Theory and Application of Mathieu Functions

THEORY AND APPLICATION OF MATHIEU FUNCTIONS BY N. W. McLACHLAN D.Be. (ENGINEERING) LONDON OXFORD AT THE CLARENDON PR...

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THEORY AND APPLICATION OF

MATHIEU FUNCTIONS BY

N. W. McLACHLAN D.Be.

(ENGINEERING) LONDON

OXFORD AT THE CLARENDON PRESS

Oxford University Prea8, Amen House, London E.G. 4 GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON BOMBAY CALOI1TTA MADRAS CAPE TOWN

Geoffrey Oumberleqe, Publisher to the University

First odit.ion 1947

Reprinted litho~raphically in Orpat Brit sin u,t the Universit.y Press, Oxford, 1051 frorn

r-orrcctcd sheets of the flrst. oditron

PREFACE THE purpose of this book is to give the theory of Mathieu functions and to demonstrate its application to representative problems in physics and engineering science. It has been written for the technologist, and is not addressed in any sense to the pure mathematician, for whom I ani not qualified to write.' Between the outlook of the two parties lies a gulf as wide as that between sinner and saint or vice versa! Although, by virtue of necessity, the technologist may occasionally deviate from the narrow path followed rigorously by the pure mathematician, it must not be forgotten that the consequences of such deviation may be practical results of considerable benefit to the community at, large. Since the pure mathematician profits by these technological advances, any criticisms of methods used by the technologist should be entirely of a constructive and helpful nature. The text is in two parts, (1) Theory, (2) Applications; but there is no need to peruse the whole of (I) before reading (2). For instance a study of the second half of Chapter IV will enable the reader to cope with most of the applications in Chapter XV. Some theory is associated with computation of parameters, and of coefficients in series. Other parts of it, not applied directly, are included in the hope that future needs will be met. The reader is expected to have an elementary knowledge of (a) Bessel functions, because they play an important role in Chapters VIII, X, XI, XIII, XVII-XVIII; (b) convergence of infinite series and integrals, and the consequences of uniform convergence, e.g. continuity of a function, term-by-term differentiation and integration of series. If these subjects are unknown, the requisite knowledge should be acquired [202, 214, 215, 218]. The chapter sequence is such as to reduce forward references to a minimum; unfortunately they are unavoidable. The original memoirs from which information was taken are indicated in [ ], which denotes the number in the reference list on pp. 373-81. The analysis herein is seldom the same as that in the original memoir. In writing the manuscript I realized that there were wide gaps which, if left unfilled, would have rendered the text discontinuous. The filling of these gaps has entailed considerable labour; in fact more than one-third of the text is new. The main part of it is in 'Additional

vi

PREFACE

Results', and in Chapters IV, V, VII-XI, XIII, XIV, XVII, Appendixes I, III. The new method of computing the coefficients in the periodic part of the series representation of fem, gem' Fe m, Gem in Chapter VII is a joint contribution with W. G. Bickley. SymboliBm. The variety of notations encountered in the literature on the subject, the introduction of new forms of solution, new multipliers, etc., necessitated a careful survey of and a decision on notation. The first item for consideration was the form to be taken by Mathieu's equation. The canonical form chosen is

~+(a-2qC082Z)Y =

0..

(1)

More than two years after this decision had been made, H. Jeffreys pointed out that the form used by Mathieu in his original memoir [130] was d2P (2) drx 2 + (R-2h 2 cos 2cx )P = o. Thus after encircling the point at issue for about three-quarters of a century and encountering many branches, we have at last returned (quite unwittingly) to the initial or starting value! The coincidence seems significant. If we take k = +ql, the arguments in the Bessel series solutions of (1), and its three other forms, have no fractional factors. Symbols used for various classes of solution have been selected to avoid confusion with other mathematical functions. The basis of the notation is 'e' for elliptic cylinder function, introduced by E. 1'. Whittaker some thirty-five years ago. He used ce, se, to signify 'cosine-elliptic', 'sine-elliptic', and associated Mathieu '8 name with these periodic functions. Herein the generic designation 'Mathieu function' applies to all solutions of (1) in its four forms, which have the appropriate multipliers and/or normalization. The functions are classified, for q real, in Appendix III. For second (non-periodic) solutions of (I) which correspond to cem(z, q), s~m(z, q), the letters f, g are used. Thus the respective second solutions are written fem(z, q), gem{z,q). When iz is substituted for z in cem(z,q), etc., we follow H. Jeffreys [104] and use capital letters. Accordingly we have Cem{z, q), Sem(z, q), Fem(z, q), Gam(z, q). There are also second solutions involving the Y- and K-BesseI functions, which take precedence over the two latter. They are designated Feym(z, q), Gay m(z, q), Fek"l(z, q), Gekm(z, q). FeYm(z, q) is a partic'Uktr linear combination of the even

PREFACE

vii

first solution cem(z, q), and the corresponding odd solution fem(z, q) of (1), in which z is replaced by iz, For wave propagation problems, combination solutions akin to the Hankel functions have been introduced. They have been allocated the symbols Me, Ne, M being for Mathieu and N the next letter of the alphabet. Owing to the war, it was not till the summer of 1946 that, as a result of correspondence with G. Blanch, I became aware that in references [23,211] Stratton, Morse, Chu, and Hutner had defined functions represented by series of Y -Bessel functions (similar to some in Chapter VIII), and also combination functions akin to Me, Ne. The Mathieu functions of positive integral order are solutions of (1) in its four forms when the parametric point (a,q) lies upon one of the characteristic curves am' bm of Figs. 8 and 11. The functions of fractional order, namely, cem+p(z, q), sem+p(z, q), 0 < fJ < 1, have been introduced to provide standard solutions when (a, q) lies witkin a stable region of Figs. 8, II. If fJ = p18, a rational fraction, alu'ay8 presumed to be in its lowest terms, the functions are periodic in z real, with period 2S1T, S ~ 2. The functions ceum+IL(±z,q) have been introduced to provide standard solutions when (a, q) lies in an unstable region of Figs. 8, 11. /L is real and positive, and 'u' signifies that the solution pertains to an unstable region. If q is negative imaginary, cem(z, q) is a complex function of z, so in Chapter III, cer., z and ceim z, have been used to denote its real and imaginary parts, respectively. Symbols in heavy type signify 'per unit area', or 'per unit length'; m, n, p, s usually represent positive integers including zero, except in §§ 3.40-3.51 where 8 is real and positive; r represents any integer; ~ means 'approximately equal to'; "" signifies 'asymptotically equal to', 'approaches asymptotically to', or 'asymptotic form'; R(z), Im(z) or Imag(z), indicate the real and imaginary parts of z, respectively; superscripts in A~~+11), B~~+-t2) signify the order ofthe function, while the subscripts denote that of the coefficient itself; the subscripts in am' bm, am+fJ indicate the order of the function of which am. etc., is the characteristic number. Wherever possible a standard summation from 0 to +00 is used. Other symbols employed are those generally found in advanced mathematical texts in English. References to Fig. 8 are to either 8 A or 8 B, whichever is the more convenient. Tables. The values of the characteristic numbers ao,... , a5 ; bl , ••• , b6 , computed by Ince, and abbreviated by him to 7 decimal places [95],

viii

PREFACE

are given in Appendix II for the range q = 0 to 40. For q imaginary, some ao computed by Goldstein and Mulholland are given in Table 4, p. 51 [57]. No other tabular values are reproduced, since they are much too extensive for inclusion here. .., An interpolable table of ao,···, a15 ; bt , ••. , bt 5, for the range q = 0-25, compiled by the National Bureau of Standards Mathematical Tables Project, became available after completion of the manuscript. By its aid, the stability chart of Fig. SA may be extended considerably for a positive. To obtain a more uniform vertical spacing (a-axis), a 1 should be plotted instead of a. There are no tabular values of Cern' Se m , Feym' Gey m' Fekm , Gek m , and this restricts the use of Mathieu functions in applications. Functions of integral order suitable for tabulation are listed in reference 8. To the list may be added an interpolable table of a, ai, q, fl, fL· This would enable a large iso-flfL chart akin to Fig. II (but much extended) to be plotted using different coloured inks, thereby rendering visual interpolation possible. If, as mentioned above, a 1 is plotted instead of a, the interpolation is approximately linear when Iql is not too large. A chart of this type would be of inestimable value for solving Mathieu equations. Acknowledgements. Prof. W. G. Bickley and Mr. T. Lewis undertook the arduous task of reading much of the manuscript. To them I express my sincere thanks for help and advice which has been invaluable. I am indebted to Prof, Bickley for additional terms in (2), (4), (6) § 2.151. My thanks are also due to Prof. T. A. A. Broadbent and Dr. J. C. P. Miller for criticizing certain sections of the manuscript. I take this opportunity of thanking Dr. Gertrude Blanch for pointing out the relations in § 4 of the additional results. She obtained (1)-(6) § 3, and (1), (2) § 8 independently. The whole of the proofs have been read and the analysis checked by Messrs. T. V. Davies and A. L. Meyers. To them I tender my best thanks for the meticulous care which they have exercised, and for their valuable suggestions. I am much indebted to Dr. L. J. Comrie for the loan of books and reprints of papers; also to him and Miss Dorothy Reynolds for checking some of the calculations in Chapter VI, and for computing the numerical data, using analysis in Chapter V, from which the important iso-f3J..L chart of Figure 11 was plotted. Sir Edmund T. Whittaker kindly loaned me a large number of reprints ofpapers by various authors, while Mrs. P. Ince kindly gave me copies of the late Prof. E. L. Ince's published works. Prof. A. L. Dixon,

PREFAOE Drs. -Iohn Dougall, I. M. H. Etherington, Harold Jeffreys, and Miss Ethel M. Harris were good enough to loan me books and memoirs, which I would not have seen otherwise. It is with pleasure that I acknowledge permission to reproduce the following diagrams and tables in the text: I. Table 4; Philosophica; Magazine; paper by S. Goldstein and H. P. Mulholland [57]. 2. Figs. 10 A, B; Institute of Radio Engineers (America); paper by F. Maginniss [123]. 3. Fig. 25; Institute of Radio Engineers (America); paper by W. L. Barrow, 21, 1182 (1933). 4. Figs. 41 A, B; Institute of Radio Engineers (America); paper by W. L. Barrow and L. J. Chu, 26, 1526,1529 (1938). 5. Fig. 43; Institute of Radio Engineers (America); paper by G. C. Southworth, 25, H08 (1937). 6. Figs. 44 A, B; Journal of Applied Physics; paper by L. J. Chu [22]. 7. }'igs. 48, 49; Physical Review; paper by P. M. Morse and P. J. Rubenstein [144]. 8. Appendix II; Mrs. P. Ince and Royal Society, Edinburgh; tables by E. L. Inee [95]. Finally, I wish to thank the Delegates of the Press for their catholicity of outlook in publishing a large book of this kind, at a time when the printing trade has not yet begun to recover from the devastation of war. N.W.~I.

LONDON

October 1946

CONTENTS I.

Historical Introduction

1

PART I. ~ THEORY OF MATHIEU FUNCTIONS

n. Functions of Integral Order

10

ITI.

Calculation of Characteristic Numbers and Coefficients.

28

IV.

General Theory: Functions of Fractional Order: Solu.. tion of Equations

57

v. VI.

VII.

VIII. IX.

Numerical Solution of Equations

104

Hill's Equation

127

Non.. Periodic Second Solutions corresponding to cem, sem , Cem , SCm

141

Solutions in Series of Bessel Functions

158

Wave Equation in Elliptical Coordinates: Orthogonality 170 Theorem

x. Integral Equations and Relations XI. XII.

Asymptotic Formulae

178 219

Zeros of the Mathieu and modifiedMathieu Functions of Integral Order 234

XIII.

Solutions in Bessel Function Product Series

243

XIV.

Miscellaneous Integrals involving Mathieu Functions .

259

PART II. APPLICATIONS OF MATHIEU FUNCTIONS XV. XVI.

Applications ofy"+(a-2qcos2z)y = 0 .

Application of the Wave Equation to Vibrational Systems 294

xvn. Electrical and Thermal Diffusion xvm, Electromagnetic Wave Guides XIX.

267

Diffraction of Sound and Electromagnetic Waves

320

333 358

OONTENTS ApPENDIX

I.

ApPENDIX II. ApPENDIX ITI. REFERENCES

Degenerate Forms of Mathieu and modified Mathieu Functions 367 Table of Characteristic Numbers am' bm 371 Classification ofv Mathieu Functions 372 A. Scientific Papers B. Books . C. Additional References

373

380 381

ADDITIONAL RESULTS

382

INDEX

395

Additional Results, p. 382 Where applicable these should be read in conjunction with the text

I 11ISTORICAI~

IN'l'I{()T>lT( ~TI0N

majority of functions used in technical and applied mat.hemat.ics have originated as the result of investigating practical problems, Mathieu functions were introduced by their originator in 1868l130], when he determined the vibrational modes of a stretched membrane having an elliptical boundary. The two-dimensional wave equation

THE

i;l2V + i;l2V +k2V ox2 oy2 1

=

0

(1)

was transformed to elliptical (confocal) coordinates, and then H!)Jit up into two ordinary differential equations. If q~ - -- ~kl 11" It being the semi-interfocal distance, and a ali arbitrary separation constant, the equations take the form

d 2v dz 2 + (a-2qcos 2z)v =-= 0,

~2V2 -(a-2qcosh2z)v

0,

(3)t

d~i~2+(a-2qCOS2Zi)V ::= O.

(4)

dz

i.e.

(2)

zr z:

Mathieu's problem the parameters a, q were real. It is evident that (3), the second of the t\VO equations into which (1), expressed in elliptical coordinates, was resolved, nHty be derived from (2) by writing ±zi for z, and vice versa. This reciprocal relation is sometimes considered to be a fluke! Equations (~), (:~) will be regarded herein as the Mathieu and modified Mathieu oquat.ions respectively, for q ,> o. For the elliptical membrane problem, tho appropriate solutions of equation (2) arc called (ordinary) Mathieu functions, being periodic ill z with period 1T or 21T. A~ a consequence of this periodicity, a has special values called characteristic numbers, The corresponding solutions of (3), for the same a as in (2), are known as modified Mathieu functions.j being derived from the ordinary type III

t

In reference ll03] this is termed the "modified l\luthieu oquation '.

+ SOB10 writers refer to thorn as ·associated ' functions, and others as "hyperbolic '

Mathieu functions, They could bo dosiguated Mu.t,hiol1 functions of imaginary argument, In view of the analogy with the derivation of the modified Bessel functions 1, K, tho torm "modified ' seems to he preferable. AIRO IIH'o in [8;"'>] \1~S tho word "associatcd ' to define an entirely different l)()t of functions. 4~UJ.

B

llISTORICAL INTRODUCTION

[Chap. I

by making the argument imaginary, The second independent sets of solutions for the same a are not needed in the membrane problem. They are non-periodic, and those for (2) tend to infinity with z real. Sometimes the solutions of (2), (3) are designated elliptic and hyperbolic cylinder functions, respectively, just as the J-, Y-Bessel functions are termed (circular) cylinder functions. Following the appearance of Mathieu's work, some ten years elapsed before anything further was published on the subject. In Kugelfunktionen [196] Heine (1878) defined the first solutions of integral order of (2) by cosine and sine series, but the coefficients were not calculated. These series fulfil the conditions for Fourier series, but the coefficients are not obtained by integration in the usual way. They have been called Fourier series by Iuany authors. Heine also gave a transcendental equation for the characteristic numbers, pertaining to the first solutions, in the guise of an infinite continued fraction. This form was used to great advantage about half a century later by Goldstein [52J, and by Ince [88, 92, O:J], for computing the characteristic numbers and the coefficients in the series. Heine also demonstrated that one set of periodic functions of integral order could be expanded in a series of Bessel functions. CL W. Hill, in a celebrated memoir, investigated the 'Mean motion of the .Lunar Perigee' [70 J by means of an extended or generalized form of Mathieu equation, namely, d 2y

dZ2+[a-2q,p(~z)JY

==

0,

(5)

where in Hill's case -- 2q¢J(2z) == 2[82 cos 2z-t 84 cos 4z+ ...], a == 80 , the fJ being known parameters, The work, done in 1877, was published in IS86. The subject of infinite determinants was introduced into analysis for the first time, and Hill's name is now associated with an equation of form (5). In 1883 G. Floquet published a general treatment of linear differential equations with periodic coefficients, of which Mathieu's and Hill's equations are cases in point [49]' Lord Rayleigh studied the classical Meldet experiment by aid of Hill's analysis in 1887 [155, 207J. He also dealt with the problem of wave propagation in stratified media, and the oscillations of strings having a periodic distribution of mass [155]. In Melde's experiment one end of a horit Pogg. Ann. 109, 193, 1860.

Chap. I]

HISTORICAL INTRODUCTION

3

zontal thread is fixed, the other being attached to the prong of a massive low-frequency tuning-fork mounted vertically. When the fork moves along the thread, and the tension is suitably adjusted) the thread vibrates at right angles to its length, i.e. transversely, at a frequency one-half that of the fork. As we shall see ill due course, this sub-harmonic of half-frequency is consistent with the periodic solutions of equation (2), of odd integral order, whose period is 21T. III 1894 Tisserand [213] showed how the solution of (5) could be obtained in the form of a Maclaurin expansion. He also described Lindstedt's method of solving (2) by aid of continued fractions, the convergence of which was investigated by H. Bruns [17]. The theory of Mathieu functions was extended by E. Sarchinger in that year [158]. The first appearance of an asymptotic formula for the modified functions in 1898 was due to R. C. Maclaurin [122]. Some years later W. Marshall published a different but more detailed analysis [128]. Neither of these authors obtained the constant multipliers, which are indispensable In numerical work. In 1922, however, Marshall produced the multiplier for his series [129]. 1). Hilbert discussed characteristic values and obtained an integral equation, with a discontinuous nucleus, for the periodic solutions of (2) in 1904 [H9J. The theory of the functions was treated ill certain respects by S. Dannaeher in 1906 [2H], while W. H. Butts extended the treatment and computed some tabular values in 1908 [19]. In that year B. Sieger published an important paper on the diffraction of electromagnetic waves by an elliptical cylinder. Amongst other topics he dealt with orthogonality, and developed integral equations by means of which he reproduced Heine's solutions in Bessel function series. Using an integral equation with a different nucleus, he derived a solution of equation (3) as a series of Bessel function products, and discussed its convergence [162]. This paper does not seem to have been known to British authors, whose contributions after 1HOS sometimes cover similar ground. It appears that, apart from Sieger's paper, the subject attracted but scant attention in the period 1~87-1912, owing possibly to a dearth of physical applications, and to analytical difficulties; for the Mathieu functions cannot be treated in a straightforward way like Bessel or Legendre functions. In 1912, however, E. T. Whittaker started the first systematic study of the subject by a paper read before the International Congress of Mathematicians [184]' Therein

4

HISTORICAL INTRODUCTION

[Chap. I

he gave an integral equation for one set of the periodic functions of integral order. A similar equation for the modified functions was published in 1908 by Sieger (supra), and Whittaker was obviously unaware of this. Next year ~1913) Whittaker published a new method of obtaining the general solution of (2) when a is IIOt a characteristic number for a function of integral order [186]' Using this method as a basis, A. W. Young, one of Whittaker's pupils, gave a treatment of general solutions and discussed the question of their stability, i.e. whether the solution tends to zero or to infinity as z ~ +00 [191]' Recurrence formulae for the Mathieu functions cannot be deduced by the direct procedure used for functions of hypergeometric type, e.g. Bessel and Legendre functions. Whittaker, however, evolved a new method, and in 1928 applied it to obtain recurrence relations for the modified Mathieu functions [187]. From 1915 till his early decease ill ]941, the chief contributor to the subject was E. L. Inoe, a pupil of Whittaker. During this period he published eighteen papers on Mathieu functions and cognate matters. In his first paper (1915) he obtained the second non-periodic solution of equation (2) when a is a characteristic number for a function of period 1T, 271, this being the first solution [80]. Following this he treated Hill's equation on the lines of [186], and obtained formulae different in character from those given by Hill [81-3]. Many aspects of the subject, including characteristic numbers, periodicity, zeros, were covered [87-97]. He introduced the stability chart (Fig. 8A) for functions of integral order in 1925 [88]. The culminating point was perhaps his almost single-handed feat in calculating the characteristic numbers, coefficients in the cosine and sine series for the first solutions of integral order, zeros of these functions, turning-points and values of the functions. The tables occupy some sixty pages of print and appeared in 1932 [95, 96]' A general study of Mathieu's equation was made by J. Dougall in three papers published between 1916 and 1926 [36-8]. As well as a gen~al solution, he obtained asymptotic expansions for the modified functions with large z, and a contour integral which, under certain conditions, degenerates to one for the J -Bessel function. Unaware of Sieger's work in 1908, Dougall derived a solution in terms of Bessel function products. The method of derivation was different from that of Sieger. Until 1921 the only known periodic solutions of Mathieu's equa-

Chap. I]

HISTORICAL INTRODUCTION

tion (2) had period 17 or 217. In that year E. G. C. Poole generalized the position and showed that with appropriate values of a for an assigned q, (2) would admit solutions having period 2817, 8 being an integer ~ 2 [150]. These solutions coexist and their aum, with the usual arbitrary constant multipliers, constitutes a fundamental system. About the same time Ince proved that two solutions of period 17 or 217 could not coexist (for the same a, q), i.e. if the first solution had period 7T or 27T, the second would be non-periodic [84]. A different proof of this was given a few years later by Z. Markovic [126]' He introduced some new integral equations of Volterra type in 19~5 L127 J. The second solution of (2), a being a characteristic number for a periodic function (first solution) of integral order, was studied by S. Dhar in 1921 using a method different from that of Ince [30]. Dhar's publications from 1921 to 1928 cover various aspects of convergence, and integral equations for tile second solution [29-35]. Using expansions ill Mathieu functions (ordinary and modified), he reproduced Rayleigh's formula'] for the diffraction of electromagnetic waves due to a long metal cylinder of elliptical cross-section [193]. In 1922 P. Humbert discussed a modified form of Mathieu equatiOll,t whose solutions he called "Mathieu functions of higher order'. He showed the relation between these and the Gegenbauer polynomials [79]. It often happens that the zeros of functions which occur in practical applications are essential in connexion with boundary conditions. The solution for a vibrating circular membrane is expressed in terms of .I-Bessel and circular function products. 'I'he zeros of the Bessel functions determine the vibrational pulsatances and location of the nodal circles, while those of the circular functions define the positions of the nodal radii. In the case of an elliptical membrane, the solution is expressed in terms of modified and ordinary Mathieu function \ producta. The zeros of the modified functions determine the vibrational pulsatances and confocal nodal ellipses, while those of the ordinary functions define a system of confocal nodal hyperbolae, When the eccentricity of the bounding ellipse tends to zero, the nodal ellipses tend to become nodal circles, and the nodal hyperbolae tend to nodal radii. Analytically, apart from constant multipliers, the ; Phil. functions.

]l.[ag.

44, 28, 1896. The analysis in Rayleigh's paper is devoid of Mathieu : Not equation (3).

