28.3.13.23 2.3593 23.29.31 2.3.1201 2.5.7 * 103 28.3.601 2.32.40 1

2.3623 22.72.37 2.11.33 1 27-3.19 2.13-281 22.3'.7.29 28.3.5.61 2.5-733 ~22.3.13.47 229 11 167 2 *3* 5'.7' 2'*3*307 9

23.5.11.17 2.19-197 20.329 13 2923,163 2.3'. 139 2211879 2.3761 2'*941 2'.3.157 22.5.13.29 2.78.11 22.3.17.37 2y3779 23.3'. 5.7 2'*3.631 23.947 2.17.223 22.7.27 1 2.3.5.li.23 2.3.7.181 2.3803 22.3.5.127 2.3.19.67 2.3821 26.239 2'.3'.7 1 2'.7.137 2'.3*5 2.32.7.61 2.5.769 2.3.1283 2.3851 22.3.843 2*3'*11.13 2.3863 ~

g

-G

3 5 7 7 3 10 2 3 3 2 5 2 2 2 2 5 2 2 5 2 6 7 2 6 2 6 7 5 2 5 3 2 3 2 2 A 5 7 2 2 2 2 3 7 2 2 2 13 13 2 3 5 2 6 2 5 2 7 2 3

3 2 ___-7 ----3 ___-2 -10 10 * l O 3 -10 3 f10 10 2 3 _____ 5 _____ 4 10 2 f10 2 4 -10 2 10 2 3 -10 5 3 -10 6 f10 7 ----4 __-_6 _____ 2 fl0 5 ----7 _____ 5 &lO 4 10 5 3 f10 4 10 3 f10 4 10 2 6 3 10 7 3 10 4 -10 2 3 -10 3 7 2 f10 3 -10 2 _____ 2 -10 13 2 _--__ 3 f10 2 10 2 2 -10 4 2 10 2 5 -10 3 -10 3 2 3 f10 17 2 10 10 3 5 10 2 10 2 6 _____ 2 10

-

2

3 17 6 2 3 5 2 3 5

P- 1

-

_____

_____ _____ _____

_____

__ __ __ __ __ _____ _____ _____ _____

_____ _____

_____

_____

__ __ __ __ __ _____ _____

7741 7753 7757 7758 7789 7793 7817 7823 7829 7841 7853 7867 7873 7877 7879 7883 7901 7907 7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081 8087 8089 8093 8101 8111 8117 8123 8147 8161 8167 8171 8179 8191 8209 8219 8221 8231 8233 8237 8243 8263 8269 8273 8287 8291 8293 8297 8311 8317 8329 8353 8363 8369 8377 8387 8389

g

22.31.5.43 7 23.3 * I. 7. I 9 10 22.7.277 2 2.P.431 3 22.3.11159 2 24.487 3 2'*977 3 2.3911 5 22. 19.103 2 26.5.72 12 22.13.15 1 2 2.32.19.23 3 2'*3.41 5 22.11.179 2 2+3*13*101 3 2.7.563 2 2'. 52.79 2 2.59.67 2 2.37.107 7 2.3.1321 3 22.3.661 2 28.31 -~ 3 22.1987 2 2.3.5'. 53 6 2.3.1 327 5 2'.3'.37 5 2a.7.11.13 3 2.31.5.89 - - . -. 14 24.3-167 5 2,4019 11 22. 3.1 1 6 1 2 2.3.17.79 3 22.2017 2

3

2.5.811 22.2029 2.31-131 2.4073 25.3*5* 17 2.3.1361 2.5.19.43 2.3.29947 2.31.5.7.13 24.33. 19 2.7.587 22-3.5.137 2.5.823 2'*3*7' 22.29.71 2-13.317 2.3'. 17 21.3.13.53 24. 11.47 2.391381 2.5.829 2'. 1049 22.2099 2.3.5.277 22*3'*7*11 23.3.347 26.31.29 2937.1 13 24.523 23.3.349 2.7.599 22.32.233

-G

- -

5 17 2 6 11 2 2 2 7 3 2 2 17 7 2 2 11 10 2 2 3 2 3 3 2 2 3 3 6 7 5 2

3

5 2 6

7 10 2 2 2 3 3 2 2 12 2 6 5 2 2 3 2 3 2c 2 3 2 2 10 5 3

7

5 2 2 5 2 3 2

17 2 6 2 2 3 3 CI

6

3

4 11 7 3 2 2 10 2 3 2 2 3 7 3 2 3 2 6

7 5 3 3 5 3 6

869

COMBINATORIAL ANALYSIS

Primitive Roots, Factorization of p-1 g,

P 8419 8423 8429 8431 8443 8447 846 1 8467 8501 8513 8521 8527 8537 8539 8543 8563 8573 8581 8597 U599 8609 8623 8627 8630 8641 8647 8663 8669 8677 8681 8689 8693 8699 8707 8713 8719 8731 8737 8741 8747 8753 8761 8779 8783 8803 8807 8819 882 1 8831 8837 8839 8849 8861 8863 8867 8887 8893 8923 8929 8933

Table 24.8

G denote the least positive and least negative.(respectively) primitive roots of p . E denotes whether 10, -10 both or neither are primitive roots. - - -

P-1 2.3.23.61 2.4211 21.72.43 2-3.5.281 2.32.7-67 2.41.103 22.32.5.47 2.3.17-83 2'.58*17 26.7.19 28.3.5-71 2.3.72.29 28.11.97 2-3.1423 2.4271 2.3-1427 21.2143 24.3.5.11.13 21.7.307 2.3.1433 25.269 2-32.479 2.19.227 22.3.7 19 28.38.5 2.3.11.131 2.61.71 22. 11* 197 29.3'. 24 1 28.5.7.31 24.39181 22.41.53 2.4349 2.3.1451 2'.3'*11' 2.3.1453 2932.5.97 25.3.7.13 29.5.19.23 2.4373 24.547 28.3.5.73 2.3.7.11.19 2.4391 2.38.163 2.7-17.37 2,4409 2%3'.5.7' 2.5.883 22.47' 2.38.49 1 24.7.79 5.5.443 2.3.7.211 2.11.13.31 2.3.1481 29.32.13*19 -2.3. i487 25.32.31 22.7.1 1.29

B

3 5 2 3 2 5

6 2 7 5 13 5 3 2 5 2 2 6 2 3 3 3 2 6 17 3

I

-G

6 2

1

2 4 2 6 4 7 5 13 2 3 4 2 4 2 6 2 2 3 2 3 6

17 2 5 2 2 2 2 2 15 15 13 13 2 2 2 3 5 7 5 5 3 5 2 4 5 5 2 2 2 3 3 3 23 23 11 22 5 2 2 4 5 2 2 3 2 2 5 7 2 2 3 2 3 3 2 2 3 9 2 3 3 2 5 5 4 2 11 11 2 2

____

8941 8951 8963 f10 8969 -10 8971 -10 10 8999 9001 -10 9007 9011 f10 9013 *lO . _ _ _ _ 9029 9041 9043 f10 9049 9059 10 -10 9067 . _ _ - - 9091 _ _ _ _ _ 9103 _ _ _ - - 9109 _ _ _ _ _ 9127 9133 9137 10 9151 -10 _ _ _ _ _ 9157 9161 9173 10 10 9181 9187 f10 _ _ _ _ - 9199 9203 _ _ _ _ _ 9209 9221 9227 10 9239 -10 9241 k10 9257 -10 10 9277 _ _ _ _ _ 9281 9283 f10 -10 9293 9311 A10 _ _ _ _ _ 9319 9323 10 9337 9341 9343 10 10 9349 9371 f10 9377 -10 _ _ _ _ _ 9391 9397 -10 9403 9413 f10 9419 10 9421 -10 9431 10 - - _ - - 9433 - - - - - 9437 _ _ _ - - 9439 9461 10

_____

_____

_____

_____

_____

_____

_____

P-1 23.3.5.149 2.52.179 2,4481 28.19.59 2.3.5.13.23

22.3.751 22.37.61 24.5.113 2.3.1 1,137 28.3.13.29 2.7.647 2.3.1511 2*32.5.101 2.3.37.41 29.3'. 11* 23 2.3'. 13' 22.3.761 24.571 2.3.52.6 1 21.3.7.109 28.5.229 22.2293 2'. 38. 5.17 2.3.1531 2.32.7.73 2.43.107 2a.1151 20.5.46 1 2.7.659 2.31.149 2'. 3.5.7.11 2'. 13.89 22*3*77J 20.5.29 2.3.7.13-17 2'*23*101 2.5.7'. 19 2.3.1553 2.59.79 28.3.389 29.5.467 2*3'*173 2**3-19*41 2-5.937 28.293 2.3.5.313 2'*3'.29 2.3.1567 22.13-181 2.17.277 22.3.5.157 2.5.23.41 28.3'. 131 23.7.337 2.3.1 13.13 22.5.11-43

g

6 13 2 3 2 7 7 3 2 5 2 3 3

7 2 3 3 6 10 3 6 3 3 6 3 2 2 3 3 2 3 2 2 19 13 3 5 3 2 2 7 3 2 5 2 5 2 2 3 3 2 3 3 2 2 7 5 2 22 3

-

II

,G

- -

6 2 3 3 4 2 7 2 4 5 2 3 6 7 4 6 5 2 10 2 6 3 2 6 3 2 2 6 2 3 3 2 3 2 13 3 5 3 4 2 2 2 3 5 2 2 2 3 3 2 2 6 3 3 2 3 5 2 7 3

-

- -

II

1-

____

9463 9467 9473 _ _ - - 9479 10 11 9491 9497 -10 9511 _ _ _ _ 9521 10 9533 _ _ _ _ 9539 9547 f10 _ _ _ _ 9551 -10 9587 _ _ _ _ 9601 10 9613 -10 9619 _ _ _ _ 9623 10 9629 9631 &lO 9643 . _ _ _ _ 9649 9661 A10 9677 9679 9689 9697 9719 -10 9721 -10 9733 -10 9739 . - - - - 9743 9749 510 -10 9767 -10 9769 9781 9787 *10 _ _ _ _ _ 9791 9803 _ - - - - 9811 9817 -10 9829 -10 9833 -10 9839 9851 9857 *10 10 9859 9871 10 9883 9887 *10 -10 9901 9907 9923 9929 _ _ _ _ _ 9931 9941 &10 -10 9949 _ _ _ _ _ 9967 - - - _ - 9973 -10 -10

____

_____ _____ _____

_____ _____ __ __ __ __ __ _____

P- 1 2.3.19.83 2.4733 28.37 2.7.677 2.5.13.73 28.1187 2.3.5.317 24.5.7.17 21.2383 2919.251 2.3.37.43 2.5'. 191 2.4793 21.3.5' 29.38.89 2.3.7.229 2.17.283 20.29.83 2.32.5.107 2.3.1607 2'.3'.67 22-3.5.7.23 22.41.59 2.3.1613 28.7.173 25.3 * 10 1 2.43.113 28.3'. 5 22,32311 2.39.541 2,4871 29.2437 2.19.257 - _. ~. 2a.3.11.37 21.3.5.163 2.3.7.233 2.5.11.89 2.132.29 - ~. 2.3'. 5 * 109 28.3.409 22.38. 7* 13 28.1229 2.4919 2.52.197 21.7.1 1 2.3.31.53 2.3.5.7.47 2.34.6 1 2.4943 22.3'. 5'. 11 2.3.13.127 2.11'941 28.17.73 2.3.5.331 22.5.7.71 22.3.829 2.3.11.151 22.33.277

B

-G

- 3 2 3 7 2 3

3 3 2 2 2

9 3 3 2 3

3

11

9 3 2 3 4 2

3 10 17 7 2 3 5 2 5 13 6 3 11 2 3 5 10 3 7 2 5 2 3 2 5 2 2 2 3 10 2 2 3 11

13 2 4 3 2 9 4 7 2 2 2 3 10 3 7 2 5 2 2 2 13 6 6 2 3 5 5 10 3 2 4 5 4 2 4 2 2 4 3 3 5 2 2 2 11

2 13 2 2 5 2 3 2 7 2 2 3

3

- -

870

COMBINATORIAL ANALYSIS PRIMES

T a l i l ~24.9 0 2 3 5 7 11

16

17

22

23

24

1993 1997 1999 2003 2011

2749 2753 2767 2777 2789

3581 3583 3593 3607 3613

4421 4423 4441 4447 4451

5281 5297 5303 5309 5323

6143 6151 6163 6173 6197

7001 7013 7019 7027 7039

7927 7933 7937 7949 7951

8837 8839 8849 8861 8863

9739 9743 9749 9767 9769

10663 10667 10687 10691 10709

11677 11681 11689 11699 11701

12569 12577 12583 12589 12601

13513 13523 13537 13553 13567

14533 14537 14543 14549 14551

19 16411 16417 16421 16427 16433

21

1229 1231 1237 1249 1259

18 15413 15427 15439 15443 15451

20

547 557 563 569 571

17393 17401 17417 17419 17431

18329 18341 18353 18367 18371

19427 19429 19433 19441 19447

20359 20369 20389 20393 20399

21391 21391 21401 21407 21419

9 10

13 17 19 23 29

577 587 593 599 601

1277 1279 1283 1289 1291

2017 2027 2029 2039 2053

2791 2797 2801 2803 2819

3617 3623 3631 3637 3643

4457 4463 4481 4483 4493

5333 5347 5351 5381 5387

6199 6203 6211 6217 6221

7043 7057 7069 7079 7103

7963 7993 8009 8011 8017

8867 8887 8893 8923 8929

9781 9787 9791 9803 9811

10711 10723 10729 10733 10739

11717 11719 11731 11743 11777

12611 12613 12619 12637 12641

13577 13591 13597 13613 13619

14557 14561 14563 14591 14593

15461 15467 15473 15493 15497

16447 16451 16453 16477 16481

17443 17449 17467 17471 17477

18379 iZ97 18401 18413 18427

19457 i9463 19469 19471 19477

20407 20411 20431 20441 20443

21433 21467 21481 21487 21491

11 12 13 14 15

31 37 41 43 47

607 613 617 619 631

1297 1301 1303 1307 1319

2063 2069 2081 2083 2087

2833 2837 2843 2851 2857

3659 3671 3673 3677 3691

4507 4513 4517 4519 4523

5393 5397 5407 5413 5417

6229 6247 6257 6263 6269

7109 7121 7127 7129 7151

8039 8053 8059 8069 8081

8933 8941 8951 8963 8969

9817 9829 9833 9839 9851

10753 10771 10781 10789 10799

11779 11783 11789 11801 11807

12647 12653 12659 12671 12689

13627 13633 13649 13669 13679

14621 14627 14629 14633 14639

15511 15527 15541 15551 15559

16487 16493 16519 16529 16547

17483 17489 17491 17497 17509

18433 18439 18443 18451 18457

19483 19489 19501 19507 19531

20477 20479 20483 20507 20509

21493 21499 21503 21517 21521

16 17 18 19 20

53 59 61 67 71

641 643 647 653 659

1321 1327 1361 1367 1373

2089 2099 2111 2113 2129

2861 2879 2887 2897 2903

3697 3701 3709 3719 3727

4547 4549 4561 4567 4583

5419 5431 5437 5441 5443

6271 6277 6287 6299 6301

7159 7177 7187 7193 7207

8087 8089 8093 8101 8111

8971 8999 9001 9007 9011

9857 9859 9871 9883 9887

10831 10837 10847 10853 10859

11813 11821 11827 11831 11833

12697 12703 12713 12721 12739

13681 13687 13691 13693 13697

14653 14657 14669 14683 14699

15569 15581 15583 15601 15607

16553 16561 16567 16573 16603

17519 17539 17551 17569 17573

18461 18481 18493 18503 18517

19541 19543 19553 19559 19571

20521 20533 20543 20549 20551

21523 21529 21557 21559 21563

21 22 23 24 25

73 79 83 89 97

661 673 677 683 691

1381 1399 1409 1423 1427

2131 2137 2141 2143 2153

2909 2917 2927 2939 2953

3733 3739 3761 3767 3769

4591 4597 4603 4621 4637

5449 5471 5477 5479 5483

6311 6317 6323 6329 6337

7211 7213 7219 7229 7237

8117 8123 8147 8161 8167

9013 9029 9041 9043 9049

9901 9907 9923 9929 9931

10861 10867 10883 10889 10891

11839 11863 11867 11887 11897

12743 12757 12763 12781 12791

13709 13711 13721 13723 13729

14713 14717 14723 14731 14737

15619 15629 15641 15643 15647

16607 16619 16631 I6633 16649

17579 17581 17597 17599 17609

18521 18523 18539 18541 18553

19577 19583 19597 19603 19609

20563 20593 20599 20611 20627

21569 21577 21587 21589 21599

26 27 28 29 30

101 103 107 109 113

701 709 719 727 733

1429 1433 1439 1447 1451

2161 2179 2203 2207 2213

2957 2963 2969 2971 2999

3779 3793 3797 3803 3821

4639 4643 4649 4651 4657

5501 5503 5507 5519 5521

6343 6353 6359 6361 6367

7243 7247 7253 7283 7297

8171 8179 8191 8209 8219

9059 9941 9067 9949 9091 9967 9103 9973 9109 10007

10903 10909 10937 10939 10949

11903 11909 11923 11927 11933

12799 12809 12821 12823 12829

13751 14741 15649 16651 16657 14747 15667 15661 16661 13759 14753 13757 13763 14759 15671 16673 13781 14767 15679 16691

17623 17627 17657 17659 17669

18583 18587 18593 18617 18637

19661 19681 19687 19697 19699

20639 20641 20663 20681 20693

21601 21611 21613 21617 21647

31 32 33 34 35

127 131 137 139 149

739 743 751 757 761

1453 1459 1471 1481 1483

2221 2237 2239 2243 2251

3001 3011 3019 3023 3037

3823 3833 3847 3851 3853

4663 4673 4679 4691 4703

5527 5531 5557 5563 5569

6373 6379 6389 6397 6421

7307 7309 7321 7331 7333

8221 8231 8233 8237 8243

9127 9133 9137 9151 9157

10009 10037 10039 10061 10067

10957 10973 10979 10987 10993

11939 11941 11953 11959 11969

12841 12853 12889 12893 12899

13789 13799 13807 13829 13831

14771 14779 14783 14797 14813

15683 15727 15731 15733 15737

16693 16699 16703 16729 16741

17681 17683 17707 17713 17729

18661 18671 18679 18691 18701

19709 19717 19727 19739 19751

20707 20717 20719 20731 20743

21649 21661 21673 21683 21701

36 37 38 39 40

151 157 163 167 173

769 773 787 797 809

1487 1489 1493 1499 1511

2267 2269 2273 2281 2287

3041 3049 3061 3067 3079

3863 3877 3881 3889 3907

4721 4723 4729 4733 4751

5573 5581 5591 5623 5639

6427 6449 6451 6469 6473

7349 7351 7369 7393 7411

8263 8269 8273 8287 8291

9161 9173 9181 9187 9199

10069 10079 10091 10093 10099

11003 11027 11047 11057 11059

11971 11981 11987 12007 12011

12907 12911 12917 12919 12923

13841 13859 13873 13877 13879

14821 14827 14831 14843 14851

15739 15749 15761 15767 15773

16747 16759 16763 16787 16811

17737 17747 17749 17761 17783

18713 18719 18731 18743 18749

19753 19759 19763 19777 19793

20747 20749 20753 20759 20771

21713 21727 21737 21739 21751

41 42 43 44 45

179 181 191 193 197

811 821 823 827 829

1523 1531 1543 1549 1553

2293 2297 2309 2311 2333

3083 3089 3109 3119 3121

3911 3917 3919 3923 3929

4759 4783 4787 4789 4793

5641 5647 5651 5653 5657

6481 6491 6521 6529 6547

7417 7433 7451 7457 7459

8293 8297 8311 8317 8329

9203 9209 9221 9227 9239

10103 10111 10133 10139 10141

11069 11071 11083 11087 11093

12037 12041 12043 12049 12071

12941 12953 12959 12967 12973

13883 13901 13903 13907 13913

14867 14869 14879 14887 14891

15787 15791 15797 15803 15809

16823 16829 16831 16843 16871

17789 17791 17807 17827 17837

18757 18773 18787 18793 18797

19801 19813 19819 19841 19843

20773 20789 20807 20809 20849

21757 21767 21773 21787 21799

46 47 48 49 50

199 211 223 227 229

839 853 857 859 863

1559 1567 1571 1579 1583

2339 2341 2347 2351 2357

3137 3163 3167 3169 3181

3931 3943 3947 3967 3989

4799 4801 4813 4817 4831

5659 5669 5683 5689 5693

6551 6553 6563 6569 6571

7477 7481 7487 7489 7499

8353 8363 8369 8377 8387

9241 9257 9277 9281 9283

10151 10159 10163 10169 10177

11113 11117 11119 11131 11149

12073 12097 12101 12107 12109

12979 12983 13001 13003 13007

13921 13931 13933 13963 13967

14897 14923 14929 14939 14947

15817 15823 15859 15877 15881

16879 16883 16889 16901 16903

17839 17851 17863 17881 17891

18803 18839 18859 18869 18899

19853 19861 19867 19889 19891

20857 20873 20879 20887 20897

21803 21817 21821 21839 21841

51 52 53 54 55

233 239 241 251 257

877 881 883 887 907

1597 l60i 1607 1609 1613

2371 2377 2381 2383 2389

3187 3191 3203 3209 3217

4001 4003 4007 4013 4019

4861 4871 4877 4889 4903

5701 5711 5717 5737 5741

6577 6581 6599 6607 6619

7507 7517 7523 7529 7537

8389 8419 8423 8429 8431

9293 9311 9319 9323 9337

10181 10193 10211 10223 10243

11159 11161 11171 11173 11177

12113 12119 12143 12149 12157

13009 13033 13037 13043 13049

13997 13999 14009 14011 14029

14951 14957 14969 14983 15013

15887 15889 15901 15907 15913

16921 16927 16931 16937 16943

17903 17909 17911 17921 17923

18911 18913 18917 18919 18947

19913 19919 19927 19937 19949

20899 20903 20921 20929 20939

21851 21859 21863 21871 21881

56 57 58 59 60

263 269 271 277 281

911 919 929 937 941

1619 1621 1627 1637 1657

2393 2399 2411 2417 2423

3221 3229 3251 3253 3257

4021 4027 4049 4051 4057

4909 4919 4931 4933 4937

5743 5749 5779 5783 5791

6637 6653 6659 6661 6673

7541 7547 7549 7559 7561

8443 8447 8461 8467 8501

9341 9343 9349 9371 9377

10247 10253 10259 10267 10271

11197 11213 11239 11243 11251

12161 12163 12197 12203 12211

13063 13093 13099 13103 13109

14033 14051 14057 14071 14081

15017 15031 15053 15061 15073

15919 15923 15937 15959 15971

16963 16979 16981 16987 16993

17929 17939 17957 17959 17971

18959 18973 18979 19001 19009

19961 19963 19973 19979 19991

20947 20959 20963 20981 20983

21893 21911 21929 21937 21943

61 62 63 64 65

283 293 307 311 313

947 953 967 971 977

I663 1667 1669 1693 1697

2437 2441 2447 2459 2467

3259 3271 3299 3301 3307

4073 4079 4091 4093 4099

4943 4951 4957 4967 4969

58G1 5807 5813 5821 5827

6679 6689 6691 6701 6703

7573 7577 7583 7589 7591

8513 8521 8527 8537 8539

9391 9397 9403 9413 9419

10273 10289 10301 10303 10313

11257 11261 11273 11279 11287

12227 12239 12241 12251 12253

13121 13127 13147 13151 13159

14083 14087 14107 14143 14149

15077 15083 15091 15101 15107

15973 15991 16001 16007 16033

17011 17021 17027 17029 17033

17977 17981 17987 17989 18013

19013 19031 19037 19051 19069

19993 19997 20011 20021 20023

21001 21011 21013 21017 21019

21961 21977 21991 21997 22003

66 67 68 69 70

317 331 337 347 349

983 991 997 1009 1013

1699 1709 1721 1723 1733

2473 2477 2503 2521 2531

3313 3319 3323 3329 3331

4111 4127 4129 4133 4139

4973 4987 4993 4999 5003

5839 5843 5849 5851 5857

6709 6719 6733 6737 6761

7603 7607 7621 7639 7643

8543 8563 8573 8581 8597

9421 9431 9433 9437 9439

10321 10331 10333 10337 10343

11299 11311 11317 11321 11329

12263 12269 12277 12281 12289

13163 13171 13177 13183 13187

14153 14159 14173 14177 14197

15121 15131 15137 15139 15149

16057 16061 16063 16067 16069

17041 17047 17053 17077 17093

18041 18043 18047 18049 18059

19073 19079 19081 19087 19121

20029 20047 20051 20063 20071

21023 21031 21059 21061 21067

22013 22027 22031 22037 22039

71 72 73 74 75

353 359 367 373 379

1019 1021 1031 1033 1039

1741 1747 1753 1759 1777

2539 2543 2549 2551 2557

3343 3347 3359 3361 3371

4153 4157 4159 4177 4201

5009 5011 5021 5023 5039

5861 5867 5869 5879 5881

6763 6779 6781 6791 6793

7649 7669 7673 7681 7687

8599 8609 8623 8627 8629

9461 9463 9467 9473 9479

10357 10369 10391 10399 10427

11351 11353 11369 11383 11393

12301 12323 12329 12343 12347

13217 13219 13229 13241 13249

14207 14221 14243 14249 14251

15161 15173 15187 15193 15199

16073 16087 16091 16097 16103

17099 17107 17117 17123 17137

18061 18077 18089 18097 18119

19139 19141 19157 19163 19181

20089 20101 20107 20113 20117

21089 21101 21107 21121 21139

22051 22063 22067 22073 22079

76 77 78 79 80

383 389 397 401 409

1049 1051 1061 1063 1069

1783 1787 1789 1801 1811

2579 2591 2593 2609 2617

3373 3389 3391 3407 3413

4211 4217 4219 4229 4231

5051 5059 5077 5081 5087

5897 5903 5923 5927 5939

6803 6823 6827 6829 6833

7691 7699 7703 7717 7723

8641 8647 8663 8669 8677

9491 9497 9511 9521 9533

10429 10433 10453 10457 10459

11399 11411 11423 11437 11443

12373 12377 12379 12391 12401

13259 13267 13291 13297 13309

14281 14293 14303 14321 14323

15217 15227 15233 15241 15259

16111 16127 16139 16141 16183

17159 17167 17183 17189 17191

18121 18127 18131 18133 18143

19183 19207 19211 19213 19219

20123 20129 20143 20147 20149

21143 21149 21157 21163 21169

22091 22093 22109 22111 22123

81 82 83 84 85

419 421 431 433 439

1087 1091 1093 1097 1103

1823 1831 1847 1861 1867

2621 2633 2647 2657 2659

3433 3449 3457 3461 3463

4241 4243 4253 4259 4261

5099 5101 5107 5113 5119

5953 5981 5987 6007 6011

6841 6857 6863 6869 6871

7727 7741 7753 7757 7759

8681 8689 8693 8699 8707

9539 9547 9551 9587 9601

10463 10477 10487 10499 10501

11447 11467 11471 11483 11489

12409 12413 12421 12433 12437

13313 13327 13331 13337 13339

14327 14341 14347 14369 14387

15263 15269 15271 15277 15287

16187 16189 16193 16217 16223

17203 17207 17209 17231 17239

18149 18169 18181 18191 18199

19231 19237 19249 19259 19267

20161 20173 20177 20183 20201

21179 21187 21191 21193 21211

22129 22133 22147 22153 22157

86 87 88 89 90

443 449 457 461 463

1109 1117 1123 1129 1151

1871 1873 1877 1879 1889

2663 2671 2677 2683 2687

3467 4271 3469 4273 3491 4283 3499 4289 3511 2297

5147 5153 5167 5171 5179

6029 6037 6043 6047 6053

6883 6899 6907 6911 6917

7789 7793 7817 7823 7829

8713 8719 8731 8737 8741

9613 9619 9623 9629 9631

10513 10529 10531 10559 10567

11491 11497 11503 11519 11527

12451 12457 12473 12479 12487

13367 13381 13397 13399 13411

14389 14401 14407 14411 14419

15289 15299 15307 15313 15319

16229 16231 36249 16253 16267

17257 17291 17293 17299 17317

18211 18217 18223 18229 18233

19273

20219

19301 19289 19309 19319

20233 20231 20249 20261

2122l 21227 21247 21269 21277

22159 22171 22189 22193 22229

91 92 93 94 95

467 479 487 491 499

1153 1163 1171 1181 1187

1901 1907 1913 1931 1933

2689 2693 2699 2707 2711

3517 3527 3529 3533 3539

4327 4337 4339 4349 4357

5189 5197 5209 5227 5231

6067 6073 6079 6089 6091

6947 6949 6959 6961 6967

7841 7853 7867 7873 7877

8747 8753 8761 8779 8783

9643 9649 9661 9677 9679

10589 10597 10601 10607 10613

11549 11551 11579 11587 11593

12491 12497 12503 12511 12517

13417 13421 13441 13451 13457

14423 14431 14437 14447 14449

15329 15331 15349 15359 15361

16273 16301 16319 16333 16339

17321 17327 17333 17341 17351

18251 18253 18257 18269 18287

19333 19373 19379 19381 19387

20269 20287 20297 20323 20327

21283 21313 21317 21319 21323

22247 22259 22271 22273 22277

96 97 98 99 100

503 509 521 523 541

1193 1201 1213 1217 1223

1949 1951 1973 1979 1987

2713 2719 2729 2731 2741

3541 3547 3557 3559 3571

4363 4373 4391 4397 4409

5233 5237 5261 5273 5279

6101 6113 6121 6131 6133

6971 6977 6983 6991 6997

7879 7883 7901 7907 7919

8803 8807 8819 8821 8831

9689 9697 9719 9721 9733

10627 10631 10639 10651 10657

11597 11617 11621 11633 11657

12527 12539 12541 12547 12553

13463 13469 13477 13487 13499

14461 14479 14489 14503 14519

15373 15377 15383 15391 15401

16349 16361 16363 l63b9 16381

17359 17377 17383 17387 17389

18289 18301 18307 18311 18313

19403 19391

20333 20341

19417 19421 19423

20347 20353 20357

21341 21347 21377 21379 21383

22279 22283 22291 22303 22307

1

2 3 4 5 6 7

871

COMBINATORIAL ANALYSIS PRIM E;S 28513 28517 28537 28541 28547

:12 29453 29473 29483 29501 29527

27527 21529 27539 21541 27551

28549 28559 28571 28573 28579

26501 26513 26539 26557 26561

27581 27583 27611 27617 27631

25519 25583 25589 25601 25603

26513 26591 26597 26627 26633

24527 24533 24547 24551 24571

25609 25621 25633 25639 25643

23603 23609 23623 23627 23629

24593 24611 24623 24631 24659

22651 22669 22679 22691 22697

23633 23663 23669 23671 23677

36 37 38 39 40

22699 22109 22717 22721 22727

41 42 43 44 45

:%I

24.9

86

97

:I8

39

IU

11

12

48

14

43

4ti

4;

32611 32621 32633 32647 32653

33617 33619 33623 33629 33637

34651 34667 34613 34679 34681

35711 35797 35801 35803 35809

36787 36791 36793 36809 36821

31831 37847 37853 37861 37871

38923 38933 38953 38959 38911

39979 39983 39989 40009 40013

41113 41117 41131 41141 41143

42083 42089 42101 42131 42139

43063 43067 43093 43103 43117

44203 44201 44221 44249 44251

45317 45319 45329 45337 45341

46451 46457 46471 46477 46489

47533 41543 47563 41569 47581

31663 31661 31681 31699 31721

32687 32693 32707 32113 32717

33641 33647 33619 33703 33713

34693 34103 34721 34729 34739

35831 35837 35839 35851 35863

36833 36847 36857 36811 36877

37819 37889 37891 31907 37951

38977 38993 39019 39023 39041

40031 40037 40039 40063 40087

41149 41161 41117 41179 41183

42157 42169 42179 42181 42187

43133 43151 43159 43117 43189

44263 44267 44269 44273 44279

45343 45361 45371 45389 45403

46499 46501 46511 46523 46549

47591 47599 41609 47623 47629

30697 30103 30701 30713 30727

31123 31721 31129 31741 31751

32719 32149 32171 32779 32783

33721 33739 33749 33751 33757

34147 34757 34759 34763 34181

35869 35879 35891 35899 35911

36881 36899 36901 36913 36919

37957 37963 37967 37987 37991

39043 19047 39019 39089 39097

40093 40099 40111 40123 40127

41189 42193 41201 42197 41203 42209 41221 41213 42223 42221

43201 43207 43223 43237 43261

44281 44293 44351 44351 44371

45413 45427 45433 45439 45481

4h559 ... 46567 46573 46589 46591

47639 47653 47657 41659 47681

29633 29641 29663 29669 29671

30157 30763 30113 30781 30803

31769 31111 31193 31799 31817

32189 32797 32801 32803 32831

33761 33769 '33773 33791 33197

34807 34819 34841 34843 34847

35923 35933 35951 35963 35969

36923 36929 36931 36943 36941

31993 37997 38011 38039 38041

39103 39107 39113 39119 39133

40129 40151 40153 40163 40169

41227 41231 41233 41243 41251

42227 42239 42257 42281 42283

43271 43283 43291 43313 43319

44381 44383 44389 44417 44449

45491 45497 45503 45523 45533

46601 46619 46633 46639 46643

47699 47701 47711 47713 47717

28657 28661 28663 28669 28687

29683 29717 29123 29141 29753

30809 30811 30829 30839 30841

31841 31849 31859 31873 31883

32833 32839 32843 32869 32887

33809 33811 33821 33829 33851

34849 34811 34877 34883 34897

35977 35983 35993 35999 36107

36973 36919 36997 37003 37013

38053 38069 38083 38113 38119

39139 39157 39161 39163 39181

40111 40189 40193 40213 40231

41263 41269 41281 41299 41333

42293 42299 42307 42323 42331

43321 43331 43391 43397 43399

44453 44483 44491 44497 44501

45541 45553 45551 45569 45587

46649 46663 46619 46681 46681

47137 47141 41143 47117 47779

27743 27749 27151 27763 27767

28697 28703 28711 28723 28729

29759 29761 29789 29803 29819

30851 30853 30859 30869 30871

31891 .31907 31957 31963 31973

32909 32911 32911 32933 32939

33857 33863 33871 13aa9

34913 34919 34939 14949

36011 36013 36011 36131 36961

37C19 37321 31039 3794') 37357

38149 39153 38157 ?5177 39a3

59131 39199 39219 33217 57227

40237 40241 40253 40277 40283

41341 41351 41351 41381 41387

42337 42349 42359 42373 42379

43403 43411 43421 43441 43451

44501 44519 44531 44533 44537

45589 45599 45613 45631 45641

46691 46103 46723 46121 46747

47791 47797 41807 47809 47819

26713 26717 26123 26129 26731

27773 27719 27791 27793 27799

28751 28153 28759 28771 28789

29833 29831 29851 29863 29867

30881 30893 30911 30931 30937

31981 31991 32003 32009 32027

32941 32951 32969 32971 32983

33911 33923 33931 33937 33941

34963 34981 35023 35027 35051

36067 36013 36083 36097 36107

37061 37087 37091 37117 37123

38189 38197 38201 38219 38231

39229 39233 39239 39241 39251

40289 40343 40351 40351 40361

41389 41399 41411 41413 41443

42391 42397 42403 42407 42409

43457 43481 43487 43499 43511

44543 44549 44563 44579 44587

45659 45667 45673 45611 45691

46151 46757 46769 46771 46807

47831 41843 47851 47869 47881

25759 25763 25711 25793 25199

26737 26759 26177 26783 26801

27803 27809 27817 27823 27821

28793 28807 28813 28811 28837

29873 29879 29881 29917 29921

30941 30949 30971 30977 30983

32029 32051 32057 32059 32063

32987 32993 32999 33013 33323

33961 33967 33997 34019 34031

35053 35059 35069 35081 35083

36109 36131 36137 36151 36161

37139 37159 37171 37181 37189

38237 38239 38261 38273 38281

39293 39301 39313 39317 39323

40367 40423 40427 40429 40433

41453 41461 41479 41491 41507

42433 42437 42443 42451 42457

43541 43543 43573 43577 43579

44617 44621 44623 44633 44641

45697 45707 45731 45151 45757

46811 46817 46819 46829 46831

47903 47911 47917 47933 41939

24781 24793 24799 24809 24821

25801 25819 25841 25847 25849

26813 26821 26833 26839 26849

21847 27851 27883 21893 21901

28843 28859 28867 28871 28819

29927 29947 29959 29983 29989

31013 31019 31033 31039 31051

32069 32077 32083 32089 32099

33029 33031 33349 33053 33011

34033 34039 34051 34061 34123

35089 35099 35101 35111 35117

36187 36191 36209 36211 36229

37199 37201 37217 37223 37243

38287 38299 38303 38317 38321

39341 39343 39359 39361 39371

40459 40471 40483 40487 40493

41513 41519 41521 41539 41543

42461 42463 42461 42473 42487

43591 43597 43607 43609 43613

44647 44651 44657 44683 44687

45763 45767 45779 45817 45821

46853 46861 46867 46877 46889

47947 47951 47963 47969 47977

23789 23801 23813 23819 23821

24841 24847 24851 24859 24677

25861 25873 25889 25903 25913

26861 26863 26879 26861 26891

27917 27919 27941 27943 27947

28901 28909 28921 28927 28933

30011 30013 30029 30047 30059

31063 31069 31079 31081 31091

32117 32119 32141 32143 32159

33073 33083 33391 33101 33113

34121 34129 34141 34147 34157

35129 35141 35149 35153 35159

36241 36251 36263 36269 36277

37253 37273 37271 37307 37309

38327 38329 38333 38351 38311

39373 39383 39391 39409 39419

40499 40507 40519 40529 40531

41549 41579 41593 41591 41603

42491 42499 42509 42533 42557

43627 43633 43649 43651 43661

44699 44701 44711 44129 44141

45823 45827 45833 45841 45853

46901 46919 46933 46957 46993

47981 48011 48023 48029 48049

22853 22859 22861 22811 22817

23831 23833 23851 23869 23873

24889 24907 24917 24919 24923

25919 25931 25933 25939 25943

26893 26903 26921 26927 26947

27953 21961 21967 27983 27997

28949 28961 28979 29009 29011

30071 30089 30091 30097 30103

31121 31123 31139 31147 31151

32173 32183 32189 32191 32203

33119 33149 33151 33161 33179

34159 34171 34183 34211 34213

35111 35201 35221 35227 35251

36293 36299 36307 36313 36319

37313 38377 39439 40543 40559 37337 37321 38393 38431 39451 39443 40577 37339 38441 39461 40583 37351 38449 39499 40591

41609 41611 41611 41621 41621

42569 42571 42577 42589 42611

43669 43691 43711 43117 43721

44753 44771 44773 44777 44189

45863 45869 45887 45893 45943

46997 47017 41041 47051 47057

48073 48019 48091 48109 48119

56 57 58 59 60

22901 22907 22921 22937 22943

23879 23887 23893 23899 23909

24943 24953 24961 24971 24977

25951 25969 25981 25997 25999

26951 26953 26959 26981 26987

28001 28019 28021 28031 28051

29021 29023 29027 29033 29059

30109 30113 30119 30133 30131

31153 31159 31117 31181 31183

32213 32233 32237 32251 32251

33181 33191 33199 33203 33211

34217 34231 34253 34259 34261

35257 35267 35219 35281 35291

36341 36343 36353 36313 36383

37361 37363 37369 31319 37397

38453 38459 38461 38501 38543

39503 39509 39511 39521 39541

40597 40609 40621 40631 40639

41641 41647 41651 41659 41669

42641 42643 42649 42667 42617

43753 43159 43777 43781 43783

44791 44809 44819 44839 44843

45949 45953 45959 45971 45979

47059 47087 47093 47111 47119

48121 48131 48157 48163 48179

61 62 63 64 65

22961 22963 22973 22993 23003

23911 23917 23929 23957 23971

24979 24989 25013 25031 25033

26003 26017 26021 26029 26041

26993 27011 27017 27031 27043

28057 28069 28081 28087 28097

29063 29071 29101 29123 29129

30139 30161 30169 30181 30187

31189 31193 31219 31223 31231

32261 32297 32299 32303 32309

33223 33247 33287 33289 33301

34267 34213 34283 34291 34301

35311 35317 35323 35321 35339

36389 36433 36451 36457 36461

37409 37423 37441 37447 31463

38557 38561 38567 38569 38593

39551 39563 39569 39581 39601

40693 40697 40699 40109 40739

41681 41687 41719 41129 41737

42683 42689 42697 42101 42703

43787 43789 43793 43801 43853

44851 44867 44879 44887 44893

45989 46021 46027 46049 46051

47123 47129 41131 4i143 47147

48187 48193 48197 48221 48239

66 67 68 69 70

23011 23017 23021 23027 23029

23977 23981 23993 24001 24007

25037 25057 25073 25087 25097

26053 26083 26099 26107 26111

27059 21061 21067 27013 27377

28099 28109 28111 20123 28151

29131 29137 29147 29153 29161

30191 30203 30211 30223 30241

31237 31241 31249 31253 31259

32321 32323 32327 32341 32353

33311 33311 33329 33331 33343

34303 34313 34319 34327 34331

35353 35363 35381 35393 35401

36469 36413 36479 36493 36491

37483 31489 37413 37iCi 37507

36b33 3ab99 38611 36b29 38t39

33619 31623 31631 33059 39007

40151 40759 40763 40771 40787

41759 41761 41171 41177 41801

42709 42119 42727 42737 42743

43867 43889 43891 43913 43933

44909 44917 44927 44939 44953

46061 46073 46091 46093 46099

47149 47161 47189 47207 47221

48247 48259 48271 48281 48299

71 72 73 74 75

23039 23041 23053 23057 23059

24019 24023 24029 24043 24049

25111 25117 25121 25127 25147

26113 26119 26141 26153 26161

21091 27103 27107 27109 27127

28163 28181 28183 28201 28211

29173 29179 29191 29201 29201

30253 30259 30269 30271 30293

31261 31271 31277 31301 31319

32359 32363 32369 32371 32377

33341 33349 33353 33359 33371

34351 34361 34361 34369 34381

35407 35419 35423 35431 35447

36523 36527 36529 36541 36551

37511 37517 37529 37531 37547

38651 38653 38669 38611 38677

39611 39679 39103 39709 39719

40801 40813 40819 40823 40829

41809 41813 41843 41849 41851

42751 42767 42773 42787 42793

43943 43951 43961 43963 43969

44959 44963 44911 44983 44987

46103 46133 46141 46147 46153

47237 47251 47269 47219 47287

48311 48313 48337 48341 48353

76 77 78 79 80

23063 23071 23081 23087 23099

24061 24071 24077 24083 24091

25153 25163 25169 25171 25183

26171 26171 26183 26189 26203

27143 21179 27191 27197 27211

28219 28229 28277 28279 28283

29209 29221 29231 29243 29251

30307 30313 30319 30323 30341

31321 31327 31333 31337 31357

32381 32401 32411 32413 32423

33391 33403 33409 33413 33427

34403 34421 34429 34439 34451

35449 35461 35491 35507 35509

36559 36563 36571 36583 36587

37549 37561 31567 37571 31573

38693 38699 38707 38711 38713

39727 39133 39149 39761 39769

40841 40841 40849 40853 40867

41863 41879 41887 41893 41897

42197 42821 42829 42839 42841

43973 43987 43991 43997 44017

45007 45013 45053 45061 45077

46111 46181 46183 46187 46199

47293 47297 41303 47309 47311

48371 48383 48397 48407 48409

81 82 83 84 85

23117 23131 23143 23159 23167

24091 24103 24107 24109 24113

25189 25219 25229 25237 25243

26209 26221 26237 26249 26251

21239 27241 21253 21259 21211

28289 28297 28301 28309 28319

29269 29287 29291 29303 29311

31341 30367 30389 30391 30403

31319 31387 31391 31393 31397

32429 32441 32443 32467 32479

33457 33461 33469 33419 33487

34469 34471 34483 34487 34499

35521 35527 35531 35533 35537

36599 36601 36629 36637 36643

31579 37589 37591 37601 37619

38723 38129 38137 38747 38749

39119 39791 39199 39821 39821

40819 40883 40897 40903 40927

41903 42853 41911 42859 41927 42863 41941 42901 42899

44021 44027 44029 44041 44053

45083 45119 45121 45127 45131

46219 46229 46237 46261 46271

47339 47351 47353 47363 41381

48413 48437 48449 48463 48473

86 87 88 89 90

23113 23189 23197 23201 23203

24121 24133 24137 24151 24169

25247 25253 25261 25301 25303

26261 26263 26267 26293 26291

27217 21281 27283 27299 21329

28349 28351 28387 28393 28403

29327 29333 29339 29347 29363

30421 30431 30449 30461 30469

31469 31471 31481 31489 31511

32491 32491 32503 32507 32531

33493 33503 33521 33529 33533

34501 34511 34513 34519 34531

35543 35569 35513 35591 35593

36653 36671 36677 36683 36691

37633 31643 37649 37657 31663

38767 38783 38191 38803 38821

39829 39839 39841 39841 39857

40933 40939 40949 40961 40973

41953 41957 41959 41969 41981

42923 42929 42937 42943 42953

44059 44011 44087 44089 44101

45137 45139 45161 45179 45181

46273 46279 46301 46301 46309

47387 41389 47407 47417 47419

48419 48481 48487 48491 48497

91 92 93 94 95

23209 23227 23251 23269 23279

24119 24181 24197 24203 24223

25307 25309 25321 25339 25343

26309 26317 26321 26339 26341

27331 21361 27367 27397 27407

28409 28411 28429 28433 28439

29383 29381 29389 29399 29401

30491 30493 30491 30509 30511

31513 31517 31531 31541 31543

32533 32531 32561 32563 32569

33541 33563 33569 33571 33581

34543 34549 34583 34589 34591

35597 35603 35617 35611 35611

36691 36709 36713 36121 36139

31691 37693 37699 31117 37747

38833 38839 38851 38861 38867

39863 39869 39811 39883 39887

40993 41011 41017 41023 41039

41983 41999 42013 42011 42019

42961 42967 42979 42989 43003

44111 44119 44123 44129 44131

45191 45191 45233 45247 45259

46321 46331 46349 46351 46381

47431 41441 41459 41491 47497

48523 48527 48533 48539 48541

96 97 98 99 100

23291 23293 23291 23311 23321

24229 24239 24247 24251 24281

25349 25357 25367 25373 25391

26351 26371 26387 26393 26399

27409 21427 27431 21437 27449

28447 28463 28411 28493 28499'

29411 29423 29429 29431 29443

30529 30539 30553 30551 30559

31547 31567 31513 31583 31601

32513 32579 32581 32603 32609'

33581 33589 33599 33601 33613

34603 34601 34613 34631 34649

35729 35731 35741 35753 35759

36149 36161 36767 36779 36781

37181 37183 31799 37811 37813

38813 38891 38903 38917 38921

39901 39929 39931 39953 39911

41041 41051 41057 41071 41081

42023 42043 42061 42011 42013

43013 43019 43031 43049 43051

44159 44171 44179 44189 44201

45263 45281 45289 45293 45307

46411 46439 46447 46441

.

47501 47507 41513 47521 87521

48563 48571 48589 48611 48593

25

26

21

28

%Y

:I0

1 2 3 4 5

22343 22349 22367 22369 22381

23327 23333 23339 23357 23369

24317 24329 24337 24359 24371

25409 25411 25423 25439 25447

26407 26417 26423 26431 26437

27457 27479 27481 27487 27509

6 1 8 9 10

22391 22397 22409 22433 22441

23371 23399 23411 23431 23447

24373 24379 24391 24407 24413

25453 25457 25463 25469 25411

26449 26459 26419 26489 26497

11 12 13 14 15

22441 22453 22469 22481 22483

23459 23473 23497 23509 23531

24419 24421 24439 24443 24469

25523 25537 25541 25561 25577

16 11 18 19 20

22501 22511 22531 22541 22543

23537 23539 23549 23557 23561

24473 24481 24499 24509 24517

21 22 23 24 25

22549 22567 22511 22573 22613

23563 23561 23581 23593 23599

26 27 28 29 30

22619 22621 22637 22639 22643

31 32 33 34 35

30577 30593 30631 30637 30643

31 31607 31627 31643 31649 31657

29531 29531 29567 29569 29573

30649 30661 30671 30611 30689

28591 28597 28603 28607 28619

29581 29587 29599 29611 29629

21641 27653 27673 27689 27691

28621 28621 28631 28643 28649

26641 26647 26669 26681 26683

21697 27701 21733 27131 27139

25657 25667 25673 25679 25693

26687 26693 26699 26701 26711

24671 24677 24683 24691 24691

25103 25711 25733 25741 25747

23687 23689 23719 23741 23143

24709 24733 24749 24763 24167

22739 22741 22751 22769 22777

23147 23753 23161 23767 23773

46 47 48 49 50

22783 22781 22807 22811 22811

51 52 53 54 55

1

872

COMBINATORIAL ANALYSIS I’RIMES

‘lallll* 24.9

69 69497 69499

70663 10667 70681 10109 10111

11 11119 71741 71161 71717 11189

72 12859 12869 12811 72883 12889

13999 14011 14021 14021 14041

75109 15133 15149 15161

67339 68413 69653 67343 68417 69661 67349 68483 69617 67369 68489 69691

70129 10153 10769 10183 70193

11807 11809 11821 71831 11843

12893 12901 12901 72911 12923

14051 14011 14011 14093 14099

75161 15169 15181 15193 15209

66211 67391 68491 69697 66293 61399 68501 69109 66.301 61409 68501 69731 66337 61411 68521 69139 66343 67421 68531 69761

10823 10841 10843 70849 10853

11849 71861 11867 11879 71881

72931 12931 12949 12953 12959

14101 14131 74143 14149 14159

15211 15211 15223 15221 15239

66347 66359 66361 66373 66311

68567 69119 68581 69809 68597 69821

10867 10817 10819 10891 10901

71887 11899 11909 11911 11933

12913 12911 12991 13009 13013

14161 14161 14111 14189 14191

15253 75269 15217 15289 15307

68611 68633 68639 68659 68669

69827 69829 69833 69841 69851

70913 10919 10921 10931 10949

71941 71947

71963

11911 11983

73019 13037 13039 13043 13061

14201 74203 14209 14219 74231

15323 15329 15337 15347 15353

68683 69859 68687 69877 68699 69899 68711 69911 68713 69929

10951 10951 10969 10919 70981

11981 71993 71999 12019 12031

13063 13079 73091 13121 73127

74251 14279 14281 74293 14291

15361 15311 15389 15391 15401

31 48941 49991 51131 52147 53233 54311 55469 56509 57551 58603 59663 60821 61987 63131 64217 65393 66509 61559 68129 69931

10991 10991 70999 71011 11023

12043 12041 12053 12013 12011

13133 13141 13181 73189 13231

14311 14311 14323 14353 14357

15403 15401 15431 15431 15419

60913 62053 63247 64301 65449 66571 61619 68191 70009 60917 62051 63211 64303 65479 66587 61631 68813 10019 60919 62071 63281 64319 65497 66593 67651 68819 10039

11039 71059 71069 71081 71089

12089 72091 72101 12103 12109

13243 13259 73217 13291 13303

14363 14311 14381 14383 14411

15503 15511 15521 15521 15533

53 51811 51821 51829 51839

54

55

56

57

62 G3 64 61 57193 58243 59369 60509 61631 62791 63823 51203 58211 59377 60521 61643 62801 63839 57221 58309 59387 60527 61651 62819 63841 57223 58313 59393 60539 61651 62821 63853 51241 58321 59399 60589 61661 62851 63851 58

59

60

65 66 67 68 65011 66107 61247 68389 65089 66109 67261 68399 65099 66137 67271 68437 65101 66161 61273 68443 65111 66169 61289 68441

48619 48623 48641 48649 48661

49661 49669 49681 49691 49111

$2 50161 50113 50171 50189 50821

6 48613 1 48611 8 48619 9 48131 10 48133

49121 49139 49141 49141 49751

50833 50839 50849 50851 50861

51859 51869 51811 51893 51899

52913 54059 55171 56249 52981 54083 55201 56263 52999 54091 55201 56261 53003 54101 55213 56269 53011 54121 55211 56299

11 12 13 14 15

48151 48751 48161 48161 48719

49183 49781 49189 49801 49801

50813 50891 50893 50909 50923

51901 51913 51929 51941 51949

53041 53051 53069 53017 53081

54133 54139 54151 54163 54161

55219 55229 55243 55249 55259

56311 56333 56359 56369

16 17 18 19 20

48781 48187 48799 48809 48817

49811 49823 49831 49843 49853

50929 51911 50951 51913 50951 51911 50969 51991 50911 52009

53089 53093 53101 53113 53111

54181 54193 54217 54251 54269

55291 55313 55331 55333 55331

56383 51349 58421 59491 60661 61813 62911 64019 56393 57367 58439 59509 60679 61819 62981 64033 56401 51313 58441 59513 60689 61837 62983 64037 56417 51383 58451 59539 60703 61843 62987 64063

21 22 23 24 25

48821 49811 50989 52021 53129 54211 55339 56431 57397 58411 59561 60127 61871 63029 64081 65269 66383 48823 49817 50993 52021 53141 54281 55343 56443 57413 58481 59567 60133 61879 63031 64091 65281 66403 48847 49891 51001 52051 53149 54293 55351 56453 51421 58511 59581 60131 61909 63059 64109 65293 66413 48851 49919 51031 52051 53161 54311 55373 56461 51451 58537 59611 60157 61921 63067 64123 65309 66431

Ni ..

1 2 3 4 5

51.

54001 55109 56191 54011 54013 54031 54049

55111 55127 55147 55163

56201 56209 56237 56239

62861 62869 62873 62897 62903

48869 49921 51041 48811 49931 51059 48883 49939 51061 48889 49943 51011 30 48907 49951 51109 48953 48913 48989 48991

49993 49999 50021 50023

51133 51131 51151 51151

52069 52081 52103 52121 52121 52153 52163 52111 52181

53113 53189 53191 53201 53231 53239 53261 53269 53219

54323 54331 54341 54361 54361 54311 54401 54403 54409

55399 55411 55439 55441 55451

58331 58363 58367 58369 58319

59407 59417 59419 59441 59443

60601 61673 60607 61681 60611 61687 60611 61703 60623 61717

51287 51301 57329 57331 56377 51347

58391 58393 58403 58411 58417

59441 59453 59461 59411 59473

60637 61729 62927 63971 60647 61751 62929 63997 60649 61151 62939 64007 60659 61781 62969 64013

63901 65123 66119 63907 65129 66191 63913 65141 66221 63929 65141 66239

60631 61723 62921 63949 65167 65111 65173 65119 65183

65203 65213 65239 65251 56431 51389 58453 59551 60119 61861 62989 64067 65267

55487 55501 55511 55529

56411 56419 56489 56501 56503 56519 56521 56531 56533

57481 51493 51503 51527 57529 51559 57511 57587 51593

58549 58561 58573 58519 58601

58613 58631 58657 58661

36 49003 50033 51169 52183 53281 54413 55541 56543 57601 58619 31 49009 50041 51193 52189 53299 54419 55541 56569 51637 58687 3 8 49019 50051 51197 52201 53309 54421 55519 56591 51641 58693 39 49031 50053 51199 52223 53323 54431 55589 56591 51649 58699 40 49033 50069 51203 52231 53321 54443 55603 56599 57653 58111

59621 59621 59629 59651 59659

60763 61949 63079 64153 60113 61961 63091 64157 60719 61967 63103 64171 60793 61979 63113 64187 60811 61981 63121 64189

59669 60859 61991 63149 59671 60869 62003 63119 59693 60887 62011 63197 59699 60889 62017 63199

59101 59123 59129 59143 59141

69539 69557

69593

63863 65119 66173 61301 68449 69623

51251 51259 51269 57211 51283

67427 67429 61433 61441 67453

67477 67481 67489 61493 48859 49921 51043 52061 53111 54319 55381 56413 57461 58543 59617 60761 61933 63073 64151 65323 66449 67499

26 21 28 29

32 33 34 35

52931 52951 52951 52963 51853 52961

64223 64231 64237 64271

65327 65353 65351 65311 65381 65407 65413 65419 65423

66457 66463 66467 66491

67511 61523 67531 61531 66499 61547 66523 66529 66533 66541

61561 61517 61579 61589

68539 69763 68543 69161

68737 68143 68749 68167

69941 69959 69991 69997

60899 62039 63211 64279 65437 66553 67601 68771 70001

60901 62047 63241 64283 65441 66569 67607 68177 10003

70

i:i

i4 75083

41 49031 42 49043 43 49051 44 49069 45 49081

50011 50081 50093 50101 50111

51211 51229 51239 51241 51251

52249 52253 52259 52261 52289

53353 54449 55609 56611 57661 58121 59753 60923 62081 63299 64327 65519 66601 67619 68821 70051 53359 54469 55619 56629 51679 58133 59111 60931 62099 63311 64333 65521 66617 61699 68863 70061 53311 54493 55621 56633 51689 58141 59119 60943 62119 63313 64313 65537 66629 67709 68879 10067 53381 54491 55631 56659 57691 58151 59791 60953 62129 63317 64381 65539 66643 61723 68881 70019 53401 54499 55633 56663 57109 58763 59191 60961 62131 63331 64399 65543 66653 61733 68891 10099

71119 11129 11143 11141 71153

12139 72161 12161 72169 12173

13309 13321 13331 13351 13361

14413 14419 14441 14449 74453

15539 15541 15553 15551 15511

46 49103 4 1 49109 48 49111 49 49121 50 49123

50119 50123 50129 50131 50141

51263 51283 51281 51301 51329

52291 52301 52313 52321 52361

53401 54503 55639 56611 5 3 4 1 1 54511 55661 56681 53419 54521 55663 56687 53431 54539 55667 56101 53441 54541 55673 56711

59809 61001 62137 63337 64403 65551 66683 67741 68897 70111

71161 71167 11171 71191 11209

72211 12221 72223 72227 12229

73363 13369 73319

73387 13411

14471 14489 14501 74509 14521

15571 15583 75611 15611 15619

51 52 53 54 55

50153 50159 50111 50201 50221

51343 51341 51349 51361

52369 52319 52381 52391

53419 53503 53501 53521

54559 54563 54511 54581

55691 55691 55711 55111

56731 56131 56141 56767

57173 51181 51781 57191

71233 11237 71249 11251 11261

12251 72253 12269 12211 72211

13421 13433 73453 13459 73471

14521 74531 74551 14561 14561

15629 15641 15653 75659 15679

56 49193 50221 51383 51 49199 50231 51401 58 49201 50261 51413 59 49201 50263 51419 60 49211 50273 51421

52433 52453 52451 52489 52501

53549 53551 53569 53591 53593

54583 54601 54617 54623 54629

55721 55133 55763 55787 55193

56773 56119 56183 56807 56809

51793 51803 51809 57829 51839

11263 71287 71293 71317 71321

12281 72301 12313 72337 12341

13411 73483 13517 73523 13529

14573 14581 14597 74609 14611

75683 15689 15103 15101 75709

61 62 63 64

52511 52511 52529 52541 52543

53591 53609 53611 53611 53623

54631 54647 54667 54613 54619

55199 55807 55813 55811 55819

56813 56821 56827 56843 56851

51841 51853 51859 51881

58979 58991 58991 59009 57899 59011

60031 60041 60077 60083 60089

61211 61223 61231 61253 61261

53629 54109 53633 54113 53639 54721 53653 54121 53651 54151

55823 55829 55837 55843 55849

56813 56891 56893 56891 56909

51901 51911 51923 51943 51947

60091 60101 60103 60107 60121

61283 61291 61297 61331

49139 49151 49169 49111 49111

49223 49253 49261 49271 65 49219

50281 50291 50311 50321 50329

57113 57119 51727 51731 51131

58171 58181 58789 58831 58889

59833 61007 62141 63347 64433 65551 66691 61151 68899 10117 59863 61021 62143 63353 64439 65563 66701 61157 68903 10121 59819 61031 62111 63361 64451 65519 66113 61759 68909 70123 59881 61043 62189 63361 64453 65581 66721 61763 68917 10139

51341 52363 53453 54541 55681 56113 51151 58897 59921 61051 62191 63317 64483 65587 66133 67117 68927 10141

51421 51431 51431 51439 51449

58901 58901 58909 58913

59929 59951 59951 59971

61057 61091 61099 61121

62201 62207 62213 62219

63389 64489 65599 66139 67783 68947 70151

63391 64499 65609 66749 61189 68963 70163 63397 64513 65611 66751 67801 68993 10111 63409 64553 65629 66763 61807 69001 70181

58921 59981 61129 62233 63419 64567 65633 66791 61819 69011 70183

58937 59999 61141 62273 63421 64517 65647 66191 61829 69019 10199

58943 60313 61151 62297 63439 64519 65651 66809 61843 69029 70201 58963 60017 61153 62299 63443 64591 65657 66821 67853 69031 10207 58961 60029 61169 62303 63463 64601 65677 66841 67867 69061 70223

67891 67901 61921 67931

67883 69067 69073 69109 69119

10229 10237 10241 70249 69127 70271

lb29 71333 71339 11341 11347

12353 72361 12379 12383 12421

13547 13553 13561 13511

73583

74623 74653 14681 14699 14101

15121 15131 75743 75161 15173

69143 10289

11353 11359 71363 71381

13589 13591 13607 73609 73613

74113 74111 14119 74729 14131

15181 75187 15193 15191 15821

13631 13643 13651

14141 14159 74761 74771 14719

15833 15853 15869 15883 15913

49291 49301 49331 493j3 49339

50333 51461 52553 50341 51413 52561 50359 51419 52567 50363 51481 52571 50317 51481 52519

63521 64661 65117 63527 64663 65719 63533 64667 65729 63541 64619 65731 63559 64693 65761

66889 66919 66923 66931 66943

61933 61939 67943 61957 67961

69149 10291 69151 10309 69163 70313 69191 10321

11 12 13 74 15

49363 49361 49369 49391 49393

50383 51503 52583 53681 54161 55811 56911 51913 59063 60133 61339 62467 63577 64709 65177 50381 51511 52609 53693 54173 55889 56921 57911 59069 60139 61343 62473 63587 64111 65789 50411 51511 52627 53699 54179 55891 56923 57991 59017 60149 61351 62477 63589 64747 65809 50417 51521 52631 53117 54181 55901 56929 58013 59083 60161 61363 62483 63599 64763 65827 50423 51539 52639 53719 54199 55903 56941 58027 59093 60167 61379 62491 63601 64781 65831

66941 66949 66959 66973 66971

61967 61979 61987 61993 68023

69193 69191 69203 69221 69233

51551 52661 51563 52613 51571 52691 51581 52691 80 49433 50503 51593 52109

53131 53159 53113 53111 53183

81 19451 50513 51599 82 49459 50521 51601 83 49463 50539 51613 84 49411 50543 51631 85 49481 50549 51631

53191 53813 53819 53831

56951 56957 56963 56983 56989

11431 11443 71453 11411 11473

12551 12559 72571 12613 12611

73699 13109 13121

14197 14821 74821 74831 14843

15931 15931 15941 75961 15919

70451 10459 70481 10481

12623 12643 12641 72649 72661

73121 73151 13757 13171 13783

14851 74861 74869 74873 74881

75983 15989 15991 15991 76001

63649 64853 63659 64811 63667 64877 63611 64819 63689 64891

67057 68099 69313 67061 68111 69311 67073 68113 69337 67079 68141 69341 67103 68147 69371

70489

71479 71483 71503 71521 11537

60317 61493 60331 61507 60337 61511 60343 61519 60353 61543

62617 62621 62633 62639 62653

63691 64901 65951 67121 63691 64919 65957 67129 63103 64921 65963 61139 63709 64927 65981 67141 63719 64937 65983 67153

68161 68171 68207 68209 68213

69379 10501 69383 70507 69389 10529 69401 10531 69403 10549

11549 11551 11563 11569 71593

12671 12613 12679 12689 12101

13819 73823 73841 73849 13859

14891 74897 14903 74923 14929

76003 16031 16039 76079 76081

58171 59239 60313 61547 62659 63127 64951 65993 67157 61169 67181 67187 61189

68219 68227 68239 68261 68219

69421 69431 69439 69457 69463

10571 10573 70583 70589 10607

11597 11633 71647 71663 71671

12101 12719 12721 12733 72739

13861 73811 73897 13901

74933 74941 14959 75011 75013

16091 16099 16103 16123 76129

50683 51161 52883 53939 55051 56131 51149 58211 59333 60443 61603 62731 63781 65021 66067 67211 50101 51169 52889 53951 55061 56149 51163 58211 59341 60449 61609 62743 63793 65029 66071 67213 50123 51181 52901 53959 55073 56167 57113 58229 59351 60457 61613 62153 63799 65033 66083 61217 50141 51197 52903 53987 55019 56171 51179 58231 59357 60493 61627 62761 63803 65053 66089 67219 50153 51803 52919 53993 55103 56179 51191 58237 59359 60497 61631 62773 63809 65063 66103 67231

68281 68311 68329 68351 68371

69467 69413 69481 69491 69493

70619 10621 70627 70639 10657

71693 11699 71707 71711 11113

12763 12761 72791 72811 12823

13939 73943 73951 73961 73913

15011 75029 15037 15041 75079

16147 16151 16159 16163 76201

49541 50599 51691 52811 53897 55001 56093 51101 49549 50627 51113 52831 53899 55009 56099 51119 49559 50641 51119 52859 53911 55021 56101 57131 49591 50651 51121 52861 53923 55049 56113 57139 49603 50671 51749 52819 53921 55051 56123 51143 49613 49627 49633 49639 49663

68059 69257 10429

73673 13619

62563 62581 62591 62597 62603

58067 58073 58099 58109 58111

59149 59159 59161 59183 59191

59201 59209 59219 59221 90 49531 50593 51683 52813 53891 54983 56081 51097 58169 59233

96 97 98 99 100

72493 12491 12503 72533 12541

61463 61469 61471 61483 61487

56993 56999 57031 51041 53849 54941 56009 51041 55967 55981 55991 56003

86 49499 50551 51641 52151 53857 54949 56039 51059 58129 87 49523 50581 51659 52769 53861 54959 56041 51073 58141 88 49529 50581 51613 52183 53881 54973 56053 51077 58151 89 49531 50591 51619 52801 53881 54979 56081 57089 58153

91 92 93 94 95

11399 71411 11413 71419 11429

67003 68041 69239 10393

54881 54901 54911 54919

58031 59101 60169 58043 59113 60209 58049 59119 60211 58057 59123 60223 58061 59141 60251

10321 70351 10373 70319 10381

63607 64783 65837 63611 64193 65839 63617 64811 65843 63629 64817 65851 63647 64849 65861

52111 52121 52121 52133 52141

55921 55927 55931 55933 55949

71389

12431 72461 72467 72469 72481

61381 62501 61403 62507 61409 62533 61411 62539 61441 62549

50441 50459 50461 50491

54829 54833 54851 54869 54811

62383 62401 62417 62423 61333 62459

63467 64609 65681 66851 63473 64613 65699 66853 63487 64621 65701 66863 63493 64627 65107 66877 63499 64633 65713 66883

66 61 68 69 70

1 6 49409 11 49411 18 49411 79 49429

59021 59023 59029 59051 59053

62311 62323 62327 62347 62351

60257 60259 60211 60289 60293

65881 65899 65921 65921 65929

67021 67033 67043 67049

58189 59243 60383 61553 62683 63737 64969 66029 58193 59263 60397 61559 62687 61743 64997 66037 58199 59213 60413 61561 62701 63761 65003 66041 58201 59281 60427 61583 62723 63773 65011 66041

68053 69247 10423 68071 69259 10439 68087 69263 10451

13681 73693

73883

873

COMBINATORIAL ANALYSIS

‘rsilit- 21.9

PHlhlES 76

7i

79

80

81

85

2 3 4 5

76231 76243 76249 76253

77369 77377 77383 77417

78497 78509 78511 78517

79631 79633 79657 79669

80747 80749 80761 80777

81839 81847 81853 81869

82913 82939 82963 82981

84137 84143 84163 84179

85247 85259 85297 85303

86389 86399 86413 86423

87559 87583 87587 87589

88811 88813 88817 88819

89891 89897 89899 89909

90997 91009 91019 91033

92179 92189 92203 92219

93199 93229 93239 93241

94379 94397 94399 94421

92 95443 95461 95467 95471 95479

6 7 8 9 10

76259 76261 76283 76289 76303

77419 77431 77447 77471 77477

78539 78541 78553 78569 78571

79687 79691 79693 79697 79699

80779 80783 80789 80803 80809

81883 81899 81901 81919 81929

82997 83003 83009 83023 83047

84181 84191 84199 84211 84221

85313 85331 85333 85361 85363

86441 86453 86461 86467 86477

87613 87623 87629 87631 87641

88843 88853 88861 88867 88873

89917 89923 89939 89959 89963

91079 91081 91097 91099 91121

92221 92227 92233 92237 92243

93251 93253 93257 93263 93281

94427 94433 94439 94441 94447

95483 95507 95527 95531 95539

96667 96671 96697 96703 96731

97859 97861 97871 97879 97883

99013 99017 99023 99041 99053

11 12 13 14 15

76333 76343 76367 76369 76379

77479 77489 77491 77509 77513

78577 78583 78593 78607 78623

79757 79769 79777 79801 79811

80819 80831 80833 80849 80863

81931 81937 81943 81953 81967

83059 83063 83071 83077 83089

84223 84229 84239 84247 84263

85369 85381 85411 85427 85429

86491 86501 86509 86531 86533

87643 87649 E7671 87679 87683

88883 88897 88903 88919 88937

89977 89983 89989 90001 90007

91127 91129 91139 91141 91151

92251 92269 92297 92311 92317

93283 93287 93307 93319 93323

94463 94477 94483 94513 94529

95549 95561 95569 95581 95597

96737 96739 96749 96757 96763

97919 97927 97931 97943 97961

99079 99083 99089 99103 99109

16 17 18 19 20

76387 76403 76421 76423 76441

77521 77527 77543 77549 77551

78643 78649 78653 78691 78697

79813 79817 79823 79829 79841

80897 80909 80911 80917 80923

81971 81973 82003 82007 82009

83093 83101 83117 83137 83177

84299 84307 84313 84317 84319

85439 85447 85451 85453 85469

86539 86561 86573 86579 86587

87691 87697 87701 87719 87721

88951 88969 88993 88997 89003

90011 90017 90019 90023 90031

91153 91159 91163 91183 91193

92333 92347 92353 92357 92363

93329 93337 93371 93377 93383

94531 94541 94543 94547 94559

95603 95617 95621 95629 95633

96769 96779 96787 96797 96799

97967 97973 97987 98009 98011

99119 99131 99133 99137 99139

21 22 23 24 25

76463 76471 76481 76487 76493

77557 77563 77569 77573 77587

78707 78713 78721 78737 78779

79843 79847 79861 79867 79873

80929 80933 80953 80963 80989

82013 82021 82031 82037 82039

83203 83207 83219 83221 83227

84347 84349 84377 84389 84391

85487 85513 85517 85523 85531

86599 86627 86629 86677 86689

87739 87743 87751 87767 87793

89009 89017 89021 89041 89051

90053 90059 90067 90071 90073

91199 91229 91237 91243 91249

92369 92377 92381 92383 92387

93401 93419 93427 93463 93479

94561 54573 94583 94597 94603

95651 95701 95707 95713 95717

96821 96823 96827 96847 96851

98017 98041 90047 98057 98081

99149 99173 99181 99191 99223

26 27 28 29 30

76507 76511 76519 76537 76541

77591 77611 77617 77621 77641

78781 78787 78791 78797 78803

79889 79901 79903 799P7 79939

81001 81013 81017 81019 81023

82051 82067 82073 82129 82139

83231 83233 83243 83257 83267

84401 84407 84421 84431 84437

85549 85571 85577 85597 85601

86693 86711 86719 86729 86743

87797 87803 87811 87833 07853

89057 89069 89071 89083 89087

90089 90107 90121 90127 90149

91253 91283 91291 91297 91303

92399 92401 92413 92419 92431

93481 93487 93491 93493 93497

94613 94621 94649 94651 94687

95723 95731 95737 95747 95773

96857 96893 96907 96911 96931

98101 98123 98129 98143 98179

99233 99241 99251 99257 99259

31 32 33 34 35

76543 76561 76579 76597 76603

77647 77659 77681 77687 77689

78809 78823 78839 78853 78857

79943 79967 79973 79979 79987

81031 81041 81043 81047 81049

82141 82153 82163 82171 82183

83269 83273 83299 83311 83939

84443 84449 84457 84463 84467

85607 85619 85621 85627 85639

86753 86767 86771 86783 86813

87869 87877 87881 87887 87911

89101 89107 89113 89119 89123

90163 90173 90187 90191 90197

91309 91331 91367 91369 91373

92459 92461 92467 92479 92489

93503 93523 93529 93553 93557

94693 94709 94723 94727 94747

95783 95789 95791 95801 95803

96953 96959 96973 96979 96989

98207 98213 98221 98227 98251

99277 99289 99317 99347 99349

36 37 38 39 40

76607 76631 76649 76651 76667

77699 77711 77713 77719 77723

78177 78887 78889 78893 78901

75197 79999 80021 80039 80051

31071 81077 81083 81097 81101

82189 82193 82207 82217 82219

83341 83357 83383 83389 83399

84481 84499 84503 84509 84521

85643 85661 85667 85669 85691

86837 86843 86851 86857 86861

87917 87931 87943 87959 87961

89137 89153 89189 89203 89209

90199 90203 90217 90227 90239

91381 91387 91393 91397 91411

92503 92507 92551 92557 92567

93559 93563 93581 93601 93607

94771 94777 94781 94789 94793

95813 95819 95857 95869 95873

96997 97001 97003 97007 97021

98257 98269 98297 98299 98317

99367 99371 99377 99391 99397

41 42 43 44 45

76673 76679 76697 76717 76733

77731 77743 77747 77761 77773

78919 78929 78941 78977 78979

80071 80077 80107 80111 80141

81119 81131 81157 81163 81173

82223 82231 82237 82241 82261

83401 83407 83417 83423 83431

84523 84533 84551 84559 84589

85703 85711 85717 85733 85751

86869 86923 86927 86929 86939

87973 87977 87991 88001 88003

89213 89227 89231 89237 89261

90247 90263 90271 90281 90289

91423 91433 91453 91457 91459

92569 92581 92593 92623 92627

93629 93637 93683 93701 93703

94811 94819 94823 94837 94841

95881 95891 95911 95917 95923

97039 97073 97081 97103 97117

98321 98323 98327 98347 98369

99401 99409 99431 99439 99469

46 47 48 49 50

76753 76757 76771 76777 76781

77783 77797 77801 77813 77839

78989 79031 79039 79043 79063

80147 80149 80153 80167 80173

81181 81197 81199 81203 81223

82267 82279 82301 82307 82339

83437 83443 83449 83459 83471

84629 84631 84649 84653 84659

85781 85793 85817 85819 85829

86951 86959 86969 86981 86993

88007 88019 88037 88069 88079

89269 89273 89293 89303 89317

90313 90353 90359 90371 90373

91463 91493 91499 91513 91529

92639 92641 92647 92657 92669

93719 93739 93761 93763 93787

94847 94849 94873 94889 94903

95929 95947 95957 95959 95971

97127 97151 97157 97159 97169

98377 98387 98389 98407 98411

99487 99497 99523 99527 99529

51 52 53 54 55

76801 76819 76829 76831 76837

77849 77863 77867 77893 77899

79087 79103 79111 79133 79139

80177 80191 80207 80209 80221

81233 81239 81281 81283 81293

82349 82351 82361 82373 82387

83477 83497 83537 83557 83561

84673 84691 84697 84701 84713

85831 85837 85843 85847 85853

87011 87013 87037 87041 87049

88093 88117 88129 88169 88177

89329 89363 89371 89381 89387

90379 90397 90401 90403 90407

91541 91571 91573 91577 91583

92671 92681 92683 92693 92699

93809 93811 93827 93851 93871

94907 94933 94949 94951 94961

95987 95989 96001 96013 96017

97171 97177 97187 97213 97231

98419 98429 98443 98453 98459

99551 99559 99563 99571 99577

56 57 58 59 60

76847 76871 76873 76883 76907

77929 77933 77951 77969 77977

79147 79151 79153 79159 79181

80231 80233 80239 80251 80263

81299 81307 81331 81343 81349

82393 82421 82457 82463 82469

83563 83579 83591 83597 83609

84719 84731 84737 84751 84761

85889 85903 85909 85931 85933

87071 87083 87103 87107 87119

80211 88223 88237 88241 88259

89393 89399 89413 89417 89431

90437 90439 90469 90473 90481

91591 91621 91631 91639 91673

92707 92717 92723 92737 92753

93887 93889 93893 93901 93911

94993 94999 95003 95009 95021

96043 96053 96059 96079 96097

97241 97259 97283 97301 97303

98467 98473 98479 98491 98507

99581 99607 99611 99623 99643

61 62 63 64 65

76913 76919 76943 76949 76961

77983 77999 78007 78017 78031

79187 79193 79201 79229 79231

80273 80279 80287 80309 80317

81353 81359 81371 81373 81401

82471 82483 82487 82493 82499

83617 83621 83639 83641 83653

84787 84793 84809 84811 84827

85991 85999 86011 86017 86027

87121 87133 87149 87151 87179

88261 88289 88301 88321 88327

89443 89449 89459 89477 89491

90499 90511 90523 90527 90529

91691 91703 91711 91733 91753

92761 92767 92779 92789 92791

93913 93923 93937 93941 93949

95027 95063 95071 95083 95087

96137 96149 96157 96167 96179

97327 97367 97369 97373 97379

98519 98533 98543 98561 98563

99661 99667 99679 99689 99707

66 67 68 69 70

76963 76991 77003 77017 77023

78041 78049 78059 78079 78101

79241 79259 79273 79279 79283

80329 80341 80347 80363 80369

81409 81421 81439 81457 81463

82507 82529 82531 82549 82559

83663 83689 83701 83717 83719

84857 84859 84869 84871 84913

86029 86069 86077 86083 86111

87181 87187 87211 87221 87223

88337 88339 88379 88397 88411

89501 89513 89519 89521 89527

90533 90547 90583 90599 90617

91757 91771 91781 91801 91807

92801 92809 92821 92831 92849

93967 93971 93979 93983 93997

95089 95093 95101 95107 95111

96181 96199 96211 96221 96223

97381 97387 97397 97423 97429

98573 98597 98621 98627 98639

99709 99713 99719 99721 99733

71 72 73 74 75

77029 77041 77047 77069 77081

78121 78137 78139 78157 78163

79301 79309 79319 79333 79337

80387 80407 80429 80447 80449

81509 81517 81527 81533 81547

82561 82567 82571 82591 82601

83737 83761 83773 83777 83791

84919 84947 84961 84967 84977

86113 86117 86131 86137 86143

87251 87253 87257 87277 87281

88423 88427 88463 88469 88471

89533 89561 89563 89567 89591

90619 90631 90641 90647 90659

91811 91813 91823 91837 91841

92857 92861 92863 92867 92893

94007 94009 94033 94049 94057

95131 95143 95153 95177 95189

96233 96259 96263 96269 96281

97441 97453 97459 97463 97499

98641 98663 98669 98689 98711

99761 99767 99787 99793 99809

76 77 78 79 80

77093 77101 77137 77141 77153

78167 78173 78179 78191 78193

79349 79357 79367 79379 79393

80471 80473 80489 80491 80513

81551 81553 81559 81563 81569

82609 82613 82619 82633 82651

83813 83833 83843 83857 83869

84979 84991 85009 85021 85027

86161 86171 86179 86183 86197

87293 87299 87313 87317 87323

88493 88499 88513 88523 88547

89597 89599 89603 89611 89627

90677 90679 90697 90703 90709

91867 91873 91909 91921 91939

92899 92921 92927 92941 92951

94063 94079 94099 94109 94111

95191 95203 95213 95219 95231

96289 96293 96323 96329 96331

97501 97511 97523 97547 97549

98713 98717 98729 98731 98737

99817 99823 99829 99833 99839

81 82 83 84 85

77167 71171 77191 77201 77213

78203 78229 78233 78241 78259

79397 19399 79411 79423 79427

80527 80537 80557 80567 80599

81611 81619 81629 81637 81647

82657 82699 82721 82723 82127

83873 83891 83903 83911 83921

85037 85049 85061 85081 85087

86201 86209 86239 86243 86249

87337 87359 87383 87403 87407

88589 88591 88607 88609 88643

89633 89653 89657 89659 89669

90731 90749 90787 90793 90803

91943 91951 91957 91961 91967

92957 92959 92987 92993 93001

94117 94121 94151 94153 94169

95233 95239 95251 95261 95261

96337 96353 96377 96401 96419

97553 97561 97571 97577 97579

98773 98779 98801 98807 98809

99859 99871 99877 99881 99901

86 87 88 89 90

77237 77239 77243 77249 77261

78277 78283 78301 78307 78311

79433 79451 79481 79493 79531

80603 80611 80621 80677 80629

81649 81667 81671 81677 81689

82729 82757 82759 82763 82781

83933 83939 83969 83983 83987

85091 85093 85103 85109 85121

86257 86263 86269 86287 86291

87421 87427 87433 87443 87473

88651 88657 88661 88663 88667

89671 89681 89689 89753 89759

90821 90823 90833 90841 90847

91969 91997 92003 92009 92033

93047 93053 93059 93077 93083

94201 94207 94219 94229 94253

95273 95279 95287 95311 95317

96431 96443 96451 96457 96461

97583 97607 97609 97613 97649

98837 98849 98867 98869 98873

99907 99923 99929 99961 99971

91 92 93 94 95

77263 77267 77269 77279 77291

78317 78341 78347 78367 78401

79537 79549 79559 79561 79579

80651 80657 80b69 80671 80677

81701 81703 81707 81727 81737

82787 82793 82799 82811 82813

84011 84017 84047 84053 84059

85133 85147 85159 85193 85199

86293 86297 86311 86323 86341

87481 87491 87509 87511 87517

88681 88721 88729 88741 88747

89767 89779 89783 89797 89809

90863 90887 90901 90907 90911

92041 92051 92077 92083 92107

93089 93097 93103 93113 93131

94261 94273 94291 94307 94309

95327 95339 95369 95383 95393

96469 96479 96487 96493 96497

97651 97673 97687 97711 97729

98887 98893 98897 98899 98909

99989 99991

96 97 98 99 100

77317 77323 77339 77347 77351

78427 78437 78439 78467 78479

79589 79601 79609 79613 79621

80681 80683 80687 80701 80713

81749 81761 81769 81773 81799

82837 82847 82883 82889 82891

84061 84067 84089 84121 84127

85201 85213 85223 85229 85237

86351 86353 86357 86369 86371

87523 87539 87541 87547 87553

88771 88789 88793 88799 88801

E9819 89821 89833 89839 89849

90917 90931 90947 90971 90977

92111 92119 92143 92153 92173

93133 93139 93151 93169 93179

94321 94327 94331 94343 94349

95401 95413 95419 95429 95441

96517 96527 96553 96557 96581

97771 97777 97787 97789 97813

98911 98927 98929 98939 98947

75

78

X2

83

84

86

87

88

89

90

91

1 76213 77359 78487 79627 80737 81817 82903 84131 85243 86381 87557 88807 89867 90989 92177 93187 94351

96587 96589 96601 96643 96661

94 97829 97841 97843 97847 97849

96 98953 98963 98981 98993 98999

94

i

25. Numerical Interpolation, Differentiation, and Integration PHILIPJ. DAVIS AND IVAN POLONSKY *

Contents Formulas 25.1. Differences . . . . . . . . , 25.2. Interpolation . . . . . . . . 25.3. Differentiation , . , , . . . 25.4. Integration . . . . . . . . . 25.5. Ordinary Differential Equations

. . . . . . . . . . . . . . . . .., . , . . . .

