!
HANDBOOK OF MAT HE.MAT1CA L F U NCT I0NS
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with Formulas, Graphs, and Mathematical Tables
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4
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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
(a
4w, 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.