6

HISTORICAL INTRODUCTION

[Chap. I

modified Mathieu functions tend to J -Bessel functions of the same order, and the ordinary functions tend to circular functions. In 1923 E. Hille published a lengthy discourse on zeros and cognate matters. He also gave another; proof of the non-periodic nature of the second solution of (2), when a is a characteristic number for the first solution of period 7T or 27T [71]. A table of zeros of eight of these latter functions was published by Ince in 1932 [96]. At the time of writing, there is no table of zeros of the modified functions, but formulae are given herein from which the larger zeros nlay be computed. During recent years the problem of frequency modulation in radio transmission has assumed importance, in particular since the frequency was raised to 45 megacycles per second or more. The subject was studied analytically by J. R. Carson in 1922, who dealt with the simple case of a resistanceless oscillatory electrical circuit having a periodically varying capacitance. An approximate differential equation for a circuit with capacitance O(t) == 0 0+°1cos 2wt, where 0 1 ~ Go,.is a Mathieu type. Values of the circuital parameters were such that the solution could, with adequate approximation, be expressed as a series of circular functions whose coefficients were J-Bessel functions. These coefficients give the relative amplitudes of the 'side-band frequencies' on each side of the central or · carrier frequency'. The latter is said to be 'frequency-modulated' [21]. In 1934 A. Erdelyi studied the problem above, when the circuit contained resistance. Solutions were obtained for stable and unstable cases, by aid of integral equations. Both solutions were better approximations than that of Carson, and revealed the fact that 'amplitude modulation' of the 'carrier' occurs as well as 'frequency modulation' [44]. In 1936 Erdelyi obtained a solution of equation (3) by aid of the Laplace integral. He derived another form of asymptotic expansion and gave relationships of the type Yl(zei m 1T ) = eP.m1TYl(Z), I-' == ni, n an

integer [47]. In the period 1932-6 W. L. Barrow treated problems on electrical circuits with periodically varying parameters, both analytically and experimentally [2, 3, 4]. Sometimes in applications the parameters of a differential equation are such that it lends itself to approximate methods of solution, In 1923 H. Jeffreys gave an analysis pertaining to approximate solutions of (3). He also obtained asymptotic formulae, The results were

Chap. I]

HISTORICAL INTRODUCTION

7

applied in all investigation of the vibrational modes of water in a lake whose plan view is elliptical, this problem being of special interest in hydrodynamics, Numerical data were obtained for a number of the lower modes, their periods and the tide heights being computed [101-4]. The making of numerical tables usually receives little encourage.. merit, while the thanks offered by the user are parsimonious rather than plentiful. We have already referred to Ince's philanthropic gesture in computing tabular values. In 1927 S. Goldstein published the results of extensive work on Mathieu functions, and included a set of tables for five of the periodic functions of integral order [52]. Following Heine and Sieger the tabulated functions are defined as sine and cosine series, the coefficients and the characteristic numbers being given for a wide range of q. A new and acceptable normalization of the functions was adopted, this being based upon their orthogohal propertios. The same publication contained additional integral equations (akin to those of Sieger and Whittaker), some asymptotic expansions for z large and q large (complete with multipliers), an asymptotic formula for the characteristic numbers,'] and formulae for the larger zeros in q of the modified functions. There is also a general discussion relating to the second solution and an extension of Heine's expansions in Bessel function series, the I-type being introduced. In additional papers Goldatein extended Jeffreys's investigation on elliptical lakes, and his own researches on characteristic numbers [53-7]' The problem of eddy currents in a straight conductor of elliptical cross-section was investigated hy l\f. J. O. Strutf in 19~7 [167]. He assumed constant current density at the surface. AR this is untrue in practice, his analysis is mainly of academic interest. In common with other boundary-condition problems, the wave equation expressed in elliptical coordinates is separable into two Mathieu equations like (~), (3), but in the above problem q is negative imaginary. Strutt has solved a variety of technical problems involving Mathieu functions, e.g. diffraction of plane waves at a slit [173]. In ,1929 he published a detailed study of the characteristic exponent p, in Hill's equation (5) [171]. A list of his other works will be found in the references at the end of the book.

t Identical with that found independently [92, 93].

by Ince using a different procedure

HISTORICAL INTRODUCTION

8

[Chap. I

When experimenting with a moving-coil loud-speaker ill 1925, the author found that under certain conditions the coil, although actuated by a sinusoidal current, mooed out of the magnetic field of the magnet. The analysis, published in 1933, involved a Mathieu equation. Its solution for the a~ial displacement of the coil was characterized by a dominant term having a period much greater than that of the driving current. Reduction ill amplitude of the latter resulted in an increase in the period of the dominant term, and vice versa [201]. Oseen's approximate differential equation is familiar in hydrodynamics. As might be anticipated from this equation, calculation of the vorticity in a viscous fluid flowing past a very long elliptical cylinder in its path necessitates the use of Mathieu functions, This subject was studied by M. Ray in 1036 [154], and by D. Mek~YJl in 1937 [137]. In 1935 '1'. Lewis published a general treatment of the problem of circular and elliptical cylinders, and a flat plate in a viscous fluid [116]. He showed that the solution given by Meksyn required modification to avoid infinite circulation of fluid. In 1941 'I', V. Davies made a detailed study of the flat plate using modified Mathieu functions, whose behaviour he investigated in the neighbourhood of the origin [28]. About fifty years ago, the transmission of electromugnetio waves within hollow metal cylinders was contemplated by J. J. Thomson.] O. J. Lodgej verified the theory qualitatively using a hollow metal cylinder having a spark gap and transmitter at one end, the other heing open. Three years later Lord l~ayleigh§ showed that there was a plurality of modes in which electromagnetic waves could he transmitted within a perfectly conducting uniform circular tube of unlimited length. Using modern electronic devices, it is eOlH!>aratively easy to incite the transmission modes of lower order within a hollow metal cylinder, 110W called a 'wave guide'. No cylinder is perfectly circular, so to ascertain the influence of deviation from circularity, L. J. Chu in 1938 investigated the propagation of electromagnetic waves within a hollow cylinder of elliptical cross-section. As might be expected, the final result entailed products of ordinary and modified Mathieu functions of integral order, the former having period 11, 21T [22].

t §

Recent Researchrs (1893). .Alug, 43, ] 25, 1897.

l~hil .

t

Proc. Roy. Institution, 14, 321, 1894.

Chap. I]

HISTORICAL INTRODUCTION

9

In 1940 W. G. Bickley published new solutions of (3) with q < 0, these being expressed as expansions in l- and K-Bessel functions [6]. He also gave integral and asymptotic formulae for these solutions. During the same year J. G. Brainerd and C. N. \\Teygandt published data regarding general solutions of equation (~), when a is not a characteristic number for functions of integral order. The data were given up to z = O·31T, using the Maclaurin method of solution (see § 4.40), with terms as far as y<21)(O). Beyond z = 0·311 the solutions, given in the form of curves, were extended by alternative means [12]' In 1943 S. Lubkin and J. J. Stoker applied Mathieu equations to investigate the stability of columns subjected to a steady pull with a periodically varying force superimposed [121]. What precedes is merely a brief survey of salient matters relating to Mathieu functions and their applications to technical problems, The work of other authors, too extensive to be examined ill detail, is given in titular form in the list of references on pp. 373-81. The occurrence of Mathieu functions in practical applications may be divided into two main categories. First there are the boundarycondition problems arising from solution of the two-dimensional wave equation (1), when expressed ill elliptical coordinates. As stated previously, this yields a pair of Mathieu equations like (2), (3).t '!'he appropriate solution of (2) is usually a periodic Mathieu function of integral order, while that of (3), in the cases of the elliptical membrane and the wave guide, is obtained by making the argument of this solution imaginary. Secondly there are what may be regarded as initial-value problems, ill which only one equation like (2) is involved. Usually a is not then a characteristic number for a function of integral order having period 11' or 21T, and the solution is general in type. It may (a) have period 2811, 8 an integer ~ 2, (b) be non-periodic but bounded as z ~ +00, (c) be non-periodic but unbounded as z -+ +00. Frequency modulation and the moving-coil loud-speaker (see Chap. XV) are examples of the class of problem in question. 'I'he majority of applications, to date, pertain to the first category and involve the wave equation. t In the analysis of Melde's experiment, the differential equation is in x and t coordinates, the solutions of the two resulting equations being circular functions in x, and periodic Mathloll functions of integral order in t.

4961

c

PAR'f I THEORY

O~.

l'IATHIEU FUNCTIONS .., II

FUN(JTIONS OF INTEGRAL ORDER

2.10. Canonical form of Mathieu's equation. In solving the problem of the elliptical membrane [130], Mathieu obtained a differential equation of the type d2y

dz2 +(a-2qcos 2z)y = O.

(1)

We shall regard this as the canonical form-ruled by general usageof the equation. At times it may be expedient to replace +q by k2 • The parameters a, q = k 2 , will be limited to real numbers, unless stated otherwise, but z is usually unrestricted. The equation is a. particular case of a linear type of the second order with periodic coefficients. Its solution takes different forms according to the values of a, q. For the present we shall confine our attention to solutions having period." or 2." in z, In the membranal problem, the displacement from the equilibrium position is a periodic function of one of the coordinates. t Thus if q is assigned, y will have period," or 211 in z, provided a has one of an infinite sequence of particular numbers.

2.11. Solution of (i) § 2.10 when q = the equation becomes d2y dz 2 +ay = 0,

o.

Under this condition (1)

and it is convenient that the solution of (1) § 2.10 should then reduce to cosmz or sinmz. Accordingly we put a = m 2 , m = 1, 2, 3,... , thereby obtaining the pairs of distinct solutions +cosz, -l-sin z: + cos 2z, +sin 2z; and so on. When m = 0, we take the respective solutions to be + 1, o. Also until and unless stated otherwise, we shall adopt the convention that the coefficient of cos mz and sin mz is unity for all q [130]. t

FQr a circular membrane the displacement at any radius depends upon C?8(m8) 8m

andis, therefore. B periodic function of (J [202, p. 27]. The displacement of an elliptical membrane is periodic iit-tbe elliptical coordinate "1 (z in (1)), as shown in § 16.11.

2.12]

FUNCTIONS OF INTEGRAL ORDER .

11

2.12. Dependence of a upon q fora periodic solution. When q :1= 0, a, q must be interrelated for the solution of (1) § 2.10 to have period

'11

or 21T, so a is a function of q. If we write

a

=

m2+CXt q+cx:aq2+cxaq3+ ...,

the desired form a = m 2 is obtained when q then reduces to d2 m 2y = 0 --.1t+ dz 2 ' whose solutions are a positive sign.

± cos mz, ± sin mz.

=

(1)

0, and the equation (2)

Conventionally we adopt

2.13. Periodic solution of the first kind. To illustrate one method of finding a particular periodic solution of (1) § 2.10, we take the case where a = m 2 = 1 when q = o. Then by (1) § 2.12 we have for q ~

°

(1)

SInce the solution is to reduce to, say, cos z when q vanishes, we assume that

(2)t C1, c2, ... being functions of z to be determined. Substituting (2) into (I) §2.10 and using the valueof e from (I) above, we get

;+...,

y" = -cosz+qci+q2C;+q3C ay ;= cos z+q(c 1+cxl cos Z)+q2(C2+CXl c1+cx2 cos z)+ +q3(Ca+CXl C2+CX2 c1+cxa cosz)+ ..., -(2qcos 2z)y = -q(cosz+cos 3z)-2q2c1cos 2z- 2q3c 2 cos 2z- ....

(3)

Equating coefficients of like powers of q to zero gives qO

q

cosz-cosz

== 0,

ci+c1-cos3z+(cxt-l)cosz = 0,

(4) (5)

of which the particular integral corresponding to (cxl-l)c08Z is the t In ref. [158] it is shown that the solutions of Mathieu's equation, having period or 211' consist entirely of cosine or sine terms, and not 8 combination of the two. In ref. [223] it is shown that if one solution is even, the other must be odd. Thus two independent even or two independent odd solutions cannot occur. The only singularity of (1) § 2.10 is an essential one at infinity due to cos 2z. Hence every solution is an integral function of z, 1T

12

FUNCTIONS OF INTEGRAL ORDER

non-periodic function term must vanish, so

!( 1- (Xl)Zsin z. (Xl

[Chap. II

Since y is to be periodic, this

= 1,

(6)

ci+c1 = cos 3z. (7) Now the particular integral of Vi;'+v = A cosrnz is -A cosmzj(rn 2 - 1) (m =1= I), so with A ::::: 1 and m = 3, while