. . . . . . . . . . . . . . .. . . . . . . . .

.. . . .. . . . . ..

. . . . . .

. . . .

..

Table 25.1. *Point Lagrangian Interpolation Coefficients (3 < n 5 8 )

..

References

.

,

n=3,4, p = n=5,6, p = n=7,8, p = -

,

., .. . .. .

[“;‘I [“;‘I E]! [“;‘I E],

, ,

- (.01) [;I1

Exact

- (.01)

10D

-

(.1)

. . . . . .

Page

877 878 882 885

896 898 900

10D

Table 25.2. *Point Coefficients for k-th Order Differentiation (15k15). . . . . . . . . , . , . . . , . . . . . , . . . . k= 1, n=3(1)6, Exact k= 2 (1)5, n=k 4-1(l)6, Exact

914

. .

915

Table 25.3. n-Point Lagrangian Integration Coefficients (3 I n 2 10) n=3(1) 10, Exact

Table 25.4. Abscissas and Weight Factors for Gaussian Integration (2In596). . , . . . . . . , , . . . . . , . , . . . . . . . n=2(1)10, 12, 15D n = 16(4)24(8)48 (16)96, 21D Table 25.5. Abscissas for Equal Weight (2In19). . . . . . . . , , . . . . n=2(1)7, 9, 10D

.

Chebyshev ,

916

Integration

. . . . . . . . .

920

Table 25.6. Abscissas and Weight Factors for Lobatto Integration (31nI10). . . . . . . . . , , . , , , . . . . . . . . . . . n=3(1)10, 8-10D

920

Table 25.7. Abscissas and Weight Factors for Gaussian Integration for Integrands with a Logarithmic Singularity (2 I n 1 4 ) . . . . . n=2(1)4, 6D

920

a

National Bureau of Standards. National Bureau of Standards. (Presently, Bell Tel. Labs., Whippany, N.J.) 875

I

NUMERICAL ANALYSIS

876

Table 25.8. Abscissas and Weight Factors for Gaussian Integration of Moments(lSn18). . . . . . . . . . . . . . . . . . . . . . k=0(1)5,

n=1(1)8,

Page

921

10D

Table 25.9. Abscissas and Weight Factors for Laguerre Integration ( 2 5 7 ~ 5 1 5 ) .. . . . . . . . . n=2(1)10, 12, 15, 12D or S

. . . . . . . . . . . . . . . .

923

Table 25.10. Abscissas and Weight Factors for Hermite Integration (257~520).. . . . . . . . . . . . n=2(1)10, 12, 16, 20, 13-15D or S

. . . . . . . . . . . . .

Table 25.11. Coefficients for Filon’s Quadrature Formula (05051) 8= O ( .Ol). 1(.1) I , 8D

924

.

,

924

25. Numerical Interpolation, Differentiation,and Integration Numerical analysts have a tendency to accumulate a multiplicity of tools each designed for highly specialized operations and each requiring special knowledge to use properly. From the vast stock of formulas available we have culled the present selection. We hope that it will be useful. As with all such compendia, the reader may miss his favorites and find others whose utility he thinks is marginal. We would have liked to give examples to illuminate the formulas, but this has not been feasible. Numerical analysis is partially a science and partially an art, and short of writing a textbook on the subject it has been impossible to indicate where and under what circumstances the various formulas are useful or accurate, or to elucidate the numerical difficulties to which one might be led by uncritical use. The formulas are therefore issued together with a caveat against their blind application.

8i,=Af(n-k) if n and Forward Differences

k are of same parity.

Central Differences

xo f o

2-1

f-1

XfJ

fo

A0

A:

fl

Formulas

21

Notation: Abscissas: xo<x~< . .; functions: j , g, . . .; values: j ( x t ) T,Y(X!) =f;. Y, ... indicate lBt,2d, . . . erivatives. If abscissas are equally spaced, xr+l-xi=h and f p = (xo+ph) ( p not necessarily integral). R , R , in icate remainders.

22

f2

x3

f3

G

A1

A1

6-4

6; 6;

64 21

fl

52

f 2

6:

A2

6312

d

Mean Differences

25.1. Differences

25.1.3

p(f

n)

=3 (fn +t +f n -1)

Forward Differences

25.1.1 Divided Differences

[xo,a]=-=

25.1.4

[ X I , 201

[zoo,211- [ x 1 , 5 2 1 -[&,

[ZO,XI,

*

. .,Xk]--

* * -,xk-l]-[xl,

* *,zk]

Tn-xk

Divided Differences in Terms of Functional Values

25.1.5

It

[&,XI,

fk

. l x n l = k-0 X - *A(xk)

877

878

NUMERICAL ANALYSIS

where T,(z)=(z-zo) and a:(z) is its derivative:

25.1.6

(2-21)

. . . (z-z,)

Remainder in Lagrange Interpolation Formula

25.2.3

25.1.7

Rn(4=a*(d "&,%

a;(zk)=(2k-20)

*

*

*

(w-zk-l)(Zk--k+l>

...

* *

.,2,,4

(Zt-2,)

Let D be a simply connected domain with a piecewise smooth boundary C and contain the points zo, , . ., z, in its interior. Let f(z) be analytic in D and continuous in D+C. Then,

25.2.4

25.2.5 (t-z)(t-zoCo). f(t)

. . (t-2,)

dt

k=O

25.1.9

Aa=hflfc"'([)

(z~<[
The conditions of 25.1.8 are assumed here. Lagrange Interpolation, Equally Spaced Abscissas

25.1.10

n Point Formula

25-2-6

f(zo+ph)=T A;(p)fk+R,-i

25.2.11 [ 5- ,

2- ,+l

8;" , . . .,Zo,. . .,2,]=h2" (2n)!

For n even,

(-;

(n-2>
Reciprocal Differences

25.1.12 25.2.7

n even.

n odd.

25.2.8

(XO<S<Xd

k has the same range as in 25.2.6. Lagrange T w o Point Interpolation Formula (Linear Interpolation)

879

NUMERICAL ANALYSIS

Lagrange Three Point Interpolation Formula

25.2.11

+

f (20 ph) =A - if-

1

--

+Aofo +Alfl+

25.2.18 RdP) = .0049hsf@'(f)= .0049A6 (O
.0071h6f'"(~)z .0071A6

RZ P(P+ 1)f l

.024hBf'"([) =.024A6

(-l
l
(-2
2
25.2.12

Rz(p)= .065hy3'([) = .065A3

Lagrange Seven Point Interpolation Formula

(1~15 1)

Lagrange Four Point Interpolation Formula

25.2.13

3

25.2.20

R6(P)=

25.2.14 R3(P) = .024h4f(4)(()= .024A4 (O
f ( z o + P h ) = C Atft+R~

25-2-19

i=-3

I

.0025h7f(7)(f)A .0025A7

(Ipl
.0046h?f'7'(5)c.O046A7

(1
.019h7f'7)(i)= .019A7

(2
<

23)

Lagrange Eight Point Interpolation Formula

l
(z-1 < E < a )

25.2.21 25.2.22

.OO1lhsf's)(f)= .0011A8

Lagrange Five Point Interpolation Formula

(O
25.2.15

f(G+Ph)=

5 AJt+R4

i=-2

-- (P"-l)P(P-2)f-z-

24 (P2- 1)(P2-4) + 4

f o-

(P-l)P(P2-4) 6 (P+ 1)P(P2-4)f1 6

f-l

R7 (PI =.

(X-3<E<X4)

Aitken's Iteration Method

Let ~ ( Z ~ Z .~ ., ,,zJ ~ ~ ,denote the unique polynomial of kth degree which coincides in value with f(s) a t xo, . . ., zr. 25.2.23

880

NUMERICAL ANALYSIS

Everett's Formula

Taylor Expansion

25.2.24

25.2.31 (2-

4

f ( x ) = j o + ( x - Z O v ; + ~ jP+

25.2.25

*

..

f

(xo+ Ph) =(1-p)jo+pf1-

P(P-1) (P--2) 3!

*;

(x-t)"

at R n = lj("+l) ( t ) n!

x1

fl

6:

6:

25.2.32

Relation Between Everett and Lagrange Coe5cients

25.2.33

Ez =Af 1

E4=AB_2

FZ=A:

F,=A:

EB=A!a F6=&

Everett's Formula With Throwback (Modified Central Difference)

25.2.34 j(so+ ph) = (1-p)jo+pfi

25.2.35 25.2.36 25.2.37

(G
Xn)

Relation Between Newton and Lagrange Coefficients

25.2.30

e)=-Atl(p)

6;=62-

+&&,o+ F&,1+R ,18464

+

R = .000451p6;( .0006ll6fl

Simultaneous Throwback

25.2.54

m=2n+l

25.2.43

(k=O,l,. 25.2.55

. .,n)

m=2n

12n-1 ak=; f, cos kx,;

br=-

r=O

(k=O,1, . . .,n)

12n-1

n

r=O

f, sin kx,

(k=O,1, . . .,n-I)

b, is arbitrary. Subtabulation

Let f(z) be tabulated width h. It is desired intervals of width hlm. differences with respect

initially in intervals of to subtabulate f(x) in Let A and designate to the original and the

final intervals respectively. Thus

ao=f

-.f(z,). Assuming that the original 5th order differences are zero,

25.2.56

-

1 1-m Ao=- m Ao+- 2m2

2- XI

(l-~n)(1-2m)~! Em3

+(l-m)(l-2m)(1-3m1 24m4

&

(For reciprocal differences, p , see 25.1.12.) Trigonometric Interpolation

25.2.52

-

1

G = s A!+

3(1-m) 2m4 &

From this information we may construct the final tabulation by addition. For m=10, sin #(Z-S+~) . . . sin $(z-zZn) sin + ( ~ - a + . . ~. sin) +(zk-x2,J

25.2.57

.045$+ .02854-- ,02066A; .009&+.007725g &= ,OOlA!-.OO135&

-$=.Ol$-

t,(z) is a trigonometric polynomial of degree n such that t,(zn)=fk (k=0,1, , . .,2n)

-A:=

Harmonic Analysis

Equally spaced abscissas q=O,

51,

-

-Ao=.lAo-

.OOOlA$ Linear Inverse Interpolation

. , ~ m - 1 , ~ m = 2 ~

Find p , givenf,(=f(G++ph)).

25.2.53

Linear

1

n

f (2)“5 a o + C (akcos kx+bb sin kz) k=l

25.2.58

p = fp -fo fl

-fo

882

NUMERICAL ANALYSIS

Quadratic Inverse Interpolation

+

+

(f1- 2O f +f- 1)p2 (fi -f- 1)p 2 (fo

-fp>

=0

Three Point Formula (Linear)

25.2.65

Inverse Interpolation by Reversion of Series OD

25.2.60

Given f ( x o + p h ) = f p = ~akpk k-0

25.2.61

. . ., h=(fP-ao)/al

p=h+c2h2+c3h3+

+ +

Bivariate Interpolation

25.2.59

f($O+Ph, Y O +

25.2.62 cz= -az/al

!?w=(1 -2)-

dfo, 0

+Pfl, o+

+

do,1 O(h2)

Four Point Formula

25.2.66 c4=-+----a4

al

5aza3 5ai a: a:

-a5

6aza4 3ai

C S = - + T + - - T + -

a,

a,

a:

21a:a3 14a: a, a:

Inversion of Newton’s Forward Difference Formula

25.2.63

ao=fo

!

(Used in conjunction with 25.2.62.) 25.2.64

Inversion of Everett’s Formula

ao=fo

s:

a,----+

-2

s: s:

s:

--3 6 +-+-+ 20 30 8:

=6,

8: 24

9

.

*

...

25.3. Differentiation Lagrange’s Formula

. ..

a,=,+ 8:

25.3.1

aa=- -4+a:+

120

(See 25.2.1.) ’

*

(Used in conjunction with 25.2.62.)

883

NUMERICAL ANALYSIS

25.3.3

25.3.10

hyi2)=At-A:+z

11

25.3.11

h3fia)=A:-l

7 A:+- 4 A:-

15 8 g+

,

.

Equally Spaced Abscissas 25.3.12

Three Points

h4ji4)=~-2g+

25.3.4

7 6g-z~;+ .,.

17

25.3.13

h‘fi”=Ai--

5 25 35 At+- A;--- 6 A:+ 2 6

. ..

Everett’s Formula

25.3.14

hf’(q+ph)

=-fo+fi-

3p2-6p+2 6

-5p4-20p3+ 120 1 5 ~ ~ f 1 0 p - 66;+ Five Points

6

f-2

- 1+2P3-5Pfo -4pa-3P2-8P+4f 2 6 -4p3+3p2- 8 p -4f1

Differences in Terms of Derivatives

25.3.16

6

For numerical values of differentiation coefficients see Table 25.2. Markoff’r Formulas

25.3.18

(Newton’s Forward Difference Formula Differentiated)

25.3.19

25.3.7 f ’ ( o o + P h1) = ~ p2p-1 o++4

25.3.20

6:

5p4-15pa+4 6: 120

25.3.15

25.3.6

f:=f

I 3p2--I

+

884

NUMERICAL ANALYSIS

25.3.26

25.3.22

0

I.

t

+

25.3.23

25.3.27

+ + -b2fo* O- =I c i l , 0 tfI , o+ fo,I +fo, axay 2h2

-2fo,o-fi,

25.3.24

-1 i

-f - I, -1)

+0(h')

25.3.28

$-;a O - k

25.3.29

25.3.25

b;2-

---

3;a (fl.I - 2 f 0 , 1

+f- 1,1+fl.

+fi,-1-2fo.

0-2f0, -i+f-1,

o t f - l,o -1)

+O(h2)

+

cf2,0-4f1,0+6f0.0- 4 f - I ,~+f2.0) 0 (ha)

+ Laplacian

25.3.30

ss5

NUMERICAL hNALYSIS

25.3.33 0 0

0

0

0

Extended Trapezoidal Rule

25.4.2 Biharmonic Operator

25.3.32 Error Term in Trapezoidal Formula for Periodic Functionlr

If f(z) is periodic and has a continuous kth derivative, and if the integral is taken over a period, then constan t 25.4.3 IErrorl S mk Modifled Trapezoidal Rule

25.4.4

886

NUMERICAL ANALYSIS

Simpson's Rule

25.4.5

(See Table 25.3 for Af(m).) Newton-Cotes Formulas (Closed Type)

(For Trapezoidal and Simpson's Rules see 25.4.125.4.6.) 25.4.13

(Simpson'e 5 3 rule)

25.4.14

(Bode's rule)

Extended Simpson's Rule

25.4.6

+32j3+7fJ Euler-Maclaurin Summation Formula

-

8f(6) (0h7 945

25.4.15

25.4.7

25.4.16

(For B2k,Bernoulli numbers, see chapter 23.) Ifj('"+"(z) and f(2k+4)(z) do not change sign for xo
25.4.8 25.4.18

(See 25.2.1.) 25.4.9

[f(z)

+

4h dz= 14175 (989fo 5888f1-928fi

+10496f3-4540f4 +1O496f a-928f 25.4.19

e+5888f7

887

NUMERICAL ANALYSIS

25.4.20

-[f(zo+ih)+f(zo-ih)l}

+R

IR(I-1h17Max If'e)(z)I, S designates the square

-32691 1346350 8592

f(l2)

(€)h13

1890 r e 5 with vertices zo+ikh(k=O, 1,2,3);hcan be complex. Chebyshev's Equal Weight Integration Formula

Newton-Coter Formulas (Open Type)

25.4.21

Abscissas: xi is the it"zero of the polynomial part of x" exp -n n n .. 4 - 5 2 6.72'

[=----- .]

(See Table 25.5 for s,.)

For n=8 and n 2 1 0 some of the zeros are complex. Remainder:

25.4.23

where [=[(z) satisfies O ~ [ < Z and O l & l r i (i=l, . . . ,n)

Integration Formulas of Gaussian Type

(For Orthogonal Polynomials see chapter 22) Gaurr' Formula

Related orthogonal polynomials: Legendre polynomials p , ( ~ P,(l)=l ), Abscissas: zf is the ithzero of P,(z) Weights: wi=2/(1-23 [pk(zr)la (See Table 25.4 for zi and w,.)

Gauss' Formula, Arbitrary Interval

Yf=(q)

b 2,-

ih *See page 11.

Z*+(T)b+a

*

888

*

NUMERICAL ANALYSIS

Related orthozonal polynomiak: P,(s),P , ( l ) = l Abscissas: xi is the Ith zero of P,(+) Weights: w,=2/( 1-2;) [P:(zr)]'

s,'

25*4*33

z l f ( x ) d z = 2 w,f(x,)

+R.

1-1

Related orthogonal polynomials: qn(Z)=4 k f 2 n

+U'$*

O)

(1-22)

(For the Jacobi polynomials PAr,o'see chapter 22.) Radau's Integration Formula

Abscissas :

25.4.31

x, is the i* zero of q,(z) Weights :

*

Related polynomials: (See Table 25.8 for q and w,.) Remainder :

Abscissas: xi is tho ithzero of

R,= (k+2n+1)(2n)! f (2n)(5) r!(k+n),!] (kf2n).

Pn-l(z) +Pn(s) 2+1

(O
25.4.34

Weights : 1

1--21

1

1

w'=n2 [P.- 1(x,) 12- 1-2, [Pn-1(2r)l2 Remainder:

Lobatto's Integration Formula

25.4.32

Abscissas: x,=l--f: where tt is the ithpositive zero of P2n+l(x). Weights: w,=2t2tw,( 2 n + l ) where W I ( ~ ~ + ~ are ) the Gaussian weights of order 2nf 1. Remainder:

25.4.35

Related polynomials: Pb-, (2) Abscissas: xi is the (i-l)Ot zero of Ph-,(z) Weights :

Related orthogonal polynomials: (See Table 25.6 for xr and wt:) Remainder :

(-1<E<1> *See page 11.

where tt is the ith positive Abscissas: x,=l-[: zero of P2n+l(x). Weights: wI=2t:wP'+l) whe.re w P + ~ are) the Gaussian weights of order 2nf 1.

889

NUMERICAL ANALYSIS

Abscissas: (2i-1)A 2n

xi=cos Related orthogonal polynomials: ~2~

Weights: A

(JCX), P2n(l)=1

Abscissas: zi=l-[: where tt is the ith positive zero of P Z n ( z ) . Weights: wi=2wPn),wiz") are the Gaussian weights of order 2n. Remaicder :

U);=-

n

25.4.40

Related orthogonal polynomials: Chebyshev Polynomials of Second Kind * [(n+l) arccos x] un (x) =sin sin (arccos x) Abscissas: Y

i lr xi=cos 1

71+

yt=a+ @-a) zi

Weights: wi=-

Related orthogonal polynomials:

-

*

i sin' -A n+l n+l A

p z n CJ 1-2) 7 PZn (1) =1 Abscissas: xi=l-[: where t t is the ithpositive zero of Pzn(x). Weights: w,=2wj2"), wjzn' are the Gaussian weights of order 2n.

b+a

y*=-+- 2

Related orthogonal polynomials: Chebyshev Polynomials of First Kind Tn(x) 9 Tn(1) ~

1

sin [ (n+ 1) arccos sin (arccos x>

un(x)=

xi =cos

(2i-1)A 2n

xi=cos

?r

Wf=-

n

25.442

J1 0

j(x,J-

A

n+l

-A Y

n+l

i sin2 -a n+l

5d z = k 1-x

wJ(x,)+R,

i-1

Related orthogonal polynomials:

25.4.39

b+a

yi=-+--z; 2

b-a 2

Related orthogonal polynomials:

* *

wi=-

Remainder:

XI

Abscissas: Weights :

Weights:

2 zi

Related orthogonal polynomials:

2 n - 1

Abscissas:

b-a

Abscissas:

-

2i-1 A xi=cos2 - 2n+l 2

Weights : wi=*See page 11.

2s 2n+1"

*

890

NUMERICAL ANALYSIS

Remainder:

Rn=- n!JT

2 fl (2n)!f'2"'(t) (-

25.4.47

CO<(<

->

Filon's Integration Formula 8

Related orthogonal polynomials: 1 Tin+l(rn

Abscissas:

2i-1 * xt=cos2 -* 2n+l 2

Weights :

Related orthogonal polynomials: polynomials orthogonal with respect to the weight function -In x Abscissas: See Table 25.7 Weights: See Table 25.7 25.4.45

Related orthogonal polynomials: Laguerre polynomials L,(z). Abscissas: xr is the it"zero of Ln(z) Weights:

For small e we have 25.4.53

28a

205

(See Table 25.9 , for xf and w t . ) Remainder :

Related orthogonal polynomials: Hermite polynomials H,(z). Abscissas: xt is the ithzero of H,(x) Weights : 2"-1n!J;; n2[H,-1(2,)12 (See Table 25.10 for xt and wt.)

207

"'

fl=7+---+2 2e2 4e4 2ea3

15

25.4.54

105 567

e4

' '

e6

+m ... ~=3-15+2104 2e2

25.4.46

I

"==-a 4725

* For certain difficulties associated with this formula, see the article by J. W. Tukey, p. 400, <'On Numerical Approximation," Ed. R. E. L,anger, Madison, 1959.

NUMERICAL ANALYSIS

n

25-4-57

CLn-l=x j%, COS (t~zt-1) 1- 1

(See Table 25.11 for a,8,y.) Iterated Integrals

25.4.58

=-

1

JZ

(n-l)!

(x-t),-If(t)dt

0

25.4.59

=-(n- 1) !

l1

t~-tf(x-(x-u)t)dt

Multidimensional Integration

Circumference of Circle

r: x2+y2=h2.

25.4.60

1

27rh -

s

1 am r j ( x , y)ds=%

2f(h cos

nn

h sin m

+O(h2"-? Circle C:x2+y21ha, 25.4.61

891

i

892

NUMERICAL ANALYSIS

( f f , f % ) 116

(f$h,O)

118

R=O(h6)

For regions, such as the square, cube, cylinder, etc., which are the Cartesian products of lower dimensional regions, one may always develop integration rules by “multiplying together” the lower dimensional rules. Thus

16+4 360 (k=1,

. . JO)

R=O(h’O)

if

is a one dimensional rule, then

becomes a two dimensional rule. Such rules are not necessarily the most “economical”.

893

NUMERICAL ANALYSIS

10181 1 0/8l Equilateral Triangle T

Radius of Circumscribed Circle=h 25.4.63

I

((-OF) h,O)

Regular Hexagon H

(-3'h fh943)

(-2)&2 h h 43)

Radius of Circumscribed Circle=h 1/12

3/60 8/60

R=o(A~)

25.4.64

894

NUMERICAL ANALYSIS

(x*,Y t> (090) 10

(Ifrh

9)o)

Wt

258/1008 12511008

R =O(h9

125/1008

Surface of Sphere 2 : x2+y2+z2=h2 25.4.65

401840

895

NUMERICAL ANALYSIS

c f , = s u m of values off a t the 6 points midway from the center of C to the 6 faces. cf,=sum of values of f at the 6 centers of the faces of C. cfw=sumof values off a t the 8 vertices of C. c f e = s u m of values off at the 12 midpoints of edges of C. cfa=sum of values of f a t the 4 points on the diagonals of each face at a distance of

J f i h from the center of the face. 2

25.4.67

Tetrahedron: 925.470

+terms of 4thorder where

V: Volume of 9-

Cf,,:Sum of values of the function at the vertices of 9-.

25.4.69

Cf6:Sum of

where fm=f(Ol

0,O).

See footnote to 26.4.62.

values of the function a t midpoints of the edges of 9-. Cf,: Sum of values of the function at the center of gravity of the faces of 9-. fm: Value of function at center of gravity of 9-.

896

NUMERICAL ANALYSIS

25.5. Ordinary Differential Equations'

First Order: y'=f(z, y) Point Slope Formula

Yn+1 =Y

25.5.1

O(h2)

+ +O(h3)

Y

25.5.2

.+hY: +

=Y%- 1 2hy:

Trapezoidal Formula

h

25-5-3

Y~+I=Y%+~ (yb+l+yb)

+O(h3)

Adams' Extrapolation Formula

25.5.4

h

+ g (55Yb- 59y:-

Y n + lnY'

+37yb-2-

1

9y:-3)

+O(h6)

Adams' Interpolation Formula

25.5.5

h

Yn+lnY'

(%b+

I

+19yb- 5Yb-

1

-2)

+0(h')

Runge-Kutta Methods Second Order

25.5.6

Yn+l=Yn+;

1 ~ n + i = ~ n + s

(k1+k2)+O(h3)

+ +

k1 =hf(zn ,Yn) ,kz= hf(2, h,Yn k,) 25.5.7 Yn+ I

~n

+kz +0(ha) Third Order

25.5.8

~

The reader is cautioned against possible instabilities especially in formulas 25.5.2 and 25.5.13. See, e.g.

[25.11], I25.121.

(k,+2

(1-4) kz

NUMERICAL ANALYSIS

Systems of Differential Equations

*

First Order: y’= f ( 2 ,y , z ) , z’=g(z, y , 2). Second Order Runge-Kutta

25.5.17

1 Yn+l=Yn+Z (kl+kz) +o(h3), 1 zn+l=zn+g (Zl+Zz)

897

+0(h3)

*See page XI.

898

NUMERICAL ANALYSIS

References Texts

(For textbooks on numerical analysis, see texts in chapter 3) [25.1] J. Balbrecht and L. Collatz, Zur numerischen Auswertung mehrdimensionaler Integrale, Z. Angew. Math. Mech. 38, 1-15 (1958). [25.2] Berthod-Zaborowski, Le calcul des intkgrales de

la forme:

L1

f(x) log x dx.

H . Mineur, Techniques de calcul numkrique, pp. 555-556 (Librairie Polytechnique Ch. BBranger, Paris, France, 1952). [25.3] W. G. Bickley, Formulae for numerical integration, Math. Gaz. 23, 352 (1939). [25.4] W. G. Bickley, Formulae for numerical differentiation, Math. Gaz. 25, 19-27 (1941). [25.5] W. G. Bickley, Finite difference formulae for the square lattice, Quart. J. Mech. Appl. Math., 1 , 35-42 (1948). [25.6] G. Birkhoff and D. Young, Numerical quadrature of analytic and harmonic functions, J. Math. Phys. 29,217-221 (1950). [25.7] L. Fox, The use and construction of mathematical tables, Mathematical Tables vol. I, National Physical Laboratory (Her Majesty’s Stationery Office, London, England, 1956). [25.8] S. Gill, Process for the step-by-step integration of differential equations in a n automatic digital computing machine, Proc. Cambridge Philos. SOC. 47, 96-108 (1951). [25.9] P. C. Hammer and A. H. Stroud, Numerical evaluation of multiple integrals 11, Math. Tables Aids Comp. 12, 272-280 (1958). [25.10] P. C. Hammer and A. W. Wymore, Numerical evaluation of multiple integrals I, Math. Tables Aids Comp. 11,59-67 (1957). [25.11] P. Henrici, Discrete variable methods in ordinary differential equations (John Wiley & Sons, Inc., New York, N. Y., 1961). [25.12] F. B. Hildebrand, Introduction to numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1956). [25.13] Z. Kopal, Numerical analysis (John Wiley & Sons, Inc., New YorL, N.Y., 1955). [25.14] A. A. Markoff, Differenzenrechnung (B. G. Teubner, Leipzig, Germany, 1896). [25.15] S. E. Mikeladze, Quadrature formulas for a regular function, SoobSE. Akad. Nauk Gruzin. SSR. 17, 289-296 (1956). [25.16] W. E. Milne, A note on the numerical integration of differential equations, J. Research NBS 43, 537-542 (1949) RP2046. [25.17] D. J. Panov, Formelsammlung zur numerischen Behandlung partieller Differentialgleichungen nach dem Differenzenverfahren (Akad. Verlag, Berlin, Germany, 1955). t25.18) R. Radau, gtudes sur les formules d’approximation qui servent A calculer la valeur d’une intBgrale dbfinie, J. Math. Pures Appl. (3) 6, 283-336 (1880).

[25.19] R. D. Richtmeyer, Difference methods for initialvalue problems (Interscience Publishers, New York, N.Y., 1957). [25.20] M. Sadowsky, A formula for approximate computation of a triple integral, Amer. Math. Monthly 47, 539-543 (1940). [25.21] H. E. Salzer, A new formula for inverse interpolation, Bull. Amer. Math. Soc. 50, 513-516 (1944). I25.221 H. E. Salzer, Formulas for complex Cartesian interpolation of higher degree, J. Math. Phys. 28,-200-203 (1949). [25.23] H . E. Salzer, Formulas for numerical integration of first and second order differential equations in the complex plane, J. Math. Phys. 29, 207-216 (1950). [25.24] H. E. Salzer, Formulas for numerical differentiation in the complex plane, J. Math. Phys. 31, 155-169 (1952). [25.25] A. Sard, Integral representations of remainders, Duke Math. J. 15,333-345 (1948). [25.26] A. Sard, Remainders: functions of several variables, Acta Math. 84, 319-346 (1951). [25.27] G. Schulz, Formelsammlung zur praktischen Mathematik (DeGruyter and Co., Berlin, Germany, 1945). [25.28] A. H . Stroud, A bibliography on approximate integration, Math. Comp. 15, 52-80 (1961). [25.29] G. J. Tranter, Integral transforms in mathematical physics (John Wiley & Sons, Inc., New York, N.Y., 1951). [25.30] G. W. Tyler, Numerical integration with several variables, Canad. J. Math. 5, 393-412 (1953).

Tables [25.31] L. J. Comrie, Chambers’ six-figure mathematical tables, vol. 2 (W. R. Chambers, Ltd., London, England, 1949). [25.32] P. Davis and P. Rabinowitz, Abscissas and weights for Gaussian quadratures of high order, J. Research NBS 56, 35-37 (1956) RP2645. [25.33] P. Davis and P. Rabinowitz, Additional abscissas and weights for Gaussian quadratures of high order: Values for n=64, 80, and 96, J. Research NBS 60, 613-614 (1958) RP2875. [25.34] E. W. Dijkstra and A. van Wijngaarden, Table of Everett’s interpolation coefficients (Elcelsior’s Photo-offset, The Hague, Holland, 1955). [25.35] H. Fishman, Numerical integration constants, Math. Tables Aids Comp. 11, 1-9 (1957). [25.36] H . J. Gawlik, Zeros of Legendre polynomials of orders 2-64 and weight coefficients of Gauss quadrature formulae, A.R.D.E. Memo (€3) 77/58, Fort Halstead, Kent, England (1958). [25.37] Gt. Britain H.M. Nautical Almanac Office, Interpolation and allied tables (Her Majesty’s Stationery Office, London, England, 1956).

NUMERICAL ANALYSIS

[25.38] I. M. Longman, Tables for the rapid and accurate numerical evaluation of certain infinite integrals involving Bessel functions, Math. Tables Aids Comp. 11, 166-180 (1957). [25.39] A. N. Lowan, N. Davids, and A. Levenson, Table of the zeros of the Legendre polynomials of order 1-16 and the weight coefficients for Gauss' mechanical quadrature formula, Bull. Amer. Math. soc.48, 739-743 (1942). [25.40] National Bureau of Standards, Tables of Lagrangian interpolation coefficients (Columbia Univ. Press, New York, N.Y., 1944). [25.41] National Bureau of Standards, Collected Short Tables of the Computation Laboratory, Tables of functions and of zeros of functions, Applied Math. Series 37 (U.S.Government Printing Office, Washington, D.C., 1954). [25.42] P. Rabinowitz, Abscissas and weights for Lobatto quadrature of high order, Math. Tables Aids Comp. 69, 47-52 (1960). [25.43] P. Rabinowitz and G. Weiss, Tables of abscissas and weights for numerical evaluation of integrals of the form OD

e- "2"f(2)dz,

Msth. Tables Aids Comp. 68, 285-294 (1959).

899

[25.44] H. E. Salzer, Tables for facilitating the use of Chebyshev's quadrature formula, J. Math. Phys. 26, 191-194 (1947). [25.45] H. E. Salzer and R. Zucker, Table of the zeros and weight factors of the first fifteen Laguerre polynomials, Bull. Amer. Math. Soc. 55, 10041012 (1949). [25.46] H. E. Salzer, R. Zucker, and R. Capuano, Table of the zeros and weight factors of the first twenty Hermite polynomials, J. Research NBS 48, 111-116 (1952) RP2294. [25.47] H. E. Salzer, Table of coefficients for obtaining the first derivative without differences, NBS Applied Math. Series 2 (U.S. Government Printing Office, Washington, D.C., 1948). [25.48] H. E. Salzer, Coefficients for facilitating trigonometric interpolation, J. Math. Phys. 27,274-278 (1949). [25.49] H. E. Salzer and P. T. Roberson, Table of coefficients for obtaining the second derivative without differences, Convair-Astronautics, San Diego, Calif. (1957). [25.50] H. E. Salzer, Tables of osculatory interpolation coefficients, NBS Applied Math. Series 56 (US.Government Printing Office, Washington, D.C., 1958).

900

NUMERICAL ANALYSIS

Table 25.1

THREE-POINT LAGRANGIAN INTERPOLATION COEFFICIENTS

P

A- I

0.00 0.01 0.02 0.03 0.04

-0.00000

Ao

A1 0.00000

-0.00495 -0.00980 -0.01455 -0.01920

1.00000 0.9 9990 0.99960 0.99910 0.99840

0.05 0.06 0.07 0.08 0.09

-0.02375 -0I02820 -0.03255 -0.03680 -0.04095

0.10 0.11 0.12 0.13 0.14

A0

A1

0.00505 0.01020 0.01545 0.02080

0.50 0.51 0.52 0.53 0.54

-0.1250 0 -0.12495 -0.12480 -0.12455 -0.12420

0.75000 0.73990 0,72960 0,71910 0.70840

0.37500 0.38505 0.3952 0 0.40545 0.41580

0.99750 0.99640 0.99510 0.99360 0.99190

0.02625 0.03180 0.03745 0.04320 0.04905

0.55 0.56 0.57 0.58 0.59

-0.12375 -0.12320 -0.12255 -0.12180 -0.12095

0.69750 0.68640 0.67510 0.66360 0.65190

0.42625 0.43680 0.44745 0.45820 0.4 6905

-0.04500 -0.04895 -0.05280 -0.05655 -0.06020

0.99000 0.98790 0.98560 0.98310 0.98040

0.05500 0.06105 0.06720 0.07345 0.07980

0.60 0.61 0.62 0.63 0.64

-0.12000 -0.11895 -0.11 780 -0.11655 -0.11520

0.64000 0.62790 0.61560 0.60310 0.59040

0.48000 0.49105 0.50220 0.51345 0.52480

0.15 0.16 0.17 0.18 0.19

-0.06375 -0.06720 -0.07055 -0.07380 -0.07695

0.97750 0.97440 0.97110 0.96760 0.96390

0.08625 0.09280 0.09945 0.10620 0.11305

0.65 0.66 0.67 0.68 0.69

-0.11375 -0.11220 -0.11055 -0.10880 -0.10695

0.57750 0.56440 0.55110 0.53760 0.52390

0.53625 0.54780 0.55945 0.57120 0.58305

0.20 0.21 0.22 0.23 0.24

-0.08000 -0.08295 -0.08580 -0.08855 -0.09120

0.96000 0.95590 0.95160 0.94710 0.94240

0.12000 0.12705 0.13420 0.14145 0.14880

0.70 0.71 0.72 0.73 0.74

-0.10500 -0.10295 -0.10080 -0.09855 -0.09620

0.51000 0.49590 0.48160 0.46710 0.4 5240

0.59500 0.60705 0.61920 0.63145 0.64380

0.25 0.26 0.27 0.28 0.29

-0.09375 -0.09620 -0.09855 -0.10080 -0.10295

0.93750 0.93240 0.92710 0.92160 0.91590

0.15625 0.16380 0.17145 0.17920 0.18705

0.75 0.76 0.77 0.78 0.79

-0.09375 -0.09120 -0.08855 -0.08580 -0,08295

0.43750 0.42240 0.40710 0.39160 0.37590

0.65625 0.66880 0.68145 0.69420 0.70705

0.30 0.31 0.32 0.33 0.34

-0.10500 -0.10695 -0.10880 -0.11055 -0.11220

0.91000 0.90390 0.89760 0.89110 0.88440

0.195 00 0.20305 0.21120 0.21945 0.22780

0.80 0.81 0.82 0.83 0.84

-0.08000 -0.07695 -0.07380 -0.07055 -0.06720

0.36000 0.34390 0.32760 0.31110 0.29440

0.72000 0.73305 0;74620 0.75945 0.77280

0.35 0.36 0.37 0.38 0.39

-0.11375 -0.11520 -0.11655 -0.11780 -0.11895

0.87750 0.87040 0.86310 0.85560 0.84790

0.23625 0.24480 0.25345 0.26220 0.27105

0.85 0.86 0.87 0.88 0.89

-0.06375 -0,06020 -0.05655 -0.05280 -0.04895

0.27750 0.26040 0.24310 0.22560 0.20790

0.78625 0.79980 0.81345 0.82720 0.84105

0.40 0.41 0.42 0.43 0.44

-0.12000 -0.12095 -0.12180 -0.12255 -0.12320

0.84000 0.83190 0.82360 0.81510 0.80640

0.28000 0.28905 0.29820 0.30745 0.31680

0.90 0.91 0.92 0.93 0.94

-0.04500 -0.04095 -0.03680 - 0.0 3255 -0.02820

0.19000 0.17190 0.15 360 0.13510 0.11640

0.85500 0.86905 0.88320 0.89745 0.91180

0.45 0.46 0.47 0.48 0.49

-0.12375 -0.12420 -0.12455 -0.12480 -0.12495

0.79750 0.78840 0.77910 0.76960 0.75990

0.32625 0.33580 0.34545 0.3552 0 0.365 05

0.95 0.96 0.97 0.98 0.99

-0.02375 -0.01920 -0.01455 -0.00980 -0.00495

0.09750 0.07840 0.05910 0.03960 0.01990

0.92625 0.94080 0.95545 0.97020 0.98505

0.50

-0.12500

0.75000

0.37500

1.00

-0.00000 A1

0.00000

1.00000

-P

A1

Ao

A- 1

P

-P

8-1

Ao

8-1

See 25.2.6. Compiled from National Bureau of Standards, Tables of Lagrangian interpolation coefficients. Columbia Univ. Press, New York, N.Y., 1944 (with permission).

901

NUMERICAL ANALYSIS

FOUR-POINT LAGRANGIAN INTERPOLATION COEFFICIENTS A:(p) A-1

=(- l)k+2

Table 25.1

P (P2- 1 ) (P - 2) (1 + k ) !(2- k ) ! ( p- k )

Az

0. 00 0. 01 0. 02 0. 03 0. 04

0.00000 -0.00328 -0,00646 -0.00955 -0.01254

00 35 80 45 40

Ao 1.00000 0.99490 0.98960 0.98411 0.97843

00 05 40 35 20

0.00000 0.01004 0,02019 0.03043 0.04076

00 95 60 65 80

0.00000 -0.00166 -0.00333 -0.00499 -0.00665

00 65 20 55 60

1.00 0.99 0.98 0.97 0. 96

0. 05 0. 06 0. 07 0. 08 0. 09

-0.01543 -0.01823 -0.02094 -0.02355 -0.02607

75 60 05 20 15

0,97256 0.96650 0.96027 0.95385 0.94726

25 80 15 60 45

0.05118 0.06169 0.07227 0.08294 0.09368

75 20 85 40 55

-0.00831 -0.00996 -0.01160 -0.01324 -0. 01487

25 40 95 80 85

0.95 0.94 0.93 0.92 0. 91

0.10 0.11 0.12 0.13 0.14

-0.02850 -0.03083 -0.03308 -0,03524 -0.03732

00 85 80 95 40

0,94050 0.93356 0.92646 0.91919 0.91177

00 55 40 85 20

0.10450 0.11538 0.12633 0.13735 0.14842

00 45 60 15 80

-0.01650 -0.01811 -0.01971 -0.02130 -0.02287

00 15 20 05 60

0.90 0.89 0.88 0. 87 0.86

0.15 0.16 0.17 0.18 0.19

-0.03931 -0.04121 -0,04303 -0.04477 -0.04642

25 60 55 20 65

0.90418 0.89644 0.88855 0.88051 0.87232

75 80 65 60 95

0.15956 0.17075 0.18199 0.19328 0.20462

25 20 35 40 05

-0,02443 -0.02598 -0.02751 -0,02902 -0.03052

75 40 45 80 35

0. 85 0.84 0.83 0.82 0.81

0.20 0. 21 0.22 0. 23 0. 24

-0.04800 -0.04949 -0.05090 -0.05224 -0.05350

00 35 80 45 40

0.86400 0.85553 0.84692 0.83818 0.82931

00 05 40 35 20

0.21600 0.22741 0.23887 0.25036 0.26188

00 95 60 65 80

-0.03200 -0.03345 -0.03489 -0.03630 -0.03769

00 65 20 55 60

0. 80 0.79 0.78 0.77 0.76

0. 25 0. 26 0.27 0, 28 0.29

-0. 05468'75 -0.05579 60 -0.05683 05 -0.05779 20 -0.05868 15

0,82031 0.81118 0.80194 0.79257 0.78309

25 80 15 60 45

0.27343 0.28501 0.29660 0.30822 0,31985

75 20 85 40 55

-0.03906 -0,04040 -0.04171 -0.04300 -0.04426

25 40 95 80 85

0.75 0.74 0.73 0.72 0.71

0.30 0. 31 0. 32 0. 33 0.34

-0,05950 -0,06024 -0. 06092 -0.06153 -0,06208

00 85 80 95 40

0.77350 0.76379 0,75398 0.74406 0.73405

00 55 40 85 20

0.33150 0.34315 0.35481 0.36648 0.37814

00 45 60 15 80

-0.04550 -0.04670 -0.04787 -0,04901 -0.05011

00 15 20 05 60

0. 70 0. 69 0. 68 0.67 0. 66

0.35 0. 36 0.37 0. 38 0.39

-0.06256 -0,06297 -0,06332 -0,06361 -0.06383

25 60 55 20 65

0,72393 0.71372 0,70342 0.69303 0.68255

75 80 65 60 95

0.38981 0.40147 0.41312 0.42476 0.43639

25 20 35 40 05

-0.05118 -0.05222 -0.05322 -0.05418 -0.05511

75 40 45 80 35

0. 65 0. 64 0. 63 0.62 0.61

0.40 0.41 0.42 0. 43 0.44

-0.06400 -0,06410 -0.06414 -0.06413 -0,06406

00 35 80 45 40

0.67200 0.66136 0.65064 0.63985 0.62899

00 05 40 35 20

0.44800 0.45958 0.47115 0,48269 0.49420

00 95 60 65 80

-0.05600 -0.05684 -0.05765 -0.05841 -0. 05913

00 65 20 55 60

0. 60 0. 59 0. 58 0.57 0.56

0.45 0. 46 0.47 0.48 0. 49

-0.06393 -0.06375 -0.06352 -0.06323 -0.06289

75 60 05 20 15

0.61806 0.60706 0.59601 0.58489 0.57372

25 80 15 60 45

0.50568 0.51713 0.52853 0.53990 0.55122

75 20 85 40 55

-0.05981 -0.06044 -0.06102 -0.06156 -0.06205

25 40 95 80 85

0. 55 0. 54 0.53 0.52 0.51

0. 50

-0.06250 00 Aa

P

A1

0.56250 00 A1

0.56250 00 Ao

-0,06250 00 A-i

0.50 P

902

NUMERICAL AN.4LYSIS

Table 25.1

FOUR-POINT LAGRANGIAN INTERPOLATION COEFFICIENTS Ai(p)=(-l)k+2-- P W - 1)(P- 2) (1 k ) !(2 - k ) ! ( p- k )

+

A-1

Ao 0.00000 00

95 60 65 80

A1 1.00000 1.00489 1.00959 1.01408 1.01836

00 95 60 65 80

0.00000 0,00338 0.00686 0,01045 0.01414

00 35 80 45 40

0.01 0. 02 0.03 0. 04

-0.04868 -0IO5809 -0.06737 -0IO7654 -0.08558

75 20 85 40 55

1,02243 1,02629 1,02992 1.03334 1.03653

75 20 85 40 55

0.01793 0.02183 0.02584 0.02995 0.03417

75 60 05 20 15

0. 05 0.06 0. 07 0. 08 0. 09

00 15 20 05 60

-0.09450 -0,10328 -0.11193 -0.12045 -0.12882

00 45 60 15 80

1.03950 1.04223 1. 04473 1.04700 1,04902

00 45 60 15 80

0.03850 0. 04293 0,04748 0.05214 0.05692

00 85 80 95 40

0.10 0.11 0.12 0.13 0.14

0,02443 0,02598 0.02751 0.02902 0.03052

75 40 45 80 35

-0.13706 -0.14515 -0.15309 -0.16088 -0.16852

25 2.0 35 40 05

1.05081 1.05235 1.05364 1.05468 1.05547

25 20 35 40 05

0. 06181 0,06681 0,07193 0.07717 0. 08252

25 60 55 20 65

0.15 0.16 0.17 0.18 0.19

1.20 1.21 1.22 1.23 1.24

0.03200 0.03345 0.03489 0,03630 0.03769

00 65 20 55 60

-0.17600 -0.18331 -0.19047 -0.19746 -0.20428

00 95 60 65 80

1.05600 1,05626 1.05627 1.05601 1.05548

00 95 60 65 80

0.08800 0,09359 0,09930 0.10514 0.11110

00 35 80 45 40

0.20 0.21 0.22 0.23 0. 24

1.25 1. 26 1.27 1.28 1.29

0,03906 0.04040 0.04171 0.04300 0,04426

25 40 95 80 85

-0,21093 -0.21741 -0,22370 -0,22982 -0.23575

75 20 85 40 55

1.05468 1.05361 1.05225 1.05062 1.04870

75 20 85 40 55

0.11718 0.12339 0,12973 0,13619 0.14278

75 60 05 20 15

0.25 0. 26 0.27 0.28 0.29

1.30 1.31 1.32 1.33 1.34

0. 04550 0,04670 0.04787 0.04901 0,05011

00 15 20 05 60

-0.24150 -0.24705 -0.25241 -0.25758 -0.26254

00 45 60 15 80

1,04650 1.04400 1.04121 1,03813 1,03474

00 45 60 15 80

0.14950 0.15634 0,16332 0,17043 0.17768

00 85 80 95 40

0.30 0. 31 0. 32 0.33 0.34

1.35 1.36 1.37 1. 38 1. 39

0.05118 0.05222 0.05322 0.05418 0.05511

75 40 45 80 35

-0.26731 -0I27187 -0.27622 -0,28036 -0,28429

25 20 35 40 05

1.03106 1.02707 1.02277 1.01816 1.01324

25 20 35 40 05

0.18506 0,19257 0.20022 0.20801 0.21593

25 60 55 20 65

0.35 0.36 0. 37 0.38 0.39

1.40 1.41 1.42 1.43 1.44

0,05600 0.05684 0.05765 0,05841 0.05913

00 65 20 55 60

-0.28800 -0.29148 -0.29475 -0.29779 -0.30060

00 95 60 65 80

1.00800 1.00243 0,99655 0.99034 0.98380

00 95 60 65 80

0.22400 0.23220 0.24054 0,24903 0.25766

00 35 80 45 40

0.40 0.41 0.42 0.43 0.44

1.45 1.46 1.47 1.48 1.49

0.05981 0.06044 0.06102 0.06156 0,06205

25 40 95 80 85

-0.30318 -0.30553 -0.30763 -0.30950 -0.31112

75 20 85 40 55

0.97693 0.96973 0.96216 0,95430 0.94607

75 20 85 40 55

0.26643 0.27535 0.28442 0,29363 0.30299

75 60 05 20 15

0. 45 0.46 0.47 0.48 0.49

1.50

0.06250 00

0.31250 00 -4-1

0. 50

P 1. 00 1. 01 1.02 1. 03 1. 04

0.00000 0.00166 0,00333 0.00499 0.00665

00 65 20 55 60

-0,00994 -0.01979 -0. 02953 -0.03916

1. 05 1. 06 1. 07 1. 08 1. 09

0.00831 0.00996 0.01160 0,01324 0,01487

25 40 95 80 85

1.10 1.11 1.12 1.13 1.14

0.01650 0,01811 0.01971 0,02130 0.02287

1.15 1.16 1.17 1.18 1.19

A2

-0.31250 00 AI

A2

0.93750 00

Ao

0. 00

-P

903

NUMERICAL ANALYSIS

FOUR-POINT LAGRANGIAN INTERPOLATION COEFFICIENTS

Table 25.1

P 1.50 1. 5 1 1.52 1.53 1.54

A-1 0.06250 0.06289 0,06323 0,06352 0.06375

00 15 20 05 60

Ao -0.31250 -0.31362 -0.31449 -0.31511 -0.31546

P (P2- 1) ( P - 2) (1 k ) !(2- k)!Cp - k ) A1 00 0.93750 00 45 0,92857 45 60 0.91929 60 15 0.90966 1 5 80 0,89966 80

1. 55 1.56 1.57 1.58 1.59

0.06393 0.06406 0.06413 0.06414 0.06410

75 40 45 80 35

-0.31556 -0.31539 -0.31495 -0,31424 -0.31326

25 20 35 40 05

0.88931 0.87859 0,86750 0.85604 0.84421

25 20 35 40 05

0,36231 0.37273 0.38331 0.39405 0.40494

25 60 55 20 65

0.55 0.56 0. 57 0.58 0. 59

1. 60 1.61 1. 62 1. 63 1. 64

0.06400 0,06383 0.06361 0.06332 0.06297

00 65 20 55 60

-0.31200 -0,31045 -0,30863 -0.30652 -0,30412

00 95 60 65 80

0.83200 0.81940 0.80643 0.79307 0.77932

00 95 60 65 80

0,41600 0,42721 0.43858 0.45012 0,46182

00 35 80 45 40

0. 60 0. 6 1 0.62 0. 63 0.64

1. 65 1.66 1. 67 1. 68 1. 69

0.06256 0.06208 0.06153 0.06092 0,06024

25 40 95 80 85

-0,30143 -0.29845 -0,29516 -0.29158 -0.28769

75 20 85 40 55

0.76518 0.75065 0,73571 0,72038 0,70464

75 20 85 40 55

0.47368 0,48571 0.49791 0,51027 0.52280

75 60 05 20 15

0. 65 0. 66 0. 67 0.68 0.69

1. 70 1. 7 1 1. 72 1.73 1.74

0.05950 0.05868 0.05779 0.05683 0,05579

00 15 20 05 60

-0.28350 -0.27899 -0.27417 -0.26904 -0.26358

00 45 60 15 80

0.68850 0.67194 0.65497 0.63759 0.61978

00 45 60 15 80

0.53550 0,54836 0.56140 0,57461 0,58800

00 85 80 95 40

0.70 0. 7 1 0.72 0.73 0. 74

1.75 1. 76 1. 77 1. 78 1.79

0.05468 0,05350 0.05224 0.05090 0.04949

75 40 45 80 35

-0.25781 -0.25171 -0.24528 -0.23852 -0.23143

25 20 35 40 05

0.60156 0.58291 0.56383 0.54432 0.52438

25 20 35 40 05

0.60156 0.61529 0.62920 0.64329 0.65755

25 60 55 20 65

0.75 0.76 0.77 0.78 0.79

1.80 1.81 1.82 1.83 1.84

0,04800 0,04642 0.04477 0.04303 0,04121

00 65 20 55 60

-0.22400 -0,21622 -0.20811 -0.19965 -0.19084

00 95 60 65 80

0.50400 0.48317 0.46191 0,44020 0.41804

00 95 60 65 80

0.67200 0.68662 0,70142 0.71641 0.73158

00 35 80 45 40

0.80 0. 8 1 0.82 0.83 0.84

1. 85 1.86 1. 87 1. 88 1.89

0.03931 0.03732 0.03524 0.03308 0.03083

25 40 95 80 85

-0.18168 -0.17217 -0,16229 -0.15206 -0.14146

75 20 85 40 55

0,39543 0.37237 0.34884 0.32486 0.30041

75 20 85 40 55

0.74693 0.76247 0.77820 0.79411 0,81021

75 60 05 20 15

0.85 0. 86 0.87 0.88 0.89

1.90 1.91 1.92 1.93 1.94

0,02850 0.02607 0,02355 0.02094 0.01823

00 15 20 05 60

-0.13050 -0.11916 -0,10745 -0.09537 -0.08290

00 45 60 15 80

0.27550 0.25011 0.22425 0.19792 0.17110

00 45 60 15 80

0.82650 0.84297 0.85964 0.87650 0.89356

00 85 80 95 40

0.90 0. 9 1 0.92 0.93 0.94

1.95 1.96 1.97 1.98 1.99

0.01543 0,01254 0.00955 0.00646 0.00328

75 40 45 80 35

-0.07006 -0.05683 -0.04321 -0.02920 -0.01480

25 20 35 40 05

0.14381 0.11603 0.08776 0.05900 0.02975

25 20 35 40 05

0.91081 0.92825 0.94589 0.96373 0,98176

25 60 55 20 65

0.95 0.96 0.97 0.98 0.99

2. 00

0.00000 00 A2

1.00000 00 A-1

1.00 -P

A:(p) =(-l)k+2

+

0.00000 00 A1

0.00000 00 Ao

A2 0,31250 0,32215 0.33196 0,34192 0.35204

00 85 80 95 40

0.50 0.51 0.52 0.53 0.54

904

NUMERICAL ANALYSIS

Table 25.1

FIVE-POINT LAGRANGIAN INTERPOLATION COEFFICIENTS

A ~ ( P :)( =- l ) k f 2

P ( P 2 - 1 ) (94) (2+ k ) !(2 -k ) !( p - k ) Ao A1

0.01 0. 02 0. 03 0. 04

A-2 0.00000 00000 0.00082 90838 0.00164 93400 0.00246 02838 0.00326 14400

A-i 0.00000 00000 -0.00659 98350 -0.01306 53600 -0. 01939 56350 -0.02558 97600

1.00000 0.99987 0.99950 0.99887 0.99800

00000 50025 00400 52025 06400

0.00000 0.00673 0.01359 0.02059 0.02772

00000 31650 86400 53650 22400

A2 0.00000 -0.00083 -0.00168 -0.00253 -0.00339

00000 74163 26600 52163 45600

0. 00 0. 01 0. 02 0. 03 0. 04

0. 05 0. 06 0. 07 0. 08 0. 09

0.00405 0.00483 0.00560 0.00635 0.00710

23438 25400 15838 90400 44838

-0.03164 -0.03756 -0.04334 -0.04898 -0.05448

68750 61600 68350 81600 94350

0.99687 0.99550 0.99388 0.99201 0.98989

65625 32400 10025 02400 14025

0.03497 0.04236 0.04987 0.05750 0.06526

81250 18400 21650 78400 75650

-0.00426 -0. 00513 -0.00600 -0.00688 -0.00777

01563 14600 79163 89600 40163

0. 05 0. 06

0.10 0.11 0.12 0.13 0.14

0.00783 0.00855 0.00926 0.00995 0.01063

75000 76838 46400 79838 73400

-0.05985 -0.06506 -0.07014 -0.07508 -0.07987

00000 92350 65600 14350 33600

0.98752 0.98491 0.98205 0.97894 0.97559

50000 16025 18400 64025 60400

0.07315 0.08115 0.08927 0.09751 0.10587

00000 37650 74400 95650 86400

-0.00866 -0.00955 -0.01044 -0.01134 -0.01223

25000 38163 73600 25163 86600

0.10 0. 11 0.12 0.13 0. 14

0.15 0.16 0.17 0.18 0.19

0.01130 0.01195 0.01258 0.01320 0.01381

23438 2'6400 78838 77400 18838

-0.08452 -0.08902 -0.09338 -0.09760 -0.10167

18750 65600 70350 29600 40350

0.97200 0.96816 0.96408 0.95976 0.95520

15625 38400 38025 24400 08025

0.11435 0.12294 0.13164 0.14045 0.14937

31250 14400 19650 30400 29650

-0.01313 -0.01403 -0.01492 -0.01582 -0.01671

51563 13600 66163 02600 16163

0. 15 0.16 0. 17 0.18 0. 19

0.20 0.21 0.22 0. 23 0. 24

0.01440 0.01497 0.01552 0.01606 0.01658

00000 17838 69400 51838 62400

-0.10560 -0.10938 -0.11301 -0.11650 -0.11984

00000

06350 57600 52350 89600

0.95040 0.94536 0.94008 0.93457 0.92882

00000 12025 56400 46025 94400

0.15840 0.16753 0.17676 0.18610 0.19554

00000 23650 82400 57650 30400

-0.01760 -0.01848 -0.01936 -0. 02024 -0.02110

00000 47163 50600 03163 97600

0.20 0.21 0.22 0. 23 0. 24

0. 25 0. 26 0. 27 0. 28 0. 29

0.01708 0.01757 0.01804 0.01849 0.01892

98438 57400 36838 34400 47838

-0.12304 -0.12609 -0.12900 -0.13176 -0.13438

68750 89600 52350 57600 06350

0.92285 0.91664 0.91020 0; 90353 0.89664

15625 24400 36025 66400 32025

0.20507 0.21470 0.22443 0; 23425 0.24415

81250 90400 37650 02400 63650

-0.02197 -0.02282 -0.02367 -0.02451 -0.02534

26563 82600 58163 45600 37163

0. 25 0.26 0.27 0.28 0. 29

0.30 0. 32 0. 33 0. 34

0.01933 0.01973 0.02010 0.02046 0.02079

75000 13838 62400 18838 81400

-0.13685 -0.13917 -0.14135 -0.14338 -0.14527

00000 40350 29600 70350 65600

0.88952 0.88218 0.87462 0.86683 0.85884

50000 38025 14400 98025 08400

0.25415 0.26422 0.27439 0.28463 0.29495

00000 89650 10400 39650 54400

-0.02616 -0.02697 -0.02776 -0.02854 -0.02931

25000 01163 57600 86163 78600

0. 30 0. 31 0. 32 0.33 0.34

0.35 0. 36 0.37 0. 38 0. 39

0.02111 0.02141 0.02168 0.02194 0.02218

48438 18400 89838 61400 31838

-0.14702 -0.14862 -0.15008 -0.15139 -0.15256

18750 33600 14350 65600 92350

0.85062 0.84219 0.83356 0.82471 0.81565

65625 90400 04025 28400 86025

0.30535 0.31582 0.32636 0; 33697 0.34765

31250 46400 75650 94400 77650

-0.03007 -0.03081 -0.03153 -0.03224 -0.03293

26563 21600 55163 18600 03163

0. 35 0.36 0. 37 0: 38 0. 39

0. 40 0. 41 0. 42 0. 43 0. 44

0.02240 0.02259 0.02277 0.02292 0.02306

00000 64838 25400 80838 30400

-0.15360 -0.15448 -0.15523 -0.15584 -0.15631

00000

94350 81600 68350 61600

0.80640 0.79693 0.78727 0.77742 0.76737

00000 94025 92400 20025 02400

0.35840 0.36920 0.38006 0.39098 0.40195

00000 35650 58400 41650 58400

-0.03360 -0.03425 -0.03487 -0.03548 -0.03607

00000 00163 94600 74163 29600

0. 40 0. 41 0. 42 0. 43 0.44

0. 45 0. 46 0. 47 0. 48 0. 49

0.02317 0.02327 0.02334 0.02339 0.02342

73438 09400 37838 58400 70838

-0.15664 -0.15683 -0.15689 -0.15681 -0.15659

68750 97600 56350 53600 98350

0.75712 0.74669 0.73607 0.72527 0.71428

65625 36400 42025 10400 70025

0.41297 0.42404 0.43516 0.44632 0.45751

81250 82400 33650 06400 71650

-0.03663 -0.03717 -0.03768 -0.03817 -0.03863

51563 30600 57163 21600 14163

0.45 0. 46 0. 47 0. 48 0. 49

0. 50

0.02343 75000 A2

-0.03906 25000 A -2

0. 50 -P

P 0. 00

0. 31

-0.15625 00000 A1

0.70312 50000 Ao

0.46875 00000 A-1

0. 07 0.08 0. 09

NUMERICAL ANALYSIS

FIVE-P 'OINT LAGRANGIAN INTERPOLATION COEFFICIENI T S P ( P 2 - 1 ) (P2-4) A ! ( p ) = ( - 1 ) k + 2 (2+ k ) !(2- k ) !( p - k )

Tab1le 25.1

P 0.50 0.51 0.52 0.53 0.54

A -2 0.02343 75000 0.02342 70838 0.02339 58400 0.02334 37838 0.02327 09400

-0.15625 -0.15576 -0.15515 -0.15440 -0.15352

00000 68350 13600 46350 77600

0.70312 0.69178 0.68027 0.66860 0.65675

50000 80025 90400 12025 76400

0.46875 0.48001 0.49131 0.50263 0.51398

00000 61650 26400 63650 42400

-0.03906 -0.03946 -0.03983 -0.04017 -0.04048

25000 44163 61600 67163 50600

0. 50 0. 51 0. 52 0.53 0. 54

0.55 0.56 0.57 0.58 0.59

0.02317 0.02306 0.02292 0.02277 0.02259

73438 30400 80838 25400 64838

-0.15252 -0.15138 -0.15012 -0.14874 -0.14723

18750 81600 78350 21600 24350

0.64475 0.63258 0.62026 0.60779 0.59516

15625 62400 50025 12400 84025

0.52535 31250 0.53673 98400 0.54814 11650 0.5595,5 38400 0.57097 45650

-0.04076 -0.04100 -0.04120 -0.04137 -0.04150

01563 09600 64163 54600 70163

0.55 0. 56 0. 57 0. 58 0. 59

0.60 0.61 0.62 0.63 0.64

0.02240 0.02218 0.02194 0.02168 0.02141

00000 31838 61400 89838 18400

-0.14560 -0.14384 -0.14197 -0.13998 -0.13787

00000 62350 25600 04350 13600

0.58240 0.56948 0.55644 0.54325 0.52994

00000 96025 08400 74025 30400

0.58240 0.59382 0.60525 0.61667 0.62808

00000 67650 14400 05650 06400

-0.04160 -0.04165 -0.04165 -0.04163 -0.04156

00000 33163 58600 65163 41600

0. 60 0. 61 0. 62 0. 63 0. 64

0. 65 0.66 0.67 0.68 0.69

0.02111 0.02079 0.02046 0.02010 0.01973

48438 81400 18838 62400 13838

-0.13564 -0.13330 -0.13085 -0.12829 -0.12562

68750 85600 80350 69600 70350

0.51650 0.50293 0.48925 0.47545 0.46154

15625 68400 28025 34400 28025

0.63947 0.65085 0.66222 0.67355 0.68486

81250 94400 09650 90400 99650

-0.04144 -0.04128 -0.04107 -0.04082 -0.04051

76563 58600 76163 17600 71163

0. 65 0. 66 0. 67 0. 68 0. 69

0.70 0.71 0.72 0.73 0.74

0.01933 0.01892 0.01849 0.01804 0.01757

75000 47838 34400 36838 57400

-0.12285 -0.11996 -0.11698 -0.11389 -0.11070

OOUOO

0.44752 0.43340 0.41918 0.40487 0.39046

50000 42025 46400 06025 64400

0.69615 0.70739 0.71860 0.72976 0.74088

00000 53650 22400 67650 50400

-0.04016 -0.03975 -0.03929 -0.03878 -0.03822

25000 67163 85600 68163 02600

0. 70

76350 17600 42350 69600

0.73 0.74

0.75 0.76 0.77 0.78 0.79

0.01708 0.01658 0.01606 0.01552 0.01497

98438 62400 51838 69400 17838

-0.10742 -0.10404 -0.10056 -0.09699 -0.09334

18750 09600 62350 97600 36350

0.37597 0.36140 0.34675 0.33203 0.31725

65625 54400 76025 76400 02025

0.75195 0.76296 0; 77392 0.78481 0; 79564

31250 70400 27650 62400 33650

-0.03759 -0.03691 -0.03617 -0.03538 -0.03452

76563 77600 93163 10600 17163

0.75 0. 76 0.77 0. 78 0. 79

0.80 0.81 0.82 0.83 0.84

0,01440 0.01381 0.01320 0.01258 0.01195

00000 18838 77400 78838 26400

-0.08960 -0.08577 -0.08185 -0.07786 -0.07379

00000 10350 89600 60350 45600

0.30240 0.28749 0.27253 0.25752 0.24246

00000 18025 04400 08025 78400

0.80640 0.81708 0.82768 0.83820 0.84863

00000 19650 50400 49650 74400

-0.03360 -0.03261 -0.03156 -0.03044 -0.02926

00000 46163 42600 76163 33600

0. 80 0. 81 0. 82 0.83 0. 84

0.85 OI86 0.87 0.88 0.89

0.01130 OI01063 0.00995 0.00926 0.00855

23438 73400 79838 46400 76838

-0.06964 -0.06542 -0.06113 -0.05677 -0.05234

68750 53600 24350 05600 22350

0.22737 0.21225 0.19709 0.18192 0.16673

65625 20400 94025 38400 06025

0.85897 0.86922 0.87936 0.88940 0.89933

81250 26400 65650 54400 47650

-0.02801 -0.02668 -0.02529 -0.02382 -0.02228

01563 66600 15163 33600 08163

0. 85 0. 86 0. 87 0. 88 0.89

0.90 0.91 0.92 0.93 0.94

0.00783 0.00710 0.00635 OI00560 0.00483

75000 44838 90400 15838 25400

-0.04785 -0.04329 -0.03868 -0.03401 -0.02929

00000 64350 41600 58350 41600

0.15152 0.13631 0.12109 0.10588 0.09068

50000 24025 82400 80025 72400

0.90915 0.91884 0.92841 0.93786 0.94717

00000 65650 98400 51650 78400

-0.02066 -0.01896 -0.01719 -0.01533 -0.01340

25000 70163 29600 89163 34600

0.90 0. 91 0. 92 0.93 0.94

0.95 0.96 0.97 0.98 0.99

0.00405 0.00326 0.00246 0.00164 0.00082

23438 14400 02838 93400 90838

-0,02452 -0.01970 -0.01483 -0.00992 -0.00498

18750 17600 66350 93600 28350

0.07550 0.06033 0.04519 0.03009 0.01502

15625 66400 82025 20400 40025

0.95635 0.96538 0.97427 0: 98300 0.99158

31250 62400 23650 66400 41650

-0.01138.51563 -0.00928 25600 -0.00709 42163 -0.00481 86600 -0.00245 44163

0.95 0: 96 0.97 0.98 0.99

1.00

0.00000 00000

1.00000 00000

0.00000 00000

1.00

A2

A-1

0.00000 00000 A1

Ao

0.00000 00000 Ao

A1

A-1

Az

A-2

0. 71 0. 72

-P

NUMERICAL ANALYSIS

FIT‘E-POINT LAGRANGIAN INTERPOLATION COEFFICIENTS

Table 25.1

4(P) P

A -2

-4-1 0.00000 00000

=(- l)k+2

P ( P 2 - 1) (P2-4) (2 k) !( 2-k ) !( p-k )

+

Ao 0.00000 00000

A1 00000

A2

91650 66400 73650 62400

0.00000 0.00254 0.00518 0.00791 0.01074

00000 60838 53400 92838 94400

1.00 1.01 1.02 1.03 1. 04

0.00501 0. U1006 0.01513 0.02023

61650 26400 63650 42400

-0.01497 -0.02989 -0.04474 -0.05953

39975 19600 77975 53600

1. OOPOO 1.00824 1.01632 1.02422 1.03194

01563 14600 79163 89600 40163

0.02535 0.03048 0.03564 0; 04080 0.04597

31250 98400 11650 38400 45650

-0.07424 -0,08888 -0.10342 -0.11787 -0.13222

84375 07600 59975 77600 95975

1.03947 1.04681 1.05396 1.06089 1.06763

81250 78400 01650 98400 15650

0.01367 0.01670 0.01983 0.02306 0.02639

73438 45400 25838 30400 74838

1.05 1.06 1. 07 1.08 1. 09

-0.00866 -0; 00955 -0.01044 -0.01134 -0.01223

25000 38163 73600 25163 86600

0.05115 0.05632 0.06150 0.06667 0.07183

00000 67650 14400 05650 06400

-0.14647 -0.16060 -0.17462 -0.18850 -0.20225

50000 73975 01600 65975 99600

1.07415 1.08044 1.08652 1.09237 1.09798

00000 97650 54400 15650 26400

0.02983 0.03338 0.03704 0.04080 0.04468

75000 46838 06400 69838 53400

1.10 1.11 1.12 1.13 1.14

1.15 1.16 1.17 1.18 1.19

-0.01313 -0.01403 -0.01492 -0.01582 -0.01671

51563 13600 66163 02600 16163

0.07697 0.08210 0.08722 0,09230 0.09736

81250 94400 09650 90400 99650

-0.21587 -0.22934 -0,24265 -0.25580 -0.26879

34375 01600 31975 55600 01975

1.10335 1.10847 1.11334 1.11796 1.12231

31250 74400 99650 50400 69650

0.04867 0.05278 0.05700 0.06135 0.06581

73438 46400 88838 17400 48838

1.15 1.16 1. 17 1.18 1.19

1.20 1.21 1.22 1.23 1. 24

-0.01760 -0.01848 -0.01936 -0.02024 -0.02110

00000 47163 50600 03163 97600

0.10240 0.10739 0.11235 0.11726 0.12213

00000 53650 22400 67650 50400

-0.28160 -0.29422 -0.30666 -0.31890 -0.33094

00000 77975 63600 83975 65600

1.12640 1.13020 1.13373 1.13697 1.13992

00000 83650 62400 77650 70400

0.07040 0.07510 0.07994 0.08490 0.08999

00000 87838 29400 41838 42400

1.20 1.21 1.22 1.23 1. 24

1.25 1.26 1.27 1.28 1.29

-0.02197 -0.02282 -0.02367 -0.02451 -0.02534

26563 82600 58163 45600 37163

0.12695 0.13171 0.13642 0.14106 0.14564

31250 70400 27650 62400 33650

-0.34277 -0.35438 -0.36576 -0.37691 -0.38781

34375 15600 33975 13600 77975

1.14257 1.14492 1.14696 1.14868 1.15008

81250 50400 17650 22400 03650

0.09521 0.10056 0.10605 0.11167 0.11743

48438 77400 46838 74400 77838

1.25 1.26 1.27 1.28 1.29

1. 30 1.31 1. 32 1.33 1.34

-0.02616 -0. 02697 -0.02776 -0.02854 -0.02931

25000 01163 57600 86163 78600

0.15015 0.15458 0.15893 0.16320 0.16738

00000 19650 50400 49650 74400

-0.39847 -0.40887 -0.41901 -0.42887 -0.43845

50000 51975 05600 31975 51600

1.15115 1.15188 1.15227 1.15232 1.15201

00000 49650 90400 59650 94400

0.12333 0.12937 0.13556 0.14189 0.14836

75000 83838 22400 08838 61400

1. 30 1.31 1.32 1.33 1.34

1.35 1. 36 1.37 1.38 1. 39

-0.03007 -0.03081 -0.03153 -0.03224 -0.03293

26563 21600 55163 18600 03163

0.17147 81250 0.17936 65650 0.18315 54400 0.18683 47650

-0.44774 -0.45674 -0.46543 -0.47381 -0.48187

84375 49600 65975 51600 23975

1.15135 1.15032 1.14891 1.14713 1.14496

31250 06400 55650 14400 17650

0.15498 0.16176 0.16868 0.17577 0.18300

98438 38400 99838 01400 61838

1.35 1. 36 1.37 1.38 1.39

1.40 1. 41 1.42 1.43 1.44

-0.03360 -0.03425 -0.03487 -0.03548 -0.03607

00000 00163 94600 74163 29600

0.19040 0.19384 0.19716 0.20036 0.20342

00000 65650 98400 51650 78400

-0.48960 -0.49698 -0.50403 -0.51072 -0.51704

00000 95975 27600 09975 57600

1.14240 1.13943 1.13607 1.13229 1.12809

00000 95650 38400 61650 98400

0.19040 0.19795 0.20566 0.21354 0.22159

00000 34838 85400 70838 10400

1.40 1.41 1.42 1.43 1.44

1.45 1.46 1.47 1.48 1.49

-0.03663 -0.03717 -0.03768 -0.03817 -0.03863

51563 30600 57163 21600 14163

0.20635 0.20913 0.21177 0.21425 0.21658

31250 62400 23650 66400 41650

-0.52299 -0.52857 -0; 53375 -0.53853 -0.54291

84375 03600 27975 69600 39975

1.12347 1.11842 1; 11293 1.10699 1.10060

81250 42400 13650 26400 11650

0.22980 0.23818 0.24673 0.25545 0.26436

23438 29400 47838 98400 00838

1.45 1.46 1.47 1. 48 1.49

1.50

-0.03906 25000

0.27343 75000

1. 50

1.00 1.01 1.02 1. 03 1. 04

0.00000 -0.00083 -0.00168 -0.00253 -0.00339

00000 74163 26600 52163 45600

1. 05 1.06 1.07 1. 08 1.09

-0.00426 -0.00513 -0.00600 -0.00688 -0.00777

1.10 1.11 1.12 1.13 1.14

A2

0: 17547 26400

0.21875 00000 A1

-0.54687 50000 Ao

1.09375 00000 A-1

A-z

-P

907

NUMERICAL ANALYSIS

FIVE-POINT LAGRANGIAN INTERPOLATION COEFFICIENTS =( - l ) k + 2

P

A -z

A-1 00000

1.50 1.51 1.52 1.53 1.54

-0.03906 -0.03946 -0.03983 -0.04017 -0.04048

25000 44163 61600 67163 50600

0.21875 0.22074 0.22257 0.22422 0.22569

1.55 1. 56 1.57 1. 58 1.59

-0.04076 -0.04100 -0.04120 -0.04137 -0.04150

01563 09600 64163 54600 70163

1. 60 1. 61 1.62 1. 63 1. 64

-0.04160 -0.04165 -0.04166 -0.04163 -0.04156

1.65 1.66 1. 67 1. 68 1. 69

Table 25.1

P (PZ- 1) (PZ-4) ( 2 +k) !( 2 - k)!( p - k )

Az

91650 66400 73650 62400

-0.54687 -0.55041 -0.55351 -0.55617 -0.55837

50000 09975 29600 17975 83600

1.09375 1.08643 1.07864 1.07036 1.06160

00000 21650 06400 83650 82400

0.27343 0.28269 0.29213 0.30175 0.31155

75000 40838 18400 27838 89400

1.50 1.51 1. 52 1.53 1.54

0.22697 0.22806 0.22896 0.22964 0.23013

81250 78400 01650 98400 15650

-0.56012 -0.56139 -0.56219 -0.56249 -0.56230

34375 77600 19975 67600 25975

1.05235 1.04259 1.03232 1.02154 1.01023

31250 58400 91650 58400 85650

0.32155 0.33173 0.34210 0.35267 0.36343

23438 50400 90838 65400 94838

1.55 1.56 1.57 1. 58 1. 59

00000 33163 58600 65163 41600

0.23040 0.23044 0.23027 0.22987 0.22923

00000 97650 54400 15650 26400

-0.56160 -0.56037 -0.55863 -0.55634 -0.55351

00000 93975 11600 55975 29600

0.99840 0.98602 0.97309 0.95962 0.94558

00000 27650 94400 25650 46400

0.37440 0.38556 0.39692 0.40848 0.42025

00000 01838 21400 79838 98400

1.60 1. 61 1. 62 1. 63 1. 64

-0.04144 -0.04128 -0.04107 -0.04082 -0.04051

76563 58600 76163 17600 71163

0.22835 0.22722 0.22584 0.22421 0.22231

31250 74400 99650 50400 69650

-0.55012 -0.54616 -0.54163 -0.53651 -0.53079

34375 71600 41975 45600 81975

0.93097 0.91579 0.90002 0.88367 0.86671

81250 54400 89650 10400 39650

0.43223 0.44443 0.45683 0.46945 0.48228

98438 01400 28838 02400 43838

1. 65 1.66 1.67 1. 68 1. 69

1. 70 1.71 1. 72 1.73 1.74

-0.04016 -0.03975 -0.03929 -0.03878 -0.03822

25000 67163 85600 68163 02600

0.22015 0.21770 0.21498 0.21197 0.20867

00000 83650 62400 77650 70400

-0.52447 -0.51753 -0.50996 -0.50176 -0.49290

50000 47975 73600 23975 95600

0.84915 0.83097 0.81217 0.79273 0.77266

00000 13650 02400 87650 90400

0.49533 0.50861 0.52210 0.53583 0.54978

75000 17838 94400 26838 37400

1. 70 1. 71 1.72 1.73 1.74

1.75 1.76 1.77 1.78 1. 79

-0.03759 -0.03691 -0.03617 -0.03538 -0.03452

76563 77600 93163 10600 17163

0.20507 0.20117 0.19696 0.19243 0.18758

81250 50400 17650 22400 03650

-0.48339 -0.47321 -0.46235 -0.45081 -0.43856

84375 85600 93975 03600 07975

0.75195 0.73058 0.70855 0.68584 0.66246

31250 30400 07650 82400 73650

0.56396 0.57837 0.59302 0.60791 0.62303

48438 82400 61838 09400 47838

1.75 1. 76 1.77 1.78 1.79

1.80 1.81 1.82 1. 83 1. 84

-0.03360 -0.03261 -0.03156 -0.03044 -0.02926

00000 46163 42600 76163 33600

0.18240 0.17688 0.17102 0.16482 0.15826

00000 49650 90400 59650 94400

-0.42560 -0.41191 -0.39750 -0.38234 -0.36642

00000 71975 15600 21975 81600

0.63840 0.61363 0.58817 0.56199 0.53510

00000 79650 30400 69650 14400

0.63840 0.65400 0.66986 0.68596 0.70232

00000 88838 37400 68838 06400

1.80 1.81 1. 82 1.83 1.84

1.85 1.86 1.87 1.88 1. 89

-0.02801 -0.02668 -0.02529 -0.02382 -0.02228

01563 66600 15163 33600 08163

0.15135 0.14407 0.13641 0.12838 0.11996

31250 06400 55650 14400 17650

-0.34974 -0.33229 -0.31404 -0.29500 -0.27515

84375 19600 75975 41600 03975

0.50747 0.47911 0.45001 0.42015 0.38953

81250 86400 45650 74400 87650

0.71892 73438 0.75290 89838 0. 77028 86400 0.78793 06838

1.85 1.86 1. 87 1. 88 1.89

1.90 1.91 1.92 1.93 1.94

-0.02066 -0.01896 -0.01719 -0.01533 -0.01340

25000 70163 29600 89163 34600

0.11115 0.10193 0.09232 0.08229 0.07184

00000 95650 38400 61650 98400

-0.25447 -0.23296 -0.21061 -0.18740 -0.16332

50000 65975 37600 49975 87600

0.35815 0.32598 0.29302 0.25927 0.22472

00000 25650 78400 71650 18400

0.80583 0.82401 0.84245 0.86117 0.88016

75000 14838 50400 05838 05400

1. 90 1.91 1.92 1. 93 1.94

1.95 1.96 1.97 1.98 1.99

-0.01138 -0.00928 -0.00709 -0. 00481 -0.00245

51563 25600 42163 86600 44163

0.06097 0.04967 0.03793 0.02574 0.01310

81250 42400 13650 26400 11650

-0.13837 -0.11252 -0.08577 -0.05811 -0.02952

34375 73600 87975 59600 69975

0.18935 0.15316 0.11614 0.07827 0.03956

31250 22400 03650 86400 81650

0.89942 0.91897 0.93880 0.95891 0.97931

73438 34400 12838 33400 20838

1.95 1. 96 1.97 1. 98 1. 99

1.00000 00000

2. 00

2.00

0.00000 00000 Az

0.00000 00000 Ai

0.00000 00000 Ao

0.00000 00000 A-i

0. 73578 93400

A-z

-P

908

NUMERICAL ANALYSIS

SIX-POINT LAGRANGIAN INTERPOLATION COEFFICIENTS

Table 25.1

A-2

4

0.02 0.03 0.04

0.00000 0.00049 0.00098 0.00146 0.00193

00000 57921 30066 14085 07725

A-I 0.00000 00000 -0.00493 33767 -0.00973 36932 -0.01440 12590 -0.01893 64224

An 1.00000 00000 0.99654 20858 0.99283 67064 0.98888 64505 0.98469 39648

0.00000 0.01006 0.02026 0.03058 0.04102

00000 60817 19736 41170 89152

A2 0.00000 00000 -0.00250 38746 -0.00501 43268 -0.00752 95922 -0.01004 78976

0.05 0.06 0.07 0.08 0.09

0.00239 0.00284 0.00328 0.00371 0.00413

08828 15335 25281 36794 48096

-0.02333 ._ ~ . .95703 . -0.02761 i i 2 7 b -0.03175 15567 -0.03576 13568 -0.03964 10640

0.98026 0.97559 0.97069 0.96555 0.96019

19531 31752 04458 66336 46604

0.05159 O;ObZ27 0.07306 0.08396 0.09496

27344

19048 27217 14464 43071

0.10 0.11 0.12 0.13 0.14

0.00454 0.00494 0.00533 0.00571 0.00608

57500 63412 64326 58827 45585

-0.04339 -0.04701 -0.05050 -0.05387 -0,05710

12500 25223 55232 09296 94524

0.95460 0.94879 0.94276 0.93652 0.93006

75000 81771 97664 53917 82248

0.10606 0.11726 0.12855 0.13994 0.15140

0.15 0.16 0.17 0.18 0.19

0.00644 0.00678 0.00712 0.00744 0.00776

23359 90995 47422 91654 22787

-0.06022 -0.06320 -0.06607 -0.06881 -0.07142

18359 88576 13273 00868 60096

0.92340 0.91652 0.90945 0.90217 0.89470

14844 84352 23870 66936 47517

0.20 0.21 0.22 0.23 0.24

0.00806 0.00835 0.00863 0.00890 0.00915

40000 42553 29786 01118 56045

-0.07392 -0.07629 -0.07854 -0.08067 -0.08269

00000 29929 59532 98752 57824

0.88704 0.87918 0.87114 0.86292 0.85452

0.25 0.26 0.27 0.28 0.29

0.00939 0.00963 0.00985 0.01006 0.01025

94141 15055 18513 04314 72328

-0.08459 -0.08637 -0.08804 -0.08960 -0.09104

47266 77876 60729 07168 28802

0.30 0.31 0.32 0.33 0.34

0.01044 0.01061 0.01077 0.01092 0.01106

22500 54844 69446 66459 46105

-0.09237 -0.09359 -0.09470 -0.09571 -0.09660

0.35 0.36 0.37 0.38 0.39

0.01119 0.01130 0.01140 0.01149 0.01157

08672 54515 84054 97774 96219

0.40 0.41 0.42 0.43 0.44

0.01164 0.01170 0.01175 0.01178 0.01180

0.45 0.46 0.47 0.48 0.49

0.01182 0.01182 0.01181 0.01179 0.01176

0.50

0.01171 87500

P 0.00 0.01

-43

0.00000 0.00033 0.00066 0.00099 0.00133

00000 32917 63334 88752 06675

1.00 0.99 0.98 0.97 0.96

-0.01256 7 4 6 0 9

0.00166 0.00199 0.00231 0.00264 0.00296

14609 10065 90557 53606 96742

0.95 0.94 0.93 0.92 0.91

75000 71904 95136 05758 64552

-0.02512 -0,02761 -0.03008 -0.03255 -0.03500

12500 05290 83968 30217 25676

0.00329 0.00361 0.00392 0.00424 0.00455

17500 0.90 13426 0.89 82074 0.88 21011 0.87 27815 0.86

0.16295 0.17457 0.18627 0.19803 0.20986

32031 68448 33805 87864 90158

-0.03743 -0.03984 -0.04224 -0.04461 -0.04695

51953 90624 23240 31332 96417

0.00486 0.00516 0.00546 0.00575 0.00604

00078 0.85 35405 0.84 31416 0.83 85746 0.82 96051 0.81

00000 59183 60264 38830 30848

0.22176 0.23370 0.24570 0.25775 0.26984

00000 76492 78536 64845 93952

-0.04928 -0.05157 -0.05383 -0.05606 -0.05826

00000 23583 48668 56760 29376

0.00633 0.00661 0.00689 0.00716 0.00743

60000 75284 39614 50719 06355

0.80 0.79 0.78 0.77 0.76

0.84594 0.83720 0.82828 0.81920 0.80996

72656 00952 52783 65536 76929

0.28198 0.29415 0.30635 0.31858 0.33083

24219 13848 20892 03264 18746

-0.06042 -0.06254 -0.06463 -0.06667 -0.06868

48047 94324 49783 96032 14711

0.00769 0.00794 0.00819 0.00843 0.00866

04297 42345 18324 30086 75510

0.75 0.74 0.73 0.72 0.71

37500 45385 64832 08458 89124

0.80057 0.79102 0.78132 0.77148 0.76150

25000 48096 84864 74242 55448

0.34310 0.35538 0.36768 0.37998 0.39229

25000 79579 39936 63433 07352

-0.07063 -0.07254 -0,07441 -0.07622 -0.07798

87500 96127 22368 48054 55076

0.00889 0.00911 0.00932 0.00953 0.00973

52500 58993 92954 52378 35295

0.70 0.69 0.68 0.67 0.66

-0.09740 -0.09809 -0.09867 -0.09916 -0,09955

19922 14176 85435 47468 14258

0.75138 0.74113 0.73075 0.72024 0.70962

67969 51552 46195 92136 29842

0.40459 0.41688 0.42917 0.44144 0.45369

28906 85248 33480 30664 33833

-0.08594 81254

0.00992 0.01010 0.01028 0.01044 0.01060

39766 63885 05783 63626 35618

0.65 0.64 0.63 0.62 0.61

80000 49786 06306 50351 82765

-0.09984 -0.10003 -0.10012 -0.10013 -0.10004

00000 19092 86132 15915 23424

0.69888 0.68802 0.67706 0.66599 0.65482

00000 43508 01464 15155 26048

0.46592 0.47811 0.49028 0.50241 0.51450

00000 86167 49336 46520 34752

-0.08736 -0.08870 -0.08998 -0.09120 -0.09234

00000 75421 90068 26598 67776

0.01075 0.01089 0.01102 0.01114 0.01125

20000 15052 19094 30487 47635

0.60 0.59 0.58 0.57 0.56

04453 16375 19546 15034 03961

-0.09986 -0.09959 -0.09923 -0.09879 -0.09826

23828 32476 64892 36768 63965

0.64355 0.63220 0,62075 0.60922 0.59762

75781 06152 59108 76736 01254

0.52654 0.53854 0.55048 0.56236 0.57418

71094 12648 16567 40064 40421

-0.09341 -0.09441 -0,09534 -0.09619 -0.09696

96484 95724 48621 38432 48548

0.01135 0.01144 0.01153 0.01160 0.01166

68984 93025 18292 43366 66877

0.55 0.54 0.53 0.52 0.51

0.01171 87500

0.50

A3

-0.09765 62500

A2

0.58593 75000 A1

0.58593 75000 A0

-0.09765 62500 8-1

-4-2

P

909

NUMERICAL ANALYSIS

SIX-POINT LAGRANGIAN INTERPOLATION COEFFICIENTS

P

A-2 00000 32917 63334 88752 06675

Ao

-4-1

A2 A3 0.00000 00000 0.00000 00000 0.00506 67067 -0,00050 41246 0.01026 69732 -0.00101 63266 0.01560 09890 -0.00153 63410 0.02106 89024 -0.00206 38925

00000 27517 86936 43870 64352

1.00000 1.00320 1.00616 1.00886 1.01130

00000 79192 33736 39545 73152

-0.04826 14844 -0.05747 62248

1.01349 1.01541 1.01707 1.01846 1.01958

11719 33048 15592 38464 81446

0.02667 08203 0.03240 68076

75000 26604 86336 24458 11752

1.02044 1.02102 1.02133 1.02136 1.02112

25000 50279 39136 74133 38552

-0.13312 -0.14066 -0ii48oi -0,15518 -0.16217

19531 19648 84505 87064 00858

1.02060 1.01979 1.01871 1.01734 1.01569

-0.16896 -0.17555 -0.18195 -0.18815 -0.19415

00000 59192 53736 59545 53152

0.05554 0.05729 0.05900 0.06065 0.06226

19922 -0.19995 69124 -0.20554 28458 -0.21092 84832 -0.21609 25385 -0.22105

0.06381 0.06531 0.06675 0.06813 0.06946

37500 08802 27168 80729 57876

0.00000 0.00249 0.00498 0.00745 0.00991

00000 55421 10068 46597 47776

1.00 1.01 1.02 1.03 1.04

0.00000 -0.00033 -0.00066 -0.00099 -0.00133

0.00000 -0.00993 -0.01972 -0.02938 -0.03889

1.05 1.06 1.07 1.08 1.09

-0.00166 14609

1.10 1.11 1.12 1.13 1.14

-0.00329 -0.00361 -0.00392 -0.00424 -0.00455

17500 13426 82074 21011 27815

0.02429 0.02661 0.02890 0.03116 0.03340

62500 43965 56768 84892 12476

-0.09276 -0.10118 -0.10942 -0.11750 -0.12540

1.15 1.16 1.17 1.18 1.19

-0.00486 -0.00516 -0.00546 -0,00575 -0.00604

00078 35405 31415 85746 96051

0.03560 0.03777 0.03990 0.04200 0.04405

23828 03424 35915 06132 99092

1.20 1.21 1.22 1.23 1.24

-0.00633 -0.00661 -0.00689 -0.00716 -0.00743

60000 75284 39614 50719 06355

1.25 1.26 1.27 1.28 1.29

-0.00769 -0.00794 -0.00819 -0.00843 -0,00866

04297 42345 18324 30086 75509

1.30 1.31 1.32 1.33 1.34

-0.00889 -0.00911 -0.00932 ..__-~ -0.00953 -0.00973

52500 58993 92954 52378 35295

1.35 1.36 1.37 1.38 1.39

-0.00992 -0,01010 -0.01028 -0.01044 -0.01060

39766 63885 05783 63626 35618

1.40 1.41 1.42 1.43 1.44

Table 25.1

86953 04535 88606 35994 43420

0.05 0.06 0.07

0.05669 0.06309 0.06963 0.07630 0.08311

12500 -0.00537 07500 70523 -0.00594 24737 64032 -0.00651 91526 90596 -0.00710 04152 47324 -0.00768 58785

0.10 0.11 0.12 0.13 0.14

16406 92448 52180 81864 68533

0.09005 0.09712 0.10432 0.11166 0.11912

30859 -0.00827 37376 -0.00886 62572 -0.00946 01668 -0.01006 49396 -0.01066

51484 78195 34747 16854 20112

0.15 0.16 0.17 0.18 0.19

1.01376 1.01153 1.00902 1.00622 1.00313

00000 64867 52536 53220 57952

0.12672 0.13444 0.14229 0.15028 0.15838

00000 -0.01126 47229 -0.01186 84332 -0.01247 04052 -0.01307 98624 -0.01367

40000 71878 10986 52443 91245

0.20 0.21 0.22 0.23 0.24

11719 13048 35592 58464 61446

0.99975 0.99608 0.99212 0.98786 0.98331

58594 47848 19267 67264 87121

0.16662 0.17498 0.18347 0.19208 0.20081

59766 78676 46029 51968 86102

-0.01428 -0.01488 -0.01548 -0.01608 -0.01667

22266 40255 39838 15514 61653

0.25 0.26 0.27 0.28 0.29

-0.22580 25000 -0.23033 30279 -0;23464 59136 -0.23873 94133 -0.24261 18552

0.97847 0.97334 0196791 0.96219 0.95617

75000 27954 43936 21808 61352

0.20967 0.21864 0.22774 0.23695 0.24628

37500 94685 45632 77758 77924

-0.01726 -0.01785 -0.01843 -0.01901 -0.01958

72500 42169 64646 33784 43305

0.30 0.31 0.32 0.33 0.34

0.07520 09929

-0.24626 -0.24968 -0.25288 -0.25586 -0.25860

16406 72448 72180 01864 48533

0.94986 0.94326 0.93636 0.92917 0.92169

63281 0.25573 32422 29248 0.26529 26976 61855 0.27496 46735 64664 0.28474 76268 42208 0.29463 99558

-0,02014 -0.02070 -0.02125 -0.02179 -0.02232

86797 0.35 57715 0.36 49379 0.37 54974 0.38 67544 0.39

-0.01075 -0.01089 -0.01102 -0.01114 -0.01125

20000 0.07616 00000 15052 0.07705 40096 19094 0.07788 20868 30487 0.07864 33273 47635 0.07933 68576

-0.26112 -0.26340 -0.26545 -0.26727 -0.26886

00000 44867 72536 73220 37952

0.91392 0.90585 0.89749 0.88885 0.87991

00000 44542 83336 24895 78752

00000 60392 62932 89215 20224

-0.02284 -0.02335 -0.02385 -0.02434 -0.02481

80000 85111 75506 43676 81965

0.40 0.41 0.42 0.43 0.44

1.45 1.46 1.47 1.48 1.49

-0.01135 -0.01144 -0.01153 -0.01160 -0.01166

68984 93025 18292 43366 66877

0.07996 0.08051 0.08100 0.08141 0.08176

-0.27021 -0.27133 -0.27221 -0.27285 -0,27326

58594 27848 39267 87264 67121

0.87069 0.86118 0.85139 0.84131 0.83095

55469 66648 24942 44064 38796

0.39918 37265

-0.02527 -0.02572 -0.02615 -0.02656 -0.02696

82578 37575 38871 78234 47286

0.45 0.46 0.47 0.48 0.49

1.50

-0.01171 87500

0.82031 25000

0.41015 62500

-0.02734

37500

0.50

-43

0.01235 96484 0.01478 75724

18359 74524 29296 75232 05223

0.08203 12500 -0.27343 75000

A2

A1

Ao

0.30464 0.31474 0.32495 0.33526 0.34568

8-1

-0.00259 -o:ooli4 -0.00368 -0.00424 -0.00480

0.00 0.01 0.02 0.03 0.04

A-2

0.08 0.09

-P

910

NUMERICAL ANALYSIS

Table 25.1

P

SIX-POINT LAGRANGIAN INTERPOLATION COEFFICIENTS

44-2

8-1

1.50 1.51 1.52 1.53 1.54

-0.01171 -0.01176 -0.01179 -0.01181 -0.01182

87500 03961 15034 19546 16375

0.08203 0.08222 0.08235 0.08240 0.08237

1.55 1.56 1.57 1.58 1.59

-0.01182 -0.01180 -0.01178 -0.01175 -0.01170

1.60 i;6i 1.62 1.63 1.64

-0.01164 -0.01157 -0.01149 -0.01140 -0.01130

80000 96219 97774 84054 54515

1.65 1.66 1.67 1.68 1.69

-0.01119 -0.01106 -0.01092 -0.01077 -0.01061

08672 46105 66459 69446 54845

1.70 1.71 1.72 1.73 1.74

-0.01044 -0.01025 -0.01006 -0.00985 -0.00963

22500 72328 04314 18513 15055

0.07154 0.07021 0.06879 0.06730 0.06573

1.75 1.76 1.77 1.78 1.79

-0.00939 -0.00915 -0.00890 -0.00863 -0.00835

94141 56045 01118 29786 42553

0.06408 0.06236 0.06056 0.05869 0.05674

95703 44224 32590 56932 13767

-0.27072 -0.26947 -0.26797 -0.26624 -0.26428

63281 09248 81855 84664 22208

0.76295 0.75066 0.73811 0.72529 0.71221

60156 90048 53530 75464 81883

0.51244 96721

0.08064 0.08008 0.07944 0.07873 0.07793

00000 12933 50268 10110 90976

-0.26208 -0.25964 -0.25697 -0.25406 -0.25092

00000 24542 03336 44895 58752

0.69888 0.68528 0.67143 0.65734 0.64299

00000 58217 86136 14570 75552

0.52416 0.53592 0.54775 0.55962 0.57155

00000 86554 25532 85377 33824

-0.24755 -0.24395 -0.24012 -0.23606 -0.23178

55469 46648 44942 64064 18796

0.58352 0.59553 0.60758 0.61967 0.63179

37891 63876 77354 43168 25427

87500 09477 55968 29404 32676

-0.22727 -0.22253 -0.21758 -0.21241 -0.20702

25000 99629 60736 27483 20152

0.55194 0.53597 0.51978 0.50338 0.48678

75000 65304 89536 91158 14952

0.64393 0.65610 0.66830 0.68050 0.69272

87500 92010 00832 75083 75124

-0.02972 -0.02949 -0.02923 -0.02893 -0.02858

02500 94834 81286 49649 87545

0.70 0.71 0.72 0.73 0.74

69141 42624 57427 18332 30604

-0.20141 -0.19559 -0.18956 -0.18332 -0.17688

60156 70048 73530 95464 61883

0.46997 0.45296 0.43575 0.41836 0.40079

07031 14848 87205 74264 27558

0.70495 0.71718 0.72942 0.74165 0.75387

60547 90176 22061 13468 20883

-0.02819 -0.02776 -0.02727 -0.02674 -0.02616

82422 21555 92045 80814 74609

0.75 0.76 0.77 0.78 0.79

-0.17024 -0.16339 -0.15635 -0.14911 -0.14168

00000 38217 06136 34570 55552

0.76608 0.77827 0.79043 0.80258 0.81469

00000 05717 92132 12540 19424

02344 09448 12617 48864 56471

0.82676 0.83879 0.85078 0.86272 0.87460

64453 98476 71516 32768 30590

-0.00454 57500 -0.00413 . . ..-. 48096 . .. . -0.00371 36794 -0,00328 25281 -0.00284 15335

0.03056 0.02777 0.02492 0.02201 0.01904

62500 85315 74368 42242 02076

-0.09330 75000 -0.08464 45304. .___ -0.07582 09536 -0.06684 11158 -0.05770 94952

08828 07725 14086 30066 57921

0.01600 0.01291 0.00976 0.00656 0.00330

67578 -0.04843 53024 -0.03900 73265 -0.02945 43732 -0.01975 80442 -0.00994

1.95 1.96 1.97 1.98 1.99

-0.00239 -0.00193 -0.00146 -0.00098 -0.00049

0.50 0;51 0.52 0.53 0.54

04453 0.08227 82765 0.08210 50350 0.08185 06306 0.08152 49786 0.08112

-0.13407 -0.12627 -0.11829 -0.11013 -0.10180

1.90 1.91 1.92 1.93 1.94

37500 40202 46566 47617 34225

25000 0.41015 62500 19629 0.42121 41848 40736 0.43235 51232 07483 0.44357 65921 40152 0.45487 60524

40234 01324 73971 68032 93898

-0.00644 -0.00608 -0.00571 -0.00533 -0.00494

A3 -0.02734 -0.02770 -0.02804 -0.02836 -0.02866

0.82031 0.80939 0.79819 0.78672 0.77497

0.04351 0.04106 0.03853 0.03594 0.03328

1.85 1.86 1.87 1.88 1.89

A2

A1

75000 07954 63936 41808 41352

1.80 1.81 1.82 1.83 1.84

2.00

Ao

12500 -0.27343 90640 -0.27337 33568 -0.27306 35567 -0.27252 91276 -0.27174

23359 45585 58826 64326 63412

0.00000 00000 -4 3

0.00000 00000

A2

07031 94848 07205 94264 07558

-0,02995 -0.03007 -0.03016 -0.03022 -0.03025

20000 0.60 36943 0.61 60826 0.62 81108 0.63 87085 0.64 0.65 0.66 0.67 0.68 0.69

0.80 0.81 0.82 0.83 0.84 78203 21015 56336 69274 44750

0.85 0.86 0.87 0.88 0.89

0.19698 25000 0.88642 12500 -0.01611 67500 0.17766 0.89817 25173 -0.01483 22067 - . . .. 04979 ... . . 0.15823 50336 0.90985 14432 -0.01347 92806 0.13871 32833 0.92145 25246 -0.01205 63882 0.11910 25752 0.93297 01724 -0.01056 19265

0.90 0.91 0.92 0.93 0.94

0.29179 0.27309 0.25425 0.23528 0.21619

99219 76248 82292 81664 40145

-0.02156 -0.02060 -0.01957 -0.01848 -0.01733

~

0.09941 0.07964 0.05981 0.03992 0.01998

03906 43648 22880 21064 19233

0.94439 0.95573 0.96696 0.97809 0.98910

87109 23776 53223 16068 52046

0.00000 00000 0.00000 00000 1.00000 00000

A1

0.55 0.56 0.57 0.58 0.59

Ao

8-1

-0.00899 -0.00735 -0.00563 -0.00383 -0.00195

42734 17875 28077 56534 86242

0.95 0.96 0.97 0.98 0.99

0.00000 00000 1.00 A-2 -P

911

NUMERICAL ANALYSIS

SIX-POINT LAGRANGIAN INTERPOLATION COEFFICIENTS

P

-4-2

-4-1

2.00 2.01 2.02 2.03 2.04

0.00000 0.00050 0.00101 0.00153 0.00206

00000 41246 63266 63410 38925

0.00000 -0.00335 -0.00676 -0.01021 -0.01371

2.05 2.06 2.07 2.08 2.09

0.00259 0.00314 0.00368 0.00424 0.00480

86953 04535 88605 35994 43420

2.10 2.11 2.12 2.13 2.14

0.00537 0.00594 0.00651 0.00710 0.00768

2.15 2.16 2.17 2.18 2.19

A1

A0

00000 0.00000 00000

A2

Table 25.1

A3

80392 42932 69214 40224

0.00000 0.01005 74108 -0.02001 0.02022 59064 -0.04005 0.03049 97755 -0.06011 0.04087 31648 -0.08017

00000 52433 52264 12080 42848

1.00000 1.01076 1.02140 1.03190 1.04226

00000 0.00000 97879 0.00204 82732 0.00416 90702 0.00638 57024 0.00868

00000 19592 90134 29427 55475

1.00 1.01 1.02 1.03 1.04

-0.01725 -0.02083 -0.02445 -0.02810 -0.03179

36328 37276 22191 69568 57264

0.05134 0.06189 0.07252 0.08323 0.09401

53906 52952 46033 37536 30179

1.05247 1.06252 1.07240 1.08211 1.09165

16016 01076 44679 78368 32752

86484 40865 37232 94406 31417

1.05 1.06 1.07 1.08 1.09

07500 24737 91526 04151 58785

-0.03551 -0.03926 -0.04304 -0.04684 -0,05066

62500 0.10485 75000 -0.20018 61847 0.11575 15021 -0.22003 31232 0.12669 29664 -0.23981 45921 0.13767 47167 -6.25951 80524 0.14868 94248 -0.27911

25000 1.10100 21346 1.11016 16864 1.11912 07492 1.12787 87448 1.13641

37500 21335 12032 36409 20324

0.02446 67500 oIolo52 i 4 8 i 4 0.03370 65686 0.03699 94615

1.10 1.11 1.12 1.13 1.14

0.00827 0.00886 0.00946 0.01006 0.01066

51484 78195 34747 16854 20112

-0.05451 -0.05837 -0.06224 -0.06612 -0.07002

08984 04576 39898 86868 16721 0.20399 00767

49219 83552 79445 24136 03092

1.14472 1.15281 1.16066 1.16827 1.17562

88672 65376 73385 34668 70208

0.04040 0.04391 0.04754 0.05129 0.05515

21953 68205 54091 00546 28726

1.15 1.16 1.17 1.18 1.19

2.20 2.21 2.22 2.23 .2.24

0.01126 0.01186 0.01247 0.01307 0.01367

40000 -0.07392 00000 71878 -0.07782 06554 10986 -0.08172 05532 52443 -0.08561 65377 91245 -0.08950 53824

0.21504 0.22606 0.23706 0.24801 0.25892

00000 -0.39424 00000 72433 -0.41289 96758 32264 -0.43137 73464 92080 -0.44966 08405 62848 -0.46773 78048

1.18272 1.18954 1.19609 1.20235 1.20832

00000 43042 17332 39865 26624

0.05913 0.06324 0.06747 0.07182 0.07631

60000 15959 18414 89394 51155

1;21 1.22 1.23 1.24

2.25 2.26 2.27 2.28 2.29

0.01428 0.01488 0.01548 0.01608 0.01667

22266 40255 39838 15514 61653

-0.09338 -0.09724 -0.10109 -0.10492 -0.10872

37891 83876 57353 23168 45427

0.26977 0.28055 0.29126 0.30188 0.31240

53906 72952 26033 17536 50179

-0,55457 94504

0.08093 0.08568 0.09057 0.09559 0.10076

26172 1.25 37145 1.26 06999 1.27 58886 1.28 16184 1.29

2.30 2.31 2.32 2.33 2.34

0.01726 0.01785 0.01843 0.01901 0.01958

72500 42169 64646 33784 43305

-0.11249 -0.11624 -0,11994 -0.12361 -0.12723

87500 12010 80832 55083 95124

0.32282 0.33312 0.34329 0.35333 0.36323

25000 41346 96864 87492 07448

-0.57114 -0.58741 -0.60337 -0.61900 -0.63429

75000 73671 52064 69817 84648

1.23748 1.24115 1.24446 1.24739 1.24994

62500 60498 13632 28571 10924

0.10607 0.11152 0.11712 0.12287 0.12878

02500 41668 57754 75053 18095

2.35 2.36 2.37 2.38 2.39

0.02014 0.02070 0.02125 0.02179 0.02232

86797 57715 49379 54974 67544

-0.13081 -0.13434 -0.13781 -0.14121 -0.14456

60547 0.37296 49219 10176 0.38253 03552 02060 0.39191 59445 93468 0.40111 04136 40883 0.41010 23092

-0.64923 -0.66380 -0.67798 -0.69177 -0.70513

52344 26752 59770 01336 99417

1.25209 1.25384 1.25519 1.25610 1.25659

65234 0.13484 11641 1.35 94976 0.14105 80685 1.36 02548 0.14743 50458 1.37 89268 0.15397 46426 1.38 55371 0.16067 94293 1.39

2.40 2.41 2.42 2.43 2.44

0.02284 0.02335 0.02385 0.02434 0.02481

80000 85111 75506 43676 81965

-0.14784 -0.15104 -0.15416 -0.15720 -0.16016

00000 25717 72132 92540 39424

0.41888 0.42743 0.43574 0.44380 0.45160

00000 16758 53464 88405 98048

-0.71808 -0.73057 -0.74260 -0.75416 -0.76522

00000 47083 82664 46730 77248

1.25664 1.25623 1.25536 1.25401 1.25219

00000 21204 15932 80027 08224

20000 49727 09894 27162 28435

1.40 1.41 1.42 1.43 1.44

2.45 2.46 2.47 2;48 2.49

0.02527 0.02572 0.02615 0.02656 0.02696

82578 37575 38870 78234 47286

-0.16302 -0.16579 -0.16845 -0.17101 -0.17345

64453 18476 51516 12768 50590

0.45913 0.46637 0.47331 0.47993 0.48623

57031 38152 12358 48736 14504

-0.77578 10156 -0.78580 79352 -0.79529 16683 -0.80421 51936 -0.81256 12829

1.24986 1.24704 1.24370 1.23983 1.23542

94141 0.20452 40859 30276 0.21245 91825 08004 0.22058 08967 17568 0.22889 20166 48077 0.23739 53552

1.45 1.46 1.47 1.48 1.49

2.50

0.02734 37500 -4 3

-0.17578

12500

0.49218 75000

-0.82031

1.23046 87500

A2

00781 43752 97708 98336 79854

A1

-0.10023 -0.12028 -0.14031 -0.16031 -0.18027

-0.29862 -0.31801 -0.33728 -0.35642 -0.37541

25000

Ao

A-1

0.01107 0.01356 0.01614 0.01881 0.02159

0.02744 22100

0.16755 0.17459 0.18181 0.18920 0.19677

0.24609 37500 -4-2

1.20

1.30 1.31 1.32 1.33 1.34

1.50

-P

912

NUMERICAL ANALYSIS

SIX-POINT LAGRANGIAN INTERPOLATION COEFFICIENTS

Table 25.1

(P-3)

k+3 P(P2-Q(P2-4)

(2+k) ! ( 3 4 ) !( p - k )

P 2.50 2.51 2.52 2.53 2.54

A-2 0.02734 37500 0.02770 40203 0.02804 46566 0.02836 47616 0.02866 34225

A- 1 -0.17578 12500 -0.17798 45173 -0.18005 94432 -0.18200 05246 -0.18380 21724

0.49218 0.49778 0.50302 0.50787 0.51233

75000 93671 32064 49817 04648

A1 -0.82031 25000 -0.82745 11996 -0.83395 95264 -0.83981 94142 -0.84501 25848

1.23046 1.22495 1.21886 1.21219 1.20492

87500 22660 39232 21734 53524

A3 0.24609 37500 0.25499 00635 0.26408 71834 0.27338 80221 0.28289 55175

1.50 1.51 1.52 1.53 1.54

2.55 2.56 2.57 2.58 2.59

0.02893 0.02919 0.02942 0.02962 0.02980

97109 26835 13812 48294 20377

-0.18545 -0.18696 -0.18831 -0.18949 -0.19051

87109 43776 33223 96068 72046

0.51637 0.51999 0.52317 0.52589 0.52815

52344 46752 39770 81336 19417

-0.84952 -0.85332 -0.85640 -0.85874 -0.86032

05469 45952 58095 50536 29742

1.19705 1.18855 1.17943 1.16966 1.15924

16797 92576 60710 99868 87533

0.29261 0.30254 0.31268 0.32305 0.33363

26328 23565 77026 17106 74461

1.55 1.56 1.57 1.58 1.59

2.60 2.61 2.62 2.63 2.64

0.02995 0.03007 0.03016 0.03022 0.03025

20000 36943 60826 81107 87085

-0.19136 -0.19202 -0.19249 -0.19277 -0.19285

00000 17879 62732 70702 77024

0.52992 0.53118 0.53193 0.53215 0.53181

00000 67083 62664 26730 97248

-0.86112 -0.86111 -0.86029 -0.85862 -0.85610

00000 63408 19864 67055 00448

1.14816 1.13639 1.12392 1.11076 1.09687

00000 12367 98532 31190 81824

0.34444 0.35548 0.36675 0.37825 0.39000

80000 64894 60574 98730 11315

1.60 1.61 1.62 1.63 1.64

2.65 2.66 2.67 2.68 2.69

0.03025 0.03022 0.03015 0.03004 0.02990

67891 12495 09704 48154 16317

-0.19273 -0.19239 -0.19183 -0.19104 -0.19002

16016 21076 24679 58368 52752

0.53092 0.52943 0.52735 0.52466 0.52133

10156 99352 96683 31936 32829

-0.85269 -0.84837 -0.84314 -0.83696 -0.82981

13281 96552 39008 27136 45154

1.08226 1.06690 1.05078 1.03389 1.01622

20703 16876 38166 51168 21240

0.40198 0.41420 0.42668 0.43940 0.45238

30547 88905 19134 54246 27520

1.65 1.66 1.67 1.68 1.69

2.70 2.71 2.72 2.73 2.74

0.02972 0.02949 0.02923 0.02893 0.02858

02500 94834 81286 49650 87545

-0.18876 -0.18725 -0.18548 -0.18346 -0.18116

37500 41335 92032 16409 40324

0.51735 0.51270 0.50736 0.50132 0.49456

25000 31996 75264 74142 45848

-0.82167 -0.81252 -0.80234 -0.79111 -0.77879

75000 96321 86464 20467 71048

0.99775 0.97846 0.95836 0.93741 0.91561

12500 87823 08832 35896 28124

0.46561 0.47911 0.49287 0.50689 0.52119

72500 23003 13114 77188 49855

1.70 1.71 1.72 1.73 1.74

2.75 2.76 2.77 2.78 2.79

0.02819 0.02776 0.02727 0.02674 0.02616

82422 21555 92044 80814 74609

-0.17858 -0.17572 -0.17257 -0,16912 -0.16535

88672 85376 53385 14668 90208

0.48706 0.47879 0.46975 0.45991 0.44925

05469 65952 38095 30536 49742

-0.76538 -0.75084 -0.73515 -0.71829 -0.70023

08594 01152 14420 11736 54067

0.89294 0.86939 0.84494 0.81958 0.79330

43359 38176 67873 86468 46696

0.53576 0.55061 0.56574 0.58116 0.59686

66016 60845 69793 28586 73228

1.75 1.76 1.77 1.78 1.79

2.80

0.02553 60000 -0.16128 -0.15687 -0.15213 -0.14706 -0.14163

00000 63042 97332 19865 46624

0.43776 0.42540 0.41217 0.39805 0.38301

00000 83408 99864 47055 20448

-0.68096 -0.66044 -0.63865 -0.61557 -0.59117

00000 05733 25064 09380 07648

0.76608 0.73789 0.70874 0.67861 0.64747

00000 96529 85132 13352 27424

0.61286 0.62915 0.64574 0.66264 0.67984

40000 65462 86454 40097 63795

1.80 1.81 1.82 1.83 1.84

2.85 2.86 2.87 2.88 2.89

0.02156 0.02060 0.01957 0.01848 0.01733

78203 21015 56335 69274 44751

-0.13584 -0.12969 -0.12316 -0.11625 -0.10895

92578 71676 96841 79968 31915

0.36703 0.35009 0.33217 0.31325 0.29330

13281 16552 19008 07136 65154

-0.56542 -0.53831 -0.50980 -0.47987 -0.44849

66406 29752 39333 34336 51479

0.61531 0.58212 0.54789 0.51259 0.47621

72266 91476 27329 20768 11402

0.69735 0.71518 0.73333 0.75180 0.77059

95234 72385 33502 17126 62087

1.85 1.86 1.87 1.88 1.89

2.90 2.91 2.92 2.93 2.94

0.01611 0.01483 0.01347 0.01205 0.01056

67500 22068 92806 63881 19265

-0.10124 -0.09312 -0.08458 -0.07562 -0.06621

62500 80498 93632 08571 30924

0.27231 0.25026 0.22711 0.20286 0.17746

75000 16321 66464 00467 91048

-0.41564 -0.38128 -0.34540 -0.30796 -0.26894

25000 86646 65664 88792 80248

0.43873 0.40014 0.36042 0.31955 0.27753

37500 35985 42432 91059 14724

0.78972 0.80917 0.82897 0.84911 0.86959

07500 92770 57594 41956 86135

1.90 1.91 1.92 1.93 1.94

2.95 2.96 2.97 2.98 2.99

0.00899 0.00735 0.00563 0.00383 0.00195

42734 17875 28077 56534 86242

-0.05635 -0.04604 -0.03525 -0.02399 -0.01224

65234 14976 82547 69268 75371

0.15092 0.12319 0.09425 0.06409 0.03268

08594 21152 94420 917.36 74067

-0.22831 -0.18604 -0.14210 -0.09647 -0.04911

61719 52352 68745 24936 32392

0.23432 0.18992 0.14430 0.09745 0.04936

44922 11776 44035 69068 12858 0.97735 34596

1.95 1.96 1.97 1.98 1.99

3.00

0.00000 00000 -43

A0

0.00000 00000 0.00000 00000

A2

A1

Az

0.00000 00000 0.00000 00000 1.00000 00000 2.00

Ao

A-1

A-2

-P

913

NUMERICAL ANALYSIS

SEVEK-POINT LAGRANGIAN INTERPOLATION COEFFICIENTS

P

A -3 0.00000 00000

0.00000

0.4

-0.00159 -0.00295 -0.00400 -0.00465

10125 68000 28625 92000

0.01409 0.02580 0.03445 0.03960

00000 18250 48000 94250 32000

0.00000 -0.06725 -0.11827 -0.15241 -0.16972

00000 64375 20000 66875 80000

1.00000 0.98642 0.94617 0.88062 0.79206

77500 60000 97500 40000

0.00000 0.08220 0.17740 0.28305 0.39603

00000 23125 80000 95625 20000

0 00000 -0'01557 -0:03153 -0.04662 -0.05940

51750 92000 15750 48000

0 00000 0'00170 0100337 0.00489 0.00609

0.5 0.6 0.7 0.8 0.9

-0.00488 -0.00465 -0.00400 -0.00295 -0.00159

28125 92000 28625 68000 10125

0.04101 0.03870 0.03291 0.02407 0.01283

56250 72000 24250 68000 78250

-0.17089 -0.15724 -0.13068 -0.09363 -0.04898

84375 80000 16875 20000 64375

0.68359 0.55910 0.42315 0.28089 0.13788

37500 40000 97500 60000 77500

0.51269 0.62899 0.74052 0 84268 0:93074

53125 20000 95625 80000 23125

-0.06835 -0.07188 -0.06835 -0 05617 -0:03384

93750 48000 65750 92000 51750

0.00683 0.00698 0.00643 0.00510 0.00295

59375

1.0 1.1 1.2 13 1:4

0.00000 0.00170 0.00337 0.00489 0.00609

00000 07375 92000 23875 28000

0.00000 -0.01349 -0.02661 -0 03824 -0:04730

00000 61750 12000 95750 88000

0.00000 00000

0.04980 0.09676 0.13719 0.16755

73125 80000 95625 20000

0.00000 -0.12678 -0.23654 -0.32365 -0.38297

00000 22500 40000 02500 60000

1.00000 1 04595 1:06444 1 05186 1:0053l

00000 35625 80000 33125 20000

0.00000 0 04648 0:10644 0.18031 0.26808

00000 68250 48000 94250 32000

0.00000 -0 00367 -0:00788 -0.01237 -0.01675

00000 00125 48000 48625 52000

1.0 11 1:2 1.3 1.4

1.5 1.6 1.7 1.8 1.9

0.00683 0.00698 0.00643 0.00510 0.00295

59375 88000 93875 72000 47375

-0.05273 -0.05358 -0.04907 -0.03870 -0.02227

43750 08000 85750 72000 41750

0.18457 0.18547 0.16813 0.13132 0.07488

03125 20000 95625 80000 73125

-0.41015 -0.40185 -0.35606 -0.27238 -0.15240

62500 60000 02500 40000 22500

0 92285 0:80371 0 64853 0145964 0.24130

15625 20000 83125 80000 35625

0 36914 0148222 0.60530 0.73543 0.86869

06250 72000 24250 68000 28250

-0 02050 -0102296 -0.02328 -0.02042 -0.01316

78125 32000 08625 88000 20125

15 1:6 1.7 1.8 1.9

2.0 0.00000 2.1 -0.00367 2.2 -0.00788 2.3 -0.01237 2.4 -0.01675

00000 00125 48000 48625 52000

0.00000 0.02739 0.05857 0.09151 0.12337

00000 08250 28000 64250 92000

0.00000 -0.09056 -0.19219 -0.29812 -0.39916

00000

0.00000 00000

0.17825 0.37273 0.57031 0.75398

77500 60000 97500 40000

0.00000 -0.25523 -0.51251 -0,75677 -0.96940

00000 26875 20000 04375 80000

100000 1:12302 1.23002 1.31173 1.35717

00000 38250 88000 54250 12000

0.00000 0.02079 0.05125 0.09369 0.15079

00000

64375 20000 16875 80000

67375 12000 53875 68000

2.0 2.1 2.2 2.3 2.4

-0.02050 -0.02296 -0 02328 -0102042 -0.01316

78125 32000 08625 88000 20125

0.15039 0.16773 0.16940 0.14810 0.09508

06250 12000 54250 88000 88250

-0.48339 -0.53580 -0.53797 -0.46771 -0.29867

84375 80000 66875 20000 64375

0 90234 0:98918 0.98296 0.84633 0.53555

37500 40000 97500 60000 77500

-1 12792 -1'20556 -1'17089 -0198739 -0.61307

96875 80000 04375 20000 26875

135351 1:28593 1 13743 0:88865 0.51770

56250 92000 64250 28000 58250

0.22558 0.32148 0.44233 0.59243 0.77655

59375 48000 63875 52000 87375

0.00000

00000

0.0

0.1 0.2 0.3

2.5 2.6 27 2:8 2.9

A-2

0.00000 00000

0.00000 00000

3.0

A-1

A0 00000

0.00000 00000

AI

A2

A3

AI

Table 25.1

A2

00000

0.00000 00000

0.00000 00000

A-1

A-2

Ao

'43

00000

0 0

07375 92000 23875 28000

0'1 012 0.3 0.4

0.5 0.6 93875 0.7 72000 0.8 47375 0.9 88000

2.5 2.6 2.7 2.8 2.9 1.00000 00000 3.0 A-3

-P

EIGHT-POIIVT LAGRANGIAN INTERPOLATION COEFFICIENTS

P

A-3

0o:

00000

A-2

A-1

64213 51200 57988 61600 14063

0.00000 0.00915 0.01634 0.02124 0.02376 0.02392

00000 96863 30400 99787 19200 57812

0.00000 0.00070 0 00135 0'00188 0:00226

00000 45912 16800 70638 30400

0.00000 -0.00652 -0.01241 -0.01721 -0.02050

00000

0.00000 00000

31512 85600 23088 04800

0.02888 0.05419 0.07408 0.08712

82412 00800. 77638 70400

0.00244 0.00239 0.00211 0.00160 0.00088

14062 -0.02197 26562

61600 57988 51200 64213

-0.02143 -0.01881 -0.01419 -0.00779

23200 34538 26400 59613

0.09228 0.08902 0.07734 0.05778 0.03145

2.0 0.00000 00000 2.1 -0.00099 61462 2.2 -0.00202 75200 2.3 -0.00300 53238 2.4 -0.00382 97600

0.00000 0.00867 0.01757 0.02592 0.03290

00000 37612 18400 96538 11200

45312 26400 35888 20800 83162

0.03759 0.03913 0.03670 0.02962 0.01743

0,00000 00000

0.00000 0.1 -0,00088 0 2. -0.00160 0'3 -0.00211 014 -0.00239 0.5 -0.00244

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

0.00000 -0.05246 -0.08988 -0.11278 -0.12220 -0.11962

00000 00213 67200 83487 41600 89062

An -1,

1.00000 0.96176 0 89886 0'81458 0:71285 0.59814

00000 70563 72000 25188 76000 45312

0.00000 00000 -0.09191 71312

A. *-1 '-2A . 0.00000 00000 0.00000 00000 0.10686 30063 -0.03037 15913 0.22471 68000 -0.05992 44800 0 34910 67938 -0.08624 99137 0'47523 84000 -0.10692 86400 0:59814 45313 -0.11962 89062

A3 00000

0.00000 0.00663 0.01284 0,01810 0.02193 0.02392

28763 09600 18337 40800 57812

A4 0.00000 00000 -0.00070 45913 -0.00135 16800 -0.00188 70638 -0.00226 30400 -0,00244 14062

1.0 0.9 0.8 0.7 0.6 0.5

00000

0.00000 00000

0.00000

84438 48000 69812 04000

0.06740 0.14902 0.24343 0.34850

98962 27200 12238 81600

0.00000 -0.01064 -0.02207 -0.03341 -0.04356

00000

-0'21846 39188 -0:24893 44000

1.00000 101108 0'99348 0'94667 0:87127

30362 74400 21288 35200

0.00099 0.00202 0.00300 0.00382

00000 61462 75200 53238 97600

0.0 -0.1 -0.2 -0.3 -0.4

51562 65600 41988 43200 26712

-0 25634 -0:24111 -0 20473 -0114981 -0.08001

76562 36000 46438 12000 11812

0 76904 0'64296 0'49721 0133707 0.16891

29688 96000 27062 52000 24938

0.46142 0.57867 0.69609 0.80898 0.91212

57812 26400 77888 04800 74662

-0.05126 -0.05511 -0.05354 -0.04494 -0.02764

95312 16800 59838 33600 02263

0.00439 0.00459 0.00432 0.00350 0.00206

45312 26400 35888 20800 83163

-0.5 -0.6 -0.7

0.00000 -0.03441 -0.06918 -0.10136 -0.12773

00000 52462 91200 13738 37600

0.00000 0.08467 0.16773 0.24238 0.30159

00000 24312 12000 58938 36000

3.00000 -0.16164 -0.30750 -0.42883 -0.51701

00000 73688 72000 65812 76000

1.00000 1.06687 1.10702 1.11497 1.08573

00000 26338 59200 51112 69600

0.00000 0.03951 0.09225 0.15928 0.24127

00000

38012 21600 21588 48800

0.00000 -0.00267 -0.00585 -0.00936 -0.01292

38662 -1.1 72800 -1.2 95388 -1.3 54400 -1.4

76562 72800 45088 17600 29512

-0.14501 -0.15002 -0.13987 -0.11225 -0.06570

95312 62400 39388 08800 88162

0.33837 0 34621 0:31946 0.25390 0.14727

89062 -0.56396 48438 44000 -0.56259 84000 51688 -0.50738 58562 08000 -0.39495 68000 83812 -0.22479 33188

1.01513 0.90015 0.73933 0.53319 0.28473

67188 74400 36762 16800 82038

0.33837 0.45007 0.57503 0.71092 0.85421

89062 87200 73038 22400 46112

-0.01611 -0.01837 -0.01895 -0.01692 -0.01109

32812 05600 72738 67200 36962

-1.5 -1.6 -1.7 -1.8 -1.9

0.00000 00000

0,00000 00000

-0 16558 08000

-0.8 -0.9

00000 -1.0

2.5 2.6 2.7 2.8 2.9

-0.00439 -0 00459 -0:00432 -0.00350 -0.00206

3.0 3.1 3.2 3.3 3.4

0.00267 0.00585 0 00936 0:01292

38662 72800 95388 54400

0.00000 -0.02238 -0.04888 -0.07796 -0.10723

00000 70762 57600 16338 32800

0.08354 0.18157 0.28827 0.39481

20162 56800 67388 34400

0.00000 -0.18415 -0 39719 -0'62605 -0:85155

17562 68000 55438 84000

0.00000 0.27184 0.57774 0 89825 1:20637

00000 30688 08000 36062 44000

-0.31138 -0.63551 -0 95353 -1:24084

38788 48800 07512 22400

1,00000 1.14174 1.27102 137732 1:44764

00000 08888 97600 21962 92800

0.00000 0.01812 0.04539 0.08432 0.13787

00000 28712 39200 58488 13600

-2.0 -2.1 -2.2 -2.3 -2.a

3.5 3.6 3.7 3.8 3.9

0 01611 0:01837 0.01895 0.01692 0.01109

32812 05600 72738 67200 36962

-0.13330 -0.15155 -0.15598 -0.13891 -0.09081

07812 71200 17788 58400 78862

0.48876 0.55351 0.56750 0.50356 0.32805

95312 29600 81738 99200 64462

-1.04736 -1.17877 -1.20148 -1.06014 -0.68695

32812 76000 12688 72000 58062

146630 1:63215 1.64647 1.43877 0.92383

85938 36000 43312 12000 71188

-1.46630 -1.59134 -1.56899 -1.34285 -0.84604

85938 97600 31862 31200 03088

1.46630 1.41453 1.27013 1.00713 0.59536

85938 31200 73412 98400 16988

0.20947 0.30311 0.42337 0.57550 0.76546

26562 42400 91138 84800 50412

-2.5 -2.6 -2.7 -2.8 -2.9

4.0

0.00000 00000

0.00000

00000

-3.0

A4

0.00000, 00000

0,00000 00000

A3

A2

00000

0,00000 00000

AI

A0

0.00000 00000

0.00000 00000

1.00000 00000

'4-1

A-2

A-3

P

914

NUMERICAL ANALYSIS

Table 2.5.2

COEFFICIENTS FOR DIFFERENTIATION

FIRST DERIVATIVE ( k = l )

*

j

-41

-40

A?

AJ

4

THIRD DERIVATIVE ( k = 3 )

A2

!k! Eror

j

do

0 1 2

-3 -1 1

0 1 2 3

-11 -2 1 -2

-

4 0 4

-1 1 3

1/3 -l/6 1/3

h3f (')

0 1 2 3

-1 -1 -1 -1

3 3 3 3

0 1 2

36 20

4

-10 -6 -2 2 6

-12 -28

0 1 2 3 4 5

-85 -35 -5 5 -5 -35

355 125 -5 -35 35 205

Four Point ( i n = 3 ) 18 -3 -6 9

-9 6 3 -18

0 1 2 3 4

-50 -6 2 -2 6

96 -20 -1 6 12 -32

0 1 2 3 4 5

-274 -24 6 -4

600 -130 -60 30 -40 150

(7)1=4)

-72 36 0 -36 72

32 -12 16 20 -96

-6 2 -2 6 50

1/5 -1/20 1/30 -1/20 1/5

6

-24

400 -120 120 40 -240 600

-150 40 -30 60 130 -600

24 -6 4 -6 24 274

-1 / 6 l)30 -1/60 6 1/60h -1 /30 lj6

-2

1

1

1

-2

1

2

1

-2

1

6 3 0 -3

-15 -6

0 1 2 3 4

35 11 -1 -1 11

-104 -2 0 16 4 -56

12 3

3

-6

12

-15

Five Point 114 6 -30 6 114

-48 -24 0 24 48

4

28 12 4 -20 -36

-

-590 -170 50 70 -110 -490

490 110 -70 -50 170 590

A:!

7/24 1/24 -1/24 1/24 7/24

-6 -2 2 6 10

4 (4)

h5f(5)

-5/l6 -1/48 1148 -1/46 1/48 5/16

35 5 -5 5 35 85

-205 -35 35 5 -125 -355

-4.j

Five Point

k!

6 (6)

0 1 2 3 4

1 1 1 1 1

0 1 2

15 10 5 0 -5 -10

-4 -4 -4

6 6 6 6 6

-4

-4

-44

*

It" Error

-I:,

k!

(it1=4)

-

-1/12 h 5 f ( 5 ) -1/24 -1/144 h 6 f(') 1/24 h 5 f ( 5 ) 1/12

1

4 4 -4 -4 -4

1

1 1 1

six Point (t)~=5)

Four Point (m=3) 0 1 2 3

-1/4 -1/12 l/lZh 1/4

FOURTH DERIVATIVE (k=4)

!!!Error

Three Point (in=2) 1

*

!!!Error k!

1 1 1 1

-3 -3 -3 -3

-41

-44

rl2

0

Aj

A4

(6)

SECOND DERIVATIVE (k= .I,

Ag

Six Point (1)1=5) h5f(5)

Six Point ())1=5) -600 240 -40 -120 120 -400

A?

Five Point (m=4)

2 -1 2 11

3

Five Point

AI

Four Point (m=3)

Three Point (in=2)

11/24 -1124 -1/24h 11/24

-3 0

3 6

*( 4 )

3

4 5

-7 0 -4 5 -2 0 5 30 55

130 80 30 -2 0 -7 0 -120

-120 -7 0 -2 0 30 80 130

55 30 5 -2 0 -4 5 -7 0

-1 0 -5 0 5 10 15

17/144 5ji44 -1/144 -1/144h 5/144 17/144

6 (6)

(tt1=4)

-56 4 16 -20 -104

-5/12 h 5 f ( 5 ) 1/24 1/ 18 0 h f (') -1/24 h 5 f ( 5 ) 5/12

11 -1 -1 11 35

FIFTH DERIVATIVE (k- 5 ) -41

A2

.l{

-44

.I-,

hk Error -

*

k!

Six Point ( ~ 5 ) 0 1 2 3 4

5

225 50 -5 0 5

-55

-770 -75 80

-5

-30 _-

305

1070 -20 -150 80 70

-780

-780 70 80

-150 -20

1070

305 -30 -5 80 -75 -770

-50 5 -5

137/360 -13/360 1/180 6 1/180h

50

-13/360

0

225

137j360

(6)

0 1 2

5

3

-1 -1 -1 -1

5 5 5

-10 -10 -10 -10

4

-1

5

-10

5

-1

5

-io

10 10 10 10 10 10

-5 -5 -5 -5 -5 -5

1

-1/48

1

-1/80

1 1 1

1

-1/240 6 1/24Ch 1/60 1/48

Compiled from W. G. Bickley, Formulae for numerical differentiation,Math. Gaz. 25, 19-27, 1941 (with permission). *See page I!.

(6)

915

NUMERICAL ANALYSIS

LAGRANGIAN INTEGRATION COEFFICIENTS

4

n m\k 3 -1

DAXm) 71 = odd 0

1

5

8

-1

251 -1 9

646 346

-264 456

106 -7 4

-1 9 11

65112 25128 -2760

-46461 46989 30819

37504 -16256 37504

-20211 7299 -6771

6312 -2088 1608

-863 271 -191

-4604594 5595358 3244786 -1752542 1638286 2631838 -216014 1909858

-5033120 1317280 -833120 2224480

3146338 -1291214 -755042 294286 397858 -142094 -425762 126286

312874 -68706 31594 -25706

-33953 7297 -3233 2497

-3

-4

-2

-3

5 -2 -1

7 -3 -2 -1

19087 -863 271

9 -4 -3 -2 -1

1070017 4467094 -33953 1375594 7297 -99626 -3233 36394

4

Table 25.3

-1

2

3

0

1

2

4

3

0

-2

-1

I1 12

1 0

720

2 1 0

60480

3 3628800 2 1 0 k\in

n = even

-4

n m\k 4 -1 0

-3

6 -2

-1 0

8 -3 -2 -1 0

36799 -1375 351 -191

0 19 13

1

2

9 -1

-5 13

1 -1

475 -2 7 11

1427 637 -9 3

-798 1022 802

482 -258 802

-173 77 -9 3

27 -11 11

139849 47799 -4183 1879

-121797 101349 57627 -9531

123133 -44797 81693 68323

-88547 26883 -20227 68323

41499 -11547 7227 -9531

-11351 2999 -1719 1879

-2

-1

3

4

5 1 0

1375 -351 191 -191

10 -4 2082753 9449717 -11271304 16002320 -17283646 13510082 -7394032 2687864 -583435 57281 3973310 -2848834 1481072 -520312 110219 -10625 6872072 -4397584 -3 -57281 2655563 3969 -2 10625 -163531 3133688 5597072 -2166334 1295810 -617584 206072 -42187 27467 -2497 462320 -141304 50315 -342136 3609968 4763582 -1166146 -1 -3969 0 2497 -28939 162680 -641776 4134338 4134338 -641776 162680 -28939 2497

2 1 0

1440

3 2 1 0

120960

4 7257600 3 2 1 0 k\m

5 4 3 2 1 0 -1 -2 -3 -4 Compiled from National Bureau of Standards, Tables of Lagrangian interpolation coefficients. Columbia Univ. Press, New York, N.Y., 1944 (with permission).

*See page

11.

U 24

916

NUMERICAL ANALYSIS Table 25.4

ABSCISSAS AND WEIGHT FACTORS FOR GAUSSIAN INTEGRATION 2= UI

Weight Factors=wi

Abscissas= i t i (Zeros of Legendre Polynomials) ixi

I1 =

n=2 0.57735 02691 89626

1.00000 00000 00000

n=3 0.00000 00000 00000 0.77459 66692 41483

0.33998 10435 84856 0.86113 63115 94053

0.65214 51548 62546 0.34785 48451 37454

.n=5 0.56888 88888 88889 0.47862 86704 99366 0.23692 68850 56189

0.00000 00000 00000 0.53846 93101 05683 0.90617 98459 38664

n=6 0.46791 39345 72691 0.36076 15730 48139 0.17132 44923 79170

0.23861 91860 83197 0.66120 93864 66265 0.93246 95142 03152 00000 51513 11855 79123

00000 77397 99394 42759

0.18343 0.52553 0.79666 0.96028

46424 24099 64774 98564

95650 16329 13627 97536

0.00000 0.32425 0.61337 0.83603 0.96816

00000 34234 14327 11073 b2395

00000 03809 00590 26636 07626

0.14887 0.43339 0.67940 0.86506 0.97390

43389 53941 95682 33666 65285

81631 29247 99024 88985 17172

0.12523 0.36783 0.58731 0.76990 0.90411 0.98156

34085 14989 79542 26741 72563 06342

11469 98180 86617 94305 70475 46719

0.88888 88888 88889 0.55555 55555 55556

n=4

0.00000 0.40584 0.74153 0.94910

wi

+xi

i

'U)

.

n ..2 7

0.41795 0.38183 0.27970 0.12948

91836 00505 53914 49661

73469 05119 89277 68870

8 0.36268 0.31370 0.22238 0.10122

37833 66458 10344 85362

78362 77887 53374 90376

0.33023 0.31234 0.26061 0.18064 0.08127

93550 70770 06964 81606 43883

01260 40003 02935 94857 61574

0.29552 0.26926 0.21908 0.14945 0.06667

42247 67193 63625 13491 13443

14753 09996 15982 50581 08688

0.24914 0.23349 0.20316 0.16007 0.10693 0.04717

70458 25365 74267 83285 93259 53363

13403 38355 23066 43346 95318 86512

n=9

n =10

n= 12

16

0.09501 0.28160 0.45801 0.61787 0.75540 0.86563 0.94457 0.98940

ixi 25098 37637 35507 79258 67776 57227 62444 02643 44083 55003 12023 87831 50230 73232 09349 91649

440185 913230 386342 748447 033895 743880 576078 932596

0.18945 Oil8260 0.16915 0.14959 0.12462 0.09515 0.06225 0.02715

06104 34150 65193 59888 89712 85116 35239 24594

5

002 1 576 55533 82492 38647 11754

538189 732081 872052 784810 892863 094852

0.07652 0.22778 0.37370 0.51086 0.63605 0.74633 0.83911 0.91223 0.96397 0.99312

65211 58511 60887 70019 36807 19064 69718 44282 19272 85991

33497 41645 15419 50827 26515 60150 22218 51325 77913 85094

333755 078080 560673 098004 025453 792614 823395 905868 791268 924786

0.15275 0.14917 0.14209 0.13168 0.11819 0.10193 0.08327 Oi06267 0.04060 0.01761

33871 29864 61093 86384 45319 01198 67415 20483 14298 40071

30725 72603 18382 49176 61518 17240 76704 34109 00386 39152

850698 746788 051329 626898 417312 435037 748725 063570 941331 118312

0.06405 0.19111 0.31504 0.43379 0.54542 0.64809 0.74012 0.82000 0.88641 0.93827 0.97472 0.99518

68928 88674 26796 35076 14713 36519 41915 19859 55270 45520 85559 72199

62605 73616 96163 26045 88839 36975 78554 73902 04401 02732 71309 97021

626085 309159 374387 138487 535658 569252 364244 921954 034213 758524 498198 360180

0.12793 0.12583 0.12167 0.11550 0.10744 0.09761 0.08619 0.07334 0.05929 0.04427 0.02853 0.01234

81953 74563 04729 56680 42701 86521 01615 64814 85849 74388 13886 12297

46752 46828 27803 53725 15965 04113 31953 11080 15436 17419 28933 99987

156974 296121 91204 6 1353 63 783 888270 275917 305734 780746 806169 663181 199547

n=

U'i

068 496285

$, 923 588867

n=24

's,

Compiled from P. Davis and P. Rabinowitz, Abscissas and weights for Gaussian quadratures of high order, J. Research NBS 56, 35-37, 1956, RP2645; P. Davis and P. Rabinowitz, Additional abscissas and weights for Gaussian quadratures of high order. Values for n=64, 80, and 96, J. Research NBS 60, 613-614,1958, RP2875; and A. N. Lowan, N. Davids, and A. Levenson, Table of the zeros of the Legendre polynomials of order 1-16 and the weight coefficients for Gauss' mechanical quadrature formula, Bull. Amer. Math. Soc. 48, 739-743, 1942 (with permission).

917

NUMERICAL ANALYSIS

Table 25.4 ABSCISSAS AND WEIGHT FACTORS FOR GAUSSIAN INTEGRATION

Abscissas= *zi (Zeros of Legendre Polynomials) *Xi

Wi

n=32

0.04830 0.14447 0.23928 0.33186 0.42135 0.50689 0.58771 0.66304 0.73218 0.79448 0.84936 0.89632 0.93490 0.96476 0.98561 0.99726

76656 19615 73622 86022 12761 99089 57572 42669 21187 37959 76137 11557 60759 22555 15115 38618

87738 82796 52137 82127 30635 32229 40762 30215 40289 67942 32569 66052 37739 87506 45268 49481

316235 493485 074545 649780 345364 390024 329041 200975 680387 406963 970134 123965 689171 430774 335400 563545

0.03877 0.11608 0.19269 0.26815 0.34199 0.41377 0.48307 0.54946 0.61255 0.67195 0.72731 0.77830 0.82461 0.86595 0.90209 0.93281 0.95791 0.97725 0.99072 0.99823

24175 40706 75807 21850 40908 92043 58016 71250 38896 66846 82551 56514 22308 95032 88069 28082 68192 99499 62386 77097

06050 821933 75255 208483 01371 ~~. .~ 099716 07253 681141 25758 473007 71605 001525 86178 712909 95128 202076 67980 237953 14179 548379 89927 103281 26519 387695 33311 663196 12259 503821 68874 296728 78676 533361 13791 655805 83774 262663 99457 006453 10559 200350

0.03238 0.09700 0.16122 0.22476 0.28736 0.34875 0.40868 0.46690 0.52316 0.57722 0.62886 0.67787 0.72403 0.76715 0.80706 0.84358 0.87657 0.90587 0.93138 0.95298 0.97059 0.98412 0.99353 0.99877

01709 46992 23560 37903 24873 58862 64819 29047 09747 47260 73967 23796 41309 90325 62040 82616 20202 91367 66907 77031 15925 45837 01722 10072

62869 09462 68891 94689 55455 92160 90716 50958 22233 83972 76513 32663 23814 15740 29442 24393 74247 15569 06554 60430 46247 22826 66350 52426

Weight Factors=wi

0.09654 0.09563 0.09384 0.09117 0.08765 0.08331 0.07819 0.07234 0.06582 0.05868 0.05099 0.04283 0.03427 0.02539 0.01627 0.00701

00885 87200 43990 38786 20930 19242 38957 57941 22227 40934 80592 58980 38629 20653 43947 86100

14727 79274 80804 95763 04403 26946 87070 08848 76361 78535 62376 22226 13021 09262 30905 09470

800567 859419 565639 884713 811143 755222 306472 506225 846838 547145 176196 680657 433103 059456 670605 096600

0.07750 0.07703 0.07611 0.07472 0.07288 0.07061 0.06791 0.06480 0.06130 0.05743 0.05322 0.04869 0.04387 0.03878 0.03346 0.02793 0.02224 0.01642 0.01049 0.00452

59479 98181 03619 31690 65823 16473 20458 40134 62424 97690 78469 58076 09081 21679 01952 70069 58491 10583 82845 12770

78424 64247 00626 57968 95804 91286 15233 56601 92928 99391 83936 35072 85673 74472 82547 80023 94166 81907 31152 98533

811264 965588 242372 264200 059061 779695 903826 038075 939167 551367 824355 232061 271992 017640 847393 401098 957262 888713 813615 191258

0.06473 0.06446 0.06392 0.06311 0.06203 0.06070 0.05911 0.05727 0.05519 0.05289 0.05035 0.04761 0.04467 0.04154 0.03824 0.03477 0.03116 0.02742 0.02357 0.01961 0.01557 0.01147 0.00732 0.00315

76968 61644 42385 41922 94231 44391 48396 72921 95036 01894 90355 66584 45608 50829 13510 72225 72278 65097 07608 61604 93157 72345 75539 33460

12683 35950 84648 86254 59892 65893 98395 00403 99984 85193 53854 92490 56694 43464 65830 64770 32798 08356 39324 57355 22943 79234 01276 52305

922503 082207 186624 025657 663904 880053 635746 215705 162868 667096 474958 474826 280419 749214 706317 438893 088902 948200 379141 527814 848728 539490 262102 838633

n=40

362033 698930 718056 061225 576736 738160 729916 404545 033678 703818 623995 905212 654674 339254 627083 530711 885906 672822 333114 860723 250461 857745 757548 118601

n~48

918

NUMERICAL ANALYSIS Table 25.4 ABSCISSAS AND WEIGHT FACTORS FOR GAUSSIAN INTEGRATION

Abscissas= JT, (Zeros of Legendre Polynomials) *Xi

Wi

n=64

0.02435 0.07299 Oil2146 0.16964 0.21742 Oi26468 0.31132 0.35722 0.40227 0.44636 0.48940 0.53127 0.57189 0.61115 0.64896 0.68523 0.71988 0.75281 0.78397 0.81326 0.84062 0.86599 0.88931 0.91052 0.92956 0.94641 0.96100 0.97332 0.98333 0.99101 0.99634 0.99930

02926 31217 28192 44204 36437 71622 28719 01583 01579 60172 31457 94640 56462 53551 54712 63130 18501 99072 23589 53151 92962 93981 54459 21370 91721 13748 87996 68277 62538 33714 01167 50417

63424 87799 96120 23992 40007 08767 90210 37668 63991 53464 07052 19894 02634 72393 54657 54233 71610 60531 43341 22797 52580 54092 95114 78502 31939 58402 52053 89910 84625 76744 71955 35772

432509 039450 554470 818037 084150 416374 956158 115950 603696 087985 957479 545658 034284 250249 339858 242564 826849 896612 407610 559742 362752 819761 105853 805756 575821 816062 718919 963742 956931 320739 279347 139457

0.01951 0.05850 0.09740 0.13616 0.17471 0.21299 0.25095 0.28852 0.32566 0.36230 0.39839 0.43387 0.46869 0.50280 0.53614 0.56867 0.60033 0.63107 0.66085 0.68963 0.71736 0.74400 0.76950 0.79383 0.81695 0.83883 0.85943 0.87872 0.89667 0.91326 0.92845 0.94224 0.95459 0.96548 0.97490 0.98284 0.98929 0.99422 0.99764 0.99955

13832 44371 83984 40228 22918 45028 23583 80548 43707 47534 34058 53708 66151 41118 59208 12681 06228 57730 98989 76443 51853 02975 24201 27175 41386 14735 14066 25676 55794 31025 98771 27613 07663 50890 91405 85727 13024 75409 98643 38226

56793 52420 41584 09143 32646 57666 92272 84511 47701 99487 81969 31756 70544 88784 97131 22709 29751 46871 86119 42027 62099 83597 35041 04605 81463 80255 63111 78213 38770 71757 72445 09872 43634 43799 85727 38629 99755 65688 98237 51630

997654 668629 599063 886559 812559 132572 120493 853109 914614 315619 227024 093062 477036 987594 932020 784725 743155 966248 801736 600771 880254 272317 373866 449949 470371 275617 096977 828704 683194 654165 795953 674752 905493 251452 793386 070418 531027 277892 688900 629880

Weight Factors=w,

0.04869 0.04857 0.04834 0.04799 0.04754 0.04696 0.04628 0.04549 0.04459 0.04358 0.04247 0.04126 0.03995 0.03855 0.03705 0.03547 0.03380 0.03205 0.03023 0.02833 0.02637 0.02435 0.02227 0.02013 0.01795 0.01572 0.01346 0.01116 0.00884 0.00650 0.00414 0.00178

09570 54674 47622 93885 01657 81828 47965 16279 05581 37245 35151 25632 37411 01531 51285 22132 51618 79283 46570 96726 74697 27025 01738 48231 17157 60304 30478 81394 67598 44579 70332 32807

09139 41503 34802 96458 14830 16210 81314 27418 63756 29323 23653 42623 32720 78615 40240 56882 37141 54851 72402 14259 15054 68710 08383 53530 75697 76024 96718 60131 26363 68978 60562 21696

720383 426935 957170 307728 308662 017325 417296 144480 563060 453377 589007 528610 341387 629129 046040 383811 609392 553585 478868 483228 658672 873338 254159 209372 343085 719322 642598 128819 947723 362856 467635 432947

0.03901 0.03895 0.03883 0.03866 0.03842 0.03812 0.03777 0.03736 0.03689 0.03637 0.03579 0.03516 0.03447 0.03373 0.03294 0.03210 0.03121 0.03027 0.02928 0.02825 0.02718 0.02607 0.02492 0.02373 0.02250 0.02124 0.01995 0.01862 0.01727 0.01589 0.01449 0.01306 0.01162 0.01016 0.00868 0.00719 0.00569 0.00418 0.00266 0.00114

78136 83959 96510 17597 49930 97113 63643 54902 77146 37499 43939 05290 31204 32149 19393 04986 01741 23217 83695 98160 82275 52357 25357 18828 50902 40261 06108 68142 46520 61835 35080 87615 41141 17660 39452 29047 09224 03131 35335 49500

56306 62769 59051 74076 06959 14477 62001 38730 38276 05835 53416 44747 51753 84611 97645 73487 88114 59557 83267 57276 00486 67565 64115 65930 46332 15782 78141 08299 56269 83725 40509 92401 20797 41103 69260 68117 51403 24694 89512 03186

654811 531199 968932 463327 423185 638344 397490 490027 008839 978044 054603 593496 928794 522817 401383 773148 701642 980661 847693 862397 380674 117903 491105 101293 461926 006389 998929 031429 306359 688045 076117 339294 826916 064521 858426 312753 198649 895237 681669 941534

n=80

919

NUMERICAL ANALYSIS

Table 25.4 ABSCISSAS AND WEIGHT FACTORS FOR GAUSSIAN INTEGRATION

Abscissas=

(Zeros of Legendre Polynomials) fxi

Weight Factors=wi wi

n=96

0.01627 0.04881 0.08129 0.11369 0.14597 0.17809

67448 29851 74954 58501 37146 68823

49602 36049 64425 10665 54896 67618

969579 731112 558994 920911 941989 602759

0.03255 0.03251 0.03244 0.03234 0.03220 0.03203

06144 61187 71637 38225 62047 44562

92363 13868 14064 68575 94030 31992

166242 835987 269364 928429 250669 663218

0.21003 0.24174 0.27319 0.30436 0.33520 0.36569

13104 31561 88125 49443 85228 68614

60567 63840 91049 54496 92625 72313

203603 012328 141487 353024 422616 635031

0.03182 0.03158 0.03131 0.03101 0.03067 0.03 029

87588 93307 64255 03325 13761 99154

94411 70727 96861 86313 23669 20827

006535 168558 355813 837423 149014 593794

0.39579 0.42547 0.45470 0.48345 0.51169 0.53938

76498 89884 94221 79739 41771 81083

28908 07300 67743 20596 54667 24357

603285 545365 008636 359768 673586 436227

0.02989 0.02946 0.02899 0.02849 0.02797 0.02741

63441 10899 46141 74110 00076 29627

36328 58167 50555 65085 16848 26029

385984 905970 236543 385646 334440 242823

0.56651 0.59303 0.61892 0.64416 0.66871 0.69256

04185 23647 58401 34037 83100 45366

61397 77572 25468 84967 43916 42171

168404 080684 570386 106798 153953 561344

0.02682 0.02621 0.02557 0.02490 0.02420 0.02348

68667 23407 00360 06332 48417 33990

25591 35672 05349 22483 92364 85926

762198 413913 361499 610288 691282 219842

0.71567 0.73803 0.75960 0.78036 0.80030 0.81940

68123 06437 23411 90438 87441 03107

48967 44400 76647 67433 39140 37931

626225 132851 498703 217604 817229 675539

0.02273 0.02196 0.02117 0.02035 0.01951 0.01866

70696 66444 29398 67971 90811 06796

58329 38744 92191 54333 40145 27411

374001 349195 298988 324595 022410 467385

0.83762 0.85495 0.87138 0.88689 0.90146 0.91507

35112 90334 85059 45174 06353 14231

28187 34601 09296 02420 15852 20898

121494 455463 502874 416057 341319 074206

0.01778 0.01688 0.01597 0.01503 0.01409 0.01312

25023 54798 05629 87210 09417 82295

16045 64245 02562 26994 72314 66961

260838 172450 291381 938006 860916 572637

0.92771 0.93937 0.95003 0.95968 0.96832 0.97593

24567 03397 27177 82914 68284 91745

22308 52755 84437 48742 63264 85136

690965 216932 635756 539300 212174 466453

0.01215 0.01116 0.01016 0.00914 0.00812 0.00709

16046 21020 07705 86712 68769 64707

71088 99838 35008 30783 25698 91153

319635 498591 415758 386633 759217 865269

0.98251 0.98805 0.99254 0.99598 0.99836 0.99968

72635 41263 39003 18429 43758 95038

63014 29623 23762 87209 63181 83230

677447 799481 624572 290650 677724 766828

0.00605 0.00501 0.00396 0.00291 0.00185 0.00079

85455 42027 45543 07318 39607 67920

04235 42927 38444 17934 88946 65552

961683 517693 686674 946408 921732 012429

I

920

NUMERICAL ANALYSIS ABSCISSAS FOR EQUAL WEIGHT CHEBYSHEV INTEGRATION

Table 25.5

Abscissas= i . r i

+ .ci

It

*

11

2

0.57735 02692

3

0.70710 67812 0.00000 00000

n 7

l'i

5

0.83249 74870 0.37454 14096 0.00000 00000

6

0.86624 68181 0.42251 86538 0.26663 54015

ir ;

9 4

0.79465 44723 0.18750 24741

0.88386 0.52965 0.32391 0.00000

17008 67753 18105 00000

0.91158 0.60101 0.52876 0.16790 0.00000

93077 86554 17831 61842 00000

Compiled from H. E. Salzer, Tables for facilitating the use of Chebyshev's quadrature formula, J. Math. Phys. 26, 191-194,1947 (with permission). Ttlblc 25.6

ABSCISSAS AKD WEIGHT FACTORS FOR LOBAT;rO INTEGRATION

;:J

j(.r)t/.r= /(,if(

-I)+:%'

+ Wnj(1)

wij(,ri)

1=2

Weight Factors=wi

Abscissas=

* .ri 3

4

1.00000 000 0.00000 000

0.33333 333 1.33333 333

1.00000 000 0.44721 360

0.16666 667 0.83393 333

6

1.00000 000 0.76505 532 0.28523 152

71';

7

1.00000 0.83022 0.46884 0.00000

000 390 879 000

0.04761 0.27682 0.43174 0,48761

904 604 538 904

8

1.00000 Oi87174 0.59170 0.20929

000 015 018 922

0.03571 0,21070 0.34112 0.41245

428 422 270 880

9

1.00000 0.89975 0.67718 0.36311 0.00000

00000 79954 62795 74638 00000

0.02777 0.16549 0.27453 0.34642 0.37151

77778 53616 87126 85110 92744

10

1.00000 0.91953 0.73877 0.47792 0.16527

00000 39082 38651 49498 89577

0.02222 0.13330 0.22488 0.29204 0.32753

22222 59908 93420 26836 97612

0.10000 000 0.54444 444 0.71111 111

1.00000 000 0.65465 367 0.00000 000

5

i.ri

II

Us;

0.06666 667 0.37847 496 0.55485 838

Compiled from Z. Kopal, Numerical analysis, John Wiley & Sons, Inc., New York, N.Y., 1955 (with permission).

Table 25.7

ABSCISSAS AND WEIGHT FACTORS FOR GAUSSIAN INTEGRATION FOR INTEGRANDS WITH A LOGARITHMIC SINGULARITY

Abscissas=.,,; r , r

..

-

I

Wi

7 1 M W 2 0.11200'.7 n".,*.,<<, 0.602277 0,281461

Kn

-."----

n

nn7~5

I1

3

~

rt

-.--

Weight Factors- wi -wi

n n k_-.m i n-.-.5iwn5 0.368997 0.391980 0.766880 0.094615

Kf,

II

n.nnni7

4

-.----

-wi

I.)

0.041448 0.245275 Oi556165 0.848982

0.383464 0.386875 0.190435 0.039225

K,, O.OOOO~

Compiled from Berthod-Zaborowski, Le calcul des integrales de la forme ji; j(.i,)log d.r. H. Mineur, Techniques de calcul numerique, pp. 555-556. Librairie Polytechnique Ch. BBranger, Paris, France, 1952 (with permission). .I(

*See page

11.