=

ct

Inclusion of

~~~z

1 --cos 3z.

(8)

8

[complementary function of (7)] in (8) would

involve a term in q sin z in the solution. By § 2.11 the coefficient of cos

cos Z is to be unity for all q; also sin z is odd. Hence the C.F. of (7) and subsequent equations is omitted.

c;+C2+CX l ct-2c·1cos 2Z+CX2COSZ == O.

q2

(9)

Substituting for ~1' C1 from (6), (8) and expressing cos 2z cos 3z as a sum of two cosines, we get

c;+c2 -

!cos 3z+~ cos 5z+ (~+ 888

(

2) COSZ

== O.

(10)

As hefore, the particular integral pertaining to (1+cx2)cos z is non'periodic, so (11)

c2w+ c2

while

1 Q I 5z. -COStlZ--COS

=

S

S

( 12)

By aid of the general formula above, the particular integral is 1

1

c2 = -64cos3z+192cos5z.

(13)

Proceeding in this way leads to the results: a3

= - 64 ~,

<X.

=

C3

= -

_1_ (~ cos 3z- ~ cos 5z + ~ cos 7z), (14) 512 3 9 1 M

1 -1536'

1 (11 1 1 1 ) c.= 4096 Ocos3z+6cos5z-12cos7z+180cos9z,

and so on. Substituting for

Cl'

(15)

cz,... in (2) gives a solution of Mathieu '8

FUNCTIONS OF INTEGRAL ORDER

2.13]

13

equation, periodic in z, with period 21T. It is denoted by cel(Z,q)t and represented by the series

ce1(z,q)

= cOSZ-~qCOS3Z+ 6

14

q'J.( -cos 3Z+~COS5Z)-

3(13

1 cos Tz) + - -1q - cos 3z- 4·-cos 5z+--512

9

18

1 q4 (11' 1 1 1 ) +O(q5). +__ -9 cos3z+-cos5z--cos7z+-cos9z 4096 6 12 180

(16)

The value of a necessary to yield this solution is found by substituting the (X into (1). Then we get

a

==

I+q_!q2 __~q3 __~q4+_11_q5+0(q6) 8 64 15~6 36864 ·

(17)

For a given q, the value of a found from (17) is called the characteristic number of the Mathieu function ce1(z, q). When q = 0, a = 1, and the r.h.s, of (16) reduces to cosz. The notation cem(z, q) signifies a cosine type of Mathieu function of order m, which reduces to a multiple of cos rnz when q = o. Since m may be any positive integer, there is an infinite number of solutions of type (16). These functions are even in z, being unchanged if -z is written for z,

2.14. Another periodic solution of the first kind. In accordance with our convention in § 2.11, if -rn 2 := 1, the second solution of (1) § 2.10 reduces to sin z when q = o. If we assume a solution (1)

and proceed as in § 2.13, we obtain a sine type of Mathieu function designated se l (z, q).t Thus set (Z, q)

=

2(.

I. 1q · Slnzsq sin 3~ z+ 64

3(1. 3

1 - 512 q

sm

3 t

z

SIll

3Z+31. sm 5z) -

7) + "94.SIn 5z + 11. ~ SIll Z +

+40196 Q4(- .!.!9 sin 3z +!6 sin 5z + -.!.. sin 7z + _1_ sin 9Z) + O(q5), 12 180 t

t

"oe ' is an abbreviation of 'c-osine-elliptic " introduced by E. T. Whittaker. "se ' is an abbreviation of 'sine-elliptic t.

(2)

FUNCTIONS OF INTEGRAL ORDER

14

[Chap. II

provided a

=

1 2 +-q 1 3 - - '1l .n4 - - 11 6) l-q--q - q5+ O(q. 8 64 1536 36864

(3)

-e

This particular value of a is the characteristic number for the Mathieu function se1(z,q), periodic in z with period 217', which reduces to sin z when q = O. Since se1(z, q) = -se1( -z, q), the function is odd in z.

1·

FIG. 1. Graphs of ce1(z, 2), cea(z, 2), ce 6(z, 2) over a period 211".

1·

271

}4'IG.

2. Graphs of 8e 1(z, 2),

S6 a(Z,

2),

8e o(z,

2) over a period 21T.

If in (16) § 2.13 we write (!1T-Z) for z and -q for q, (2) above is reproduced, and vice versa. Since the values of a differ for the two periodic functions cel , se l , these functions are not solutions of the same equation except when q = o. The graphs of these functions for q = 2 are shown in Figs. 1, 2.

FUNCTIONS OF INTEGRAL ORDER

2.15]

15

2.150. Periodic solution of (1) § 2.10 of order m. To derive this, we assume for the cosine type of function that y = cos 1UZ+qCt(Z)+q2C2(Z)+q3C3(Z)+ ...

(1)

a = m2+CXt q+CX2q2+cxaq3+ ....

(2)

and

The solution obtained is either of the class ce2n(z, q) or ce21~+t(z, q), according as m in (2) is even or odd. If m = 2, we find that

1(2 8 3

) . 3841

ce2(z,q) = cos2z---q -cos4z-2 +--q2cos6z-

3(

1 1 43 40) + ---q -cos8z+-cos4z+512

45

27

3

I q4 (1. _.-cosI0z+~--cos6z 40)93) +O(q5) + __ 4096

a === 4

with

540

540

'

7

63 4 ~ q2 _ 10 02401 Q6_ O( 8) q +12 13H24 +796 26240 t- q.

(3)

(4)

Then a reduces to 4 and cC 2 to cos 2z when q = 0, which is in accordance with our convention in § 2.11. When q =1= 0, the function has a constant term whose value is (5)

The graph of ce2(z, 2) is shown in Fig. 3. This function is periodic in z with period 1T. In the particular case of rn == 0 in (1), (2), we obtain ceo(z, q)

=

1-

~q cos 2z + 312 q2 cos 4z - -d~q3(~ cos 6z-7 cos 2Z) + +

with

73~28q4(COS 8z--320cos4z)+O(q5),

(6)

(7)

Then a = 0 and ceo = 1 for q = o. Here again the function has a, constant term. In general this holds for all the functions ce 2n , but not for ce2n +1 • To derive the sine type function of order m, replace cos mz in (1) by sin mz and c,(z) by 8r(Z). The solution obtained is either se2n+l or

[Chap. II

}'UNCTIONS OF INTJ4JORAL ()RDER

16

sesn+z,t according as m is odd or even. If m se 2(z,q)

=

=

2, we have

· I. 4 1 2 • 6 sln2zQ s l u Z+3R4 Q 8111 z-

f2

a( 1

1

· 8

)j.

-51~q· 45 s1n z-27 sU1

4-) + Z

1 q4 (1 . 1 +-4096 - ---- SIB 540

(l) +o(q

:37. ()Z---SIll lZ

a

with

::=

'I 4 __ q 1~

5 2+__

13R24

540 6

q~+()(q

\)

5) ,

).

(8)

(9)

re2(t,Z) 1·

/ r p4(%.2)

I

I

teo(:%,~

rr

}4~IG.

FlO.

4.

3. Graphs of ('eo(.: , 2), ceo(z, 8), C't'z(z, 2). (·t\~(z, 2) over a period TT. For q finite ,,' 0, coo> H. AR q --'? -1-co, cc o(m17', q) ~ O. :FIG. 4. Graphs of S8s(Z, 2), sc4(z, 2), ~e6(z, ) over a period 17'.

When q = 0, a reduces to 4 and se 2 (z, 0) :=:: sin 2z. The function se 2(z, q) is periodic in z with period 1T. This is shown graphically in ~'ig. 4.

2.151. Formulae for a. We desiguato the characteristic numbers for ce/n(z,q) and sel1&(z,q) by anI' bm respectively. Then _

1

ao - -"2 q

2

7

+12s q

4

20 6 68()H7 8 O( 10) -2:304 Q +i~~ 74:3US Q + q ·

b1 = l-q- !q2+~q3 B 64

al

~_.q4 __!.! q5+

1536

_~ q6 + 589824

(1)

36864

55

9437184

q7 _

= write -q for q in (2). t The signiflcanoe of 2n -+ 2 in place of 2n is that

205 q8+0( 9) 11324620S q.

(2) (3)

roprosents tho number of real zeros of eel•• ce ••+1• 86 1'1+1' se."+1 in the open interval 0 < Z < }17' (800 Chap. XII). 1l

FUNCTIONS O}4' INTEGRAL ORDEli.

2.16]

17

+ 13:24 if -796 : : 240 q6 + 45 8:4~:~2400 q8+ O(qlO}. 2

b2 = 4 - 112 q2

(4) _ 4 a2

-

~ q2_ 763 4 _~~~2~~!_ q6_ 16690 68401 Q8+0( 10) +12 13824 Q+79626240 458647142400 Q. (5)

b - 9+~q2_~q3+~q4+_5_Q5 16

3 -

64

20480

'16384

.~.~_q6+

235 92960

+-~?~--- - q'+O(q8}. 1048 5i600

aa

=

b -

write -q for q in (6).

4 -

(7)

317 4 10049 8 O( 8) 64000 Q -1-272WOOOOoq + q.

(8)

16+ 1 q2+ 433 q4 .__5_~_~!.._._ q6-t-O(o8) 30 8 64000 - 2721H 00000 1 •

(B)

16

4 -

a -

1

2

+30 Q

-S

= 25+..!. q2+ ~__ if--- ~- --

blS

(6)

48

7 74144

1 47456

q5+

3?_ q8+0(q').

8918 13888

at) == write -q for q ill (10). b6

== 36+~q2+ __1~~

a

-=

70

36

6

~

(II)

q4 _

439 04000

2

187

(10)

4

+ 70 q + 4a9 04000 q

___~8__~_~63~. q6-t- O(q8). U293 59872 00000

( 12)

_ 6+0( 8)

( 13)

tl743617

_

1 9293

59872 00000 q q .

When m ~ 7, the following formula is correct as far as and including the term in q6:

am,bm

= m2+2(m~_I}q2+~f2(m2~~~;~12~-4)q4+ 91n 4 + 58m 2 + 29

6

+ 64(mi = -f )5(m 2-:'=-4)(rn2=-"9)q +....

(14)

These formulae may he used to calculate a when q is small enough and of either sign. For equal accuracy, q nlay increase with increase in m, It must not be inferred from (14) that when m ~ 7, am == bm • Series (12), (13) for be, u'e are identical 'tt]) to O(q4.), but not thereafter.] An extension of (14) to higher orders of q would show the same behaviour. As Iql ~ 0, a,n ---» bop but for Iql > 0, (am-b m ) ~j::. 0, t For m > 3, the series for am' l)m are identical up to qm-1 or qm -2 according as sn is odd or even. 4961

D

18

}4'UNCTIONS

O~' INT}4~Gl~AL

ORDER

[Chap. II

although it is very small near q = O. Under this condition the characteristic curves for cem, se m are substantially coincident (see Fig, 8). In (14) when m 2 is lunch greater than any subsequent term, am' bm are almost constant, so the characterietic curves are practically parallel to the q-axis.

2.152. Remarks on the periodic Mathieu functions of integral order. 1. The functions ce2n , n == 0, 1, 2,... , have period 11'. There is a constant term in the series, which is a function of q. As a consequence, ceo is never negative. although oscillatory. 2. The functions ce2n 1-1' n == 0, 1, 2,..., have period 21T. There is no constant term in the series. 3. The functions SC 2u +1' n ==.:- 0, 1, 2, ... , have period 21T. There is no constant term in the series. 4. The functions se 2n + 2 , n. == 0, 1,2,... , have period er. 'I'here is no constant term ill the series. 5. All the above functions have n real zeros ill 0 < z < iTT.

2.153. Dynamical illustration. The differential equation may be written d2 (1) dt; +ay = (2qcos2t)y.

Referring to Fig, 5 (A), if a ::::.-: s[m; 2q = JolIn, (I) becomes lny" +sy

--=-:=

(fo cos 2l)y,

(2)

which is the differential equation of the mechanism illustrated there, provided yjl <{ 1, w :::::: 1. A mass m. is axed to a massless helical spring of stiffness s, and driven along the linear axis of the spring by a force (focos 2t)y, which depends upon the displacement y. If m, 8, fo have the appropriate values, y may take any of the forms ce 2n , ce2n +1 , se2n +1 , se2n +2 • Suppose Y == cet(t, q), then the motion is described by (16) § 2.13, and the component cost is a sub-harmonic of cos 2t in (2) above. This remark applies also if m; 8, fo are such that y =--= se1 (t , q), by virtue of the component sint in (2) § ~.]4. Hence in any system (dynamical, electrical, etc.) represented by (~), if the parameters a, q have certain interrelated values, there will be a subharmonic of 10 cos 2t. In l\ny event, so long as the parameters are those for a function having period 1T, 27T, the displacement (or its equivalent) ha« an infinite number of harmonic components corre-

FUNCTIONS OF IN'l'EGRAL ORl)EH,

2.15]

19

sponding to the various terms in the series, e.g. (16) § 2.13, (2) § 2.14, (3) §2.150. Additional comment is made in §§ 15.40, 15.41.

J; :(ficosz wl.).'!

t

(A)

Mass",

o~

_-___-

Fired pm

gUIde r,.; _ .....~~fl:'t.JJII,:~',~

l---a--..-..

----.. i,; : If; cos2ldf,

Joinc-.. . -.. . . _

'l/IIIIHf!tz,rrrJ!!j'I///I7!i:!'u'!/flll.~'"rJfI'17/rtI!lm!lln'/,·Zlz7!!l!f!l, gUide (12

"- ..

(B)

Mass 171 FIG. 5. (A) Schematic diagram (plan) of mechanism illuat.rat.ing Mathieu's equation.

(n) Curves illustrating, motion of m and the point B whon a sub-harmonic of pulsatance W oe0urs.

.2.16. Mathieu functions of fractional order.] We define a function of order v (real but either rational or irrational and positive), to be one which aatisfles Mathieu '8 equation and reduces to cos vz, sin vz when q === o. The normalization in this section corresponds to that of § 2.11, so the coefficient of cos vz, sin vz is unity for all q. If we assume that 00 (1) cev(z,q) = cosvz+ ~ qrcr(z), r--=l 00

sev(z, q) = sin vz+ l,q'"Sr(z),

(2)

r=l

co

a

= v2

+ r=l ~ oc,qr,

(3)

t Although the title of this chapter implies functions of integral order only, those of fractional order are introduced here because their expansions under the normalizotion oj ~ 2.11 are obtained in ascending powers of q by a procedure akin to that used for the former, A different definition is given ill § 4.71 under the normalization of § 2.21.

20

FUNCTI()NS O}4""

INT~~GRAL

ORDER

[Chap. II

and proceed analytically as in § 2.13, the following results are obtained: 1 [COS(v-t- 2)z cos(v- 2)Z] cCv(z,q) = C08VZ- 4Q (v+l) "..,- -(;=1) +

2( cos(v+-4)z + cos(v-4)z ]

1

+ 32 q 1

- 128

==

(v-='i)(v-2j-

2-4v+ 7)cos(v- 2)z + q3[(v2+4v-t--7)COS(v+ 2)z (v -(~- f)(v-+ j )3('~~2)- - (v+ 1)(v- fj3(v- 2)-

+ sCv (z,q )

(v+l)(v-t2)

COR(V"+-O)z ._ cos(v-6)z 3(v+ l)(v+ 2)(v+:1) :3(v-l)(v- 2)(v- 3)

+ -!.-q2[ 32

_

Hin(v- 2)Z] ~ (v-I)

. I [Sill(V t· 2)z SlllVZ--q 4 _ (v-t I)

2

1

2

sin(v+4)z

_(v-t-l)(v+~) ( 5v

2

+

]+...,

(4)

+

~in(!'=.!J:_] _ ..., (5)

(v-l)(v-2)

-t-7)

4

a --- v +2(vil_-l)q +:32(IJ2 - -1)3(v2 - 4) Q + Bv4 -t-- 58v 2 -t-29 6 L 5( 2 )( 2 - q -1••• v --4 v -B)

+ b4(v"---I)' -..

<)

•

(6)

These formulae arc suitable for computation when q2j2(v2 --- 1) ~ v2 , v > o. a has the same value for both solutions for any q, so (4), (5) coexist, and are linearly independent. Thus the complete solution with two arbitrary constants is

y

=== A cev(z, q)

+R sev(z, q),

(7)

provided v is non-integral. (4), (5) constitute a fundamental system, Since (6) is even in q, the characteristic curves, showing the relationship between a, q, for cCv(z, q), sev(z, q) are symmetrical about the a-axis. Put v == (1n-t-fi) , tn being a positive integer, and 0 < f1 < 1. If fi == 1)/8, a rational fraction in its lowest terms, the functions have period 27T8 in z, 8 ~~. If f3 is irrational, the functions are nonperiodic, but they are hounded and tend neither to zero nor infinity as z ~ +00.

2.17. Preferred form of series for cem(z,q), sem(z, q). The form of the series given hitherto precludes the application of standard convergence tests. In [180] it is shown that by constructing a 'fonction majorante', the series for ceo(z,q) converges if 14q12 < 1. This is the only case which has been treated. Generally the series appear

O}4~ INT~~GRAL

FUNCTIONS

2.17]

ORDER

21

to converge if q is small, but diverge when it is large enough. The preferred form of the series is given below [52, 88, 95, 162, 196] for q positive:

Oharacteristic number <X)

ce 2n(z, q)

!

==

r-O <X)

!

ce 2n +1 (z, q) --

A (2n) 2r cos 2rz

(a 2n )

(I)

A (2n +1 ) cosrzr ,( ') t-1)z 2r+l

(a2n~1)

(2)

B~;'+11) sill(2r-t-I)z

(b2n +1 )

(3)

t- '») ~ Z

(b211 +2)

(4)

r=O 00

!

se2n +1(z, q) ===

r=O co

se2n+ 2

(Z, q) --- !

B(2n+2) • ('> 2r+2 SIll ~r-

r=O

In these series A, Bare functions of q. This will be evident if we express (16) §2.13 in the form (2)\ with n ::~-= O. Then ce 1(z, q)

== A 1 COS z -t A 3 COS 3z--1- .•4:> cos 5z+...,

(5)

where under the convention of § 2.11

Al As

==:

:::=:;

.t1 3

1 for all q' ,

..:::::: _

..1 q ( 8

1 o2 1+ __S1 q __L 192

_~q2(1_t-_!q+_I_q2_t_....) Itl2

6

128

I

'

and so

1.

11 q3 + 4608

••. ) •

'

011.

When the A, B are obtained by the method used in § 3.13, and the solutions are normalized in accordance with § 2.21, their convergence is readily demonstrated. This is left until § 3.21.

2.18. Changing the sign of q. If in (1) § 2.10 we write ±(!1T±Z) for Z, it becomes y"+(a+2qcos2z)y == O. (1) Consequently the solutions of (1) §2.10 with period tr, 27T, q being negative, are obtained if the above substitution is made in (1)-(4) §2.17. Hence, standardizing on (-~7T--Z), we get [52, 95J: ce2n(z, -q) == (-l)n ce2n (! 1T - Z, q)

==

00

(-1)71,

!

(-1)rA~;.n)cos2rz

r=O

ce 2n t-l(Z, -q) = (-l)n se 2n -t 1 (-l7T - z, q) co

=; (-I)n ~ (_.~ l)r B~;'t+il) cos(2r-t-l}z r=O

(b2 11 f 1)' (3)

FUNCTIONS OF INTEGRAL ORDER

22

[Chap. II

00

(-l)n

:=

!

r=:O

~e211.+2(z, -q)

(-I)"A~;+il)8in(2r+l)z

=

(-I)"'se 21l -t 2(! 1T- z, q)

=

(-I)1t

00

!

r=O

(U 2n +1),

(4)

V'

(-1)rB~~+~2)sin(2r+2)z

(b2n +2 ) . (5)

Since ce 2tt +1 (z, -q), se2n +1(z, -q) entail cosine and sine series, respectively, they are derived from Re 2n +1(z,q) and ce2n +1(z,q). Accordingly the A, B, and the characteristic numbers are interchanged, i.e. (A,a) and (B, b) accompany each other. The multiplier (-I)1t ensures that when q:=.:- 0, the functions reduce to -l-cos raz or +sinmz as the case may be. Any linear differential cquat.iou of the second order ha« two linearly independent solutions, that i~ to say, they are not proportional. All solutions of integral order obtained hitherto are of the first kind. The second solutions require separate consideration, and being of less practical importance than the first kind, are treated in

Chapter VII. 2.19. Orthogonality of the periodic functions cent' sem • Let Yl' Y2 be solutions of y"+(a-2qcos2z)y.=::: 0, (I) for the same value of q, hut usually different values of a, e.g, at, a 2 ,

Then !/~+(al-2qcos 2z)Yt

:::::=:

0

and

y;+(a 2 -- 2q cos 2Z)Y2

= o.

(2)

Multiplying the first of these equations by Y2' the second by Yl' and subtracting, gives y~ 1!2-Y; Yl == (a 2-a t )Yl Y2·

Integrating both sides of (:3) between the limits Zs

Zs

(3) %1' %2'

Zs

f Y2 dy~ - f Yl d.1J~ = (a2-al ) f YlY2 ds, oZ.

Z'1

\ve get • (4)

Zl

which after integration by parts becomes z.

[Y~Y2-Y~Yl];: = (as-a l ) f YlYS dz.

(5)

z,

Eor a git'cn q, the

f1,

corresponding to Yl -= cem(z, q), Y2 :..:.::; cep(z, q),

23

FUNCTIONS OF INTEGRAL Oll,!>ER

2.19J

m =F Pt are different. These functions have period 'IT or 27T, so with %1 = 0, Z2 = 217', the l.h.s. of (5) vanishes. Hence, if m =1= p, 2".

I o When m

::.=::

cem(z, q)cep(z, q) dz

=

(6)

O.

2n, an even integer, from (1) § 2.17, we have

211'

2n

o

0

2n

I ce~(z, q) dz = I r~o [A~~n) cos 2rz]2 dz = Jo f [A~~)]2 cos22rz de, ex>

00

(7)

0

since by § 3.21 the series concerned are absolutely and uniformly convergent, term-by-term integration being permissible, The other integrals vanish by virtue of orthogonality of the circular functions. Thus 2n

f ce~n(z,q) dz = 21T[A~2n)]2+1TJl[~~~·n)]2.

(8)

o

The results given below may be established

I ce~n+1(z,q) dz

ill

a similar manner:

~

21r

=

1T

Jo[A~~+11)J2.

(!I)

(m -j: p).

( 10)

o 211

f sem(z, q)sel'(z, q) dz =-~ 0

o 2n

J' .-:1" o

o...!H2

211 t 1

(~(1) (l~,:., lor, ~

m

f celll(z, q)sep(z,q) dz -- -

r8<2n+1),]2 2r -I-1 •

,~

--- ,.,.,

k r-

#I

( II )

0

21r

0,

In,

1), positive integers.

(l~)

o 21T

f se 2n2 +2(Z, q) dz -_ o

00

1T

~

1-2) 2 [B (2n 2r + 2 ] •

( 13)

r=O

These results arc valid for all values of q, and lead us to t he important conclusion that cem(z, q), Hem(z, q) are orthogonal. Thi« fundamental property is used in S§ ..... BO, 1H.llO.

2.191. A fundamental identity. In (5) § 2.19 let 1/1' .'12 be independent solutions of (I) §~.] n for the same a, q. Theu with Zt = 0, Z2 = Z, we get the Wronskian Yl(Z)Y~(Z)-Y2(Z)Y~(z)-== Yl(O)y~(O)- !J2(O)y~(O) == c2 ,

a constunt.

(I)

FUNCTIONS

24

If Yl(O) = y~(O)

=

INTEGRAL ()l-tI)ER

O~.,

1, and Y2(O) = y~(O)

=

0, c2

(Chap. II

= 1 and we obtain

Yl(Z)Y~(Z)-Y2(Z)Y~(z)== 1.

(2)

For a given pair of independens solutions, c will ill general vary with q. 2

2.20. Orthogonality of functions of rational fractional order. Since Yl' Y2 in (5) § ~.lB are any solutions for the same q, the formula is valid for functions of rational fractional order v. Then by § 2.16 the period is 2178, so with Zl =-= 0, Z2 :::::: 2178 in (5) § 2.19, we have 2ns

2ns

o

0

f ce~(z, q)ce,.(z, q) dz = f se~(z, q)se,.(z, q) dz 27TS

f ce~(z,q)se,.(z,q) dz = 0,

=

(1)

o

provided v ::j:: ft in the first t\VO integrals. Also 2ns

2ns

o

o

f ce~(z, q) dz =F- 0, f se~(z, q) dz =F- o.

(2)

Hence the periodic functions of (rational) fractional order are orthogonal. The value of these integrals is given in § 4.72.

2.21. Normalization of cem(z, q), sem(z, q). In § 2.11 we introduced a convention that the coefficients of cos mz, sin 111Z should be unity for all values of q. Thus in (1)-(4) §2.17 \\te should then have A~~) == B~n) zzz: 1 for all q. It will be shown later that constancy of these coefficients leads to undesirable consequences. Accordingly we now introduce an improved convention based UpOll § 2.19 [52, 95J. We shall make definite or normalize the functions cem(z, q), sem(z, q) by the stipulation that 2n

; Jce~.(z,q) dz 1 •

=;;. f se~n(z,q) 2~

1

o

dz

= 1,

(1)

0

for all real values of q. 1'hell by (J) above and (8) § 2.19, we have CX)

2[A~21l)J2-t-

l

[A~;n)]2 === 1.

(2)

r..:.l

Now ceo(z,O) :::.:: I by our convention of § 2.11, so to normalize in

It'UNCTIONS O}4' INTEGRAL

2.21]

OI~DER

accordance with (1) above, its constant term must be q = 0, since 211

25 A~O)

= 2- 1 for

lI

dz = 2.

(3)

o

For ce 2n +1 , se 2n +1 , se 2n +2 we have, respectively, [52] 00

1 = ~ [A~~+1l)]2 r=O

=

~

00

~ [B~~+il)]2 =

r=O

~ [B~~,.t2)]2.

(4)

r=O

The normalization (1) entails a mean square value of 1/2 over the interval (0, 21T), which is the same as that for the circular functions. It is convenient in applications when tile orthogonal properties of the Mathieu functions are used. The functions of fractional order may be normalized in like manner, but this is postponed till § 4.72.

2.22. Relationships. The following are easily established from what precedes: (1)

(2) (3)

From these relationships it. is seen to be sufficient to give tabular values of the ordinary Mathieu functions in the interval 0 ~ z ~ t1T for q > o. When q < 0, the appropriate values may be derived from the foregoing by aid of (2)-(5) § 2.18. Also

ce2n +1 (r + ! )1T

:=

se2n +1(r1T)

=

se2n +2(! r1T)

=

O.

(4)

When z ~!1T (-I)n ce2 n(z, q),

(-1)" ce2n+l(z, q)/cos z,

(-1)11 se 2n + 1(z, q),

(-I)n se 2fl +2(z, q)/cos z,

(5)

are positive.

2.23. Transformation of y"+(a-2q cos 2z)y = }O.

Substitute x

=

o.

u(z), a continuous function of z, and we get

d2y o« dy dU)2-+----~(a-2qcos2z)y= o. (-dz du dz du 2

.111

2

(1)

FlTNCTIONS OF INTEGRAL ORDER