*

921

NUMERICAL ANALYSIS

ABSCISSAS AND WEIGHT FACTORS FOR GAUSSIAN INTEGRATION OF MOMENTS

J,'

xkf(z)d~

5

2=

1w i f ( x i )

Abscissas = x i

Weight Factors =wi k=l

k =O n

Xi

Wi

Table 25.8

Xi

k =2 Wi

Wi

Xi

1

0.50000 00000 1.00000 00000

0.66666 66667 0.50000 00000

0.75000 00000 0.33333 33333

2

0.21132 48654 0.50000 00000 0.78867 51346 0.50000 00000

0.35505 10257 0.18195 86183 0.84494 89743 0.31804 13817

0.45584 81560 0.10078 58821 0.87748 51773 0.23254 74513

3

0.11270 16654 0.27777 77778 0.50000 00000 0.44444 44444 0.88729 83346 0.27777 77778

0.21234 05382 0.06982 69799 0.59053 31356 0.22924 11064 0.91141 20405 0.20093 19137

0.29499 77901 0.02995 07030 0.65299 62340 0.14624 62693 0.92700 59759 0.15713 63611

4

0.06943 0.33000 0.66999 0.93056

18442 94782 05218 81558

0.17392 0.32607 0.32607 0.17392

74226 25774 25774 74226

0.13975 0.41640 0.72315 0.94289

98643 95676 69864 58039

0.03118 0.12984 0.20346 0.13550

09710 75476 45680 69134

0.20414 0.48295 0.76139 0.95149

85821 27049 92624 94506

0.01035 0.06863 0.14345 0.11088

22408 38872 87898 84156

5

0.04691 0.23076 0.50000 0.76923 0.95308

00770 53449 00000 46551 99230

0.11846 0.23931 0.28444 0.23931 0.11846

34425 43352 44444 43352 34425

0.09853 0.30453 0.56202 0.80198 0.96019

50858 57266 51898 65821 01429

0.01574 0.07390 0.14638 0.16717 0.09678

79145 88701 69871 46381 15902

0.14894 0.36566 0.61011 0.82651 0.96542

57871 65274 36129 96792 10601

0.00411 0.03205 0.08920 0.12619 0.08176

38252 56007 01612 89619 47843

6

0.03376 0.16939 0.38069 0.61930 0.83060 0.96623

52429 53068 04070 95930 46932 47571

0.08566 0.18038 0.23395 0.23395 0.18038 0.08566

22462 07865 69673 69673 07865 22462

0.07305 0.23076 0.44132 0.66301 0.85192 0.97068

43287 61380 84812 53097 14003 35728

0.00873 0.04395 0.09866 0.14079 0.13554 0.07231

83018 51656 11509 25538 24972 03307

0.11319 0.28431 0.49096 0.69756 0.86843 0.97409

43838 88727 35868 30820 60583 54449

0.00183 0.01572 0.05128 0.09457 0.10737 0.06253

10758 02972 95711 71867 64997 87027

7

0.02544 0.12923 Oi29707 0.50000 0.70292 0.87076 0.97455

60438 44072 74243 00000 25757 55928 39562

0.06474 0.13985 0.19091 0.20897 0.19091 0.13985 0.06474

24831 26957 50253 95918 50253 26957 24831

0.05626 0.18024 0.35262 0.5 4715 0.7342 1 0.88532 0.97752

25605 06917 47171 36263 01772 09468 06136

0.00521 0.02740 0.06638 0.10712 0.12739 0.11050 0.05596

43622 83567 46965 50657 08973 92582 73634

0.08881 0.22648 0.39997 0.58599 0.75944 0.89691 0.97986

68334 27534 84867 78554 58740 09709 72262

0.00089 0.00816 0.02942 0.06314 0.09173 0.09069 0.04927

26880 29256 22113 63787 38033 88246 65018

0.04463 39553 0.00329 51914 0.07149 10350 0.00046 85178 0.01985 50718 0.05061 42681 0.10166 67613 0.11119 05172 0.14436 62570 0.01784 29027 0.18422 82964 0.00447 45217 0.23723 37950 0.15685 33229 0.28682 47571 0.04543 93195 0.33044 77282 0.01724 68638 0.40828 26788 0.18134 18917 0.45481 33152 0.07919 95995 0.49440 29218 0.04081 44264 0.59171 73212 0.18134 18917 0.62806 78354 0.10604 73594 0.65834 80085 0.06844 71834 0.76276 62050 0.15685 33229 0.78569 15206 0.11250 57995 0.80452 48315 0.08528 47692 0.89833 32387 0.11119 05172 0.90867 63921 0.09111 90236 0.91709 93825 0.07681 80933 0.98014 49282 0.05061 42681 0.98222 00849 0.04455 08044 0.98390 22404 0.03977 89578 Compiled from H. Fishman, Numerical integration constants, Math. Tables Aids Comp. 11, 1-9,1957 (withpermission). 8

922

NUMERICAL ANALYSIS

ABSCISSAS AND WEIGHT FACTORS FOR GAUSSIAN INTEGRATION OF MOMENTS

Table 25.8

s,'

xkf(z)dz =

5

r=l

Wif(Xi)

Weight Factors =wi

Abscissas =xi

k=4

k.= 3

n

9

Xi

Wi

k=5 Xi

Wi

Wi

1

0.80000 00000 0.25000 00000

0.83333 33333 0.20000 00000

0.85714 28571 0.16666 66667

2

0.52985 79359 0.06690 G 4 9 8 0.89871 34927 0.18309 47502

0.58633 65823 0.04908 24923 0.91366 34177 0.15091 75077

0,63079 15938 0.03833 75627 0.92476 39617 0.12832 91039

3

0.36326 46302 0.01647 90593 0.69881 12692 0.10459 98976 0.93792 41006 0.12892 10432

0.42011 30593 0.01046 90422 0.73388 93552 0.08027 66735 0.94599 75855 0.10925 42844

0.46798 32355 0.00729 70036 0.76162 39697 0.06459 66123 0.95221 09767 0.09477 30507

4

0.26147 0.53584 0.79028 0.95784

77888 64461 32300 70806

0.00465 0.04254 0.10900 0.09379

83671 17241 43689 55399

0.31213 0.57891 0.81289 0.96272

54928 56596 15166 39976

0.00251 0.02916 0.08706 0.08124

63516 93822 77121 65541

0.35689 0.61466 0.83107 0.96658

37290 93899 90039 86465

0.00153 0.02142 0.07205 0.07164

44797 84046 63642 74181

5

0.19621 0.41710 0.64857 0.84560 0.96943

20074 02118 00042 51500 57035

0.00152 0.01695 0.06044 0.10031 0.07076

06894 73249 49532 65045 05281

0.23979 0.46093 0.68005 0.86088 0.97261

20448 36745 92327 63437 44185

0.00069 0.01021 0.04402 0.08271 0.06235

69771 05417 44695 27131 52986

0.27969 0.49870 0.70633 0.87340 0.97519

31248 98270 38189 27279 38347

0.00036 0.00672 0.03376 0.07007 0.05572

97155 96904 77450 13397 81761

6

0.15227 0.33130 0.53241 0.72560 0.88161 0.97679

31618 04570 15667 27783 66844 53517

0.00056 0.00708 0.03052 0.06844 0.08830 0.05508

17109 53159 61922 32818 09912 25080

0.18946 0.37275 0.56757 0.74883 0.89238 0.97898

95839 11560 23729 64975 51584 52313

0.00021 0.00372 0.01995 0.05223 0.07464 0.04920

94140 67844 62647 99543 91503 84323

0.22446 0.40953 0.59778 0.76841 0.90135 0.98079

89954 33505 90484 36046 07338 72084

0,00010 0.00218 0.01396 0.04148 0.06445 0.04446

13258 79257 96531 63470 88592 25560

7

0.12142 0.26836 0.44086 0.61860 0.78025 0.90636 0.98176

71288 34403 64606 40284 35520 25341 99145

0.00022 0.00314 0.01531 0.04099 0.06975 0.07655 0.04400

99041 75964 21671 51686 00981 65614 85043

0.15324 0.30632 0.47654 0.64638 0.79771 0.91421 0.98334

14389 65225 00930 93025 66898 99006 38305

0,00007 0.00144 0.00892 0.02854 0.05522 0.06602 0.03975

70737 70088 69676 78428 48742 18459 43870

0.18382 0.34080 0.50794 0.67036 0.81258 0.92085 0.98466

87683 75951 05240 34101 84660 64173 74508

0.00003 0.00075 0.00566 0.02095 0.04510 0,05790 0.03624

11046 53838 04137 92982 49816 76135 78712

8

0.09900 0.22124 0.36912 0.52854 0.68399 0.82028 0.92409 0.98529

17577 35074 39000 54312 32484 39497 37129 34401

0.00010 0.00148 0.00785 0.02363 0.04745 0.06736 0.06618 0.03592

24601 56841 50738 15807 43798 18394 20353 69468

0.12637 0.25552 0.40364 0.55831 0.70600 0.83367 0.92999 0.98646

29744 90521 12989 66758 95429 15420 57161 31979

0.00002 0.00059 0.00407 0.01490 0.03471 0.05491 0.05800 0.03275

97092 89500 79241 99334 99507 00973 05653 28699

0.15315 0.28726 0.43462 0.58451 0.72512 0.84518 0.93504 0.98746

06616 44039 74067 85666 64097 94879 35075 05085

0.00001 0.00027 0.00233 0,01004 0.02648 0.04588 0.05153 0.03009

05316 83586 53415 46144 53011 56532 42238 26424

923

NUMERICAL ANALYSIS

ABSCISSAS AND WEIGHT FACTORS FOR LAGUERRE INTEGRATION

E

Jome-zf(x)dx-z = 1 w l f ( x l )

S,"g(x)dx-

wi

Wiezi

wi

Xi

n=2

n=9 1.53332 603312 4.45095 733505

n=80.41577 45567 83 2.29428 03602 79 6.28994 50829 37

0.15232 0.80722 2.00513 3.78347 6.20495 9.37298 13.46623 18.83359 26.37407

22277 00227 51556 39733 67778 52516 69110 77889 18909

32 42 19 31 77 88 92 92 27

0.13779 0.72945 1.80834 3.40143 5.55249 8.33015 11.84378 16.27925 40422 08 21.99658 29.92069 787360 324237 690092 635435

34705 45495 29017 36978 61400 27467 58379 78313 58119 70122

40 03 40 55 64 64 00 78 81 74

0.11572 0.61175 0.57353 55074 23 1.51261 1.36925 259071 2.83375 2.26068 459338 4.59922 3.35052 458236 6.84452 4.88682 680021 9:62iji 7.84901 594560 13.00605 17.11685 22.15109 28.48796 37.09912

21173 74845 02697 13377 76394 54531 b8424 49933 51874 03793 72509 10444

58 15 76 44 18 15 57 06 62 97 84 67

78120 17403 54120 95262 27217 66274 62266 85680 24821 77083 88994 42267 91697 33064 55726

17 02

1.07769 285927 2.76214 296190 5.60109 462543

n=4 0.32254 1.74576 4.53662 9.39507

76896 11011 02969 09123

19 58 21 01

0.26356 1.41340 3.59642 7. 08581 12.64080

03197 30591 57710 00058 08442

18 07 41 59 76

5 w ,erig(xi>

2=1

Weight Factors = w,

Abscissas =x, (Zeros of Laguerre Polynomials) Xi

0.83273 2.04810 3.63114 6.48714

n=5 0.67909 1.63848 2.76944 4.31565 7.21918

91238 38 243845 630582 508441

n=10

n=12 n-6 0.22284 1.18893 2.99273 5.77514 9.83746 15.98287

66041 21016 63260 35691 74183 39806

79 73 59 05 83 02

0.19304 1.02666 2.56787 4.90035 8.18215 12.73418 19.39572

36765 48953 67449 30845 34445 02917 78622

60 39 51 26 63 98 63

n=7

-5 1i58654 643486 -81 3.17031 547900

n=8 0.17027 0.90370 2.25108 4.26670 7.04590 10.75851 15.74067 22i86313

96323 17767 66298 01702 54023 60101 86412 17368

05 99 66 88 93 81 78 89

Table 2S.9

0.49647 1.17764 1.91824 2.77184 3; 84124 5.38067 8.40543

75975 40 306086 978166 863623 912249 820792 248683

0,09330 0.49269 1.21559 2.26994 3.66762 5.42533 7.56591 0.43772 34104 93 10,12022 1.03386 934767 13.13028 1.66970 976566 16;65440 2.37692 470176 20.77647 3.20854 091335 25.62389 4.26857 551083 31.40751 5.81808 336867 38.53068 8.90622 621529 48.02608

7041 51 14 13 19 76 30 49 29 54 86 86

(-

I

-

1) 2.64731 371055 275873 011320 222117 811546 354187 592663 585682 387654 103052 163504 746743

n=15

1:

1-

1 2.18234 1 3.42210 112.63027 1 1.26425 2j4.02068

1-

6\6:45992 676202

885940 177923 577942 818106 649210

16 i:48302 705111 (I2011.60059 490621

0.39143 0.92180 1.48012 2.08677

11243 1 6 50285 29 790994 080755

3. 59162 4. 64876 6.21227 9.36321

606809 600214 541975 823771

0.35400 0.83190 1.33028 1.86306 i; 45025 3.12276 3.93415 4.99241 6.57220 9.78469

97386 07 23010 44 856175 390311 555808 415514 269556 487219 248513 584037

0.29720 0.69646 1.10778 I: 53846 1.99832 2.50074 3.06532 3.72328 4.52981 5; 59725 7.21299 10.54383

96360 44 29804 31 139462 423904 760627 576910 151828 911078 402998 846184 546093 74619

0.23957 0.56010 0.88700 1.22366 1.57444 1.94475 2.34150 2.77404 3.25564 3.80631 4.45847 5.27001 6.35956 8.03178 11.52777

81703 11 08427 9 3 82629 1 9 440215 872163 197653 205664 192683 334640 171423 775384 778443 346973 763212 21009

Compiled from H. E. Salzer and R. Zucker, Table of the zeros and weight factors of the first fifteen Laguerre polynomials, Bull. Amer. Math. Soc. 55, 1004-1012, 1949 (with permission).

924

NUMERICAL ANALYSIS

Table 2 5 . 1 0

ABSCISSAS AND WEIGHT FACTORS FOR HERMITE INTEGRATION m

J-

J'_"ooe-q(X)dx=2,$ = 1 W&i>

Weight Factors=wi

Abscissas=+xi (Zeros of Hermite Polynomials) *xi

*Xi

W p t

Wi

g(x)dx= 1,$ = I wi d g ( x i ,

a,

Wi

n=2 0.70710 67811 86548 (-1) 8.86226 92545 28 1.46114 11826 611

n=3 0 00000 00000 00000 0 1.18163 59006 04 1.18163 59006 037 1:22474 48713 91589 1-112.95408 97515 09 1.32393 11752 136

0.34290 1.03661 1.75668 2.53273 3.43615

13272 08297 36492 16742 91188

23705 89514 99882 32790 37738

6.10862 63373 61108 94455 57467 28552

0. 31424 0,94778 1.59768 2.27950 3. 02063 3.88972

03762 83912 26351 70805 70251 48978

54359 40164 52605 01060 20890 69782

5 70135 2:60492 5.16079 3 90539 8: 57368 2.65855

0.27348 0.82295 1.38025 1.95178 2.54620 3.17699 3.86944 4.68873

10461 14491 85391 79909 21578 91619 79048 89393

3815 4466 9888 1625 4748 7996 6012 0582

5.07929 2 80647 8'38100 1: 28803 9.32284 2.71186 2.32098 2.65480

0, 24534 0.73747 1.23407 1.73853 2.25497 2.78880 3.34785 3,94476 4. 60368 5.38748

07083 37285 62153 77121 40020 60584 45673 40401 24495 08900

009 454 953 166 893 281 832 156 507 112

n=4

1 8.04914 09000 55 1.05996 44828 950 0.52464 76232 75290 1.65068 01238 85785 11218.13128 35447 25 1.24022 58176 958

n=5

9.45308 72048 29 0.94530 87204 829 3.93619 32315 22 0.98658 09967 514 1.99532 42059 05 1.18148 86255 360

0.00000 00000 00000 0.95857 24646 13819 2.02018 28704 56086

{I 1

1

n=7

0.00000 0.81628 1.67355 2.65196

00000 78828 16287 13568

00000 1 8.10264 61755 58965 111 4.25607 25261 67471 2 5.45155 82819 4 9.71781 24509 35233

0.38118 1.15719 1.98165 2.93063

69902 37124 67566 74202

07322 46780 95843 57244

0.00000

ooopo

0.72355 1.46855 2.26658 3.19099

10187 32892 05845 32017

Table 25.11

n=12

n-6

0.43607 74119 27617 1 7.24629 59522 44 0.87640 13344 362 1 1.57067 32032 29 0.93558 05576 312 1.33584 90740 13697 2.35060 49736 74492 (-3)4.53000 99055 09 1.13690 83326 745

1:

68 01 13 95

0.81026 0.82868 0.89718 1.10133

46175 73032 46002 07296

568 836 252 103

6255 32581 83007 07221

82 49 41 14

0.76454 0.79289 0.86675 1.07193

41286 00483 26065 01442

517 864 634 480

61 26 38 37 26

0.72023 0.73030 0.76460 0.84175 1.04700

52156 24527 81250 27014 35809

061 451 946 787 767

n=9 21560 55900 27394 42755 77263

23626 31026 85615 05846 70435 16843

53 23 48 81 33

0.68708 0.70329 0.74144 0.82066 1.02545

18539 63231 19319 61264 16913

513 049 436 048 657

0.62930 0.63962 0.66266 0.70522 88 0.78664 56 0.98969

78743 12320 27732 03661 39394 90470

695 203 669 122 633 923

66 85 99 51 42 38 65 11

0.54737 0.55244 0.56321 0.58124 0.60973 0.65575 0.73824 0.93687

s2O5O 19573 78290 72754 69582 56728 56222 44928

378 675 882 009 560 761 777 841

06 28 00 46 38 63 32 69 73 34

0.49092 0.49384 0.49992 0.50967 0.52408 0.54485 0.57526 0.62227 0.70433 0.89859

15006 33852 08713 90271 03509 17423 24428 86961 29611 19614

667 721 363 175 486 644 525 914 769 532

25 42 88 29

n = 16

n=8

-1)6.61147 -1 2.07802 -211. 70779 -4)l. 99604

W,&

n=10

47901 45852 41398 11535 00862 00925 08448 74740

n=20 66960 50536 20602 20887 33422 63601 64785 93707 09922 36455

Compiled from H. E. Salzer, R. Zucker, and R. Capuano, Table of the zeros and weight factors of the first twenty Hermite polynomials, J. Research NBS 48,111-116,1952,RF'2294 (with permission). COEFFICIENTS FOR FILON'S QUADRATURE FORMULA

e

B

Y

0.00 0.01 0.02 0.03 0.04

a 0.00000 0.00000 0,00000 0.00000 0,00000

000 004 036 120 284

0.66666 0.66668 0.66671 0.66678 0.66687

667 000 999 664 990

1.33333 1.33332 1.33328 1.33321 1.33312

333 000 000 334 001

0.05 0.06 0.07 0.08 0.09

0.00000 0.00000 0.00001 0.00002 0.00003

555 961 524 274 237

0.66699 0.66714 0.66731 0.66751 0.66774

976 617 909 844 417

1.33300 1.33285 1.33268 1.33248 1.33225

003 340 012 020 365

0.1 0.2 0.3 0.4 0. 5

0.00004 0.00035 0.00118 0.00278 0.00536

438 354 467 012 042

0.66799 0.67193 0.67836 0.68703 0.69767

619 927 065 909 347

1.33200 1.32800 1.32137 1.31212 1.30029

048 761 184 154 624

0. 6 0.7 0.8 0.9 1. 0

0.00911 0.01421 0.02076 0.02884 0.03850

797 151 156 683 188

0.70989 0.72325 0.73729 0.75147 0.76525

111 813 136 168 831

1.28594 1.26913 1.24992 1.22841 1.20467

638 302 752 118 472

See 25.4.47.

26. Probability Functions MARVINZELENl AND NORMAN C. SEVERO*

Contents Page

....................

Mathematical Properties 26.1. Probability Functions: Definitions and Properties 26.2. Normal or Gaussian Probability Function . . . 26.3. Bivariate Normal Probability Function . . . . 26.4. Chi-Square Probability Function . . . . . . . 26.5. Incomplete Beta Function . . . . . . . . . . 26.6. F-(Variance-Ratio) Distribution Function . . . 26.7. Student's t-Distribution . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.......................

927 927 931 936 940 944 946 948

Numerical Methods 26.8. Methods of Generating Random Numbers and Their Applications.. . . . . . . . . . . . . . . . . . . . . . . 26.9. Use and Extension of the Tables . . . . . . . . . . . . .

949 953

References

961

..........................

949

.

9 66

Table 26.2. Normal Probability Function for Large Arguments (5Iz5500). . . . . . . . . . . . . . . . . . . . . . . . . -log Q(z), ~=5(1)50(10)100(50)500, 5D

972

Table 26.3. Higher Derivatives of the Normal Probability Function ( 0 5 ~ 1 5 ). . . . . . . . . . . . . . . . . . . . . . . . . . n=7(1)12, z=0(.1)5, 8s

974

Table 26.1. Normal Probability Function and Derivatives (0 5 x 1 5 ) . P(z), Z(z), Z(l)(z), 15D Z 2 ) ( z ) , 10D; Z(')(Z), n=3(1)6, 8D z=0(.02)3 P(z), 10D; Z(S), 10s; P)(z),n=1(1)6, 8s ~=3(.05)5

z(')(x),

Table 26.4. Normal Probability Function-Values P(z)andQ(z) . . . . . . . . . . . . . . . Q(z)=0(.001).5, 5D

Table 26.5. Normal Probability Function-Values P(z) and Q(z) . . . . . . . . . . . . . . . . Q( z) =0(.OO 1).5 , 5 D a

of Z(z) in Terms of

.........

975

of z in Terms of

........

976

National Bureau of Standards. (Presently, National Institutes of Health.) National Bureau of Standards. (Presently, University of Buffalo.)

925

PROBABILITY FUNCTIONS

926

Table 26.6. Normal Probability Function-Values Values of P ( z ) and Q(z) . . . . . . . . . . . . Q(z)=0(.0001).025, 5D Q ( ~ ) = I O - ~ , m=4(1)23, 5D

of z for Extreme

........

Table 26.7. Probability Integral of x*-Distribution, Incomplete Gamma Function, Cumulative Sums of the Poisson Distribution . . . . . .

Page

977

978

x2= .001(.001).01( .01).1(.1)2(.2) 10(.5)20(1)40(2)76 v=1(1)30,

5D

Table 26.8. Percentage Points of the X2-Distribution-Values of x2 in TermsofQandv . . . . . . . . . . . . . . . . . . . . . . . Q(x21v)=.995, .99, .975, .95, .9, $75, .5, .25, . l , .05, .025, .01, .005,

984

,001, .0005, ,0001 v= 1(1)30( 10) 100,

5-6s

Table 26.9. Percentage Points of the F-Distribution-Values of F in Termsof Q, vl, V2 . . . . . . . . . . . . . . . . . . . . . . . Q(FIv1, Y 2 ) = . 5 , -25, .l, .05, ,025, -01, .005, .OO1

986

vl=1(1)6, 8, 12, 15, 20, 30, 60, ~2=1(1)30,40, 60, 120, a, 3-5s

Table 26.10. Percentage Points of the t-Distribution-Values of t in Terms of A and Y . . . . . . . . . . . . . . . . . . . . . . . A(tlv)=.2, .5, .8, .9, .95, .98, .99, .995, .998, ,999, ,9999, .99999, .999999 v=1(1)30, 40, 60, 120, a, 3D Table 26.11. 2500 Five Digit Random Numbers

. . . . . . . . . .

990

99 1

The authors gratefully acknowledge the assistance of David S. Liepman in the preparation and checking of the tables and graphs and the many helpful comments received from members of the Committee on Mathematical Tables of the Institute of Mathematical Statistics.

26. Probability Functions Mathematical Properties a 26.1. Probability Functions : Definitions and Properties Univariate Cumulative Distribution Functions

A real-valued function F(z) is termed a (univariate) cumulative distribution function (c.d.f.) or simply distribution function if i) F(z) is non-decreasing, i.e., F(zl)5 F(Q) for 51

5x2

ii) F(z) is everywhere continuous from the right, i.e., F(z)=lim F(z+e) cto+

iii) F(-=)=O, F ( = ) = l . The function F(z) signifies the probability of the event “ X S z ” where X is a random variable, i.e., P r { X < z }= F ( z ) , and thus describes the c.d.f. of X. The two principal types of distribution functions are termed discrete and continuous. Discrete Distributions: Discrete distributions are characterized by the random variable X taking on an enumerable number of values , . ., z - ~ , ZO, zl, . . . with point probabilities

Pn=Pr{X=Zn}20

where the summation is over all values of z for which sn_O is termed the domain of the random variable X . A discrete distribution of a random variable is called a lattice distribution if therc, exist numbers a and b#O such that every possible value of X can be represented in the form a+bn where n takes on only integral values. A summary of some properties of certain discrete distributions is presented in 26.1.19-26.1.24. Continuous Distributions. Continuous distributions are characterized by F(s) being absolutely continuous. Hence F ( z ) possesses a derivative F’(z)=.f(z) and the c.d.f. can be written

26.1.2

F ( z ) = P r { X _ < z=Sl?(t)dt. )

The derivative f(z) is termed the probability density junction (p.d.f.) or frequency junction, and the values of z for which f(z)>O make up the domain of the random variable X . A summary of some properties of certain selected continuous distributions is presented in 26.1.25-26.1.34.

which need only be subject to the restriction

Multivariate Probability Functions

CPn=1*

The real-valued function F(zl, g, . . . s,) defines an n-variate cumulative distribution function if

The corresponding distribution function can then be written

26.1.1

F ( z )=Pr { X
-*3?

* Comment on notation and conventions.

a. We follow the customary convention of denoting a random variable by a capital letter, i.e., X,and using the corresponding louer case letter, i.e., z, for a particular value t h a t the random variable assumes. b. For statistical applications it is oftcn convcnicnt to have tabulated the “upper tail arca,” l-F(z), or the c.d.f. for 1x1, F ( z ) - F ( - z ) , instead of simply tho c.d.f. F ( z ) . We use the notation P to indicate the c.d.f. of X, Q = I - P t o indicnte the “uppet tail area” and A = P-Q to denote the c.d.f. of In particular we use P ( z ) , Q(z), and A ( r ) to denote the corresponding functions for thc iiormtrl or Gaussian probability function, see 26.2.2-26.2.4. When these distributions dcpcnd on other pnrirrnetcrs, say 01 and 02, u c indic:itc this by n titing P(rleI,et), Q(zlel,e l ) , or A(r(eI,&). For csirniplc the chisquare distribution 26.4 depends on the parameter U and the tabulated function is I\ ritten Q ( x 2 I v ) .

1x1.

i) F(zl, z2, . . . 2,) is a non-decreasing function for each 5; ii) F ( q , s2, . . , 2,) is continuous from the right in each xi; i.e., F(z1, 32, . . . 2,) =lim F ( z l , . . ., z;+c, . . ., 2), a++

. . t n ) = O when any X I = - = ; * F(=,a , ., ., =)=1. F(sl,s2, . . ., s,) assigns nonnegative prob-

iii) F ( q , zz,. iv)

ability to the event t l < X I ~ z 1 + h l , ~2<X2<~2+hzr . . tn<XnO for

+

k=l,

2,

. . ., n.

927

I

928

PROBABILITY FUNCTIONS

The joint probability of the event Xl<sl, vectors (zl, sz,. . ., 5,) and continuous distribuX z < z 2 , , . ., X f l < z , is F ( q , s2,. . . s,). A d o - tions are characterized by absolute continuity of gous to the one-dimensional case, discrete distribu- F(s1, s2,. . ., s,). tions assign all probability to an enumerable set of Characteristics of distribution functions: Moments, characteristic functions, cumulants Continuous distributions 26.1.3

nth moment

26.1.4

mean

about origin

wiJ-:

r*flz)&

26.1.5

cantral moment

26.1.6

ntb

26.1.7

eted value operator for eT he function g(z)

26.1.8

characteristicfunction of X

26.1.9

charakteristic runction or e(x)

26.1.10

inversion formula aattlce dtstrlbutions only)

Relation of the Characteristic Function to Momenta About the Origin

I

Coefficients of Skewnese and EXWM

26.1.15

(skewness)

26.1.11 1-0

Cumulant Function

26.1.16

(excess)

Occasionally coefficients of skewness and excess (or kurtosis) are given by 26.1.17 Relation of Central Moments to Momenta About the Origin

26.1.14

I 26.1.18

81=Y:=(

89=YZ

3)

+3=f

a

(skewness)

P4

(excess or kurtosis)

Some one-dimensional discrete distribution functions Domain

Name 26.1.19 Single mint or degenerate

z=c (c a mnstant)

26.1.20 Binomial

s=r. for 8-0, 1, 2,

26.1.21 Hypergeometric

b =8,

for

r=O, 1,.

Point Probabilities p=1

..

Restrictions on Parameters

Skewness 7 1

Variance

Excess

Characteristic function

--
Cumulants

G=O for r>l

.I=&

_. n

..min (n, NI)

26.1.22 Polsson

O<m<-

'NI\

Complicated

cm(e"-l)

K,=m

for r=l, 2, 26.1.23

Negative binomial

re=#, for r=O, 1.2,

...,m

.. .

m=nP da. r+r=PQ dQ

for r l l 26.1.24 Qoometric

&=a, for r=O, 1.2,

..

_. m

P(1-P)'

2--p G

P

U--,

1-P

P

r,+1=-(1-p)

da. -

dp' r>l

Some one-dimensional continuous distribution functions

Name

Domain

-

Probability Densitj Function J(z)

Restrictions on ParSmeteIS

-

26.1.25

Error function

O
26.1.26

Normal

--<m<-

Skewness Mean

Characteristic function

71

-P -

0

1 -

0

0

&ha

m

+ I

0

0

c

2

c"I-BI

tI

2hl

Cumulants

1 2hz r.=O for n>2 Nl=O,

#2=-

.'I

inD-

O
Cauchy

m

--
O-<m

26.i.m Exponential

--
-<-

--

Laplme. or double exponential

--
26.1.30

Extreme-Value.' (Fisher-Tippett Type. I or doubly exponential)

--
26.1.31

Pearson Type I11

--
26.1.29

o
oa<

o
O
26.1.32

Oamma distribution

Kp<-

26.1.33

Beta distribution

1
--<m
(Euler's constant) =.67721 66649

not defined

not defined

not defined

P-tB

c

2

6

a

w

0

3

p+YB

1.3

2.1

Q+@

2 -

WP

I6

P

P a

WJ<-o+b

26.1.34

4y

-

not defined

m

A' 12

2 6

WP

2(a-6) (a+b+2)

See footnote 6.

0

-11

not defined

(I-*)--

r1=m, S"+l==O

... *See pa@2

11.

931

PROBABILITY FUNCTIONS

Inequalities for distribution functions IF(") . . , denotes the c.d.f.of the random variable Xand t denotes n positive constant; further m is always assumed to he finite and all expectations are assumed to exist.) Inequality

26.1.35

P r ( u ( X ) I t ) <E[u(X)l/t

26.1.36

P r ( X 2 t )Smlt m

(i) P t ( X < O ) =O (ii) E ( X ) = m

F(t) 21-7 26.1.37

P r ( ( X - m l 2 t u ) Il/P F(m+tu)-F(m-tu)

26.1.38

P r ( IX=?iil2tF)

21--01

(i) E(X)=rn (ii) E ( X - m ) l = Z

*

(i) E ( X J =ml (ii) E ( Xi - nti)3= U: (iii) E([X,-mmi][X,-mi])=O(i#n

1 nP

(i) E ( X - m ) 9 = Z (ii) F(z) is a continuous c.d.f. (iii) F(z) is unimodal at ZOO

(i) (ii) (iii) (iv)

E(X-m)*=u9 F(z) is a continuous c.d.f. F(z) is unimodal at zoo

m=zo

(i) E ( X - m ) a = $ (ii) E ( X - m ) 4 = ~ 4

Is such that F'(m)>F'(z) for t#zo.

020

26.2. Normal or Gauasian Probability Function 1 26.2.1 Z(z) e-z2/a

26.2.8

=Fi

26.2.2 P(x)=26.2.3 26.2.4

e - t a / ~ d t = J ~ w~ ( t ) d t

sw

Q(z) =-

S,

e+/adt =

Jz;;r

&S-* e-ta/adt=J'sZ(t)dt

A(z)=-1

The corresponding probability density function

~ ( dt t )

"

is 26.2.9

+

26.2.5

P ( 4 Q ( 4 =1

26.2.6 26.2.7

A(x)=2P(x)-l

P(- 4=Q(4

I and is symmetric around m ,i.e.

Probability Integral w i t h Mean m and Variance Z

A random variable X is said to be normally distributed with mean m and variance a* if the probability that X is less than or equal to x is given by

z

(y)

The inflexion points of the probability density function are at m f a . 'See page n.

932

PROBABILITY FUNCTIONS

Polynomial and Rational Approximations7 for P(r) and Z(x)

Power Series (x 2 0 )

o<x< 26.2.16 26.2.11

Asymptotic Expansions (x>O)

26.2.12

Q(x)=z(.i(

1

1.3

,

26.2.11

.

X

+(-1)"1

*

3 . . . (2n-')}+&

t=-

X2n

1 1+pz

where R,=(-l)"+'l.

3 . . . (2n+l)Jm2

$$dt

which is less in absolute value than the first neglected term.

26.2.18

26.2.13 &(X).UT

(')

p=.23164 19 bl= .31938 1530 b4=-1.82125 5978 b2=-.35656 3782 ba= 1.33027 4429 b3= 1.78147 7937

{ 1-'

P(x)=l-- 1 (1+c~2+c2~+c3xa+c424)-~+€(x) a2

2'4-2 a + (x2+2) ( ~ ~ $ 4 )

2

.

where a l = l , a 2 = l , a3=5, a4=9, a,=129 and th6 general term is an=col * 3 . . . (2n-1) +2~11*3 . . . (2n-3) . . . +2"-lcn4 f22c2l ' 3 . . . (2n-5)

le(z)1<2.5x 10-4 ~1=.196854 ~3=.000344 C2=. 115 194 C, =.O 19527 26.2.19

P(z)= I-- 1 (1 + d l ~ + d 2 ~ ~ + d 3 $ 2

+

and ca is the coefficient of t"+ in the expansion of ttt-1) * . . ( t - n f l ) . Continued Fraction Expansions

26.2.14

Q(x)=Z(x)

1 1 2 3 4 {----x+ x+ x+ x+ x f ' } * *

26.2.15

1 Q(x)=Z-Z(2)

{-----x 3+ 552

1-

2x2 3X24X2.. 7+ 9-

*}

+d4x4 +d52' +de~')-~'+ t(2) Ie(s))<1.5x10-' dl= .O4986 73470 dr= .00003 80036 dz=.O2114 10061 dg=.OOO04 88906 d3= .OO327 76263 da= .OOOOO 53830 26.2.20

b>O)

(3

2 0)

+

+

Z(x) = (ao u2x2+a4x4 sex')-' 4-e (2) ie(x)1<2.7x 10-3 a0=2.490895 a4= -.024393 a'= ,178257 a2=1.466003

7 Based on approximations in C . Hastings, Jr., Approximations for digital computers. Princeton Univ. Press, Princeton, N.J., 1955 (with permission).

PROBABILITY FUNCTIONS

933

26.2.25

26.2.21

Z ( X= ) (bo+ b,2+

b 4 d + b6Z6

+b8z8+

+

blrj2')-'

E(X)

(e(z)1<2.3X 10-4

b6=

b0=2.50523 67 b2=1.28312 04

.13064 69 b8=-.02024 90

,22647 18

bq=

.00391 32

blo=

Rational Approximations 1 for x g where Q(x,)=p

O
I.5

26.2.22

See Figure 26.1 for error curves. ,006,004

l4P)1<3X10-* ao=2.30753

bl =,99229

- ,004 - ,006 -

a, = .27061

bz= .0448 1

- ,008

-

Pp-Pw

FIGURE26.1. Error

26.2.23 zp=t-

-

curves for distribution.

bounds

on

normal

1+d,t +d,t2+d,t3 Derivatives of the Normal Probability Density Function

Ia(p)I<4.5X 10-4 ~0=2.515517 CI=

,802853

da= ,189269

cZ=

.010328

d3= ,001308

z'"'(5)=dym d" Z ( 5 )

26.2.26

d1=1.432788

Differential Equation

Bounds UBeful as Approximations to the Normal Distribution Function

26.2.24 1 1 -+-

P, (2)=2 2 (1 -e -a2/*) $

(X>O)

(- I)"/%!

P(4I P, (2,= 1- (4+2')'-2 2

(2 -4 -za/a ). e

for m=2r, r=Ol 1, .

27") (0) =

(x> 1.4)

I

o

for odd m>O

..

934

PROBABILITY FUNCTIONS

Relation of P ( z ) and Z ( d ( z ) to Other Functions

Relation

Function

erf 2 = 2 ~ ( s J z ) - 1

26.2.29

Error function

26.2.30

Incomplete gamma function (special case)

26.2.31

Hermite polynomial

26.2.32

I(

26.2.33

H h function

26.2.34

I(

26.2.35

Tetrachoric function

26.2.36

Confluent hypergeometric function (special case)

26.2.37

(L

26.2.38

(I

26.2.39

l(

26.2.40

Parabolic cylinder function Repeated Integrals of the Normal Probability Integral

26-2-41

I,(z)=

Lrn

I,-l(t)dt

(n>O)

where I-1(x)=2(x) 26.2.42

I-,(5)

d =(-z)

n-1

Z(2)= (- 1)n - l Z ( n -

1) ( 2 )

(n2-1) 26.2.43

(&+2

~--n)I.(x)=O

26.2.44

(n+ l)I,+I(X)

+ d , ( 2 ) -178-1

(2)=0

(n> - 1)

935

PROBABILITY FUNCTIONS

Asymptotic Expansion for the Inverse Function of an Arbitrary Distribution Function

26.2.45

Let the cumulative distribution function of It

Y = C Y,be denoted by F ( y ) . Then the (Cornishi- 1

Fisher) asymptotic expansion with respect to n for the value of yp such that F(y,) = 1- p is Asymptotic Expansions of an Arbitrary Probability Density Function and Distribution Function

Let Y t (i=1,2,

. . ., n) be n

Pp-mfuw

26.2.49

where

w=z+ [Tlhl(X)1

+[Yzhz(4

independent random variables with mean m f , variance U:, and higher cumulants K,, Then asymptotic expansions with respect to n for the probability density and cumulative distribution function of

,.

+YVhl

(d1

+YlYZhlZ(d +$hllI(dl

+[Y3h3(4

+

[Y4h4(2)fr%Z (2)f Y 1 7 3 h 1 3 ( 2, +$YZhl12 (2)

+rZ~1111(41+

and Q(x)=p,

~

~

Kr-

~

r=3,4,. =

~

..

~

' ' * j

Ka

26.2.47

26.2.50

hl(z)=g1 He2(z) 1

h z ( 2 ) = a He3(d bl(d=-z 1 h3(2)=,,,

26.2.48

1

We&)

+He1(41

[He4(x)I

hI2(z)=-- 1 [He&)+He&)] 24 hlll(r)=&

[12~e,(z)+lg~ez(x>1

where

1

Terms in brackets are terms of the mme order with respect to n. When the Y,have the same distribution, then mf=m, ui=u2, Kr, f=Kr and

+

h1111(2) =-7776 [252He5(z) 832He3(2) +227Hel

I

(2)

Terms in brackets in 26.2.49 are terms of the same order with respect to n. The He,(z) are the Hermite polynomials. (See chapter 22.)

936

PROBABILITY FUNCTION8 r-7

26.2.51 In the following auxiliary table, the polynomial functions h,(z),h2(2) .

. . hllll(z)are tabulated for

p = .25, .1, .05, .025, .01, .005, .0025, .001, .0005. Auziliary coeficients 0 jor use with Cornish-Fisher asymptotic expansion. 26.2.49 P

1151

.10

1.28155 .I0706 -.07249 .06106

-.W .14644 -.11629 ,00227 .a776

,

,01088

.05

1.64486 .28428 -.ON18 -.01878 -.04828 .17532 -.11800 -.01082 .05986 ,09462

-.1W --.39517 .25823 ,08586

1.95996 .47358 .W72 -.14607 -.04410 .I0210 -.02937 -.02357 ,09659 .161oB -.SS58

.31624

2.32631 ,73532 .23379 -.87634 -.00152 17621 .25195 -.03176 ,07888 ,18058 -.32821 .07286

-.

2.5758a ,93915 ,38012 59171 .owl0 -. 53531 .59757 02821 -. 01226

-.

2.80703 1.14667 .67070

-_ 89880

-.

*

-.

05366

.3M186 46534

-.1.m17498 45

-1.39199

8.09022 1.42491 .a4331

-1.21025 .80746 -1.88355 1.86787 ,04591 69080 -.7w 4.29304 -3. 32708

-.

8.29058

1.63793 1.07EO -1,52234 ,46059

-2.71243 2.62887 .loeXl -1.03555 -1.80531 7.23307 -5.40702

8 From R. A. Fisher. Contributions to mathematical statistics. Paper 30 (with E.A. Cornish) Extralt de la Revue de 1'Institute International de Statistique 4. 1-14 (1937) (with permission).

26.3. Bivariate Normal Probability Function 26.3.1

means and variances (mz, mu) and (4, U;) and correlation p if the joint probability that X is less than or equal to h and Y less than or equal to k is given by

=L = I m Z ( z ) d z swmZ(w) dw,

(q?),-(--),

k- mu

w=@) The probability density function is

26.3.12

where

26.3.8 26.3.9

L(-h, k,p)+L(h, k, --p)=Q(k) L(-h, -k, p)-L(h,k, p)=P(k)-Q(h)

26.3.10

Circular Normal Probability Dendty Function

* 2[L(h,k, p)+L(h, k, -~)+P@)-&@)l-l 26.3.13 Probability Function With Means m,, my, Variance8 d , d , and Correlation p

The random variables X,Y are said to be distributed as a bivariate Normal distribution with *See page 11.

p)

937

PROBABILITY FUNCTIONS

Special Values of L(h, k, p )

26.3.14 26.3.15

I

I

L (h,k , 0) =Q (h)Q(k) L(h,k , - 1 ) = O

L(h,k, - 1 ) = P ( h ) - Q ( k )

26.3.17

L(h,k , I ) = & @ )

(klh)

26.3.18

L ( h , k ,I ) = Q ( k )

(k>h)

1 .J(O,O, p ) = z +

26.3.20

(h+k>O)

26.3.16

26.3.19

L(h, k, p ) as a Function of L(h, 0, p )

(

(h+kSO)

arc sin 2?r

( 4 - h ) (sgn k ) ) +L k’01Jh2-2phk+k2 if hk>O or hk=O

-

p

andh+k>O otherwise

where sgn h = 1 if h 1 0 and sgn h = - 1

P

FIGUBE26.2. L(h, 0,

14

p)

for O < h < l

and - 1 < p < O .

Values for h
0, p ) .

if h
938

PROBABILITY FUNCTIONS

P

FIGURE26.3. L(h, 0,

p)

for O l h l l and O I p I l .

Values for h
0, p ) .

939

PROBABILITY FUNCTIONS P

FIGURE26.4.

L(h, 0,p ) for h 2 1 and - 1


Values for h
51. 0 , p.)

940

PROBABILITY FUNCTIONS

Approximation to P(RI 2, rJ)

Integral Over an Ellipse With Center at (mz, mu)

26.3.25 26.3.21

A pproz:'mation

2RZ

4+RZ exp-where A is the area enclosed by the ellipse

Condition

21.2 4+Rz

26.3.26

P(x1)

R>1

26.3.27

P(xJ

R>5

2,=

Integral Over an Arbitrary Region

26.3.22

R, r both large

xZ= R - \ n

*

Inequality

26.3.28

where A*@,t ) is the transformed region obtained from the tkansformation where

Ph-k>O,

O< ~ < l *

Series Expansion

26.3.29 ull

Integral of the Circular Normal Probability Function With Parameters m+=m,=O, o = l Over the Triangle Bounded by y= 0, y =ax, x= h

26.4. Chi-Square Probability Function 26.4.1

26.3.23

1

1

=-+L(h,o, 4 p ) - L ( O , 0,p ) - 2 Q(h) where p=--

26.4.2

&(x21v)=l-P(x2(Y)

a

in2 Relation to Normal Distribution

Integral of Circular Normal Distribution Over an Offset Circle With Radius R o and Center a Distance ru From (mz*

mu)

26.3.24

Let X,, X z , . , ,,X , be independent and identically distributed random variables each following a normal distribution with mean zero and unit variance. Then X2=&X: 1-1 is said to follow the chi-square distribution with v degrees of freedom is given by P(xZlv). and the probability that P<x2

where P(R2(2,r2) is the c.d.f. of the non-central x2 distribution (see 26.4.25) with v = 2 degrees of freedom and noncentrality parameter 1.2.

Cumulanta

26.4.3

K,+1=2%!Y

*Se? page 11.

.

(%=0,1, , .)

941

PROBABILITY FUNCTIONS

Approximationr to the Chi-Square Dirtribution for Large U

Serier Expanrions

26.4.4

26.4.13 A pprozimdion

Q(x2lv)= Q ( z ~ ) ,

Condition

x ~ = ~ - J % T(v>IOO)

26.4.14 (v odd) and

x=fi

26.4.5 -

-

{ +k

Q(x21~)=J%Z(x)1

r-1

Q(x2(v)= Q ( x J ,

m

52-

26.4.15

"-3 I

i)

(Xziv) 113 -( 1 -

2.4.

..(2~)

Q(x2lv) Q(zz+h")1

hv=-

60 V

(v>30)

hao

(v>30)

(v even)

26.4.6 2

*

-3. 6 -3.0

--.0118

-1.6

+.OOO1

-.m7

-. 0033

-2. b -2.0

{ "2 (v+2)

X2'

(v+4).

.( v + 2 ~ )

-.0010

-1.0 -.6 .O

+.OOO2

1.0

-.OO08

4-1.6

+.OO08

.m

+.a

2.0 2.5

-.OOO3

3.0 3.6

-.OOO6 +.m2 .0017

.CO43

.m

Approximations for the Inveme Function for Large

U

If Q(x:(v)=p and Q(zp)=l-P(zp)=p, then Approzimatwn

26.616

xi=:

26.4.17

X;=V

Recurrence and Differential Relation.

{Z , + ~ % T } ~

Condition (v> 100)

(u>301

where hv is given by 26.4.15. Continued Fraction

Relation to Other Functionr

26.4.19

Incomplete gamma function

r (4

Arymptotic Dirtribution for Large Y

26.4.11 P(X2(v) -P(z)

26.4.20

xL2z

v=%,

y(a,=P(X21v),

Pearson's incomplete gamma function

where z=-X2-v

Jz;

Asymptotic Expanaionr for Large x'

v=2(p+l), 26.4.21

x2=2u

Poisson distribution

26.4.12

Q(xzlV ) -Q(x21v-2)

=e -m

(c- 1) !

&q7

942

PROBABILITY FUNCTIONS

26.4.22

Non-Central x4 Distribution Function

Pemon Type I11 26.4.25

2a b

v=

26.4.23

+2, x2= 2b (x+ a )

Incomplete moments of Normal distribu-

where A 2 0 is termed the non-centrality parameter.

tion

Relation of Non-Central xa Distribution With v=2 to the Integral of Circular Normal Distribution (a*= 1) Over an Offset Circle Having Radius R and Center a Distance r = f i From the Origin. (See 26.3.24-26.3.27.)

26.4.26

x2=2, v=n+1 26.4.24

Generalized Laguerre Polynomials

g(x, y , 0)dxdy=P(x2=R2Jv=2, A)

JJA

x= x2/2 a =v/2 1

Approximations to the Non-Central

a=v+h Approximating Function

b=-

x1

Distribution

h v+x

Approximation

26.4.27

x2 distribution

26.4.29

Normal distribution

2x’2 3

P ( X ’ ~ A)~ P =P(x), ,

x=[-] 1+b

-[&-l]’

Approximations to the Inverse Function of Non-Central X* Distribution

I f Q(xL21vl A>=p, Q(~”,v*)=p,and &(x,)=p then Approximating Variable

26.4.30

x2

26.4.31

Normal

26.4.32

Normal

Approximation to the Inverse Function

+

xi2 = (1 b)x;

PROBABILITY FUNCTIONS

943

h

I h 3

3

b

5I

6

3

'4

5

+

5p

i U

*

17

U

TPI

*Id: I

T

I

h

3 I

b

s

f

5I

b

f

3l&

5;t I

9

NI&

NIO,

+

5

Y

It

3

Ts

'4

5

4 I3

-f I

E'

b

3 1

h 3

0'

I

4

i

k2 W

*

U

944

PROBABILITY FUNCTIONS

26.5. Incomplete Beta Function 26.5.1

Continued Fractions

26.5.8

26.5.2 ],(U,

b ) =1-Z]--t ( b ,a) Relation to the Chi-Square Distribution

If Xi and X i are independent random variables following chi-square distributions 26.4.1 with v 1 and v2 degrees of freedom respectively, then

x:+x:is said to follow a beta distribution with and u2 degrees of freedom and has the distribution function

u1

Best results are obtained when z<--

a- 1 a+b-2

Also the 4m and 4 m+ 1 convergents are less than ZJa, b) and the 4 m f 2 , 4 m + 3 convergents are greater than Z,(u, 6). 26.5.9

26.5.3

*

x
el=l

(a+m-l)(b-m) . eZm= (a+2m -2) (a+2m - 1) 1-2

-

m(a+b-l+m) ezm+l= (a+2m--l)(a+2m)

Series Expansions (O<x
z

1-2

26.5.4 Recurrence Relations

26.5.10

+

I&, b ) =xZr(a- 1 ,b ) (1-x)Z,(a, b- 1) 26.5.11

26.5.5

26.5.12

26.5.13

26.5.14

26.5.6 1-I,(u, b) =ZI -r(6, U )

26.5.15 26.5.7 1-Z&,

b)=ZI-Z(b,

= (1-r)

26.5.16

0,)

afb-1

-0

(a+!-l) a

(L)’ 1-x

(integer a)

945

PROBABILITY FUNCTIONS Asymptotic Expansions

-r

26.5.17

and y is taken negative when z<

F(b,Y) 1-I,(a, a) =I1-, ( b,a) -

a- 1 a+b-2 ~

(b)

Approximations

+

If (a b - 1) ( 1-z) 2 . 8

26.5.20 [ (b-3)(b-2)(5b+7)

( b + l +Y)

I I ( a , b) = Q(x'Iv)

+E,

/c(<5X10-3 if a+b>6

y=-Nln

b 1 N=a+--2 2

z,

x2= (U+ b - 11(1-Z)(3-Z)- (1-5 ) ( b- 1), v=2b

26.5.18

r(a,w) e-"uY I,(% b) "r(a,

(a-1-w) 2b

+(.{

If (a+b-l)(l-z)

26.5.21

2.8

(-"-)

w=b 1--2 26.5.19

a3(1+y2/2)+ l+az al=z (b-a)[(a+b-2)(a-l)(b-l)]-% 3

...

+

1

Approximation to the Inverse Function

26.5.22

If I,,(a, b ) = p and Q(yP)=pthen U

XU

=:a+bel"

a- 1

h=?(=+-) 1 2a

1 1 2b-1

E ) ,~ : ~

-l

6

Relationr to Other Functions and Distributions

Function

Relation

1

1

26.5.23

Hypergeometric function

zqqz F(a, 1-b;

26.5.24

Binomial distribution

5 c ) p ' (1-p)

26.5.25 26.5.26

( L

Negative binomial distribution

#-a

n-8

a + l ; z)=Iz(a, b )

=I, (a, n-'a

Student's distribution

26.5.28

F-(variance-ratio) distribution

I, n-a)

c)pa(l-p)n-a=Ip(a, n-a+l)-I,(a+

5 r+8~pnq'=I,(a, -l) i-a

26.5.27

+1)

v l 1 1 Z [ l - A ( t l ~ ) ] = I, ~ (2t2)p

n) V

z=- v + t 2 v2

*See page 11.

*

*

946

PROBABILITY FUNCTIONS

26.6. F-(Vgriance-Ratio) Distribution Function

26.6.5

26.6.1

Q(F 1 ~,14=1- (1-X) '1"

2.4

x2+.

..

P(FIV1, y z ) = ( F 2 0) 26.6.2

26.6.6

where v2

x=vz+

vi

F

Relation to the Chi-Square Distribution

If X,Z and Xi are independent random variables following chi-square distributions 26.4.1 with v1 and v2 degrees of freedom respectively, then the

26.6.7

distribution of F=-x,Z'vl is said to follow the

x;lv2

variance ratio or F-distribution with v1 and v2 degrees of freedom. The corresponding distribution function is P(FIvl,va).

+

+

' '

.

(v1+vz-2) * * (v1+2) 2 - 4 . . . (v2-2)

(A)?] 1-x

Statistical Properties

26.6.3

'z{e+sine[cose+-

mean:

a

2 cos3 e+ 3

..,+

third central moment:

moments about the origin:

I

(v2+1)(vZ+3).. . (Y]+Vz-4) sin'1-3 3 . 5 . . * (v1-2)

characteristic function:

Oforvl=l

Series Expansions

x=26.6.4

*See page

n.

v2

vz+viF

where

*

e

for v2>l

PROBABILITY FUNCTIONS

947

then

26.6.9

Relation to Student’s t-Distribution Function (See 26.7)

26-6-10 Q(F 1 ~= 1 1, ~ 2 = ) 1 - A ( t 1 v2)

t=@ Approximation to the Inverse Function

Limiting Forms

26.6.16

26.6.11

lim Q(FIv,,Y ~ ) = Q ( X * ~ V ~ ) ,

x2=vlF

us-) m

If &(FpIvl,v Z ) = p ,then

Fp= ezW where w is given by 26.5.22, with u1= 2

26.6.12

b, vz=2a

XI=^

lim Q(FIvl,v2)=P(xZlv2),

v2

v,+m

Approximations

26.6.13

F Q(FIvi,~

2 -) & ( a ! ) ,

a!=

--v2-2vz

(vl and v2 large)

26.6.14

and A 2 0 is termed the non-centrality parameter. Relation of Non-Central F-Distribution Function to Other Functions

Function

26.6,18

F-distribu tion

26.6.19

Non-central tdistribution

26.6.20

Incomplete Beta function

Relation

~-

26.6.21

Confluent hypergeometric function

-*See page

n.

P(F’lv1,v2,A)=

Z

L

2e-wz

X

PROBABILIT

948

FUNCTIONS

is called Student’s t-distribution with v degrees of

Series Expansion

26.6.22

freedom. The probability that

- will be less 4 %

in absolute value thnn n fixed constant t is 26.7.1

where

To=1 “A1

1-x

T ,=- 2 i [ ( v i + ~ 2 - 2 i + A ~Tl) 1 -=x

+

1-5)

Ti - 2 1

x=-

vIF’+vz Limiting Forms

(O
- 9 -

where

v2

I

U

v+ iz

Stat istioal Properties

26.6.23

26.7.2

-

lim P(F’Jvl,v2,A) =P(X’~IV,N, ”I+

(2’ ;) ,

=l-I,

X”=vlF’,

v=vl

mean:

m=O

variance:

u2=-

26.6.24

lim P(F’Iv,,vl,X)=&(x21v),

vI-+m

wherc A / v l - w z as vl+

x2=----vz(1F‘+c2)

skc wness: yI=0

w.

Approximations to the Non-Central F-Distribution

26.6.25

P(F’Ivl,v2,A)= P ( x l ) ,

(vl and v2 largc)

cxccss:

y2=-

moments:

wherc F’--

V

v-2

v2 (U1 +A)

P5=

6 v-4 1 * 3 . . , (2n-l)P (v-2)(v-4) . . . (v-2n)

pZn+l=O

characteristic function:

Series Expansions

26.7.3

I

26.7. Student’s t-Distribution

If X is a random variable following a normal distribution with mean zero and variance unity, and xz is a random variable following an independent chi-square distribution with U degrees of X

freedom, then the distribution of the ratio =

4X’b

26.7.4

Ia

(v>l and odd) (v=l’ cos2 e+-

. . . (v-3) ‘ 2 . 4 . 6 . . . (v-2) 1.3.5

1-3 2.4

e+ . . .

(80s~

’1

(v

even)

*

949

PROBABILITY FUNCTIONS Approximation for Large v

Aeymptotic Expansion for the Inverse Function

If A(tp(v)=l-2p and &(s,)=p, then

t(1-$)

26.7.8 A(t(v)= 2 P ( ~ ) - l ,

X=

2v Non-Central t-Distribution

26.7.9

P(t'Iv, 6)= g3(4

=&

g4 (z)=

(3z7+19z6+171-15z)

m o

(79z8+776z7$1482z5-1920~-945~) Limiting Dietribution

26.7.6

&J:te-I*/qs=A(t)

lim A(tlv)=U+=

where 6 is termed the non-centrality parameter.

Approximation for Large Values of t and v I 5

A(tlv)= l - 2

26.7.7

{;+&}

26.7.10

1

d

3

4

a,

-3183

b,

.OOOO

,4991 .0518

1.1094 -.0460

3.0941 -2.756

U

Approximation to the Non-Central t-Dietribution

5

9.948 -14.05

t'(l-L)-6

P(t'(v,6) = P ( z )

where 2= (1+9(

Numerical Methods 26.8. Methods of Generating Random Numbers and Their Applications ' Random digits are digits generated by repeated Experience has shown that the congruence independent drawings from the population 0, 1, method is the most preferable device for genersting 2, . . ., 9 where the probability of selecting any random numbers on a computer. Let the sequence digit is one-tenth. This is equivalent to putting of pseudo-random numbers be denoted by { Xn} , 10 balls, numbered from 0 to 9, into an urn and n=O, 1, 2, . . . . Then the congruence method drawing one ball at a time, replacing the ball of generating pseudo-random numbers is after each drawing. The recorded set of numbers X,+l=aX,+b(mod T ) forms a collection of random digits. Any group of where b and T are relatively prime. The choice n successive random digits is known as a random number . of T is determined by the capacity and base of the Several lengthy tables of random digits are computer; a and b are chosen so that: (1) the reavailable (see references). However, the use of sulting sequence { X , } possesses the desired starandom numbers in electronic computers has retistical properties of random numbers, (2) the sulted in a need for random numbers to be genperiod of the sequence is as long as possible, and erated in a completely deterministic way. The (3) the speed of generatlon is fast. A guide for numbers so generated are termed pseudo-random choosing a and b is to iiiake the correlation numbers. The quality of pseudo-random numbetween the numbers be near zero, e.g., the correlabers is determined by subjecting the numbers to tion between X. and Xn+sis several statistical tests, see [26.55], [26.56]. The 1-6 7 ba (1 -f)+ b purpose of these statistical tests is to detect any properties of the pseudo-random numbers which PI= are different from the (conceptual) properties of a, random numbers. where aa=a' (mod T ) O The authors wish to cxpress their appreciation to b,=(l+a+a2+ . . . +a'-')b (mod 2") Professor .J. W. Tukey who mnde many penetrating and helpful suggestions in this section. I e I
950

PROBABILITY FUNCTIONS

When T= log, b need only be not divisible by 2 or 5, and a = l (mod 20). The most convenient choices for n are of the form a = F + l (for binary computers) and a=1OS++1 (for decimal computers). This results in the fastest generation of random numbers as the operations only require a shift operation plus two additions. Also any number can serve as the starting point to generate A sequence of random digits. A good summary of genernting pseudo-random numbers is [26.51]. Below are listed various congruence schemes and their properties. a= 1 (mod 4).

which occur in

Xn+,=a,Xn+b, (mod T ) When a is chosen so that a = P 2the , correlation p 1 T-11'. The sequence defined by the multiplicative congruence method will have R full period of T numbers if b is relatively prime to T (i) a= 1 (mod p ) if p is a prime factor of T (ii) a=l (mod 4) if 4 is a factor of T. (iii)

Consequently if p=2g, b need only be odd, and

I

v

Congruence methods for generating random numbers Xmtl=aX.+b(mod T), T and b relatively prime

-

ib

--

_-

Period

xo

T=to

to

OlXo
26.8.2

T=tQ

t 0-.

relatively prime to

26.8.3

T=tofl

(varies)

relatively prime to

T=W, Xo unknown; a-z7+1, b=l; T=2c7, a 4 + 1 . b=2974108625 8473. XO-76293 94531 25. T=24o 242 Xo=l. a=517(8=2) T=2:6' Xd=1, TLZIQ,Xo=l-Z-IQ, ,5478126193; a=Ll1(8=2) T=21a( Xo=lf a=515(8=2) T=2:i+l, Xo=10,987,654,321; a=23; period= 106 T=lOa+l, Xo=47,594,118; a=23; perlod=5.8XlW

26.8.4

T- 101

5.10r-1

relatively prime to

T-1010 Xo-1; a-7 T-1011: XO-1; a=71:

10x0 given

T=lOq

5.10P-r

T T T

relatively prime to T

is the startlng polnt for random numbers when statistical tests were made.

When the numbers are generated using a congruence scheme, the least significant digits have short periods. Hence the entire word length cannot be used. If one desired random nuriibers with as many digits as possible, one would have to modify the congruence schemes. One way is to generate the numbers mod T f 1. 'Phis unfortunately reduces the period. Generation of Random Deviates Let { X } be a generated sequence of independent random numbers having the domain (0, T ) . Then { U }= { T - l X ) is a sequence of random deviates (numbers) from a uniform distribution on the interval (0, 1). This is usually a necessary preliminary step in the generation of random deviates having a given cumulative distribution function F ( y ) or probability density function f ( y ) , Below are summarized some general techniques

~

'See page

Spzcial cases for which random numbers have passed statistical tests for randomness 10

26.8.1

26.8.5

*

a

T

n.

for producing arbitrary random deviates. (In what follows { U }will always denote a sequence of random deviates from a uniform distribution on the interval (0, l).) 1. Inverse Method

The solutions { g ] of the equations { u = F ( y ) } form a sequence of independent random deviates with cumulstive distribution function F ( y ) . (If F ( y ) has a discontinuity a t y=yo, then whenever U is such that F(yo-O)
Let Y be ii discrete rirndom variable with point probabilities p , = P r { Y = y , J for i=1, 2, . . . .

951

PROBABILITY FUNCTIONS

The direct way to generate. Y is to generate { U! and put Y = yr if p1+p2+

*

*

-

..

+Pt-l
*

+pt.

However, this method requires complicated machine programs that, take t'oo long. An alternative way due to Marsaglia [26.53] is simple, fast, and seems to be well suited to highspeed computa.tions. Let p , for i = l , 2, , . . , n be expressed by k decimal digits as pt=.b1t82t . . . akt where the 6's are the decimal digits. (If the domain of the random variable is infinite, it is necessary to truncate the probability distribulion at p,.) Define

Po=O, P,= 10-'

2a,,

for r = l , 2,

. . . , k, and

II,=qIO'P,, s = l , 2, . . . , k. 8

r-

Number the computer memory locations by 0, . . , , IIk-l. The memory locations are divided into k mutually exclusive sets such that the sth set consists of memory locations II,_,, 11,-1+1, . . . , & - l . The information stored in the memory locations of the sth set consists of y1 in locations, yz in locationg, . . . , yn in 68, locations. Denote the decimal expansion of the uniform deviates generated b y the computer by U =+dld2d3. . . and finally let o { m } be the contents of memory location m. Then if 1, 2,

c P t 5 U < 5 Pt

8-1

I-0

i-0

DIlt, r --

v=a{ dldz . . . r l , + ~ ~ ~ - ~ - l O ' This method is perhaps the best all-around method for generating random deviat,es from R discrete distribution. In order to illustrate this method consider the problem of generating deviates from the DinomiRl distrihution with point probabilities PI=@) p'(1-p)"-'

for n = 5 nnd p = . 2 0 . The point probabilities to 4 D are Value of Random Variable Point Probabilities 0 2)o=O.3277 1 pi= .4096 pz= ,2048 2 3 ?),a= .0512 p4= -0064 4 5 pb= a0003

and thus Po=O, Pl=.9, P2=.07, P,=.027, P4=.0030 from which IIo=O, 111=9, 112=16, 113=43, I14=73. The 73 memory locations are divided into 4 mutually exclusive sets such that Set 1 2 3 4

Memory Locations 0, 1, . . . , 8 9, 10, . . . , 15 16, . . . , 42 43, . . , 72

.

Among the nine memory locations of set 1, zero is stored 610=3 times, 1 is stored 611=4 times, 2 is stored a12=2 times; the seven locations of set 2 store 0 azO=2 times and 3 8 ~ 3 ~times; 5 etc. A summary of the memory locations is set out below: Value of Random Variable 0 1 2 3 4 5 Frequency Frequency Frequency Frequency

(set (set (set (set

3 2 7 7

1) 2) 3) 4)

4 0 9 6

2

0 4 8

0 5 1 2

0 0 6 4

0 0 0 3

Then to generate the random variables if put

0Iu<.9 .9 5u<.97 .975u<.997 .9975u<1.000

y=a { dl } y =a { dldz-8 1 } y =a { dld2d3-954 }

y = a { dld2d3d4-99273

3. Generating a Continuous Random Variable

The method for generating deviates from a discrete distribution can be adapted to random variables having a continuous distribution. Let F(y) be the cumulative distribution function and assume that t8hedomain of the random variable is (a,b) where the interval is finite. (If the domain is infinite, it must be truncated at (say) the points a and b.) Divide the interval (b-a) into n sub-intervals of length A (nA=b-a) such that the boundary of the ith interval is ( y ~ yt) , where yt=a+iA for i=O, 1, . , . , n. Now define a discrete distribution having domain

{}*Y={. with point probabilities pt=F(yt)--F(yt-l). Finally, - . let W be a random variable having a uniform distribution on

(-$

(

done by setting W=A U--

%)a

1)

This can be Then random

952

PROBABILITY FUNCTIONS

deviates from the distribution function F ( y ) , can be generated (approximately) by setting y= z+w

. This is simply an approximate ( 3 decomposition of the continuous random variable =z+A

U--

into the sum of a discrete and continuous random variable. The discrete variable can be generated quickly b y the method described previously. The smaller the value of A the better will be the approximation. Each number can be generated by using the leading digits of U to generate the discrete random variable 2 and the remaining digits forming a uniformly distributed deviate having (0,l) domain. 4. Acceptance-Rejection Methods

I n what follows the random variable I' will be assumed to have finite domain (a,b ) . If the domain is infinite, it must be truncated for computational purposes a t (say) the points a and b. Then the resulting random deviates will only have this truncated domain. a) Let f be the maximum of f(y). Then the procedure for generating random deviates is: (1) generate a pair of uniform deviates Ul, U p ; (2) compute a point y=a+(b--a)u2 in (a, b ) ; (3) if u,
viates having the probability density function f(y) is: (1) generate U,, U,,U3;(2) define z=a

+(b-a)u3; (3) if both ul< fl ( z ) and u2< f&)

fl

5-1

k

c P5 [cs,-t,+l)f,I-' j=1

b) Let F(y) be such that ~f(y)=fl(y)f2(y) where the domain of y is (a, b ) . L e t f l a n d f 2 be the maximuin of fl(y) and f2(y) respectitely. Then the procedure for generating random de-

f2

take z as the random deviate; otherwise take another sample of three uniform deviates. The acceptance ratio of this method is [ (b-a)flf2]-' and can be increased by dividing (a, b ) into subintervals as in the previous case. c) Let the probability density function of Y be f(y)

=Jab w ,

(.It

I@), (a
Let g be the maximum of g(y, t ) . Then -the procedure for generating random deviates having the probability density function f(y) is: (1) generate Ul, U,, U3; (2) define s=a+(p-a)u2; z=a+

(b--a)ua;(3) if ul< g ( z ' s ) ~take z as the random

9 deviate; otherwise take another sample of three. The acceptance ratio for this method is [ (b-u)g]-' and can be increased by dividing t>hedomain of t and y into sub-domains. 5. Composition Method

Let gz(y) be a probability density function which depends on the parameter z; further let H ( z ) be the cumulative distribution function for z. In order to generate random deviates Y having the frequency function

f ( y ) = J r n SL(YMH(Z) -0)

one draws a deviate having the cumulative distribution function H ( z ); then draws a second sample having the probability density function gp(y). 6. Generation of Random Deviates From Well Known Distributions

f(y)dy=p,; further let the

maximum ofJ(y) in t h e j t h interval bef5. Then to generate random deviates from f(y), generate n pairs of deviates (ulS,u2Js=1, 2, , . ,, n. Assign [ n p J such pairs to the j t h interval and If u1,< f(y,)/f, compute y,=t1-1+ ( f , - t l - l ) ~ l s . accept y, as a deviate. The acceptance ratio of this method is

--)

a. Normal distribution

(1) Inverse method: The inverse method depends on having U convenient approxiination t o the inverse function r=P-'(u) where

u=(27r)-1/2

j-:e-%t,

Two ways of performing this operation are to (i) use 26.2.23 with t=

(In U'>"' -1

or (ii) approximate

z=P-'(u) piecewise using Chebyshev polynomials, see [26.54]. ( 2 ) Sum of uniform deuiates: Let Ul, U,, . . . l U , be sequence of n uniform deviates. Then

953

PROBABILITY FUNCTIONS

will be distributed asymptotically as a normal random deviate. When n= 12, the mctximum errors made in the normal deviate are 9X10-3 for jXI<2, 9 x 1 0 - ' for 2
x*,=x,c az8;ylns 8PO

as the normal deviate where azs are suitable coefficients. These coefficients may be crtlculated using (say) Chebyshev polynomials or simply by making the asymptotic random deviate agree with the correc normal deviate at certain specified points. When n=12, the maximum error in the normal deviate is 8 X 1 0 - 4 using the coefficients * ~ 0 = 9 . 8 7 4 6 * U S = (-7) -5.102 * a2 = (- 3)3.9439 * u H = (-7)1.141 * U , = (- 5)7.474 ( 3 ) Direct method: Generate a pair of uniform deviates (Ul, U 2 ) . Than Xl = (- 2 In U1)1/2 cos 27rUz, Xz=(-2 In U1)1/2 sin 27rUZ will be a pair of independent normal random deviates with mean zero and unit variance. This procedure can be modified by calculating cos 27rU and sin 27rU using an acceptance rejection method; e.g., ( 1) generate ( Ul, Uz); (2) if (2 Ul - I ) * (2 Uz- 1) I1 generate a third uniform deviate U3, otherwise reject the pair and start over; (3) calculate u:-u; UlUZ (* yl=(-ln u3)'lZ -t yz= f2(-1n u3)lI2-

+

U:

*

+

U;

4-U ;

U?

random). Both y1 and yz are the desired random deviates. ( 4 ) Acceptance-rejection method: 1 ) Generate a pair of uniform deviates (Ull U z ) ;2 ) compute x = -1n ul; 3) if e-H(z-1)' >uz (or equivalently ( ~ - 1 ) ~-2s (In uz)accept x, otherwise reject the

pair and start over. The quantity will be the required normal deviate with mean zero and unit variance. b. Bivariate normal distribution

Let { X I ,X * } be a pair of independent normal deviates with mean zero and unit variance. Then { Xl, pXl ( 1 - p2)'/zXz} represent a pair of deviates from a bivariate normal distribution with zero means, unit variances, and correlrttiori coefficient p.

+

c. Exponential distribution

method: Since F ( z ) = e - z / e , X = In U will be a deviate from the exponential distribution with parameter 0. ( 2 ) Acceptance-rejection method: 1 ) Generate a pair of independent uniform deviates ( U,, U ] ); 2) if Ul
-0

+

Then X=Y+min(Ul, U,, , , . , U,) will follow an exponential distribution, The average value of n is 1.58 so that one needs, on the average, only 1.58 U ' S from which the minimum is selected.

26.9. Use and Extension of the Tables U= of Probability Function Inequalities

Example 1. Let X be a random variable with finite mean and variance equal to m and U', respectively. Use the inequalities for probability functions 26.1.37, 40, 41 to place lower bounds on

A(t)= F ( f )--F( -t) =P {'X;m' -I t } for t=1(1)4. *See page 11.

Lower bounds on A ( t ) = F ( t ) - F ( - t ) 4 0 .7500 .8889 .9375

1=1

d

S

Remarks

no knowledge of F ( t ) ;26.1.37 .5556 .8889 .9506 .9722 F(t) is unimodal and continuous; 26.1.40 0 3 1 8 2 .9697 .9912 F(t) is such that p r = 3 ; 26.1.41

954

PROBABILITY FUNCTIONS

It is of interest to note that the standard normal distribution is unimodal, has mean zero, unit variance p4=3, is continuous, and such that

A(t)=P(t)-P(-f)

values we can write r=zo+.O1 and a two-term Taylor series is P(z)=P(xo) Z(zo)10+. Thus one need only multiply Z(zo) by 10+ and add the result to P(xo).

+

Calculation of P ( x ) for x Approximate

=.6827, ,9545, .9973, and ,9999

for t = l , 2, 3 and 4 respectively. Interpolation for P ( x ) in Table 26.1

Example 2. Compute P ( z ) for x=2.576 to fifteen decimal places using a Taylor expansion. Writing x=xo+e we have e2

P(Z)

=p(z0)+Z(zo)e+z(l)(zoo)

Example 4. Using Table 26.1, fmd P ( z ) for z=1.96, when there is a possible error in z of f 5 X 10-! This is an example where the argument is only known approximately. The question arises as to how many decimal places one should retain in P(z). If Ax and AP(z) denote the error in z and the resulting error in P ( z ), respectively, then

a p ( z )=Z(z)AZ

Taking z0=2.58 and 8= -4X 10-a we calculate the successive terms to 16D +.99505 99842 42230 5 72204 35976 2952 57449 8 63097 1439

-

6 6 8 4 9

,99500 24676 84265 7

Hence AP( 1.960) = 3 X 10-4 which indicates tntt.c P(1.960) need only be calculated to 4D. Therefore P( 1.960) = ,9750. Inverse Interpolation for P(x)

Example 5. Find the value of z for which P ( z )= .97500 00000 00000 using Table 26.1 and determining as many decimal places as is consistent with the tabulated function. For inverse interpolation the tabulated function P(z) may be regarded as having a possible error of .5X1O-l5. Hence

The result correct to 17D is P(2.576) =,99500 24676 84264 98 Calculation for Arbitrary Mean and Variance

Example 3.

Find the value to 5D of

using 26.2.8 and Table 26.1. This represents the probability of the random variable being less than or equal to .5 for a normal distribution with mean m=l and variance a2=4. Using 26.2.8 we have

(*5i

P{X<.5}=P ')=P( - 25)

Since P(-z)=l-P(z), we have P(--.25)=1-P(.25)=1-.59871=.40129

where a two-term Taylor series was used for interpolation. Note that when interpolating for P ( z ) for a value of z midway between the tabulated

Let P(zoX correspond to the closest tabulated value of P(z). Then a convenient formula for inverse interpolation is

where

If only the first two terms (i.e., z=zo+t) are used, the error in

2

will be bounded by

X10-4 and the

true value will always be greater than the value thus calculated. With respect to this example, A x = ~ O - ' ~ and thus the interpolated value of z may be in error by one unit in the fourteenth place. The closest value to P(x)= ,97500 00000 00000 is P(xo)=.97500 21048 51780 with z0=1.96. Hence using the preceding inverse interpolation formulas with

955

PROBABILITY FUNCTIONS

t=-.00003

60167 31129

and carrying fifteen decimals we have the successive terms 1.96000 00000 00000

+

-

+ -

.00003

60167 31129 12 71261 68 0

1 are the mean and vari4v ance to terms of order U-' of Jz;z" (see 26.4.34).

where (2v-l)t and 1--

The values of

y1 and yz for

42x2 are

4- 1.95996 39845 40064 Edgeworth Asymptotic Expansion

Thus we obtain

Example 6. Find the Edgeworth asymptotic expansion 26.2.49 for the c.d.f. of chi-square. Method 1 . Expansion for xz

Let For numerical examples using these expansions see Example 12.

Q(XZ1 v ) =1-F(t)

where

Calculation of L(h,k,p )

Example 7.

Since the values of

y1 and yz 26.4.33

+

are

Jh2- 2 phk k2= d.oS=. 3

y1=2 &/v, yz=

w,

we obtain, by using the first two bracketed terms of 26.2.49

Find L(.5, .4,.8). Using 26.3.20

L(.5, .4, .8)=L(.5,0,0)+L(.4, 0, -.6)

Reference to Figure 26.2 yields L(.5,0,0)+L(.4,0, -.6)=.16+.08=.24

The answer to 3D is L(.5, .4, .8) =.250. L"

A

The Edgeworth expansion is an asymptotic expansion in terms of derivatives of the normal distribution function. It is often possible to transform a random variable so that the distribution of the transformed random variable more closely approximates the normal distribution function than does the distribution of the original random variable. Hence for the same number of terms, greater accuracy may be achieved by using the transformed variable in the expansion. Since the distribution of is more closely approximated by a normal distribution than x2 itself (as judged by a comparison of the values of rl and yz), we would expect that the Edgeworth asymptotic expansion of would be superior to that of xz. Method 2. Expansion for &$. Let

Calculation of the Bivariate Normal Probability Function

Example 8. Let X and Y follow a bivariate normal distribution with parameters m2=3, mv=2, u,=4, uV=2, and p=-.125. Find the value of P,{X>2, Y 1 4 } using 26.3.20 and Figures 26.2, 26.3.

Since P, { X > h , Y >k} =L have P { X > 2 , Y > 4 } =L(-.25, 26.3.20 L(-.25,

1, -.125)=L(-.25,0,

Using

,969) + L ( l , 0, .125)-g

4%

42x2

1, -.125).

1

Figure 26.2 only gives values for h>O, however, using the relationship 26.3.8 with k=O, L(-h, 0, p ) =+L(h, 0, - p ) and thus L(-.25,0, .969) Therefore L(-.25, 1, -.125) =$--L(.25,0, -.969). =-L(.25,0,

-.969)+L(l,O,

.125)=-.01+.09=.08.

The answer to 3D is L(- .25, l., -.125) =.080.

956

PROBABILITY FUNCTIONS

Integral of a Bivariate Normal Distribution Over a Polygon

For the following two configurations we define

Example 9. Let the random variables X and Y have a bivariate normal distribution with parameters mz=5, a,=2, my=9, uy=4, and p = . 5 . Find the probability that the point (X, Y) be inside the triangle whose vertices are A= (7,8), B = ( 9 , 13), and C = ( 2 , 9 ) . When obtaining the integral of a bivariate normal distribution over a polygon, it is first necessary to use 26.3.22 in order to transform the variates so that one deals with a circular normal distribution. The polygon in the region of the transformed variables is then divided’ into configurations such that the integral over any selected configuration can be easily obtained. Below are listed some of the most useful configurations.

FIQTJRE 26.8

I

> x

nn

FIGURE26.5

Y

m

FIQURE26.6 X

FIWRE26.9

Using the circularizing transformation 26.3.2 2 for our example results in 1

2-5

y-9

“Z(T+T) t=-i(T-T) 1 x-5 1’

See 26.3.23 for definition of V ( h ,k).

y-9

957

PROBABILITY FUNCTIONS

The vertices of the triangle in the (s,t) coordinates become A=(&/4, -5/4), B=(&, -1) and C=(

-$,i)

*

These points are plotted below.

From the figure it is seen that the desired probability is the sum of the probabilities that the point having the transformed variables as coordinates is inside the triangles AOB, AOC, and BOC.

=[;+L(i.31,

1 -[a+L(l

0,-.76)-L(o,

.31, 0,-.14)-L(O,

zL(1.31, 0,-.76)-L(O,

o , - - . 7 6 b 51 &(1.31)] 0,- .U) -1 &(1.31)] 2

0,-.76)

-L(1.31, 0,-.14)+L(O, 0,-.14) = .OO- -11- .04+ .23=.08

AAOC

=[;+L(.14,

0,-.99)-L(O,

0,-.99)-2 1 &(.14)]

+ [ z1+ ~ ( . i 4 ,

o , - i ) - ~ ( o , 0 , - 1 ) - ~ 1 &(.i4)] = .01+ .02= .03

FIGURE26.10

g(s, t,O)dsdt='V

For these three triangles we have h

AAOB

kl

8/14

ki

$6

ABOC

=[z+L(.48, 1

0,--.97)-L(O,

0,-.97)-- 21 &(.48)]

+[i+L(.48,

0,-.96)-L(O,

0,-.96)-2 1 &(.48)] =.05+.04=.09

Thus adding all parts, the probability that X and Y are in triangle ABCis =.08+.03+.09=.20. The answer to 3D is ,211. From the graph it is seen that the probability over AOB may be found in the same manner as that over Figure 26.8, and over AOC and BOC the probabilities may be found as that over Figure 26.9. Hence

Calculation of a Circular Normal Distribution Over an Offset Circle

Example 10. Let X and Y have a circular normal distribution with u=lOOO. Find the probability that the point ( X , Y) falls within a circle having a radius equql to 540 whose center is displaced 1210 from the mean of the circular normal distribution. In units of U, the radius and displacement from

540

the center are, respectively, R=-=.54 1000

and r

=-l2l0 1.21. The problem is thus reduced to

1000-

and consequently using 26.3.23 and Figure 26.2

finding the probability of X and Y falling in a circle of radius R=.54 displaced r=1.21 from the center of the distribution where u = l .

958

PROBABILITY FUNCTIONS

Since R<1, the approximation 26.3.25 is used. This results in

For this example Ax2= ? ~ 5 X 1 0 and - ~ %=25. This results in

=z1 (-

.076) ( f 5) 10-4= rt 2X 10-6

AQ(x'~v)

as the possible error in Q(xzlv).