26

If u = 2k C08 Z, q equation as

=

/.,.2, (1)

gives the transformed version of Mathieu's

(u2_4k2)~2y2 +u dy + (u 2du

2°. Let Y obtain

du

= u(z)f(z), u, f

a - 2k 2)y =

Takef(z) = eICZ , then/'//=

1<,1"//= /(2, so

2zJu =

(2)

o.

(3)

(3) becomes

u"+2/(u'+[(a+K2)-2qcos2z]u

=

o.

being continuous functions of z, then we

UN +2(f'/f)u' +[a+(!"/f)-2qcos

The solution of (4) is u equation.

[Chap. II

=

O.

(4)

ye- KZ , where y is a solution of Mathieu's

2.30. Modified Mathieu functions of the first kind of integral order t q positive. If in (1) § 2. lOwe write iz for z, the equation becomes

~~-(a-2qC08h2Z)Y = o.

(1)

This is the canonical form of the second equation found by Mathieu in his analysis of the elliptical membrane. For the values of a corresponding to cem(z, q), sem(z,q) the first solutions of (1) are derived by substituting iz for z in (1)-(4) §2.17. Thus we have 00

Ce2n(z, q) = ce2n (iz, q)

=!

A~~) cosh 2,Z

(2)

r=O 00

=!

Ce2n +1(z, q)

=

ce2n +1(i z, q)

Sean+1(z,q)

=

-ise2n+l(iz,q)

B~~'+11)8inh(2r+l)z

(b2n+1 ) ,

(4)

Sean+2(Z, Q)

= -ise2n +l (i z, q) = ~ B&~~:t2)sinh(2r+2)z

(b2n +2 ) .

(5)

A~;'~11) cosh(2r+ l)z

r~O

00

=!

1'=0 00

1'=0

These are defined to be modified Mathieu functions of the first kind of integral order'] for q > o. (2), (5) are periodic in z, with period tri, whilst (3), (4) are also periodic with period 21Ti. 2.31. Changing the sign of q. Equation (1) § 2.30 then becomes

d2y dz2 - (a+ 2q cosh 2z)y ==

o.

(1)

This is obtained if z in (1) §2.18 is replaced by iz, or if (lm+z) is t The use of capital letters to denote the modified Ma.thieu functions is due to H. Jeffreys [104, p. 459].

FUNCTIONS OF IN'l"EGRAL ORDER.

2.31]

27

written for z in (1) § 2.30. Changing the arguments in either (2)-(5) § 2.18 or in (2)-(5) § 2.30 leads to the following solutions of (1): Cesn(z, -q) =- cetn.(iz, -q)

=

ex>

(-l)nr~ (-l)rA~n)cosh 2rz

(2)

Cesn+1(z, -q) . ce sn +1(iz, -q)

= Se2n +1(z, -q)

00

!

Bg+1 1) cosh(2r+ l)z

(b2n +1),

(3)

(-I)"'! (_])rA~~'+11)sinh(2r+l)z (a 2n +1 ) ,

(4)

(-I)'1l

r=O

(-I)r

= - i se2n +1(iz, -q) =

GO

r=O

Sesn+s(z, -q)

= - i se2n +2(iz, -q) 00

= (--I)n! (-1)rB~~t+~2)sinh(2r+2)z (b2n +S) .

(5)

r=O

(2), (5) have period 7Ti, while (3), (4) have period 27Ti in z. The interchange of the coefficients A, B, and characteristic numbers a 2n +1 , b2n +1 , in (3), (4) §§ 2.30, 2.31, should be observed-see remarks in § 2.18, The multiplier (-l)n ensures that when q = 0, the functions reduce to cosh mz, or sinh mz, as the case may be. (2)-(5) may be regarded as modified Mathieu functions of the first kind for q negative.

+

+

III

CALCULATION OF

(~IfARACTERISTIC NUl\IBERS

AND COE.f.i"}"ICIENTS

3.10. Recurrence relations for the coefficients. If each series (1)-(4) § 2.17 is substituted in turn in y"+ {a-2q cos 2z)y = 0, and the coefficients of cos 2rz, cos(2r+ 1 )z, sin(2r+ 1 )z, sin(2r+ 2)z equated to zero for r = 0, 1, 2, ... , the following recurrence relations are obtained [52, 88]: aA o-qA 2 = 0 ) for ce 2n{z, q), (a-4)A 2-q(A 4+2A o) ~ ot (1) (a-4r2)A2r-q(A2r+2+A2r-2) = 0 r ~ 2.

(a-I-q)A1-qA a == 0 } for ce 2n +1(z, q), (a-(2r+l)2]A2r+l-q(A2r+a+A2r-l) = 0 r ~ 1. (a-l+q)B1-qB a = 0 } for se2n +1(z,q),

= (a-4)B 2- qB 4, = (a-4r2)B2r-q(B2r+2+B2r-2) =

[a-(2r+l)2JB2r+l-q(B2r+s+B2r-l)

0

r ~ 1.

0 } for se2n+2(z,q),

0

r ~ 2.

(2)

(3)

(4)

For simplicity the superscripts 2n, 2n+ 1, 2n+ 2 have been omitted from the A and B. The respective recurrence relations for ce 2n(z, -q), eto., Ce 2n(z, q),..., Ce2n {z, -q), ... are identical with those above.j To ensure convergence of the series in §§2.17, 2.30 it is necessary that Am ~ 0, B m ~ 0 as m ~ +00.

3.11. Calculation of a and the A, B. If q is small enough, the formulae for a given in § 2.151 may be used. In general, however, a must be calculated using one of the methods illustrated below. We adopt the normalization of § 2.21, thereby entailing coefficients which differ from those obtained using the normalization of § 2.11 [52, 88]. We commence with the function ceo(z, 8), i.e. q = 8, which is well outside the ra.nge of the formulae in Chapter II. First we derive an infinite continued fraction. From the second formula in (1) § 3.10, we have (1)

t

Note the factor 2 for A o•

: Using q or -q as the case may be.

3.11]

CHARACTf~RISTIC NU)IB}4~RS

AND

CO(4~~'.£t'ICI)"~NTS

29

Writing 1,'0 = A 2/A o, l' 2 === A 4/A 2 , then 'l'0t'2 = A 4/A o. Dividing (1) by A o and making these subatitutions, gives (4-a)v o+ q(1Jot'2+ 2) == 0, (2) 80

111

-Vo

:=:

~q/[1-1{a-ql)2)].

(3)

the same way from the third formula in (I) § 3.10, with V 2r - 2

we get so (r ~ 2)

=

A2r/A2r-2'

V 2r

== A 2r f-2/ A 2r ,

(4r2-a)v2r-2+q(v2rv2r'-2-r 1) :::::: 0, -t'2r--2 == (q/4r 2)j[1-- (1/4r 2)(a - q l'2r)

J,

(4)

which may be regarded as an alternative form of recurrence relation. Substituting r == 2 in (4) yields - V2 =

(q/I6)/[I-- k(a-qt'4)]'

(5)

and on inserting this in (3) we get -t'o

== !q/{1-!a-(q2/64)/f l -

n (a -- ql'4)]}.

(6)

Putting r = 3 in (4), we obtain a formula for 1'4' which is now substituted in (6). Proceeding thus we ultimately get the infinite continued fraction -V = ~ 6)q2 !;~6q2 !'il~~_ q2/ 16r2<;-1)2 .... (7) o l-la-l--R-a- 1--3'6a-1-6~I-a 4r 2 From the first formula in (1) § 3.10, V o === A 2/A o = alq, so (7) becomes - !q2

f.q 2

6~i1q2

2!~iq2

l a - l ---=i:!i - 1- 6 J- a- ... ' a == l - l;;[a - I - 11f 36 _ 4

(8)

which is in effect a transcendental equation for u"n- Since a, q are finite, the denominator of the general term approaches unity as r ~ +00, while the numerator tends to zero. Hence (8) is convergent. 3.12. Computation of ao forq == 8. We commence by finding an approximate value of a o by trial and error. Neglecting all members on the r.h.s, of (8) § 3.11 except the first, we get

a o = -32/(I-!ao),

+

(1) a o = 2±(132)1 ~ -9-5 or 13·5. Substituting +13-5 in the first three members on the r.h.s. of (8) §3.11 gives the following:

so

o+6q2/ (I -

:t\-a) = 0·178, ~\q2/(1-tra-O-]78) = --43-6,

-!q2/(I-!a+43·6)

~

-0·776.

}

(2)

CAI..CULATION

30

O~..

CHARACTERISTIC

[Chap. III

The great discrepancy between +13·5 and -0·776 indicates that we chose the wrong root of (1). Suppose we start with a o = -10, then to a first approximation 32 1 0·111 a o = -~5-1-625-1.278

==

-11-2_

(3)

This is numerically greater than -10, so we now use the average, namely, -10-6, and repeat the calculation. Before doing so, we shall obtain an estimate of the number of members from (8) § 3_11 to give a reasonable approximation, The general component is [q2JI6r2(r-

1)2J/[1- 4~2(a-qt'2r)J.

(4)

and as r increases, we see from (4) § 3.11 that 1'2r-2 decreases, while in (4) the term Iq1'2rl/4r2 <{ 1, provided It is large enough. Taking r == 6 in (4) §:3.11, and neglecting qt'2r/4r2, we get -1,'10 ~ (1/18)/[I+(10·6/144)J

==

0·052,

(5)

so Iqv12 //4r2 < Iqt"101/4 X 36 ~ 0-0029_ We shall presume, that for the purpose of illustrating the method of calculation, the accuracy will be adequate if (8) § :1·11 is terminated at the fifth member on the r.h.s, Thus assuming that V 12 Inay be neglected, we get [88]

r= 5

[q2/16r 2 (r - I )2]/{1- a/4r2 )

-== (1/100)/[1+(10'0/100)] == 0-00904 0, r

.:=;

4

(1/36)/[1+(IO·fj/64)-Oo00904]

r === 3

(1/9)/[I+(lO·Oj:36)-O·0240~]

=

1/[l+(IO·6/16)-OoOS746] -32/[I+(lO·6j4)--O·n:l4BO 4]

r

2

r = 1

== == == := a o ==

0·02402 0, 0-08746 0, 0-63490 4, -10·61326 O.

(6)

The trial value of 00 was - 1n-t), RO (6) differs from this in the fourth significant figure. Taking the average] gives ao "= -10-60663 o. If this iR used instead of ---10·6 and the calculation repeated, the new value' of a o to six significant figures is - t 0-6067. A more accurate result may be obtained by using this value of ao' and repeating the computation with r == 7. To seven decimal places

ao == -10·60672 92. It is apposite to remark that a formula of the type (7) § 2.15 could

t

This procedure is not always expedient.

3.12]

NUMBERS AND COEFFICIENTS

31

not be used in the present case where q = 8. Even for q = 0·8 the accuracy would be 10"".

3.13. Computation of coefficients for ceo(z, 8). From (5) § 3.12, so using (4) §3.11, with ao == -10·6 to demonstrate the procedure, we obtain

1'10

~ -0-052,

r = 5

-vs = (2/25)/[1+(10'6/100)-(2 X 0·052/25)] ~ 0·0726

r=

4

= 3

= -As/A"

=

2

(1)

(2)

-v, = (2/9)/[1+(10'6/36)-(2 X 0-108/9)] ~ 0-175 ==

r

-Alo/A s,

-v. = (2/16)/[1+(10'6/64)-(2 X 0'0726/16)] ~ 0·108

r

=

-As/A.,

(3)

0·318 = -A 4/A 2 •

(4)

-va = (2/4}/[1+(10·6/16)-(2XO-175/4)] ~

By (3) § 3.11

-vo == 4/[1+(10·6/4)-2 X 0-318] ~

1-327 = -A 2/A o•

(5)

Normalizing the function in accordance with the convention of§ 2.21, we have 1 == 2A~·+A~+A~+A:+ ...,

so

l/A~ ==

From (5), -A 2

2--t-(A2/Ao)2+(A4/Ao)2+(A6/Ao)2+ ....

== l'~27Ao,

(6)

(7)

so by (4)

A 4 == 1·327 X O·318A o ~ O·422A o.

(8)

In the same way we find that As == -O'0739A o, } As =:-: 0'00798A o, A l o = -O-00058A o'

(9)

Using these numerical values in (7) leads to I/A~

=

3·9445,

so

A o = 0-5035,

(10)

the positive sign being chosen. A more accurate value may be obtained by using a o = -10-60672 92, starting at r = 6 and working to, say, eight decimal places. Then A o = 0-5037681, so our approximate computation errs in the fourth significant figure, We can now

CALCUI~ATION

32

[Chap. III

OF CHARACTERISTIC

calculate the other A, using A o at (10), and obtain

A2 A4

~

-0,668

~

0·212

0-21197 32,

A6

~

-0-037

-0·03707 27,

A8

~

0-004

0·00400 65,

A l o ~ -0,00029

-0-66791 64,t (11)

-O-OOO~9 09.

Hence to a first approximation the first and periodic solution (cosine) of (12) y"+(-10·6... -16cos2z)y = 0, under the normalization of § 2'21, is cc o(z,8) == 0'504-0-668 cos 2z+0-212 cos 4z-

-0,037 cos 6z+0'004cos 8z-0'0003 cos IOz+... .

The graph of (13) is shown in Fig. 3. Checking the solution. To check the solution at z we have from the differential equation (12) y"(nTT)

=

=

0,

'IT, ••• ,

(16+ 10·6... )y(n1T).

(13)

nn, (14)

From (13), ce~(n1T,8) ~

4xO·668-16XO·212+36xO·037-04 >~ O'O()4-t-IOO '< 0·0003

==--==

0-388,

(15)

0-391.

(16)

and (26·6 ... }ceo(n7T, 8)

~

26·O... (O·504-()·668+0·212-O'O:~7 +U'OO4-0'OO(3) ===

The results in (15), (16) agree within 1 per cent., which is sufficient for Ilurposes of illustration. The solution may be checked at any other point, e.g. z = !1T. Remarks on the coefficients in (13). There are two salient features: (a) beyond a certain term the coefficients decrease rapidly with increase in r, (b) they alternate in sign, (a) entails the rapid convergence of the series. For 0 ~ q < :i'2, tabular values show that IAol is the largest coefficient, being 2- 1 at q = o. When q > 3·2 approximately, IA2 1 takes precedence, e.g. in (13). The relative values of some of the early coefficients in the series for ce2(z,2), se s(z,2), ce5(z, 2} are depicted ill Fig. 6.

t

The figures in this column are correct to seven decimal places [95]_

NUMBERS AND COEFFICIENTS

3.14]

33

3.14. Evaluation of a 2n for ce 2n(z,q), n > O. Writing (r+ I) for r in (4) §3.11 and multiplying both sides by q/4r2 , we get [52]

-q4V~ = r

[q2/16r2(r+I)2J/[I- (J. 2 (a- QV 2r +2>] 4 r+ 1)

(1)

q2/16r 2(r+l)2

q2/16(r+l)2(r+2)2

q2/16(r+2)2(r+3)2

l-a/4(r+ 1)2-

-1-=-a74(r-+2)2~-

l-aJ4(r+3)2 (2)

= E 2r •

1·0

o·s

Cl'z

'1=2

L

B~)

(3) 0(3)1

B1 FIG.

3

6. Relative magnitudes of the coefficionts in tho series for cCz(z, 2), sca(z, 2), ce 6(z, 2). For q = 2, tho coefficient of functional order is the greatest.

If a = a2n , the characteristic number for cc 2n(z, q), q =/:= 0, none of the denominators in the components of (2) vanish. Then the C.~'. is convergent, since the denominator of the general term approaches unity and the numerator tends to zero as r -+ +00. We shall now derive an alternative terminating e.F. to represent the l.h.s. of (I). After division by 4r2 , the recurrence formula prior to (4) § 3.11 may be written

_qv2r = 1- a 4r 2

4r

=

2

+ q/4r

2

(3)

V 2r- 2

a q2J 16r 2(r - l )2 1- 41=2 + qV2r_~/4-(r~1)2

(r

~ 2).

(4)

Now write (r-l) for r in (4), thereby obtaining a formula for -qv2r _ 2/4(r-l)2. Substituting this in the third member 011 the r.h.s, of (4), we get

-[q2J16r2(r-l)2] 4961

q2/16(r-I)2(r-2)2]

/[

1-[a/4(r-l)2]+---------------2-· qVsr - . /4(r- 2) F

(5)

[Chap. III

CALCULATION OF CHARACTERISTIC

34

Continuing this process leads to the terminating C.F. qV 2r

- 4r 2 -

a q2jI6r2(r-I)2 q2jI6(r-l)2(r-2)2 q2J64 1- --2 - - _ . - . - -- ------- - - - -- · 4r l-aj4(r-l)2-; l-a/4(r-2)2 _···+qv2j 4 (6)

Since V o = aJq by (I) §3.10, (2) §3.11 may be written !qv 2 = -(1-1a+q2/2a).

(7)

Substituting from (7) into (6) gives

_qv 2r - 1--~--F2 4r2 -

4r 2

(8)

r'

where F

2r

= q2/16r2(r- 1t2

[2/~~(r-l)2(!'-_2~)2

l-aj4(r-l)2-

l-aj4(r-2)

...

-

t;

q2/64 -1-aJ4+2a

(9)

For the recurrence relations (I) § 3.10 to be consistent, (2) and (8) must be equal, so E 2r == l-a/4r 2 - F2r , (10) and for convenience in computation we may write

G2r == I-a/4r2-E2r-F2r (r ~ I). (II) The appropriate value of a corresponds to G2r = o. To obtain a 2 , a 4 , ••• for ce 2 , ce 4 ,. •• we use (11) with r = 1,2,3,... initially. 3.15. Evaluation of a4 for ce4(z, 8). To start the computation we have to find a trial value of a4 • By (9) § 2.151, a4 = 16 when q = 0, and increases with increase in q, so let us assume that a = 20 for q = 8. III the r.h.s, of (4) § 3.14 take r = 2, substitute for iqv2 from (7) § 3.14, and we get

l-lija-[(q2/64)/(I-la+q2j2a)] === l-r\a-F4

(r == 2).

(I)

The first two members in (2) § 3.14 give E

r-J

4, -

1/9

1/36

(2)

l-a/36-1-aJ64·

Then (1) is approx_ 0·166 while (2) is approx. 0·271, so

0, = 0·166-0·271 ~ -0-105. Examination of the formulae in § 3·14 indicates that at

< 20, so we now try a4 = 19 and obtain 0·297 for (1) and 0-26 for (2), i.e. G.. ~ +0·037. Thus the value of at lies between 19 and 20_ By linear interpolation (see Fig. 7 A), at ~ 19+x = 19·26_ Using this value in (1), (2) yields 0-261, 0·263 respectively, so Gf, ~ -0·002.

NUMBERS AND COEFFICIENTS

3.16]

36

A second linear interpolation gives a4 ~ 19·253. If now, with this value of a 4 , we take r = 7 ill (2), (9) § 3.14 and repeat the computation, the value of a4 correct to seven decimal places is found to be 19·25270 51.

(A)

0·3

L~~_ _-----~O'271(2) 0·2.5

0·2

0·2

0-166(1) 0·15--------------0·15 19 20 Values of a

(B)

0·263 (2)

(1)0·262

0'262

(2) Q·l61 Values of a

19'25 FIG.

19'26

7 (A, n). Hlustrating linear interpolat ion.

3.16. Evaluation of coefficients for ce4 (z, 8). By the first recurrence relation in (1) § 3.10 Vo

=

A 2/A o == a4 /q = 19'25271/8

~

2·4066.

(1)

From (7) § 3.14, v2 = A 4/A 2

== =

-4(1-!a- q2J2a}/q -(1-4-8132+1·6624)/2 = 1·0754.

(2)

(3) § 3.14 may be expressed in the form v 2r

=

1 -(a-4r 2 ) -

q

1 -

V 2r- 2

(r ~ 2),

(3)

80

[Chap. III

CALCULATION OF CHARACTERISTIC

36 V,

= As/A, =

2-4066-2-0-9298

(r = 2)

= -0-5232, Vs

=

(4)

As/As = 2-4066-4-5+1-9113 (r

=

3)

= -0-18"21, Vs

= A10/As =

2·4066-8+5·4915

(5)

(r

=

4)

= -0·1019.t

(6)

As the computation proceeds, it becomes evident that V 2r is the small difference between two comparatively large numbers. For reasonable accuracy, therefore, it is imperative to use more decimal places than we have had hitherto. However, our purpose is merely to illustrate the method of computation, not to attain high accuracy, so we shall now evaluate the coefficients in accordance with the normalization of § 2.21_ Following the routine in § 3.13, we find that A 2 ::c 2-4066A o,

Then by §2.21

At = 2·589A o, A 6 == -1·355A o, As = O·247A o, A 10 = -0·0245A o•

(7)

+

I/A~ =

2+(A2/Ao)2+(A4/Ao)2+(A6/Ao)2+ ... +(A 10IAo)2+ ... ~ 16-392, (8) A o = 0-247. (9) 80 We now calculate A 2 = 0·595, At = 0-639, A 6 = -0-335, (10) As = 0·061, A 10 = -0-00606. The signs of the coefficients are in accordance with the remarks at the end of § 3.33_ Inserting these coefficients in (1) § 2.17, the first solution of y"+(19-2527_ .. -16cos2z)y = 0, (11)

to

a first approximation, is the periodic function

cet (z, 8) = 0-247 +0-595 cos 2z+0-639 cos 4z-0-335 cos 6%+0-061 cos 8z-0-0061 coslOZ+

t

The value of t', correct to live decimal places is 0-09971.

.

(12)

3.16]

37

NUMBERS AND COEFFICIENTS

Oheck on the solution. By (14) § 3.13, we have y"(n7T) ~ (16-19'253)y(n7T) = -3·253y(nn). (13) Now ce;(n1T) 8) = -3,85, and -3,253 ce4 (n1T, 8) = -3'9, so the agreement is of the order 1 per cent. To better this, it would be necessary to give the values of the coefficients to more than three decimal places.

3.20. Characteristic numbers and coefficients for ce211 +1 , se2n +1 , se2n +2 • These may be calculated in a similar way to those for ce2 1P using appropriate formulae derived from (2)-(4) §3.10. For ce2n+ 1 , with V 2r+1 = A~~~~l)/A~~+il), VI =

(a-l-q)fq,

(I)

-V 2r- 1 = [q/(2r+ 1 )2]/[I-a/{2r+ 1 )2+qv2r+I/(2r+ 1)2] For se 2n +1 , with V 2r +1 = B~;~~l)/B~;'+il), VI = (a-l+q)/q, -V 2r - 1 = as at (2) above. For se 2n +2 , with V 2r = B~~~~2) / B~;n+2), V2

-V2r- 2

=

(r ~ 2).

(a-4)jq,

(2)

(3) (4)

(5) (r ~ 3).

= (q/4r2)/[I-a/4r2+qv2r/4r2]

(6)

3.21. Convergence of series for cern(z, q), sem(z, q). The third relation at (I) §3.10 is a linear difference equation. Writing (r+l) for r, we get [a-4(r+ 1)2JA2r+2-q(A2r+4+A2r} = O. (1) Dividing throughout by A 2r and putting v2r = A 2r t-2/A2r, gives [a-4(r+ 1)2]v2r-Q(v2r+2v2r+ 1) = 0, (2)

V2r+2+~ =

so Then

'1.12r+2+~ -+ V

v 2r

-00

(3)

[a-4(r+l)2]q-l.

monotonically as r -+

+00,

and it is evident

2r

that V 2r cannot: (1) oscillate boundedly, (2) tend to a unique finite limit other than zero. Hence either V 2r ~ 0 or /v 2r l ~ 00. Now for convergence of the series representing ce2n(z, q), A 2r ~ 0 as r ~ +00, so v2r must also tend to zero. Consequently, as r ~ +00, one solution of (I) tends to zero, and the other to infinity. In the present instance the latter solution is inadmissible. Thus in (2), when r is large enough, V2r+2V2r ~ I, a ~ 4(r+I)2, and with a, q finite

Iv2r1 "" q/4(r+ 1)2 ~ 0

as

r~

+00.

(4)

38

CALCULATION OF CHARACTERISTIC

[Chap. III

If z is real, [cos 2rzl ~ 1, so it follows by the' M' test that the series for ce 2n (z, q) is absolutely and uniformly convergent. Moreover, it represents a continuous function for all z real. In like manner the series for ce2n +1(z, q), sem(z, q) may be proved to have the above properties. Hence all ordinary Mathieu functions are continuous for z rea], and the series may be differentiated or integrated term by term provided the resulting series are uniformly convergent. This can be proved to be true. 'I'hese conclusions apply to the series for q < o.

3.22. Convergence of series for

ce~J:)(z, 'I), se~)(z, q).

'I'he pth

derivative of ce2n(z, q) is

tiP dzP [ce2n(z, q)]

(-I)llJ cos 2rz

00

= r~ (2r)PA 2r (-I)i
} p even, p odd.

(1)

Thus the ratio of the coefficients of the (r+ 1)th and rth terms is

IU~:ll = (r~lrIV2rl = (1+lfr)PJ v2rl·

(2)

Then by (4) § 3.21,

lur+1/url -7 0

as

r -+ -1-00,

(3)

so all derivatives of ce2n(z, q) are absolutely and uniformly convergent for z real. '!'he same conclusion may he reached in connexion with ce 2n +1 (z, q), sem(z, q) by similar analysis. This, and proof of the validity of term-by-term integration, are left as exercises for the reader.

3.23. Convergence of series for Cem(z, q), Sem(z, q), and derivatives. The series are merely those for cem(z, q), sem(z, q) with zi written for z. Hence, by (2) § 2.30 and (4) § 3.21, u r +1 1

l

q

u,.. ......, 4(r+l)2

cosh(2r+2)z

cosh2rz

·

(1)

In § 3.21 we proved that the series for ce2n(z, q) is absolutely and uniformly convergent for all z real. Hence the series for Ce2n(z, q) conforms to this when z is imaginary, i.e. R(z) = o. Now when R(z) ::j= 0, from (1),

IU~:ll""'" qe2/zlf4(r+ 1)2 ~ 0

as r ~ +00

(2)

provided R(z) is finite. Hence the series for Ce2n (z, q) is absolutely convergent under this condition, By applying the' M' test it may

NUMBERS AND COEFFICIENTS

3.23]

39

be proved uniformly convergent. Hence the series is absolutely and uniformly convergent in an)" finite region of the z-plane or in any closed interval of z real. 'I'he same result may be reached for the three remaining functions, and for the pt.h derivatives of all four functions (see § 3.22).

3.24. Characteristic curves for cem(z, ±q), sem(z, ±q), q real. There are tabular values of ao, a l , ..• , as and bl , bz,..., bl in the range q = 0-40, given in Appendix 2 [95]. When these are plotted using cartesian coordinates, the chart of Fig. 8 A is obtained [88]. It is symmetrical about the a-axis. a Zn ' b2n +2, the characteristic curves for ce2n , se 2n +2 respectively, are symmetrical about this axis, but a2n +l , b2n +1 corresponding to ce 2n +l , se2n +1 are asymmetrical. Never.. theless the symmetry of the diagram is maintained, since the curves for the functions of odd order, i.e, a 2n t t , b2n +1 are mutually sym.. metrical, e.g. aa, ba. When q is positive and large enough, the curves a 2n , b2n +1 approach and are mutually asymptotic, but have no independent linear asymptote. A similar remark pertains to am' bm +1 • When q is negative and large enough, a 2n "-I a 2n +l , and b2n +1 ~ b2n +2 • It should be noted that of the corresponding pairs of functions one has period 1T, the other period 21T• Excepting the curve ao, which is tangential to the q-axis at the origin in Fig. 8 A, B, all curves intersect this axis twice, one value of q being positive, the other negative. Thus each characteristic has two real zeros in q. For the functions of even order ce2n , se 2n +2 , the zeros are equal but opposite. The zeros of ce2n+l are equal but opposite to those of se2n +l • Since Mathieu's equation has only one solution of type ce., or Re n l (q i= 0), two characteristic curves cannot intersect. Approximate values of q ~ 0 for which a is zero are given in Table 1. TABLE

1. Zeros in q ~ 0 of am' bm

eliaracteristic number

ao

bI

Value of q

0

0·88

For

1n

-

at, bl

a 2 , b3

7·5~

21·28

a""bm I 1 , 1n

>3

O·86(21n-t-l)lt

> 3, the q interval between the zeros of am and Gm+1 is 6·88(m+ 1).

t This formula was obtained by taking several terms of (I) § 11.44, putting a and solving for q.

=0

CALCULATION OF CHARACTERISTIC

40

[Chap. III

40 36

35

30

25 t:3

c..a V> ~

20

:3

~

16 15

10 9

5

4

35

that the continued fractions for computing the a (see §§3.11, 3.14, 3.15) may be expressed as the quotient of two integral functions. The denominator of this quotient cannot vanish if the value of a is a characteristic number for a periodic Mathieu function of integral

3.25J

Fig. 8 (a), regions q .> am' a m+1 ;

NUMBERS AND COEFFICIENTS

41

Hlustrat.ing variation in the parameter a in stable and unstable n. am' bm f 1 are mutually asymptot.ic and ~ ---rJ) as q ~ +00. bm , bm+1 are mutually asymptotic and-e- -00 as q-> -00 (see § 12.30). () and 'I '-:

from the recurrence relations, the coefficients Am' B m are continuous'] functions of q (see Fig, 9). 3.26. Additional comments on normalization. We are now able to. comment upon the normalization of § 2.11, where the coefficient of cosmz in cem(z,q), and that of sinmz in sem(z,q) is unity for all values of q. We shall show that under this convention the remaining coefficients are infinite for certain values of q =1= 0 [52]. Consider ce 2(z, q) whose characteristic number is a i . Then (1) § 3.10

q/a2

gives

t 4961

=

A oIA2 •

And single-valued. G

(1)

CALCULATION OF CHARACTERISTIC

42

[Chap. III

°

Now by §3.24, a 2 = for qz ~ ±21·28, so A o/A 2 is then infinite, and since As = 1, A o = 00. Also with a 2 = 0 in the second formula in (1) § 3.10 we have 2qAo+4A 2*-qA, = 0, (2) and since A o = 00, A 2 = 1, q ::/= 0, we get A, = -00, and so 011. Thus when qz ~ ±21·28, a 2 = 0, and all the coefficients except A 2 become infinite. A similar conclusion may be reached for the coefficients of any Mathieu function of integral order, excepting ceo, cet, set, se 2 • We shall now examine these singular cases. For ceo, (1) § 3.10 gives A o/A 2 = q/a o. (3) Since A o = 1 for all q, and ao never vanishes in q > 0, A 2 is always finite. It is shown in § 12.30 that a o I"-' -2q as q -)- ±oo, so A o/A2 ~ -l as q ~ +00. By employing the second recurrence formula for ceo, we can demonstrate that A, is finite, and so on. }"or eel' (2) § 3.10 gives

A t/A 3 = q/(at-I-q),

(4)

and since At = 1 for all q, while by Appendix 2 (at-l-q) =I- 0 in q > 0, A 3 cannot become infinite. Since at I"-' -2q as q -)- +00, A I /A3 ~ -1. The functions se., se 2 are amenable to similar treatment. Apart from the above exceptions, it appears-although we have not given a general proof-that for functions of order 2n, 2n+ I there are n values of q > 0, and an equal number for q < 0, for which all coefficients, save that of the same order as the function, become infinite. The positive and negative q are equal numerically. We see, therefore, that the convention of § 2.11 is inadmissible for general purposes, although it is sometimes useful when q is small, and series for ce m , se m are given in ascending powers of q (§ 2.13 et seq.) , Accordingly we shall now enumerate the advantages of the convention of § 2.21, which is based upon the orthogonal properties of the Mathieu functions. 3.27. Advantages of the normalization of § 2.21. 1°. By (2), (4) § 2.21, none of the coefficients is infinite for any q. 2°. When q -+ 0, it is shown in § 3.32 that the coefficient of the same order as the function tends to unity,'] while the remainder tend to zcro.t Hence oem(z, q) ~ cos mz, sem(z, q) ~ sin mz, and these t Except A~O) for ceo which as shown in § 2.21 is 2-l. In t.his section m /' o.

t

As stated in § 3.32, this is independent of the normalization,

3.27]

NUMBERS AND COEFFICIENTS

43

are the formal solutions of y" +ay = 0, a = m 2 , which is Mathieu's equation with q = o. The vanishing of the coefficients entails A2n/A~, A2n + 1/A1 , etc., being infinite, but this is inconsequential. 3°. The integrals at (1) § 2.21 are independent of q (real). 4°. The mean square value of the function for the interval (0-211) is the same as that of cos mz, sin mz, i.e. !.

3.30. Properties of the A, B under the normalization of § 2.21. Referring to (12) § 3.16, we see that (a) beyond a certain term the 1·0 0·8

0·2

A(2) 10

-0,6

-0·8 FIG.

9. Illustrating A~2>, ..., A\~l as Ringle-valued continuous functions of q ~ When q --. O. A o o: 'I» A z -~ I, At C~. q, AI cc q2, and so on.

o.

A decrease rapidly in numerical value with increase in r, (b) after the term in cos 4z, they alternate in sign, (c) IA 4 \ is the largest, but this depends upon the value of q, e.g, IA 2 > /A 4 / when q = 9. The values of the A, B vary with q, and by § 3.25 the variation is continuous in q ~ o. This is illustrated in Fig. 9 by data from [52, 95]. In the range 0 < q ~ 40, tabular data show thatA~2), A~3), A~4), A~6), B~4), B~S), B~6) each have one zero in q. B~3) has a zero in q > 40. 1

44

CALCULATION OF CHARACTERISTIC

[Chap. III

1°. Coe.lficient8 in series/or ceo, ce1 , se1 , sese For 000, (1) § 3.10 gives

A,/A o = ao/g. (1) From Fig. 8, ao is negative in q > 0, so the r.h.s, of (1) is negative also. By § 3.27, A o, As are finite, 80 neither has a zero in q > o. From the first two relations in (1) § 3.10, we get A.a/A 2 = [(a o-4)/q]-2Q/ao. (2) Thus A 4/A 2 will not vanish in q > 0 provided a o(ao- 4) > 2q2 . Tabular values show that A../A 2 does not vanish in 0 < q ~ 1600 [52]. Beyond this range we take ao '" -2(Q_ql) from (7) § 12.30 and find that the above inequality is. satisfied. Hence A 4 has no zero in q > o. Similarly it Dlay be shown that the remaining A have no zeros in q > o. This conclusion is valid for the coefficients in the series for ce1 , sel , Be2 • It is confirmed up to q = 1600 by tabular values in [52]. 2° • Real zeros

0' AC2n) ~

2r'

A(2n+l)

2'"+ l '

B(2n+l) BC2n+2) 2r+ l ' 2r* 2

in q

>

0, n >. 0

}'or celn, (1) § 3.10 gives

A 2/A o = a2n/q. (1) When n > 0, a 2n vanishes once in q > 0 (see Fig. 8), and since by § 3.27 A o is finite, and by (1) above it has no zero in q > 0, A 2 = 0 when as" = o. With q large enough, by (7) § 12.30 a2n ' " -2q+(8n+2)ql, 80 AI/A o ~ -2 as q ~ +00, and A o never vanishes.in q > o. From (I) §3.IO we obtain A 4/A 2 = _(2q2+4a2n-a~n)/qa2n. (2) Then at the tuo intersections of 2q2+4a21l-a~n' and the curve a 2n (q

>

0), we get

A t/A 2

= o.

(3)

Hence At vanishes twice in q > o. Proceeding thus we find that As vanishes thrice in q > 0, and so on. The coefficients for other functions may be treated in a similar way. The zeros of some of the A ill ce 2 are given in Table 2. The second zero of A~2), the second and third of A~2), and all four of A~2) lie beyond q = 1600. Tabular data indicate that the coefficient of equal order with the function has the smallest zero in q.

2 ri-« zero in

TABLE

I

Coefficient q (approx.) A~) I--~~' , A~) 373·6 A~~u A~~)

I

1352·8

> 1600

45

NUMBERS AND COEFFICIENTS

3.30]

Although we have not given a general proof, under the normalization of§2.21 it appears that ifn > 0, A~~n), A~~'t11), B~~+il), B~~'t~2) each has r real zeros in q > o. These correspond to the infinities under the normalization of § 2.11 (see § 3.26). It may be shown that when r = 0 none of the above coefficients has a zero in q > o.

3.31. Numerical check on § 3.30. The values of q at the intersections of the straight lines or curves for a null numerator, e.g. as in 20 § 3.30, where 2q2+4a2n-a~n = 0, and the appropriate characteristic curves am' b,n (Fig. 8), were calculated for several functions and are given in Table 3. They are ill satisfactory agreement with the zeros in q of the coefficients A, B computed by interpolation from tabular values. The limited range of q in the tables permitted checking of the first zeros only. TABLE

3. Approximate zeros in 0

-

A~)

-------m

q

4-r-

< q~ B~)

for cem(z, q)

I

I

--2 3 21·28 15·6

5

17·4

20·7

40 of A~), B~) for sOm(z, q) --~--

3

4

5

6

-:;..- - - -31·4 - -31·8 -40 37·3

3.32. Behaviour of coefficients as q ~ o. From (1) § 3.10 for cet(z, q}

(I)

and since A 2 is finite, A o ~ 0 as q ~

o.

Also

(a 2 - 4)A 2 == q(2A o+ A . ).

(2)

Substituting for a 2 from (5) § 2.151, and for A o from (1) into (2), gives [i52q2+ 0 (q4)]A 2 == .~q2A2+qA., (3) 80

when q is small

A4

~

Again by (1) §3.10 with r

-l2qA 2 -~ 0 as q -» O.

=

(4)

2,

(a 2 - 16)A 4 == q(A 2 + A 6 ) ,

(5)

and by (5) § 2.151 and (4) above

-h A 2[-

12+ 0 (q2)] = A 2+A 6 •

(6)

Thus when q ~ 0, A 6 ~ 0, and so on for the other coefficients. Now using (2) § 2.21, it follows from above that A~) ~

1

as q ~ O.

(7)

CAI~CULATION

46

OF CHARACTERISTIC

[Chap. III

In a similar way it may be demonstrated that in (1)-(4) § 2.17, 1, B~) ~ 1 when q ~ 0, while all the other A, B tend to zero. The latter result depends upon the recurrence relations, and is independent of the normalization, A~n) ~

3.33. Form of coefficients when q is small. By solving Mathieu's equation as shown in § 2.13 et seq. or alternatively, the following results may be derived [215]: A(O) 2r

(-I)r[ __ ~ tr - 2r(:~r+4)tr+2+0(tr-t4)]A(O} (r!)2 [( 1"-t- I ) !]2 0 ,

==

(r ~ 1);

t == l q 4

(1) AU)

-

2r+1 --

(1)r[ 1 tr r! (r -t-Tf!

+[(r +r 1) !]2 tr+ + 4(r 1

1 1)! (r

r 2 + 2) ! t + + ·

+ O(tr+3)JAil); A(2) o

A (2)

= [t-~t3+ 1363 t5+ OW )] A (2). 3 216 2 , (-I)r[

-

(-1)'[

2r+2 B(l)

2r-ll -

(3)

t,+r{47r 222r+ 247) tr-t 2+ O(tr+4)]A~2). (4\ rl (r+ 2)! 18(r+2)! (r+3)! 2' f

=

~ __

2+

1

r! (r+ I)!

t'--- _ r

_tr-J1+

[(r+ 1)!]2'

I

4(r--I)! (r+2)!

tr-t2+

+ OW+ Bi1>; 3

B(2) .

==:

(_1),[ __2_

r! (r+ 2)!

2r+2

(2)

)]

t' _ r(r+ 1)(7r+23) tr-t2+0(tr+4)]B!2). 18(r+ 2)! (r+ 3)!

(5)

(6)

2

Either by obtaining formulae similar to (1)-(6) or by using recurrence relations, and the expansions of a in terms of q, it may be deduced that if q is small enough, A~;"\ . 4~;~~_t-ll), IJk~l_Vll), 13~;~.~2), arc O(qll- r) 01' o(/r--II) according as n 5 1". '1'0 illustrate this point, \\'C find for ct"£J(:' q) that when q is small enough, the recurrence relations givp A(5)

1

,-..J

-

A (5) 7

00

Since

~

'"

r.:::O

.~q2A(5)

A(5)

__

:3H4 r-J

-

[A 2r+l (5) ]2

-

3

&'

_

~IA (fJ) ~4 '1

5'

~ _~_qA(5) 16

-

5'

A~5) ~ 1, u

and

I , we obtain

A (5) 5

~

13

I - _ _.(12 460~

1

+O(q4).

(7)

3.33]

47

NUl\IBERS AND COEFFICIENTS

From [2M] we get the general formulae (1' ~

0,111

>

0),

(m ) (l)r 'In! tr A ",.+2r ~ "r. ( 'In +)' r. '

A
and

1""-.1

m-2r -

(8)

(1n-r-I)!t r r! (m-I)! ·

(9)

The results in (7) agree with those obtained from (8), (9). When q is not too large, we infer from (8) that the A of order greater than m are alternately negative and positive, while (9) shows that all those of order less than m are positive. A~) is positive. These statements are borne out by tabular values, from which they are seen to be invalid when q exceeds a certain value [95].

3.34. Behaviour of coefficients as q ~ +00. We use the recurrence relations in § 3.10, and a -2q from § 12.30. Then for ce 2 /P aA o-qA 2 == 0, (1) t'w'

+00.

(2)

(a-4)A 2-q(A 4+2A o) = 0, substituting for a, A 2 ,

(3)

A 2/A o ~ -2 as q ~

80

Also 80

-4A o+A 4+2A o = 0,

giving A 4/A o ~ 2 as q ~

+00.

(4)

In general we find that A~;n)/Ab2n) ~

(-I)r2

as q-)-

+00.

(5)

By (2) § 2.21 it follows that for n ~ 0, all the A(2n) ~ 0 as q ~ +00, in a manner compatible therewith, Similar analysis yields the same conclusion for the A(2n tl), B<m). The results corresponding to (5) are, for n ~ 0, A 2r +1/A1 ~ (-I)r(2r+I); B 2r +1/B1 ~ (-I)r;

B 2r +2/B2 -+ (-l)r(r+l).

(6)

Numerical ill'ltstration. When q = 1600, from [52], A 2/A o ~ -}·95 for ceo, and -1·75 for ce 2 , as against -2 as q ~ +00. Using a ~ -2q+(8n+2)ql from § 12.30 in the recurrence relations, we obtain -1·95, -1·75 for q = 1600. For eel' As/AI ~ -2·85, and with the better asymptotic form of a we get -2·85. As q ~ +00, the ratio ~ - 3. The accuracy of the values calculated in the above way decreases with increase in the order of the function. A~~n) as a [unction. of q. By § 3.33 it is O(q1J ) in the neighbourhood of q = 0, with p ~ 0; by § 3.30 it has r real zeros in q > 0; and from

48

CALCULATION OF CHARACTERISTIC

[Chap. III

above it tends to zero as q ~ +00. Thus we surmise that in general is an alternating non-periodic function of q which ultimately tends to zero asymptotically. The ~me conclusion pertains to the other coefficients. A~;n)

3.35. Behaviour of coefficients as n -+ +00. When a is very large, positive, and much greater than q, a 2n " " 4n 2• From (1) § 3.10 AojA a = qja

"-I

qj4n 2 -+ 0

as n

~

+00.

(1)

Hence A o ~ O. From (1) § 3.10 A ajA 4

=

q/(a-4) -+ 0

as n -+

+00.

(2)

Hence A 2 4- O. In like manner it can be shown that all the A -+ 0 except A~~n) which tends to unity. '!'he same conclusion applies to the coefficients for the other functions of integral or fractional order.

3.40. Solution of y"+(a-2qcos2z)y == 0, when q is negative imaginary. In the problem of eddy current effects in § 17.10, q takes the form -is, s being real and positive. In numerical work it is preferable to work with q = +is, and then obtain the solution for q = -i8 by writing (t1T-Z) for z. The value of a is real or complex according to conditions. For q III oderate, the series for am' bm in § 2.151 are convergent, and may be used for computation. The series for a 2n, ban proceed in powers of q2, while those for a 2n +1 , b2n +1 proceed in powers of q. Hence a 2rp b2n are real, but a 2n +1 , b2n +1 are complex. When 8 > 8 0 , 80 depending upon n and the function, a 2n , b2n are complex. In the sections following, the method of calculating characteristic numbers and coefficients is exemplified. 3.41. Calculation of a o and coefficients in the series for ceo(z, -O·16i). We take 8 = 0·16 in (1) § 2.151, so with q = is _ 1 2 7 4 29 6+ aO-28+12Ss-t-23048 ...

(1)

= 0'0128+0·00003 584+°.0821

=

0'01283 6

to six decimal places.

(2)

By (1) § 3.10, Vo =

so

As/A o = aojis

= -0·01283 6ijO·16 = - 0'08022 5i,

A 2 = -0'08022 5iA o

o

(3)

49

NUMBERS AND COEE'}4'TCIENTS

3.41]

From (2) §3.II, V2

= (ao-4-2q!vo)/q

=

(4)

[0·01283 6-4+(0·32i/O·08022 5i)]/O'16i

= -0,01011 2i.

(5)

'rhus A,

=

-0,01011 2iA 2

=

-0·01011 2 X 0·08022 5A o

=== -0,00081 lAo'

(6)

Hence cCo(z, +0'16i) ~ A o[ I - O'0802i cos 2z-0'0008 cos 4z+ ...], and by writing (11T-Z) for z on the r.h.s, of (7),t we get

(7)

ceo(z, -0'16i) ~ Ao[I+O·0802icos2z-0·0008cos4z+ ... ].

(8)

An approximate check on the A may be obtained by using the formula (6) § 2.150, namely,

ceo(z,q)

~ 1-(~q-l~8q3+···)c08 2Z+U2q2_u552Q4+ ...)C084z-

v4

(1l~2 q3_ ...)C08 6z+ (73~28 q4_ ...)C088z+... .

= A 6/A 4 may be calculated

form

V2,.

=

(9)

by aid of (4) §3.11 expressed in the

(a-4r 2 - q/v2r _2)/Q (r

~

2),

(10)

but the numbers involved are such that unless each is given to more decimal places than we have used above, a large error will occur. It is preferable to write (10) in the form V2r-2

=

q/(a-4r2-qv2r ) ,

(11)

and as an approximation to assume that V 2r vanishes to an adequate number of decimal places. Then with r = 3 in (11), we get v~ !:.= O·16i/(0·0128-36)

=

-0,00444 6i,

(12)

80

.As = -0,00444 6iA 4 = 0·00444 6i X 0·00081 lAo,

=

0·00000 36iA o•

(13)

To the same number of decimal places this value is obtained from (9), i.e, -q3/1152, with q = O·16i. The value of As in (7) is given ·at (13), but in (8) its sign will be negative. t When the A are either real or imaginary, but not complex, (8) is obtained by changing the sign of i in (7). 4M1

H

CALCULATION OF CHARACTERISTIC

50

[Chap. III

3.42. a 2 and the A in the series for ce 2(z, -0·16·i). From (5) §2.151 we have, with q = is, _ 4 a2

-

5 2 763 4 10 02401 e+ -128-13824.1-796262408 ...

(1)

= 4-0·01066 667 -0·00003 617-0.0621

(2)

= 3·98929 7.

Proceeding as in § 3.41, we find that ce 2(z, 0·16i)

= A o[ I -

25i cos 2z-0·328 cos 4z+0·00164i cos 6z+ ... ] (3)

and ce2(z, -0·16i)

= A o[I + 25i cos 2z-0·328 cos 4z-0·00164i cos 6z+ ... ]. (4)

When q is moderate and real, IA 2n l is the largest coefficient in the series for ce2n(z,q). This is true if q is moderate but imaginary, as is exemplified by the A in (4).

3.43. Calculation of a o and the A in ceo(z, -4·8i). Here 8 = 4·8, which is too large for (1) § 3.4i to give an adequately accurate result. Instead of using this formula, we obtain an approximation to ao by means of two asymptotic formulae. The first is (1) § 11.44, and for brevity we designate the r.h.s. by I:n. The second is in reference [56], and takes the form 4m

-t 5(2)!ql(m+ l )e- 4ql(l + c1 q- l+ c2 q- l + ... )::= ~m'

2 -, bm+1-am "", 111,.

(1)

7T

where C1 == -0·225, -0·849, --1·765, - 2·925 correspond respectively to m = 0, 1, 2, 3, and the value of Iql is large enough for terms involving c2 , c3 ' ••• to be neglected. In reference [57] it is explained that ~n-!~m is a suitable approximation to a for starting a computation, provided 8 is large enough, We take q = is, 8 real and positive, and finally write (!7T-Z) for z in the ceo(z, 4·~i) series, thereby obtaining that for ceo(z, -4-8i). In the formulae for rm , ~,n we take ±i = e±l1Ti, i±i == e±t1Ti, and so on. Values for m == 0 are given in Table 4 [57]. We commence the computation by assuming that the tabular value of a o for 8 == 4·8 is an adequate approximation, and apply the recurrence relation (4) § 3.11 in the form Vir-I

t

q/(a-4r2-qv2r )

(r ~ 2).

(2)

NUMBERS AND COEFFICIENTS

3.43]

TABLE 8

1·6 3·2 4·8 6·4 8·0 9·6 11·2 12·8 14·4 16·0

l~

I

4

~o

1·526240- 1·384916, 2·2688!) 6 - 3·8fi346 8,: 2·83878 8 - 6·48850 8i 3·31939 2-- 9·211468i 3·74247 2-11'99053 6i 4,124860-14,80972 o: 4·47642 8-17'65937 2i 4,803600-20,53316 8'i 5·11084 8-23·42672 Oi 5·40142 4-26·33682 Oi

51

l~--l~o

::::: a o

- -()'5fi --- 0·68; 1·80624 0 -- I·U4491 6i -0·2848 f- O·2132i 2·4112;) 6 - :J·960068i + 0·0356 + 0·1528i 2·82098 8 -- 6·£>6490 8i +0·07016 -f-0'02744i 3·28431 2- 9·22518 8i -f- 0·03416 8--0'01775 6i 3·72538 8--11'98165 8i -~-0'00656 0--0-01960 8; 4-12158 0--14'79991 6i -0,00432 4-0'0107] 2; 4·478£>90-17·66472 8i -0,00574 4-0'()0336 8i 14'80647 2-20'53148 4i - 0·00393 2 0·00033 Oi 5·11281 4 -- 23·42688 5i -0,00178 0-f-0'OOI48 8i 5·40231 4- 26·33756 4i

+

To illustrate the procedure, high accuracy is waived: we take V 6 and a o = 2·S21-6·565i in (2), thereby obtaining V4

=

4-Si/{2·821-6-565i-36)

=

-(O-0276+0-139i).

Applying (2) with r V2

-4·Si/(33-179+6·565i) (3)

2, we have

= 4-Si/[2·S21-6·565i-16+4-Si(0-0276+0-139i)]

= = Also

=

=

= 0,

Vo

-4-Si/(13·85+6·433i)

-(O·132+0·285i).

= 2q/(ao-4-qv2 )

(4) (5)

= 9-6i/[2·821-6·S65i-4+4·Si(O·132+0·285i)] = -9·6i/(2·547+5-93Ii) = -(1·367+0·588i).

(6)

We can now calculate the coefficients A 2 , A 4 , ••• in terms of A o, but before doing so, we shall investigate the accuracy of the assumed value of ao. 3.44. Closer approximations to ao and the v. The first recurrence relation in (I) §3.10 may be written L = qvo-ao = 0. (I) Thus if a o and V o are calculated very accurately, the l.h.s. of (1) should be correspondingly small. Moreover, the magnitude of L will be an index of the error incurred due to the assumed value of ao and in taking Ve = o. The numerical results in § 3-43 are not accurate enough to permit the calculation of L_ However, we can show the method of obtaining a closer approximation to a o and the v symbolically.

CALCULA1.'ION

O~'

CHARACTERISTIC

[Chap. III

Since the v are functions of ao, by (1) we have, with q constant, L(ao) = qvo-ao'

(2)

Then if Sao is small enough, by Taylor's theorem

L(ao+~ao) ~ L(ao)+~aoo~(ao).

(3)

(lao

We presume that L(a o+8ao) ~ 0, so we have Newton's well-known approximation

Sao

= L(ao)/[ - o~~:ol

(4)

Our next step is to derive an expression for -oL(ao)/Bao in terms of VO' v2 , v" .... Then from (2)

BL __ 1 o(qvo) - Ba o - ca • o V o = 2q/(a o- 4- qv 2 ) ,

Also, by (2) §3.11, 80

oVo

(6)

-2q 0 ( .------2 a - 4- qv2 ) , (aO-4-qv2 ) oao o

Ba o

_ o(qvo) = tV~[l- B(QV 2 ) ] Bao Bao

and, therefore,

(5)

Again, by (2) § 3.43,

.

(8)

v2 = q/(ao-16-qv4 ) ,

(9)

o(qv2 ) = V~[l- O(qV4 ) ] . Bao Bao Substituting the r.h.s. of (10) into that of (8),yields 80

_

_ o(qvo)

Ba o

In general

= !V~[I+V~{l- O~qV')}]. oao

- o(QV2r- 2) =