The answer to 5D is ,06870.

Calculation of

1v)

Interpolation for Q ( x a

Example 11. Find Q(25.298120) using the interpolation formula given with Table 26.7. Taking x2=25, 8=.298 and applying the interpolation formula results in Q(25.298120)=: { Q(25)16)0'+ Q(25)18) (46-28') 1 8

+~ ( 2120)~3-40+ 5 0') I

=- { (.06982)(.088804)

Outside the Range of Table 26.7

Q(xalv)

Example 12. Find the value of Q(84172). Since this value is outside the range of Table 26.7 we can approximate Q(84172) by (1) using the Edgeworth expansion for Q(x21v) given in Example 6, (2) the cube root approximation 26.4.14, (3) the improved cube root approximation 26.4.15 or (4) the square root approximation 26.4.13. The results of using all four methods are presented below: 1. Edgeworth expansion

+(.12492)(1.014392)

-I- (.20143) (6.896804) }

The successive terms of the Edgeworth expansion for the distribution of chi-square result in 1-&(84172)=.841345

=.19027

.oooooo

A less accurate interpolate may be obtained by setting $ equal to zero in the above formula. This results in the value .19003. The correct value to 6D is &(25.298120)=.190259. On the other hand if x2=25.298 is assumed to have an error of f5X10-4, then how large an error arises in &(x2~v)? Denoting the error in x2 by Ax2 and the resulting error in Q(x21v) by AQ(x21v), we then have the approximate relationship

.001120 2442465

Hence Q(84 172) = ,15754. The successive terns of the Edgeworth expansion for the distribution of result in

4%'

1- Q(84172) == ,842544 - .000034 -.000138

342372

Hence Q(84 172)=. 15764. Using 26.4.8 we can write

2. Cube root approximation 26.4.14

Using the cube root approximation we have Q(84172)=Q(z>

and

where

For practical purposes it is sufficient to evaluate the derivative to one or two significant figures. Consequently we can write

LmJ This

results

in

Q(84~72)=Q(1.0046)=1-

P(1.0046) =.15754. 3. Improved cube root approximation 26.4.15

where x," is the closest value to tabulated. Hence

xz for which

Q is

The improved cube root approximation involves calculating a correction factor h, to 2. Linearly interpolating for h,, (which appears below 26.4.15) with z= 1.0046 results in hso= -.0006 and hence

959

PROBABILITY FUNCTIONS

2. Cube approximation 26.4.17

60 h~~~--(-.0006)=-.00049 '- 72

Taking x.,,=2.32635 we have

Thus x2=144

Q(84172)= Q(1.0046- .0005) = Q(1.0041)

{

[1-&]

= 1-P( 1.004 1)= .15766 4. Square root approximation 26.4.13

Using the square root approximation we have &(84/72)=&(z) where ~=7'-)-%'2(72)-

From the table for he, we obtain using linear interpolation with z =2.33 (approximately)

1: 1.0032.

l (.0012)=.00049 he,=.0012 and thus h l o = f144

This results in

Hence

&(84/72)=Q(1.0032) = 1-P(1.0032)= .15788

The value correct to 6 D is &(84172)=.157653. Generally the improved cube root approximation will be correct with a maximum error of a few units in the fifth decimal and is recommended for calculations which are outside the range of Table 26.7. Calculation of

x2

for Q ( x * I v ) Outside the Range of Table 26.8

Example 13. Find the value of x2 for which &(x2)144)=.01. Since v=144 is outside the range of Table 26.8, we can compute it by using (1) the Cornish-Fisher asymptotic expansion 26.2.50, for x2, (2) the cube approximation 26.4.17, (3) the improved cube approximation 26.4.18, or (4) the square approximation 26.4.16. We shall compute the value by all four methods. 1. Cornish-Fisher asymptotic expansion 26.2.50

The Cornish-Fisher asymptotic expansion for X' with v=144 can be written as X2- 144

8 +

~

(5)

442

(2)+2h11(~)]

16JZ

+12hn(Z)+6h112(2) +4h1111

1

(2)

Hence using the auxiliary table following 26.2.51 with p = .01 we have 144.0000 39.4794 2.9413 0242 -. 0019 0002

-.

+.

Xz= 186.395

J&1= -

186.394

4. Square approximation 26.4.16

x'=- 21 [2.32635+42(144) -1]*= 185.616 The correct answer to 3D is X2=186.394. Generally the improved cube approximation will give results correct in the second or third decimal for v>30. Calculation of the Incomplete Gamma Function

Example 14. Find the value of

niltking use of 26.4.19 and Table 26.7. Using 26.4.19 we have

3 (2.5, .9) =- JT[1- .87607]=. 16475 4

[%(z) 2 +%z(z> +2h111jz)I+-iijj- [30h*(~)

+gh22(2)

+ (2.32635-.00049)

~ ( 2 . 5.9) , = r(2.5)P(1.815)=r(2.5)[1-&(1.815)1

+124524- 4h1 +m[3h2

*sec Dam 11.

3. Improved cube approximation 26.4.18

Poisson Distribution

Example 15. Find the value of m for which 3 1-0

mf

e-m -=.99

i!

using 26.4.21 and Table 26.8. From Table 26.8 with v=2c=8 and Q=.99 we have x2= 1.646482. Hence m=x2/2=.823241. Inverse of the Incomplete Beta Function

Example 16. Find the value of 2 for which I z ( l O , 6)=.10 using Table 26.9 and 26.5.28. Using 26.5.28 we have

*

960

PROBABILITY FUNCTIONS

20

I, (10,6) =&(Fi12,20)=.10 whew z = 2 w F From Table 26.9 the upper 10 percent point of F with 12 and 20 degrees of freedom is F=1.89. Hence X=

2o -.469 204- 12(1.89)-

[ (10.5) (.60)J1'3= 1.8469, ~

wI=

i = [16(.4)]1'3=

1.8566

3[(1.8469) (.98942)-(1.8566) (.99306)]=-.0668 [ ( l . ~ ~ ~ (1.8566)' ) ' -l- 16

*

Y=

1

and interpolating in Table 26.1 gives P ( - .0668) = 1-P(.0668) =.47336

The correct value to 4 D is x=.4683.

The answer correct to 5D is I.eo(16, 10.5)=.47332.

Calculation of Iz(a, b) for a or b Small Integers

Example 17. Calculate IalO(3,20). Values of Iz(ujb ) for small integral a or b can conveniently be calculated using 26.5.6 or 26.5.7. Using 26.5.6 we have

-

21576 (. 110390X 10-') = .620O40 .216450X10-3 Binomial Distribution

Example 18. Find the value of p which satisfies

g *

Find the value of F for which

Example 20.

Q(F(7, 20) =.05 using Table 26.9.

Interpolation in Table 26.9 is approximately linear when the reciprocals of the degrees of freedom (vl, v2) are used as the interpolating variable. For this example it is only necessary to interpolate with respect to l/vl. Thus linear interpolation on l / v l results in F=2.51 which is the correct interpolate. Calculation of F for Q(FIYI,Y~)>.~O

(5so)py-'=.95,

p=l-p

using 26.5.24 and Table 26.9. Combining 26.5.24 and 26.5.28 we have

where vl=2(n-u+1),

Example 21. Find the value of F for which &(F /4 ,8 )=.90 using 26.6.9 and Table 26.9. Table 26.9 only tabulates values of F for which Q(FIvl, v2)=p where p=.500, .250, .loo, ,050, .025, .010, ,005, .001. However making, use of Table 26.9 we can find the values of F, for which p = .75, -9, .95, .975, .99, ,995, .999. For this example we have

vZ=2(a), andp=

U

a

+ (n-a+

1)F

Hence

v2)=. 10 and referring to the table for which &(Flvl, 1 gives F,lo(8, 4)=3.95 and thus F.M,(~,S)=m =l-&(F160,42)=.95

Harmonic interpolation on v2 in the table for which (3(FIv1,vz)=.05 results in F=1.624 for 42 =.301. 42+60(1.624) The correct answer to 4 D is p=.3003. vl=6O, v2=42, and thus p =

Approximating the Incomplete Beta Function

Example 19. Find 1,80(16,10.5) using 26.5.21. Values of Z ( a , b ) can conveniently be calculated with good accuracy using the iipproximntion given by 26.5.20 or 26.5.21. For this exaniple (a+b-l)(l-x)=10.20 which is greater than .8 and hence 26.5.21 will be used. Thus 'See page 11.

Interpolation for F i n Table 26.9

=.253. Calculation of Q(FIv1,ut) for Small Integral u1 or

Example 22.

Compute

Q(2.514, 15)

YZ

using

26.6.4.

Values of Q(FJv,,v2) can be readily computed for small v1 or v2 using the expansions 26.6.4 to 26.6.8 inclusive. We have using 26.6.4

+ l5

'= 15 4 (2.50)

=.60

and

[ ';"

I

Q(2.5014,15)=(.6)7.5 1+-- (.4) =.086 735

961

PROBABILITY FUNCTIONS

y=3.7190 (i.e., Q(3.7190) =.0001). stituting in 26.5.22 gives

Approximating Q(FI.1, Y Z )

Calculate &(1.714110,40) using

Example 23. 26.6.15.

h=2

The approximation given by 26.6.15 will result in a maximum error of .0005. For this example we have

-&)-(

(1.714) ll3 (1 5=

[&+(1.714)2/3

W-3.7190

9 (40)

Q( 1.714110,40) z Q( 1.2222) = 1-P( 1.2222) =. 1108

and thus F=e"=7.23.

On the other hand the approximation given by

5=

I

Ill+;

(1.714)

and interpolating in Table 26.1 gives

5=

[(&)(3.71)]1'3 [1 -"I-9(10) [ [i G+& [(&)

Q(1.714)10, 40) z Q(1.2210) = 1-P(1.2210) =.1112 Calculation of F Outside the Range of Table 26.9

Example 24. Find the value of F for which Q(F(10,20)= .0001 using 26.6.16 and 26.5.22. 2

The correct answer is

Example 25. Compute P(3.71(3,10,4) using the approximation 26.6.27 to the non-central Fdistribution. Using 26.6.27 with v1=3, u2=10, X=4, F'=3.71 we have

=1.2210

For this problem we have a=?=lO,

.9889

Approximating the Non-Central F-Distribution

4Wl

.n

12.2143

F=7.180.

26.6.14 which is usually less accurate results in (1.714) -

*

( 12.21434-1.8052)

W=

The correct value to 5 D is Q(1.714/10,40)=.11108.

(g)

1+19 1-'=12.2143

[19

1 1 -( 9-173) ['*8052+*8333-3 (12.2143) 2 l

Interpolating in Table 26.1 results in

4 1 2 (40) - 11

Hence sub-

1

-;S2] = .675

and interpolating in Table 26.1 gives

b=5=5 2 '

P(3.71/3,10,4)=P(.675)=.750

p=.OOOl. The value of the normal deviate which cuts off .0001 in the tail of the distribution is

The exact answer is P(3.7113,10,4)=.745.

References Texts (26.1) H. Cram&, Mathematical methods of statistics (Princeton Univ. Press, Princeton, N.J., 1951). [26.2] A. Erdblyi e t al., Higher transcendental functions, vols. I, 11, 111. (McGraw-Hill Book Co., Inc., New York, N.Y., 1955). [26.3] W. Feller, Probability theory and its applications, 2d ed. (John Wiley & Sons, Inc., New York, N.Y., 1957). [26.4] R. A. Fisher, Contributions to mathematical statistics, Paper 30 (with E. A. Cornish), Moments and cumulants in the specification of distributions (John Wiley &Sons, Inc., New York, N.Y., 1950). [26.5] C. Hastings, Jr., Approximations for digital computers (Princeton Univ. Press, Princeton, N.J., 1955). [26.61 M. G . Kendall and A. Stuart, The advanced theory of statistics, vol. I, Distribution theory (Charles Griffin and Co. Ltd., London, England, 1958).

Tables

I ~

General Collwtionm

[26.7] R. A. Fisher and F. Yates, Statistical tables for biological, agricultural and medical research (Oliver and Boyd, London, England, 1949). [26.8] J. Arthur Greenwood and H. 0. Hartley, Guide to taSles in mathematical statistics (Princeton Univ. Press, Princeton, N.J., 1962). (Catalogues a large selection of tables used in mathematical statistics). [26.9] A. Hald, Statistical tables and formulas (John Wiley & Sons, Inc., New York, N.Y., 1952). [26.10] 1).B. Owen, Handbook of statistical tables (Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962). [26.11] E.

S.Pearson and H. 0. Hartley (Editors). Biometrika tables for statisticians, vol. I (Cambridge Univ. Press, Cambridge, England, 1954).

962

PROBABILITY FUNCTIONS

[26.12] K. Pearson (Editor), Tables for statisticians and biometricians, parts I and I1 (Cambridge Univ. Press, Cambridge, England, 1914, 1931). Normal Probability Integral end Derivativea

[26.13] J. R. Airey, Table of H h functions, British Association for the Advancement of Science, Mathematical Tables I (Cambridge Univ. Press, Cambridge, England, 1931). [26.14] Harvard University, Tables of the error function and of its first twenty derivatives (Harvard Univ. Press, Cambridge, Mass., 1952). P ( z ) - - f , Z ( r ) , Z(n)(z), n=1(1)4 for z=0(.004) 6.468, 6D; Z(n)(z), n=5(1) 10 for r=0(.004) 8.236, 6D; Z(n)(z), n=11(1)15 for z=0(.002) 9.61, 7s; Z(n)(z), n=16(1)20 for 2=0(.002) 10.902, 75 or 6D. [26.15] T. L. Kelley, The Kelley Statistical Tables (Harvard Univ. Press, Cambridge, Mass., 1948). z for P(z)=.5(.0001).9999 and corresponding values of Z(z), 8D. [26.16] National Bureau of Standards, A guide to tables of the normal probability integral, Applied Math. Series 21 (U.S. Government Printing Office, Washington, D.C., 1951). [26.17] National Bureau of standards, Tables of normal probability functions, Applied Math. Series 23 (U.S. Government Printing Office, Washington, D.C., 1953). Z(z) and A ( z ) for z=O(.OOOI) 1(.001)7.8, 15D; Z(z) and 2[1-P(z)] for z=6(.01)10, 7s. [26.18] W. F. Sheppard, The probability integral, British Association for the Advancement of Science, ’ Mathematical Tables VII (Cambridge Univ. Press, Cambridge, England, 1939). A (z)/Z(z) for z=0(.01)10, 12D; z=O(.l)lO, 24D. Bivariate Normal Probability Integral

[26.19] Bell Aircraft Corporation, Table of circular normal probabilities, Report No. 02-949-106 (1956). Tabulates the integral of the circular normal distribution over an off-set circle having its center a distance r from the origin with radius R ; R=0(.01)4.59, r=0(.01)3, 5D. [26.20] National Bureau of Standards, Tables of the bivariate normal distribution function and related functions, Applied Math. Series 50 (U.S.Government Printing Office, Washington, D.C., 1959). L(h, Ic, p ) for h, k=0(.1)4, p=0(.05).95 (.Ol)l, 6D; L(h, k, - p ) for h, Ic=O(.l)A, p =O( .05) .95 (.O 1) 1where A is such that L< .5 * 10-7, 7D; V ( h , ah) for h=O(.Ol)4(.02)4.6(.1)5.6, a, 7D; V(ah, h) for a = . l ( . l ) l , h=0(.01)4(.02)5.6, m, 7D. (26.211 C. Nicholson, The probability integral for two variables, Biometrika 33, 59-72 (1943). V ( h ,ah) for h=.1(.1)3, ah=.1(.1)3, m , 6D. [26.22] D. B. Owen, Tables for computing bivariate normal probabilities, Ann. Math. Statist. 27, 1075-1090 1 arctan a - V ( h , ah) for (1956). T(h, a)=2s a=.25(.25)1, h=0(.01)2(.02)3; a=0(.01)1, a , h=0(.25)3; a = . l , .2(.05).5(.1).8, 1, m , h= 3(.05)3.5(.1)4.7, 6D.

t26.231 D. B. Owen, The bivariate normal probability function, Office of Technical Services, U.S. Department of Commerce (1957). T ( h , a)= 1 - arctan a-V(h, ah) for a=0(.025)1, m ; h= 2* 0(.01)3.5(.05)4.75, 6D. [26.24] Tables VIII and IX, Part I1 of [26.12]. L(h, k, p ) for h, k=0(.1)2.6, p=-1(.05)1, 6D for p>O and 7D for p
Integral,

[26.25] G. A. Campbell, Probability curves showing Poisson’s exponential summation, Bell System Technical Journal, 95-113 (1923). Tabulates

values of m=-

X2

2

for which Q(x21v)=.000001,

2D ; .0001, .01, 3D ; .l, -25, .5, .75, .9,

4D ;

.99, .9999, 3D; .999999, 2D for c=!=l(l)lOl. 2 [26.26] Table I V of [26.7]. Tabulates values of x2 for &( xzIu) = .001, .01, .02, .05, .l, .2, .3, .5, .7, -8, .9, .95, .98, .99 and ~ = 1 ( 1 ) 3 0 , 3D or 3s. [26.27] E. Fix, Tables of noncentral ~ 2 Univ. , of California Publications in Statistics 1, 15-19 (1949). Tabulates X for P(x’2lu, X) =.1(.1).9, Q ( X ’ ~ ~ V ) = .01, .05; ~=1(1)20(2)40(5)60(10)100, 3D or 35. [26.28] H. 0. Hartley and E. S. Pearson, Tables of the x2 integral and of the cumulative Poisson distribution, Biometrika 37, 313-325 (1950). Also reproduced as Table 7 in [26.11]. P(X2lu) for v=1(1)20(2)70, x2=0(.001).01(.01).1(.1)2(.2)10 (.5)20( 1)40(2) 134, 5D. [26.29] T. Kitagawa, Tables of Poisson distribution (Baifukan, Tokyo, Japan, 1951). e-mm’/s! for m=.001(.001)1(.01)5, 8 0 ; m=5(.01)10, 7D. [26.30] E. C. Molina, Poisson’s exponential binomial limit (D. Van Nostrand Co., Inc., New York, N.Y., m

1940). e-mm*/s! and P ( X ~ ~ Ye-mmf/j! ) = Z for j=c

m= x2/2= 0 (. 1)16(1) 100, 6D ; m= 0(.001) .01 (.01)3, 7D. [26.31] K. Pearson (Editor), Tables of the incomplete r-function, Biometrika Office, University College (Cambridge Univ. Press, Cambridge, England, 1934). Z(u,p) for p = -1(.05)0(.1)5(.2)50, u=O(.l) Z(u,p)=l to 7D; p=-l(.O1)-.75, u=0(.1)6, 5D; ln[l(u,p)lu~+1],p = - 1(.05)0 (.l)lO, u=O(.1)1.5, 8 D ; [zP.+l r(p+l)]-ly(p,z), p = - 1(.01)-.9,~=0(.01)3, 7D. [26.32] E. E. Sluckii, Tablitsy dlya vyEioleniya nepolnoI r-funktsii i funktsii veroyatnosti ~ 2 . (Izdat. Akad. Nauk SSSR, Moscow-Leningrad, U.S.S.R., 1950). r(x2,v)=(i

x2)-’”

P (x+),

9(tIv)=

Q ( x ~ ~ un(t, ) , z)=Q(x21v) where t = (2x2)*- (2~)3, r(x2,v), x2=0(.05)2(.1) 10, u=0(.05) 2(. 1)6; Q(x21v), x2=0(.1)3.2, u=0(.05)2( .1)6; xz=3.2 (. 2) 7 (.5) 10(1)35, v =0 (. 1).4(.2) 6 ; 9( t ,v ) , t=-4(.1)4.8, v=6(.5)11(1)32; l I ( t , ~ ) :t = -4.5 (.1)4.8, z=0(.02).22(.01).25, 5D.

z= b/2)-*.

963

PROBABILITY FUNCTIONS Inaomplste Beta Function, Binomial Dimtribution

[26.33] Harvard University, Tables of the cumulative binomial probability distribution (Harvnrd Univ. Press, Cambridge, Mass., 1955). 03

(J")p'(l-p)n-'

for p=.01(.01).5, 1/16, 1/12,

a=c

1/8, 1/6, 3/16, 5/16, 1/3, 318, 5/12, 7/16, n= 1( 1)50 (2) 100( 10)200 (20)500 (50) 1000, 5D.

[26.42] M. Merrington and C. M. Thompson, Tables of percentage points of the inverted beta ( F ) distribution, Biometrika 33, 73-88 (1943). Tabulates values of F for which & ( F l u l , v 2 ) = .5, .25, .1, .05, .025, .01, .005; vl=l(l)lO, 12, 15, 20, 24,30,40,60, 120, -;Yz=1(1)30,40,60, 120, W .

[ 26.431 P. C. Tang, The power function of the analysis of variance tests with tables and illustrations of their use, Statistical Research Memoirs 11, 126149 and tables (1938). P(F'IYI,Y Z , 6) for U I = 1(1)8, ~,=2(2)6(1)30,60, a n d + = d m = 1(.5)3(1)8 where Q(F'lv,, vZ)=.O1, .05, 3D.

[26.34] National Bureau of Standards, Tables of the binomial probability distribution, Applied Math. Series 6 (U.S. Government Printing Office, Washington, D.C., 1950). (:)p*(l--p)n-a and

2

(:)p*(l-

p)n-'

7D. l26.351 K. Pearson (Editor), Tables of the incomplete beta function, Biometrika Office, University College (Cambridge Univ. Press, Cambridge, England, 1948). Z,(a,b) for z=.01(.01)1; a,b=.5(.5)11(1)50, a>b, 7D. 126.361 W. H. Robertson, Tables of the binomial distribution function for small values of p , Office of Technical Services, U S . Department of Commerce (1960).

5 (")

Student'm t and Non-Central t-Dimtributionm

for p = .Ol(.Ol) .5, n = 2(1)49,

a=e

[ 26.441 E. T. Federighi, Extended tables of the percentage points of Student's t-distribution, J. Amer. Statist. Assoc. 64, 683-688 (1959.) Values of

[26.45]

p' ( 1-p)n-a for p=.001(.001).02, n=2(1)

a=O

[26.46]

lOO(2)200 (10)500(20) 1000; p = .021(.001) .05, n=2(1)50(2)100(5)200(10)300(20) 600(50) 1000, 5D. [26.37] H I G. Romig, 50-100 Binomial tables (John Wiley %I Sons, Inc., New York, N.Y., 1953). [26.47] p=.01(.01).5 and n=50(5)100, 6D. I26.381 C. M. Thompson, Tables of percentage points of the incomplete beta function, Biometrika 32, 151-181 (1941). Also reproduced as Table 16 in [26.11]. Tabulates values of z for which Iz(a, b) =.005, . O l , ,025, .05, .l, .25, .5; 2a=1(1)30, 40, 60, 120, a ;2b=1(1)10, 12, 15, 20, 24, 30, 40, 60, 120, 5D. [26.39] U.S. Ordnance Corps, Tables of the cumulative binomial .probabilities, ORDP 20-1, Office of Technical Services, Washington, D.C. (1952).

*

5 (r)pa(l-p)n-a

for p=.01(.01).5 and n = l

a -E

(1)150, 7D. F (Varianae-Ratio) and Non-Central F DLtribution

1 [l-A(tlv)]=.25XlO-n, 2 .1X 10-", n=0(1)6, .05X 10-n, n=0(1)5, U = 1(1)30(5)60(10)100, 200, 500, 1000, 2000, 10000, a ; 3D. Table I11 of [26.7]. Values o f t for which A(tlu)= .1(.1).9, .95, .98, .99, .999 and U = 1(1)30, 40, 60, 120, a ; 3D. N. L. Johnson and B. L. Welch, Applications of the noncentral t-distribution, Biometrika 31, 362-389 (1939). Tabulates a n auxiliary function which enables calculation of 6 for given t'andp,ort' for given 6 a n d p where P(t'Iv,S)= p = ,005, .01,.025, .05, .1(. 1) .9, .95, ,975, .99, .995. J. Neyman and B. Tokarska, Errors of the second kind in testing Student's hypothesis, J. Amer. Statist. Assoc. 31, 318-326 (1936). Tabulates 6 for P(t'lv,l)= .01,.05, .1(.1).9; Y = 1(1)30, m ; Q ( t ' 1 ~=.01, ) -05. 1 Table 9 of [26.11].P(tlv)=- [l+A(tlv)l for t = 2 0(.1)4(.2)8; Y = 1(1)20, 5 D ; t=0(.05)2(.1)4, 5; v=20(1)24, 30, 40, 60, 120, a , 5D. t for which Q ( t l v ) =

l26.48)

-

[26.49] G. S. Resnikoff and G. J. Lieberman, Tables of the noncentral t-distribution (Stanford Univ. Press, Stanford, Calif., 1957). bP(t'/v,6)/Pt' and P(t' I v,6) for v=2(1)24(5)49, 6 = &l z, whereQ(z,)= p = .25, .15, .l, .065, .04, .025, .01, .004, .0025, .001 and t'/& covers the range of values such that throughout most of the table the entries lie between 0 and 1, 4D.

Random Numbem and Normal D e d a t a 126.401 Table V of [26.7]. Tabulates values of F and 1 [26.50] E. C. Fieller, T. Lewis and E. S. Pearson, CorZ=- In F for Q ( F l v , , 4 = . 2 , .l, .05, .01, .001; related random normal deviates, Tracts for 2 Computers 26 (Cambridge Univ. Press, Camul=1(1)6, 8, 12, 24, 03; ~z=1(1)30,40, 60, 120, bridge, England, 1955). a , 2D for F, 4D for Z . [26.51] T. E. Hull and A. R. Dobell, Random number [26.41] E. Lehmer, Inverse tables of probabilities of errors generators, Soc. Ind. App. Math. 4, 230-254 of the second kind, Ann. Math. Statist. 16, (1962). 388-398 (1944). . $ = J X / ( U ~ + ~for ) ~~=1(1)10, [26.52] M. G. Kendall and B. Babington Smith, Random 12, 15, 20, 24, 30, .40, 60, 120, 0); vz=2(2)20, sampling numbers (Cambridge Univ. Press, Y Z , +) 24, 30, 40, 60, 80, 120, 240, and P(F'IY~, Cambridge, England, 1939). =.2, .3 where Q(F'lv,, ~ 1 ) = . 0 1 ,.05, 3D or 3s. *See page U.

964

PROBABILITY FUNCTIONS

[26.53]G. Marsaglia, Random variables and computers, Proc. Third Prague Conference in Probability Theory 1962. (Also aa Math. Note No. 260, Boeing Scientific Research Laboratories, 1962). [26.54] M. E. Muller, An inverse method for the generation of random normal deviates on large scale computers, Math. Tables Aids Comp. 63. 167174 (1958).

[26.55]Rand Corporation, A million random digits with 100,000 normal deviates (The Free Press, Glencoe, Ill. 1955). [26.56]H. Wold, Random normal deviates, Tracts for Computers 25 (Cambridge Univ. Press, Cambridge, England, 1948).

PROBABILITY FUNCTIONS

966 Table 26.1

NORMAL PROBABILITY FUNCTION AND DERIVATIVES Z(.,.)

P(T)

I'

2")( T )

0.00 0.02 0.04 0.06 0.08

0.50000 0.50797 0.51595 0.52392 0.53188

00000 16902 52831 54107 13988

0.39894 0.39886 0.39862 0.39822 0.39766

22804 24999 32542 48301 77055

01433 23666 04605 95607 11609

0.00000 -0.00797 -0.01594 -0.02389 -0.03181

00000 72499 49301 34898 34164

00000 98473 68184 11736 40929

0.10 0.12 0.14 0.16 0.18

0.53982 78372 77029 0.54775 84260 20584 0.55567 00048 05907 0.56355 94628 91433 0.57142 37159 00901

0.39695 0.39608 0.39505 0.39386 0.39253

25474 02117 17408 83615 14831

77012 93656 34611 68541 20429

-0.03969 -0,04752 -0.05530 -0,06301 -0.07065

52547 96254 72437 89378 56669

47701 15239 16846 50967 61677

0.20 0.22 0.24 0.26 0.28

0.57925 0.58706 0.59483 0.60256 0.61026

97094 44226 48716 81132 12475

39103 48215 97796 01761 55797

0.39104 0.38940 0.38761 0.38568 0.38360

26939 37588 66151 33691 62921

75456 33790 25014 91816 53479

-0.07820 -0.08566 -0.09302 -0,10027 -0.10740

85387 88269 79876 76759 97618

95091 43434 30003 89872 02974

0.30 0.32 0.34 0.36 0.38

0.61791 0.62551 0.63307 0.64057 0.64802

14221 58347 17360 64332 72924

88953 23320 36028 17991 24163

0.38138 0.37903 0.37653 0.37391 0.37115

78154 05261 71618 06053 38793

60524 52702 33254 73128 59466

-0.11441 -0.12128 -0.12802 -0.13460 -0.14103

63446 97683 26350 78179 84741

38157 68865 23306 34326 56597

0.40 0.42 0.44 0.46 0.48

0.65542 0.66275 0.67003 0.67721 0.68438

17416 72731 14463 18897 63034

10324 51751 39407 49653 83778

0.36827 0.36526 0.36213 0.35889 0.35553

01403 26726 48824 02910 25285

03323 22154 13092 33545 05997

-0.14730 -0.15341 -0.15933 -0.16508 -0.17065

80561 03225 93482 95338 56136

21329 01305 61761 75431 82879

0.50 0.52 0.54 0.56 0.58

0.69146 0.69846 0.70540 0.71226 0.71904

24612 82124 14837 02811 26911

74013 53034 84302 50973 01436

0.35206 0.34849 0.34481 0.34104 0.33717

53267 25127 80014 57886 99438

64299 58974 39333 30353 22381

-0,17603 -0.18121 -0.18620 -0.19098 -0.19556

26633 61066 17207 56416 43674

82150 34667 77240 32997 16981

0.60 0.62 0.64 0.66 0.68

0.72574 0.73237 0.73891 0.74537 0.75174

68822 11065 37003 30853 77695

49927 31017 07139 28664 46430

0.33322 0.32918 0.32506 0.32086 0.31659

46028 39607 22640 38037 29077

91800 70765 84082 71172 10893

-0.19993 -0.20409 -0.20803 -0,21177 -0,21528

47617 40556 98490 01104 31772

35080 77874 13813 88974 43407

0.70 0.72 0.74 0.76 0.78

0.75803 0.76423 0.77035 0.77637 0.78230

63477 75022 00028 27075 45624

76927 20749 35210 62401 14267

0.31225 0.30785 0.30338 0.29887 0.29430

39333 12604 92837 24057 50297

66761 69853 56300 75953 88325

-0.21857 -0.22165 -0.22450 -0.22714 -0.22955

77533 29075 80699 30283 79232

56733 38294 79662 89724 34894

0.80 0.82 0.84 0.86 0.88

0.78814 0.79389 0.79954 0.80510 0.81057

46014 19464 58067 54787 03452

16604 14187 39551 48192 23288

0.28969 0.28503 0.28034 0.27561 0.27086

15527 63584 38108 82471 39717

61483 89007 39621 53457 98338

-0.23175 -0.23372 -0.23548 -0.23703 -0.23836

32422 98139 88011 16925 02951

09186 60986 05281 51973 82537

0.90 0.92 0.94 0.96 0.98

0.81593 0.82121 0.82639 0.83147 0.83645

98746 36203 12196 23925 69406

53241 85629 61376 33162 72308

0.26608 0.26128 0.25647 0.25164 0.24680

52498 63012 12944 43410 94905

98755 49553 25620 98117 67043

-0.23947 -0.24038 -0,24108 -0.24157 -0.24187

67249 33971 30167 85674 33007

08879 49589 60083 54192 55702

1.00

0.84134 47460 68543

0.24197

07245 19143

-0.24197

07245 19143

00000 83137 34368 21826 13720

[(331

967

PROBABILITY FUNCTIONS

NORMAL PROBABILITY FUNCTION AND DERIVATIVES X 0.00 0.02 0.04 0.06

2(2)(x)

2(31(~)

z(4)(x)

Z(j)(X)

0.08

-0.39894 -0.39870 -0.39798 -0.39679 -0.39512

22804 29549 54570 12208 26322

0.00000 0.02392 0.04780 0.07159 0.09523

000 856 928 445 664

1.19682 1.19563 1.19204 1.18607 1.17774

684 029 400 800 897

0.00000 -0.11962 -0.23891 -0.35754 -0.47516

000 684 887 249 649

-5.98413 -5.97575 -5.95066 -5.90893 -5.85073

a93 325 742 151

0.10 0.12 0.14 0.16 0.18

-0.39298 -0.39037 -0.38730 -0.38378 -0.37981

30220 66567 87267 53315 34631

0.11868 0.14190 0.16483 0.18744 0.20967

881 445 771 353 776

1.16708 1.15410 1.13884 1.12136 1.10169

019 144 890 503 a39

-0.59146 -0.70610 -0.81878 -0.92919 -1.03701

327 997 968 252 674

-5.77625 -5.68577 -5.57961 -5.45815 -5.32182

460 399 395 435 a95

0.20 0.22 0.24 0.26 0.28

-0.37540 -0.37055 -0.36528 -0.35961 -0.35353

09862 66169 98981 11734 15588

0.23149 0.25286 0.27372 0.29405 0.31380

727 011 555 426 836

1.07990 1.05604 1.03017 1.00237 0.97272

350 063 556 941 834

-1.14196 -1.24376 -1.34214 -1.43683 -1.52759

980 938 434 568 737

-5.17112 -5.00657 -4.82876 -4.63831 -4.43591

356 387 317 979 441

0.30 0.32 0.34 0.36 0.38

-0.34706 -0.34021 -0.33300 -0i32545 -0.31755

29121 78003 94659 17909 92592

0.33295 0.35144 0.36926 0.38637 0.40274

156 923 849 828 947

0.94130 0.90818 0.87347 0.83725 0.79963

327 965 711 919 298

-1.61419 -1.69641 -1.77405 -1.84692 -1.91485

723 762 617 643 840

-4.22225 -3.99809 -3.76420 -3.52140 -3.27051

716 459 646 244 871

0.40 0.42 0.44 0.46 0.48

-0.30934 -0.30083 -0.29202 -0.28294 -0.27361

69179 03372 55692 91055 78339

0.41835 0.43316 0.44716 0.46033 0.47264

488 939 995 566 779

0.76069 0.72055 0.67932 0.63709 0.59398

880 987 193 291 256

-1.97769 -2.03531 -2.08758 -2.13440 -2.17570

904 269 144 537 278

-3.01241 -2.74796 -2.47807 -2.20363 -1.92557

439 802 382 810 548

0.50 0.52 0.54 0.56 0.58

-0.26404 89951 -0.25426 01373 -0.24426 90722 -0.23409 38293 -0.22375 26107

0.48408 0.49464 0.50430 0.51306 0.52090

982 748 874 383 525

0.55010 0.50556 0.46048 0.41496 0.36913

207 372 050 574 279

-2.21141 -2.24148 -2.26589 -2.28463 -2.29771

033 307 443 613

801

-1.64480 -1.36224 -1.07881 -0.79543 -0.51298

520 740 949 249 749

0.60 0.62 0.64 0.66 0.68

-0.21326 -0.20264 -0.19191 -0,18109 -0.17020

37459 56463 67607 55308 03472

0.52782 0.53382 0.53890 0.54306 0.54630

777 841 643 327 259

0.32309 0.27696 0.23085 0.18486 0.13911

457 332 017 483 528

-2.30516 -2.30703 -2.30336 -2.29426 -2.27980

783 091 981 388 875

-0.23237 +0.04554 0.31990 0.58988 0.85469

218 255 583 999 355

0.70 0.72 0.74 0.76 0.78

-0.15924 -0.14826 -0.13725 -0.12624 -0.11524

95060 11670 33120 37042 98497

0.54863 0.55005 0.55058 0.55023 0.54901

016 386 359 127 073

0.09370 0.04874 +o.0043:: -0.03944 -0.08247

741 473 808 465 882

-2.26011 -2.23531 -2.20553 -2.17094 -2.13170

583 162 714 715 944

1.11354 1,36570 1.61045 1.84714 2.07512

405 074 709 311 746

0.80 0.82 0.84 0.86 0.88

-0.10428 -0.09337 -0.08253 -0.07177 -0.06110

89590 79110 32179 09916 69120

0.54693 0.54402 0.54030 0.53578 0.53049

765 952 551 644 467

-0.12468 -0.16597 -0.20625 -0.24546 -0.28351

324 047 697 336 458

-2.08800 -2.04002 -1.98796 -1.93204 -1.87248

401 228 617 726 587

2.29381 2.50267 2.70117 2.88887 3.06536

943 061 643 745 044

0.90 0.92 OI94 0.96 0.98

-0.05055 -0.04013 -0.02985 -0.01972 -0.00977

61975 35759 32587 89163 36558

0.52445 0.51768 0.51022 0.50209 0.49332

403 968 310 689 478

-0.32034 -0.35587 -0.39005 -0.42282 -0.45413

003 378 463 627 732

-1,80951 -1.74335 -1.67426 -1,60247 -1.52824

008 486 103 436 456

3.23025 3.38325 3.52407 3.65250 3.76836

923 538 854 673 628

-0.48394 145

-1.45182

435

3.87153 159

1.00

0.00000 00000

0,48394 145

[ P (- x)= 1-P ( x )

(-4)3 6

Z( -.)=Z(x)

]

[‘“‘3

ziq-+(

-

421

968

PROBABILITY FUNCTIONS

NORMAL PROBABILITY FUNCTION AND DERIVATIVES

Table 26.1

1.00 1.02 1.04 1.06 1.08

0.84134 0.84613 0.85083 0.85542 0.85992

47460 57696 00496 77003 89099

68543 27265 69019 36091 11231

0.24197 0.23713 0.23229 0.22746 0.22265

Z(4 07245 19520 70047 96324 34987

19143 19380 43366 57386 51761

-0.24197 -0.24187 -0.24158 -0.24111 -0.24046

07245 45910 88849 78104 57786

19143 59767 33101 04829 51902

1.10 1.12 ~.~ 1.14 1.16 1.18

0.86433 0.86864 OI87285 0.87697 0.88099

39390 31189 68494 55969 98925

53618 57270 37202 48657 44800

0.21785 0.21306 OI20830 0.20357 0.19886

21770 91467 77900 13882 31193

32551 75718 47108 90759 87276

-0.23963 -0.23863 -0.23747 -0.23614 -0.23465

73947 74443 08806 28104 84808

35806 88804 53704 17281 76986

1.20 1.22 1.24 1.26 1.28

0.88493 0.88876 0.89251 0.89616 0.89972

03297 75625 23029 53188 74320

78292 52166 25413 78700 45558

0.19418 0.18954 0.18493 0.18037 0.17584

60549 31580 72809 11632 74302

83213 91640 63305 27080 97662

-0.23302 -0.23124 -0.22932 -0.22726 -0.22508

32659 26528 22283 76656 47107

79856 71801 94499 66121 81008

1.30 1.32 1.34 1.36 1.38

0.90319 0.90658 0.90987 0.91308 0.91620

95154 24910 73275 50380 66775

14390 06528 35548 52915 84986

0.17136 0.16693 0116255 0.15822 0.15394

85920 70417 50552 47903 82867

47807 41714 25534 70383 62634

-0.22277 -0.22035 -0.21782 -0.21518 -0.21244

91696 68950 37740 57149 86357

62150 99062 02216 03721 32434

1.40 1.42 1.44 1.46 1.48

0.91924 OI92219 0.92506 0.92785 0.93056

33407 61594 63004 49630 33766

66229 73454 65673 34106 66669

0.14972 oIi4556 0.14145 0.13741 0.13343

74656 41300 99652 65392 53039

35745 37348 24839 82282 51002

-0.20961 -0I20670 -0.20370 -0.20062 -0.19748

84518 10646 23499 81473 42498

90043 53034 23768 52131 47483

1.50 1.52 1.54 1.56 1.58

0.93319 0.93574 0.93821 0.94062 0.94294

27987 45121 98232 00594 65667

31142 81064 88188 05207 62246

0.12951 0.12566 0.12187 0.11815 0.11450

75956 46367 75370 72950 48002

65892 89088 32402 59582 59292

-0.19427 -0.19101 -0.18769 -0.18432 -0.18091

63934 02479 14070 53802 75844

98838 19414 29899 92948 09682

1.60 1.62 1.64 1.66 1.68

0.94520 0.94738 0.94949 0.95154 0.95352

07083 00442 38615,45748 74165 25897 27737 33277 13421 36280

0.11092 0.10740 0.10396 0.10058 0.09728

08346 60751 10953 63684 22693

79456 13484 28764 27691 31467

-0.17747 -0.17399 -0.17049 -0.16697 -0.16343

33354 78416 61963 33715 42124

87129 83844 39173 89966 76865

1.70 1.72 1.74 1.76 1.78

0.95543 Oi95728 0.95907 0.96079 0.96246

45372 37792 04910 60967 20196

41457 08671 21193 12518 51483

0.09404 0;09088 0.08779 0.08477 0.08182

90773 69790 60706 63613 77759

76887 16283 10906 08022 92143

-0.15988 -0Ii5632 -0.15276 -0.14920 -0.14565

34315 56039 51628 63959 34412

40708 08007 62976 02119 66014

1.80 1.82 1.84 1.86 1.88

0.96406 0.96562 0.96711 0.96855 0.96994

96808 04975 58813 72370 59610

87074 54110 40836 19248 38800

0.07895 0.07614 0.07340 0.07074 0.06814

01583 32736 68125 03934 35661

00894 96207 81657 56983 01045

-0.14211 -0.13858 -0.13506 -0.13157 -0.12810

02849 07581 85351 71318 99042

41609 27097 50249 29989 69964

1.90 1.92 1.94 1.96 1.98

0.97128 0.97257 0.97381 0.97500 0.97614

34401 10502 01550 21048 82356

83998 96163 59548 51780 58492

0.06561 0.06315 0.06076 0.05844 0.05618

58147 65614 51689 09443 31419

74677 35199 54565 33451 03868

-0.12467 -0.12126 -0.11788 -0.11454 -0.11124

00480 05979 44277 42508 26209

71886 55581 71856 93565 69659

2.00

0.97724 98680 51821

P(X)

X

Z(') (x)

0.05399 09665 13188

-0.10798 19330 26376

";a"]

969

PROBABILITY FUNCTIONS

Table 26.1

NORMAL PROBABILITY FUNCTION AND DERIVATIVES X

2 3 5 ) (XI

Z ( 6 )(x)

1.00 1.02 1.04 1.06 1.08

0.00000 0.00958 0.01895 0.02811 0.03704

00000 01309 54356 52466 95422

0.48394 0.47397 0.46346 0.45243 0.44091

145 745 412 346 805

-0.48394 -0.51219 -0.53886 -0.56392 -0.58734

145 739 899 521 012

-1.45182 -1.37346 -1.29343 -1.21197 -1.12934

435 846 272 312 487

3.87153 3.96192 4.03951 4.10431 4.15639

159 478 497 754 308

1.10 1.12 1.14 1.16 1.18

0.04574 0.05420 0.06240 0.07035 0.07803

89572 47909 90139 42718 38880

0.42895 0.41656 0.40379 0.39067 0.37723

094 552 549 467 697

-0.60909 -0.62916 -0.64755 -0.66424 -0.67924

290 776 390 543 129

-1.04580 -0.96159 -0.87697 -0.79217 -0.70744

155 420 050 397 317

4.19584 4.22282 4.23751 4.24014 4.23098

622 430 585 894 941

1.20 1.22 1.24 1.26 1.28

0.08544 0.09257 0.09942 0.10598 0.11226

18642 28784 22822 60955 09995

0.36351 0.34954 0.33536 0.32099 0.30647

629 639 083 285 534

-0.69254 -0.70416 -0.71411 -0.72240 -0.72907

515 524 427 928 143

-0.62301 -0.53910 -0.45594 -0.37373 -0.29268

100 399 161 571 993

4.21033 4.17853 4.13593 4.08295 4.02000

894 305 896 339 029

1.30 1.32 1.34 1.36 1.38

0.11824 0.12393 0.12932 0.13442 0.13923

43285 40598 88019 77819 08305

0.29184 0.27712 0.26234 0.24754 0.23275

071 083 695 965 873

-0.73412 -0.73760 -0.73953 -0.73995 -0.73889

591 168 132 087 953

-0.21299 -0.13484 -0.05841 +0.01613 0.08864

916 911 584 459 645

3.94752 3.86600 3.77593 3.67781 3.57216

847 921 384 128 556

1.40 1.42 1.44 1.46 1.48

0.14373 0.14795 0.15187 0.15550 0.15884

83670 13818 14187 05559 13858

0.21800 0.20331 0.18870 0.17422 0.15988

319 117 986 548 325

-0.73641 -0.73255 -0.72735 -0.72087 -0.71315

957 600 645 087 137

0.15897 0.22698 0.29255 0.35556 0.41593

463 486 386 954 103

3.45953 3.34046 3.21550 3.08522 2.95017

335 152 469 283 891

1.50 1.52 1.54 1.56 1.58

0.16189 0.16467 0.16716 0.16939 0.17134

69946 09400 72298 02982 49831

0.14570 0.13172 0.11794 0.10440 0.09111

730 067 528 190 010

-0.70425 -0.69422 -0.68313 -0.67103 -0.65798

193 823 742 785 890

0.47354 0.52834 0.58025 0.62921 0.67518

871 425 051 147 208

2.81093 2.66805 2.52210 2.37361 2.22315

657 791 132 937 681

1.60 1.62 1.64 1.66 1.68

0.17303 0.17447 0.17565 0.17658 0.17728

65021 04284 26667 94284 72076

0.07808 0.06535 0.05292 0.04080 0.02902

827 359 202 829 592

-0.64405 -0.62928 -0.61375 -0.59751 -0.58062

073 410 011 005 516

0.71812 0.75802 0.79486 0.82863 0.85934

810 588 211 352 661

2.07124 1.91841 1.76517 1.61201 1.45942

871 857 671 862 351

1.70 1.72 1.74 1.76 1.78

0.17775 0.17799 0.17801 0.17782 0.17743

27562 30597 53128 68955 53495

0.01758 +0.00650 -0.00421 -0.01456 -0.02452

718 315 632 254 804

-0.56315 -0.54516 -0.52670 -0.50785 -0.48864

647 459 954 061 614

0.88701 0.91167 0.93333 0.95206 0.96790

729 051 988 725 228

1.30785 1.15774 1.00953 0.86361 0.72036

296 966 633 469 463

1.80 1.82 1.84 1.86 1.88

0.17684 0.17607 0.17511 0.17399 0.17270

83546 37061 92921 30717 30539

-0.03410 -0.04329 -0.05208 -0.06047 -0.06846

647 263 243 285 193

-0.46915 -0.44942 -0.42952 -0.40949 -0.38940

342 853 621 971 073

0.98090 0.99113 0.99865 1.00356 1.00592

203 045 794 087 110

0.58014 0.44328 0.31010 0.18087 +0.05587

345 526 045 536 197

1.90 1.92 1.94 1.96 1.98

0.17125 0.16966 0.16793 0.16606 0.16407

72766 37866 06209 57874 72476

-0.07604 -0.08323 -0.09001 -0.09640 -0.10238

873 327 655 044 771

-0.36927 -0.34918 -0.32915 -0.30925 -0.28950

924 347 976 250 408

1.00582 1.00336 0.99863 0.99173 0.98276

548 537 613 666 891

-0.06467 -0.18054 -0.29155 -0.39754 -0.49836

219 414 530 137 204

2.00

0.16197 28995

P(-5)

-0.10798 193

= 1- P ( z )

0.97183 740

-0.26995 483

['-:"I

Z(-5) = Z ( x )

-0.59390 063

[);-( z

'I

970

PROBABILITY FUNCTIONS

Table 26.1

NORMAL PROBABILITY FUNCTION AND DERIVATIVES

2.00 2.02 2.04 2.06 2.08

0.97724 0.97830 0.97932 0.98030 0.98123

98680 51821 83062 32353 48371 33930 07295 90623 72335 65062

0.05399 0.05186 0.04980 0.04779 0.04586

09665 13188 35766 82821 00877 35071 95748 82077 10762 71055

-0.10798 -0.10476 -0.10159 -0.09846 -0.09539

19330 26376 44248 99298 21789 79544 71242 57079 10386 43794

2.10 2.12 2.14 2.16 2.18

0.98213 0.98299 0.98382 0.98461 0.98537

55794 69773 26166 36652 12692

37184 52367 27834 16075 24011

0.04398 0.04216 0.04040 0.03870 0.03706

35959 61069 75539 68561 29102

80427 61770 22860 47456 47806

-0.09236 -0.08939 -0.08647 -0.08360 -0.08079

55515 21467 21653 68092 71443

58897 58953 94921 78504 40218

2.20 2.22 2.24 2.26 2.28

0.98609 0.98679 0.98745 0.98808 0.98869

65524 86502 06161 92744 45385 64054 93745 81453 61557 61447

0.03547 0.03394 0.03246 0.03103 0.02965

45928 07631 02656 19322 45848

46231 82449 43697 15008 47341

-0.07804 -0.07534 -0.07271 -0.07013 -0.06761

41042 84942 09950 21668 24534

61709 65037 41882 05919 51938

2.30 2.32 2.34 2.36 2.38

0.98927 0.98982 0.99035 0.99086 0.99134

58899 78324 95613 31281 81300 54642 25324 69428 36809 74484

0.02832 0.02704 0.02581 0.02463 0.02349

70377 41601 80995 46882 65754 71588 12693 06382 09853 58201

-0.06515 -0.06275 -0.06041 -0.05812 -0.05590

21868 15909 07866 97955 85451

05683 48766 03515 63063 52519

2.40 2.42 2.44 2.46 2.46

0.99180 0.99223 0.99265 0.99305 0.99343

24640 75404 97464 49447 63690 44652 31492 11376 08808 64453

0.02239 0.02134 0.02032 0.01935 0.01842

45302 94843 07148 99923 83557 38226 62767 31737 33106 46862

-0.05374 -0I05164 -0.04960 -0.04761 -0.04568

68727 07623 45300 57813 11880 01271 64407 60073 98104 04218

2.50 2.52 2.54 2.56 2.58

0.99379 0.99413 0.99445 0.99476 0.99505

03346 74224 22582 84668 73765 56918 63918 36444 99842 42230

0.01752 0.01667 0.01584 0.01505 0.01430

83004 01008 75790 96163 51089

93569 37381 25361 27377 94150

-0.04382 -0.04200 -0.04025 -0.03855 -0.03690

07512 33921 86541 10200 28507 24416 26177 98086 71812 04906

2.60 2.62 2.64 2.66 2.68

0.99533 88119 0199560 35116 0.99585 46986 0.99609 29674 0.99631 88919

76281 51879 38964 25147 90825

0.01358 0.01289 0.01223 0.01160 0.01099

29692 21261 15263 01351 69366

33686 07895 51278 13703 29406

-0.03531 57200 -0103377 73704 -0.03229 12295 -0.03085 63594 -0.02947 17901

07583 02686 67374 02449 66807

2.70 2.72 2.74 2.76 2.78

0.99653 0.99673 OI99692 0.99710 OI99728

30261 59041 80407 99319 20550

96960 84109 81350 23774 77299

0.01042 0.00987 0.00934 0.00884 0.00836

09348 14423 11537 94751 66383 67612 64543 98237 96891 54653

-0.02813 -0.02684 -0.02560 -0.02441 -0.02326

65239 95383 97891 62141 77358

98941 21723 27258 39135 49935

2.80 2.82 2.84 2.86 2.88

0.99744 0.99759 0.99774 0.99788 0.99801

48696 88175 43233 17949 16241

69572 25811 08458 59596 45106

0.00791 0.00748 0.00707 0.00667 0.00630

54515 28725 11048 93237 67263

82980 25781 86019 39203 96266

-0.02216 -0.02110 -0.02008 -0.01910 -0.01816

32644 32344 17005 22701 19378 76295 28658 94119 33720 21246

2.90 2.92 2.94 2.96 2.98

0.99813 0.99824 0.99835 0.99846 0.99855

41866 99616 98430 71324 89387 65843 18047 88262 87580 82660

0.00595 OI00561 0.00529 0.00499 0.00470

25324 59835 63438 28992 49575

19776 95991 65311 13612 26934

-0.01726 -0.01639 -0.01557 -0.01477 -0.01402

23440 86721 12509 89816 07734

3.00

0.99865 01019 68370

0.00443 18484 11938

-0.01329 55452 35814 ";971

=s'

P(x)

- m

Z(/)dt

17350 00294 64014 72293 30263

97 1

PROBABILITY FUNCTIONS

Table 26.1

NORhIAL PROBABIIJTY FUNCTION AND DERIVATIVES

2.00 2.02 2.04 2.06 2.08

0.16197 0.15976 0.15744 0.15504 0.15255

28995 05616 79574 27011 22841

Z(3)(x) -0.10798 193 -0.11318 748 -0.11800 948 -0.12245 372 -0.12652 667

-0.26995 -0.25064 -0.23160 -0.21287 -0.19448

483 297 454 345 137

0.97183 0.95904 0.94451 0.92833 0.91062

740 873 117 417 795

-0.59390 -0.68406 -0.76878 -0.84800 -0.92169

063 360 007 114 927

2.10 2.12 2.14 2.16 2.18

0.14998 0.14734 0.14464 0.14188 0.13907

40623 52442 28800 38519 48644

-0.13023 -0.13358 -0.13659 -0.13925 -0.14158

543 762 143 550 892

-0.17645 779 -0.15882 997 -0.14162 297 -0.12485 967 -Oil0856 076

0.89150 0.87107 0.84943 0.82671 0.80301

307 003 890 890 811

-0.98986 -1.05251 -1.10968 -1.16141 -1.20777

750 862 436 446 570

2.20 2.22 2.24 2.26 2.28

0.13622 0.13333 0.13041 0.12746 0.12450

24365 28941 23633 67648 18090

-0.14360 -0.14530 -0.14670 -0.14781 -0.14863

115 204 170 055 922

-0.09274 -0.07742 -0.06262 -0.04834 -0.03460

478 816 527 844 801

0.77844 0.75309 0.72708 0.70050 0.67346

311 866 743 969 314

-1.24885 -1.28473 -1.31554 -1.34140 -1.36245

097 823 947 971 589

2.30 2.32 2.34 2.36 2.38

0.12152 0.11853 0.11554 0.11255 0.10957

29919 55915 46652 50482 13521

-0.14919 -0.14949 -0.14955 -0.14937 -0.14896

851 939 294 032 273

-0.02141 -0.00876 +0.00331 0.01484 0.02581

241 819 989 882 724

0.64604 0.61833 0.59044 0.56243 0.53440

257 976 323 808 589

-1.37883 -1.39070 -1.39823 -1.40159 -1.40097

587 730 661 796 220

2.40 2.42 2.44 2.46 2.48

0.10659 0.10363 0.10069 0.09778 0.09488

79642 90478 85430 01675 74192

-0.14834 -0.14751 -0.14650 -0.14530 -0.14394

137 744 207 633 118

0.03622 0.04607 0.05536 0.06411 0.07231

539 505 942 307 187

0.50642 0.47856 0.45090 0.42350 0.39643

453 812 689 717 129

-1.39654 -1.38851 -1.37705 -1.36239 -1.34470

584 010 991 299 892

2.50 2.52 2.54 2.56 2.58

0.09202 0.08919 0.08639 0.08363 0.08091

35776 17075 46618 50852 54185

-0.14241 -0.14074 -0.13893 -0.13700 -0.13494

744 579 674 058 742

0.07997 0.08710 0.09371 0.09981 0.10541

287 428 533 624 808

0.36973 0.34348 0.31771 0.29247 0.26781

759 039 001 277 102

-1.32420 -1.30109 -1.27556 -1.24781 -1.21804

833 199 010 146 284

2.60 2.62 2.64 2.66 2.68

0.07823 0.07560 0.07301 0.07047 0.06798

79028 45843 73197 77809 74610

-0.13278 -0.13052 -0.12818 -0.12575 -0.12326

711 927 326 818 282

0.11053 0.11517 0.11935 0.12308 0.12638

277 293 186 341 196

0.24376 0.22036 0.19764 0.17563 0.15434

323 399 415 084 760

-1.18644 -1.15321 -1.11853 -1.08259 -1.04556

824 833 985 509 139

2.70 2.72 2.74 2.76 2.78

0.06554 0.06315 0.06082 0.05854 0.05631

76800 95904 41838 22966 46165

-0.12070 -0.11809 -0.11543 -0.11274 -0.11001

569 501 869 431 916

0.12926 0.13173 0.13382 0.13554 0.13690

232 965 945 741 942

0.13381 0.11404 0.09506 0.07686 0.05946

449 817 206 640 846

-1.00761 -0.96890 -0.92961 -0.88988 -0.84986

072 932 727 829 942

2.80 2.82 2.84 2.86 2.88

0.05414 0.05202 0.04996 0.04795 0.04600

16888 39229 15987 48727 37850

-0.10727 020 -0.10450 406 -0.10172 706 -0.09894 520 -0,09616 416

0.13793 0.13862 0.13902 0.13911 0.13894

149 969 007 867 142

0.04287 0.02708 +0.01209 -0.00209 -0.01549

262 053 127 857 465

-0.80970 -0.76951 -0.72943 -0.68959 -0.65008

080 553 954 143 248

2.90 2.92 2.94 2.96 2.98

0.04410 0.04226 0.04048 0.03875 0.03707

82652 81389 31340 28865 69473

-0,09338 928 -0.09062 562 -0.08787 791 -0.08515 058 -0.08244 776

0.13850 0.13782 0.13691 0.13578 0.13446

412 240 166 706 347

-0.02810 -0.03993 -0.05100 -0.06132 -0.07091

482 892 863 737 012

-0.61101 -0.57249 -0.53459 -0.49740 -0.46100

661 036 292 627 520

3.00

0.03545 47873

-0.07977 327

0.13295 545

X

Z(2)(5)

Z(4)(x)

[(-65)lI

P( -2)

= 1- P(2)

Z( --x) =Z(x)

Z(5)(x)

-0.07977 327

w21

Z(")(-X) =(-1).Z(.)(2)

Z(6)(x)

-0.42545 745 [(-:)71

972

PROBABILITY FUNCTIONS

Table 26.1

NORMAL PROBABILITY FUNCTION AND DERIVATIVES

3.20

0.99931 28621

3.25 3.30 3.35 3.40 3.45

0.99942 0.99951 0.99959 0.99966 0.99971

3.50 . _. .

0.99976 - . . . . - 73709 .- . - .

3.55 3.60 3.65 3.70

0.99980 0.99984 0.99986 0.99989

73844 08914 88798 22003

3.75 3. 80 3.85 3.90 3.95

0.99991 0.99992 0.99994 0.99995 0.99996

15827 76520 09411 19037 09244

4.00 4.05 4.10 4.15 4.20

0.99996 0.99997 0.99997 0.99998 0.99998

a3288 43912 93425 33762 66543

4.25 4.30 4.35 4.40 4.45

0.99998 0.99999 0.99999 0.99999 0.99999

93115 14601 31931 45875 57065

4.50 4.55 4.60 4.65 4.70

0.99999 0.99999 0.99999 0.99999 0.99999

66023 73177 78875 83403 86992

4.75 4.80 4.85 4.90 4.95

0.99999 0.99999 0.99999 0.99999 0.99999

89829 92067 93827 95208 96289

5.00

0.99999 97133

29750 65759 59422 30707 97067

(-6)1.48671 9515

(-6)-7.43359 76

[(-76)31 Table 26.2 NORMAL PROBABlLITY FUNCTION FOR LARGE ARGUMENTS I'

-log Q ( a )

2'

,r

-log Q ( . E )

5 6 7 8 9

6.54265 9.00586 11.89285 15.20614 18.94746

15 16 17 18 19

50.43522 57.19458 64.38658 72.01140 80.06919

25 26 27 28 29

137.51475 148.60624 160.13139 172.09024 164.48283

20 21 22 23 24

88.56010 97.48422 106.84167 116.63253 126.85686

30 31 32 33 34

197.30921 210.56940 224.26344 238.39135 252.95315

-log

Q(X)

[(-;I 51

[(-32)51

From E. S. Pearson and H. 0. Hartley (editors), Biometrika tables for statisticians, vol. I. Cambridge Univ. Press, Cambridge, England, 1954(with permission). Known error has been corrected.

973

PROBABILITY FUNCTIONS

NORMAL PROBABILITY FUNCTION AND DERIVATIVES

Table 26.1

2 (2)(x)

2(6)(x)

X 3.00 3.05 3.10 3.15 3.20

-2 3.54547 -2 3.16305 1-21 2.81273 -2 2.49317 -2)2.20289

87 (-2)-7.97732 71 50 -2 -7.32336 28 12 1-21 -6.69403 89 71 -2 -6.09312 50 75 (-21-5.52345 55

3.25 3.30 3.35 3.40 3.45

(-2)1.94027 -2 1.70362 1-211.49118 2 1.30122 {12{1.13198

72 07 76 34 62

(-2)-4.98701 (-21-4I48505 (-2)-4,01812 2 -3.58625 (121-3.18893

97 (-1)1.03869 27 -2 9i68981 87 1-218.98716 07 -2 8.28958 82 (-217.60587

82 (-1) -1.38096 - - -. 20 -1.40361 85 \Ill-1.40345 19 1 -1.38395 84 1111-1.34845

14 -

( - 2 ) -7.05366 66

-1

69 (-2 -2.12970 34 00 1-2 +2.07973 - - - - - - 11 -76 -2 5.60664 85 27 1-21 8.49222 78

03 (-2)-2.82531 02 69 2 -2.49416 18 71 (121-2.19403 56 46 2 -1.92328 53 16 [12{-1.68013 34 3.75 3.80 3.85 3.90 3.95

(-3)4.60578 (-3 3.92316 (-313.33297 3 2,82289 11312,38395

I

-9.53712 78 -1) 1.52468 79 -8.77684 95 [-1 1.51237 96 -8.02840 11 (-1 1.47814 11

11

I

67 22 42 17 27 05 65 82 86

(-2 -5.94206 (-21-5.31924 (-2 -4.73847 -2 -4.20116 1-2 -3.70770

i

20 82 30 64 95

2 -1.60082 16 95 60 [121-1.37210 59 19 (-2 -1.17154 20 97 1-31-9.96506 67 43 -3 -8.44460 51 4.75 4.80 4.85 4.90 4.95

-4)1.08448 1-51 8.73070 -5 7.00939 -5 5.61204 (-5 4.48098

75 -4 -4.67351 25 32 1-41 -3.81045 28 74 -4 -3.09767 67 87 -4 -2.51088 57 88 -4 -2.02933 60

5.00

(-5)3.56812 68 (-4)-1.63539 15 (-4)7.10651 93 (-3)-2.89910 31

(-2) 1.09422 56

NORMAL PROBABILITY FUNCTION FOR LARGE ARGUMENTS

Table 2 6 . 2

r

-log Q ( x )

L

35 36 37 38 39

267.94888 283.37855 299.24218 315.53979 332.27139

45 46 47 48 49

40 41 42 43 44

349.4 3701 367.03664 385.07032 403.53804 422.43983

50 60 70 80 90

-log Q ( r ) 441.77568 461.54561 481.74964 502.38776 523.45999 544.96634 783.90743 1066.26576 1392.04459 1761.24604

r 100 is0 200 250

300

-log Q ( X ) 2173.87154 4888.38812 8688,58977 13574.49960 19546.12790

350 400 450 500

26603,48018 34746.55970 43975.36860 54289.90830

974

PROBABILITY FUNCTIONS

HIGHER DERIVATIVES OF THE NORMAL PROBABILITY FUNCTION

Tahle 26.3

ZCV(1)

I

0.0 0.00000 00 0.1 ( 0) 4.12640 51 0.2 ( 0) 7.88604 35

0.3 0.4

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

2(9)

[

1.40908 1.39704 1.27812 1) 1.06929 0) 7.94982

0 65 30 0 14 ( 1 1 -2.46111 11 69 1 -2.97666 59 72

0 4.83941 03+1.65937 0 -1.31434 0 -3.85379 01-5.79719

45 85 07 20 45

[ [

1

[ [( [

1 -3.19401 11-3.11962 1 -2.78951 1 -2.26227 11-1.61006

36 40 64 70 61

0)-9.09001 66 0)-2.30231 38 ( 0 +3.67230 04 01 8.41240 57 1 1.16856

03 44 07 26 49

0 -7.05769 71

01-7.62276 0 -7.54545 01-6.92967 0 -5.91207

[

I

I [

I [

/

Z(lC ( I )

(r)

39 42 ( 56 ( 42 37

[

?:

1.9

1,

Z(8J (I ) 4.18889 4.00211 3.46206 2.62702 1.58584

1) 1) ( 1 1 1.09518 61 ( 11 11 1.30711 60 ( 1)

-1.14961 -1.07710 -9.05305 1 -6.58548 11-3.68086

02 05 52 60 24

0 -6.77518 03

11+2.10408 1 4.39889 1 6.02399 l / 6.89184

36 22 37 82

1 [ [

1

Z(IU(T)

Z(12)( 1 . )

131 1.01729 46 (3 1.98042 89 3 1.14847 09 (2)+6.22581 20 11+1.72666 2 1.25426 2 2.14046 2 2.74183 2) 3.01027

73 91 31 89 69 (

2 2.94236 21 2.57621 2 1.98269 2 1.25293 dr4.84200

40 2)-2.26484 60 3 -3.01011 58 24 i2)-4.93791 72 131-2.29066 27 77 2)-6.77812 94 3 -1.36759 19 01 2 -7.65280 28 2 -3.83358 74 76 121-7.56972 92 [2/+5.27141 25

[ [

(3) 1.60633 92 (3) 1.19573 79 2.76469 29 +2.51533 48

/

( 1 -2.50848 12 ( 11-2.36048 69

3.0 ( 0 1.75501 20 ( 0 -2.28683 38 3.1 ( 01 1.49720 05 ( 0)-2.80440 64 0 -2.96904 52 1.20591 21 9.12450 33 03-2.86200 69 6.39748 51 0 -2.56761 03

1

11-2.04053 83 1 -1.61917 24 1 -1.16080 0 1

1 [

+3.85905 2.45855 11 3.82142 1 4.49758 1 4.58182

05 2 2.41200 50 73 I21 1.72126 20 44 25 18

0 -7.17959 44 [ -101-3.28394 42 -1.46351 84

1

f 1)-2.53474 56

Ojr2.14502 0 0 0 3.61188 70

02 27

[:)

+4.35697 68 8.87625 64

1 5.88418 05 1) 3.23557 28

(0) 2.21617 27

5.0 (-2)-3.73166 60 (-1) 1.09987 51 (-1)-2.51404 27 (-1) 2.67133 76 ( 0 ) 1.17837 39 (0)-8.83034 08 Z ( ,-) =-

1

42;

P-W

(1 n

Z(n)( r ) =- - Z ( I ) rl I 7b

110,~ ( I ) = (- 1)“z(n)( I )/ Z (I )

Z(70(-

= (-

1)n Z ( 4( I - )

975

PROBABILITY FUNCTIONS

NORMAL PROBABILITY FUIVCTION-VALUES

OF Z(x) IK TERMS OF P ( x ) AND

Q(x)

Table 26.1

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0,010

0.00000 0.02665 0.04842 0.06804 0.08617

0.00337 0.02896 0.05046 0.06992 0.08792

0.00634 0.03123 0.05249 0.07177 0.08965

0.00915 0.03348 0.05449 0.07362 0.09137

0.01185 0.03569 0.05648 0.07545 0.09309

0.01446 0.03787 0.05845 0.07727 0.09479

0.01700 0.04003 0.06040 0.07908 0.09648

0.01949 0.04216 0.06233 0.08087 0.09816

0.02192 0.04427 0.06425 0.08265 0.09983

0.02431 0.04635 0.06615 0.08442 0.10149

0.02665 0.04842 0.06804 0.08617 0.10314

0.99 0.98 0.97 0.96 0.95

0.09

0.10314 0.11912 0.13427 0.14867 0.16239

0.10478 0.12067 0.13574 0,15007 0,16373

0.10641 0.12222 0.13720 0.15146 0.16506

0.10803 0.12375 0.13866 0.15285 0.16639

0.10964 0.12528 0.14011 0.15423 0.16770

0.11124 0.12679 0.14156 0.15561 0.16902

0.11284 0.12830 0.14299 0.15698 0.17033

0.11442 0.12981 0.14442 0.15834 0.17163

0.11600 0.13133 0.14584 0.15970 0.17292

0.11756 0.13279 0.14726 0.16105 0.17421

0.11912 0.13427 0.14867 0.16239 0.17550

0.94 0.93 0.92 0.91 0.90

0.10 0.11 0.12 0.13 0.14

0.17550 0.18804 0.20004 0.21155 0.22258

0.17678 0.18926 0.20121 0.21267 0.22365

0.17805 0.19048 0.20238 0.21379 0.22473

0.17932 0.19169 0.20354 0.21490 0.22580

0.18057 0.19293 0.20470 0.21601 0.22686

0.18184 0.19410 0.20585 0.21712 0.22792

0.18309 0.19530 0.20700 0.21822 0.22898

0.18433 0.19649 0.20814 0.21932 0.23003

0.18557 0.19768 0.20928 0.22041 0.23108

0.18681 0.19886 0.21042 0.22149 0.23212

0.18804 0.20004 0.21155 0.22258 0.23316

0.89 0.88 0.87 0.86 0.85

0.15 Oil6 0.17 0.18 0.19

0.23316 0.24331 0.25305 0.26240 0.27137

0.23419 0.24430 0.25401 0.26331 0.27224

0.23522 0.24529 0.25495 0.26422 0.27311

0.23625 0.24628 0.25590 0.26513 0.27398

0.23727 0.24726 0.25684 0.26603 0.27485

0.23829 0.24823 0.25778 0.26693 0.27571

0.23930 0.24921 0.25871 0.26782 0.27657

0.24031 0.25017 0.25964 0.26871 0.27742

0.24131 0.25114 0.26056 0.26960 0.27827

0.24232 0.25210 0.26148 0.27049 0.27912

0.24331 0.25305 0.26240 0.27137 0.27996

0.84 0.83 0.82 0.81 0.80

0.20 0.21 0.22 0.23 0.24

0.27996 0.28820 0.29609 0.30365 0.31087

0.28080 0.28901 0.29686 0.30439 0.31158

0.28164 0.28981 0.29763 0.30512 0.31228

0.28247 0.29060 0.29840 0.30585 0.31298

0.28330 0.29140 0.29916 0.30658 0.31367

0.28413 0.29219 0.29991 0.30730 0.31436

0.28495 0.29295 0.30067 0.30802 0.31505

0.28577 0.29376 0.30142 0.30874 0.31574

0.28658 0.29454 0.30216 0.30945 0.31642

0.28739 0.29532 0.30291 0.31016 0.31710

0.28820 0.29609 0.30365 0.31087 0.31778

0.79 0.78 0.77 0.76 0.75

0.25 0.26 0.27 0.28 0.29

0.31778 0.32437 0.33065 0.33662 0.34230

0.31845 0.32501 0.33126 0.33720 0.34285

0.31912 0.32565 0.33187 0.33778 0.34341

0.31979 0.32628 0.33247 0.33836 0.34395

0.32045 0.32691 0.33307 0.33893 0.34449

0.32111 0.32754 0.33367 0.33950 0.34503

0.32177 0.32817 0.33427 0.34007 0.34557

0.32242 0.32879 0.33486 0.34063 0.34611

0.32307 0.32941 0.33545 0.34119 0.34664

0.32372 0.33003 0.33604 0.34175 0.34717

0.32437 0.33065 0.33662 0.34230 0.34769

0.74 0.73 0.72 0.71 0.70

0.30 0.31 0.32 0.33 0.34

0.34769 0.35279 0.35761 0.36215 0.36641

0.34822 0.35329 0.35808 0.36259 0.36682

0.34874 0.35378 0.35854 0.36302 0.36723

0.34925 0.35427 0.35900 0.36346 0.36764

0.34977 0.35475 0.35946 0.36389 0.36804

0.35028 0.35524 0.35991 0.36431 0.36844

0.35079 0.35572 0.36037 0.36474 0.36884

0.35129 0.35620 0.36082 0.36516 0.36923

0.35180 0.35667 0.36126 0.36558 0.36962

0.35230 0.35714 0.36171 0.36600 0.37001

0.35279 0.35761 0.36215 0.36641 0.37040

0.69 0.68 0.67 0.66 0.65

0.35 0.36 0.37 0.38 0.39

0.37040 0.37412 0.37757 0.38076 0.38368

0.37078 0.37447 0.37790 0.38106 0.38396

0.37116 0.37483 0.37823 0.38136 0.38423

0.37154 0.37518 0.37855 0.38166 0.38451

0.37192 0.37553 0.37888 0.38196 0.38478

0.37229 0.37588 0.37920 0.38225 0.38504

0.37266 0.37622 0.37951 0.38254 0.38531

0.37303 0.37656 0.37983 0.38283 0.38557

0.37340 0.37693 0.38014 0.38312 0.38583

0.37376 0.37724 0.38045 0.383413 0.38609

0.37412 0.37757 0.38076 0.38368 0.38634

0.64 0.63 0.62 0.61 0.60

0.40 0.41 0.42 0.43 0.44

0.38634 0.38875 0.39089 0.39279 0.39442

0.38659 0.38897 0.39109 0.39296 0.39457

0.38684 0.38920 0.39129 0.39313 0.39472

0.38709 0.38942 0.39149 0.39330 0.39486

0.38734 0.38964 0.39169 0.39347 0.39501

0.38758 0.38985 0.39187 0.39364 0.39514

0.38782 0.39007 0.39206 0.39380 0.39528

0.38805 0.39020 0.39224 0.39396 0.39542

0.38829 0.39049 0.39243 0.39411 0.39555

0.38852 0.39069 0.39261 0.39427 0.39568

0.38875 0.39089 0.39279 0.39442 0.39580

0.59 0.58 0.57 0.56 0.55

0.45 0.46 0.47 0.48 0.49

0.39580 0.39694 0.39781 0.39844 0.39882

0.39593 0.39703 0.39789 0.39849 0.39884

0.39605 0.39713 0.39796 0.39854 0.39686

0.39617 0.39723 0.39803 0.39859 0.39883

0.39629 0.39732 0.39809 0.39862 0.39890

0.39640 0.39741 0.39816 0.39866 0.39891

0.39651 0.39749 0.39322 0.39870 0.39892

0.39662 0.39758 0.39828 0.39873 0.39893

0.39673 0.39766 0.39834 0.39876 0.39894

0.39683 0.39774 0.39839 0.39879 0.39894

0.39694 0.39781 0.39844 0.39882 0.39894

0.54 0.53 0.52 0.51 0.50

0,000

I.(.,,)

0.03 0.01 0.02 0.03 0.04

0.05 0.06 0.07 0.08

0.010 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 Linear interpolation yields an error no greater than 5 units in the fifth decimal place.

Compiled from T. L. Kelley, The Kelley Statistical Tables. Harvard Univ. Press, Cambridge, Mass., 1948 (with permission).

976

PROBABILITY FUNCTIONS

Table 26.5

NORMAL PROBABILITY FUNCTION-VALUES

OF z IN TERMS OF P ( z ) AND Q(x)

0.004

0.005

0.006

0.007

0.008

0.009

0.010

2I22621 1.99539 1.83842 1.71689

2.65207 2I19729 1.97737 1.82501 1.70604

2.57583 2.17009 1.95996 1.81191 1.69540

2.51214 2.14441 1.94313 1.79912 1.68494

2.45726 2.12007 1.92684 1.78661 1.67466

2.40892 2.09693 1.91104 1.77438 1.66456

2.36562 2.07485 1.89570 1.76241 1.65463

2.32635 2.05375 1.88079 1.75069 1.64485

0.99 0.98 0.97 0.96 0.95

1.62576 1.53820 1.46106 1.39174 1.32854

1.61644 1.53007 1.45381 1.38517 1.32251

1.60725 1.52204 1.44663 1.37866 1.31652

1.59819 1.51410 1.43953 1.37220 1.31058

1.58927 1.50626 1.43250 1.36581 1.30469

1.58047 1.49851 1.42554 1.35946 1.29884

1.57179 1.49085 1.41865 1.35317 1.29303

1.56322 1.48328 1.41183 1.34694 1.28727

1.55477 1.47579 1.40507 1.34076 1.28155

0.94 0.93 0.92 0.91 0.90

1.27587 1.22123 1.17000 1.12168 1.07584

1.27024 ._ 1.21596 1.16505 1.11699 1.07138

1.26464 iIZi072 1.16012 1.11232 1.06694

1.25908 ~ _ _ ~1.25357 _ _ ~ 1.24808 ~ ~ 1.24264 1.20553 1.20036 I 3 9 5 2 2 i i i 9 0 1 2 1.15522 1.15035 1.14551 1.14069 1.10768 1.10306 1.09847 1.09390 1.06252 1.05812 1.05374 1.04939

1.23723 i.18504 1.13590 1.08935 1.04505

1.23186 1.22653 i.18000 i.17499 1.13113 1.12639 1.08482 1.08032 1.04073 1.03643

0.89 0.88 0.87 0.86 0.85

1.03643 0.99446 0.95416 0.91537 0.87790

1.03215 0.99036 0.95022 0.91156 0.87422

1.02789 0.98627 0.94629 0.90777 0.87055

1.02365 0.98220 0.94238 0.90399 0.86689

1.01943 0.97815 0.93848 0.90023 0.86325

1.01522 0.97411 0.93458 0.89647 0.85962

1.01103 0.97009 0.93072 0.89273 0.85600

1.00686 0.96609 0.92686 0.88901 0.85239

1.00271 0.96210 0.92301 0.88529 0.84879

0.99858 0.95812 0.91918 0.88159 0.84520

0.99446 0.95416 0.91537 0.87790 0.84162

0.84 0.83 0.82 0.81 0.80

0.20 0.21 0.22 0.23 0.24

0.84162 0.80642 0.77219 0.73885 0.70630

0.83805 0.80296 0.76882 0.73556 0.70309

0.83450 0.79950 0.76546 0.73228 0.69988

0.83095 0.79606 0.76210 0.72900 0.69668

0.82742 0.79262 0.75875 0.72574 0.69349

0.82390 0.78919 0.75542 0.72248 0.69031

0.82038 0.78577 0.75208 0.71923 0.68713

0.81687 0.78237 0.74876 0.71599 0.68396

0.81338 0.77897 0.74545 0.71275 0.68080

0.80990 0.77557 0.74214 0.70952 0.67764

0.80642 0.77219 0.73885 0.70630 0.67449

0.79 0.78 0.77 0.76 0.75

0.25 0.26 0.27 0.28 0.29

0.67449 0.64335 0.61281 0.58284 0.55338

0.67135 0.64027 0.60979 0.57987 0.55047

0.66821 0.63719 0.60678 0.57691 0.54755

0.66508 0.63412 0.60376 0.57395 0.54464

0.66196 0.63106 0.60076 0.57100 0.54174

0.65884 0.62801 0.59776 0.56805 0.53884

0.65573 0.62496 0.59477 0.56511 0.53594

0.65262 0.62191 0.59178 0.56217 0.53305

0.64952 0.61887 0.58879 0.55924 0.53016

0.64643 0.61584 0.58581 0.55631 0.52728

0.64335 0.61281 0.58284 0.55338 0.52440

0.74 0.73 0.72 0.71 0.70

0.30 0.31 0.32 0.33 0.34

0.52440 0.49585 0.46770 0.43991 0.41246

0.52153 0.49302 0.46490 0.43715 0.40974

0.51866 0.49019 0.46211 0.43440 0.40701

0.51579 0.48736 0.45933 0.43164 0.40429

0.51293 0.48454 0.45654 0.42889 0.40157

0.51007 0.48173 0.45376 0.42615 0.39886

0.50722 0.47891 0.45099 0.42340 0.39614

0.50437 0.47610 0.44821 0.42066 0.39343

0.50153 0.47330 0.44544 0.41793 0.39073

0.49869 0.47050 0.44268 0.41519 0.38802

0.49585 0.46770 0.43991 0.41246 0.38532

0.69 0.68 0.67 0.66 0.65

0.35 0.36 0.37 0.38 0.39

0.38532 0.35846 0.33185 0.30548 0.27932

0.38262 0.35579 0.32921 0.30286 0.27671

0.37993 0.35312 0.32656 0.30023 0.27411

0.37723 0.35045 0.32392 0.29761 0.27151

0.37454 0.34779 0.32128 0.29499 0.26891

0.37186 0.34513 0.31864 0.29237 0.26631

0.36917 0.34247 0.31600 0.28976 0.26371

0.36649 0.33981 0.31337 0.28715 0.26112

0.36381 0.33716 0.31074 0.28454 0.25853

0.36113 0.33450 0.30811 0.28193 0.25594

0.35846 0.33185 0.30548 0.27932 0.25335

0.64 0.63 0.62 0.61 0.60

0.40 0.41 0.42 0.43 0.44

0.25335 0.22754 0.20189 0.17637 0.15097

0.25076 0.22497 0.19934 0.17383 0.14843

0.24817 0.22240 0.19678 0.17128 0.14590

0.24559 0.21983 0.19422 0.16874 0.14337

0.24301 0.21727 0.19167 0.16620 0.14084

0.24043 0.21470 0.18912 0.16366 0.13830

0.23785 0.21214 0.18657 0.16112 0.13577

0.23527 0.20957 0.18402 0.15858 0.13324

0.23269 0.20701 0.18147 0.15604 0.13072

0.23012 0.20445 0.17892 0.15351 0.12819

0.22754 0.20189 0.17637 0.15097 0.12566

0.59 0.58 0.57 0.56 0.55

0.45 0.46 0.47 0.48 0.49

0.12566 0.10043 0.07527 0.05015 0.02507

0.12314 0.09791 0.07276 0.04764 0.02256

0.12061 0.09540 0.07024 0.04513 0.02005

0.11809 0.09288 0.06773 0.04263 0.01755

0.11556 0.09036 0.06522 0.04012 0.01504

0.11301 0.08784 0.06271 0.03761 0.01253

0.11052 0.08533 0.06020 0.03510 0.01003

0.10799 0.08281 0.05768 0.03259 0.00752

0.10547 0.08030 0.05517 0.03008 0.00501

0.10295 0.07778 0.05266 0.02758 0.00251

0.10043 0.07527 0.05015 0.02507 0.00000

0.54 0.53 0.52 0.51 0.50

0 h) 0.000

0.001 0.002 0.003 3.09023 _._ .-_2.87816 - _ _--. 2.74778

0.00

m

0.01 0.02 0.03 0.04

2.32635 2.05375 1.88079 1.75069

2.29037 2.03352 1.86630 1.73920

2.25713 2.01409 1.85218 1.72793

0.05 0.06 0.07 0.08 0.09

1.64485 1.55477 1.47579 1.40507 1.34076

1.63523 1.54643 1.46838 1.39838 1.33462

0.10 0.11 0.12 0.13 0.14

1.28155 1,22653 1.17499 1.12639 1.08032

0.15 0.16 0.17 0.18 0.19

~

~

~

0.010 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.000 P ( z ) For Q(r)>0.007,linear interpolation yields an error of one unit in the third decimal place; five-point interpolation is necessary to obtain full accuracy.