Bao

V~r-2[ 1- O(QV2r ) ] Bao

(7)

,

r:» 1

(10)

(11) (12)

so by inserting the expression for -o(qv4}/oaO in (11), and repeating the process, we are led to o) _ 1 2[1+ 2+ 2 2+ 2 V42 V82+ ] -o(qv -- "!vo V2 v2v, V:a •••• Ba o Accordingly, by (5), (13)

(13)

;:r---

ei. 2 2 2+ V 2 2 V42+ ••• ] -oa - = 1+!"1[V O+VOV 2 OV:a

(14)

= 1 + 2~1l[A~+A~+A:+ ...].

(15)

o

o

NU~[BERS

3.44J

H~oo

- 2AX fJL = 2AX+ Bao

53

ANI) COEJ4'FICIENTS

i

r=l

~

f

A~r = !W ce~n(z,q) dz.

(16)t

o We calculate L from (2) using adequately accurate values of vo, a o. Let the result be L = xo+iyo. Now calculate -oL/oao from (14) using the v. Let -oL/8ao = x1+iYl. Then from (4)

Sao ~ (xo+iYo)/(x1+iYl) = x 2+iY2' (17) so a o+3ao ~ (2·82098 8+x 2 )- (6-56490 8-Y2)i, (18) which is a closer approximation to the characteristic number than that in § 3.43. This new value of ao is used to recalculate the v, as in § 3.43. The process may be repeated, but the number of decimal places required for increasing accuracy ,,,iII increase at each repetition, owing to decrease in the real and imaginary parts of L_

3.45. The series for ceo(z, --:t·8t). For this we use the v calculated in § 3.43. Then

A 2 = voA o, A 4 = vov2A o, and we find approximately that

As

=

V OV 2

v4 A o,

A 2 = -(1'367+0.5~8i)Ao, } A 4 = (O·OI3+0-468l)A o,

(1)

As = (O-065-0-015i)A o· Thus to a first approximation, we obtain

ceo(z,4·8i) = Ao[I-(1-367+0-588i)cos2z+ +(0-013+0-468i)cos 4z+(O-065-0-015i)cos 6z+

].

(2)

Writing (!w-z) for z in (2) leads to ceo(z, -4-8i) = Ao[I+(1-367+0-588i)cos2z+ (O-OI3+0-468i)cos 4z- (O-06~-O-015i)cos6z+

]_

(3)

+

It may be remarked that (3) is not (2) with the sign of i changed (see footnote to § 3.41). 3.46. a 2 and the A for ce2(z, -4-8i). It is found by computation [57] that when 8 > 1-4688 , ao is complex, and its conjugate is a 2 By applying § 3_44 to the value of ao in Table 4, we obtain a o = 2-82120 8-6-56266 8i, (1) as = 2-82120 8+6-56266 8i_ (2)

-t

so

t ~

This has not been equated to unity because q is imaginary (see § 2.21). No formal proof of this has yet been published,

64

CALCULATION OJ4' CHARACTERISTIC

[Chap. III

The v for ce 2 are obtained from those for ceo by merely altering the signs of the real parts--a procedure whose validity is easily established. Thus from (3), (4), (tj) § 3.4:}, we get

ro == 1·:J67 - O·588i,

} (3)

O·132-0·285i,

'1'2

=

1,'4

== O·0276-0·139i,

these data being approximate for purposes of illustration. The A Inay be calculated from formulae .at the beginning of § 3.45, but they can be found immediately by changing the signs of the real and imaginary parts in alternate A in (I) § 3.45. Thus for ce2 we have A 2 == (1·367--0·588i)A o, A 4 == (O·OI3-0·468i)A o,

)

(4)

As = -(O·065+0·015i)A o·

Hence we find that ce2 (z, 4·8i )

=

A o[ I + (1·367-O·588i)cos 2z+

+(O·OI3-0·468i)cos 4z-(O·065+0·015i)cos 6z+ ... ],

(5)

and, therefore, writing (!7T-Z) for z, ce2(z, -4·8i) = Ao[l-(1·367-0·588i)cos2z+ +(O·OI3-0·468i)cos 4z+(O·065+0·015i)cos 6z+ ... ].

(6)

3.47. Characteristic numbers for other functions when 8 is positive. When 8 > 8 0 > 0 and the charaoterietio numbers are complex, numerical values indicate the following results [57]:

ao is conjugate to a 2 , " al " bt , as" " b s' " b2 " b4• If 8 is large enough, numerical values indicate that [57]: a o "" bl' as " - I b4 , !(aO+b 1 ) "" r o, !(a s+b4 ) "" r 1 , bl-aO "" ~o, b4 - aS "" ~l·

See § 3.43 for meaning of symbols.

55

NUMBEltS AND COEFFICIENTS

3.47]

It may be remarked that when 8 varies fro III 0·16 to 1·44, a o varies from 0·012836 to 1·689280, whilst a 2 varies fro 111 3-989297 to 2-481324_ Thus the variation in (ao+a z) is from 4-002132 to 4-170604, so (ao+a z) is in the neighbourhood of 4 when 8 is in the range 0-16 ~ 8 ~ 1-44, i.e, the two a are real. This is readily explained by aid of (I) § 3.41 and (1) § 3.42. From these formulae when s is moderate

4+(-~-~-)82+(_? 2 12 12~

(ao+ a", ) = ..

4+ l~s2-13~24 8

=

4

-

._~-~~ -)84 -

13824

•• _

(1)

(2)

••• ,

so the dominant term is 4. Numerical data show that as 8--)+ 1·4688... , a o ~ (2-0), and a 2 ~ (2+0)_ Since the series for a o, a z at (1) § 3.41, (1) § 3.42 proceed in powers of 8 2, all the terms are real. The fact that a o, a 2 are both complex when s > 1·4688... indicates, therefore, that the series cannot hold then, i.e. they appear to be divergent.

3.50. cer, cei, ser, sei functions. In § :1.40 et seq. we have seen that cem(z, -is) is a complex function of z, so we nlay put

cem(z, -i8) == cer

(1)

1I l z + i c e i mz.

Writing the complex coefficients in polar form, we take ~4~n) === IA~~l' lei~p. 00

Thus, if ce2n (z, +is} =

l

r=O

A~;n) cos 2rz,

by (2) § 2.18 we have

00

ce2n (z, -is) = (-l)n l

r=O

=

(-1)"A~~n'cos2rz

00

(-l}n l (-I)rIA&~n)leiCPaf'cos2rz

(2)

r-O

.= (-I)nL~ (-I)rIA~~n)lcos1l2rco82rz+ +i r~o(-1 )rIA~~n)1 sin 1l2r cos 2rz],

(3)

so <X)

ceranz

=

(-l)n l (-I)rIA~n)lcosq>2rco82rz

=

(-l)nIA~2n)1 {1-lvolcos'P2cos 2z+

(4)

r=O

+ Ivova[cos rp.

C08 4z-lvovav.ICOSrp6 cos

6z+ ...}, (5)

56

NU~IBERS

AND COEFFICIENTS

[Chap. III

and 00

cei2n z = (-I)n ~ ( -1)r IA~~n) [sin ~2r cos 2rz

(6)

r-O

=

(-1)11IAh2n ) I {1-lvolsfn f1>2 cos 2z+

+

f·v ov2lsinq>4 cos 4z-lvov2 v41sin (j)6 cos 6z+ ...}. (7) These series are absolutely and uniformly convergent ill any finite region of the z-plane. The series for cer 2n +1 , cei2n +1 , ser m , seim may be found by procedure akin to that above. TIle modified functions Carm , Ceim , etc., are derived from the foregoing by writing iz for z.

3.51. Bessel series for cer, cei, ser, sci. \Vriting k == (is)l == itl in (2) § 8.20, using the complex A from § 3.50, and the relation 12r{u ) = (-I)r J2r (i u ), we obtain ce2n (z , is) == ~~2n(~:.i8) A(2n) o

~

~

A 2r (2n)j; i2i i l sin z) 2r\ •

(1)

r=O

Writing (!1T-Z) for z gives ce2n (z , -is)

==

(-l)n

~e2n.{~,i8) ~ A(2n)J, (2i1lcosz) • A(21l) ~ 2r 2r o

(2)

,'=0

By [202, pp. 121, 122], we have J 2r(i i u ) == ber., u--1-i bei 2r 7t == M2r(u)ei82,.(U). (3) Let ce2n (O, is)jA o == E2 n ei t/l ;ha, and use (3) in (2) ; then we find that ce 211 (z,

--is) =-= (-I)nE2n

f

~

\A<2n)IAt 2r 2r ei(82rtCP~r1.p2") ,

(4)

r=O

(5)

so

and

cei2n z

=

00

(-1 )1tE21l, ~ IA~n) IM2r sin«(l2r+cp2r+tP2n),

(6)

r=O

the argument of M, (I being 2l cos z. . These series have the same convergence properties as in § 3.50. Series for the other functions may be derived in a similar way, while those for Cerm , etc., may be obtained if iz be written for z. By replacing k~ in (3) § 2, Appendix 1, with -ikf, it will be found that when the fundamental ellipse tends to a circle, Cer, Cei, etc., degenerate to constant multiples of the ber and bei functions,

IV GENERAL THEORY: FUNCTIONS OF FRACTIONAL ORDER: SOLUTION OF EQUATIONS

4.10. Introduction. The discussion in this and in the next eight sections applies to any linear differential equation of the second order with single-valued periodic coefficients, e.g. (2), (5) Chapter I, these being Mathieu's and Hill's equations respectively. Let Yl(Z), Y2(Z) be allY two solutions which constitute a fundamental system. The complete solution is (1)

A, B being arbitrary constants. If we write (Z+l7) for z, then Yl(Z+l7), Y2(Z+11) are solutions also, 1T being the period of the coefficients. In accordance with the theory in. [49], we have Yl(Z+1T) Y2(Z+1T)

= =

<Xl Yl(Z)+ CX 2Y2(Z),

(2)

/31Yl(Z)+P2YS(Z),

(3)

where ~l' ~2' PI' P2 are constants determinable from the conditions at, say, Z = o. It is convenient in this discussion to choose the initial conditions YI(O) = 1, Y~(O) = 0, Y2(O) = 0, y;(O) = 1. Substituting the first and third of these in (2), (3) yields = YI(1T),

<Xl

PI

=

Y2(1T).

(4)

Differentiating (2), (3) with respect to z, we get Y~(Z+l7)

and

~lY~(Z)+(X2Y~(Z),

(5)

Y~(Z+l7) = PlY~(Z)+P2Y~(Z);

(6)

=

so insertion of the second and fourth conditions leads to (X2

= Y; (11),

PI = Y~(1T).

(7)

If Yl were an even function of z, and y" an odd one, on writing z = -17 in (3) we should get Y2(O)

Since • Hl

PI =1= 0, it

=

0

= PIYl(1T)-P2YS(1T) =

= ~I' i.e. YI(1T) = 1/;("').

follows that

~1(CXI-fJ2).

(8)

(Xl

J

(9)

GENERAL THEORY: FUNCTIONS OF

fj8

[Chap. IV

4.11. Relationship between y(z) and Y(Z+1T). From (1)-(3) §4.10 we have

Y(Z+1T)

==

AYt(Z+1T)+BY2fz+1T)

==:

A{a:l Yl(Z)+~2Y2(Z)}+ B{,81Yt(Z)+,s2Y2(Z)}

=

(Aa:t+B,8t)Yt(Z)+(Acx2+B,82)Y2(Z).

(1)

(2)

Thus, if cp, a constant, can be found such that (A();l+ B fJl ) == rpA, and (A Cl:2 + B,82) == ~B, then by (1) § 4.10, (2) above may be written in the form (3)

Y(Z+1T) == cpy(z).

This necessitates

-AlB == fil/(~l-'f», and ACl:2+B(P2-CP) == 0, or -AlB == (,82-cp)/Ot2. Accordingly, if A, B are non-zero, we must have A«();1-q»+B,81

== 0,

or

(al-'fJ)(fJ2-~) == a2fJt'

or

~2_(Cl:l+fJ2)q>+(alfJ2-a:2fJt)

(4) (5)

(6)

=- O.

(7)

Since £Xl" •• ' fJ2 were determined in §4.10, the two values of q> which satisfy (7) may be expressed in terms of Yl(1T), Y~(1T), Y2(1T), Y~(1T).

4.12. Introduction of the index p,. Let q> = eIJ.'", where J.L is a number dependent upon the parameters in the equation, i.e. a, q in (1) § 2.10. Also take ep(z) == e-PZy(z), both of these being definitions. Writing (Z+1T) for Z gives ep(Z+1T} = e-ft(Z+7T)Y(Z+7T) (1) == e-,..,(Z+7Tkn!(z),

==

by (3) § 4.11,

e-ItZy(z)

==

4>(z).

(2)

Hence q,(z) is periodic in z with period 'TT. Since y(z) is a solution of the type of differential equation under consideration, it follows that ep.zeP(z) is a solution.

4.13. Complete solution of equation. For our specific purpose we fix attention upon the equations d2y dz 2 (a-2q cos 2z)y = 0, (1)

+

and

d 2y dz2+ [a- 2q.p(z)]y = O.

(2)

.p(z) is an even, differentiable] function of z, periodic therein with

t

See footnote on p. ] 27.

~.13]

FRACTIONAL ORDER: SOLUTION OF EQUATIONS

period

'IT,

59

00

C.g.

.p(Z)

=

!82r cos 2rz.t r-1

By virtue of periodicity

1T,

we

take 00

eP(Z) =

!

C2r e2rzi ,

(3)

C2re-2rl:i.

(4)

r=-~

00

eP(-Z)

and

= I

r=-~

Then by §4.12, if JL = a+ip, (x, P real, eJ.Lzt!>(z) is a formal solution of (1) or of (2). Since both of these equations are unchanged.j if -z be written for z, e-p,zq,( -z) is an independent solution, provided either ex =f. 0, or when 0: = 0, fJ is non-integral, Hence the complete solutions of equations of type (I), (2) Inay he expressed in the form 00

y(z) == Aep,z

!

r:::. -00

C2r

e2rzi

+ Be

00

!

-JLZ

r·:-OCJ

C2r

e- 2rz ; ,

(5)

A, B being arbitrary constants. The above analysis is based 011 [215]' 4.14. Stability of solutions, z real. (a) A solution is defined to be unstable if it tends to ±oo as Z~ +00.

(b) A solution is defined to be stable if it tends to zero or remains bounded as z -)- +00. (c) A solution with period 'IT, 21T is said to he neutral, but Inay he regarded as a special case of a stable solution. Cases (a), (b) are non-periodic.§ In (5) § 4.13, since the sigma terms are both periodic in z, the stability depends upon eft Z , i.e, upon fL. It may, in general, have any real, imaginary, or complex value. If real and positive> 0, eF ~ +00 with z. Hence the A member of (5) §4.13 tends to ±oo as z ~ +00, while the B member approaches zero. Thus the first part of the solution is unstable, but the second is stable, so the complete solution is unstable. If f' is real and negative, instability arises from the B member. When p, == ifJ, fJ non-integral (5) §4.13 gives the stable solution 00

y(z) == A

If

f3

!

Cz, r==-IX>

e<2r+fJ>zi+ B

co

!

r:.--IX>

C2r

e-(2r+{J>zi.

(1)

is a rational fraction pis, p and s being prime to each other,

t In § 6.20 the 82r must be such that cp(z), cp'(z), ••• are uniformly r-onvergont and, therefore, t/J, cp', ••• differentiable term by term. In 0. prru-t ir-al upplk-at.ion the number-of terms would usually be finite. It may sometimes be expedient for cjI(z) to have additional properties spec-ified in § 6.10, with w = 1. t \Vith t/J as above. § Except 88 in (1) when the period is 281T.

GENERAL THEORY: FUNCTIONS OF

80

[Chap. IV

the ~ members are both periodic, with period 2S1T. When fJ is irrational, the solution is oscillatory but bounded and non-periodic, i.e, it never repeats itself exactly at any interval. The ~ members in (1) are independent solutions of (I), (2) §4.13. If p, = a+i{J, where 0:, fJ are real and non-zero, (5) §4.13 assumes the form y(z)

=

ex:>

Aecx.z

ClO

!

r=-oo

c 2r e(2r+fl)zi + B e - otz

!

r=-oo

c 2re--(2f'+fJ).i,

(2)

and from what precedes, this is an unstable typo. In numerical work it is possible, however, to arrange that IL shall be either real or imaginary (according as the solution is unstable or stable), but not complex; also that 0 < f3 < I for convenience (see §4.70). Recapitulation. (a) The solution is unstable if J.L is any real number (b) The solution is unstable if p, == o:+i{J, (c) The solution is stable and periodic

* o.

10:1 > 0, l,8i > o. if JL = ifJ, fJ a rational

fraction. (d) The solution is stable but non-periodic if p, = i{J, In (a)-(d) the complete solution is implied,

,8 irrational.

4.15. Alternative forms of solution. In accordance with the theory of linear differential equations, we may choose as a first solution Yl(Z)

= lA{eltZr=~aoC2re2rzi+e-l£Zr=~ao c2r e- 2TZi} = lAco(eJLW+e-ItZ)+lAeltZ{

i

r=l

(1)

(C2re2rZi+c_2re-2r:ci)} +

+ lAe- ltZ{ r~l (c2r e- 2r:ci+ c_2r e2rzi)} (2) co

= Acocoshp,z+A !

{c2rcosh(p,+2ri)z+C_2rCosh(fL-2ri)z}

r~·l

=

ex>

A

!

+ 2ri)z.

C2r cosh (p,

"=-00

As in § 4.10 suppose Yl(O) (Xl

=

=

I, then A

Yl(1T) = boshf'1T,

for all finite values of /L.

(3)

= 1/ ~ C2r and, therefore, (4)

4.15J

FRACTIONAL ORDER: SOLUTION OF EQUATIONS

61

Starting with & negative sign within the r.h.s, of (1), we obtain a second independent solution, namely, «:>

Ya(z) = B

I r=

(5)

c2rsinh(I'+2ri)z.

-00

As in §4.1O, let

Y~(O) = 1, then B = 1/~ (f'+2ri)cir' so

PI = Y~(1T) =

y~(n1T) =

coshp,n1T, (6) provided (5) is uniformly convergent for differentiation of the series to be permissible (see § 4.77). From (4), (6) we get COShP1T,

and

=

=

fJ2'

(7)

Yl(n1T) = y~(n1T) = cosh p,n1T.

(8)

Yl(1T)

y;(tr),

or

(Xl

Again, from (3) CX)

y;(z) == A

I

r=-oo

(J-t+2ri)czr sinh(J.L + 2ri )z,

provided (3) is uniformly convergent (see § 4.7'7) . Then with z ==

'IT,

en

(X2

= Y~(1T) = A sinhJL1T I (,u+2ri)c2r -00

A· = Balnh j.L1f. y;(n1T)

Also

Writing z =

'Tt

(9)

= ~sinhfLn1T.

(10)

in (5) leads to PI = Y2(1T) =

Also

Y2(n1T)

~ sinh un,

(11)

= ~ sinhfLn1T,

(12)

so CXZPl = Y~(11')Y2(1T) From (4), (6) cxlfJ2

=

== Y~(71")Yl(1T) =

sinh21L1T,

cosh2JL1T,

By (13), (14) (Xl~2-Ct2fJl = Y~(1T)Yl(1T)-Y~(1T)Y2(1T) = 1,

and y~(n1r)Ul(n1T)-yi(n1T)Y2(n1T)= 1. Substituting from (4), (6), (15) into (7) §4.11, we find that fPl = ef.l.",

t

zsz:

e-",.11,

CPt'1J2

= I,

q:>l

= 1/f.P2.

These are (2) § 2.191 with z = ." and n1T, respectively.

(16)t (16)t

GENERAL THEORY: FUNCTIONS OF

62

[Chap. IV

These results are valid for JL = cx+ifJ, ex =/:; 0, or if ex = 0, fJ is nonintegral: also Yl(Z) is even in z, whilst Y2(Z) is odd therein.. This restriction is removed in §4. I 52.

4.151. Extension of§4.11. Reverting now to (3) §4.11, ifYl(z) be a solution of either (I), (2) § 4.13, then Yl(Z±1T) = e±/L1TYl(Z).

(I)

Also, if Y2{Z) is a linearly independent solution, Y2(Z±1T)

From (1)

e±I-UT

COSh} · h J-L7T

Helice

If z

=

SIn

==

e":f/L 1TY2(Z).

(2)

= Yl{Z±1T)/Yl(Z).

(3)

(4)

= [Yl(Z+7T)±Yl{Z-1T)]/2Yl(Z).

0, and Yl (r1T) :1-= 0, (4) gives

COsh} . } J.L1T == RIn 1

(5)

[Yl(1T)±Yl( -1T)]/2Yl(O).

Similarly (2) yields the formulae

cosh) · I Jp-1T SIn 1

when

Z

-==

7T,

~

==

[Y2{Z-7T)±Y2(Z+1T)]/2Y2(Z),

(6)

:::=

[Y2(O)±Y2{27T)]/2Y2(7T),

(7)

provided Y2(O):f; O, Since ep(z) has period 7T, Y2(O) Y2(n1T) == 0, RO (7) would then be indeterminate.

==

0 entails

4.152. A more general case. From § 4.12 the solution of (I), (2) § 4. ] 3 has the form (1)

where for convenience we shall take 21T as the period of ~(z). Then = q,(7T) , and

q,( -1T)

y( -1T) so

== e-fL1Tq,(7T) ,

Y(TT)

=

efL1T~(1T),

(2)

Y(1T)--e 2fL1TY(-TT) == o.

(3)

Y'(1T)_p2fL1TY'(-1T) = O.

(4)

Similarly we obtain Let !II(z) and lJ2(Z) be any two linearly independent solutions, and substitute y == AYl(Z)+B Y2(Z) in (3), (4). Then we get

A[Yl(1T)-e2fL1TYl( -1T)]+B[Y2(-7T)-e2/L 1TY2( -1T)]

and

=

0

(5)

A[Y~(1T)-e2/&1Ty~(-1T)] 1- B[Y~(1T)-e2/L1ry~( -1T)] = O.

(6)

4.15]

FRACTIONAL ORDER: SOLUTI()N Olt' EQUATIONS

63

For A, B non-zero, we must have [Yl(1T)-e2~1TYl(-1T)][Y~(1T)-e2~1Ty~(-11')]-[Y2(1T)-e2~1TY2(-1T)][Y~(1T)-e2~1Ty~(-11')] = 0,

D e2~1T+ 1 == 0

e4~1T

i.e,

C2

(7) (8)

'

where D = Yl(-1I')Y~(1I')+Yl(1T)Y;( -1T)-Y2( -1T)Y~(1T)-Y2(1T)Y~(-11'),

and Yl(1T)Y~(1T)-Y2(1T)Y~(1T)

=

Yl( --1T)Y~( -1T)-Y2{ -IT)Y~(-11') == c2

(see §2.191). Solving (8) leads to cosh 2ft1T = D/2c 2 •

(9)

It may be remarked that conditions respecting oddness and evenness of the solutions have not been imposed. This is useful in connexion with problems like that in § 6.40 where the solutions are neither odd nor even. The preceding formulae are valid if 'l~1T is written for 'IT, n ~ 2. If we assign the initial conditions

== Y;( --11')

Yl(-'IT)

Y2( -'IT)

= I,

=

y~( -'IT)

cosh 2fL'IT = [Yl(7T)+Y~(1T)]/2,

(9) becomes

=

0, (10)

and, as before, 11' may be replaced by nsr,

4.153. Functions having period 21T. Apart from § 4.152 the analysis in this chapter has been based upon a solution of the type eJLzep(z), with ep(z)

=

00

~ c2re2rzi, r=-oo

which has period

n,

It is preferable

that I-' should be real or imaginary (fL = ifJ, 0 < fJ < 1), but not complex. To obtain this objective when the parametric point (a, q) lies in certain regions of the (a, g)-plane (see Fig. 8), which are specified later in § 4.70, it is essential that 4>(z) has period 21T. The results in §§ 4.10-4.151 are applicable if for 'IT we write 211', and take ~(z)

~

--

00

k~ c2r+l e(2r+l}tl., r=-oo

(I)

also in all infinite series we change 2r to (2r+l). The relation corresponding to (4), (8) §4.15 is now Yl(n1T)

=

y~(nlT)

=

(-I)n CoShf'n1T.

(2)

64

GENERAl.. THEORY: )4'USCTIONS OF

4.16. Form of solution when 1L = ifJ, and it becomes

= ip,

< p<

0

[Chap. IV

I. In (3) § 4.15

write JL

00

Yl(Z) = A

I r-:..

c2; cos(2r+ ,B)z,

(1)

-00

while the same substitution in (5) § 4.15 gives the linearly independent

solution Y2(Z)

=

00

B

I r==

c2r sin(2r+ ,B)z,

(2)

-00

the i being dropped, since it is merely a constant multiplier. Hence the complete solution of

yH+(a-2qcos2z)y

=

c2rcos(2r+,s)z+B

I

0,

(3)

c2r sin(2r+ fJ )z,

(4)

takes the form 00

y(z) = A

00

I

r= -00

r=-oo

and this constitutes a fundamental system. Corresponding to (4), (8), (13) §4.15, we have Yl{n1T)

=

y;(n1T) = cos fin1T

y~(n1T)Y2(n1T) =

and

-sin 2fJnTT.

(5)

(6)

These are readily checked by using (I), (2) and the initial conditions Yl(O) = y~(O) = 1, Y2(O) = y~«() == o. Additional forms of solution. Replacing 2r by (2r+-}) in (1), (2), (4) in accordance with § 4.15:3, leads to the solutions (independent of each other) CD

Yl(Z)

=

A

I r-=-

c08(2r+ 1+,8)z,

(7)

c2r +1sin(2r-t- 1+,8)z,

(8)

C2r +1 -00

00

and

Y2(Z)

= B I

r=-oo

giving for the complete solution 00

00

I C2r +1 cos(2r+ l + fJ )z+ B r=-oo I C2r +1 sin{2r+ l + fJ )z. r=-oo

y(z)=A

(9)

Corresponding to (2) §4.153, we have Yt(n1T)

=

y~(n1T) =

(-I)ncosfJn1T,

(IO)

which can be confirmed by using (7), (8). It, may be remarked that the A, B are quite arbitrary, and need not have the values obtained in §4.15, using the initial conditions stated there, i.e. any appropriate initial conditions may be chosen.

4.17]

FRACTIONAL ORDER: SOLUTION 014" EQUATIONS

65

4.17. Solution when,B = 0 or 1. Consider any point (a,q), q > 0, in a stable region of Fig. 8A near a Zn but not upon it. Then (3) §4.16 has two independent coexistent solutions, namely, (1), (2) § 4.16. On a Zn ' fJ = 0 and (2) §4.16 ceases to be a solution.'] Substituting either (1) or (2) §4.16 into (3) §4.16 and equating the coefficient of cos(2r+,B)z or sin(2r+,B)z to zero for r = -00 to 00, we obtain the recurrence relation [a-(2r+,B)2]c2r-Q(c2r+2+C2r-a)

With fJ

=

=

O.

(1)

0, (I) becomes (a-4r2)c2r-q(cZr+2+c2r_2)

= o.

(2)

Writing -r for r yields

= o.

(a-4r2)c-2r-Q(c-2r-2+c-2r+2)

(3)

Then (2), (3) are compatible provided Co =F 0 and c2r = c- 2r , r Hence, when fJ = 0 and a = a 2n , (1) § 4.16 may be written Yt(z)

= A [co+2J/2r cos 21Z]'

.

~

I.

(4)

which is a constant multiple of ce2n(z, q) as defined at (1) § 2.17. When ~ = 1, (a,q) is on b2n+1, and (1) §4.16 ceases to be a solution, The recurrence relation is now [a-(2r+l)2]clr-Q(c2r+2+C2r_2)

= o.

(5)

Writing -(r+ 1) for r in (5) gives [a-(2r+I)2]c-2r-z-Q(C-2r+C-2r-«)

=

O.

(6)

Then (5), (6) are compatible if c2r == -C- 2r- 2, e.g. Co = -c- 2, c2 = -c- 4 • Hence (2) § 4.16 may be written co

Y2(Z) = 2B I

7=0

C2r +1 sin(2r+ 1 ).z,

(7)

where C2r +1 has been substituted for c2r • This is a constant multiple of se 2n +1 (z, q) defined at (3) § 2.17. With P = 0 in (7), (8) §4.16, Y2(Z) is not a solution since (a,q) is on a 211. +1- Then &8 above it can be shown that Yl(Z)

=

00

2A

I

r=O

C2r +1 cos(2r+

l)z,

(8)

which is a constant multiple of ce2n+1(z, q). Similarly with fJ = 1 in t 800 § 2.13 et seq. The characteristic numbers for ce fl l' se". are different. K

~u

GENERAL THEORY: FITNCTIONS

66

O~'

[Chap. IV

(7), (8) §4.16, Yl(Z), ceases to be a solution, since (a,q) is on b:an+2.

Then we find that

co

Ys(z) = 2B

I c2r +s sin (2r + 2)z, r-O

(9)

which is a constant multiple of se 2n +2(z, q). --

4.18. Changing the sign of q. Except in § 4.17, the sign of q has not been specified. Assuming, however, that the various solutions are for q > 0, those for the same a but q (l1T-z) is written for z.

<

0 may be derived if

4.19. General relationships between the solutions. We commence with (2), (3) §4.10, and use the values of (Xl' (X2' PI' /32 and the initial conditions stated therein. Then if Yl' Y2 are independent solutions of (1) or of (2) § 4.13, Yl(1T+Z)

= =

Yl(1T)Yl(Z)+Y~(1T)Y2(Z)

= =

Yl(1T)Yl(Z)-Y~(1T)Y2(Z)

(3)

Y2(1T)Yl(Z)-Yl(1T)Y2(Z).

(4)

(1)

and Y2(1T+Z) Y2(1T)Yl(Z)+Yl(1T)Y2(Z). (2) Writing -z for z, and postulating that Yl is even and Y2 odd in z, (1), (2) yield Yl(1T-Z)

and

Y2(1T-Z)

Formulae (2), (4) are analogous to those for the circular functions. Four additional relationships are obtained if (1)-(4) are differentiated with respect to z, Writing (Z-1T) for z in (1), (2), and (1T+Z) for z in (3), (4) leads to Yl(Z)

= =

Yl(1T)Yl(1T=FZ)-Y~(1T)Y2(1T=fZ)

(5)

and Y2(Z) ±Y2(1T)Yl(1T=FZ)=FYl(1T)Y2(1T=FZ). (6) Differentiating (3), (4) with respect to z and using (2) § 2.191, we obtain Y;(1T) = Yl(Z)Y~(1T-Z)+yi(z)Yl(1T-Z.)

and Y2(1T) = Ya(Z)Y~(1T-Z)+Y~(Z)Y2(1T-Z). The following may be derived from (1)-(4): Yl(Z)Y.(1T-U)±Y2(Z)Yl(1T-U) = Yl(U)YS(1T±Z)-Y2(U)Yl(1T±Z)

(7)

(8) (9)

and Yt(Z)Y2(1T+U)±Y2(Z)Yl(1T+U) = Yt(U)Y2(1T±Z)+Y2(1t)Yl(1T±Z). (10)

If Yl and/or YI is periodic, the above relationships may be simplified. It may be remarked that if Yl is periodic in Z with period 1T or 21T,

~.19]

·FRACTIONAL ORDER: SOLUTION OF EQUATIONS

is non-periodic. But if Yl has period 281T, period (see §4.71).

'UI

8 ~

67

2, Ys also has this

4.20. Determination of JL when: q is small. Dividing (1) 1° § 4.750 by [(2r-iJL)2- a] and writing elf' = q/[(2r-iJL)I- a] leads to With If tions:

= ... -2,

C2r+'2r(C2r+2+C2r-I) = O. (1) -1,0,1,2,... we obtain the system of linear equa-

· O+'-tC-e+ c-t +'-tC-2+ 0 + · 0+ 0 +f-2C-t+ C- 2 +f-2 cO+ ·

0+

0

·

0+

0

·

0+

0

+ + +

0

+ goc- 2 +

0

+

0

0

+

0

+

0 0

+

0 0 0

+

+ +

0 0 0

+ . + .

+ . + esc o + +e 0 + . + 0 +"c + c" +e,cs+ . Co

+goc2+ C2

=

o.

2c,+

2

(2)

For (1) 2° §4.70 to be a solution of Mathieu's equation, (2) must be a consistent system of equations, i.e. they must be satisfied simultaneously, so the eliminating determinant for the c (variables of the equations) must vanish. Thus we get the infinite determinant

I· d(iJL) =

'-t

e.,

0

0

0

0

1

~-2

0

0

0

0

eo IT] eo

0

0

0

0 0

'21

,,,0

1

0

e.,

0

0 0

0

'20

1

g"

=0.

When expanded this constitutes an equation for JL. If in the write p, = 0, the determinant d(O) is obtained.

(3)

g we

4.21. Convergence of (3) § 4.20. An infinite determinant is absolutely convergent if (a) the product of the diagonal elements is absolutely convergent, (b) the sum of the non-diagonal elements is absolutely convergent. (a) is satisfied, since the product is unity, while for (b) we have to demonstrate the convergence of

I

r--CX)

/1/[(2r-il£)2- a]I =

!4 L., ~ 1~Il/[(l- i21JL)2r--a>

a

4r 2]

I.

(1)

GENERAL THEORY: FUNCTIONS OF

68

When

[Chap. IV

Irf is large enough, the terms under the sigma sign on the r.h.s.

rrr-

1
<

•

QO

2. Now the series I l/r11 is known

of (1) are each

<

to converge,

(3) § 4.20 converges absolutely for all finite a and /-L,

80

p

,

1

er vanishes, i.e. =/:: ±a1.

provided no denominator of one Qf the (2r-ip.)

4.22.

~(iJL)

as a function of the complex variable.

(a) (i) Writing -p, for fL and - r for r leaves ~(iJL) unaltered, since (2r-iJ.£r~ = (-2r+i,up~. (ii) r takes all integral values from -00 to +00, so that transposition of the elements with +r and - f leaves d{iJL) unaltered. Hence ~(ifJ-) = ~(-iJL), so that d{ip,) is an even function of p,. Further, if a, q are real, so also is a{iJL). (b) [2r+i(JL+2i)]2 = [2(r-l)+i,u]2, so replacing (r-l) by r in the second bracket gives (2r+ifL)2, the determinant is unchanged and a(iJL) = ~[i(JL+2i)]. Helice Ll is periodic in p, with period

2i, so its behaviour everywhere is deducible from that in the infinite strip of the JL-plane, 0 ~ Im(JL) ~ 1. (c) The only singularities of 6. are simple poles which occur when' [(2r-ip,)2- a] = 0, i.e. JL = i(a i-2r) and -i(a i + 2r ). Now the function X(ip,) = IJ(cosip,1T-cos1Ta1) has simple poles at these values of p"t while its period is that of ~(if'). (d) From (c) it follows that the function .p(iJ-t) = d(iJ-t)-OX(iJ-t)

(1)

will have no singularities if C is suitably chosen, so by Liouville's theorem it must be a constant. (e) To determine C we proceed as follows: When f' ~ 00, all the , in (3) § 4.20 tend to zero, and the diagonal elements alone remain. Thus lim ~(iJL) = I. Also lim X(iJL) = 0, and, there~~OO

fore, .p(iJL)

=

p~oo

I. Then GX(iJL)

= ~(iJ-t)-l,

0 =

or

[~(iJ-t)-l ]/X(ip,).

(2)

X(u) = l/(l-cos1Ta i ) ,

When IL:X:: 0, while the value of (3) §4.20 is d(O), the er being qj(4r l-a), a =1= 412, to avoid an infinity. Hence

0

by (2)

t

= [Ll(O)-I][I-coswal].

We assume that a =1= 4r t ,

80

p, =

0 is not a pole.

(3)

4.22]

FRACTIONAL ORDER: SOLUTION OF EQUATIONS

69

With this value of 0, (1) is devoid of singularities in the p,-plane. The preceding analysis is based upon [215].

4.23. Solution of (3) § 4.20. This determinantal equation is satisfied if JL be such that A(ifL) = o. Substituting this for a in (2) § 4.22 and equating to (3) § 4.22, we have I/X(ip,)

=

[1-~(O)][I-cosmJi].

(1)

Now l/X(iJL) = cosip,'TT-cos'TTai , so by (1) we get COSiJl1T

or

sin 2 I iJ-L1T

=

cos7Tal+[I-~(O)][l-coS?Tal]

= l-A{O)[I-cos7Ta l]

(2)

=

(3)

L\(O)sin2 t1Tal (a =1= 4r 2).

This result is obtained in [213] without recourse to the complex variable. The values of I-' which satisfy (3) above, satisfy (3) §4.20 also. Hence ~ is determined if L\(O) can be evaluated. If q is small,'] by aid of [213] we find that ~(O) ~ 1- 7Tq2cot!7Ta1 , (4) 4a i (a - l ) so (2) may be written COSiP.7T

~

cos7Tai+[1Tq2sin7Tal/4al(a-I)]

(a =1= m 2 ) .

(5)

4.24. The roots of (3) § 4.20. The determinant being an even function of ip., if iJlo is a root of A(iJL) == 0, so is -ifto. Since a linear D.E. of the second order has no more than two independent solutions, ±iJ-Lo are the only distinctive roots. Now the non-diagonal elements are ~2r = q/[(2r-ifL)2-a], and if we write ilL = (ip,o-2r) in the elements on either side of the origin where r = 0, ~2r above is obtained. The result of treating all non-diagonal elements in this way is merely to shift the origin, and to leave ~(i,.,.) unchanged. It follows that the roots are ip, = ±iJ-Lo-2r,r being any integer. If fLo is real, the roots occur in conjugate complex pairs il-'o-2r, -ifLo-2r. If fLo is imaginary (= ifJ), the roots occur in the real pairs ±fJ-2r. For fLo = cx+ifJ, ifLo = iex-f3. Since r is any integer, we have the conjugate pair iex-fJ-2r, and -iCl:+fJ-2(r+fJ) = -icx-fJ-2r, provided fJ is an integer. Hence JL may be complex if its imaginary part is integral. t It suffices if the term in

ql in (4) is small compared with unity.

GENERAL THEORY: FUNCTIONS OF

70

[Chap. IV

Recapitulation. (a) If ILo is real and r = 0, IL = ±ILo is real. (b) If ILo is real and r :f= 0, IL is complex. (c) If ILo is imaginary, IL is imaginary for all r. (d) If ILo = a.+ifJ, fJ an integer, ~ is complex for all r.

As shown in § 4.70 the form of solution may be chosen to make IL either real or imaginary (according to the position of a, q in the

plane, Fig. 8), but not complex. These remarks apply to all real values of a, q finite. The determinantal method of finding IL is unsuitable for numerical evaluation unless q is small. When q exceeds, say, qo, the methods exemplified in Chapter V should be used to get accurate results.

4.30. Alternative form of general solution [186]. For the first solution we take

Yl(Z) = ePep(z, a),

(1)

c/J being periodic in z with period 1T or 21T, and a a new parameter. When (a, q) lies between ao, b2 in Fig. 8, we shall obtain the form

ep(z, a)

=

sin(z-O')+8a sin(3z-a)+8s sin(5z-a)+ ... + +ca cos(3z-a)+c s cos(5z-O')+ ... ,

(2)

being dependent upon a, q, as also are the c, 8. ep has no term in cos(z-o') as its inclusion would introduce a non-periodic term, as shown below. For convenience the coefficient of sin(z-O') is taken as unity for all q. Since a, a, q are interrelated, we assume that a

also

a = l+q!1(O')+q2j2(a)+qa!3(U)+ ... ;

(3)

IL = qgl(a)+q2g2(a)+q3ga(a)+ ... ,

(4)

there being no term devoid of q, and we take cp(z, a) = sin(z-a)+qh1(z, a)+q2h2(z, a)+q3ha(z, a)+ ... , (5) where!, g are functions of a, the h being functions of Z, a, periodic in z. Substituting the r.h.s, of (1) into y"+(a-2qcos 2z)y = 0 gives ellz[ep"+2ILep'+(IL 2+a-2qcos2z)cp]

and by the aid of (3)-{5) we get -sin(z-a)+qh~+q2k;+qak;+ ... ~

=

0,

+

+2(qUl+q2g2+qag3+···)[cos(z-a)+qk~+q2h~+qah;+ ... ]+

+[{qgl+q 2U2+qaga+ ... )2+ 1+q!1+q2f2+ ... -2q cos 2z] X X

[sin(z-u)+qk1+q2k 2+···]

=0. (6)

~'1RACTIONAL

4.30]

'raking 2 cos 2zsin(z-a)

ORDER: SOLUTION OF EQUATIONS

71

= =

[sin(3z-a)-sin(z+a)] [sin(3z-a)-sin(z-a)cos 2a-cos(z-a)sin 2a], and equating coefficients of gO, q, q2, ... to zero we obtain qO: -sin(z-O')+sin(z-a) = 0 identically; (7) q: hi+hl+/tsin(z-a)+2g1cos(z-a)-2sin(z-a)cos2z = o. Thus h; +h 1 + (2Yl -l-sin 2a}cos(z-0') + +(/l+COS 2a)sin(z-a)-sin(3z-a) = O. (8) Now the particular integrals corresponding to C08.(z-a), sin(z-a) are -lzsin(z-a) and -1Zc08(z-a). These are non-periodic and tend to ±oo as z -+ +00. Since eP is to be periodic, the coefficients of cos(z-a), sin(z-a) must be zero. Hence we must have gl = -!sin2a'!1 = -cos2a, leaving h~+hl = sin(3z-a), so q2:

hi = -lsin(3z-a). (9) h;+h2+ 2g2 cos(z-u) - !gl cos(3z-a)+ (g~+12)sin(z-a)+ +lsin(3z-a)cos2a+lsin(3z-a)cos2z = 0,

so h;+h2+2g2cos(z-a)+icos(3z-a)sin2a+(!sin22a+t+!2)sin(z- a)+ +lsin(3z-a)cos 2a+lsin(5z-a) = O. (10)

To avoid non-periodic terms we take g2 = 0, 12 = -1+1c084a, leaving the equation h;+h 2 ~ cos(3z-a)sin 20'+ 1sill(3z-a)cos 2a+lsin(5z-a) = 0, of which the particular integral is

+

13 = [3 sin 2a c08(3z-a)+c08 2asin(3z-a)+! sin(5z-a)]/64. (11) Proceeding in this way, on substituting for the various functions in (3), (4) we obtain

a = l-q cos 2a +

~ q2(-

I+

~ cos 4a) +

:4

q3 cos 2a+

+..!.q"(!- !! cos 4a) - ~ q5 (~ cos 20'- ~ cos 60') + 16 3 32 32 9 128 1 (893 9181 35.) +S192q8 -27+ 216 cos 4a-"4cos 8a -...

= t

l-qcos 2a-JL2+ qss.

(12) (13)t

Thi« result was obtained b)" substituting (1) in the differential equation y. + (a -- 2q <'08 2z)y = 0, using , at (2), and equating to zero the coefficient of sin(.::-a).

GENERAL THEORY: }t'UNCTIONS O}4'

72

JL =

1.

3

[Chap. IV

3,.4. sln4a-

•

-2 Q SID 2a+ 12S q3 SID 2a-I024~

1 5 (137 · 2 9..) - 4096 Q ysm a- 2sm6a I

6 (337 S11 ·1

+ 16384 q

27

4

15.

+ )

0'-4-s1n8O'

+...

(14)

1Q2 - - I q3 cos2O'+--q 1 4 ( --+-cos40'155 82 ) = -192 1152 4096 54 27 .'.

(18)

C5

7 3· 2 II 4' = -2304 q 81n O'+27648q SIn 40'- ...

(19)

87

1 3 1 4 It) = -9216 q +49152 q C08 ...O'- ...

(20)

C7

= 4 42368 ~

86

35

n4. n SIn ~O'- •••

(21)

(22)

It may be remarked that in (2) the coefficients of cos(3z-O'), cos(5z-O'), ... are expressed in terms of sin zc, sin 40', ..., while those of sin(3z-O'),... are expressed in terms of cos 20', cos 40', .... If in (12) we write --0' for 0', it is unchanged because a is an even function of o; Making this change in (14) alters the sign of It, since it is an odd function of o, 'rhus referring to (I), we see that the

t

This result was obtained by substituting (1) in the differential equation 2z)y = 0, using ; at (2), and equating to zero the coefficient of

v' + (a- 2q cos ('o~(z

- a).

'.30]

FRACTIONAL ORDER: SOLUTION OF EQUATIONS

73

second independent solution of Mathieu's equation may take the form YI = e-~t/J(1" -u). (23) constitute a fundamental system, and the complete solution, YI' Y2 with two arbitrary constants, is [186]

y

=

AeP-2cP(1" a)

+Be-p,zep(1"

(24)

-a).

4.31. Degeneration of (1) § 4.30 to cel(1" q) and se 1(z, q). When -11T, 11' = 0, Ca = Cs = ... = 0, and the series for a becomes

a =

that for a l at (3) § 2.151. Also 1

83

and so

OD,

I

2

= -sq- 64 q

1 3 11 4 -1536 q +36864 q - ...,

(I)

while sin[(2n+l)z+I1T]

=

cos(2n+l)z.

(2)

Substituting in (I), (2) §4.30, we obtain (16) §2.13, the series for ce 1 (z, q). Similarly, when a = 0, (I), (2) §4.30 yield (2) §2.14, the series for sel(z, q).

4.32. Useful range of formulae in § 4.30. The series for a, 11', and the coefficients are similar in type to those in § 2.13 et seq. Such series probably diverge when q, real and positive, exceeds a certain value. The first term of (12) § 4.30 is unity, so if this series were used to compute a5 for ce 5 (z, 0·4), which exceeds 25, cos 2a would be correspondingly large, and low accuracy would ensue. This remark applies also to the series for p, and the coefficients. Moreover, for computation the formulae in § 4.30 may be used when q is moderate and the point (a, q) lies between the characteristics for ceo and se 2 in Fig. 8. If (a, q) were between the characteristics for se 2 , ce 2 , a more suitable form of solution would be obtained by taking

c/>(z, a) =

8in(2z-0')+8~8in(4z-a)+86sin(6z-a)+ ...

+

+c.cos(4z-a)+c6cos(6z-a)+... = sin(2z-a)+qk1 (z, a)+q 2k 2(z, 0')+ ... ,

(1) (2)

there being no term in cos(2z-a , to avoid a non-periodic term of the type z sil1(2z-a) in the solution. If we asaume that a = 4+q!l(a)+q 2f 2{a )+ ... , 4961

L

(3)

'74

GENERAL THEORY: FUNCTIONS OF

[Chap. IV

we find that 11(a) = 0, while /2(a)

so

a

= l-! cos 2a,

(4)

= 4+!q2(!-lcos2u)+ ....

(5)

Owing to the presence of 4 in (5),t" the coefficients of the terms in powers of q will be less than those in (12) §4.30 if a is, say, 4·6. In general the solution preferably has the forms at 1° (3), (6), 2° (2), (4) §4.70, the corresponding series for a being of type (3) above, the leading term for ce2n , se 2n being 4n 2 , while that for ce2n +1 , se 2n +1 is (2n+I)2.

4.40. Maclaurin series solution. If y is a solution of y"+(a-2qcos2z)y = 0,

(1)

then by Maclaurin's theorem y(z)

=

I I I

y(O)+zy'{O) + 2! Z2y "(0) + 31 Z3y "'(O)+ ... + nizny
the derivatives being obtained by successive differentiation of (1) and insertion of z = o. Although the procedure is simple, the solution 80 obtained has serious drawbacks: 1. The form, a power series in z, is markedly inferior in applications to those given previously. If (2) were periodic, its periodicity would be difficult to establish. Also it is unsuitable for interpreting the physical behaviour of a system, e.g, stability and instability. 2. If convergent, the rate of convergence is very slow and in many cases the number of terms needed to obtain accuracy for even moderate values of z, e.g. 117, would be prohibitive: see [12]. 3. The convergence is difficult to establish.

4.50. Relation between solutions in § 4.153 and § 4.30. The respective forms are Y-1 (z) --

ClO

eP.~

~

~

r--oo

CIr+l e(2r+l)ei

(I)

and Yl(Z) = elt z [ sin(z-u)+sa sin(3z-a)+s5 sin(5z-a)+ ...

+

+ca cos(3z-a)+c5 cos(5z-a) +...],

(2)

4.50]

FRACTIONAL ORDER: SOLUTION OF EQUATIONS

75

JL having the same value in both cases (see § 4.70). Omitting the

common factor elU , we expand the r.h.s. of (2) and obtain

=

tP(z)

sinz cos a-COS Z sina+ss(sin 3z cos a-COS 3z sin ej-l-

+s5(sin 5z cos a-COs 5z sin a)+ ... +cs(cos 3z cos a+sin 3z sin a)+

+c5( cos 5z cos a+sin 5z sin a)+

(3)

3z+s6sin5z+ +cacos3z+c,cos5z+ )-sin a(cos Z+S3 cos 3z+s5 cos 5z+ -c3 sin 3z-cs sin 5z- ) (4)

= cosa(sinz+sasin

ex>

=

! [K2T+1 cos(2r+l)z+L2r+1 si n (2r + r=O

(5)

l )z ],

where K 2r +1

==

L 2r+l =

(C2r + l

cos a-8 2r+l Sill a)

(8 2r +1 cos a+c 2r+l

sin u)

(r (r

> >

0),

K 1 == -sina;

0),

L1

==

cos a.

Writing the circular functions ill (5) in exponentials, we get ex>

4>(z)

Now

=:

!!

r=O

00

~

k

r=-ex>

[(K2T+l-iL2r+l)e(2T+l)zi+ (K2r+l +iL2r+1)e-(2r+l)zi].

c-2r+l e(2r+l)zi ==

ex>

~

k

r=O

[c-2r+l e(2r+l)zi+ c--2r-l e-<2r+l)Zi] •

(6)

(7)

Hence by (6), (7) the two forms of solution are identical, provided C2r+ 1 = !Zo(K2r+l-iL2r+l)

and

0-2r-l

==

!Zo(K2r+l +iL2r+l),

(8)

Zo being a

constant. It follows from (8) that C2T+ l ZOI and C- 2r - 1 ZOI are conjugate complex numbers, while Ic 2r +ll = Ic- 2r - ll.t By adding the two parts of (8), we get

Zo K 2r+1 =

02r+l +C- 2r - 1 ,

(9)

and by subtracting, we have

ZO L 2r +l

== i(C2r+l-C-2r-l)'

(10)

Writing r = 0 in (9), (10), and substituting the values of K l , L 1 from above, leads to Zo = -(C1 +c_1 )/sin a = i(ct-c_1)/cos a . (1 I) t Write then '.,+1 Zot

Zo =

Moe i S"

= Mol PI'+l ei ,

C.r+l = PI"+l e""u+l, O_I,_IZ01 = ¥O-lPI'+l e'.

By virtue of the conjugate property tPl"+I- 80 =

.... (tP-Ir-l -

8.) - 2Bfr,

80 80 = i(tP.'+l +"'-lr-l~+87r, 8 being integral. C2r+1 Ziit and C_21 _ 1 Zol are complex conjugate only if a is real, i.e, when (a, q) lies in an unstable region of Fig. 8.

GENERAL 'rHEORY: FUNCTIONS OF

76

[Chap. IV

By squaring and adding both sides of (9), (10) we obtain

z; = and, with r

=

2(c2r +1 c_ 2r - 1 )1/(K Ir+l + L Ir+l)i,

(12)

2(c1 c_ 1 )1.

(13)

Zo =

0,

We have proved, therefore, that, ~ve for

constant multiplier, the solutions in §§4.153, 4.30 are identical. A similar conclusion applies a,

co

in connexion with the solution ePA

I c.. elrd

and its alternative form.

r=-oo

These results are to be expected, since the solution of Mathieu's equation must be unique.

4.60. Division of the (a, q)-plane into stable and unstable regions. Fig. 8 shows the 'charted' ·part of the plane, i.e, that portion for which the characteristic curves am, bm for the Mathieu functions of integral order have been computed. Consider the region in Fig. 8 B lying between a l and b2 • On the curve aI' 0' = -i'7T in (12) §4.30. Now if 0' = -i1T+iO, cos 20' = -cosh 20, whatever the sign of 8, so if q > 0 is fixed, (12) § 4.30 shows that a increases with increase in until bl is reached. On this curve from [191] we have

°

a

=

4+!q2(!-lcos2O')+...

(1)

with 0' = o. If in (1) we write o = iO, a decreases with increase in 8 until a1 is regained. Thus between ai' b2 with q > 0, a is complex or imaginary according as (12) § 4.30 or (1) is used. If we take (J' = -tn"+i8, then sin 20' == -isinh20, and (14) §4.30 shows that JL is imaginary and, therefore, by § 4.14 the solution is stable. Starting from b2 , where a = 0 in (1), if for fixed q > 0, 0' (real) decreases, then a increases until a 2 is reached where o = -!1T. Since o is real in the intervening region, the series for,." (like (14) § 4.30) may be shown to contain sin 20', sin 40', etc. Thus u is real, and by § 4.14 an unstable solution is obtained for any (a, q) in the region. Now consider the region between- a 2 , bs with fixed q > o. On the former 0' = -11T in (1), so the term in cos 20' is positive. If we take (J = (-i1T+i8), then cos 20' = -cosh 28, and by increasing 8 from zero in (1), ba will ultimately be reached. Using the procedure suggested in §4.32, we obtain [191]

a = 9+ 1\q2_ a\qScos 20'+ ... (2) for the region in question, and on bs, 0' = o. Taking a = ifJ in (2) and increasing 8 from zero, at will eventually be regained. Thus between at, ba, a is complex or imaginary according as (1) or (2) is

4.60]

FRACTIONAL ORDER: SOLUTION OF EQUATIONS

77

used. Since the series for JL contains sin 2a, sin 4a, etc., f' is imaginary in the region, so the solution of Mathieu's equation for an assigned (a, q) is stable. In this way the (a, g)-plane, for q positive, may be divided into zones or regions in which the solution of Mathieu's equation corresponding to a point (a, q) is either stable or unstable. On the characteristic curves for the functions of integral order, the first solution is neutral, but the second solution treated in Chapter VII is unstable. When q is negative, by writing -q for q in the above series and using a similar argument, the plane may be divided up as illustrated in Figs. 8, 11. The series for the a hold if Iql < qo' but the results may be established for the whole range of q. Summary. (1) When (a,q), q > 0 lies between am, bm+1 in Figs. 8, 11, /L is imaginary, and the two solutions of Mathieu's equation are stable. (2) When (a, q), q > 0 lies between bm , am' JL is real provided the appropriate form of solution is taken (see 2° § 4.70), and the complete solution of Mathieu's equation is unstable. Apparatus for illustrating stability is described in [201] chap. XIV, also in § 15.40 et seq.