Compiled from T. L. Kelley, The Kelley Statistical Tables. Harvard Univ. Press, Cambridge, Mass., 1948 (with permission).

977

PROBABILITY FUNCTIONS NORMAL PROBABILITY FUNCTION-VALUES

OF

z

FOR EXTREME VALUES O F P ( z ) AND Q(z)

Tnhle 26.6

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.0010

m

3.09023 2.87816 2.74778 2.65207

3.71902 3.06181 2.86274 2.73701 2.64372

3.54008 3.03567 2.84796 2.72655 2.63555

3.43161 3.01145 2.83379 2.71638 2.62756

3.35279 2.98888 2.82016 2.70648 2.61973

3.2 9 0 5 3 2.96774 2.80703 2.69684 2.61205

3.23888 2.94784 2.79438 2.68745 2.60453

3.19465 2.92905 2.78215 2.67829 2.59715

3.15591 2.91124 2.77033 2.66934 2.58991

3.12139 2.89430 2.75888 2.66061 2.58281

3.09023 2.87816 2.74778 2.65207 2.57583

0.999 0.998 0.997 0.996 0.995

0.005 0.006 0.007 3.008 0.009

2.57583 2.51214 2.45726 2.40891 2.36562

2.56897 2.50631 2.45216 2.40437 2.36152

2.56224 2.50055 2.44713 2.39989 2.35747

2.55562 2.49489 2.44215 2.39545 2.35345

2.54910 2.48929 2.43724 2.39106 2.34947

2.54270 2.48377 2.43238 2.38671 2.34553

2.53640 2.47833 2.42758 2.38240 2.34162

2.53019 2.47296 2.42283 2.37814 2.33775

2.52408 2.46765 2.41814 2.37392 2.33392

2.51807 2.46243 2.41350 2.36975 2.33012

2.51214 2.45726 2.40891 2.36562 2.32635

0.994 0.993 0.992 0.991 0.990

0.010 0.011 0.012 0.013 0.014

2.32635 2.29037 2.25713 2.22621 2.19729

2.32261 2.28693 2.25394 2.22323 2.19449

2.31891 2.28352 2.25077 2.22028 2.19172

2.31524 2.28013 2.24763 2.21734 2.18896

2.31160 2.27677 2.24450 2.21442 2.18621

2.30798 2.27343 2.24140 2.21152 2.18349

2.30440 2.27017 2.23832 2.20864 2.18078

2.30085 2.26684 2.23526 2.20577 2.17808

2.29733 2.26358 2.23223 2.20293 2.17540

2.29383 2.26034 2.22921 2.20010 2.17274

2.29037 2.25713 2.22621 2.19729 2.17009

0.989 0.988 0.987 0.986 0.935

0.015 0.016 0.017 0.018 0.019

2.17009 2.14441 2.12007 2.09693 2.07485

2.16746 2.14192 2.11771 2.09467 2.07270

2.16484 2.13944 2.11535 2.09243 2.07056

2.16224 2.13698 2.11301 2.09020 2.06843

2.15965 2.13452 2.11068 2.08798 2.06630

2.15707 2.13208 2.10836 2.08576 2.06419

2.15451 2.12966 2.10605 2.08356 2.06208

2.15197 2.12724 2.10375 2.08137 2.05998

2.14943 2.12484 2.10147 2.07919 2.05790

2.14692 2.12245 2.09919 2.07702 2.05582

2.14441 2.12007 2.09693 2.07485 2.05375

0.984 0.983 0.932 0.981 0.980

0.020 0.021 0.022 0.023 0.024

2.05375 2.03352 2.01409 1.99539 1.97737

2.05169 2.03154 2.01219 1.99356 1.97560

2.04964 2.02957 2.01029 1.99174 1.97384

2.04759 2.02761 2.00841 1.98992 1.97208

2.04556 2.02566 2.00653 1.98811 1.97033

2.04353 2.02371 2.00465 1.98630 1.96859

2.04151 2.02177 2.00279 1.98450 1.96685

2.03950 2.01984 2.00093 1.98271 1.96512

2.03750 2.01792 1.99908 1.98092 1.96340

2.03551 2.01600 1.99723 1.97914 1.96168

2.03352 2.01409 1.99539 1.97737 1.95996

0.979 0.978 0.977 0.976 0.975

($(.I,)

0.000 0.001 0.002 0.003 0.004

0.0010 0.0009 0.0008 0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0.0000 P(t) For 0( I )>0.0007, linear interpolation yields an error of one unit in the third decimal place; five-point interpolation is necessary to obtain full accuracy.

(2 (,?)

I

Q(7)

I

‘I)(!)

I

0 (4

I

( -4) 1. 0

3.71902

( - 9) 1.0

5.99781

( -14) 1.0

7.65063

( -19) 1.0

9.01327

( - 5 ) 1.0

4.26489

( -10) 1.0

6.36134

( -15) 1. 0

7.94135

( -20) 1.0

9.26234

( -6) 1.0

4.75342

(-11) 1.0

6.70602

( -16) 1. 0

8.22208

(-21) 1.0

9.50502 9.74179 9.97305

( -7) 1.0

5.19934

(-12) 1.0

7.03448

(-17) 1.0

8.49379

( -22) 1.0

( -8) 1.0

5.61200

( -13) 1. 0

7.34880

( -18) 1.0

8.75729

(-23) 1.0

P(,)=l-((l(,)=J: Z ( / ) r / / ~

Compiled from T. L. Kelley, The Kelley Statistical Tables. Harvard Univ. Press, Cambridge, Mass., 1948 (with permission) for Q(.r)>(-9)1.

978

PROBABILITY FUNCTIONS

Table 26.7

PROBABILITY INTEGRAL OF x~-DISTRIBUTION,INCOMPLETE GAMMA FUNCTION CUMULATIVE SUMS OF THE POISSON DISTRIBUTION

7n=0.0005

0.002 0.0010

0.003 0.0015

0.004 0.0020

0.005 0.0025

0.006 0.0030

0.007 0.0035

0.008 0.0040

0.0045

0.010 0.0050

0.97477 0.99950 0.99999

0.96433 0.99900 0.99998

0,95632 0.99850 0.99996

0.94957 0.99800 0.99993

0.94363 0.99750 0.99991

0.93826 0.99700 0.99988

0.93332 0.99651 0.99984 0.99999

0.92873 0.99601 0.99981 0.99999

0.92442 0.99551 0.99977 0.99999

0.92034 0.99501 0.99973 0.99999

0.02 0.010

0.03 0.015

0.04 0.020

0.05 0.025

0.06 0.030

0.86249 0.98511 0.99863 0.99989 0.99999

0.08 0.040

0.09 0.045

0.88754 0.99005 0.99925 0.99995

0.10 0.050

0.84148 0.98020 0.99790 0.99980 0.99998

0.82306 0.97531 0.99707 0.99969 0.99997

0.07 0.035 0.79134 0.96561 0.99518 0.99940 0.99993

0,77730 0.96079 0.99412 0.99922 0.99991

0.76418 0.95600 0.99301 0.99902 0.99987

0.75183 0.95123 0.99184 0.99879 0.99984

0.99999

0.99999

0.99999

0.99998

0.9 0.45

0.50

X2=0.001 v

1 2 3 4 V

5

x2=0.01 m=0.005 0.92034 0.99501 0.99973 0.99999

0.80650 0.97045 0.99616 0.99956 0.99995

6 V

x2=0.1 m=0.05

0.2 0.10

0.3 0.15

0.4 0.20

0.5 0.25

0.6 0.30

0.7 0.35

0.8 0.40

0.009

1.0

1 2 3 4 5

0.75183 0.95123 0.99184 0.99879 0.99984

0.65472 0.90484 0.97759 0.99532 0.99911

0.58388 0.86071 0.96003 0.98981 0.99764

0.52709 0.81873 0.94024 0.98248 0.99533

0.47950 0.77880 0.91889 0.97350 0.99212

0.43858 0.74082 0.89643 0.96306 0.98800

0.40278 0.70469 0.87320 0.95133 0.98297

0.37109 0.67032 0.84947 0. 93845 0.97703

0.34278 0.63763 0.82543 0.92456 0.97022

0.31731 0.60653 0.80125 0: 90980 0.96257

6 7 8 9

0.99998

0.99985 0.99997

0.99950 0.99990 0.99998

0.99885 0.99974 0.99994 0.99999

0.99784 0.99945 0.99987 0.99997 0.99999

0.99640 0.99899 0.99973 0.99993 0.99998

0.99449 0.99834 0.99953 0.99987 0.99997

0.99207 0.99744 0.99922 0.99978 0.99994

0.98912 0.99628 0.99880 0.99964 0.99989

0.98561 0.99483 0.99825 0.99944 0.99983

0.99999

0.99998

0.99997 0.99999

0.99995 0.99999

1.7 0.85

1.8 0.90

1.9 0.95

10 11 12 iL= V

1 2 3 4 5 6 7 8 9

10

11 12 13

14 15 16

1.1

m = 0.55

1.2 0.60

1.3 0.65

1.4 0.70

1.5 0.75

1.6 0.80

2.0 1.00

0.29427 0.57695 0.77707 0.89427 0.95410

0.27332 0.54881 0.75300 0.87810 0.94488

0.25421 0.52205 0.72913 0.86138 0.93493

0.23672 0.49659 0.70553 0.84420 0.92431

0.22067 0.47237 0.68227 0.82664 0.91307

0.20590 0.44933 0.65939 0.80879 0.90125

0.19229 0.42741 0.63693 0.79072 0.88890

0.17971 0.40657 0.61493 0.77248 0.87607

0.16808 0.38674 0.59342 0.75414 0.86280

0.15730 0.36788 0.57241 0.73576 0.84915

0.98154 0.99305 0.99753 0.99917 0.99973

0.97689 0.99093 0.99664 0.99882 0.99961

0.97166 0. 98844 0.99555 0.99838 0.99944

0.96586 0: 98557 0.99425 0.99782 0.99921

0.95949 0.98231 0.99271 0.99715 0.99894

0.95258 0: 97864 0.99092 0.99633 0.99859

0.94512 0.97457 0.98887 0.99537 0.99817

0.93714 0.97008 0.98654 0.99425 0.99766

0.92866 0.96517 0.98393 0.99295 0.99705

0.91970 0. 95984 0.98101 0.99147 0.99634

0.99992 0.99998 0.99999

0.99987 0.99996 0.99999

0.99981 0.99994 0.99998 0.99999

0.99973 0.99991 0.99997 0.99999

0.99962 0.99987 0.99996 0.99999

0.99948 0.99982 0.99994 0.99998 0.99999

0.99930 0.99975 0.99991 0.99997 0.99999

0.99908 0.99966 0.99988 0.99996 0.99999

0.99882 0.99954 0.99983 0.99994 0.99998

0.99850 0.99941 0.99977 0.99992 0.99997

0.99999

0.99999

Compiled from E. S. Pearson and H. 0. Hartley (editors), Biometrika tables for statisticians, vol. I. Cambridge Univ. Press, Cambridge, England, 1954 (with permission).

979

PROBABILITY F U N C T I O N S

PROBABILITY INTEGRAL OF xz-DISTRIBUTION, INCOMPLETE GAMMA FUNCTION Table 26.7 CUMULATIVE SUMS OF THE POISSON DISTRIBUTION

x2=2.2 m=l.l

2.4 1.2

2.6 1.3

2.8 1.4

3.0 1.5

3.2 1.6

1 2 3 4 5

0.13801 0.33287 0.53195 0.69903 0.82084

0.12134 0.30119 0.49363 0.66263 0.79147

0.10686 0.27253 0.45749 0.62682 0.76137

0.09426 0.24660 0.42350 0.59183 0.73079

0.08327 0.22313 0.39163 0.55783 0.69999

6 7 8 9 10

0.90042 0.94795 0.97426 0.98790 0.99457

0.87949 0.85711 0.93444 0.91938 0.96623 0.95691 0.98345 0.97807 0.99225 0.98934

0.83350 0.90287 0.94628 0.97170 0.98575

0.80885 0.78336 0.88500 0.86590 0.93436 oI 92Ii9 0.96430 0.95583 0.98142 0.97632

11 12 13 14 15

0.99766 0.99903 0.99961 0.99985 0.99994

0.99652 0.99850 0.99938 0.99975 0.99990

0.99503 0.99777 0.99903 0.99960 0.99984

0.99311 0.99680 0.99856 0.99938 0.99974

16 17 18 19 20

0.99998 0.99999

0.99996 0.99999

0.99994 0.99998 0.99999

0.99989 0.99996 0.99998 0.99999

3.4 1.7

0.07364 0.06520 0.20190 0.18268 0.36181 0.33397 0.52493 0.49325 0.66918 0.63857

4.4 2.2

4.6 2.3

0.75722 0.84570 01 90681 0.94631 0.97039

0.73062 0.70372 0.82452 - - - - - 0.-.m- -2-5- n0.89129 0.87470 0.93572 0.92408 0.96359 0.95592

0.67668 n. - .7. .7 9. 7- ~ 0.85712 0.91141 0.94735

0.99073 0.98781 0.99554 0.99396 0.99793 0.99711 0.99907 0.99866 0.99960 0.99940

0.98431 0.99200 0.99606 0.99813 0.99913

0.98019 0.98962 0.99475 0.99743 0.99878

0.97541 0.98678 0.99314 0.99655 0.99832

0.96992 0.98344 0.99119 0.99547 0.99774

0.99983 0.99974 0.99993 0.99989 0.99997 0.99995 0.99999 0.99998 0.99999

0.99961 0.99983 0.99993 0.99997 0.99999

0.99944 0.99921 0.99975 0.99964 0.99989 0.99984 0.99995 0.99993 0.99998 0.99997

0.99890

5.0 2.5

4.8 2.4

4.0 2.0 0.04550 0.13534 0.26146 0.40601 0.54942

0.99999

x?=4.2 n~=2.1

3.8 1.9

0.05778 0.05125 0.16530 0.14957 0.30802 0.28389 0.46284 0.43375 0.60831 0.57856

21 22

V

3.6 1.8

5.2 2.6

0.99999

5.6 2.8

5.4 2.7

I

0.99948 . . . . .-

0.99976 0.99989 0.99995 0.99998 0.99999

5.8 2.9

6.0

3.0

0.02846 0.02535 0.09072 0.08209 0.18704 0.17180 0.30844 0.28730 0.44077 0.41588

0.02259 0.07427 0.15772 0.26739 0.39196

0.02014 0.06721 0.14474 0.24866 0.36904

0.01796 0.06081 0.13278 0.23108 0.34711

0.01603 0.05502 0.12176 0.21459 0.32617

0.01431 0.04979 0.11161 0.19915 0.30622

0.59604 0.70864 0.79935 0.86769 0.91625

0.56971 0.68435 0.77872 0.85138 0.90413

0.54381 0.65996 0.75758 0.83431 0.89118

0.51843 0.63557 0.73600 0.81654 0.87742

0.49363 0.61127 0.71409 0.79814 0.86291

0.46945 0.58715 0.69194 0.77919 0.84768

0.44596 0.56329 0.66962 0.75976 0.83178

0.42319 0.53975 0.64723 0.73992 0.81526

0.98887 0.99414 0.99701

0.97509 0.98614 0.99254 0.99610

0.94898 0; 97002 0.98298 0.99064 0.99501

0.94046 0; 96433 0.97934 0.98841 0.99369

0.93117 0;95798 0.97519 0.98581 0.99213

0.92109 0: 95096 0.97052 0.98283 0.99029

0.91026 0: 94327 0.96530 0.97943 0.98816

0.89868 0; 93489 0.95951 0.97559 0.98571

0.88637 0.92583 0.95313 0.97128 0.98291

0.87337 0.91608 0.94615 0.96649 0.97975

1'6 17 18 19 20

0.99851 0.99928 0.99966 0.99985 0.99993

0.99802 0.99741 0.99902 0.99869 0.99953 0.99936 0.99978 0.99969 0.99990 0.99986

0.99666 0.99828 0.99914 0.99958 0.99980

0.99575 0.99777 0.99886 0.99943 0.99972

0.99467 0.99715 0.99851 0.99924 0.99962

0.99338 0.99639 0.99809 0.99901 0.99950

0.99187 0.99550 0.99757 0.99872 0.99934

0.99012 0.99443 0.99694 0.99836 0.99914

0.98810 0.99319 0.99620 0.99793 0.99890

21- . 22 23 24 25

0.99997 0.99999 0.99999

0.99995 0.99998 0.99999

0.99993 0.99997 0.99999 0.99999

0.99991 0.99996 0.99998 0.99999

0.99987 0.99994 0.99997 0.99999 0.99999

0.99982 0.99991 0.99996 0.99998 0.99999

0.99975 0.99988 0.99994 0.99997 0.99999

0.99967 0.99984 0.99992 0.99996 0.99998

0.99956 0.99978 0.99989 0.99995 0.99998

0.99943 . ... . 0.99971 0.99986 0.99993 0.99997

0.99999

0.99999 0.99999

0.99998 0.99999

3 4 5

0.04042 0.12246 0.24066 0.37962 0.52099

0.03594 0.03197 0.11080 0.10026 0.22139 0.20354 0.35457 0.33085 0.49337 0.46662

6 7 8 9 10

0.64963 0.75647 0.83864 0.89776 0.93787

0.62271 0.73272 0.81935 0.88317 0.92750

11 12 13 14 15

0.96370 -- -

6: 97955

0-.95672 -- -

26 27

,+=; (x2-x;)

Interpolation on

//J="-"">o

x2

+'I

Q ( X ~ / ~ ) = Q ( X ~ ~ V O - ~ ) [ ~ $ ] + Q ( X ~(xiI~o)[l-++i ~~~-~)[~-~*]+Q

Double Entry Interpolation

Q (x'IV)=Q ( X ~ ~ V ~ - ~ ) [ ~ + ' ] + Q +Q ( x ; l v o ) [ l - u ' z - + + $

(X;'V~-~)[+-+'-W+]+Q

(xi1

[i

~a-l)[;W'-;

,-,+I

+2+lc+]+Q ( ~ ~ ~ ~ ~ + l )

E+W+

1

980

PROBABILITY FUNCTIONS

Table 26.7 PR(JBABILITY INTEGRAL OF X*-DISTRIBUTION, INCOMPLETE GAMMA FUKCTION CUMULATIVE SUMS OF THE POISSON DlSTRIBl'TlOK

x2 = 6.2 m=3.1 0.01278 0.04505 0.10228 0.18470 0.28724

V

6.6 3.3 0.01141 0.01020 0.04076 0.03688 0.09369 0.08580 0.17120 0.15860 0.26922 0.25213

6.8 3.4 0.00912 0.03337 0.07855 0.14684 0.23595

7.0 3.5 0.00815 0.03020 0.07190 0.13589 0.22064

7.2 3.6 0.00729 0.02732 0.06579 0.12569 0.20619

7.4 3.7 0.00652 0.02472 0.06018 0.11620 0.19255

7.6 3.8 0.00584 0.02237 0.05504 0.10738 0.17970

7.8 3.9 0.00522 0.02024 0.05033 0.09919 0.16761

8.0 4.0 0.00468 0.01832 0.04601 0.09158 0.15624

6.4

3.2 - .-

6 7 8 9 10

0.40116 0.51660 0.62484 0.71975 0.79819

0.37990 0.49390 0.60252 0.69931 0.78061

0.35943 0.47168 0.58034 0.67869 0.76259

0.33974 0.45000 0.55836 0.65793 0.74418

0.32085 0.42888 0.53663 0.63712 0.72544

0.30275 0.40836 0.51522 0.61631 0.70644

0.28543 0.38845 0.49415 0.59555 0.68722

0.26890 0.36918 0.47349 0.57490 0.66784

0.25313 0.35056 0.45325 0.55442 0.64837

0.23810 0.33259 0.43347 0.53415 0.62884

11 12 13 14 15

0.85969 0.90567 0.93857 0.96120 0.97619

0.84539 0.89459 0.93038 0.95538 0.97222

0.83049 0.88288 0.92157 0.94903 0.96782

0.81504 0.87054 0.91216 0.94215 0.96296

0.79908 0.85761 0.90215 0.93471 0.95765

0.78266 0.84412 0.89155 0.92673 0.95186

0.76583 0.83009 0.88038 0.91819 0.94559

0.74862 0.81556 0.86865 0.90911 0.93882

0.73110 0.80056 0.85638 0.89948 0.93155

0.71330 0.78513 0.84360 0.88933 0.92378

16 17 18 19 20

0.98579 0.99174 0.99532 0.99741 0.99860

0.98317 0.99007 0.99429 0.99679 0.99824

0.98022 0.98816 0.99309 0.99606 0.99781

0.97693 0.98599 0.99171 0.99521 0.99729

0.97326 0.98355 0.99013 0.99421 0.99669

0.96921. 0.98081 0.98833 0.99307 0.99598

0.96476 0.97775 0.98630 0.99176 0.99515

0.95989 0.97437 0.98402 0.99026 0.99420

0.95460 0.97064 0.98147 0.98857 0. 99311

0.94887 0.96655 0.97864 0.98667 0.99187

21 22 23 24 25

0.99926 0.99962 0.99981 0.99990 0.99995

0.99905 0.99950 0.99974 0.99987 0.99994

0.99880 0.99936 0.99967 0.99983 0.99991

0.99850 0.99919 0.99957 0.99978 0.99989

0.99814 0.99898 0.99945 0.99971 0.99985

0.99771 0.99873 0.99931 0.99963 0.99981

0.99721 0.99843 0.99913 0.99953 0.99975

0.99662 0.99807 0.99892 0.99941 0.99968

0.99594 0.99765 0.99867 0.99926 0.99960

0.99514 0.99716 0.99837 0.99908 0.99949

0.99998 0.99997 0.99996 0.99994 0.99992 0.99990 0.99999 0.99999 0.99998 0.99997 0.99996 0.99995 0.99999 0.99999 0.99999 0.99998 0.99998 0.99999 0.99999 0.99999 0.99999 2 x =8.2 8.4 8.6 8.8 9.0 9.2 m -4.1 4.2 4.4 4.3 4.5 4.6 0.00419 0.00375 0.00336 0.00301 0.00270 0.00242 0.01657 0.01500 0.01357 0.01228 0.01111 0.01005 0.04205 0.03843 0.03511 0.03207 0.02929 0.02675 0. 08452 0.07798 0.07191 0.06630 0.06110 0.05629 0.14555 0.13553 0.12612 0.11731 0.10906 0.10135

0.99987 0.99993 0.99997 0.99998 0.99999

0.99983 0,99991 0.99996 0.99998 0.99999

0.99978 0.99989 0.99994 0.99997 0.99999

0.99973 0.99985 0.99992 0.99996 0.99998

9.4 4.7 0.00217 0.00910 0.02442 0.05184 0.09413

9.6 4.8 0.00195 0.00823 0.02229 0.04773 0.08740

9.8 4.9 0. 00175 0.00745 0. 02034 0.04394 0.08110

10.0 5.0 0.00157 0.00674 0.01857 0.04043 0.07524

26 27 28 29 -. 30

V

5 6 7 8 9 10

0.22381 0.31529 0.41418 0.51412 0.60931

0.18514 0.26734 0.35945 0.45594 0.55118

0.17358 0.25266 0.34230 0.43727 0.53210

0.16264 0.23861 0.32571 0.41902 0.51323

0.15230 0.22520 0.30968 0.40120 0.49461

0.14254 0.21240 0.29423 0.38383 0.47626

0.13333 0.20019 0.27935 0.36692 0.45821

0. U465 0.18857 0.26503 0.35049 0.44049

11 12 13 14 15

0.69528 0.67709 0.65876 0.64035 0.76931 0.75314 0.73666 0.71991 0.83033 0.81660 0.80244 0.78788 0.87865 0.86746 0.85579 0.84365 0.91551 0.90675 0.89749 0.88774

0.62189 0.70293 0.77294 0.83105 0.87752

0.60344 0.68576 0.75768 0.81803 0.86683

0.58502 0.66844 0.74211 0.80461 0.85569

0.56669 0.65101 0.72627 0.79081 0.84412

0.54846 0.63350 0.71020 0.77666 0.83213

0.53039 0.61596 0.69393 0.76218 0.81974

16 17 18 19 20

0.94269 0.96208 0.97551 0.98454 0.99046

0.93606 0.95723 0.97207 0.98217 0.98887

0.92897 0.95198 0.96830 0.97955 0.98709

0.92142 0.94633 0.96420 0.97666 0.98511

0.91341 0.94026 0.95974 0.97348 0.98291

0.90495 0.93378 0.95493 0.97001 0.98047

0.89603 0.92687 0.94974 0.96623 0.97779

0.88667 0.91954 0.94418 0.96213 0.97486

0.87686 0.91179 0.93824 0.95771 0.97166

0.86663 0.90361 0.93191 0.95295 0.96817

21 22 23 24 25

0.99424 0.99659 0.99802 0.99888 0.99937

0.99320 0.99593 0.99761 0.99863 0.99922

0.99203 0.99518 0.99714 0.99833 0.99905

0.99070 0.99431 0.99659 0.99799 0.99884

0.98921 0.99333 0.99596 0.99760 0.99860

0.98755 0.99222 0.99524 0.99714 0.99831

0.98570 0.99098 0.99442 0.99661 0.99798

0.98365 0.98958 0.99349 0.99601 0.99760

0.98139 0.98803 0.99245 0.99532 0.99716

0.97891 0.98630 0.99128 0.99455 0.99665

26 27 28 29 30

0.99966 0.99981 0.99990 0.99995 0.99997

0.99957 0.99977 0.99987 0.99993 0.99997

0.99947 0.99971 0.99984 0.99991 0.99996

0.99934 0.99963 0.99980 0.99989 0.99994

0.99919 0.99955 0.99975 0.99986 0.99993

0.99902 0.99944 0.99969 0.99983 0.99991

0.99882 0.99932 0.99962 0.99979 0.99988

0.99858 0.99917 0.99953 0.99973 0.99985

0.99830 0.99900 0.99942 0.99967 0.99982

0.99798 0.99880 0.99930 0.99960 0.99977

Q

0.21024 0.29865 0.39540 0.49439 0.58983

0.19736 0.28266 0.37715 0.47499 0.57044

-7

w j / j ! ( v even,

c=

i v , ?U=

a

x2)

981

PROBABILITY FUNCTIONS

Table 26.7 PROBABILITY INTEGRAL OF x2-DISTRIBUTION, INCOMPLETE GAMMA FUNCTION CUMULATIVE SUMS OF THE POlSSON DISTRIBUTION

~2= 1 0 . 5 11.0 v m = 5.25 5.5

1 2 3 4 5

0.00119 0.00525 0.01476 0,03280 0.06225

6 7 8 9

0 10511

11.5 12.0 5.75 6.0

12.5 13.0 6.25 6.5

0.00091 0.00409 0.01173 0.02656 0.05138

0.00070 0.00318 0.00931 0.02148 0.04232

0.00053 0.00248 0.00738 0.01735 0.03479

0.00041 0.00193 0.00585 0.01400 0.02854

0.00031 0.00150 0.00464 0.01128 0.02338

10

0.08838 0.13862 0.23167 0.20170 0.31154 0.27571 0.39777 0.35752

0.07410 0.11825 0.17495 0.24299 0.31991

0.06197 0.10056 0.15120 0.21331 0.28506

0.05170 0.08527 0.13025 0.18657 0.25299

11 12 13 14 15

0.48605 0.57218 0.65263 0 72479 0: 78717

0.44326 0.52892 0.61082 0.68604 0.75259

0.40237 0.48662 0.56901 0.64639 0.71641

0.36364 0.44568 0.52764 0.60630 0.67903

16 17 18 19 20

0.83925 0.88135 0.91436 0.93952 0.95817

0.80949 0.85656 0.89436 0.92384 0.94622

0.77762 0.82942 0.87195 0.90587 0.93221

21 22 23 24 25

0.97166 0.98118 0.98773 0.99216 0.99507

0.96279 0.97475 0.98319 0.98901 0.99295

26 27 28 29 30

0.99696 0.99815 0.99890 0.99935 0.99963

0.99555 0.99724 0.99831 0.99899 0.99940

v

x2=15.5 m = 7.75

0: 16196

16.0 8.0

13.5 14.0 6.75 7.0

0.00018

0.00014 0.00071

15.0 7.5

0.00367 0.00907 0.01912

0.00291 0.00730 0.01561

0.00230 0.00586 0.01273

0.00011 0.00055 0.00182 0.00470 0; 01036

0.04304 0.07211 0.11185 0.16261 0.22367

0.03575 0.06082 0.09577 0.14126 0.19704

0.02964 0.05118 0.08177 0.12233 0.17299

0.02452 0.04297 0.06963 0.10562 0.15138

0.02026 0.03600 0.05915 0.09094 0.13206

0.32726 0.40640 0.48713 0.56622 0.64086

0.29333 0.36904 0.44781 0.52652 0.60230

0.26190 0.33377 0.40997 0.48759 0.56374

0.23299 0.30071 0.37384 0.44971 0.52553

0.20655 0.26992 0.33960 0.41316 0.48800

0.18250 0.24144 0.30735 0.37815 0.45142

0.74398 0.80014 0.84724 0.88562 0.91608

0.70890 0.76896 0.82038 0.86316 0.89779

0.67276 0.63591 0.73619 0.70212 0.79157 0.76106 0.83857 0.81202 0.87738 0.85492

0.59871 0.66710 0.72909 0.78369 0.83050

0.56152 0.63145 0.69596 0.75380 0.80427

0.52464 0.59548 0.66197 0.72260 0.77641

0.95214 0.96686 0.97748 0.98498 0.99015

0.93962 0.95738 0.97047 0.97991 0.98657

0.92513 0.94618 0.96201 0.97367 0.98206

0.90862 0.93316 0.95199 0.96612 0.97650

0.89010 0.91827 0.94030 0.95715 0.96976

0.86960 0.90148 0.92687 0.94665 0.96173

0.84718 0.88279 0.91165 0.93454 0.95230

0.82295 0.86224 0.89463 0.92076 0.94138

0.99366 0.99598 0 997 9 0: 99816 0.99907

0.99117 0.99429 0.99637 0.99773 0.99860

0.98798 0.99208 0.99487 0.99672 0.99794

0.98397 0.98925 0.99290 0.99538 0.99704

0.97902 0.98567 0.99037 0.99363 0.99585

0.97300 0.98125 0.98719 0.99138 0.99428

0.96581 0.97588 0.98324 0.98854 0.99227

0.95733 0.96943 0.97844 0.98502 0.98974

17.5 8.75

18.0 9.0

16.5 17.0 8.25 8.5

0.00024

14.5 7.25

0. ooii7 0.00091

18.5 19.0 9.25 9.5

19.5 20.0 9.75 10.0

0.00008 0.00043 0.00144 0.00377 0.00843

0.00006 0.00034 0.00113 0.00302 0.00684

0.00005 0.00026 0.00090 0.00242 0.00555

0.00004 0.00020 0.00071 0.00193 0.00450

0.00003 0.00016 0.00056 0.00154 0.00364

0,00002 0.00012 0.00044 0.00123 0.00295

0.00002 0.00010 0.00035 0.00099 0.00238

0.00001 0.00001 0.00008 0.00006 0.00027 0.00022 0.00079 0.00063 0.00192 0.00155

0.00001 0.00005 0.00017 0.00050 0.00125

6 7 8 9 10

0.01670 0.03010 0.05012 0.07809 0.11487

0.01375 0.02512 0.04238 0.06688 0.09963

0.01131 0.02092 0.03576 0.05715 0.08619

0.00928 0.01740 0.03011 0.04872 0.07436

0.00761 0.01444 0.02530 0.04144 0.06401

0.00623 0.01197 0.02123 0.03517 0.05496

0.00510 0.00991 0.01777 0.02980 0.04709

0.00416 0.00819 0.01486 0.02519 0.04026

0.00340 0.00676 0.01240 0.02126 0.03435

0.00277 0.00557 0.01034 0.01791 0.02925

11 12 13 14 15

0.16073 0.21522 0.27719 0.34485 0.41604

0.14113 0.19124 0.24913 0.31337 0.38205

0.12356 0.16939 0.22318 0.28380 0.34962

0.10788 0.14960 0.19930 0.25618 0.31886

0.09393 0.13174 0.17744 0.23051 0.28986

0.08158 0.11569 0.15752 0.20678 0.26267

0.07068 0.10133 0.13944 0.18495 0.23729

0.06109 0.08853 0.12310 0.16495 0.21373

0.05269 0.07716 0.10840 0.14671 0.19196

0.04534 0.06709 0.09521 0.13014 0.17193

16 17 18 19 20

0.48837 0.55951 0.62740 0.69033 0.74712

0.45296 0.52383 0.59255 0.65728 0.71662

0.41864 0.48871 0.55770 0.62370 0.68516

0.38560 0.35398 0.45437 0.42102 0.52311 0.48902 0.58987 0.55603 0.65297 0.62031

0.32390 0.38884 0.45565 0.52244 0.58741

0.29544 0.35797 0.42320 0.48931 0.55451

0.26866 0.24359 0.32853 0.30060 0.39182 0.36166 0.45684 0.42521 0.52183 0.48957

0.22022 0.27423 0.33282 0.39458 0.45793

21 22 23 24 25

0.79705 0.83990 0.87582 0.90527 0.92891

0.76965 0.81589 0.85527 0.88808 0.91483

0.74093 0.79032 0.83304 0.86919 0.89912

0.71111 0.76336 0.80925 0.84866 0.88179

0.68039 0.73519 0.78402 0.82657 0.86287

0.64900 0.70599 0.75749 0.80301 0.84239

0.61718 0.58514 0.67597 0.64533 0.72983 0.70122 0.77810 0.75199 0.82044 0.79712

0.55310 0.61428 0.67185 0.72483 0.77254

0.52126 0.58304 0.64191 0.69678 0.74683

26 27 28 29 30

0.94749 0.96182 0.97266 0.98071 0.98659

0.93620 0.95295 0.96582 0.97554 0.98274

0.92341 0.94274 0.95782 0.96939 0.97810

0.90908 0.93112 0.94859 0.96218 0.97258

0.89320 0.91806 0.93805 0.95383 0.96608

0.87577 0.90352 0.92615 0.94427 0.95853

0.85683 0.88750 0.91285 0.93344 0.94986

0.81464 0.85107 0.88200 0.90779 0.92891

0.79156 0.83076 0.86446 0.89293 0.91654

1 2 3 4 5

0.83643 0.87000 0.89814 0.92129 0.94001

982

PROBABILITY FUNCTIONS

Table 26.7 PROBABILITY INTEGRAL OF x2-DISTRIBUTION, INCOMPLETE GAMMA FUNCTION CUMULATIVE SUMS OF THE POISSON DISTRIBUTION i 2 = 21 Y

1

V? = 10.5

22 11.0

23 11.5

24 12.0

25 12.5

26 13.0

27 13.5

28 14.0

29 14.5

30 15.0

0.00001 0.00004 0.00013 0.00034

0,00001 0.00003 0.00008 0.00022

0.00002 0.00005 0.00014

0.00001 0.00003 0.00009

0,00001 0.00002 0.00001 0.00006 0.00004

0.00001 0.00002

0,00001 0.00002

0.00080

0.00034 0.00076 0.00155 0.00297 0.00535

0.00022 0; 00050 0.00105 0.00204 0.00374

0.00015 0.00009 0.00006 0.00004 0; 00033 0: 00022 0; 00015 0,00010

0.00071 0.00140 0.00260

0.00047 0.00095 0.00181

0.00032 0.00065 0.00125

0.00021 0.00044 0.00086

2 3 4 5

0.00001 0.00003 0.00011 0.00032 0.00081

0.00002 0.00007 0.00020 0.00052

6 7 8 9 10

0.00184 0; 00377 0.00715 0.01265 0.02109

0.00121 0; 00254 0.00492 0.00888 0.01511

0.00336 0.00620 0.01075

0.00052 0; 00114 0.00229 0.00430 0.00760

11 12

13 14 15

0.03337 0.05038 0.07293 0.10163 0.13683

0.02437 0.03752 0.05536 0.07861 0.10780

0.01768 0.02773 0.04168 0.06027 0.08414

0.01273 0.02034 0.03113 0.04582 0.06509

0.00912 0.01482 0.02308 0.03457 0.04994

0.00649 0.01073 0.01700 0.02589 0.03802

0.00460 0.00773 0.01244 0.01925 0.02874

0.00324 0.00553 0.00905 0.01423 0.02157

0.00227 0.00394 0.00655 0.01045 0.01609

0.00159 0.00279 0.00471 0.00763 0.01192

16 17 18 19 20

0.17851 0.22629 .. 0.27941 0.33680 0.39713

0.14319 0I 18472 0.23199 0.28426 0.34051

0.11374 0; 14925 0.19059 0.23734 0.28880

0.08950 0.11944 0.15503 0.19615 0.24239

0.06982 0.09471 0.12492 0.16054 0.20143

0.05403 0.07446 0.09976 0.13019 0.16581

0.04148 0.05807 0.07900 0.10465 0.13526

0.03162 0.04494 0.06206 0.08343 0.10940

0.02394 0.03453 0.04838 0.06599 0.08776

0.01800 0.02635 0.03745 0.05180 0.06985

21 22 23 24 25

0.45894 0.52074 0.58109 0.63873 0.69261

0.39951 0.45989 0.52025 0.57927 0.63574

0.34398 0.40173 0.46077 0.51980 0.57756

0.29306 0.24716 0.20645 0.34723 0.29707 0.25168 0.40381 0.35029 0.30087 0.46160 0.40576 0.35317 0.51937 0.46237 0.40760

0.17085 0.21123 0.25597 0.30445 0.35588

0.14015 0.17568 0.21578 0.26004 0.30785

0.11400 0.14486 0.18031 0.22013 0.26392

0.09199 0.11846 0.14940 0.18475 0.22429

26 27 28 29 30

0.74196 0.78629 0.82535 0.85915 0.88789

0.68870 0.73738 0.78129 0.82019 0.85404

0.63295 0.68501 0.73304 0.77654 0.81526

0.57597 ..- 0.63032 0.68154 0.72893 0.77203

0.46311 0.51860 0; 57305 0.62549 0.67513

0.40933 0.46379 0I 51825 0.57171 0.62327

0.35846 0.41097 Ol46445 0.51791 0.57044

0.31108 0.36090 oI41253 0.46507 0.51760

0.26761 0.31415 . ~-~ 0.36322 0.41400 0.46565

32 16.0

33 16.5

34 17.0

36 18.0

37 18.5

38 19.0

39 19.5

Y

x2=31 ni = 15.5

0; 00171

0.51898 0; 57446 0.62784 0.67825 0.72503

35 17.5

40 20.0

0.00001 0.00001 0.00003 0.00002 0.00006 0.00004 0.00014 0.00009 0.00030 0.00020

0.00001 0.00003 0.00006 0.00013

0.00001 0.00002 0.00001 0; 00004 0; OOOO? 0.00009 0.00006

0.00001 0.00002 0.00004

0.00001 0.00003

0.00001 0.00002

0.00001

0.00001

11 12 13 14

0.00059 0.00110 0.00197 0.00337 0.00554

0.00040 0.00076 0.00138 0.00240 0.00401

0.00027 0.00053 0.00097 0.00170 0.00288

0.00019 0.00036 0.00068 0.00120 0.00206

0.00012 0.00025 0.00047 0.00085 0.00147

0.00008 0.00017 0.00032 0.00059 0.00104

0.00006 0.00012 0.00022 0.00041 0.00074

0.00004 0.00008 0.00015 0.00029 0.00052

0.00003 0.00005 0,00011 0.00020 0.00036

0.00002 0.00004 0.00007 0.00014 0.00026

15 16 17 18 19

0.00878 0.01346 0.01997 0.02879 0.04037

0.00644 0.01000 0.01505 0.02199 0.03125

0.00469 0.00739 0.01127 0.01669 0.02404

0.00341 0.00543 0.00840 0.01260 0.01838

0.00246 0.00397 0.00622 0.00945 0.01397

0.00177 0.00289 0.00459 0.00706 0.01056

0.00127 0.00210 0.00337 0.00524 0.00793

0.00090 0.00151 0.00246 0.00387 0.00593

0.00064 0.00109 0.00179 0.00285 0.00442

0.00045 0.00078 0.00129 0.00209 0.00327

20 21 22 23 24

0.05519 0.07366 0.09612 0.12279 0.15378

0.04330 0.05855 0.07740 0.10014 0.12699

0.03374 0.04622 0.06187 0.08107 0.10407

0.02613 0.03624 0.04912 0.06516 0.08467

0.02010 0.02824 0.03875 0.05202 0.06840

0.01538 0.02187 0.03037 0.04125 0.05489

0.01170 0.01683 0.02366 0.03251 0.04376

0.00886 0; 01289 0.01832 0.02547 0.03467

0.00667

0.00500

0.01411 0.01984 0.02731

0.01081 0.01537 0.02139

25 26 27 28 29

0.18902 0.22827 0.27114 0.31708 0.36542

0.15801 0.19312 0.23208 0.27451 0.31987

oI 16210 0;

0.13107

0.08820 _ _ . - - _0.07160 _

0.19707 0.23574 0.27774

0.10791 13502 0.16605 0.20087 0.23926

0.11165 0.13887 0.16987 0.20454

0.09167 0.11530 0.14260 0.17356

0.05774 0.07475 0.09507 0.11886 0.14622

0.04626 0.06056 0.07786 0.09840 0.12234

0.03684 0.04875 0.06336 0.08092 0.10166

0.02916 0.03901 0.05124 0.06613 0.08394

30

0.41541

0.36753

0.32254

0.28083

0.24264

0.20808

0.17714

0.14975

0.12573

0.10486

10

0; 00981 0.00744 . ... . .

983

PROBABILITY FUNCTIONS

PROBABILITY INTEGRAL OF x~-DISTRIBUTION,INCOMPLETE GAMMA FUNCTION CUMULATIVE SUMS OF THE POISSON DISTRIBUTION

52

54

56

26

27

28

0.00001 0.00001 0.00002 0.00004 0.00007

0.00001 0.00001 0.00002 0.00003

0.00001 0.00001 0.00002

0.00001

0.00022 0.00036 0.00059 0.00092 0.00142

0.00011 0.00019 0.00031 0.00050 0.00078

0.00006 0.00010 0.00016 0.00027 0.00043

0.00003 0.00005 0.00009 0.00014 0.00023

0.00001 0.00003 0.00004 0.00007 0.00012

0.00001 0.00001 0.00002 0.00004 0.00006

0.00373 0.00540 0.00768 0.01072 0.01470

0.00213 0.00314 0.00455 0.00647 0.00903

0.00120 0.00180 0.00265 0.00384 0.00545

0.00066 0.00102 0.00152 0.00224 0.00324

0.00036 0.00056 0.00086 0.00129 0.00189

0.00020 0.00031 0.00048 0.00073 0.00109

0.00011 0.00017 0.00026 0.00041 0.00062

0.01983

0.01240

0.00762

0.00460

0.00273

0.00160

0.00092

0.00001 0.00001 0.00003

0.00001 0.00001

0.00001

19

0.00023 0.00040 0.00067 0.00111 0.00177

0.00011 0.00020 0.00034 0.00058 0.00094

0.00005 0.00010 0.00017 0.00030 0.00050

0.00003 0.00005 0.00009 0.00015 0.00026

0.00001 0.00002 0.00004 0.00008 0.00013

20 21 22 23 24

0.00277 0.00421 0.00625 0.00908 0.01291

0.00151 0.00234 0.00355 0.00526 0.00763

0.00081 0.00128 0.00198 0.00299 0.00443

0.00043 0.00069 0.00109 0.00167 0.00252

25 26 27 28 29

0.01797 0.02455 0.03292 0.04336 0.05616

0.01085 0.01512 0.02068 0.02779 0.03670

0.00642 0.00912 0.01272 0.01743 0.02346

30

0.07157

0.04769

0.03107

15 16 17 18

46 23

50

0.00001 0.00002 0.00003 0.00006

10 11 12 13 14

44 22

25

x2=42 m=21 0.00001 0.00002 0.00003 0.00006 0.00012

V

48 24

x2 = 62

64

m=31 0.00001 0.00001 0.00002 0.00003 0.00006

32

21 22 23 24 25

0.00001 0.00001 0.00002 0.00003

0.00001 0.00001 0.00002

0.00001

26 27 28 29 30

0.00009 0.00014 0.00023 0.00035 0.00052

0.00005 0.00008 0.00012 0; 00019 0.00029

0.00003 0.00004 0.00007 0,00011 0.00016

0.00001 0.00002 0.00004 0.00006 0.00009

V

66 33

68

70 35

34

0.00001 0.00001 0.00002 0.00003 0.00005

;( x2-x '0)

$,=-

Interpolation on

x2

Double Entry Interpolation

Table 26.7

72

74

76

36

37

38

0.00001 0.00001 0.00002 0.00003

w=v--v~>o

0,00001 0.00001 0,00001 0.00001

58 29

60

30

984

PROBABILITY FUNCTIONS

Table 26.8

''

PERCENTAGE POINTS OF THE X2-DISTRIBUTION-VALUES OF X2 IN TERMS OF Q AND Y

0.995 0.99 0.975 0.95 -5) 3.92704 -4 1.57088 -4 9.82069 ( -3)3.93214 2 -2 1.00251 -2 2.01007 1-215.06356 0.102587 {-2{7.17212 0.114832 0.215795 0.351846 0.484419 0.710721 4 0.206990 0.297110 5 0.411740 0.554300 0.831211 1.145476 '\

0.9 0.0157908 0.210720 0.584375 1.06362? 1.61031

0.75 0.101531 0.575364 1.212534 1.92255 2.67460

0.454937 1.38629 2.36597 3.35670 4.35146

5.34812 7.84080 6.34581 9.03715 7.34412 10.2188 8.34283 11.3887 9.34182 12.5489

0.5

0.25

1.32330 2.77259 4.10835 5.38527 6.62568

6 7 8 9 10

0.675727 0.989265 1.344419 1.734926 2.15585

0.872085 1.239043 1.646482 2.087912 2.55821

1.237347 1.68987 2.17973 2.70039 3.24697

1.63539 2.16735 2.73264 3.32511 3.94030

2.20413 2.83311 3.48954 4.16816 4.86518

3.45460 4.25485 5.07064 5.89883 6.73720

11 12 13 14 15

2.60321 3i07382 3.56503 4.07468 4.60094

3.05347 3757056 4.10691 4.66043 5.22935

3.81575 4.40379 5.00874 5.62872 6.26214

4.57481 5.22603 5.89186 6.57063 7.26094

5.57779 6.30380 7.04150 7.78953 8.54675

7.58412 8.43842 9.29906 10.1653 11.0365

10.3410 11.3403 12.3398 13.3393 14.3389

13.7007 14.8454 15.9839 17.1170 18.2451

16 17 18 19 20

5.14224 5.69724 6I26481 6.84398 7.43386

5.81221 6.40776 7.01491 7.63273 8.26040

6.90766 7.56418 8.2 3075 8.90655 9.59083

7.96164 8.67176 9.39046 10;1170 10.8508

9.31223 10.0852 10.8649 1176509 12.4426

11.9122 12.7919 13.6753 14.5620 15.4518

15.3385 16.3381 17.3379 18.3376 19.3374

19.3688 20.4887 21.6049 22.7178 23.8277

21 22 23 24 25

8.03366 8i64272 9.26042 9.88623 10.5197

8.89720 9.54249 10.19567 10.8564 11.5240

10.28293 10.9823 11.6885 12.4011 1371197

11.5913 12.3380 13.0905 13.8484 14.6114

13.2396 14.0415 14.8479 15.6587 16.4734

16.3444 17.2396 18.1373 19.0372 19.9393

20.3372 21.3370 22.3369 23.3367 24.3366

24.9348 26.0393 27.1413 28.2412 29.3389

26 27 28 29 30

11.1603 11.8076 12.4613 13.1211 13.7867

12.1981 12.8786 13.5648 14.2565 14.9 535

13.8439 14.5733 15.3079 16.0471 16.7908

15.3791 16.1513 16.9279 17.7083 18.4926

17.2919 18.1138 18.9392 19.7677 20.5992

20.8434 21.7494 22.6572 23.5666 24.4776

25.3364 26.3363 27.3363 28.3362 29.3360

30.4345 31.5284 32.6205 33.7109 34.7998

40 50 60 70 80

20.7065 27.9907 35.5346 43.2752 51.1720

22.1643 29.7067 37.4848 45.4418 53.5400

24.4331 32.3574 40.4817 48.7576 57.1532

26.5093 34.7642 43.1879 51.7393 60.3915

29.0505 37.6886 46.4589 55.3290 64.2778

33.6603 42.9421 52.2938 61.6983 71.1445

39.3354 49.3349 59.3347 69.3344 79.3343

45.6160 56.3336 66.9814 77.5766 88.1303

90 100

59.1963 67.3276

61.7541 70.0648

65.6466 74.2219

69.1260 77.9295

73.2912 82.3581

80.6247 90.1332

89.3342 99.3341

98.6499 109.141

X

-2.5758

-2.3263

-1.9600

-1.6449

-1.2816

-0.6745

0.0000

0.6745

From E. S. Pearson and H. 0. Hartley (editors): Biometrika tables for statisticians, vol. I. Cambridge Univ. Press, Cambridge, England, 1954 (with permission) for Q > 0.0005.

985

PROBABILITY FUNCTIONS

PERCENTAGE POINTS OF THE xz-DISTRIBUTION- -VALUES OF X2 IN TERMS OF Q AND Y Y\

1 2 3 4 5

Q

0.1

2.70554 4.60517 6.25139 7.77944 9.23635

Table 26.8

0.001

0.0005

0.0001

3.84146 5.99147 7.81473 9.48773 11.0705

5.02389 7.37776 9.34840 11.1433 12.8325

6.63490 9.21034 11.3449 13.2767 15.0863

7.87944 10.5966 12.8381 14.8602 16.7496

10.828 13.816 16.266 18.467 20.515

12.116 15.202 17.730 19.997 22.105

15.137 18.421 21.108 23.513 25.745

0.05

0.025

0.01

0.005

6 7 8 9 10

10.6446 12.0170 13.3616 14.6837 15.9871

12.5916 14.0671 15.5073 16.9190 18.3070

14.4494 16.0128 17.5346 19.0228 20.4831

16.8119 18.4753 20.0902 21.6660 23.2093

18.5476 20.2777 21.9550 23.5893 25.1882

22.458 24.322 26.125 27.877 29.588

24.103 26.018 27.868 29.666 31.420

27.856 29.877 31.828 33.720 35.564

11 12 13 14

is

17.2750 18.5494 19.8119 21.0642 22i3072

19.6751 21.0261 22.3621 23.6848 24.9958

21.9200 23.3367 24.7356 26.1190 27I4884

24.7250 26.2170 27.6883 29.1413 30.5779

26.7569 28.2995 29.8194 31.3193 3218013

31.264 32.909 34.528 36.123 37.697

33.137 34.821 36.478 38I109 39.719

37.367 39.134 40.871 42.579 44.263

16 17 18 19 20

23.5418 24.7690 25.9894 27.2036 28.4120

26.2962 27.5871 28.8693 30.1435 31.4104

28.8454 30.1910 31.5264 32.8523 34.1696

31.9999 33.4087 34.8053 36.1908 37.5662

34.2672 35.7185 37.1564 38.5822 39.9968

39.252 40.790 42.312 43.820 45.315

41.308 42.879 44.434 45.973 47.498

45.925 47.566 49.189 50.796 52.386

21 22 23 24 25

29.6151 30.8133 32.0069 33.1963 34.3816

32.6705 33.9244 35.1725 36.4151 37.6525

35.4789 36.7807 38.0757 39.3641 40.6465

38.9321 40.2894 41.6384 42.9798 44.3141

41.4010 42.7956 44.1813 45.5585 46.9278

46.797 48.268 49.728 51.179 52.620

49.011 50.511 52.000 53.479 54.947

53.962 55.525 57.075 58.613 60.140

26 27 28 29 30

35.5631 36.7412 37.9159 39.0875 40.2560

38.8852 40.1133 41.3372 42.5569 43.7729

41.9232 43.1944 44.4607 45.7222 46.9792

45.6417 46.9630 48.2782 49.5879 50.8922

48.2899 49.6449 50.9933 52.3356 53.672 0

54.052 55.476 56.892 58.302 59;703

56.407 57.858 59.300 60.735 62I162

61.657 63.164 64.662 66.152 67.633

40

51.8050 63.1671 74.3970 85.5271 96.5782

55.7585 67.5048 79.0819 90.5312 101.879

59.3417 71.4202 83.2976 95.0231 106.629

63.6907 76.1539 88.3794 100.425 112.329

66.7659 79.4900 91.9517 104.215 116.321

73.402 86.661 99.607 112.317 124.839

76.095 89.560 102.695 115.578 128.261

82.062 95.969 109.503 122.755 135.783

113.145 124.342

118.136 129.561

124.116 135.807

128.299 140.169

137.208 149.449

140.782 153.167

148.627 161.319

50

60 70 80 90 100

x

107.565 118.498 1.2816

1.6449

1.9600

2.3263

2.5758

3.0902

3.2905

3.7190

986

PROBABILITY FUNCTIONS

Table 26.9 PERCENTAGE POINTS OF THE F-DISTRIBUTION -VALUES OF F IN TERMS OF Q, v,, 9

Q(F(vi,q)=0.5 y2\v1 1 2 1.50 1 1.00 2 0.667 1.00 3 0.585 0.881 4 0 549 0.828 5 (528 0.799

3

4

5

1.71 1.13 1.00 0.941 0.907

1.82 1.21 1.06 1.00 0.965

1.89 1.25 1.10 1.04 1.00

0.780 0.767 0.757 0.749 0.743

0.886 0 871 01860 0.852 0.845

0.942 0.926 0.915 0.906 0.899

0.739

-

6

8

12

15

20

30

60

1.94 1.28 1.13 1.06 1.02

2.00 1.32 1.16 1.09 1.05

2.07 1.36 1.20 1.13 1.09

2.09 1.38 1.21 1.14 1.10

2.12 1.39 1.23 1.15 1.11

2.15 1.41 1.24 1.16 1.12

2.17 1.43 1.25 1.18 1.14

2.20 1.44 1.27 1.19 1.15

0.977 1.00 0.960 0.983 0.948 0.971 0.939 0.962 0.932 0.954

1.03 1.01 1.00 0.990 0.983

1.06 1.04 1.03 1.02 1.01

1.07 1.05 1.04 1.03 1.02

1.08 1.07 1.05 1.04 1.03

1.10 1.08 1.07 1.05 1.05

1.11 1.09 1.08 1.07 1.06

1.12 1.10 1.09 1.08 1.07

0.893 0,888 0.885 0.881 0.878

0.926 0.921 0.917 0.914 0.911

0.948 0.943 0.939 0.936 0.933

0.977 0.972 0.967 0.964 0.960

1.01 1.00 0.996 0.992 0.989

1.02 1.01 1.01 1.00 1.00

1.03 1.02 1.02 1.01 1.01

1.04 1.03 1.03 1.03 1.02

1.05 1.05 1.04 1.04 1.03

1.06 1.06 1.05 1.05 1.05

6 7 8 9 10

0.515 0 506 0'499 01494 0.490

11 12 13 14 15

0 486 0'484 0'481 0'479 01478

0:731 0.729 0.726

0.840 0.835 0.832 0.828 0.826

16 17 18 19 20

0 476 0'475 0'474 0'473 01472

0 724 0'722 01721 0.719 0.718

0.823 0 821 01819 0 818 01816

0.876 0.874 0.872 0.870 0.868

0.908 0.906 0.904 0.902 0.900

0.930 0.928 0.926 0.924 0.922

0.958 0.955 0.953 0.951 0.950

0.986 0.983 0.981 0.979 0.977

0.997 0.995 0.992 0.990 0.989

1.01 1.01 1.00 1.00 1.00

1.02 1.02 1.02 1.01 1.01

1.03 1.03 1.03 1.02 1.02

1.04 1.04 1.04 1.04 1.03

21 22 23 24 25

0 471 0'470 0'470 0'469 01468

0 716 01715 0 714 0'714 01713

0 815 01814 0.813 0.812 0.811

0 867 0:866 0.864 0.863 0.862

0 899 01898 0.896 0.895 0.894

0.921 0.919 0.918 0.917 0.916

0.948 0.947 0.945 0.944 0.943

0.976 0.974 0.973 0.972 0.971

0.987 0.986 0.984 0.983 0.982

0.998 0.997 0.996 0.994 0.993

1.01 1.01 1.01 1.01 1.00

1.02 1.02 1.02 1.02 1.02

1.03 1.03 1.03 1.03 1.03

26 27 28 29 30

0.468 0 467 0'467 01466 0.466

0.712 0 711 0'711 0:710 0.709

0.810 0.809 0.808 0.808 0.807

0.861 0.861 0.860 0.859 0.858

0.893 0.892 0.892 0.891 0.890

0.915 0.914 0.913 0.912 0.912

0.942 0.941 0.940 0.940 0.939

0.970 0.969 0.968 0.967 0.966

0.981 0.980 0.979 0.978 0.978

0.992 0.991 0.990 0.990 0.989

1.00 1.00 1.00 1.00 1.00

1.01 1.01 1.01 1.01 1.01

1.03 1.03 1.02 1.02 1.02

40 60 120

0 463 0'461 01458 0.455

0.705 0.701 0.697 0.693

0.802 0 798 01793 0.789

0.854 0.849 0.844 0.839

0 885 0:880 0.875 0.870

0.907 0.901 0.896 0.891

0.934 0.928 0.923 0.918

0.961 0.956 0.950 0.945

0.972 0.967 0.961 0.956

0.983 0.978 0.972 0.967

0.994 0.989 0.983 0.978

1.01 1.00 0.994 0.989

1.02 1.01 1.01 1.00

2

3

4

5

20

30

60

7.50 3.00 2.28 2.00 1.85

8.20 3.15 2.36 2.05 1.88

8.58 3.23 2.39 2.06 1.89

8.82 3.28 2.41 2.07 1.89

8.98 3.31 2.42 2.08 1.89

9.19 3.35 2.44 2.08 1.89

9.41 3.39 2.45 2.08 1.89

9.49 3.41 2.46 2.08 1.89

9.58 3.43 2.46 2.08 1.88

9.67 3.44 2.47 2.08 1.88

9.76 3.46 2.47 2.08 1.87

9.85 3.48 2.47 2.08 1.87

10

1.62 1.57 1.54 1.51 1.49

1.76 1.70 1.66 1.62 1.60

1.78 1.72 1.67 1.63 1.60

1.79 1.72 1.66 1.63 1.59

1.79 1.71 1.66 1.62 1.59

1.78 1.71 1.65 1.61 1.58

1.78 1.70 1.64 1.60 1.56

1.77 1.68 1.62 1.58 1.54

1.76 1.68 1.62 1.57 1.53

1.76 1.67 1.61 1.56 1.52

1.75 1.66 1.60 1.55 1.51

1.74 1.65 1.59 1.54 1.50

1.74 1.65 1.58 1.53 1.48

11 12 13 14 15

1.47 1.46 1.45 1.44 1.43

1.58 1.56 1.55 1.53 1.52

1.58 1.56 1.55 1.53 1.52

1.57 1.55 1.53 1.52 1.51

1.56 1.54 1.52 1.51 1.49

1.55 1.53 1.51 1.50 1.48

1.53 1.51 1.49 1.48 1.46

1.51 1.49 1.47 1.45 1.44

1.50 1.48 1.46 1.44 1.43

1.49 1.47 1.45 1.43 1.41

1.48 1.45 1.43 1.41 1.40

1.47 1.44 1.42 1.40 1.38

1.45 1.42 1.40 1.38 1.36

16 17 18 19 20

1.42 1.42 1.41 1.41 1.40

1.51 1.51 1.50 1.49 1.49

1.51 1.50 1.49 1.49 1.48

1.50 1.49 1.48 1.47 1.47

1.48 1.47 1.46 1.46 1.45

1.47 1.46 1.45 1.44 1.44

1.45 1.44 1.43 1.42 1.42

1.43 1.41 1.40 1.40 1.39

1.41 1.40 1.39 1.38 1.37

1.40 1.39 1.38 1.37 1.36

1.38 1.37 1.36 1.35 1.34

1.36 1.35 1.34 1.33 1.32

1.34 1.33 1.32 1.30 1.29

21 22 23 24 25

1.40 1.40 1.39 1.39 1.39

1.48 1.48 1.47 1.47 1.47

1.48 1.47 1.47 1.46 1.46

1.46 1.45 1.45 1.44 1.44

1.44 1.44 1.43 1.43 1.42

1.43 1.42 1.42 1.41 1.41

1.41 1.40 1.40 1.39 1.39

1.38 1.37 1.37 1.36 1.36

1.37 1.36 1.35 1.35 1.34

1.35 1.34 1.34 1.33 1.33

1.33 1.32 1.32 1.31 1.31

1.31 1.30 1.30 1.29 1.28

1.28 1.28 1.27 1.26 1.25

26 27 28 29 30

1.38 1.38 1.38 1.38 1.38

1.46 1.46 1.46 1.45 1.45

1.45 1.45 1.45 1.45 1.44

1.44 1.43 1.43 1.43 1.42

1.42 1.42 1.41 1.41 1.41

1.41 1.40 1.40 1.40 1.39

1.38 1.38 1.38 1.37 1.37

1.35 1.35 1.34 1.34 1.34

1.34 1.33 1.33 1.32 1.32

1.32 1.32 1.31 1.31 1.30

1.30 1.30 1.29 1.29 1.28

1.28 1.27 1.27 1.26 1.26

1.25 1.24 1.24 1.23 1.23

40 60 120

1.36 1.35 1.34 1.32

1.44 1.42 1.40 1.39

1.42 1.41 1.39 1.37

1.40 1.38 1.37 1.35

1.39 1.37 1.35 1.33

1.37 1.35 1.33 1.31

1.35 1.32 1.30 1.28

1.31 1.29 1.26 1.24

1.30 1.27 1.24 1.22

1.28 1.25 1.22 1.19

1.25 1.22 1.19 1.16

1.22 1.19 1.16 1.12

1.19 1.15 1.10 1.00

m

S\Vi 1 1 5.83 2 2.57 3 2.02 4 1.81 5 1.69 6 7 8 9

m

0 735

Q(F'vi,y)=0.25 6 8 12

15

01

Compiled from E. S.. Pearson and H. 0. Hartley (editors), Biometrika tables for statisticians, vol. I. Cambridge Univ. Press, Cambridge, England, 1954 (with permission 1.

987

PROBABILITY FUNCTIONS

v24

Table 26.9 PERCENTAGE POINTS OF THE F-DISTRIBUTION -VALUES OF F IN TERMS OF Q, U,, U, Q ( F \ v , . h )=0.1 m 30 60 20 15 8 12 6 4 5 3 2 1 63.33 62.79 62.26 61.74 61.22

2 3 4 5

39.86 8.53 5.54 4.54 4.06

49.50 9.00 5.46 4.32 3.78

53.59 9.16 5.39 4.19 3.62

55.83 9.24 5.34 4.11 3.52

57.24 9.29 5.31 4.05 3.45

58.20 9.33 5.28 4.01 3.40

59.44 9.37 5.25 3.95 3.34

60.71 9.41 5.22 3.90 3.27

9.42 5.20 3.87 3.24

9.44 5.18 3.84 3.21

9.46 5.17 3.82 3.17

9.47 5.15 3.79 3.14

9.49 5.13 3.76 3.10

6 7 8 9 10

3.78 3.59 3.46 3.36 3.29

3.46 3.26 3.11 3.01 2.92

3.29 3.07 2.92 2.81 2.73

3.18 2.96 2.81 2.69 2.61

3.11 2.88 2.73 2.61 2.52

3.05 2.83 2.67 2.55 2.46

2.98 2.75 2.59 2.47 2.38

2.90 2.67 2.50 2.38 2.28

2.87 2.63 2.46 2.34 2.24

2.84 2.59 2.42 2.30 2.20

2.80 2.56 2.38 2.25 2.16

2.76 2.51 2.34 2.21 2.11

2.72 2.47 2.29 2.16 2.06

11 12 13 14 15

3.23 3.18 3.14 3.10 3.07

2.86 2.81 2.76 2.73 2.70

2.66 2.61 2.56 2.52 2.49

2.54 2.48 2.43 2.39 2.36

2.45 2.39 2.35 2.31 2.27

2.39 2.33 2.28 2.24 2.21

2.30 2.24 2.20 2.15 2.12

2.21 2.15 2.10 2.05 2.02

2.17 2.10 2.05 2.01 1197

2.12 2.06 2.01 1.96 1.92

2.08 2.01 1.96 1.91 1.87

2.03 1.96 1.90 1.86 1.82

1.97 1.90 1.85 1.80 1.76

16 17 18 19 20

3.05 3.03 3.01 2.99 2.97

2.67 2.64 2.62 2.61 2.59

2.46 2.44 2.42 2.40 2.38

2.33 2.31 2.29 2.27 2.25

2.24 2.22 2.20 2.18 2.16

2.18 2.15 2.13 2.11 2.09

2.09 2.06 2.04 2.02 2.00

1.99 1.96 1.93 1.91 1.89

1.94 1.91 1.89 1.86 1.84

1.89 1.86 1.84 1.81 1.79

1.84 1.81 1.78 1.76 1.74

1.78 1.75 1.72 1.70 1.68

1.72 1.69 1.66 1.63 1.61

21 22 23 24 25

2.96 2.95 2.94 2.93 2.92

2.57 2.56 2.55 2.54 2.53

2.36 2.35 2.34 2.33 2.32

2.23 2.22 2.21 2.19 2.18

2.14 2.13 2.11 2.10 2.09

2.08 2.06 2.05 2.04 2.02

1.98 1.97 1.95 1.94 1.93

1.87 1.86 1.84 1.83 1.82

1.83 1.81 1.80 1.78 1.77

1.78 1.76 1.74 1.73 1.72

1.72 1.70 1.69 1.67 1.66

1.66 1.64 1.62 1.61 1.59

1.59 1.57 1.55 1.53 1.52

26 27 28 29

2.91 2.90 2.89 2.89 2.88

2.52 2.51 2.50 2.50 2.49

2.31 2.30 2.29 2.28 2.28

2.17 2.17 2.16 2.15 2.14

2.08 2;07 2.06 2.06 2.05

2.01 2.00 2.00 1.99 1.98

1.92 1.91 1.90 1.89 1.88

1.81 1.80 1.79 1.78 1.77

1.76 1.75 1.74 1.73 1.72

1.71 1.70 1.69 1.68 1.67

1.65 1.64 1.63 1.62 1.61

1.58 1.57 1.56 1.55 1.54

1.50 1.49 1.48 1.47 1.46

2.84 2.79 2.75 2.71

2.44 2.39 2.35 2.30

2.23 2.18 2.13 2.08

2.09 2.04 1.99 1.94

2.00 1.95 1.90 1.85

1.93 1.87 1.82 1.77

1.83 1.77 1.72 1.67

1.71 1.66 1.60 1.55

1.66 1.60 1.55 1.49

1.61 1.54 1.48 1.42

1.54 1.48 1.41 1.34

1.47 1.40 1.32 1.24

1.38 1.29 1.19 1.00

1

30 40 60 120 m

988

PROBABILITY FUNCTIONS

PERCENTAGE POINTS OF THE F-DISTRIBUTION-VALUES OF F IN TERMS OF Q, V,, V? Q(F(4,q) =0.025

Table 26.9

3

4

5

6

8

12

15

20

30

647.8 38.51 17.44 12.22 10.01

799.5 39.00 16.04 10.65 8.43

864.2 39.17 15.44 9.98 7.76

899.6 39.25 15.10 9.60 7.39

921.8 39.30 14.88 9.36 7.15

937.1 39.33 14.73 9.20 6.98

956.7 39.37 14.54 8.98 6.76

976.7 39.41 14.34 8.75 6.52

984.9 39.43 14.25 8.66 6.43

993.1 39.45 14.17 8.56 6.33

1001 39.46 14.08 8.46 6.23

1010 39.48 13.99 8.36 6.12

1018 39.50 13.90 8.26 6.02

6 7 8

9 10

8.81 8.07 7.57 7.21 6.94

7.26 6.54 6.06 5.71 5.46

6.60 5.89 5.42 5.08 4.83

6.23 5.52 5.05 4.72 4.47

5.99 5.29 4.82 4.48 4.24

5.82 5.12 4.65 4.32 4.07

5.60 4.90 4.43 4.10 3.85

5.37 4.67 4.20 3.87 3.62

5.27 4.57 4.10 3.77 3.52

5.17 4.47 4.00 3.67 3.42

5.07 4.36 3.89 3.56 3.31

4.96 4.25 3.78 3.45 3.20

4.85 4.14 3.67 3.33 3.08

11 12 13 14 15

6.72 6.55 6.41 6.30 6.20

5.26 5.10 4.97 4.86 4.77

4.63 4.47 4.35 4.24 4.15

4.28 4.12 4.00 3.89 3.80

4.04 3.89 3.77 3.66 3.58

3.88 3.73 3.60 3.50 3.41

3.66 3.51 3.39 3.29 3.20

3.43 3.28 3.15 3.05 2.96

3.33 3.18 3.05 2.95 2.86

3.23 3.07 2.95 2.84 2.76

3.12 2.96 2.84 2.73 2.64

3.00 2.85 2.72 2.61 2.52

2.88 2.72 2.60 2.49 2.40

16 17 18 19 20

6.12 6.04 5.98 5.92 5.87

4.69 4.62 4.56 4.51 4.46

4.08 4.01 3.95 3.90 3.86

3.73 3.66 3.61 3.56 3.51

3.50 3.44 3.38 3.33 3.29

3.34 3.28 3.22 3.17 3.13

3.12 3.06 3.01 2.96 2.91

2.89 2.82 2.77 2.12 2.68

2.79 2.72 2.67 2.62 2.57

2.68 2.62 2.56 2.51 2.46

2.57 2.50 2.44 2.39 2.35

2.45 2.38 2.32 2.27 2.22

2.32 2.25 2.19 2.13 2.09

21 22 23 24 25

5.83 5.79 5.75 5.72 5.69

4.42 4.38 4.35 4.32 4.29

3.82 3.78 3.75 3.72 3.69

3.48 3.44 3.41 3.38 3.35

3.25 3.22 3.18 3.15 3.13

3.09 3.05 3.02 2.99 2.97

2.07 2.84 2.81 2.78 2.75

2.64 2.60 2.51 2.54 2.51

2.53 2.50 2.47 2.44 2.41

2.42 2.39 2.36 2.33 2.30

2.31 2.27 2.24 2.21 2.18

2.18 2.14 2.11 2.08 2.05

2.04 2.00 1.97 1.94 1.91

26 27 28 29 30

5.66 5.63 5.61 5.59 5.57

4.27 4.24 4.22 4.20 4.18

3.67 3.65 3.63 3.61 3.59

3.33 3.31 3.29 3.27 3.25

3.10 3.08 3.06 3.04 3.03

2.94 2.92 2.90 2.88 2.87

2.73 2.71 2.69 2.67 2.65

2.49 2.47 2.45 2.43 2.41

2.39 2.36 2.34 2.32 2.31

2.28 2.25 2.23 2.21 2.20

2.16 2.13 2.11 2.09 2.07

2.03 2.00 1.98 1.96 1.94

1.88 1.85 1.83 1.81 1.79

40 60 120

5.42 5.29 5.15 5.02

4.05 3.93 3.80 3.69

3.46 3.34 3.23 3.12

3.13 3.01 2.89 2.79

2.90 2.79 2.67 2.57

2.74 2.63 2.52 2.41

2.53 2.41 2.30 2.19

2.29 2.17 2.05 1.94

2.18 2.06 1.94 1.83

2.07 1.94 1.82 1.71

1.94 1.82 1.69 1.57

1.80 1.67 1.53 1.39

1.64 1.48 1.31 1.00

2

3

4

5

m

60

-

2 1 2 3 4 5

Q(F~V,,vz)=0.01 K\Vl 1 1 4052 2 98.50 3 34.12 4 21.20 5 16.26

20

30

60

4999.5 99.00 30.82 18.00 13.27

5403 99.17 29.46 16.69 12.06

5625 99.25 28.71 15.98 11.39

5764 99.30 28.24 15.52 10.97

5859 99.33 27.91 15.21 10.67

5982 99.37 27.49 14.80 10.29

6106 99.42 27.05 14.37 9.89

6157 99.43 26.87 14.20 9.72

6209 99.45 26.69 14.02 9.55

6261 99.47 2h.50 13.84 9.38

6313 99.48 26.32 13.65 9.20

6366 99.50 26.13 13.46 9.02

6

8

12

15

m

6 7 8 9 10

13.15 12.25 11.26 10.56 10.04

10.92 9.55 8.65 8.02 7.56

9.78 8.45 7.59 6.99 6.55

9.15 7.85 7.01 6.42 5.99

8.75 7.46 6.63 6.06 5.64

8.47 7.19 6.37 5.80 5.39

8.10 6.84 6.03 5.47 5.06

7.72 6.47 5.67 5.11 4.71

7.56 6.31 5.52 4.96 4.56

7.40 6.16 5.36 4.81 4.41

7.23 5.99 5.20 4.65 4.25

7.06 5.82 5.03 4.48 4.08

6.88 5.65 4.86 4.31 3.91

11 12 13 14

9.65 9.33 9.07 8.86

7.21 6.93 6.70 6.51 6.36

6.22 5.95 5.74 5.56 5.42

5.67 5.41 5.21 5.04 4.89

5.32 5.06 4.86 4.69 4.56

5.01 4.82 4.62 4.46 4.32

4.74 4.50 4.30 4.14 4.00

4.40 4.16 3.96 3.80 3.67

4.25 4.01 3.82 3.66 3.52

4.10 3.86 3.66 3.51 3.37

3.94 3.70 3.51 3.35 3.21

3.18 3.54 3.34 3.18 3.05

3.60 3.36 3.17 3.00 2.87

16 17 18 19 20

8.53 8;40 8.29 8.18 8.10

6.23 6-11 6.01 5.93 5.85

5.29 5.18 5.09 5.01 4.94

4.77 4.67 4.58 4.50 4.43

4.44 4.34 4.25 4.17 4.10

4.20 4.10 4.01 3.94 3.87

3.89 3.79 3.71 3.63 3.56

3.55 3.46 3.37 3.30 3.23

3.41 3.31 3.23 3.15 3.09

3.26 3.16 3.08 3.00 2.94

3.10 3.00 2.92 2.84 2.78

2.93 2.83 2.75 2.67 2.61

2.75 2.65 2.57 2.49 2.42

21 22 23 24 25

8.02 1.95 7.88 7.82 7.77

5.78 5.72 5.66 5.61 5.57

4.87 4.82 4.76 4.72 4.68

4.37 4.31 4.26 4.22 4.18

4.04 3.99 3.94 3.90 3.85

3.81 3.76 3.71 3.67 3.63

3.51 3.45 3.41 3.36 3.32

3.17 3.12 3.07 3.03 2.99

3.03 2.98 2.93 2.89 2.85

2.88 2.83 2.78 2.74 2.70

2.72 2.67 2.62 2.58 2.54

2.55 2.50 2.45 2.40 2.36

2.36 2.31 2.26 2.21 2.17

26 27 28 29 30

7.72 7.68 1.64 7.60 7.56

5.53 5.49 5.45 5.42 5.39

4.64 4.60 4.57 4.54 4.51

4.14 4.11 4.07 4.04 4.02

3.82 3.78 3.75 3.73 3.70

3.59 3.56 3.53 3.50 3.47

3.29 3.26 3.23 3.20 3.17

2.96 2.93 2.90 2.87 2.84

2.81 2.78 2.75 2.73 2.70

2.66 2.63 2.60 2.57 2.55

2.50 2.47 2.44 2.41 2.39

2.33 2.29 2.26 2.23 2.21

2.13 2.10 2.06 2.03 2.01

40 60 120

7.31 7.08 6.85 6.63

5.18 4.98 4.79 4.61

4.31 4.13 3.95 3.78

3.83 3.65 3.48 3.32

3.51 3.34 3.17 3.02

3.29 3.12 2.96 2.80

2.99 2.82 2.66 2.51

2.66 2.50 2.34 2.18

2.52 2.35 2.19 2.04

2.37 2.20 2.03 1.88

2.20 2.03 1.86 1.70

2.02 1.84 1.66 1.47

1.80 1.60 1.38 1.00

m

989

PROBABILITY FUNCTIONS

Tuble 26.9

PERCEFTAGE POlNTS OF THE F-DlSTRlBL"~lO\-\ ALUES OF F I N TERMS OF Q, Y,,Y, Q(Flv,,v?)=0.005 vz v1

1

1 16211 2 198.5 3 55.55 4 31.33 5 22.78 6 7 8 9 10

4

5

6

8

12

5;47

9.59 7.75 6.61 5.83 5.27

9.36 7.53 6.40 5.62 5.07

9.12 7.31 6.18 5.41 4.86

8.88 7.08 5.95 5.19 4.64

5.05 4.72 4.46 4.25 4.07

4.86 4.53 4;27 4.06 3.88

4.65 4.33 4.07 3.86 3.69

4.44 4.12 3.87 3.66 3.48

4.23 3.90 3.65 3.44 3.26

4.10 3.97 3.86 3.76 3.68

3.92 3.79 3.68 3.59 3.50

3.73 3.61 3.50 3.40 3.32

3.54 3.41 3.30 3.21 3.12

3.33 3.21 3.10 3.00 2.92

3.11 2.98 2.87 2.78 2.69

4.01 3.94 3.88 3.83 3.78

3.60 3.54 3.47 3.42 3.37

3.43 3.36 3.30 3.25 3.20

3.24 3.18 3.12 3.06 3.01

3.05 2.98 2.92 2.87 2.82

2.84 2.77 2.71 2.66 2.61

2.61 2.55 2.48 2.43 2.38

3.73 .. 3.69 3.65 3.61 3.58

3.33 3.28 3.25 3.21 3.18

3.15 3.11 3.07 3.04 A01

2.97 2;93 2.89 2.86 2.82

2.77 2.73 2.69 2.66 2.63

2.56 2.52 2.48 2.45 2.42

2.33 2.29 2.25 2.21 2.18

3.35 3.13 2.93 2.74

2.95 2.74 2.54 2.36

2.78 2.57 2.37 2.19

2.60 2.39 2.19 2.00

2.40 2.19 1.98 1.79

?.18 1.96 1.75 1.55

1.93 1.69 1.43 1.00

14.54 12.40 11.04 10.11 9.43

li.ii 10.88 9.60 8.72 8.08

12.03 10.05 8.81 7.96 7.34

11.46 9.52 8.30 6.87

11.07 9.16 7.95 7.13 6.54

10.57 8.68 7.50 6.69 6.12

10.03 8.18 7.01 6.23 5.66

6.88 6.52 6.23 6.00 5.80

6.42 6.07 5.79 5.56 5.37

6.10 5.76 5.48 5.26 5.07

5.68 5.35 5.08 4.86 4.67

5.24

5.64 5.5c 5.37 5.27 5.17

5.21 5.07 4.96 4.85 4.76

4.91 4.78 4.66 4.56 4.47

4.52 4.39 4.28 4.18 4.09

5.09 5.02 4.95 4.89 4.84

4.68 4.61 4.54 4.49 4.43

4.39 4.32 4.26 4.21 4.15

4.79 4.74 4.70 4.b6 4.62

4.38 4.34 4.30 4.26 4.23

4.37 4.14 3.92 3.72

3.99 3;76 3.55 3.35

4

5

21 22 23 24 25

9.83 9.73 9.63 9.55 9.48

6.89 6.81 6.73 6.66 6.60

26 27 28 29 30

9.41 9.34 9.28 9.23 9.18

6.54 6.49 6.44 6.40 6.35

40 60 120

8.83 8.49 8.18 7.88

6.07 5.79 5.54 5.30

2 ( 5 ) 5.000 999.0 148.5 61.25 37.12

5.73 5.65 5.58 5.52 5.46

4.98 .~ 4.73 4.50 4.28

3 ( 5 ) 5.404 999.2 141;l 56.18 33.20

( 5 ) 5.625 999.2 137.1 53.44 31.09

3.71 3.49 3.28 3.09

, v2) = 0.,001 Q(FIv~ 12 6 8

( 5 ) 5.764 999.3 134.6 51.71 29.75

( 5 ) 5.859 999.3 132.8 50.53 28.84

m

25465 199.5 41.83 19.32 12.14

24426 199.4 43.39 20.70 13.38

7.51 7.35 7.21 7.09 6.99

60 25253 199.5 42.15 19.61 12.40

23925 199.4 44.13 21.35 13.96

10.58 10.38 10.22 10.07 9.94

30 25044 199.5 42.47 19.89 12.66

23437 199.3 44.84 21.97 14.51

16 17 18 19 20

20 24836 199.4 42.78 20.17 12.90

23056 199.3 45.35 22.46 14.94

7.... A7

15 24630 199.4 43.08 20.44 13.15

22500 199.2 46.19 23.15 15.56

12.23 11.75 11.37 11.06 10.80

v24 1 1 (5)4.053 2 998.5 3 167.0 4 74.14 5 47;18

3 21615 199.2 47.47 24.26 16.53

11 12 13 14 15

m

~

18.63 16.24 14.69 13.61 12.83

2 20000 199.0 49.80 26.28 18.31

( 5 ) 5.981 999.4 130.6 49.00 27.64

( 5 ) 6.107 999.4 128.3 47.41 26.42

15 ( 5 ) 6.158 999.4 127.4 46.76 25.91

20 (5)6.209 999.4 126.4 46.10 25.39

30

60

(5)6.261 999.5 125.4 45.43 24.87

( 5 ) 6.313 999.5 124.5 44.75 24.33

m

( 5 ) 6.366 999.5 123;5 44.05 23.79

6 7 8 9 10

35.51 29.25 25.42 22.86 21.04

27.00 21.69 18.49 16.39 14.91

23.70 18.77 15.83 13.90 12.55

21.92 17.19 14.39 12.56 11.28

20.81 16-21 13.49 11.71 10.48

20.03 15.52 12.86 11.13 9.92

19.03 14.63 12.04 10.37 9.20

17.99 13.71 11.19 9.57 8.45

17.56 13.32 10.84 9.24 8.13

17.12 12.93 10.48 8.90 7.80

16.67 12.53 10.11 8.55 7.47

16.21 12.12 9.73 8.19 7.12

15.75 11.70 9.33 7.81 6.76

11 1; 13 14 15

19.69 18.64 17.81 17.14 16.59

13.81 12.97 12.31 11.78 11.34

11.56 10.80 10.21 9.73 9134

10.35 9.63 9.07 8.62 8;25

9.58 8.89 8.35 7.92 7.57

9.05 8.38 7.86 7.43 7.09

8.35 7.71 7.21 6.80 6.47

7.63 7.00 6.52 6.13 5.81

7.32 6.71 6.23 5.85 5.54

7.01 6.40 5.93 5.56 5.25

6.68 6.09 5.63 5.25 4.95

6.35 5.76 5.30 4.94 4.64

6.00 5.42 4.97 4.60 4.31

16 17 18 19 20

16.12 15.72 15.38 15.08 14.82

10.97 10.66 10.39 10.16 9.95

9.00 8.73 8.49 8.28 8.10

7.94 7.68 7.46 7.26 7.10

7.27 7.02 6.81 6.62 6.46

6.81 6.56 6.35 6.18 6.02

6.19 5.96 5.76 5.59 5.44

5.55 5.32 5.13 4.97 4.82

5.27 5.05 4.87 4.70 4.56

4.99 4.78 4.59 4.43 4.29

4.70 4.48 4.30 4.14 4.00

4.39 4ii8 4.00 3.84 3.70

4.06 3.85 3.67 3.51 3.38

21 22 23 24 25

14.59 14.38 14.19 14.03 13.88

9.77 9.61 9.47 9.34 9.22

7.94 7.80 7.67 7.55 7.45

6.95 6.81 6.69 6.59 6.49

6.32 6.19 6.08 5.98 5.88

5.88 5.76 5.65 5.55 5.46

5.31 5i19 5.09 4.99 4.91

4.70 4.58 4.48 4.39 4.31

4.44 4.33 4;23 4.14 4.06

4.17 4.06 3.96 3.87 3.79

3.88 3.78 3.68 3.59 3.52

3.58 3.48 3.38 3.29 3.22

3.26 3.15 3.05 2.97 2.89

26 27 28 29 30

13.74 13.61 13.50 13.39 13.29

9.12 9.02 8.93 8.85 8.77

7.36 7.27 7.19 7.12 7.05

6.41 6.33 6.25 6.19

5.80 5.73 5.66 5.59 5.53

5.38 5.31 5.24 5.18 5.12

4.83 4.76 4.69 4.64 4.58

4.24 4.17 4.11 4.05 4.00

3.99 3.92 3.86 3.80 3.75

3.72 3.66 3.60 3.54 3.49

3.44 3.38 3.32 3.27 3.22

3.15 3.08 3.02 2.97 2.92

2.82 2.75 2.69 2.64 2.59

40 60 120

12.61 11.97 11.38 10.83

8.25 7.76 7.32 6.91

6.60 6.17 5.79 5.42

5.70 5.31 4.95 4.62

5.13 4.76 4.42 4.10

4.73 4.37 4.04 3.74

4.21 3.87 3.55 3.27

3.64 3.31 3.02 2.74

3.40 3.08 2.78 2.51

3.15 2.83 2.53 2.27

2.87 2.55 2.26 1.99

m

*See page 11.

2.23 1.89 1.54 1.00

990

PROBABILITY FUNCTIONS

Taldc 26.1 0 PEHCE~TNE i

w w s OF ‘rim ~-I)ISTRIBI T I O Y - \

A I J ISS 01:I I\

~ I ~ I I ~ oI IyS,.I A ~ YD

0.5

0.8

0.9

0.95 0.98

0.99

5

0.325 0.289 0.277 0.271 0.267

1.000 0.816 0.765 0.741 0.727

3.078 1.886 1.638 1.533 1.476

6.314 2.920 2.353 2.132 2.015

12.706 4.303 3.182 2.776 2.571

31.821 6.965 4.541 3.747 3.365

63.657 9.925 5.841 4.604 4.032

127.321 14.089 7.453 5.598 4.773

318.309 22.327 10.214 7.173 5.893

636.619 31.598 12.924 8.610 6.869

6366.198 99.992 28.000 15.544 11.178

63661.977 316.225 60.397 27.771 17.897

5 7 8 9 10

0.265 0;263 0.262 0.261 0.260

0.718 0.711 0.706 0.703 0.700

1.440 1.415 1.397 1.383 1.372

1.943, 1.895 1.860 1.833 1.812

2.447 2;365 2.306 2.267 2.228

3.143 2.998 2.896 2.821 2.764

3.707 3.499 3.355 3.250 3.169

4.317 4.029 3.833 3.690 3.581

5.208 4.785 4.501 4.297 4.144

5.959 .. . 5.408 5.041 4.781 4.587

9.082 7.885 7.120 6.594 6.211

11.215 9.782 8.827 8.150

15.764 13.257 11.637 10.516

11 12 13 14 15

0.260 0.259 0.259 0.258 0.258

0.697 0.695 0.694 0.692 0.691

1.363 1.356 1.350 1.345 1.341

1.796 1.782 1.771 1.761 1.753

2.201 2.179 2.160 2.145 2.131

2.718 2.681 2.650 .2.624 2.602

3.106 3.055 3.012 2.977 2.947

3.497 3.428 3.372 3.326 3.286

4.025 3.930 3.852 3.787 3.733

4.437 4.318 4.221 4.140 4:073

5.921 5.694 5.513 5.363 5.239

7.648 7.261 6.955

-.. --

9.702 9.085 8.604

A in6

R >lR w.111

6.502

7.903

16 17 18 19 20

0.258 0.257 0.257 0.257 0.257

0.690 0.689 0.688 0.688 0.687

1.337 1.333 1.330 1.328 1.325

1.746 1.740 1.734 1.729 1.725

2.120 2.110 2.101 2.093 2.086

2.583 2.567 2.552 2.539 2.528

2.921 2.898 2.878 2.861 2.845

3.252 3.223 3.197 3.174 3.153

3.686 3.646 3.610 3.579 3.552

4.015 3.965 3.922 3.883 3.850

5.134 5.044 4.966 4.897 4.837

6.330 6.184 6.059 5.949 5.854

7.642 7.421 7.232 7.069 6.927

21 22

0.257 0.256

0.686 0.686

1.323 1.321

1.721 1.717

2.080 2.074

2.518 2.508

2.831 2.819

3.135 3.119

3.527 3.505

3.819 3.792

4.784 4.736

6 76.9

A

26 27 28 29 30

0.256 0.256 0.256 0.256 0.256

0.684 0.684 0.683 0.683 0.683

1.315 1.314 1.313 1.311 1.310

1.706 1.703 1.701 1.699 1.697

2.056 2.052 2.048 2.045 2.042

2.479 2.473 2.467 2.462 2.457

2.779 2.771 2.763 2.756 2.750

3.067 3.057 3.047 3.038 3.030

3.435 3.421 3.408 3.396 3.385

3.707 3;690 3.674 3.659 3.646

4.587 4;558 4.530 4.506 4.482

5.415 5.373 5.335 5.299

6.286 6.225 6.170 6.119

40 60 120

0.255 0.254 0.254 0.253

0.681 0.679 0.677 0.674

1.303 1.296 1.289 1.282

1.684 1.671 1.658 1.645

2.021 2.000 1.980 1.960

2.423 2.390 2.358 2.326

2.704 2.660 2.617 2.576

2.971 2.915 2.860 2.807

3.307 3.232 3.160 3.090

3.551 3.460 3.373 3.291

4.321 4.169 4.025 3.891

5.053 4.825 4.613 4.417

5.768 5.449 5.158 4.892

v \ A 0.2 1 2 3 4

m

0.995 0.998 0.999

0.9999 0.99999 0.999999

*

636619.772 999.999 130.155 49.459 28.477

13 555 -_.---

m

5 4 6-1 _.

*

From E. S. Pearson and H. 0. Hartley (editors), Biometrika tables for statisticians, vol. I. Cambridge Univ. Press, Cambridge, England, 1954 for A 0.999, from E. T. Federighi, Extended tables of the percentage points of Student’s t-distribution, J. Amer. Statist. Assoc. 54, 683-688 (1959) for A 0.999 (with permission).

n47

.

ano

A ‘Zc.3

I.>>-

*

991

PROBABILITY FUNCTIONS

2500 FIVE DIGIT RANDOM NUMBERS

Table! 26.11

53479 97344 66023 99776 30176

81115 70328 38277 75723 48979

98036 58116 74523 03172 92153

12217 91964 71118 43112 38416

59526 26240 84892 83086 42436

40238 44643 13956 81982 26636

40577 83287 98899 14538 83903

39351 97391 92315 26162 44722

43211 92823 65783 24899 69210

69255 77578 59640 20551 69117

81874 19839 09337 31151 67619

83339 90630 33435 58295 52515

14988 71863 53869 40823 03037

99937 95053 52769 41330 81699

13213 55532 18801 21093 17106

30177 60908 25820 93882 64982

47967 84108 96198 49192 60834

93793 55342 66518 44876 85319

86693 48479 78314 47185 47814

98854 63799 97013 81425 08075

61946 04811 05763 73260 54909

48790 64892 39601 56877 09976

11602 96346 56140 40794 76580

83043 79065 25513 13948 02645

22257 26999 86151 96289 35795

11832 43967 78657 90185 44537

04344 63485 02184 47111 64428

95541 93572 29715 66807 35441

20366 80753 04334 61849 28318

55937 96582 15678 44686 99001

42583 27266 49843 29316 30463

36335 27403 11442 40460 27856

60068 97520 66682 27076 67798

04044 23334 36055 69232 16837

29678 36453 32002 51423 74273

16342 33699 78600 58515 05793

48592 23672 36924 49920 02900

25547 45884 59962 03901 63498

63177 41515 68191 26597 00782

75225 04756 62580 33068 35097

28708 13183 60796 13486 34914

84088 50652 76639 46918 94502

65535 94872 30157 64683 39374

44258 28257 40295 07411 34185

33869 78547 99476 77842 57500

82530 55286 28334 01908 22514

98399 33591 15368 47796 04060

26387 61565 42481 65796 94511

02836 51723 60312 44230 44612

36838 14211 42770 77230 10485

28105 59231 87437 29046 62035

04814 45028 82758 01301 71886

85170 01173 71093 55343 94506

86490 08848 36833 65732 15263

35695 81925 53582 78714 61435

03483 71494 25986 43644 10369

57315 95401 46005 46248 42054

63174 34049 42840 53205 68257

71902 04851 81683 94868 14385

71182 65914 21459 48711 79436

38856 40666 40588 78237 98247

80048 43328 90087 86556 67474

59973 87379 37729 50276 71455

73368 86418 08667 20431 69540

52876 95841 37256 00243 01169

47673 25590 20317 02303 03320

41020 54137 53316 71029 67017

82295 94182 50982 49932 92543

26430 42308 32900 23245 97977

87377 07361 32097 00862 52728

69977 39843 62880 56138 90804

78558 23074 87277 64927 56026

65430 40814 99895 29454 48994

32627 03713 99965 52967 64569

28312 21891 34374 86624 67465

61815 96353 42556 62422 60180

14598 96806 11679 30163 12972

79728 24595 99605 76181 03848

55699 26203 98011 95317 62582

91348 26009 48867 39264 93855

09665 34756 12157 69384 93358

44672 50403 73327 07734 64565

74762 76634 74196 94451 43766

33357 12767 26668 76428 45041

67301 32220 78087 16121 44930

80546 34545 53636 09300 69970

97659 18100 52304 67417 16964

11348 53513 00007 68587 08277

78771 14521 05708 87932 67752

45011 72120 63538 38840 60292

38879 58314 83568 28067 05730

35544 60298 10227 91152 75557

99563 72394 99471 40568 93161

85404 69668 74729 33705 80921

04913 12474 22075 64510 55873

62547 93059 10233 07067 54103

78406 02053 21575 64374 34801

01017 29807 20325 26336 83157

86187 63645 21317 79652 04534

22072 12792 57124 31140 81368

Compiled from Rand Corporation, A million random digits with 100,000 normal deviates. The Free Press, Glencoe, Ill., 1955 (with permission).

992

PROBABILITY FUNCTIONS

Table 26.11

2500 FIVE DIGIT RANDOM NUMBERS

26687 60675 45418 69872 03765

74223 75169 98635 48026 86366

43546 24510 83123 89755 99539

45699 15100 98558 28470 44183

94469 02011 09953 44130 23886

82125 14375 60255 59979 89977

37370 65187 42071 91063 11964

23966 10630 40930 28766 51581

68926 37664 64421 66745 97992 93085 85962 77173 18033 56239

84686 91512 10737 54870 48967

57636 49670 49307 19676 49579

32326 32556 18307 58367 65369

19867 85189 22246 20905 74305

71345 28023 22461 38324 62085

42002 88151 10003 00026 39297

96997 62896 93157 98440 10309

84379 95498 66984 37427 23173

27991 29423 44919 22896 74212

21459 38138 30467 37637 32272

91430 92564 41734 25251 91657

79112 03685 29567 47476 12199 '77441 78110 54178 11563 66036

05411 62804 92415 78241 28523

23027 73428 63542 09226 83705

54735 04535 42115 87529 09956

91550 86395 84972 35376 76610

06250 12162 12454 90690 88116

18705 59647 33133 54178 78351

18909 97726 48467 08561 50877

00149 53250 25587 01176 83531

84745 73200 17481 12182 15544

63222 84066 56716 06882 40834

50533 59620 49749 27562 20296

50159 61009 70733 75456 88576

60433 38542 32733 54261 47815

04822 05758 60365 38564 96540

49577 06178 14108 89054 79462

89049 80193 52573 96911 78666

16162 26466 39391 88906 25353

19902 98866 96516 78705 99417 56171 77699 57853 32245 83794

32805 25556 19848 93213 99528

61091 35181 24352 27342 05150

91587 29064 51844 28906 27246

30340 84909 64047 49005 29843 68949 03791 72127 57958 31052 65815 21637 48263 62156 62469

67750 50506 08366 49385 97048

87638 45862 43190 75406 16511

12874 72753 63899 41910 16255 43271 75553 30207 41772 18441

66469 45484 26540 41814 34685

13782 55461 41298 74985 13892

64330 66518 35095 40223 38843

00056 82486 32170 91223 69007

73324 74694 70625 64238 10362

13193 09724 01050 83100 08814

19466 76490 44225 92041 66785

09270 85058 80222 83901 36303

01245 17815 8'572 88028 57833

81765 71551 62758 56743 77622

06809 36356 14858 25598 02238

10561 97519 36350 79349 53285

1 00 8 0 54144 23304 47880 77316

'I7 482 05471 82273 06902 51132 70453 77912 40106

83169 21065 52020 38456

27373 63812 84305 92214

68609 29860 02897 54278

91543 14415 82465 27306 91960

63886 33816 07781 39843 82766

60539 96334 78231 87674 09938 66874 05634 96368 02331 08797

20804 96473 72128 72022 33858

72692 08944 44451 25098 99685 84329 01278 92830 21847 17391

02870 29296 14530 40094 53755

74892 50679 08410 31776 58079

22598 07798 45953 41822 48498

59284 10428 65527 59688 44452

96108 96003 41039 43078 10188

91610 71223 79574 93275 43565

07483 21352 05105 31978 46531

37943 78685 59588 08768 93023

96832 55964 02115 84805 07618

87275 94155 26488 37073 a3835

82013 93110 76394 34547 139575

59804 49964 91282 88296 55956

78595 27753 03419 68638 93957

60553 85090 68758 12976 30361

14038 12096 77677 69303 89575 66469 50896 10023 47679 83001

03920 07865 66407 73012 84125

15444 12091 35510 94805 33446 56780 50661 18523 12910 60934 95472 66323 97835 27220 35056

36690 58317 23422 04492 18402 36279 83235 50602 53403 18401 42736 77811 66681 05785 07103

08573 22791 03171 77538 63072

993

PROBABILITY FUNCTIONS

2500 FIVE DIGIT RANDOM NUMBERS

Table 26.11

55034 25521 85421 61219 20230

81217 99536 72744 48390 03147

90564 43233 97242 47344 58854

81943 48786 66383 30413 11650

11241 49221 00132 39392 28415

84512 06960 05661 91365 12821

12288 31564 96442 56203 58931

89862 21458 37388 79204 30508

00760 88199 57671 05330 65989

76159 06312 27916 31196 26675

95776 07603 00645 62950 79350

83206 17344 17459 83162 10276

56144 01148 78742 61504 81933

55953 83300 39005 31557 26347

89787 96955 36027 80590 08068

64426 65027 98807 47893 67816

08448 31713 72666 72360 06659

45707 89013 54484 72720 87917

80364 79557 68262 08396 74166

60262 49755 38827 33674 85519

48339 05842 25855 25272 73003

69834 08439 02209 16152 29058

59047 79836 07307 82323 17605

82175 50957 59942 70718 49298

92010 32059 71389 98081 47675

58446 32910 76159 38631 90445

69591 15842 11263 91956 68919

56205 13918 38787 49909 05676

95700 41365 61541 76253 23823

86211 80115 22606 33970 84892

81310 10024 84671 29296 51771

94430 44713 52806 58162 94074

22663 59832 89124 21858 70630

06584 80721 37691 33732 41286

38142 63711 20897 94056 90583

00146 67882 82339 88806 87680

17496 25100 22627 54603 13961

51115 45345 06142 00384 55627

61458 55743 05773 66340 23670

65790 67618 03547 69232 35109

42166 78355 09552 15771 13231

56251 67041 51347 63127 99058

60770 22492 33864 34847 93754

51672 51522 89018 05660 36730

36031 31164 73418 06156 44286

77273 30450 81538 48970 44326

85218 27600 77399 55699 15729

14812 44428 30448 61818 37500

90758 96380 97740 91763 47269

23677 26772 18158 20821 13333

50583 99485 54676 99343 35492

03570 57330 39524 71549 40231

38472 10634 73785 10248 34868

73236 74905 48864 76036 55356

67613 90671 69835 31702 12847

72780 19643 62798 76868 68093

78174 69903 65205 88909 52643

18718 60950 69187 69574 32732

99092 17968 05572 27642 67016

64114 37217 74741 00336 46784

98170 02670 36934 56851 05464

25384 86155 42879 12778 28892

03841 56860 81637 24309 14271

23920 02592 79952 73660 23778

47954 01646 07066 84264 88599

10359 42200 41625 24668 17081

70114 79950 96804 16686 33884

11177 37764 92388 02239 88783

63298 82341 88860 66022 39015

99903 71952 68580 64133 57118

15025 95610 09026 81431 21431

20237 08030 40378 99955 59335

63386 81469 05731 52462 58627

71122 91066 55128 67667 94822

06620 88857 74298 97322 65484

07415 56583 49196 69808 09641

94982 01224 31669 21240 41018

32324 28097 42605 65921 85100

79427 19726 30368 12629 16110

70387 71465 96424 92896 32077

95832 99813 77210 13268 44285

76145 44631 31148 02609 71735

11636 43746 50543 79833 26620

80284 99790 11603 66058 54691

17787 86823 50934 80277 14909

97934 12114 02498 08533 52132

12822 31706 09184 28676 81110

73890 05024 95875 37532 74548

66009 28156 85840 70535 78853

27521 04202 71954 82356 31996

70526 88386 83161 50214 97689

45953 11222 73994 71’721 29341

79637 25080 17209 33851 67747

57374 71462 79441 45144 80643

05053 09818 64091 05696 13620

31965 46001 49790 29935 23943

33376 19065 11936 12823 49396

13232 68981 44864 01594 83686

85666 18310 86978 08453 37302

86615 74178 34538 52825 95350

994

PROBABILITY FUNCTIONS

Table 26.11

2500 FIVE DIGIT R ANDOM NUMB ERS

12367 38890 80788 02395 73720

23891 30239 55410 77585 70184

31506 34237 39770 08854 69112

90721 22578 93317 23562 71887

18710 74420 18270 33544 80140

89140 22734 21141 45796 72876

58595 26930 52085 10976 38984

99425 40604 78093 44721 23409

22840 10782 85638 24781 63957

08267 80128 81140 09690 44751

61383 39161 80907 09052 33425

17222 44282 74484 65670 24226

55234 14975 39884 63660 32043

18963 97498 19885 34035 60082

39006 25973 37311 06578 20418

93504 33605 04209 87837 85047

18273 60141 49675 28125 53570

49815 30030 39596 48883 32554

52802 77677 01052 50482 64099

69675 49294 43999 55735 52326

72651 04142 85226 54888 33258

69474 32092 14193 03579 51516

73648 83586 52213 91674 82032

71530 61825 60746 59502 45233

55454 35482 24414 08619 39351

19576 32736 57858 33790 33229

15552 63403 31884 29011 59464

20577 91499 51266 85193 65545

12124 37196 82293 62262 76809

50038 02762 73553 28684 16982

75973 90638 65061 64420 27175

15957 75314 15498 07427 17389

32405 35381 93348 82233 76963

82081 34451 33566 97812 75117

02214 49246 19427 39572 45580

57143 11465 66826 07766 99904

33526 25102 03044 65844 47160

47194 71489 97361 29980 55364

94526 89883 08159 15533 25666

73253 99708 47485 90114 25405

32215 54209 59286 83872 83310

30094 58043 66964 58167 57080

87276 72350 84843 01221 03366

56896 89828 71549 95558 80017

15625 02706 67553 22196 39601

32594 16815 33867 65905 40698

80663 89985 83011 38785 56434

08082 37380 66213 01355 64055

19422 44032 69372 47489 02495

80717 59366 23903 28170 50880

64545 39269 29763 06310 97541

29500 00076 05675 02998 47607

13351 55489 28193 01463 57655

78647 01524 65514 27738 59102

92628 76568 11954 90288 21851

19354 22571 78599 17697 44446

60479 20328 63902 64511 07976

57338 84623 21346 39552 54295

52133 30188 19219 34694 84671

07114 43904 90286 03211 78755

82968 76878 87394 74040 47896

85717 34727 78884 12731 41413

11619 12524 87237 59616 66431

97721 90642 92086 33697 70046

53513 16921 95633 12592 50793

53781 13669 66841 44891 45920

98941 17420 22906 67982 96564

38401 84483 64989 72972 67958

70939 68309 86952 89795 56369

11319 85241 54700 10587 44725

87778 96977 43820 57203 49065

71697 63143 13285 83960 72171

64148 72219 77811 40096 80939

54363 80040 81697 39234 06017

92114 11990 29937 65953 90323

34037 47698 70750 59911 63687

59061 95621 02029 91411 07932

62051 72990 32377 55573 99587

62049 29047 00556 88427 49014

33526 85893 86687 45573 26452

94250 68148 12208 88317 56728

84270 81382 97809 89705 80359

95798 82383 33619 26119 29613

13477 18674 28868 12416 63052

80139 40453 41646 19438 15251

26335 92828 16734 65665 44684

55169 30042 88860 60989 64681

73417 37412 32636 59766 42354

40766 43423 41985 11418 51029

45170 45138 84615 18250 77680

07138 21188 02154 90953 80103

12320 64554 12250 85238 91308

01073 55618 88738 32771 12858

19304 36088 43917 07305 41293

87042 24331 03655 36181 00325

58920 84390 21099 47420 15013

28454 16022 60805 19681 19579

81069 12200 63246 33184 91132

93978 77559 26842 41386 12720

66659 75661 35816 03249 92603

995

PROBABILITY FUNCTIONS

2500 FIVE DIGIT RASDORI NUMBlERS

Tal~ l e26.11

92630 79445 59654 31524 06348

78240 78735 71966 49587 76938

19267 71549 27386 76612 90379

95457 44843 50004 39789 51392

53497 26104 05358 13537 55887

23894 67318 94031 48086 71015

37708 00701 29281 59483 09209

79862 34986 18544 60680 79157

76471 66751 52429 84675 24440

66418 99723 06080 53014 30244

28703 68108 99938 91543 42103

51709 89266 90704 73196 02781

94456 94730 93621 34449 73920

48396 95761 66330 63513 56297

73780 75023 33393 83834 72678

06436 48464 95261 99411 12249

86641 65544 95349 58826 25270

69239 96583 51769 40456 36678

57662 18911 91616 69268 21313

80181 16391 33238 48562 75767

17138 28297 09331 31295 36146

27584 14280 56712 04204 15560

25296 54524 51333 93712 27592

28387 21618 06289 51287 42089

51350 95320 75345 05754 99281

61664 38174 08811 79396 59640

37893 05363 60579 08089 82711 57392 87399 51773 15221 96079

44143 94999 25252 33075 09961

42677 78460 30333 97061 05371

29553 23501 57888 55336 10087

18432 22642 85846 71264 10072

13630 63081 67967 88472 55980

05529 08191 07835 04334 64688

02791 89420 11314 63919 68239

81017 67800 01545 36394 20461

49027 55137 48535 11196 89381

79031 54707 17142 92470 93809

50912 32945 08552 70543 00796

09399 64522 67457 29776 95945

34101 53362 82975 54827 25464

81277 44940 66158 84673 59098

66090 60430 84731 22898 27436

88872 22834 19436 08094 89421

37818 14130 55790 14326 80754

72142 96593 69229 87038 89924

67140 23298 28661 42892 19097

50785 56203 13675 21127 67737

21380 92671 99318 30712 80368

16703 15925 76873 48489 08795

67609 44921 33170 84687 71886

60214 70924 30972 85445 56450

41475 61295 98130 06208 36567

84950 51137 95828 17654 09395

40133 47596 49786 51333 96951

02546 86735 13301 02878 35507

09570 35561 36081 35010 17555

45682 76649 80761 67578 35212

50165 18217 33985 61574 69106

15609 63446 68621 20749 01679

00475 25993 92882 25138 84631

02224 38881 53178 26810 71882

74722 68361 99195 07093 12991

14721 59560 93803 15677 83028

40215 41274 56985 60688 82484

21351 69742 53089 04410 90339

08596 40703 15305 24505 91950

45625 37993 50522 37890 74579

83981 03435 55900 67186 03539

63748 18873 43026 62829 90122

34003 53775 59316 20479 86180

92326 45749 97885 66557 84931

12793 05734 72807 50705 25455

61453 86169 54966 26999 26044

48121 42762 60859 09854 02227

74271 70175 11932 52591 52015

28363 97310 35265 14063 21820

66561 73894 71601 30214 50599

75220 88606 55577 19890 51671

35908 19994 67715 19292 65411

21451 98062 01788 62465 94324

68001 68375 64429 04841 31089

72710 80089 14430 43272 84159

40261 24135 94575 68702 92933

61281 72355 75153 01274 99989

13172 95428 94576 05437 89500

63819 11808 61393 22953 91586

48970 29740 96192 18946 02802

51732 81644 03227 99053 69471

54113 86610 32258 41690 68274

05797 10395 35177 25633 16464

43984 14289 56986 89619 48280

21575 52185 25549 75882 94254

09908 09721 59730 98256 45777

70221 25789 64718 02126 45150

19791 38562 52630 72099 68865

51578 54794 31100 57183 11382

36432 04897 62384 55887 11782

33494 59012 49483 09320 22695

79888 89251 11409 73463 41988

27. Miscellaneous Functions IRENE A. STEGUN

Contents 27.1. Debye Functions TL=

so tndt n. ...

. . . . . . .

,

. . . . .

Page

998

1(1)4, ~ = 0 ( . 1 1.4(.2)5(.5) ) 10, 6D

. . . . . .

999

. . . . . . . . . . . . . .

999

27.2. Planck’s Radiation Function z-s(el’z- l ) - l . . . . Z= .05 (.005). 1(.01).2 (.02).4 (.05).9(.1)1.5(.5)3.5, 3D %,,I, f(%ax>, 9-10s 27.3. Einstein Functions

x2ez (ez-

2

,

.

, ,

2 -ez- 1

In (l-e-z),

FT’

1)s’

. .

In (1-e-3

~ = 0 ( . 0 5 )1.5(.1)3(.2)6, 5D

27.4. Sievert Integral

so”

e-zuea+&

.

,

. .... . .. . . . . . . 6D

~ = 0 ( . 11 ) (.2)3(.5)10, 8= 10’(10’)60’(15’)90’, 27.5. f,,,(z)=Sm trne-‘’-:

1000

d t and Related Integrals

0

f,,,(~),

m=1, 2, 3; ~=0(.01).05,.1(.1)1, 4D

fS(i~),

~=0(.2)8(.5)15(1)20, 4-5D

.. . . . . . . .

1001

. . . . . . .

1003

f(x)+ln x, x=0(.05)1 f(x), ~=1(.1)3(.5)8, 4D 27.7. Dilogarithm (Spence’s Integral)f(x)=-l

x=0(.01).5,

f(8),

1004

. . . . . . . . .

1005

. . . . . . .

1006

.

8=0°(10)150

8= 15’ (1’) 30’ (2’) 90’ (5’) 180°,

6D

27.9. Vector-Addition Coefficients (j1j2m1m21j1j2jm).

Algebraic Expressions for j2=1/2, 1,3/2,2 Decimal Values for j2=l/2, 1, 3/2, 5D 1

. t.

9D

27.8. Clausen’s Integral and Related Summations

f(e)+e In 0,

. .

~In t d

National Bureau of Standards.

27. Miscellaneous Functions 27.1. Debye Functions

I

Series Representations

1

Jrn %=n!l(n+l).

27.1.3

27.1.1

tndt

Relation to Riemann Zeta Function (see chapter 23)

+5(2k+n) ( 2 k ) ! ] lm=xn[n-2(n+l) B2kxU'

k-1

( I 4 < 2 r , n 2 1) (For Bernoulli numbers Bzn,see chapter 23.) 27.1.2

I27.11 J. A. Beattie, Six-place tables of the Debye energy and specific heat functions, J. Math. Phys. 6, 1-32 (1926).

:J'&

z.[L l2

Debye Functions

0. 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1. 0 1. 1 1. 2 1. 3 1. 4 1. 6 1. 8 2. 0 2. 2 2. 4 2. 6 2. 8 3. 0 3. 2 3. 4 3. 6 3. 8 4. 0 4. 2 4. 4 4. 6 4. 8 5. 0 5. 5 6. 0 6. 5 7. 0 7. 5 8. 0 8. 6 9. 0 9. 5 10. 0

1.000000 0.975278 0.951111 0.927498 0.904437

1.000000

0. 960555

0.903746 0. 873322

0.857985 0. 824963

0. 777505 0. 758213 0. 739438 0.721173 0. 703412 0. 669366 0. 637235 0.606947 0. 578427 0. 551596 0. 526375 0. 502682 0.480435 0.459555 0. 439962 0. 421580 0. 404332 0. 388148 0. 372958 0. 358696 0. 345301 0. 332713 0. 320876 0. 294240 0. 271260

0. 205239 0. 193294 0. 182633 0. 173068 0. 164443

0.792924 0. 761859 0. 731759 0. 702615 0. 674416 0. 647148 0. 620798 0. 595351 0. 570793 0. 524275 0. 481103 0. 441129 0.404194 0. 370137 0. 338793 0. 309995 0.283580 0.259385 0.237252 0. 217030 0. 198571 0. 181737 0. 166396 0. 152424

0. 733451 0. 707878 0. 683086 0. 659064 0. 635800 0. 613281 0. 570431 0. 530404

0. 396095 0. 368324 0. 342614 0. 318834 0. 296859 0. 276565 0. 257835 0. 240554 0.224615 0. 209916

0.095241 0.077581 0. 063604

0. 147243 0. 126669

0.074269 0.066036

1

0.030840 0. 026200 0.022411 0.019296

['-;I"] 998

65.

Widerstandes reiner Metalle von der Temperatur, Ann. Physik. (5) 16, 530-540 (1933). 20 r' t'dt 4x ~ l J oet-1 ez-1' x =O( .I) 13(.2) 18(1)20 (2)52(4)80, 48.

---

Table 27.1

321,2=0(.01)24,

x o ey-1' z - e z - 1 [27.2] E. Griineisen, Die Abhangigkeit des elektrisohen

0.922221 0. 884994 0.848871 0.813846 0. 779911 0. 747057 0.715275 0.684551 0. 654874 0. 626228 0.598598 0. 571967 0. 546317 0. 497882 0. 453131 0. 411893 0. 373984 0. 339218 0. 307405 0. 278355 0. 2518Z9 0. 227742 0. 205915 0. 186075 0. 168107 0. 151855 0. 137169 0. 123913 0. 111957 0.101180 0.091471 0. 071228 0.055677 0.043730 0.034541 0.027453 0.021968 0.017702 0.014368 0.011747 0.009674

2

f (2)

5

f(2)

2

f (2)

5

f (2)

0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 0.090 0.095 0.100

0.007 0.025 0.074 0. 179 0.372 0.682 1. 137 1.752 2. 531 3. 466 4. 540

0. 10 0. 11 0. 12 0. 13 0. 14 0. 15 0. 16 0. 17 0. 18 0. 19 0. 20

4.540 6.998 9.662 12.296 14.710 16. 780 18. 446 19.692 20. 539 21.025 21.199

0. 20 0. 22 0. 24 0. 26 0. 28 0. 30 0. 32 0. 34 0. 36 0. 38 0. 40

21.199 20.819 19.777 18.372 16.809 16.224 13.696 12. 270 10.965 9.787 8. 733

0. 40 0. 45 0. 60 0. 55 0. 60 0. 66 0. 70 0. 75 0. 80 0. 85 0. 90

8.733 6.586 5.009 3.850 2.995 2.356 1.875 1.508 1.225 1.005 0.831

2

0. 9 1. 0 1. 1 1. 2 1. 3 1. 4 1. 5 2. 0

2. 5 3. 0 3. 5

f (2) 0.831 0.582 0.419 0.309 0.233 0. 178 0. 139 0.048 0.021 0.010 0.006

[ 27.31 Miscellaneous Physical Tables, Planck's radiation

functions and electronic functions, MT 17 (U.S. Government Printing Office, Washington, D.C., 1941).

&&=SoA

RI=c,A-6(ec~/XT-l)-1,

~~=[.06(.001) Yc.005) .4(.01) .6(.02) 1(.05)2]cm KO.

, NO-A (T=lOOOo K) Tab& II: RA, R ~ A NA, for A= [.5(.Ol) 1(.05) 4(,1)6( .2) 10(.5)201 microns.

RAdX,

Tab& III: N A for X=[.25(.05)1.6(.2)3(1)10] microns, T=[1000°(6000)35000 K and 6000' K].

Einstein Functions 2

e'--l 0. 00 0. 06 0. 10 0. 16 0. 20 0. 26 0. 30 0. 36 0. 40 0. 46 0. 60 0. 66 0. 60 0. 66 0. 70 0. 76 0. 80 0. 86 0. 90 0. 96 1. 00 1.06 1. 10 1. 16 1. 20 1. 26 1. 30 1. 36 1. 40 1. 46 1. 60

1.00000 0.99979 0.99917 0.99813 0. 99667 0.99481 0.99263 0.98986 0.98677 0.98329 0. 97942 0. 97617 0.97063 0.96662 0. 96016 0.96441 0.94833 0.94191 0.93616 0.92807 0.92067 0.91298 0.90499 0.89671 0.88817 0. 87937 0. 87031 0.86102 0.86161 0.84178 0. 83186

I I 1

I I

1.00000 0.97621 0. 96083 0. 92687 0.90333 0.88020 0.86749 0.83619 0.77076 0.76008 0. 72982 0. 70996 0.69060 0.67144 0. 66277 0. 83460 0.61661 0.69910 0.68198 0. 66623 0.64886 0.63286 0.61722 0.60194 0. 48702 0.47246 0.46824 0.44436 0.43083

In (1-e-m)

-0a

-3.02063 -2. 36217 - 1.97118 -1. 70777 - 1.60869 - 1.36023 - 1.21972 -1. 10963 - 1.01608 -0.93276 -0. 86026 -0.79687 -0.73824 -0.68634 -0.63936 -0. 69662 -0.66769 -0.52184

-o*48897 -0.-. 46868 - - - .-0.43069 -0.40477 -0.38073 -0.36838 -0. 33768 -0.31818 -0.30008 -0.28316 -0.26732 -0.26248

1

Table 27.3 5

-e --In (1-e-m)

1 0 )

I

I I 1

I I

3.99684 3. 30300 2. 89806 2. 61110 2.38888 2.20771 2.06491 1.92293 1.80690 1.70360 1.61036 1. 62669 1. 44820 1.37684 1.31079 1.24939 i.igm9 1. 13844 1.08809 1.04066 0.99692 0.96363 0. 91368 0. 87660 0.83962 0.80620

0.77263 0.74139 0. 71168 0.68331

1000

MISCELLANEOUS FUNCTIONS

Einstein Functions

Table 27.3 L

I

I

xae*

X

(ez- 1)'

0. 81143

1. 6 1. 7 1. 8 1. 9 2. 0

0.79035 0. 76869 0. 74657 0. 72406 0. 70127 0. 67827

2. 1 2. 2 2. 3 2. 4 2. 5 2. 6 2. 7 2. 8 2. 9 3. 0 3. 2 3. 4 3. 6 3. 8 4. 0 4. 2 4 4 4. 6 4. 8 5. 0 5. 2 5. 4 5. 6 5. 8 6. 0

0. 58589 0. 56307

0.54049 0. 51820 0. 49627

0. 45363

0.27264

0. 11683 0. 10247

0.08968

X e*- 1

I

--In (1-e-a)

0.40475

-0. 22552 -0. 20173

0.33416 0. 31304 0. 29304 0.27414 0. 25629 0. 23945 0. 22356

-0. 16201 -0. 14541

0. 37998 0. 35646

0.49617 0.45845 0. 42367 0. 39158 0. 36194 0.33455 0. 30921

-0. 13063 -0. 11744 -0. 10565

0.08695 0.07463

0.06394 0.05469 0.04671 0.03983 0.03392 0.02885 0.02450 0.02078 0.01761 0.01491

-0.01511 -0.01235 -0.01010 -0.00826 -0.00676 -0.00553 -0.00463 -0.00370 -0.00303 -0.00248

0. 13598 0. 11739 0. 10113

0.63027

0. 58171 0. 53714

-0.18068

-0.09510 -0.08565 -0.07718 -0.06957 -0.06274 -0.05659 -0.05107 -0. 04162 -0.03394 -0.02770 -0.02262 -0.01849

0. 20861 0. 19453 0. 18129 0. 16886 0. 15719

X ez- 1

In (1-e-8)

I 0.22545 0. 20826

0. 10958 0.09311

0.07905 0.06705 0.05681 0.04809 0.04068

0.02065 0.01739

[27.4] H. L. Johnston, L. Savedoff and J. Belzer, Contributions to the thermodynamic functions by a Planck-Einstein oscillator in one degree of freedom, NAVEXOS p. 646, Office of Naval Research, Department of the Navy, Washington, D.C. (1949). Values of x*e*(e*- l)-*,x(e*- l)-l, --In (1-e-") and z(e*-l)-l-ln (1-e-*) for z=0(.001)3(.01) 14.99, 5D with first differences.

27.4. Sievert Integral e - z aeo

*d4

1 ' 3 * 5 . . , (2k-1) 2.4.6... (2k) (For E2k+2(2), see chapter 5.) oro=

1,ak=

Relation to the Integal of the Bessel Function KO@)

27.4.3 Relation to the Error Function

27.4.1

2655

301035

--1 0 2 4+--. ~ 3~ 2 7 6 8 ~ ~ (For erf, see chapter 7.)

(For Ki,(s),see chapter 11.)

*

.}

1001

MISCELLANEOUS FVNCTIONB (27.51 National Bureau of Standards, Table of the Sievert integral, Applied Math. Series- (U.S. Government Printing Office, Washington, D.C. In press). x=O( .01) 2( .02) 5(.05) 10, 8=Oo (1') 90°,

'

Sievert Integral

I

30'

I

0.523599 0.471456 0.424515 0.382255 0.344209

loo

20'

0. 0 0. 1 0. 2

0. 3 0. 4

0.174533 0. 157843 0. 142749 0. 129099 0. 116754

0.349066 0.315187 0.284598 0.256978 0.232040

0. 5 0. 6 0. 7 0. 8 0. 9

0. 105589 0.095492 0.086361 0. 078103 0.070634

0.209522 0. 189191 0. 170833 0. 154256 0. 139289

0.309957 0.279118 0.251353 0.226354 0.203845

::: 1

0. 125775 0. 102553

0.083620 0.068183 0.055597

0. 183579 0. 148899 0. 120780

1. 8

0.063880 0.052247 0.042733 0.034951 0.028587

2. 0 2. 2 2. 4 2. 6 2. 8

0.023381 0.019123 0.015641 0.012793 0. 010463

4. 0 4. 5 5. 0

0.008558 0. 005178 0.003132 0.001895 0.001147

1. 0 1. 2

I

:::1 5. 5 6. 0 6. 5 7. 0 7. 5

t:: 9. 0

9. 5 10.0

I

s," I

s,'

e-=

+

I

40'

50' 0.872665 0. 777323 0. 692565 0. 617194 0.550154

0.625886 0.561159 0. 503165 0.451198 0.362893 0.325486 0.291957 0.261901

0.490508 0.437428 0.390178 0.348109 0. 310642

I

60'

1

1.047198 0.923778 0.815477 0. 720366 0. 636769

1.308997 1. 123611 0.968414 0.837712 0.727031

1.228632 1. 023680 0.868832 0.745203

I

0.563236 0.498504 0.441478 0.391204 0.346851

0.632830 0. 552287 0.483134 0.423535 0.371996

0.643694 0.558890 0.487198 0.426062 0.373579

goo 1. 570796

0.097979 0.079488

0. 122667 0.098829

0.045335 0.036967 0.030145 0.024582 0.020045

0.064492 0. 052329 0.042463 0. 034460 0.027968

0.079644 0.064201 0.051766 0.041750 0.033680

0.089954 0.071979 0.057635 0.046179 0. 037024

0.095342 0.075797 0.060342 0.048100 0.038387

0. 097108 0. 076905 0.061040 0.048541 0.038667

0. 097121 0.0769 11 0.061043 0.048542 0.038668

0.016347 0.009817 0.005896 0.003542 0.002127

0.022700 0.013477 0.008005 0.004756 0.002828

0.027177 0.015912 0.009330 0.005478 0,003221

0.029702 0.017164 0.009951 0.005787 0.003374

0.030670 0.017576 0.010128 0.005862 0.003407

0.030848 0.017634 0.010147 0.005869 0.003409

0.030848 0.017634 0. 010147 0. 005869 0.003409

0.001896 0.001117 0.000659 0.000389 0.000230

0.001972 0.001155 0. 000678 0.000399 0.000235

0.001986 0.001162 0.000681 0.000400 0.000235

0. 001987

0.001162 0.000681 0.000400 0.000235

0.001987 0.001162 0.000681 0.000400 0. 000235

0.000136 0.000081 0.000048 0.000028 0.000017

0.000139 0. 000082 0. 000048 0. 000029 0.000017

0. 000139

0.000139 0.000082 0.000048 0.000029 0.000017

0.000139 0.000082 0.000048 0.000029 0.000017

I

0.000056 0.000034 0.000021 0.000012 0.000008

0.000100 0.000060 0.000036 0.000022 0.000013

1

I

1

0.001682 0.001001 0. 000596 0. 000355 0.000211 0. 000126 0.000075 0.000045 0.000027 0.000016

=lm 27.5.4

a g '- ( m-1)jl+2fm=0

. . .)

0.000082 0.000048 0.000029 0.000017

2 j l ( z ) = 2 (akIn z + b k ) b k -0

-2 b, -

(m23)

-2ak-Z ak=k(k-l) (k-2)

bk=

G=a,.=O

@=-bo

bo= 1

Recurrence Relation

2JlR=(m-l)f,-,+zf,-*

I

Power Series Reprewntations

and

t"e-'"?dt

(m=1,2,

0.328286

I

Differential Equations

27.5.3

I

0. 254889 0. 199051 0. 156156 0. 122961

I

J&=-fm-l

75O

0.327288 0.254485 0. 198885 0. 156087 0. 122932

9

27.5.2

I

0.307694 0.242523 0. 191533 0. 151541 0. 120105

Related Integrals m=O, 1 , 2 , . 27.5.1

3D.

Table 27.4

0. 404629

I

lac*d+, ~=3O0(lo)9O0,A=0(.01).5,

&I

0. 698132

1

e-A

0. 277267 0. 221027 0. 176336 0. 140792 0. 112497

0. 001278 0.000768 0.000461 0.000277 0.000167

27.5. &(z)

9D.

0. 234956 0. 189138 0. 152298

0.000694 0.000420 0. 000254 0.000154 0.000093

I

[27.6] R. M. Sievert, Die v-Strahlungsintensitht an der Oberflache und in der nachsten Umgebung von Radiumnadeln, Acts Radiologica 11, 239-301 (1930).

I

(For y, see chapter 6.)

2(3P- 6k+ 2 )ak k(k-1) (k-2)

1002

MISCELLANEOUS FUNCTIONS Asymptotic Representation

27.5.5

27.5.11

2 f , ( ~ )=1-&%+.6342;c2+.590828-.1431~' -.01968~"+.00324~"+.000188~' ... -xa In ~(1-.08333~+.0013892'-.0000083X"+.

. .)

27.5.12

] ( A cos 0--B sin e)

g2(x)=-Gy'2iexp[-2(3) 3 x 27.5.6 2j 2 ( x ) = @ - x + 2

f i x2-.32252-.1477x'+.03195x' 2

+,00328~'- .000491~'- .00002351, . . +x3 In x(~-.01667;c2+.000198x4-

. . .)

27.5.7

2f 3 ( X ) =1-$Zf$2

.2954$f .1014S4+ .02954Xs

- .00578x6- .00047~'+.000064~~ ... -x4 In ~ ( ~ 0 8 3 3.00278$+ , 0 0 0 0 2 5 ~ ~ .- . .)

-aK ao=l a3= --.017879

Aoymptotic Representation

27.5.8

(:r3+.] ,

.

(x+-)

al=.972222 &=.148534 a4=.004594 as= -.000762

[27.7] M. Abramowitz, Evaluation of

the integral

J* e-s*-s/udu, J. Math. Phys. 32,188-192 v=3

(;)

[27.8] H. F a x h , Expansion in series of the integral

2/8

exp [-z(tht-*)]Pdt, Ark. Mat., Astr., Fys.

Jvw

16, 13, 1-57 (1921). [27.9] J. E. Kilpatrick and M. F. Kilpatrick, Discrete energy levels associated with the LennardJones potential, J. Chem. Phys. 19, 7, 930-933 (1951). (27.101 U. E. Kruse and N. F. Ramsey, The integral

1 uo=l, al=r2 (3m9-3m-1) l2(k+2)uk+$=- (12k2+36k-3m2-3m+25)u,+, +3(m-2k)

(1953).

(2k+3-m)(2k+3+2m)ak (k=O, 1 , 2

. . .)

s,-

-y*+r ( 40 (1951). ya exp

'3

dy, J. Math. Phys. 30,

[27.11] 0. Laporte, Absorption coefficients for thermal neutrons, Phys. Rev. 52, 72-74 (1937). [27.12] H. C. Torrey, Notes on intensities of radio fiequency spectra, Phys. Rev. 59, 293 (1941). [27.13] C. T. Zahn, Absorption coefficients for thermal neutrons, Phys. Rev. 62, 67-71 (1937).

s,

W

yne'r-zlfidy

for n=O,

i, 1; z=O(.Ol).l(.l)l.

1003

MISCELLANEOUS FUNCTIONS

fl(z)

Z

-0.00 0.01 0.02 0.03 0.04 0.05

.fi(Z)

0. 5000 0.4914 0. 4832 0. 4753 0. 4676 0. 4602

0. 4431 0. 4382 0. 4333 0. 4285 0. 4238 0. 4191

f3(z)

fi(Z)

f3(z)

0. 5000 0. 4956 0. 4912 0. 4869 0. 4826 0. 4784

0. 1 0. 2 0. 3 0. 4 0. 5

0. 4263 0. 3697 0. 3238 0. 2855 0. 2531

0.3970 0. 3573 0. 3227 0. 2923 0. 2654

2

0.4580 0. 4204 0. 3864 0. 3557 0. 3278

0. 6 0. 7 0. 8 0. 9 1.0

fie4

f&)

f3 (4

0.2255 0. 2015 0. 1807 0. 1626 0. 1466

0.2415 0. 2202 0. 2011 0. 1839 0. 1685

0. 3025 0. 2793 0. 2584 0. 2392 0. 2215

50000 49019 46229 41950 36543

0.00000 0.08754 0. 16933 0. 24139 0. 30136

4.0 4. 2 4. 4 4. 6 4.8

-0. -0. -0. -0. -0.

2626 2552 2441 2299 2132

0.0430 +O. 0094 -0.0214 -0.0490 -0.0734

8. 0 8. 5 9. 0 9. 5 10. 0

0.06078 0.07562 0. 08221 0.08191 0.07626

-0.09808 -0.07131 -0.04496 -0.02082 -0.00010

1. 0 1. 2 1. 4 1. 6 1. 8

0. 30366 0. 23746 0. 16972 0. 10288 $0.03892

0. 34805 0. 38122 0. 40127 0. 40910 0. 40592

5. 2 5. 4 5. 6 5.8

5.0

-0. -0. -0. -0. -0.

1945 1745 1536 1322 1108

-0.0944 -0. 1120 -0. 1263 -0. 1374 -0. 1455

10. 5 11. 0 11. 5 12. 0 12. 5

0.06684 0.05507 0. 04224 0.02937 0.01727

$0. 01654 0.02889 0.03707 0.04146 0.04259

2.0 2. 2 2. 4 2. 6 2. 8

-0.02062 -0.0746 -0. 1221 -0. 1629 -0. 1966

0. 39314 0. 3722 0. 3448 0. 3122 0.2759

6.0 6. 2 6. 4 6. 6 6. 8

-0.0896 -0.0691 -0.0493 -0,0307 -0.0132

-0. 1507 -0. 1533 -0. 1535 -0. 1515 -0. 1476

13. 0 13. 5 14. 0 14. 5 15. 0

$0.00650 -0.00259 -0.00982 -0.01517 -0.01872

-0. -0. -0. -0.

0. 2371 0. 1971 0. 1569 0. 1173

i:7. 4

$0. 00286 0.01749 0.03061 0.04220

-0. -0. -0. -0.

16. 0 17. 0 18. 0 19. 0

-0. 02118

0. 0 0. 2 0. 4 0. 6 0. 8

0. 0. 0. 0. 0.

2233 2432 2565 2639

=so t+x m

27.6. f(x)

7. 6

~

dt

ln x+e-z2[J;;

+O. 00921 -0.00022 -0.00650 -0.00965

27.6.4

Power Series Representation

j(x)=-e-z2

-0.01906 -0. 01435 -0. 00879

Asymptotic Representation

e-t2

27.6.1

14211 13518 12709 11805

0.04109 0.03758 0.03268 0.02696 0.02089

5k!(2k+l)

6[,+s+,,a+m+ 1 1 1.3 1.3.5 *I

f(d-7j-

x2k+l

*

*

k-0

-Em k-1

x2k

A-_

k!2k 2

1 1

1

2! 3!

--[>+T+T+g+ 2 s x x

*

*

.I

(x-+m)

27.6.2 (-l)k+(k+1)x2k k! (-2)kx2k+1 1 . 3 . 5 . . . (2k+l) (For y and the digamma function +(x), see chapter 6.) =-e-z2

1

In

+J;;z

" f 2 kc =O

Relation to the Exponential Integral

1 e-"'Ei (zz)+$e-z2 Jz e% 0 2 (For ~i ($) see 5 ; e-z2 e t 2 d t , see chapter 74

27.6.3 f(x)=--

[27.14] A. ErdBlyi, Note on the paper "On a definite integral" by R. H. Ritchie, hfath. Tables Aids Comp. 4, 31, 179 (1950). [27.15] E. T. Goodwin and J. Staton, Table of

Quart. J. Mech. Appl. Math. 1, 319 (1948). z=0(.02)2(.05)3(.1) 10. Auxiliary function for 2 = 0(.01) 1. [27.16] R. H. Ritchie, On a definite integral, Math. Tables Aids Comp. 4,30, 75 (1950).

1004

MISCELLANEOUS FUNCTIONS

j(z)=Im

Table 27.6

t+z dt ff

z

f(z)+lnz

z

f(z)+lnz

Z

f(z)

Z

f(4

Z

0.00 0.05 0. 10 0. 15 0. 20

-0.2886 -0.2081 -0. 1375 -0,0735 -0.0146

0. 50 0. 55 0. 60 0.65 0.70

0.2704 0. 3100 0.3479 0.3842 0.4192

1. 0 1. 1 1. 2 1. 3 1. 4

0.6051 0.5644 0.5291 0.4980 0.4705

2. 0 2. 1 2. 2 2. 3 2. 4

0.3543 0. 3404 0.3276 0.3157 0.3046

3. 0 3. 5 4. 0 4. 5 5. 0

0. 2519 0.2203 0. 1958 0. 1762 0. 1602

0.25 0. 30 0. 35 0. 40 0. 45

+O. 0402 0.0915 0. 1398 0. 1856 0.2290

0.75 0.80 0.85 0. 90 0.95

0.4529 0.4854 0. 5168 0. 5472 0. 5766

1. 5 1. 6 1. 7 1. 8 1. 9

0.4460 0.4239 0.4040 0.3860 0.3695

2. 5 2. 6 2. 7 2. 8 2. 9

0. 2944 0.2848 0. 2758 0. 2673 0.2594

5. 5 6. 0 6. 5 7. 0 7. 5

0. 1468 0. 1356 0. 1259 0. 1175 0.1102

0. 50

0. 2704

1.00

0. 6051

2. 0

0.3543

3. 0

0. 2519

8. 0

0. 1037

f(z)

“-911

“-“13 Compiled from E.T. Qoodwin and J. Staton, Table of

:;-

du, Quart. J . Mech. Appl. Math. 1,318 (1048) (with permission).

Relation to Debye Functions

27.7. Dilogarithm (Spence’s Integral for n =2)

27.7.1 [27.17] L. Lewin, Dilogarithms and associated functions (Macdonald, London, England, 1958).

Series Expansion m

27.7.2

f ( z ) =k=1 C (-l)k

(x-l)k k2

Functional Relationships

27.7.3

f(z>+j(1-z)=-ln

z In (1--2)+f 6

(12z>0)

[27.18] K. Mitchell, Tables of the function -logkl-il‘ dill with an account of some properties of this and related functions, Phil. Mag. 40, 351-368 (1949). ~ = - l ( , O l ) l ; ~=0(.001).5, 9D. [27.19] E. 0. Powell, An integral related to the radiation integrals, Pliil. Mag. 7, 34, 600-607 (1943).

27.7.4

1

(122>0)

f(l.-2)+j(l+r)=~f(l-22)

27.7.5 f ( z ) + j ( i ) = - i

(In 2)’

(05~11)

27.7.6

j ( z + l ) - j(s)=-In

K

(22x20)

~

J B d y ,

z=O(.Ol)2(.02)6,

7D.

[27.20] A. van Wijngaarden, Polylogarithms, by the Staff of the Computation Department, Report R24, Mathematisch Centrum, Amsterdam, Holland m

1 z In (z+I)---12 2f(x? ??a

(22x20)

(1954). F , ( z ) = x

h-nzh

for z=z= -l(.Ol)l;

h-1

z=iz,

10D.

for 2=0(.01)1; z=ei*a/* for a=0(.01)2,

1005

MISCELLANEOUS FUNCTIONS

Dilogarit hm fO=-J gIn1d t I

II

I

Table 27.7

II

f (2) 0.00 0. 01 0.02 0.03 0.04

1. 64493 1. 58862 1. 54579 1.50789 1. 47312

4067 5448 9712 9041 5860

0. 0. 0. 0. 0.

10 11 12 13 14

1. 29971 1. 27452 1. 25008 1. 22632 1.20316

4723 9160 7584 0101 7961

0.20 0.21 0.22 0.23 0.24

1.07479 1. 05485 1. 03527 1.01603 0.99709

0. 05 0.06 0.07 0.08 0.09

1.44063 1.40992 1. 38068 1. 35267 1. 32572

3797 8300 5041 5161 8728

0. 15 0. 16 0. 17 0. 18 0. 19

1. 18058 1. 15851 1. 13693 1. 11580 1.09510

1124 6487 6560 8451 3088

0.25 0.26 0.27 0.28 0.29

0. 10

1. 29971 4723

0. 20

1.07479 4600

0.30

f (2)

2

1 I

1 I 0.40 0. 41 0.42 0.43 0.44

0. 72758 6308 0. 71239 0.69736 0. 68247 0. 66774

5042 1058 9725 6644

2689 6024 3992 0483 9737

0.45 0. 46 0.47 0.48 0.49

0.65315 0.63870 0.62439 0.61021 0.59616

7631 8705 6071 6108 5361

0.72758 6308

0. 50

0. 58224 0526

0.30 0.31 0.32 0.33 0. 34

0.88937 0.87229 0.85542 0.83877 0.82233

7624 1733 7404 6261 0471

0. 97846 9393 0. 96012 6675 0.94205 7798 0.92425'0654 0.90669 4053

0.35 0.36 0.37 0.38 0.39

0. 80608 0.79002 0. 77415 0. 75846 0. 74293

0.88937 7624

0.40

4600 9830 7934 0062 9088

f (2)

2

['-p-j

[(- 4) 11 27.8. Clausen's Integral and Related Summations 27.8.1

f(e) =-

se (2 a) In

sin

0

dt

=$k2 sin ke

-1

(0 I 6 5 4

Summable Series

27.8.6

2

-

n-1

n

---In

n~ + c 2=--n 6 COS

n=l

Series Representation

-

27.8.2

j(e)=-e

111

(-1)k-1

(2 sin S) re 2'T

e2

(o<e<24 (05e52T)

p+1

lel+e+ck-1 (2k)! B2' 2k(2k+1)

@so<;) 27.8.3

f(?r-e)=e

--

In 2--C

k-1

(-1)k-1 (2k)! B2'(21-1)

Functional Relationship

p + 1

2k(2k+l) [27.21] A. Ashour and A. Sabri, Tabulation of the function sin n8 $(e) =E7 9 Math. Tables Aids Comp. 10, OD

n-1

Relation to Spence's Integral

27.8.5

54, 57-65 (1956). [27.22] T. Clausen, Uber die Zerlegung reeller gebrochener Funktionen, J. Reine Angew. Math. 8 , 298-300 (1832). z=O0(lo)180°, 16D. [27.23] L. B. W. Jolley, Summation of series (Chapman Publishing Co., London, England, 1925). [27.24] A. D. Wheelon, A short table of summable series, Report No. SM-14642, D o u g h Aircraft Co., Inc., Santa Monice, Calif. (1953).

1006

MISCELLANEOUS FUNCTIONS

Table 27.8

Clausen’s Integral

eo

11

15 16 17 18 19 20

0.612906 0. 635781 0.657571 0.678341 0.698149 0.717047

-II

/I

2 3 4 5 6 7 8 9 10

0. 104735 0. 122199 0.139664 0.157133 0.174607

21 22 23 24 25

0.735080 0.752292 0. 768719 0.784398 0. 799360

42 44 46 48 50

11

0. 192084 0.209567 0.227055 0.244549 0.262049

26 27 28 29 30

0.813635 0.827249 0.840230 0.852599 0.864379

52 54 56

1

12 13 14 15

I

I

30 32 34 36 38 40

0.000000 0.017453 0.034908 0.052362 0.069818 0.087276

0

*

58

60

I-I

1

62 60 64 66 68 70

1. 014942 1. 014421 1.012886 1. 010376 1.006928 1. 002576

90 95 100 105 110 115

0.915966 0.883872 0.848287 0.809505 0.767800 0. 723427

0.966174 0.977020 0.986357 0.994258 1.000791

72 74 76 78 80

0.997355 0.991294 0.984425 0.976776 0.968375

120 125 130 135 140

0.676628 0.627629 0.576647 0. 523889 0.469554

1.006016 1.009992 1.012773 1.014407 1.014942

82 84 86 88 90

0.959247 0. 949419 0. 938914 0. 927755 0.915966

145 150 160 170 180

0.413831 0.356908 0. 240176 0. 120755 0.000000

0.864379 0.886253 0.906001 0.923755 0. 939633 0.953741

“-911

[(-44)1]

y,

Math. Tables Aids Comp. 10.54,57-85 (1956) (with PermiMIOn).

Compiled from A. Ashour and A. Sabri, Tabulation ofthe function $(.9)n=l

27.9. Vector-Addition Coefficients

(Wigner coefficients or Clebsch-Gordan coefficients) Definition

27.9.1 (j l

+jz- Jl!(j+

jl

-AY (j+jz -j l ) !(2j+ 1)

(j+jl

27.9.8

Conditions

27.9.2 27.9.3

n

j1,

ja, j=+n or+g

+jz+1)!

(n=integer)

27.9.9

jl+jz+j=n

lmll Sj1, lmzl I j a , Iml I j ( ~ l ~ z m l m 2 ~ ~ l ~ z ~ ml+m2 m)=0 Zm Special Values

27.9.4 27.9.5 27.9.6 27.9.7

-j1

+ja

+j n 2

ml, mz, m = f n or f-

27.9.10

~ l O m l O ~ ~ l O ~ mj)s(ml, ) = ~ ~m) l,

27.9.11

cilj,OOIj&jO)=O

27.9.12

( j ~ l m l m l ~ j j l j m ) = O 2jl+j=2n+ 1

jl+jhj=2n+ 1

I

1007

MISCELLANEOUS FUNCTIONS

Table 27.9.1

1

SNOLL3Nfld Sfl03lwTI33SIW

8001

j=

*=2

*=1

*=O

1

jl-

j1-2

j=

j1+2

jl-2

0

I

5 *=-1

*=-2

1010

MISCELLANEOUS FUNCTIONS

Table 27.9.5 [By use of symmetry relations,. coefficients may be put in standard form j , l j , S j and m>O] m2

1 1 1 1 m

j1

j

-1JH

(jhmmlj~jzjm)

4% 4%

0.70711 0. 70711 1.00000

I

jZ= 1 -1 0 0 0 1 0 1 0 1 1 1 1 0 %

1 1 1 1

1 1 1 1 1

E # :#r 0 0 0 1 1 2

-1 0 1 0 1 1

0 1 1 2 0 0 0

1 1

H

1 1 1 1 1

H

i8

H

EH E E

P

1

1 1 1 1

4%

4%

2 2 2 2 2 2

1 1 1 1

%

- 4% 4% - 4%

EK 2

2 2 2 2 2 2 2 2 2 2 2 ~

% -- 4 4% 4% -4 % 4 x 6

4

6

4%

H4 3

4%

- 4%

4% - 4%

$Eo

4 x 0

4% 4%

0.70711 0.00000 -0.70711 0. 70711' -0.70711 0. 81650 0.57735 1.00000 0.40825 0.81650 0.40825 0. 707 11 0.70711 1.00000

*

0. 73030 -0.25820 -0. 63246 0. 63246 -0. 77460 0. 70711 0. 707 11 0.86603 0. 50000 1.00000 0. 50000 0. 50000 -0. 50000 -0.50000 0. 707 11 0.00000 -0. 70711 0. 70711 -0. 70711 0.54772 0.77460 0.31623 0.77460 0. 63246 1. OOOOO

Compiled from A Bimon Numerical tables of the Clebsch-Oordan mfUcients Oak Ridge 'Nstionh Laboratory Report 1718, Oak Rldge, Tenn. (1954)' (with Oermission).

[27.25] E. U. Condon and G. A. Shortley, Theor of atomic s ectra (Cambridge Univ. Press, h m bridge &gland, 1935). L27.261 M. E. hose Elementary theory of an ular momemtum (John Wiley & Sons, Inc., r$ew York, N.Y., 1955). L27.271 A. Simon, Numerical tables of the Clebsch-Gordan

coefficients, Oak Ridge National Laboratory Report 1718, Oak Rid e, Tenn. (1954). c j j ; mlmm) for a t ang&,r moments
b.

10

*See page n.

28. Scales of Notation S. PEAVY,' A. SCHOPF'

Contents page

. . . . . . . . . . . . . . . . . . Numerical Methods . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . Table 28.1. 2*n in Decimal, n=0(1)50, Exact . . . . . . . . . . . Table 28.2. 2" in Decimal, z=.OOl(.OOl) .01(.01).1(.1).9, 15D . . .

1012

. . . . . . . 10 in Decimal, n=1(1)10, 10D . . . .

1017

Representation of Numbers.

Table28.3. 10*" in Octal, n=0(1)18,

Exact or 20D

1013 1015 1016 1017

Table 28.4.

n log,, 2, n log,

Table 28.5.

Addition and Multiplication Tables, Binary and Octal S c a l e s . . . . . . . . . . . . . . . . . . . . . 1017

Table28.6. Mathematical Constants in Octal Scale

........

1017

1017

The authors acknowledge the assistance of David S. Liepman in the preparation and checking of the tables.

National Bureau of Standards. worker, National Bureau of Standards, from The Ameriaan University (deceased). 1

* Guest

1011

28. Scales of Notation Representation of Numbers Any positive real number x can be uniquely represented in the scale of some integer b>l as x=(Am

. . . AIAo.a-la-a . .

.)(b),

Integers X = ( A ,

(I) b-scale arithmetic. and define

. . . AI&)(b) Convert 8 to the b-scale

--

Xb =X1+AJb,

where every A, and a-j is one of the integers 0, 1, , . ., b-1, not all Ail a-j are zero, and Am>O if z > 1. There is a one-tosne correspondence between the number and the sequence

x,fi=x2+Z;/T,

m

x=A,bm+

. . . +Alb+A,+C 1

a-jb-j

where the infinite series converges. The integer b is called the base or radix of the scale. The sequence for z in the scale of b may ter. . . = O for some minate, i.e., a-.-,=a-n-a= n > 1 so that x=(Am

Scale Binary Ternary Quaternary Quinary Senary Septenary

1 1 Bese

XI, Xa, , . ., X;;; the quotients (in the b-scale) where X , X , , . . ., XE-~,respectively are divided by in the b-scale. Then convert the remainders to the &scale,

8 Octal 9 Nonary 10 Decimal 11 12 16

(11) &scale arithmetic. Convert b and Ao, A,, . . ., Am to the &scale and define, using arithmetic operations in the h a l e , Xm-l=Amb +Am-1,

Scale

Undenary Duodenary Hexadecimal

General Conversion Methods

Any number can be converted from the scale of b to the scale of some integer T#b, Z>l, by using arithmetic operations in either the b-scale or the 8-scale. Accordingly, there are four methods of conversion, depending on whether the number to be converted is an integer or a proper fraction. 1012

-1

Xm-a=Xm-lb +Am-2,

Names of Scales

I

z,zi, . . ,, A;;; are the remainders and

. , . A1Ao.a-la-2 . . . a-,,)(b);

then z is said to be a finite b-adic number. A sequence which does not terminate may have the property that the infinite sequence a-l, a-2, . , , becomes periodic from a certain digit a-.(n,>l) on; according as n=l or n>l the sequence is then said to be pure or mixed recurring. A sequence which neither terminates nor recurs represents an irrational number.

' Base

where

Proper fractionm ~=(O.a-ia-a

. . .)(b)

To convert a proper fraction x, given to n digits in the b-scale, to the scale of Z b such that inverse conversion from the &.cale may yield the same n rounded digits in the b-scale, the representation of x in the b-scale must be obtained to rounded digits where n satisfies T;>bn. (111) b-scale arithmetic, Convert 'b to the b-scale and define

a

n

-

zb= 51+EL1 d=Zq

+z1,

-

z;,J=q+a!.,

I

1013

SCALES OF NOTATION

-

where E-1, Zla, , . .,a-;; are the integral parts and xl, x2, . . ., G the fractional parts (in the b-scale) of the products x5, d,. . ., G-IT, respectively. Then convert the integral parts to the b-scale,

(IV) &scale arithmetic, Convert b and a-l,

. ., a-, to the $-scale and define, using arithmetic operations in the &scale,

a-3, ,

z-,+~=a-,/b

+a-,,++l,

Numerical Methods Example 2. Convert X=(2531)(8) to the decimal scale. By (I) we have T= 10= (12)(8)and hence, using octal arithmetic, 2531/12=210+11/12 210/12=15+6/12 15/12=1+3/12 1/12=0+1/12

The examples are restricted to the scales of 2, 8, 10 because of their importance to electronic computers. Note that the octal scale is a power of the binary scale, In fact, an octal digit corresponds to a triplet of binary digits. Then, binary arithmetic may be used whenever a number either is to be converted to the octal scale or is given in the octal scale and is to be converted to some other scale. Decimal 1 2 3

4

5

6

7

8

9

10

Octal

4

5

6

7

10

11

12

1 2 3

Binary 1 10 11 100 101 110 111 1000 1001 1010 Example 1. Convert X= (1369)(10)to the octal scale. By (I) we have b = l O , T=8(10) and SO, using decimal arithmetic, 1369/8=171+1/8,

+ 2 1/8=2 +518, 2/8 =0 +2/8 ;

171/8=2 1 3/8,

then

X= (2531)(8).

By (11) we have b=(12)(8) and Aa=1(8), A, =3(s), A1=6(8), Ao=(ll)c8). Hence, using octal arithmetic, X2= 1* 12 3 = (15)(a),

+

Xi= 15 12+6= (2 10) (Q, X=210.12+11=

(2531)(8).

Using binary arithmetic we have, by (11), b= (1010)(a)and Aa= I(a),Aa= (11)(a), AI= (11O)(a), Ao(IOOl)(a). Thus Xa= 1* 1010 11= (1101)(a), X,=llOl*1010+110=(10 001 000)(2), X=10 001 OO0*1O1O+1001=(10 101 011 001)(2), whence, on converting to the octal scale, X= (2531)o).

+

Thus, converting to the decimal scale,

-Ao=(11)(8)=9, A1=6(8)=6, A2=3(8)=3, A,=l,

I

and so

X= (1369)(10). By (11)we have T = l O , and the octal digits of X are unchanged in the decimal scale. Hence, using decimal arithmetic, x2=2 * 8 +5= (21)(10)) X1=21* 8+3= (171) (10)) 171-8 1= ( 1369)(10).

x=

+

Using binary arithmetic we have, by (11), b= 8= (1000)(2) and Ao= 1,A,=( 11)(a), Aa=( 101)(2), Aa=(10)(2). Then, Xa=lO*lOOO+lOl=

(10 1Ol)(a),

X,=lO 101.1000+11=(10 101 011)(2), X=lO 101 011.1OOO+l=(lO 101 011 001)(2), whence, on converting to the decimal scale,

X= (1369)clo). Observe that in both examples above, octal arithmetic is used as an intermediate step to convert. according to (11), the given number to the binary scale. If, instead, the given number is first convert,ed to the binary scale, then binary arithmetic may be applied directly to convert, according to (I), the given number from the binary scale to the scale desired.

1014

BCALES OF NOTATION

Alternatively, we can apply (III)with 8= (IOIO)(a), For example, in converting X=(2531)(8) to the decimal scale, we find first X = (10101011001)(~~ using binary arithmetic: and then obtain, using (I) with 8= 10= (lOlO)(a), (0.010 110 11).1010=11+(0.100 011 l), (0,100 011 1)~1010=101+(0.100 Oll), 10 101 011 001/1010=10 001 000+1001/1010, (0.100 011).1010=101+(0.011 l l ) , 10 001 000/1010=1101+110/1010, 1101/1010= l + l l / l O l O , (0.011 11)~1010=100+(0.101 1). l/lOlO=O+

l/lOlO.

Thus, on converting to the decimal scale,

Ao= (1001)(2) =9, AI= (I 10)(2)=6, As= (11)(2) =3, Aa= 1, whence

X= (1369)(10).

Example 3. Convert x= (0.355)c10,to the binary scale. We first convert to the octal scale, using decimal arithmetic. By (111),we find with 8=8 (0.355) -8=2+0.840, (0.080) .8=0+0.640 (0.840) -8=6+0.720, (0.640) *8=5+0.120 (0.720) *8=5+0.760, (0.120) *8=0+0.960 (0.760) .8=6+0.080, (0.960) *8=7+0.680

whence x=(O.26560507 . . verting to the binary scale, x=(O.OlO

Thus, on con-

110 101 110 000 101 000 111

. . .)(2)*

In order that inverse conversion of x from the binary to the decimal scale yield again x to the given number n of decimal digits, we must round x in the binary scale to at least 5 digits where 5 is chosen such that 2;>10n. As a working rule, we 10 may take 5 1- n. Hence, to obtain x= (0.355)(lm 3 by inverse conversion, x must be rounded in the 10 binary scale to E 23 = 10 digits. 3 Thus, x=(O.OlO 110 110 O)@)* To carry out the inverse conversion we can first convert to the octal scale, X = (0.266)(8),

and then apply (IV) with b=8, using decimal arithmetic: X-2=6/8 6 =6.75, ~-1=6.75/8+2=2.84375, x =2.843751a=0.355 46875.

Converting the integral parts to the decimal scale, we find U-,= (11)(2)=3, E-a=Z-a= (101)(2)=5, (IOO)(2)=4, and thus X= (0.3554) (10)

-

Note that the fractional part in any step is the unconverted remainder. Thus, to round at any step, it is only necessary to ascertain whether the unconverted portion to be neglected is greater or less than 3; i.e., whether, in the binary scale, the first neglected digit is 1 or 0. Example 4. Convert x= (3.141593)(l0).10-' to the binary scale. The desired representation is x = ( I . u - ~ u - ~ . . a-n)(2)*2-' where n and k are such that inverse conversion from the binary scale to the decimal scale will produce x to the same given 15 decimal digits. Accordingly, by the rule stated in Example 3, n 10 and k are to be chosen so as to s a t i s f y n f k 2 3 . 1 5 =50. From Table 28.1 we find 2-20<(3.141593)(l~). 10-'<2-28

Thus, we must take k=29 and, consequently, choose n>21. The conversion on a desk calculator thus proceeds as follows, First, we obtain by use of Table 28.1 2 % ~ (1.686 = 629 899) (10)

Then, for convenience's sake, we convert this number to the octal scale, using the method of Example 3 and rounding aa required, to at least 7 octal (=21 binary) digits. We find 2 ? ~ = (1.537 4337)(8,. Hsnce X = (1.537 433 7)(8)* 2-20 and, consequently, x=(1. 101 011 111 100 011 011 lll)(2)*2-m.

1015

SCALES OF NOTATION

To convert x back to the decimal scale we only need to obtain from Table 28.1 the various powers of 2 which appear in the above representation and sum them. However, since 2-"'=2-"'+l-2-"' for any real constant m, it is more convenient to reduce fmt the binary representation of x to the form

We first compute, using 4.1.19 and Table 4.1, -log10 x 83.44295=277 .05764 log, 2---30103' log,, 2- .30103 and find from Table 4.1, .05764=log10 1.1419. Hence

+

log, x =277 log'' 1'1419=277+logz 1.1419 log10 2 and so x= (1.14 19) (lo) .2277. Now we apply the methods of Example 3 to obtain (1.14 19) (lo) = (1.1 10516),8) where octal notation is used for the sake of convenience. To round such that inverse conversion will yield the same decimal digits of x, observe that the last non-zero decimal digit of x is 3 . low. Table 28.4 shows that 2265<1080<2u16. Hence, in the binary scale, x must be a binary integer times 2aeb;i.e., (l.110516)(8)must be rounded to 4 octal (=12 binary) digits. As a result, x= (1.1105)(g).2277=(11105)(8)~ 2 % ~ = ( 1 001 001 000 101)(1)2265

x=2-~-2-81-2-88-2-89+2-42-2-4b~2-~

and then sum these powers of 2. (Note that the number of summands is thereby decreased from 16 to 7.) From Table 28.1 we have +2-28= $3.725 -2-81= - ,465 -2-83, - .116 -2-89, - .oo 1 +2-42=+ .ooo -2-45=- .ooo -2-m= - ,000 X=

290 66 1 415 818 227 028 000

298 287 322 989 374 422 888

3.141 592 764

*

10-9

40-9

.10-9 40-9

-

10-9 e10-Q -10-9 .10-9

Nine decimal digits are used for sufficient accuracy reserve. Hence, rounding to seven significant figures, we find

Conversion back to the decimal scale proceeds as follows, we write log,, x=log,, 2 log2 x =10g102{265+10gz (11105) (a)}

X = (3.14 1593) (10) .1O-'.

To convert a number such as

x= (€1(10) *

=265 log10 2+10g,o (11105)(8).

to the binary scale, where k is a positive integer so large that Table 28.1 cannot be used, apply the following device: Compute

Hence, converting (1 1105)(8, to the decimal scale by any of the methods of Example 2, we obt a b log10 ~ = 2 6 5log10 2+10g10 4677 which yields, using Table 4.1

where k is the quotient and x1 the remainder, the division being carried out in the decimal scale. Then find 7=1OZ1, i.e., xl=loglo q , so that * log,, x=k+--log'' '-k+log, log10 2

log10 x =83 A4292

Thus, by Table 4.1, we find, rounded to four significant figures, X = (2.773)(10)* 10".

q

whence

References

x= (v> (lo)2L. Now convert (v),,~)to the binary scale by any of the methods described above. A similar device may be used to convert to the decimal scale a binary number that is outside the range of Table 28.1. Example 5. Convert x= (2.773)(ro).1099 to the binary scale.

I

[28.1] J. Malengreau, Etude des 6criturea binairea, Bibliothhque Sci. 32 Mathhmatique. adition Griffon, NeuchAtcl, Suisae (1958). [28.2] D. D. McCracken, Digital computer programming (John Wiley & Sons, Inc., New York, N.Y., 1957). [28.3] R. K. Richards, Arithmetic operation in digital commters (D. Van Nostrand Co., Inc., New York, N.<, 1955).

1016

SCALES OF NOTATION

Table 28.1

2*n

IN DECIMAL

2n

n

2 -n

1 2 4

0 1 2

1.0 0.5 0.25

8 16 32

3 4 5

0.125 0.0625 0.03125

64 128 256

6 7 8

0,01562 5 0.00781 25 0.00390 625

512 1024 2048

10

4096 8192 16384

12 13 14

0.00024 41406 25 0.00012 20703 125 0.00006 10351 5625

32768 65536 1 31072

15 16 17

0.00003 05175 78125 0.00001 52587 89062 5 0,00000 76293 94531 25

2 62144 5 24288 10 48576

ia 19 20

0.00000 38146 97265 625 0,00000 19073 48632 8125 0.00000 09536 74316 40625

20 97152 41 94304 83 88608

21 22 23

0.00000 04768 37158 20312 5 0.00000 02384 18579 10156 25 0.00000 01192 09289 55078 125

167 77216 335 54432 671 08864

24 25 26

0,00000 00596 04644 77539 0625 0,00000 00298 02322 38769 53125 0.00000 00149 01161 19384 76562 5

1342 17728 2684 35456 5368 70912

27 28 29

0,00000 00074 50580 59692 38281 25 0.00000 00037 25290 29846 19140 625 0.00000 00018 62645 14923 09570 3125

10737 41824 21474 83648 42949 67296

30 31 32

0.00000 00009 31322 57461 54785 15625 0,00000 00004 65661 28730 77392 57812 5 0.00000 00002 32830 64365 38696 28906 25

85899 34592 1 71798 69184 3 43597 38368

33 34 35

0.00000 00001 16415 32182 69348 14453 125 0.00000 00000 58207 66091 34674 07226 5625 0.00000 00000 29103 83045 67337 03613 28125

6 87194 76736 13 74389 53472 27 48779 06944

36 37 38

0.00000 00000 14551 91522 83668 51806 64062 5 0,00000 00000 07275 95761 41834 25903 32031 25 0.00000 00000 03637 97880 70917 12951 66015 625

54 97558 13888 109 95116 27776 219 90232 55552

39 40 41

0.00000 00000 01818 98940 35458 56475 83007 8125 0.00000 00000 00909 49470 17729 28237 91503 90625 0.00000 00000 00454 74735 08864 64118 95751 95312 5

439 80465 11104 879 60930 22208 1759 21860 44416

42 43 44

0.00000 00000 00227 37367 54432 32059 47875 97656 25 0.00000 00000 00113 68683 77216 16029 73937 98828 125 0.00000 00000 00056 84341 88608 08014 86968 99414 0625

3518 43720 88832 7036 87441 77664 14073 74883 55328

45 46 47

0.00000 00000 00028 42170 94304 04007 43484 49707 03125 0.00000 00000 00014 21085 47152 02003 71742 24853 51562 5 0.00000 00000 00007 10542 73576 01001 85871 12426 75781 25

28147 49767 10656 56294 99534 21312 112589 99068 42624

48 49 50

0.00000 00000 00003 55271 36788 00500 92935 56213 37890 625 0.00000 00000 00001 77635 68394 00250 46467 78106 68945 3125 0.00000 00000 00000 88817 84197 00125 23233 89053 34472 65625

9 11

1017

SCALES OF NOTATION

2“ IN DECIMAL

2”

X

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

1.00069 1.00138 1,00208 1.00277 1.00347 1.00416 1.00486 1,00556 1.00625

2”

x’

33874 72557 16050 64359 17485 75432 38204 05803 78234

Talde 28.2

2”

t

0.1 0. 2 0. 3 0.4 0. 5 0. 6 0. 7 0. 8 0.9

0. 01 0. 02 0. 03 0.04 0. 05 0. 06 0. 07 0. 08 0. 09

62581 11335 79633 01078 09503 38973 23785 98468 97782

10+ri IN OCTAL

3 46 575 7 346

10”

n

1 12 144 1 750 23 420

0 1

303 641 113 360 545

2

3 4

240 100 200 400 000

000 146 015 406 032

1.000 0.063 0.005 0.000 0.000

000 314 341 111 155

000 631 217 564 613

10”

000 463 270 570 530

000 146 243 651 704

00 31 66 77 15

5

0.000 002 476 132 610 706 64

6 7 8 9

0.000 0.000 0.000 0.000

000 000 000 000

206 015 001 000

157 327 257 104

364 745 143 560

055 152 561 276

37 75 06 41

n

1 0 ~ 1 02

n

0.30102 99957 o:bo% 99913 0.90308 99870 1.20411 99827 i;&ii 99783

1

5

3

4

5

log2

3.32192 6.64385 9.96578 13.28771 16.60964

10-“

n

1 16 221 2 657

112 351 432 411 142

402 035 451 634 036

762 564 210 520 440

000 000 000 000 000

10 11 12 13 14

0,000 000 000 006 676 331 66 0.000 000 000 000 537 657 77

34 327 434 157 5 432 127 6 1 405 553

724 115 413 164

461 760 542 731

500 200 400 000

000 000 000 000

15 16 17 18

0.000 0.000 0.000 0.000

0.000 000 000 000 043 136 32 0,000 000 000 000 003 411 35 0,000 000 000 000 000 264 11 000 000 000 000

n log2 10 IN DECIMAL

n logl0 2 ,

n

Talde 2 8 . 3

10-”

10

n

80949 61898 42847 23195 04744

A

n

log10

Table 28.4

2

n

1. 80617 _. . .- 99740

7

2.10720 2.40823 2.70926 % 01029

8 a 10

log2

19.93156 23; 25349 26.57542 29.89735 33.21928

99696 99653 99610 99566

10 85693 66642 47591 28540 09489

‘1al)le 2 8 . 5

ADDITION AND MULTIPLICATION TABLES

Addition

000 000 000 000 000 000 000 000 000 000 000 000

Multiplication

Binary Scale o + o = 0 0 + 1 = 1 + 0 = 1 1 + 1 = 1 0

o x o = o

0 x 1 - l x o . o 1 X l . l

Octal Scale

4

05

06

07

10

11 1 2 1 3

5

12

17

24

31

36

5

06

07

10

11 1 2 1 3 1 4

6

1 4 22

30

36

44

52

6

07

10

11 12 1 3 1 4 1 5

7

1 6 25

34

43

52

61

7

1 0 11 1 2

43

13 14 15 16

MATHEMATICAL CONSTANTS IN OCTAL SCALE I-

= (0.24276 301556)

4;In log2

(1.61337 611067)(0 (1.11206

I=

e-

(3.11037 552421)

404435)(~)

(1.51544 163223)(,)

dms (3.12305

407267)(a)

e- I

-

de=

(2.55760 5 2 1 3 0 5 ) ( ~ ) (0.27426 530661)(8)

( 1.51411 230704)

7=

In log2

’I

-

7-

Table 28.6 (0.44742 147707) -( 0.43127 233602) -( 0.62573 030645)

( 8)

(0.33626 754251)

d2=

e=

( I. 34252 166245)

In 2 =

( 0.54271 0 2 7 7 6 0 ) ( ~ )

logz 10-

(3.24464 741136)

In 10-

(2.23273 0 6 7 3 5 5 ) ( ~ )

l o g 1 0 elog2

(8)

( 1.32404 746320)

022 001 000 000

01 63 14 01

29. Laplace Transforms Contents

1020

29.3. Table of Laplace Transforms

1021

29.4. Table of Laplace-Stieltjes

. . . . . . . . . . . . . . . . Transforms . . . . . . . . . . . .

1029

29.2. Operations for the Laplace Transform

_.--

Page

. . . . . . . . . . . . . . . . . . . . . . . .

29.1. Definition of the Laplace Transform.

References

..........................

1020

1030

29. Laplace Transforms 29.1. Definition of the Laplace Transform

1

function of s in the half-plane ~ s > s o .

One-dimensional Laplace Transform

f(s)=-EP{ F ( t )1 = J m

29.1.1

Two-dimensional Laplace Transform

29.1.2

e-"F(t)clt 0

f ( u , ~ ) = L f { F ( z , y =/ J)m}J0m

F(t) is a function of the real variable t and s is a complex variable. F ( t )is called the original function andf(s) is called the image function. If the integral in 29.1.1 converges for a real s=so, i.e.,

u(t)=

limJB e-aotF(t)dt

E+

A

m

exists, then it converges for all s with 9 s > s o , and the image function is a single valued analytic

e-UZ-oYF(x,y)dxdy

Definition of the Unit Step Function

29.1.3 A-10

0

I

{

0

4 1

(tO)

I n the following tables the factor u(t) is to be understood as multiplying the original function F(t).

29.2. Operations for the Laplace Transform Original Function F ( t )

lm

e-s'F(t)dt

F(t)

29.2.1

Image Function f(e)

Inversion Formula

29.2.2 Linearity Property

Integration

29.2.6

1

8 -f(s)

29.2.7

1

82 -f(s>

Convolution (Faltung) Theorem

lf

29.2.8

Fl(t-r)F2(r)dl=Fl*F2 -tF(t)

29.2.9 29.2.10

(-I)"t"F(t)

f ' (8)

Differentiation

P' (8)

I Ada ted by permission from R. V. Churchill, Operational mathelhatics, 2d ed., McGraw-Hill Book Co., Inc., New York, N.Q., 1958.

1020

1021

LAPLACE TRANSFORMS

Original Function F ( t )

Image Function j ( s ) Integration

1

t F(t)

29.2.11

JW

f(x)dx

Linear Transformation

29.2.12

eafF(t)

f(s-a>

Translation

29.2.15

F(t - b)U ( t -6)

e-

( b>0)

Periodic Functions

e-"F(t)dt

F(t +a) =F(t)

29.2.16

l-e-aa

F(t +a) =- F( t )

29.2.17

Half-Wave Rectification of F ( t ) in 29.2.17

29.2.18

F(t)

5 (-l)Ru(t--na) n PO

Full-Wave Rectification of F ( t ) in 29.2.17

as

lF(t)I

29.2.19

f(s) coth 2 Heaviside Expansion Theorem

29.2.20

p ( s ) a polynomial of degree<m 29.2.21

eat

5 p('-n)(a) tn-' RI1

(r-n)!

(n-l)!

24s)

(s-a>l p ( s ) R polynomial of degree
29.3. Table of Laplace Traneforrn~~,~

For a comprehensive table of Laplace and other integral t8ransfonusscc [29.9]. For n tnblc of twodimensional Laplace transforms see [29.11]. 1'(U

29.3.1

-1 S

29.3.2

-1 S2

1

t

The numbers in bold type in t h cf ( s ) and F ( t ) columns indicate the chapters in which the properties of thc rcspcctivc higher mathematical functions are given. * Adapted by permission from R. V. Churchill, Operational Innthemntics, 2d. etl., McCraw-Hill Book Co., IIIC., SC\\ York, N. Y., 1958.

1022

LAPLACE TRANSFORMS

f(4 29.3.3

1 5"

(n=1,2,3,. * .)

5-(n+4)

*)

1.3.5.. . (2n-l)h

6

(k>O)

Sk

1

tk-l

e-at

1

te-"'

(5+(q2

(n=1,2,3,

29.3.10 29.3.11

J;;t

8-312

29.3.8 29.3.9

1

z

29.3.5

29.3.7

(n- 1)!

1

29.3.4

29.3.6

tn-1

(%=1,2,3, . . .)

(k>O)

(5+a>k

tn-le-at

. . .)

(n- l)!

6

tk-le-at

a-b

29.3.14

1

(5+a)(s+b)(5+c)

- ( b- c )

e

+ (c-a) (a-

(a, b, c distinct constants) 1

29.3.15

s2+a2

29.3.16

s2+a2

29.3.17 29.3.18 29.3.19 29.3.20 29.3.21

S

1 sa- a2 S

s2-a2

1

- sin at a

cos at 1

- sinh at a cosh at

5(s2fa2)

1 (1-cos at) a2

1 s2(s2+a2)

1 (at-sin at) a3

1

1

+

(s2 a')

+

e-bf (a- b) e-cf

a) (b-c)

1 - (sin at-at cos a t ) 2a3

(c-a)

LAPLACE TRANSFORMS

1023 F (t)

29.3.22 29.3.23 29.3.24

1 (sin at+at cos at)

S2

+

(s2 a=)=

2a

s2- a= (sZ+a=)2

t cos at

29.3.25

(a2# bZ)

29.3.26

COS

at-cos

bt

b2-a2 e-ar

29.3.27 29.3.28

2at sin at

S

(s*+a2)2

e-"l

sin bt

cos

bt

3a2 -I-a3 s3

29.3.29 29.3.30 29.3.31 29.3.32

4aa s4+4a4 S

s'+qa4 1 -

sin at cosh at-cos at sinh at 1 sin at sinh at 2a2

1 (sinh at-sin at)

s4-a4

2a3

S -

1 (cosh at-cos at) 2aa

s4-a4

+

29.3.33

( 1 a2t*) sin at -at cos at

29.3.34

22

29.3.35 29.3.36 29.3.37 29.3.38

1 -&+U

7

fi s-a=

7

29.3.39

29.3.44

7 1 6(8-a*)

7

1024

LAPLACE TRANSFORMS

f (4

F (0

7

29.3.41 29.3.42

eaat[b--a

29.3.43

,azt

erf aJt]-beb2t erfc b f i

erfc a d

7 7

29.3.44

7

29.3.45

7

29.3.46 29.3.47

(1-8)"

22

gn+*

(1-8)" 8"+t

22

29.3.48

9

29.3.49

9

1

I

I I

29.3.50

10

29.3.51

9

j I

I

29.3.52 9

29.3.53

I I

1

- e-r'lv(3at)

29.3.54

29.3.55

29.3.56

29.3.57

ay

1

9

9

J82+a2 avJ,(at)

9

6,lO

1025

LAPLACE TRANSFORMS

f (8) 29.3.58

29.3.59

(4W-s)' (S-@=aV

dsZ--aZ

(k>O)

kak t Jk(at)

9

(y>-l)

avI,(at)

9

f 29.3.61

-1 e - k s

29.3.62

1 e-kd -

S

62

u(t-k)

(t -k )~ (-kt )

6

29.3.64

5u(t-nk)

29.3.65

n =O

0

X

-1

Zk

a,

a"-'u(t-nk)

29.3.66

n=l

*;-f

0

29.3.67

29.3.68

29.3.69

1

- tanh ks S

1

s(l+e-ka) 1

tanh ks

u(t)+2

2

2k

k

5 (-l)"u(t--nk)

n-1

U

(-l)"u(t-nk)

t U ( t)

L

ty-L+

+2 5 (-1 )"( t-2nk)U (t -2nk) n=l

0

t 29.3.70

1

s sinh ks

2

5 u[t-(2n+l)k]

n-0

1

2

61

81

'nn t

5 (-l)"u[t-(2n+l)k] n=O

4k

lYz 0

s cosh ks

Zk

r

1 1 2

29.3.71

31

I

L

~

3k

~

5k

0

h

3L

Sk

7k

1026

LAPLACE TRANSFORMS

f (8)

F(1)

-81 coth k8

29.3.72

u(t)+2

5 u(t-2nk)

n-1

0

Zk

'

41

1

29.3.74

(sa+ 1 ) ( 1-e-rr)

1 -L

29.3.75

-e 8

1 -& cos 2 4 G

i e4

29.3.76

1

29.3.77

cosh 2@

1

z a

29.3.78

sin 2 m

1

29.3.79

83/a

1

*

e'

sinh 2 & t

29.3.81 -k&

29.3.82

e

29.3.83

-e

1

-k&

8

(00) ( k 2 0)

29.3.84

k

erfc

1exp

fl 29.3.85 29.3.86

29.3.87

29.3.88

LS i e - 4 1 e-kdi Ql+l* ~1 s 2

-kfi e

(k2O) (n=O, 1,2, . . .; k 2 0 )

(n=O, 1 , 2 , , . .; k>O)

7

G

2$

(-3

exp (-:)-k ?r

(4t)i" i" erfc

k -

erfc - = k 24 2 4

i erfc k

7

2 4

7

2 4

22

7

1027

LAPLACE TRANSFORMS

I ~

f (8)

i

F(t)

k

-k d m J52+a2

29.3.92

29.3.94

( k 2 0)

29.3.95

e-

(k20) (k>O)

k 6 - e - k w

1

-Ins

29.3.98

S

1

29.3.99

-In s

(k>O)

In s -

(a>()>

Sk

29.3.100

s-a

29.3.101 29.3.102 29.3.103

I

e -etI0(+aJt '-k2)U (t -k )

9

I

J o ( aJt2_k2) U (t -k )

9

Jw)

J o( a ak

J 1(aJt2- k2)U(t-k )

9

-1, ak Jt2-k2

(aJt2--k2)u(t-k )

9

(5:y

J,

-7-ln

t(7=.57721 56649

d

n

U (t-k )

9

. . . Euler's constant) 6

92+1

In s

cos t Si (t)-sin t Ci ( t )

5

s In s

-sin t Si (t)-cos t Ci ( t )

5

s2+1

1

- In ( l f k s ) S

(k>O)

29.3.104

s+a In s+b

29.3.105

- In (l+k2s2)

29.3.106

7

e - k ( w - r )

.JW

7

1

S

-S1 In (s2+aZ)

(9

5

(k>O)

(a>())

2 In a-2 Ci (at)

5

I

1028

LAPLACE TRANSFORMS

f (8) 29.3.107

F (t)

1

- In (s2+a2)

4a [at In a+sin

(a>O)

62

s2+a2 In -

29.3.109

In 82

2t (1-cosh

29.3.110

k arctan -

1

29.3.111

1 k - arctan -

29.3.112 29.3.113

-t (1-cos at)

S2

a2

S

S

1.S

at)

sin kt

Si (kt)

S

ekaazerfc ks

5

2

29.3.108

s2-

at--at c i (at)]

(k>O)

erfc ks

(k>O)

eklaZ

7

-exp(-&) 1 kfi

7

erf

at

5

7

I-

29.3.114

er* erfc &

(k>O)

29.3.115

1 3 erfcfi

(k20)

29.3.116 29.3.117

29.3.118

q1 ekaerfc &

(k20)

k

erf

z

1!

k

Ge8erfc 4 -

7

7

-@&-k) 1 1

J

m

7

rt Lsin 2 k 4

7

-1

-2k6

m u(t-k)

29.3.119

9

29.3.120

9

&exp(-Z)

9

iJt(t+zk)

29.3.121

-81 ekaKl(ks)

(k>O)

29.3.122 29.3.123

9 1

29.3.124

r e +Io (ks)

(k>O)

9

[u(t)-~(t-2k)] J@Ei

29.3.125

e-lUll(ks)

(k>O)

9

k-t [u(t>- ~ ( t - z k ) ] r k d m

~

1029

LAPLACE TRANSFORMS

f (5) 29.3.126

easEl(as)

29.3.127

1 --se"El(as) a

29.3.128

a'-neaaEn(a.s)

29.3.129

[;-Si(8)]

F (t) (a>O)

(a>O)

5

-

5

-

1 t+a

1

(t+# 1

(a>O;n=O, 1 ' 2 , . . .) 5

cos s+Ci(s) sin s

( t +a> "

5

-

1 t2+l

29.4. Table of Laplace-Stieltjes Transforms

W)

4s) n m

29.4.1 29.4.2

e-ka

(k>O)

u(t-k)

29.4.3 29.4.4

n=O

1 l+e-ka

(D

n=O

29.4.5

2

29.4.6

2

29.4.7 29.4.8

29'4'9 29*4g10

tanh ks 1

sinh (ks+a)

e-hs sinh ( k s f a ) sinh (hs+b) sinh (ks+a)

u(t-nk) (-l)"u(t-nk)

5 w[t-(2n+l)k] n=O

5 (-l)nu[t-(2n+l)k]

n PO

u(t)+2

(k>O)

5 (-1)"u(t-%k)

n-1

(k>o)

2 e-czn+1)uu[t-h-(2n+l)k] n=O m.

2

h>O)

m

C e-(2n+1'a{ebu[t+h-(2n+l)k]

(O
transform see [29.7]. I n practice, Laplace-Stieltjes transforms are often written as ordinary Laplace transforms involving Dirac's delta function 6 ( t ) .

This "function" may formally be considered as

n-0

-ehbu[t-h-

dt, so tllat J" du(t)=J:m

(2n+ l ) k l }

s(t)dt=

-m

I

The correspondence 29.4.2, for instance, then assumes the form

e-ka=lm

e-"'b(t-k)dt.

Adapted by permission from P. M. Morse and H. Feshbach, Methods of theoretical physics, vols. 1, 2, McCrawHill Book Co., Inc., New York, N.Y., 1953.

1030

LAPLACE TRANSFORMS

References Texts

[29.1]H. S. Carslaw and J. C. Jaeger, Operational methods in applied mathematics, 2d ed. (Oxford Univ. Press, London, England, 1948). [29.2]R. V. Churchill, Operational mathematics, 2d ed. (McGraw-Hill Book Co., Inc., New York, N.Y., Toronto, Canada, London, England, 1958). [29.3] G. Doetsch, Handbuch der Laplace-Transformation, vols. 1-111 (Birkhauser, Basel, Switzerland, 1950; Basel, Switzerland, Stuttgart, Germany, 1955, 1956). [29.4] G. Doetsch, Einfiihrung in Theorie und Anwendung der Laplace-Transformation (Birkhauser, Basel, Switzerland, Stuttgart, Sermany, 1958). [29.5]P.M.Morse and H. Feshbach, Methods of theoretical physics, vols. I, I1 (McGraw-Hill Book Co., Inc., New York, N.Y., Toronto, Canada, London, England, 1953). [29.6] B. van der Pol and H. Bremmer, Operational

calculus, 2d ed. (Cambridge Univ. Press, Cambridge, England, 1955). I29.71 D. V. Widder, The Laplace transform (Princeton Univ. Press, Princeton, N.J., 1941).

Tables

[29.8]G. Doetsch, Guide to the applications of Laplace transforms (D. Van Nostrand, London, England; Toronto, Canada; New York, N.Y.; Princeton, N.J., 1961). [29.9]A. ErdBlyi et al., Tables of integral transforms, vols. I, I1 (McGraw-Hill Book Co., Inc., New York, N.Y., Toronto, Canada, London, England, 1954). [29.10]W. Magnus and F. O)erhettinger, Formulas and theorems for the special functions of mathematical physics (Chelsea Publishing Co., New York, N.Y., 1949). [29.11] D. Voelkeq and G. Doetsch, Die zweidimensionale Laplace-Transformation (Birkhauser, Basel, Switzerland, 1950).

Subject Index Page

__ ____ ___________ _ - __ _ _ _- - _ __ __________ __ _ _ _ _ ___ _ _ ____ _ _ _ _ _ _ _ _ _ _ _ _ _ ___

822 510 804 810 803 804 809 804 804 804 805 804 805 805 804 806 805 806 804

generating function for--- -inequalities for- - - - - - - - - - - - - - - - - - - - - - - - integrals involving--- - - - - - - - - - - - - - - . - multiplication theorem for-- - - relations with Euler polynomials. - - - special values of----------------------------_ symbolic operations - _ -symmetry relations _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Bessel functions as parabolic cylinder functions _ _ - _ _ - - - _ _ _ 692,697 ___ definite integrals _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 485 modified_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _374, _ _ _509 _ notation for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _358 __ of fractional order _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 437 of the first kind _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 358,509 of the second kind _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 358,509 of the third kind - - _ _ _ _ _ _ - _ - _ _ _ _ _ _ _ - _ _358,510 _____ orthogonality properties of 485 representations in terms of Airy functions - - - - 447 spherical- - -------- --437, ‘509 Bessel functions of half-integer order--- - - - - - - - _ 437, 497 ieros and associated values 467 zeros of the derivative and associated values- 468 Bessel functions, integrals479,485 asymptotic expansions- - - - - - - - - - - - _- - - _ 480,482 computation of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _488 _ convolution type _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 485 __ Hankel-Nicholson type 488 involving products of-484 polynominal approximations- - - - - - - - 481,482 recurrence relations-- - - - _ - _- - - - - - - - - - - - - - 480,483 reduction formulas- - - - - - - - - - _- - - - - 483 repeated _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _482 ___ simple _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _480 __ tables of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _492 ___ Weber-Schafheitlin type --- - - - - - 487 Bessel functions J , ( z ) ,Y,(z) 358, 379, 381, 385 addition theorems for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 363

_______

______ _ _ _ _ _ _ _ _______ _ __ ____ _____

___________________

__

___________________ __ __

_______ _______-________ ______________________

__ _____ _ _ _ _ _ _ _ _ _ __ _____ _ _ _ _ _ _ _ _ ___________

1031

INDEX

Bessel functions J v ( z ) , Yv(z)-Continued Page analytic continuation of_-- - - - - - - _ _ - - - - 361 ascending series for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 360 asymptotic expansion for large arguments- - 364 asymptotic expansions for large orders--------365 asymptotic expansions for zeros- - - - - - 371 asymptotic expansions in the transition reglon for large orders____-_-----_----____________ 367 asymptotic expansions of modulus and phase for . . . . . . . . . . . . . . . . . . . . . . 365 large arguments connection with Legendre functions--- - 362 continued fractions for- - - - - - - - - - - - - - - - - _ _ _ _ 363 derivatives with respect to order 362 358 differential equation- - - - - - - - - - - - - - - - - - - - - - - - 362 differential equations for products- 361 formulas for derivatives- _ _ _ - - - - - - - - - - - - generating function and associated series- - - - 361 graphs of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _359,373 _____ in terms of hypergeometric functions--. 362 360 integral representations of.. - - - - - - - - - - - - - _ _ _ _ limiting forms for small arguments_ _ _ _ _ _ _ _ _ _ _ _ _ 360 modulus and phase _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _365 __ multiplication theorem for-- - - - - - - - - - 363 Neumaiin’s expansion of an arbitrary function, 363 notation _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _358 __ other differential equations- - - - - - - - - - - - 362 polynomial approximations - - - - - - - - - - - - - - - 369 recurrence relations _ _ _ _ _ _ 361 recurrence relations for cross-products---- - - - 361 relations between- - - - - - - - - - - - - - - - - - - - - - - - - - 358 tables o f - - - _ _ - _ - - - - - _ - - - - - - - - - - - _ _ _ _ _ _ _ - _ _ _ _ 390 358 uniform asymptotic expansions for large orders-362 upper b o u n d s - - - - - - - - - - - - - - - - - - - - - - - - - - - - _ - 360 Wronskian relations- - - - - - - - - - - - - - - - - - - - - - - zeros of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _370 __ 372 zeros, complex- - 370 zeros, infinite products for 371 zeros, McMahon’s expansions for-- - - - - - - - - - 374 zeros of cross products of zeros, tables of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 371,409 ,414 371 zeros, uniform expansions of- - - -__ --_ 881 Bessel’s interpolation formula- - - - - - - - - - - - - - - - - -

__

_____

__ __ ___ ________

_ _ _ __ __ ______________ ___________ __ ___ __ ________

__

_

__

__

____

_

__ _ __

____ .................... __ _

_

_______________ ____________ .................... __ _____________________ _

__ __ _ _ __

795 11 958 940 941 941 941 940 942 941 940 941 941 943 785 71,91 72 72 76 75 78 74 77 74 75 72 72 77 75 75 75 74 72 72 72 76 72 74 73 72 74 74 73 142 936 957 1005 1006

I

INDEX

1033

883 877 877 877 877 883 877 878 896 450 896 896 896 896 896 896 896 896 897 882 883 882 883 882 914 258 1004 927 935 928 927

1034

INDEX Page

927 93 1 927 930 929 827 840 21

INDEX

1035

INDEX Page

recurrence formulas _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 263, 944 relation to other functions . . . . . . . . . . . . . . . . . . . . 945 relation to the binomial expansion _ _ _ _ _ _ _ _ _ _ _ _ _ 263 relation to the +distribution- - - - - - - - - - - - - - - - 944 relation to the hypergeometric function- - - - - _ _ 263 series expansion for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 944 __ symmetry relation--- _ _ _ _ _ - - _ -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 263 Incomplete gamma function _ _ _ _ - _ _ _230,260,486, __ 509 as a confluent hypergeometric function-- - - - - - - 262 asymptotic expansions of.. - - - - - - - _ -- - - - - - - - 263 computation of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _959 ___ continued fraction for _ _ _ _ _ _ - - - - - - - - - - - _ _ _ _ _263 __ definite integrals.. - - - - - - - - - - - - - - - - - - - - - - - - - - 263 derivatives and differential equations- - - - - - - - - 262 graph o f - - - _ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 261 Pearson’s form of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _262 ___ 262 262 262 978 13 11 11 11 11 11 11 11 11

__

956 77 71 86 82 88 12 69 12 885 886 12 887 886 890 887 887 891 886 888 891 886 888 886 885 878 879 881 882 880

1037

INDEX

Page

quarter periods _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _569 ___ reciprocal parameter-- _ _ _ _ _ _ _ _ _ _ - _ _ _ _ - _ _ 573 relations between squares of the functions- - - - - 573 relation to the copolar trio . . . . . . . . . . . . . . . . . . . . 570 relation with Weierstrass functions- - - -.- - - - - - 649 series expansions in terms of the nome q- - - - - - - 575 special arguments.. - - - _ _ _ _ - - _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ 571 Jacobi’s e ta function 577 Jacobi’s polynomials- - _ _ - - - - - - - - - - - - - - - - - - - - - 561, 773 (see orthogonal polynomials) coefficients for- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _793 __ graphs of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _773,776 _____ Jacobi’s theta function (see theta functions) - - - - - 576 Jacobi’s zeta function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 578, 595 addition theorem for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 595 __ calculation by use of the arithmetic-geometric mean-----------------------------------578 graph o f _ _ - - - _ _ _ _ _ _ - _ - _ - - - - - - - - - - - - - - - - - - - -595 Jacobi’s imaginary transformation - - - - - - - - - - - - 595 q-series for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 595 relation to theta functions_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 578, 595 specialvalues------------------------------595 table o f - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 619

__

__

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ r _

K Kelvin functions--- . . . . . . . . . . . . . . . . . . . . . 379,387,509 ascending series for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 379 __ ascending series for products of--- - - - - - - - - - - - - 381 asymptotic expansions for large arguments- - - 381 asymptotie expansions for large zeros- - - - - - - - - 383 asymptotic expansions of modulus and phase- - 383 asymptotic expansions of products-- - - - - - - - - - - 383 definitions-- - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - 379 differential equations--- - - - - - - - - - - - - - - - - - - - - - 379 expansions in series of Bessel functions-381 graphs o f - - _ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 382

____

.

_______

382 380 384 380 380 379 430 384 381 822 504 16

882 886 915 878 900 878 14 890 923

INDEX

INDEX Page

896 14 11 11

997 826 374 376 375 377 377 377 374 376 376 374 377 376 375 377 377 377 378 376 375 416 378 375 377 722 734 83,498 445 443 453 443 445 445 443 444 445 445 445 444 444 443 469 443 498 498 499 498 498 498 498 498 498 501 16 928 891

1039

1040

INDEX

Orthogonal polynomials-Continued page limit relations _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _787 ____ of a discrete variable _ _ _ _ . _ _ _ _ _ 788 _______________ orthogonality relations- - - - - - - - - - - - - - - - - - - - - 774 powers of z in terms of _ _ _ _ - _ _ _ _ _ _ _ _ 793, _ _ _ 794-801 _ recurrence relations, miscellaneous- - - - - - - - - - 773, 782 recurrence relations with respect to degree n- _ _ 782 Rodrigues’ formula- - - - - - - - - - - - - - - - - - - - - - - - 773, 785 special values of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _777 ___ sum f o r m u l a s - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 785 tables of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _795, _ _ _796, _ 800, 802 zeros o f - _ _ _ _ - - - _ - _ - - - - - _ - - - - - - - - - - - - - - - - - - - 787

-

-

P Parabolic cylinder functions V ( a ,z), V ( a ,2) -_- - - 300, 509, 685, 780 asymptotic expansions of-- _ _ - - - - - - - _ _ 689 computation of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 697 connection with Bessel functions _ _ _ _ _ _ _ _ _ _ _ _ 692,697 connection with confluent hypergeometric func691 tions-----_-----_-----_-----------__-connection with Hermite polynomials and functions----__-----__--------_---___-____-69 1 connection with probability integrals and Dawson’s integral- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 69 1 Darwin’s expansions- - - - - - - - - - - - - - - - - - - - - - - - 689 differential equation- - - - - - - - - - - - - - - - - - - - - - 686 expansion8 for a large, z moderate_ _ _ _ _ _ _ _ 689 expansions for z large, a moderate 689 expansions in terms of Airy functions- - - - - - 689 integral representations of- _ _ - - - - - - - - - - - 687 modulus and phase.. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 690

- __ __

-

-

-

__

____ _____________ -_ _ __ __ -

Page

825 262 260

984 986 990 999 78, 92 256 896 509 959 978 16 260 260 260 260 260

INDEX Page

16 14 68 15 16 14 74 774 774 1000

INDEX

Page Spheroidal wave functions- Continued Page characteristic values for- - - 753, 756 Thiele’s interpolation formula-- - - - - - - - - - - - - - - 881 differential equations - --_ _ _ _ __ - 753 I Toroidal functions_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 336 evaluation of coefficients for Toronto function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _509 __ 755 Trapezoids1 rule --_ _ -----_ _ 885 expansions f o r _ _ _ _ _ - - - _ - - _ _ - _ - _ _ _ _ - - _ - _ _ - - - _755 joining factors for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 757 Triangle inequality- - - - -11 normalization of _ _ _ _ _ _ _ _ _ _ _ - - _ _ _ _ _ _ _ _ _ _755 _ _ _ _ _Trigamma _ function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 260 tables of _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ - - _ _ _ _ _ _ _ _ 766 _ _ _ _ _ _ _(see _ polygemqna functions) 71 functions- - - - - - - - - - - - - - - - - - - - table of eigenvalues of- _ _ _ _ - - - _ - _ - _ _ _ _ _ _760 _ _ _ _ Trigonometric _ (see circular functions) table of prolate joining factors 769 Stirling numbers- - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 824 Truncated exponential function- - - 70,262 table of the first kind _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 833 U table of the second kind _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 835 Ultraspherical polynomials-- - - - - - - - - - Stirling’s formula -------- --257 774 Struve’s functions.. - - - - - _ _ - -----495 (see orthogonal polynomials) coefficients for and r” in terms of _ _ _ _ _ _ _ _ _ _ 794 asymptotic expansions for large orders - - - - - - - 498 asymptotic expansions for large zI - - 497,498 graphs o f _ _ _ _ _ - - _ _ _ _ - - - - _ _ - - - - - - - - - - - - - - - - 776 computation of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 499 Unit step function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1020 ___ differential equation- - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 496 V graphs of _ _ _ _ - - - _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _496 _______ integral representations of - - - _ _ _ _ _ _ - 496,498 Variance--_----_---------_____________----928 integrals _ - _ _- -- - - - - - - - - - - - - - - - - - --- - __ -- - - 497,498 Variance-ratio distribution function--- - - - - - 946 modified-___--------_____________________-_498 (see F-distribution function) power series expansion for 496,498 1006 Vector-addition coefficients- - - - - - - - - - - - - - - - - recurrence relations-- - - _ _ - - - _ _ - 496,498 W relation to Weber’s function- - - - - 498 special properties of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ *497 Wallis’ formula _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 258 ___ tables o f _ _ - - - _ _ - - - - - - - _ _ _ - - - - - - - - - - - _ - - - - - - - 501 Wave equation in prolate and oblate spheroidal coordinates----752 Student’s 1-distribution _ _ _ _ - --948 approximations t o - - - - - - - - - - - - _ - - - _ - _ - - - - - _ _ 949 Weber’s function498 asymptotic expansion of--- _ _ - 949 relation to Anger’s function 498 limiting distribution- relation t o Struve’s function 498 949 non-central-----------------------_-----_--949 Weierstrass elliptic functions--- - - --- --- 627 series expansions for- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 948 addition formulas for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 635 statistical properties of-- - -- -- 948 case A=O---------------_-_________________, 051 Subtabulation_ _ _- - - - - - - - - - - - _ 881 computation of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _663 __ Summable series _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _1005 __ conformal mapping of 642,654,659 Summation of rational series 264 definitions- - - - _ _ - - - - - - - - - - 629 Sums of positive powers _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 813 derivatives of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _040 __ Sums of powers-- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 804 determination of periods from given invariants- 665 I Sums of reciprocal powers807,811 determination of values at half-periods, etc., from Systems of differential equations of first order----897 064 given periods _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ differential equation- - - - - - - - - - - - 629,640 T discriminant- - - - - - - - _ _ _ _ - - - - --- 029 Taylor expansion------_,---_-------------_---880 equianharmonic case _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 652 Taylor’s formula _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 14 expressing any elliptic function in terms of %nd Tesseral harmonics- - - - - - - - - - - - - - - - - - 332 8 ’ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _651 _______ Tetrachoric functions- - - - - - - - - - - - - - - - - - 934 fundamental period parallelogram.. - -629 fundamental rectangle- - - - _ -- - - - - - - - - - 630 Tetragamma function 260 (see polygamma functions) homogeneity relations _ _ _ _ -- ----631 Theta functions 576 integrals _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _641 ___ . . invariants- - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 629 addition of quarter-periods- - - - - - - - - --577 calculation by use of the arithmetic-geometric Legendre’s relation- - 634 mean___--__------_____________________ 577,580 lemniscatic case- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 658 expansions in terms of the nome q 576 maps of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 642,654,659 Jacobi’s notation for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 577 __ multiplication formulas-- _ _ _ - - - - - - - - - - - - - - - - - 635 logarithmic derivatives of - ---- - - 576 other series involving 8,9’’ 639 pseudo-lemniscatic case - - - - - - - - - - - - - - - - - - 662 logarithms of sum and difference _ _ _ _ _ _ _ _ _ _ _ _ _ _ 577 Neville’s notation for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 578,582 reduction formulas 031 relation with complete elliptic integrals 649 relations between squares of the functions - - - - - 576 relations with Jacobi’s elliptic functions- - - 049 relation to Jacobi’s zeta function 578 650 relation with Weierstrass elliptic functions-- - - - 650 relations with theta functions

________________ _____ _ _ ____ ____ __________________

__

_____ _ _ _ _ _ _ _

_______ _ _ _ _ _ _ _ _ _ _ _ ___ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _____

________________

_______

____ __ ____ __ _______ ____

I _____ _ _ _ _

__

__ _

_____ _-

__

__

______

_

_

__

__________________ __ __________ _________ ___

_ _ _ _ _ _ _ _ _ _ _ __ __ ________ ____ __ _____________ __________ ______ __ ______ _ _ _ _ _ _ _ _ _ _ __ _ _ ....................

_____________________________

___________________ __________________ - -_ _ _ _

__________________ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _

___________________

_

__ ____ __ __ ___ __ _________ ________________ ___________ . . . . . . . . . . . . . . . . . . . . __ ____ _____________ _____ __ _ _ _ _ _ _

_ _ ___ _ _ _ _ _ _ __

__ _____

____ ____ ___

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_______________ ________

_______________

_

_ _

_______. . . . . . . . . . . . . . . . . . . .

______________

_________ _-__

_ _ _ c _ _ _ _ _ _ _ _ _ _ _ _ _

INDEX

1043

Index of Notations Page

Psge

__

256 = r ( a + n ) / r ( a ) (Pochhammer's symbol) - - _ _ Ei(z) exponential integral- _ _ _ - - _ _ _ _ _ _ _ _ _ _ - - - 228 228 722 characteristic value of Mathieu's equation- E l ( z ) exponential integral- - _ _ - - - - - _ _ _ _ _ _ _ _ - A ( z ) =2P(z) - 1 normal probability function- -.. 931 E [ g ( X ) ] expected value operator for the function Ai(z) Airy function _ _ _ _ _ _ _ _ _ _446 ___.. g (.z_ ) _ __ _ __ _ __ _ __ _ - -_ - - -_ - - -_ - - -_ - - -_ - - -_ - - - - - - - - ~ - -928 228 A.G.M. arithmetic-geometric mean.- - - - - - - - - - - Ein(z) modified exponential integral- - - - - - - - - - - 571 E , Euler number _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _804 ___ am z amplitude of the complex number z_ _ _ _ _ _ _ _ 16 E,(z) Euler polynomial _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 804 antilog antilogarithm (log-') _ _ - - - - - - - - - - -.- .- .89 79 arcsin z , arccos z inverse circular functions. - - - - 228 E,(z) exponential integral-_-.- _ _ - - - - _ _ _ _ _ _ _ - - erf z error function _ _ _ _ _ _ _ _ _ _297 _____._ arctan z, arccot z erfc z complementary error function _ _ _ _ _ _ _ _ _ _ _ _ 297 arcsec z, arccsc z exp z = e * exponential function- - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 69 arcsinh z, arccosh z inverse hyperbolic functions-. 86 exsec A , exsecant A - - . - _ _ _ _ _ _ _ _ _ 78 _ _ _ - _ arctanh z, arccoth z arcsech z, arccsch z joining factors for Mathieu functions- - - 735 fc,,, fo., F ( a , b ; c; z) hypergeometric function-- _ _ _ _ _ _ _ _ _ 556 arg z argument of z-___.----------.._.------- 16 F(q\a) elliptic integral of the first k i n d - - - - - - - - 589 722 b , ( q ) characteristic value of Mathieu's equation.. B , Bernoulli number- - _ _ _ _ _ _ _ _ _ .._ Coulomb _ _ wave __ function ___ (regular) _ ___ ---538 804 F_ L ( T_ , ~ )_ FPP fundamental period parallelogram- - - - - - - - 629 804 B,(z) Rernoulli polynomial-- - - - - - - - - - -..- - - - - ,Fm(alr. . ., a,; bl, ., b,; z) generalized hyperber,z, beiA, Kelvin functions- - - - - - - - - -.-.- - - - 3 79 556 geometric function .__ _ _ - _ _ _ _ _ _ _ _ - - _ _ _ _ _ _ _ - Bi(z) Airy function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _446 ___ cd, sd, nd Jacobian elliptic functions__._. -. - _ _ -. 570 82, g3 invariants of Weierstrass cAlliptic functions- 629 c.d.f. cumulative distribution function- - - - - - - - -. 927 Be,,, q 0 . , joining factors for Mathieu functions--740 cel(z, q) Mathieu function _ _ . _ _ _ _ _ _ _ _ normal _ _ _ _ _ function--g(z,_ y, _ p) _ bivariate probability 936 725 _ _ cn Jacobian elliptic function- - - - - - - - - - - - - - - - - - Gi(z) related Airy function . . . . . . . . . . . . . . . . . . . . 448 569 Cn, Dn, Sn integrals of the squares of Jacobian G L ( T *,p ) Coulomb wave function (irregular or loga. r i t h m i c ) _ _ _ _ _ - - - - - - - - - - - - - - - _ - - . . - - - - - - - - - - 538 elliptic functions- - - - - - - - - - - -.- - - - - - - - - - - - 576 cs, ds, ns Jacobian elliptic functions- - - _ _ _ _ - - - - 570 G',(p, q, z) Jacobi polynomial- _ _ _ _ _ - - _ _ _ _ _ _ _774 ___ Gudermannian __ ___---77 300 C(z) Fresnel integral _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _. . _ _ _ _ _gd(z) _ h t l ( z ) spherical Bessel function of the third kind437 774 C,(z) Chebyshev polynomial of the second kind-hav A haversine A _ - - _ _ _ _ _ _ _ _ _ _ _ _ - _ - - _ _ _ _ 78 ____ 262 C(z, a) generalized Fresnel integral- - - - - -. - - - - H,(z) Struve's function- - - - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 496 C e , ( z , q) modified Mathieu function-. ._ _ _ _ _ _ _ _ _ 732 CI(Z), Cz(z) Fresnel integrals _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Hi(z) related Airy function . . . . . . . . . . . . . . . . . . . . 448 300 Cn(")(z) ultraspherical (Gegenbauer) polynomial---- ______ - 775 774 H e , ( z ) Hermite polynomial H!"'(z) Bessel function of the third kind (Hankel) 358 Chi(z) hyperbolic cosine integral- - -.-.- - - - - - - - 23 1 23 1 Ci(z) cosine integral- - - _ _ - - - -.- _ _ - - _ _ _ _ - _ _ _ _ Hh,(z) Hh (probability) function _ _ _ _ _ _ _ _ _ _ _ _ _ 300,891 Cin(z) cosine integral-- - - - __ _ _ _ -.- - - - _ _ _ _ - - - - 231 775 H,(z) Hermite polynomial- - - - - - - - - - - - - - - - - - - Cinh(z) hyperbolic cosine integral- - - - - - - - - - - - - H(m, n , z) confluent hypergeometric function- - - 695 231 colog cologarithm I&) modified Bessel function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 374 89 covers A , coversine A _ _ _ _ - - - . _ _ _78 __ m_ Z ,_ + g_ ( z_ ) modified _ _ _spherical ___ Bessel _ _function of the first kind _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _443 __ dc, nc, sc Jacobian elliptic functions- - _ _ _ - - -.- _ _ 570 dn=A( q) delta amplitude (Jacobian elliptic func&&Z-"+(z) modified spherical Bessel function I __ of the second kind _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 443 t i o n ) - - - - - - - - - - _ _ _ _ _ - - - - - - - - - - - - - - - - - - - - - - 569 D,(z) parabolic cylinder function (Whittaker's I(?/,p ) incomplete gamma function (Pearson's f o r m ) _ - _ _ _ _ _ - - - - - _ - - - - - - - - - - - - - - - - - - - - - - - 262 f o r m ) - - - - . _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 687 ____Zz(a, b ) incomplete beta function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 263 el, e2, c3 roots of a polynomial (Weierstrass form)-629 e s exponential function- _ _ - - - - - - - - - - -. --------9 z imaginary part of z(=y) . . . . . . . . . . . . . . . . . . . . 16 69 in erfc z repeated integral of the error function--299 262 e,(z) truncated exponential function- - - - - - - -.- - j , ( z ) spherical Bessel function of the first kind--437 E ( p\a) elliptic integral of the second kind- - - - - 589 E ( a , z ) parabolic cylinder function- _ _ ._ _ . _ _ - - - - - 693 Jy(z) Anger's function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _498 E.(z) Weber's function _ _ _ _ . _ _ _ _ _ _ _ _ _ _ _ _ __._ J ,_ ( z_ ) _Bessel function of the first kind _ _ _ _ _ _ _ _ _ _ _ 358 498 E , ( m ) ( z ) Weber parabolic cylinder function---- - -. 509 k modulus of Jacobian elliptic functions--- - - - - 590 E(m) complete elliptic integral of the second k' complementary modulus-- - - - - - - - - - - - - - - - - - 590 k . ( z ) Bateman's function _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ 510 ___ 590 (a), a,(q)

__

-

..

_____ _ _ _ _

_____

1044

_______

__

__

1045

INDEX OF NOTATIONS

Page Pare q(n) number of partitions into distinct integer 483 repeated integrals of K o ( z ) - - - _ _ _ _ _ _ _ _ _ _ _ summands--------_----------------------825 Kn+x(z ) modified spherical Bessel function of the third kind _ _ _ _ _ _ _ _ 443 _ _ . _ _associated _ _ _Legendre _ _ _ _ _ _ Q:(z) function of the second 374 332 K , ( z ) modified Bessel function-- - _ _ - _ _ - - - - - - kind--____-.----____------------^-------334 590 K ( m ) complete elliptic integral of the first kind-Qn(z) Legendre function of the second kind---- - 16 379 k e r j , k e i z Kelvin functions _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 9 2 2 real part of z(=z) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 228 753 li(z) logarithmic integral _ _ _ _ - - _ -- - - - -.- - - - - - - - R?L(c, E ) radial spheroidal wave function------. . 824 lim l i m i t - _ _ _ _ - _ - . - - - - - - - - - - - - - - - - - - - - - - - - - - 13 S,(m) Stirling number of the first kind _ _ _ _ _ _ _ _ _ _ _ 68 824 &(”’) Stirling number of the second kind- - - - - - - log,@ common (Briggs) logarithm- - - - _ _ _ _ _ - - - - 725 67 Be.(z, q) Mathieu function . . . . . . . . . . . . . . . . . . . . . log,z logarithm of z t o base n _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 569 In z (=log& natural, Naperian or hyperbolic sn Jacobian elliptic function- - - - - - - - - - - - - - - - - logarithm_.___^----_____________________-300 68 S(z) Fresnel integral_ _ _ _ _ - - _ _ - - - - - - _ _ - - - _ _ _ _ - f l ~ ( t ])= j ( s , Laplace transform- - - - - - - - - - - - - - - 1020 S , ( z ) , S2(z) Fresnel integrals_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 300 733 L ( h , k, p ) cumulative hivariate normal probaSe,(z, q) modified Mathieu function _ _ _ _ _ _ _ _ _ _ _ _ 262 bility function _ _ _ _ - 936 S(z, a) generalized Fresnel integral- - - - - - - - - - - 231 775 Shi(z) hyptrbolic sine integral- - - - - - - - - - - - - - - L,(z) Laguerre polynomial- - - - - - - - - - - - - - - - - - 231 775 L,(d (5) generalized Laguerre polynomial-- _ _ .- - Si(z) sine integral- - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ L.(z) modified Struve function _ _ _ _ __ _ _ _._ _ _ _ _ _ _ 498 S,(z) Chebyshev polynomial of the first kind- - - - 774 231 928 m=pl’ mean---_-------.-.------------------Sih(z) hyperbolic sine integral- - - - - - - - - - - - - - - - ni parameter (elliptic functions) _ _ _ - - - - - - _ _ - - - - 753 569 S;i(c, 7) angular spheroidal wave function-- - - - - . . si(z) sine integral_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _232 ___ ml complementary parameter-. - - - - - - - - - - - - - - - - 569 71 M ( a , b , z ) Kummer’s confluent hypergeometric sin z, cos z, tan z circular functions _ _ _ _ _ _ _ _ _ _ _ _ _ 72 _ _ _ _ f u n c t i o n _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _504 _ _ _ _ _ _ _ cot _ _ -2,- sec z, csc z _ _ _ _ _ _ . _ _ _ _ 83 733 sinh z, cosh z, tanh z hyperbolic functions _ _ _ _ _ _ _ Mc,(i)(z, q) modified Mathieu function--. _ _ - - 83 733 coth z, sech z, csch z _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Ms,O)(z,q ) modified Mathieu function _ _ _ _ _ _ _ _ _ _ 505 509 T(m,n , t ) Toronto function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ M . , , ( z ) Whittaker function- - - - - - _ _ - - - - - - _ _ - - 774 n characteristic of the elliptic integral of the third T.(z) Chebyshev polynomial of the first kind---590 kind______________..-.-------------^-^-^T:(z) shifted Chebyshev polynomial of the first k i n d _ - _ _ _ _ _ _ _ _ _ - - _ _ _ - - - - - - - - - - - - - - - - - - -774 -. 15 O(v,j =U,,, U, is of the order of U,, (u,/u. is bounded) un V(a,b, z) Kummer’s confluent hypergeometric o(u,)=u,,, lim -=0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 259 function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _504 ___ n+mVn 363 O,,(z) Neumann’s polynomial- - - _ _ - _ _ - - - _ _ _ _ - _ _ V.(z) Chebyshev polynomial of the second kind- 774 p ( n ) number of partitions . . . . . . . . . . . . . . . . . . . . . 825 x ( z ) shifted Chebyshev polynomial of the second 629 g ( z ) Weierstrass elliptic function--- - - - - _ _ _ _ - - kind--____-_-^---___--------------------774 ph z phase of the complex number z _ _ _ _ _ _ _ _ _ _ _ _ 16 687 V ( a ,z) Weber parabolic cylinder function--- - - - P ( a , z ) incomplete gamma function- - - - - - _ _ - - - - - 260 vers A, versine A _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 78 P ( x a [ v ) probability of the Xz-distribution_---- 262, 940 687 V(a,z) Weber parabolic cylinder function- - - - P ! ( z ) associated Legendre function of the first w ( z ) error function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 297 kind_^__________________________________332 692 W ( a ,z) Weber parabolic cylinder function- - - - - P(z) normal probability function--- - - - - _ -- - - - - - 931 505 W K . ~ ( ZWhittaker ) function P,,(z) Legendre function (spherical polynomials) - 333,774 W{f(z), g(r)I(=f(z)g’(z)-f’(z)g(z)) Wronskianrel a t i o n - _ _ _ _ _ _ _ - - - _ _ _ - - - - - - - - - - - - - - - - - - - -505 -774’ P:(x) shifted Legendre polynomial-- - - - - - - - - - - Pn(al@)(z)Jacobi polynomial--- - - - - - _ _ _ _ _ 774 [ ZO,ZI, ,zk] divided difference _ _ _ _ _ _ _ - - - - - _ _877 _ 437 Pr{ X _
Ki.(z)

__

-

_ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _

-

__

-

___________________

______

...

--

Notation modular angle (elliptic function) - - - - - _ _ _ _ _ _ _

a

a.(z)

=J”

- Greek Letters Page

~

__

Page

joining factor for spheroidal wave functions- -

757

@(ulm) Jacobi’s theta function _ _ _ _ - - - - - - - - - - 577 s,, nth c u m u l a n t - - - - - - - - - . . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _928 ___

590 228

tne-sldt _ _ _ _ _ . _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~

K ~ J

OD

~,,(z)

=J‘I

tne-aldt

=E(2k+ 1)-

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _228 ___

m

+ 1) _ _ _ _ - - - - - - - - - - - - - - - - -

A(n)

n-

------ ------- ------- -------

k -0

807

characteristic value of the spheroidal wave equation__-___-_-___-------______ 753 B.(n, b) incomplete beta function _ _ _ _ _ _ _ _ _ _ _ _ _ _ 263 595 Au(p\a) Heuman’s lambda function- - - - - - _ _ - - - B ( z , w ) beta function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 258 mean difference- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 877 y Euler’s constant _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _255 _ _ _ p ( n ) Mobius function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _82 _6 260 ?(a, 2) incomplete gamma function (normalized) p,, nth central moment _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _928 _ Pa p’,, nth moment about the origin _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 928 yl=coefficient of skewness _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 928 U3 231 r(z) number of primes 52 . . . . . . . . . . . . . . . . . . . . *,,(z) = (2-20) (2-21) * (2-2.) 878 Ir4 y2=--3 coefficient of excess _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 928 U1 II(n; p\a) elliptic integral of the third kind- - - - - 590 r(z) gamma function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 255 n(z) factorial function _ _ _ _ _ _ _ _ _ _ _ 255 r ( a , z) incomplete gamma function _ _ _ _ _ _ _ _ _ _ _ _ _ 260 p correlation coefficient- - - - - _ _ _ _ _ _ _ _ 936 822 6ii Kronecker delta ( = 0 if i # k ; = 1 if i = k ) - - - - ~,,(ZO,Z,, .,z,,) reciprocal difference _ _ _ 878 877 6:(fn) central difference- - - - _ _ _ _ _ - - _ _ _ _ _ _ _ _ _ _ _ p,,(u, 2) Poisson-Charlier function- - - - - - - - - - - 509 A difference o p e r a t o r _ _ _ _ _ - - - - - - - . . - - - - - - - - - - - - - 822 U standard deviation _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ 298 A discriminant of Weierstrass’ canonical form--- 629 U’ v a r i a n c e - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 928 A(fn) forward difference- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 877 629 u ( z ) Weierstrass sigma function- - - - - - - - - - - - - - - Ax absolute error _ _ _ _ . _ _ _ _ _ _ _ ._ __ __ _ _ _ _ _ _ _ _ _ 14_ _ U k(n) divisor function __ -__ --- - - 827 f(z) Riemann zeta function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 807 934 s,,(z) tetrachoric function- - - - - - - - - - - - - - - - - - - - f(z) Weierstrass zeta function _ _ _ _ _ _ _ - _ _ - _ _ _ - _ _ - 629 p = a m U, amplitude- - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 569 578 Z(ulm) Jacobi’s zeta function-- _ _ _ - - _ _ - - _ _ - _ _ _ _ 826 p(n) Euler-Totient function- - - - - - - - - - - - - - - - - - m = E_ ( ei 8-X) - characteristic function of X -- - - - 928 q(n) (-1)j-lk-n _ . _ _ _ _ _ _ _ _ _ _ _ 807 _ - _ _p(t) __ 504 +(a; b ; z) confluent hypergeometric function- - - - k-1 Weierstrass elliptic function - - - - - - 631 q., = f(w.) $(z) logarithmic derivative of the gamma 577 H ( u ) , Hl(u) Jacobi’s et a function- - - - _ _ _ _ _ _ _ _ _ _ function__-------------------------------258 a,,(%) theta function- - - - - - - - - - - - - - - - - - - - - - - - - - 576 504 * ( a ; c; z) confluent hypergeometric function----578 9,(c\a) ,OS( e\ a), Neville’s notation - - - - - - - - - - - - 629 o., period of Weierstrass elliptic functions- - - - - for 510 w.,,(z) Cunningham function -------- -----s,,(s\a),s,(c\a) theta functions B(n)

(- 1)k(2k

=

k-0

-.n

A,,

807

. .

..

_____________ ___ __ ____ _________ ___

______

________ -

=r,

-

___

-

_

Miscellaneous Notations determinant _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [ad] column matrix _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ vn Laplacian operator- - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ :A forward difference operator _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

page

. . . . . . . . _. ._ ._ ._ ._ ._ ._._. _. _. _. _. _

<x> nearest integer t o z z complex conjugate of z (=z-iy)

[aik]

a

partial derivative _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

883

I

______

complex number (Cartesian form) (polar form) . . . . . . . . . . . . . . . . . . . . . . . . . . . J z I absolute value or modulus of z _ _ _ _ _ _ _ _ _ _ _ _ - _ 2 overall summation _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Page

222 16 16 16 16 822 755 807 826

(m,n) greatest common divisor _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

822 437

.

______

(n;n ~ nl; , . ., n,) mu!tinomial coefficient [z] largest; integer 52 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

1046

823 66

r

Cauchy’s principal value of the integral------

=

approximately equal---- - asymptotically equal 5 1 inequality, inclusion unequal_--------_----------------_--_----

J-

-<, >, #

___

__- __ _____ __

_______________________

______________

228 14 15 10 12

A CATALOGUE OF SELECTED DOVER BOOKS I N ALL FIELDS O F INTEREST

A CATALOGUE OF SELECTED DOVER BOOKS IN ALL FIELDS OF INTEREST

LEATHER TOOLING AND CARVING, Chris H. Groneman. One of few books concentrating on tooling and carving, with complete instructions and grid designs for 39 projects ranging from bookmarks to bags. 148 illustrations. I l l p p . 77/8 x 10. 23061-9 Pa. $2.50 NUTTALL,A PICTURE MANUSCRIPT FROM ANCIENT MEXICO, as first THECODEX edited by Zelia Nuttall. Only inexpensive edition, in full color, of a pre-Columbian Mexican (Mixtec) book. 88 color plates show kings, gods, heroes, temples, sacrifices. New explanatory, historical introduction by Arthur G . Miller. 96pp. 113b 8 1 ~ 23168-2 Pa. $7.50

AMERICANPRIMITIVE PAINTING, Jean Lipman. Classic collection of an enduring American tradition. 109 plates, 8 in full color-portraits, landscapes, Biblical and historical scenes, etc., showing family groups, farm life, and so on. 8Opp. of lucid text. 83/8 x 11%. 22815-0 Pa. $4.00 WILLBRADLEY: HISGRAPHIC ART,edited by Clarence P. Hornung. Striking collection of work by foremost practitioner of Art Nouveau in America: posters, cover designs, sample pages, advertisements, other illustrations. 97 plates, including 8 in full color and 19 in two colors. 97pp. 93/8 x 12%. 20701-3 Pa. $4.00 22120-2 Clothbd. $10.00 SKETCHBOOK OF JAN FAUST, Jan Faust. 101 bitter, horrifying, THEUNDERGROUND black-humorous, penetrating sketches on sex, war, greed, various liberations, etc. Sometimes sexual, but not pornographic. Not for prudish. 1Olpp. 6% x 9%. 22740-5 Pa. $1.50 GIRL AND HERAMERICA, Charles Dana Gibson. 155 finest drawings of THEGIBSON effervescent world of 1900-1910: the Gibson Girl and her loves, amusements, adventures, Mr. Pipp, etc. Selected by E. Gillon; introduction by Henry Pitz. 144pp. 8% x 1 1 ~ 8 . 21986-0 Pa. $3.50

STAINED GLASS CRAFT, J.A.F. Divine, G . Blachford. One of the very few books that tell the beginner exactly what he needs to know: planning cuts, making shapes, avoiding design weaknesses, fitting glass, etc. 93 illustrations. 115pp. 22812-6 Pa. $1.50

CATALOGUE OF DOVER BOOKS

150 MASTERPIECES OF DRAWING, edited by Anthony Toney. 150 plates, early 15th century to end of 18th century; Rembrandt, Michelangelo, DKrer, Fragonard, Watteau, Wouwerman, many others. 150pp. 83/73 x 11%. 21032-4 Pa. $3.50 ACE OFTHE POST^, Hayward and Blanche Cirker. 70 extraordinary THEGOLDEN posters in full colors, from Maitres de I’Affiche, Mucha, Lautrec, Bradley, Cheret, 22753-7 Pa. $4.95 Beardsley, many others. 93/8 x 12%. 21718-3 Clothbd. $7.95

SIMPLICISSIMUS, selection, translations and text by Stanley Appelbaum. 180 satirical drawings, 16 in full color, from the famous German weekly magazine in the years 1896 to 1926.24 artists included: Grosz, Kley, Pascin, Kubin, Kollwitz, plus Heine, Thijny, Bruno Paul, others. 172pp. 8% x 12%. 23098-8 Pa. $5.00 23099-6 Clothbd. $10.00 Aubrey Beardsley. 157 plates, 2 in color: THEEARLY WORKOF AUBREYBEARDSLEY, Manon Lescaut, Madame Bovary, Morte d’Arthur, Salome, other. Introduction by 21816-3 Pa. $3.50 H. Marillier. 175pp. 8% x 11. THELATERWORKOF AUBREYBEARDSLEY, Aubrey Beardsley. Exotic masterpieces of full maturity: Venus and TannhPuser, Lysistrata, Rape of the Lock, Volpone, Savoy material, etc. 174 plates, 2 in color. 176pp. 8%x 11. 21817-1 Pa. $3.75 OF WILLIAM BLAKE, William Blake. 92 plates from Book of Job, Divine DRAWINGS Comedy, Paradise Lost, visionary heads, mythological figures, Laocoijn, etc. Selection, introduction, commentary by Sir Geoffrey Keynes. 178pp. 8% x 11. 22303-5 Pa. $3.50

LONDON: A PILGRIMAGE, Gustave Dad, Blanchard Jerrold. Squalor, riches, misery, beauty of mid-Victorian metropolis; 55 wonderful plates, 125 other illustrations, full social, cultural text by Jerrold. 191pp. of text. 81/73 x 11. 22306-X Pa. $5.00 THECOMPLETE WOODCUTS OF ALBRECHT D ~ E Redited , by Dr. W. Kurth. 346 in all: Old Testament, St. Jerome, Passion, Life of Virgin, Apocalypse, many others. Introduction by Campbell Dodgson. 285pp. 8% x 12%. 21097-9 Pa. $6.00 THEDISASTERS OF WAR,Francisco Goya. 83 etchings record horrors of Napoleonic wars in Spain and war in general. Reprint of 1st edition, plus 3 additional plates. Introduction by Philip Hofer. 97pp. 93/73 x 8%. 21872-4 Pa. $2.50 OF HOGARTH, William Hogarth. 101 of Hogarth’s greatest works: ENGRAVINGS Rake’s Progress, Harlot’s Progress, Illustrations for Hudibras, Midnight Modern Conversation, Before and After, Beer Street and Gin Lane, many more. Full com22479-1 Pa. $6.00 mentary. 256pp. 11 x 14. 23023-6 Clothbd. $13.50

ART, Franz Boas. Great anthropologist on ceramics, textiles, wood, PRIMITIVE stone, metal, etc.; patterns, technology, symbols, styles. All areas, but fullest on Northwest Coast Indians. 350 illustrations. 378pp. 20025-6 Pa. $3.50

CATALOGUE OF DOVER BOOKS HOUDINI ON MAGIC,Harold Houdini. Edited by Walter Gibson, Morris N. Young. How he escaped; expose‘s of fake spiritualists; instructions for eye-catching tricks; other fascinating material by and about greatest magician. 155 illustrations. 280pp. 20384-0 Pa. $2.50 OF THE NUTRITIONAL CONTENTS OF FOOD,U.S. Dept. Of Agriculture. HANDBOOK Largest, most detailed source of food nutrition information ever prepared. Two mammoth tables: one measuring nutrients in 100 grams of edible portion; the other, in edible portion of 1 pound as purchased. Originally titled Composition of 21342-0 Pa. $4.00 Foods. 19Opp. 9 x 12.

PRESERVING AND FREEZING, U.S. Dept. of COMPLETE GUIDETO HOMECANNING, Agriculture. Seven basic manuals with full instructions for jams and jellies; pickles and relishes; canning fruits, vegetables, meat; freezing anything. Really good recipes, exact instructions for optimal results. Save a fortune in food. 156 il22911-4 Pa. $2.50 lustrations. 214pp. 6l/6 x 9%. THEBREAD TRAY,Louis P. De Gouy. Nearly every bread the cook could buy or make: bread sticks of Italy, fruit breads of Greece, glazed rolls of Vienna, everything from corn pone to croissants. Over 500 recipes altogether. including buns, 23000-7 Pa. $3.50 rolls, muffins, scones, and more. 463pp. CREATIVE HAMBURGER COOKERY, Louis P. De Gouy. 182 unusual recipes for casseroles, meat loaves and hamburgers that turn inexpensive ground meat into memorable main dishes: Arizona chili burgers, burger tamale pie, burger stew, burger corn loaf, burger wine loaf, and more. 120pp. 23001-5 Pa. $1.75

J. George Frederick and Jean Joyce. Probably LONGISLAND SEAFOOD COOKBOOK, the best American seafood cookbook. Hundreds of recipes. 40 gourmet sauces, 123 recipes using oysters alone! All varieties of fish and seafood amply represented. 324pp. 22677-8 Pa. $3.00 THEEPICUREAN. A COMPLETE TREATISE OF ANALYTICAL AND PRACTICAL STUDIES IN CULINARY ART,Charles Ranhofer. Great modern classic. 3,500 recipes from master chef of Delmonico’s, turn-of-the-century America’s hest restaurant. Also explained, many techniques known only to professional chefs. 775 illustrations. 22680-8 Clothbd. $17.50 1183pp. 65/6 x 10.

THE

THEAMERICANWINECOOKBOOK,Ted Hatch. Over 700 recipes: old favorites livened up with wine plus many more: Czech fish soup, quince soup, sauce Perigueux, shrimp shortcake, filets Stroganoff, cordon bleu goulash, jambonneau, 22796-0 Pa. $2.50 wine fruit cake, more. 314pp. DELICIOUS VEGETARIAN COOKING, Ivan Baker. Close to 500 delicious and varied recipes: soups, main course dishes (pea, bean, lentil, cheese, vegetable, pasta, and egg dishes), savories, stews, whole-wheat breads and cakes, more. 168pp. U S 0 22834-7 Pa. $1.75

CATALOGUE OF DOVER BOOKS

THEART DECOSTYLE, ed. by Theodore Menten. Furniture, jewelry, metalwork, ceramics, fabrics, lighting fixtures, interior decors, exteriors, graphics from pure French sources. Best sampling around. Over 400 photographs. 183pp. @/E x 11% 22824-X Pa. $4.00 THE GENTLEMAN AND CABINET MAKERSDIRECTOR, Thomas Chippendale. Full reprint, 1762 style book, most influential of all time; chairs, tables, sofas, mirrors, cabinets, etc. 200 plates, plus 24 photographs of surviving pieces. 249pp. 97/8 x 123~ 21601-2 Pa. $5.00 PINEFURNITURE OF EARLY NEWENGLAND, Russell H. Kettell. Basic book. Thorough historical text, plus 200 illustrations of boxes, highboys, candlesticks, desks, etc. 477pp. 7718 10u. 20145-7 Clothbd. $12.50 ORIENTAL RUGS, ANTIQUE AND MODERN, Walter A. Hawley. Persia, Turkey, Caucasus, Central Asia, China, other traditions. Best general survey of all aspects: styles and periods, manufacture, uses, symbols and their interpretation, and identification. 96 illustrations, 11in color. 320pp. 6% x 9%. 22366-3 Pa. $5.00 DECORATIVE ANTIQUEIRONWORK, Henry R. d’Allemagne. Photographs of 4500 iron artifacts from worlds finest collection, Rouen. Hinges, locks, candelabra, weapons, lighting devices, clocks, tools, from Roman times to mid-19th century. Nothing else comparable to it. 420pp. 9 x 12. 22082-6 Pa. $8.50 Catherine Christopher. THECOMPLETE BOOKOF DOLLMAKINGAND COLLECTING, Instructions, patterns for dozens of dolls, from rag doll on up to elaborate, historically accurate figures. Mould faces, sew clothing, make doll houses, etc. Also collecting information. Many illustrations. 288pp. 6 x 9. 23066-4 Pa. $3.00 ANTIQUEPAPER DOLLS:1915-1920, edited by Arnold Arnold. 7 antique cut-out dolls and 24 costumes from 1915-1920, selected by Arnold Arnold from his collection of rare children’s books and entertainments, all in full color. 32pp. 9% x 12% 23176-3 Pa. $2.00 ANTIQUEPAPER DOLLS:THEEDWARDIAN ERA,Epinal. Full-color reproductions of two historic series of paper dolls that show clothing styles in 1908 and at the beginning of the First World War. 8 two-sided, stand-up dolls and 32 complete, two-sided costumes. Full instructions for assembling included. 32pp. 9% x 12%. 23175-5 Pa. $2.00

A HISTORY OF COSTUME, Car1 KBhler, Emma von Sichardt. Egypt, Babylon, Greece up through 19th century Europe; based on surviving pieces, art works, etc. Full text and 595 illustrations, including many clear, measured patterns for reproducing historic costume. Practical. 464pp. 21030-8 Pa. $4.00

EARLY AMERICANLOCOMOTIVES, John H. White, Jr. Finest locomotive engravings from late 19th century: historical (1804-18741, main-line (after 1870), special, foreign, etc. 147 plates. 200pp. llY8 x 8%. 22772-3 Pa. $3.50

CATALOGUE OF DOVER BOOKS

GREETING CARDS,Ed Sibbett, Jr. 16 cards to cut and color. Three say “Happy Chanukah,” one “Happy New Year,” others have no message, show stars of David, Torahs, wine cups, other traditional themes. 16 envelopes. 8% x 11. 23225-5 Pa. $2.00

JEWISH

AUBREYBEARDSLEY GREETING CARD BOOK,Aubrey Beardsley. Edited by Theodore Menten. 16 elegant yet inexpensive greeting cards let you combine your own sentiments with subtle Art Nouveau lines. 16 different Aubrey Beardsley designs that you can color or not, as you wish. 16 envelopes. 64pp. 8% x 11. 23173-9 Pa. $2.00

RECREATIONS IN THE THEORY OF NUMBERS, Albert Beiler. Number theory, an inexhaustible source of puzzles, recreations, for beginners and advanced. Divisors, perfect numbers. scales of notation, etc. 349pp. 21096-0 Pa. $2.50 AMUSEMENTS IN MATHEMATICS, Henry E. Dudeney. One of largest puzzle collections, based on algebra, arithmetic, permutations, probability, plane figure dissection, properties of numbers, by one of world’s foremost puzzlists. Solutions. 450 illustrations. 258pp. 20473-1 Pa. $2.75

MATHEMATICS, MAGICAND MYSTERY, Martin Gardner. Puzzle editor for Scientific American explains math behind: card tricks, stage mind reading, coin and match tricks, counting out games, geometric dissections. Probability, sets, theory of numbers, clearly explained. Plus more than 400 tricks, guaranteed to work. 135 illustrations. 176pp. 20335-2 Pa. $2.00 BESTMATHEMATICAL PUZZLES OF SAMLOUD,edited by Martin Gardner. Bizarre, original, whimsical puzzles by America’s greatest puzzler. From fabulously rare Cyclopedia, including famous 14-15 puzzles, the Horse of a Different Color, 115 more. Elementary math. 150 illustrations. 167pp. 20498-7 Pa. $2.00 MATHEMATICAL PUZZLES FOR BEGINNERS AND ENTHUSIASTS, Geoffrey Mott-Smith. 189 puzzles from easy to difficult involving arithmetic, logic, algebra, properties of digits, probability. Explanation of math behind puzzles. 135 illustrations. 248pp. 20198-8 Pa. $2.00 BIGBOOKOF MAZESAND LABYRINTHS, Walter Shepherd. Classical, solid, and ripple mazes; short path and avoidance labyrinths; more -50 mazes and labyrinths in all. 12 other figures. Full solutions. 112pp. 8% x 11. 22951-3 Pa. $2.00 AND PUZZLES, Maxey Brooke. 60 puzzles, games and stunts - from COINGAMES Japan, Korea, Africa and the ancient world, by Dudeney and the other great puzzlers, as well as Maxey Brooke’s own creations. Full solutions. 67 illustrations. 94PP. 22893-2 Pa. $1.25

HANDSHADOWS TO BE THROWN UPON THE WALL,Henry Bursill. Wonderful Victorian novelty tells how to make flying birds, dog, goose, deer, and 14 others. 21779-5 Pa. $1.00 32pp. 6% x 9%.

CATALOGUE OF DOVER BOOKS

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BUILDYOUR OWNLOW-COSTHOME,L.O.Anderson, H.F. Zornig. U.S.Dept. of Agriculture sets of plans, full, detailed, for 11 houses: A-Frame, circular, conventional. Also construction manual. Save hundreds of dollars. 204pp. 11 x 16. 21525-3 Pa. $5.95 HOUSE,L.O. Anderson. Comprehensive, easy to How TO BUILDA WOOD-FRAME follow U.S.Government manual: placement, foundations, framing, sheathing, roof, insulation, plaster, finishing - almost everything else. 179 illustrations. 223pp. 7718 x 10%. 22954-8 Pa. $3.50 MASONRY AND BRICKWORK, U.S. Department of the Army. Practical CONCRETE, handbook for the home owner and small builder, manual contains basic principles, techniques, and important background information on construction with concrete, concrete blocks, and brick. 177 figures, 37 tables. 200pp. 6% x 9%. 23203-4 Pa. $4.00 BOOKOF QUILTMAKING AND COLLECTING, Marguerite Ickis. Full inTHESTANDARD formation, full-sized patterns for making 46 traditional quilts, also 150 other patterns. Quilted cloths, lame', satin quilts, etc. 483 illustrations. 273pp. 6718 x @/. . 20582-7 Pa. $3.50 '

101 PATCHWORK PATTERNS, Ruby S. McKim. 101 beautiful, immediately useable patterns, full-size, modern and traditional. Also general information, estimating, quilt lore. 124pp. 7718 x 10%. 20773-0 Pa. $2.50 KNITYOUR OWNNORWEGIAN SWEATERS, Dale Yarn Company. Complete instructions for 50 authentic sweaters, hats, mittens, gloves, caps, etc. Thoroughly modem designs that command high prices in stores. 24 patterns, 24 color photographs. Nearly 100 charts and other illustrations. 58pp. 8318 x 11%. 23031-7 Pa. $2.50 IRON-ON TRANSFER PATTERNS FOR CREWEL AND EMBROIDERY FROM EARLY AMERICAN SOURCES, edited by Rita Weiss. 75 designs, borders, alphabets, from traditional American sources printed on translucent paper in transfer ink. Reuseable. Instructions. Test patterns. 24pp. 8% x 11. 23162-3 Pa. $1.50 AMERICANINDIAN NEEDLEPOINTDESIGNS FOR PILLOWS, BELTS,HANDBAGS AND OTHER PROJECTS, Roslyn Epstein. 37 authentic American Indian designs adapted for modern needlepoint projects. Grid backing makes designs easily transferable to canvas. 48pp. 8% x 11. 22973-4 Pa. $1.50

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CHARTED FOLKDESIGNS FOR CROSS-STITCH EMBROIDERY, P4aria Foris & Andreas Foris. 278 charted folk designs, most in 2 colors, frail Danube region: florals, fantastic beasts, geometrics, traditional symbols, more. Border and central patterns. 77pp. 8% x 11. US0 23191-7 Pa. $2.00 Prices subject to change without notice. Available at your book dealer or write for free catalogue to Dept. GI, Dover Publications, Inc., 180 Varick St., N.Y., N.Y. 10014. Dover publishes inore than 150 books each year on science, elementary and advanced inatheinatics, I)iology, inusic, art, literary history, social sciences and other areas.

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CALCULUS REFRESHENER FOR TECHNICAL MEN,A. Albert Klaf. (20370-0) $4.00 PRACTICAL STATISTICS, Russell Langley. ( 22729-4) $4.00 CHANCE,LUCKA N D STATISTICS, Horace C. Levinson. (21007-3) $2.50 UNIVERSITYALGEBRA, D. E. Littlewood. (62715-2) $3.50 H O W Do You USE A SLIDEKULE?, Arthur A. Merrill. . ( ”Y62 0) $ 1 .OO MATHEMATICALEXCURSIONS, Helen Abbott Merrill. (20350-6) $1.75 THE GENTLEART OF MATHEMATICS, Dan Pedoe. (22949-1) $2.50 _--

GEOMETRIC EXERCISES I N PAPERFOLDJNG, T. Sundara Row. f 7 ) (21 594-6) $2.00 GREATIPAS IN INFORMATION THEORY,LANGUAGE AND CYBERNETICS, Jagjit Singh. (21694-2) $3.50 APPLIED MATHEMATICSFOR RADIOAND COMMUNICATIONS ENGINEERS, Car1 E. Smith. (60142-0) $3.00 TEACHYOURSELF THE SLIDERULE,Burns Snodgrass. (20684-X) Clothbound $2.50 H O W To CALCULA? E QUICKLY, Henry Sticker. (20295-X) $2.50 THENUMBER SYSTEM,Hugh A. Thurston. (61848-X) $2.00 CONCEPTS AND METHODS OF ARITHMETIC,Marvin C. Volpel. (61 237-6) $3.00 I *

aperbound unless otherwise indicated. Prices subject to change ithout notice. Available at your book dealer or write for freo catalogues to Dept. TF4, Dover Publications, Inc., 180 Varick Street, New York, N. Y. 1001$
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