4.70. Form of solution for different regions of (a, q)-plane. Certain advantages accrue by assuming different forms of solution corresponding to various regions in which the point (a, q) may lie in Figs. 8, 11. By adopting the convention given below, f' will be either real as in 2°, or imaginary as in 1°, but never complex. It will then have the same value for the two forms of solution in 2° (real), and the three forms in 1° (imaginary). z is assumed to be real. 10 • Stable solution, q small and positive. When (a, q) lies between a 2n , b1n +1 , for the first solution we take Yl(Z) = eifJlJ

co

I

C2r

e2rzi

= eiP-cp(z)2,.

(1)

"1:1-00

co

or Yl(Z) = or

I clrcos(2r+,B)z ,.a-oo

Yl(Z) = eifJe[sin(2nz-a)+82sin(2z-a)+8~8in(4z-0')+ ...

(2)

+

+C. cos(2z-a)+ct cos(4z-0')+ ...],

(3)

there being no term in cos(2nz-a), e.g. if n = 1, c1 = 0, for the reason given in § 4.30. In these series fJ is real and 0 < fJ < 1. If

GENERAL 'fHEORY: FUNCTIONS OF

78

[Chap. IV

n ~ 1, formulae for u, ifJ = p., and the coefficients c2r 821' may be developed as shown in § 4.30. When (a, q) lies between a2n+1 , b2n+2, for the first solution we take Yl(Z)

= eif3z I

or

Yl(Z)

= I

or

Yl(Z)

=

ex>

"'=-a)

C2r+1 e(2r+l)zi

"

= ei {3zc!>(z)2r+l

(4)

co

"'=-CD

C2r +1 cos(2r+

(5)

l+,B)z

ei,8tf8in{(2n+l)z-u}+f~lsin(z-u)+83sin(3z-u)+ ... +

+C1 cos(z-u}+cacos(3z-u)+ ...],

(6)

there being no term in cos{(2n+l)z-u}, e.g. if n = 1, ca = 0, for the reason given in § 4.30. If n ~ 1, formulae for a, if3 = p., and the \ coefficients c2r +V 8 2r +1 may be developed as shown in § 4.30. The second solution is obtained by writing -z for z in (I), (4); sin for cos in (2), (5); -{J for {J, -a for a in (3), (6). In each case the two solutions are linearly independent, provided f3 is in the range o < f3 < 1, and constitute a fundamental system, When the initial conditions are specified, all forms of solution yield an identical result, since the solution is unique. 2°. Unstable solution, q small and positive. When (a, q) lies between b2n +2 , a2n + 2 for the first solution we take Yl(Z)

=

eP.~

co

I r==

=

C2r e2rzi

(1)

ep.zep(Z)2r

-00

or

Yl(Z)

==

ep.z[ sin(2nz-u) -+-8 2 sin(2z-u)-t-84 sine4z-a)+ ... +

+c 2 cos(2z-u)+c. cos( 4z-u)+ ...],

(2)

where p. (real) > o. See remarks below (3) 1°. When (a, q) lies between b2n +1 , a 2n +1 , for the first solution we take

Y1(z) -- eP.Z

co

k~ c2r+l e(2r+l)zi -r=-oo

or

Yl(Z)

==

efLZ,J..(z) 'f' 2r+l

(3)

ep.z[sin{(2n+ I )z-a}+81 sin{z-u)+S3 8in(3z~a)+ +

+C1 cos(z-u)+cscos(3z-a)+

]. (4)

See remarks below (6) 1°. The second solution is derived by making the substitutions stated in 1°. The period of t!>(Z)2.,. is e while that of t!>(Z)2r+l is 2",. Observe remark at end of }O. 3°. Stable 8olution, q moderate and positive. See (1), (2), (4), (5) in 1°. The form at (3), (6) 1° is usually unsuitable for computation when q > 0·4 approx. I

4.70]

FRACTIONAL ORDER: SOLUTION OF EQUATIONS

70

4°. Unstable solution, q moderate and positive. As at (1), (3) in 2°. 5°. Any soluiion, q negative. In the solution for q > 0, write (!1T-Z) for z,

4.71. Mathieu functions of real fractional order (134). In § 4.16 (1), (2) represent coexistent solutions of Mathieu's equation, i.e. for an assigned value of q they have the same value of a. Consider the region in Fig. II between the curves am' bm +1 with m = 1. Take any line parallel to the a-axis and terminating on these curves. Hereon 0 ~ 13 ~ 1. If for any assigned 13, say 0,8, the a are calculated for q increasing from zero in small steps, and the points plotted, the characteristic curve fJ = 0·8 is obtained. Lying between those for eel' se 2 , we shall define it as that for the Mathieu function of real fractional order (1+,8). In general, if (a,q) lies between the curves am' b"l+l' the order of the function will be (m+,B), and the value of a on any curve is that for cem+{3(z, q) and sem+l3(z, q). Moreover, by computing a series of curves at intervals of, say, {J. = 0·1, we can plot an iso-fJ chart,'] of the type depicted in Fig. 11. For q > 0, 0 < ,8 < 1, fJ real, we adopt the definitions [134]

r=~coA~~1t+,8)C08(2r+,8)Z,}

ce2n+p(z,q) =

ex>

se21t +I3(z, q)

=

ce21t +1+P(z, q)

~ A~;n+,8> sin(2r+fJ)z,

r=-oo

coexistent solutions with (1) a = a 2n +!3 : (a, q) between a 2n and b2n +1 ; (2)

(2 +1+,8> }coeXistent solutions == r=s-oo ~ A2r~1 cos(2r+ 1+ {3)z, ith _ . WI a - a + +13' ex>

00

(Q)

se2n +1 +/l(z, q) == ~ A2~+11+t-' sin(2r+l+{3)z, r=-oo

q

< o.

(3)

2n 1

(a, q) between a 2n +1 db (4)

an

2n+2'

Writing (!1T-Z) for z in (1), (2), we obtain

ce2n+I3(!1T-z, q)

==

00

cos !fJ1T ~ (-1)'A~~n+l3> cos(2r+fJ)z+ r=-co

00

+sinl-f37T ~ (-I)rA~~n+P)sin(2r+f3)z,

(5)

r=-oo

se2n +,8(! 1T - z, Q)

== sinlfJ1T

co

! (-l)rA~~n+l3)co8(2r+fJ)zr--oo

-COSlP1T

00

~ (-I)rA~;n+p)sin(2r+,8)z.

(6)

r=-oo

t Using an argument similar to that in § 3.25, it may be shown that the iso-p curves are single-valued and continuous.

GENERAL THEORY: FUNCTIONS OF

80

[Chap. IV

If we adopt the definitions OOtn+p(z, -q)

= (-I)n

f

(-ItA~+P)co8(2r+,8)z,}coeXistent

r--oo eo

I r--oo

seSf~+~(Z, -q) = (-I)'"

(7) solutions with (-I)rA~+,8) sin(2r+,B)z, a = a 2n +p, (8)

each function is a linearly independent solution of Mathieu's equation, and when q ~ 0 it degenerates to the appropriate form given in §4.73. The functions corresponding to (3), (4) are obtained by writing (1+,8) for fJ in (7), (8) and using A~~11+P) for Agn+p>. If the tabular values of (1), (2) are known, those of (7), (8) may be calculated therefrom by aid of the following relationships derived from (5)-(8): cesn+p(z, -q) = (-I)n[ cos iP1T ce2n +p(!'n"- z, q)+sin !P1T se1n +p(! 1T- Z, q)]

(9)

and sesn+p(z, -q)

=

(-I)n[ sin 1,81T ce1n+fJ(!'n"-z, q)-cos 1,81Tseln+~(l1T-z, q)].

(IO)

A similar remark applies concerning (3), (4).

The relationships for functions of order (2n+I+{3) are derived from (9), (10) as described above with regard to (7), (8). If the functions are defined as ClO

cegn+p(z, +q)

=

sec IfJ1T

I

eo

se1n+p(z, +q) = sec!,81T

A~n+p) cos[ (2r+ ,8)z - ifJ1T] ,

r=-oo

I

(11)

A~~n+p)sin[(2r+p)z-iP1T],

r--oo

then for q < 0, the substitution (!1T-Z) for Z yields (11), but with (-l)r within the sigma sign (see (7) and (8)). Although the functions defined thus are well suited for tabulation, since ce2n+p(z, -q) is then equal to ce2n+,9(!1T-Z, q), they have the following disadvantages: (a) they are neither odd nor even, (b) they do not reduce to cos(m+p)z, sin(m+p)z when q -+ 0, (e) they are more complicated than (1)-( 4:), (7), (8). These comments apply also to (5), (6). As in § 3.21, all the above series may be proved absolutely and uniformly convergent in any real closed interval ZI ~ Z ~ %1' or in any closed rectangle of the e-plane (see § 4.77). For the pairs of functions (1), (7); (2), (8); etc. the iso-p curves are symmetrical about the a-axis of Fig. 11. No two of these inter-

4.71]

FRACTIONAL ORDER: SOLUTION OF EQUATIONS

81

sect; for if they did, the equation would have more than two independent solutions corresponding to the point of intersection (a, q), which is impossible. If P = pis, a rational fraction less than unity, p, 8 being prime to each other, and z is real, the function has period 281T, 8 ~ 2. When fJ is irrational, the function is not periodic, and tends neither to zero nor to infinity as z -+ +00. By appropriate choice of (a, q) the function may have any real period 2811. An argument akin to that in § 3.25 may be used to show that the characteristic numbers and coefficients are continuous in q.t Modified functions of fractional order. These are solutions of (1) §§ 2.30, 2.31, respectively, being defined as follows:

ce ,Sem+{J

(z, ±q)

= . ce

- ~ sem+fJ

(iz, ±q)

(am+fJ)·

(12)

The series representations may be derived from above by writing iz for z.

4.711. Behaviour of the coefficients as

f3

~

0 and 1. Consider a point (a, q) in a stable region of Figs. 8 or 11 between a2n and b2n +1 ; then 0 < fJ < 1. With q > 0 fixed, let a 2n be approached so that 8 ~ o. Then by § 4.17 celn+~(z,q) ~ ce2n(z,q),

se2n+~(Z,q) -+ 0;

(1)

so we must have +/l>~ A(2n+!J> ~ J-A (2n) A (2n -2r 2r 2 2r ,

Similarly, as

fJ -+ 1,

se2n+/3(z, q) ~ se2n +1(z, q),

ce2n+~(Z, q) ~ 0;

(3)

b2n+l·

(4)

_A(2n+~) ~ A(2n+/l) ~ _lB(2n+l). a Q ~ -2r-2 2r I 2r+l '2n+,.,

For (a, q) between a 2n +1 and b2n +2 , as fJ ~ 0, ce2n +1 +p(z, q) ~ ce2n +1(z, q), +l +/l>~ A(2n+l+fJ> ~ A (2n -2,.-1 - r 21'+1

As

0;

(5)

1

(6)

ce 2n + 1+p(Z, q) ~. 0;

(7)

se2n +1+p(z, q)

4-

~

a2

1A(2n+l).

}

2r+l

a

Q

'2n+l+~

n+ •

f3 ~ 1 se2n +1+/3(Z, q) ~ se 2n +2(Z, q),

_A(2n+l+~) ~ A(2n+l+!J> ~ lB(2n+2) -2r-3 2r+l )- tT+2 ,

A_I --* 0; a 2n +1 +/3 ~ b2n +2 • (8)

r

~ 0

except in (2), where r ~ 1.

t 4961

They are single-valued also. M

GENERAL THEORY: j."UNCTIONS Oli'

82

[Chap. IV

4.72. Normalization of oom+{J(z, q), sem+{J(z, q). 10 • fJ = p]«, a rational fraction in its lowest terms. Since the functions have period 281T, we take

Jce~+IJ(z,q) 281T

~

S1T

Jse~+{J(z,q) 2811'

dz

=

~

1,

81T

o

dz = 1.

(1)

0

Inserting the series (1)-(4) §4.71 in these integrals leads to 00

I

[A~~n+{J)]2

r=-oo

00

= I

r=-oo

[A~~'+il+fJ)]2 = 1.

(2)

2°. fJ an irrational number, 0 < fJ < 1. If in (1), 8 ~ +00, (2) follows, so we normalize accordingly. Comparison of (1), (2) §4.16 and (1), (2) §4.71 shows that we may write A~;.n+fJ) = Kc 2r , where K is a constant. Then from (2) we get

K2r~coCt. =

1,

80

K

=

1/[~ct.t

(3)t

If the c are found as in § 5.20 et seq., K may be computed, e.g. § 5.311.

4.73. Form of solution when q = o. At the intersection of an iso-fJ curve with the a-axis in Fig. 11, q = o. Then a = m 2+Lla = (m+fi)2, m being the order of the function whose characteristic curve intersects the a-axis at the point a = m 2 • Thus the differential equation reduces to (1) y"+(m+fi)2y = 0, whose formal solutions are cos(m+!3)z, sin(m+,B)z. By analysis akin to that in § 3.32 it can be shown that as q ~ 0, all the A in (1), (2) §4.71 except A~;n+{J) tend to zero. By §4.72 A~;n+fJ) = 1 when q = o. A similar conclusion applies to (3), (4) § 4.71. Hence, when q = 0, the functions of fractional order become cos(m+,B)z, sin(m+,B)z, so (1) is satisfied.

4.74. Formula for fJ [134]. When the parametric point lies in a stable region of Figs. 8 or 11, formula (6) §2.16 may be adapted to calculate f3. When q > 0, if the curves bounding the region are am (lower), bm+1 (upper), m ~ 0, we take v = m+f3. Then from (6) § 2.16

v2 - a-

1

2(v2-1)

q2

~5v2t7~_ q4_ 32(v 2-1)3(v2-4) 4+58v2+29

9v

t

6+0( 8) q.

- 64(v 2-1)5(v2-4)(v2 - 9) Q This formula is valid when fJ is a rational fraction.

(1)

4.74]

FRACTIONAl..

ORD~JR:

SOLUTION OF EQUATIONS

83

>

This formula is usable under the condition that fal Iq2j2(v2-1)I, and that the ratio of each of the terms in q2r to its predecessor is small. For 8 first approximation we have v 2 = a. Inserting this in the term in q2 and omitting the others, the second approximation is v2 = a- q2j2(a-I). (2) Substituting from (2) into the second term on the r.h.s. of (1) and in the third and fourth, yields the third approximation, namely,

v2 = a

(a-I)

2

(00+7)

2

v = a-[2(a_l)2_ q2]q -32(a-l)3(a-4)tf2

9a + 58a+ 29 -- '+O( 8) 64(a-I)5(a-4)(a-9) q q .

(3)

Since v 2 = (m+,B)2, we obtain (whatever the sign of q) ,-.."J

f3 -

[

(a-I) a-[2{a_l)2_ q2]q

2

(5a+7)

-3~i(a-=-1)3(a-4)q

4

'

-

9a2+58a+29

-64(a-l)5(a-4)(a-9)q

6]1

-m,

(4)

provided no denominator vanishes. If VI = (m+,8I) calculated from (4) is fairly accurate, a closer result may be obtained by substituting VI for v on the r.h.s. of (1), and recalculating v. The accuracy obtained from (4) increases with decrease in q, a being assigned. Broadly, for moderate accuracy we must have raJql ~ 1. Thus, if a is large enough, q also may be large, e.g. if a = 1000, q might be 50. For lower accuracy the term in q6 may be omitted. When a, q are such that adequate accuracy cannot be obtained with (4), the procedure described in §§5.11-5.14, 5.32 may be employed.

4.750. Conjugate properties of the coefficients, JL real, (a, q) in an unstable region. 1°. When (a,q) lies between b2t" and a 2 ,t' the recurrence relation is

[a-(2r-ip,)2]C2r-q(C2r+2+C2r-2)

=

O.

(1)

Writing -r for r, (1) becomes

[a- (2r+ip)2]c-2r-Q(C-2r-2+C-2r+2) =

o.

(2)

Since a, q, J-L are real, it follows from (I), (2) that C2r and C-2r expressed in terms of Co (real) are conjugate complex numbers.

84

GENERAL

THf~ORY:

FUNCTIONS OF

[Chap. IV

2°. When (a, q) lies between b2n +1 and a 2n +1 , the recurrence relation is [a-(2r+l-iJL)2]C2r+l-Q(Car+3+C2r_l)

= o.

(1)

Writing -(r+l) for r in (1) givea[a- (2r+ 1 +ip)2]c-2r-l-Q(C-Ir-s+c-sr+l) = O. (2) Suppose that the C2r+1 are expressed in terms of C1, and the C- 2r - 1 in terms of C- 1• Then c2r +1 and (cl/c-I)c-2r-lt are conjugate. In §4.751 it is shown that Ic 2r +1 1 = IC- 2r - 1 1, so Icll = Ie-If. The conjugate property of the coefficients is useful for checking purposes in numerical solutions. If this species of checking is waived, computation of C2r , r = 1, 2,... , in terms of Co, and C2r +1, r = -1, 1, 2,... , in terms of C1 is sufficient. 4.7.51. Alternative form of sotutlon in unstable region. 1°. By 2° §4.70, when (a,q) lies between b2n +1 and a 2n +l , the form of solution 00 YI(Z)

=

elJ-Z

!

C2r +1 e(2r+l)zi

(1)

r=-oo

ensures the reality of J-L. If J.L > 0, YI --+ 0 as Z --+ -00. Since a, q are real, the c are complex. Then it may be shown that if z is real, save for a constant complex multiplier, say Zo, there is a real solution which tends to zero as z ~ -00. Let

C2r +1

= Z od 2r +l , c1

Zo = ei 8o, d2r+1

= 1,

=

rZor

= I

P2r+l ei q,2r+l

(2)

for all r. We shalt now demonstrate that d2r +1 and d_2r - 1 are conjugate complex numbers. Substituting from (2) into (1) leads to 00

Zi)lYI(Z) =

eP.Z

! P2r+I{cos[(2,,+I)Z+cPsr+l]+i sin[(2r+ I)Z+CP2r+l]}. r=-oo (3)

For a real solution, the imaginary part of the r.h.s, of (3) must vanish identically. Hence P2r+18in[(2r+1)Z+¢Sr+I]-P-2r-18in[(2r+ l)Z-,p-Ir-l]

= 0,

(4)

and, therefore,

and q,2r+l = -(4)-2r-1+ 2811 ), (5) 8 being an integer. It follows that dSr+1 and a-ar-l are conjugate, 80 (3) may be written ,. P2r+l

=

P-2r-l'

ClO

Yl(Z)

t

=

2Zoe~! P2r+l c08[(2r+ 1)Z+~2r+l].

r=O By 'lid of 1°, 4.751, it may be shown that (cl/c_l)

= e-li'o.

(6)

4.75]

FRACTIONAL ORDER: SOLUTION O}4' EQUATIONS

85

The second independent solution is 00

Y2(Z) -.:.. Yl( -Z) = 2Zo e- p.z ! P2rf-l cos[(2r+ 1)Z-ep2rf-ll·

(7)

r=O

Then (6) -+ 0 as z -? -00 if J.t > 0, while (7) -+ ±oo as z ~ , Determination of ZOo For all integral values of r CZ,+l == Z od 2r +1 , with IZol == I. Since d 2r +1 and d_ 2r - 1 are conjugate, it follows that Ic2r +11 =

IC-

1·

Thus we may write

2r - 1 C2r +1 == P2r+l ei .p2r+ 1, C- 2r-1

=

C2r+l/d2r+l == ei (t/12r + l - eP2r 11)

Zo == ei 80

==

C -2r- lid - 2r- I

==

-00.

(8) (9)

==

P2r+l ei .p- 2r-l.

Then ( 10)

ei(t/J -2.--1 +t/J2r 11 +2811) ,

(11)

by (5). Hence

+

200 = (tP2r+l-¢J2r+l) -to (tP- 2r-l +¢J2r+l 2S1T) , so (}0 = !( .p2r+ I +.p -2r-l) 81T• ( 12) It follows from the footnote in § 4.50 that Zo == M o'Zo, i.e. Zo and Zo have equal angles. 2°. When (a,q) lies between b'}m' a 2n , by 2° §4.70 p, is real if

+

00

Yl(Z) == ellZ

!

c2re2rzi.

(1)

r=-oo

Take c2r = P2r ei eP2" Co == 2po .real, then for a real solution it may be shown that P2r = P-2r' cP2r === -(cP-21+ 2 81T) , so if r ~ 1, c2r and c- 2r are conjugate. Also

Yt(z) and

Y2(Z)

=

= 2el-'z{p~+ %l2rCOS(2rz+~2r)},

Yt( -z) = 2e-I-'z{po+

J/2r cos(2rz-~2r)}'

(2) (3)

In this case Zo = 1. 4.752. Example illustrating analysis in § 4.751. We shall consider the solution of y"+(1-0-32cos2z)y = 0, (I) obtained in § 5.33. The parametric point a = 1, q = Oe16, lies in an unstable region between bl and ai- From 2° § 5.33 we have C1 = 1, c- 1 = (5 10-2+O 998i ) ; Cs = -(J·994X IO-2+1·198x IO-3i), (2) c- 3 = -(2-38 X 10-3 + ]-978 X IO-2i); Cs = (1·325x lO-4+1·24x lO-5'i), e95X

C- 6

=

e

(2 e024 X IO-5+1e313X IO-ti)_

(J}4~N}4~ItAL

86

THEORY: l4'UNC1'IONS OF

[Chap. IV

1°. We find that Icll ~ Ic- 11, Ical ~ IC-al, Ic 5 1~ Ie- 5 1. (a) «PI = 0, «P-l = tan-1(O·998/ 0·0595) = SO·56°_ Hence by (12), 1° § 4.751 with ,8 = 0, ~o_

eo = 1(<
(1)

«P3 = tan- 1(-1-198/-19·94) = 183-44°, ~-3 = tan- 1 (- 1·978/ - 0·238) = 263-12°.

Hence by (12), 1° §4.751 with

eo = (c)

43·28°.

8

= -I,

!(cPs+¢t-s)-180° = 43-28°.

(2)

rPs = tan- 1 (0·124/ 1·325) = 5·37°, ~-5 = tan-1 (1-313/0-2024) = 81.25°_

Hence by (12), 1° § 4.751 with

eo =

8

= 0,

1(<
=

43-31°.

(3)

The agreement at (1), (2), (3) is satisfactory for the purpose of illustration. Zo = e43"28°i = O·728+0-686i

3°.

/Zol

and

= 1_

d1 = c1 e-4ao28°i = e-4So28°i d_ = c- e-43·28°' = e 43'280 ' 1

}

1

conjugate.

= /c s/e140"16i J . d-3 -- c- s e-43 -_ ICs /e219'84i conjugate. do = C5 e- 4S"28i = /cs/e-S7091i J . conjugate. d-5 -- C- 5 e-43"28i -_ /Cs Ie37 ds

=

Cs e-4So28i

028i

097i

(1) .

(2) (3)

As an exercise, the reader may check the remark in §4.750 that C1,.+1 and (C1/C-1)C- Ir- 1 are conjugate. 5°_ PI = 1, cPl = -43-28°; Pa = 2XIO-2, cPa = 140-16°; (1) P5 = I-33x 10-', c/>6 = -37-91°. By 10 §5.33 J.L = 0·08_ 6°_ Two approximate independent solutions of (1) §4_752 are Yl(Z)

and

~ 2Z0el'8{Jt2r+l cos[(2r+l)Z+tI>llr+1J}

(1)

4.75]

FRACTIONAL ORDER: SOLUTION O}4'

"'~QUATIONS

87

By combining these linearly independent solutions, we obtain the

even solution 1

-z (Yl +Yt) = 4 0

_

Yl ~ cosh fLz

2

! P2r+l 008(2r+ l)z cos tP2r+lr=O 2

-sinh JLz ~ P2r+l sin(2r+ l)z sin 4>2r+l'

,-0

and the odd solution l (YI-y,J = 4Z 0

ii2 ~ sinh J.LZ

(3)

f P2J'+1 cos(2r+ l)z cos 4>21'+1-

r=O

2

-coshJ.LZ I

r=O

P2,.+l sin (2r+

1)zsin cP2r+l .

(4)

In practical applications (3), (4) have the disadvantage that both. tend to infinity with z. 4.753. Formula for fL when a < o. p, may be calculated for (a, q) below a o in Figs. 8, 11, by means of (4) §4.74 with 'Itt = o. Then JL = - ifJ. The remarks on accuracy in the last paragraph of § 4.74 apply here also. As shown later, (7) § 4.91 may be used when (a, q) lies in an unstable region between bl1l. and am' m > o.

4.760. The iso-p. curves in Fig. 11. Consider the segment of any line between the curves bm, am' parallel to the a-axis, i.e. in an unstable region. The convention of §4.70 ensures the reality of /L. On b,n and amt fL = 0, while at SODle point on the segment of the line, ft attains a maximum value. This is true also for points on the segment of a line parallel to the q-axis and terminating on am. Thus if the turning-point (nearest the a-axis) on an iso-r, curve is q = qOt there are two values of a for any q > qo on the curve. Moreover, the coefficients in the solutions corresponding to the two a are different, except at the turning-point. No two iso-u curves intersect for the reason stated in § 4.71 regarding the same property of iso-p curves. The iso-,u curves are asymptotic to the characteristic curves bm , a,n which bound the unstable region where they lie, and they have no linear asymptotes. If the numerical data were available, families of iso-j, curves akin to those depicted in Fig. 11 could be plotted. Tabulation of a, a l , q, JL would permit the value of the latter being found immediately, or by interpolation. The solution of an equation with (a, q) in a , charted' unstable region would then be completed by computing the coefficients in (1) or (3), 2° §4.70, using the procedure in 2° §5.3:3 and § 5.34.

88

GENERAL THRORY: FUNCTIONS OF

[Chap.

I\~

4.761. Functions of order m+fL, q positive. When (a, q) lies in an unstable region between b2n and a 21l' we define the functions of order (2n+JL) by aid of (2), (3), 2° §4.751. Thus " ceuzlI+,.(±z,q) = Ke±"z{po+ J/2rCOS(2rz±q,2r)}

(b211 +,.). (1,2)

}'or the region between b2n +1 , a 2n +1 we use (6), (7) 1° §4.751, so ClO

ceu 2n +1 +1£(± z , q)

=

Xl e±1£3 !

P2r+1 cos(2r+ I)Z±4>2r+l]

r=O

(b2n +1 +1£ )' (3, 4)

where K and K 1 are normalizing constants defined in § 4.764. The above forms of solution are preferable in applications to those at (3), (4), 6° § 4.752 for the following reasons: (i) One solution tends to zero, the other to infinity with z, whereas the even and odd solutions both tend to infinity with z. (ii) In numerical work, tables of eZ , e- Z are better to use than those of sinh z, cosh z. (iii) Analytical work using (1)-(4) is likely to be simpler than that involving the even and odd solutions. It is of interest to remark that if we substitute Zol(A~~'til-il-L)+A~~r+!lil£» = P2r+l COSep2r+l'

(5)

and -iZOl(A~~'t11-ifL)-A~~:_11-'iJL» == P2r+l sin 4>2r+l' in the r.h.s. of

(6)

i

.ce} (z, q) = A~~~11-il-L). C?s(2r+ l-ijL)z, (7, 8)t se 2n+l-i/L r= -co ~ SIn . the r.h.s. of (3), (4), 6° § 4.752 are reproduced, with r = 0 to +00. Since JL is real, 2n+ I-ijL is complex, so (7), (8), and the corresponding functions derived by writing 2n, 2r for (2n+ 1), (2r+ I), respectively, may be designated functions of complex order. Discrimination between solutions for the same q but different a. We refer to § 4.760. On the upper and lower parts, and at the turningpoint of an iso-jL curve, we use the symbols 1.

ceum+il' ceum+~' ceum+ii' respectively. (9) For example, if (a, q) lies on the upper part of the curve between b2n and a2n , we write ."

ceuzn+;:&(±z,q)

= Ke±iiz{po+ r~l2rC08(2rz±~2r)}'

(10, II)

t The r.h.s. of (7), (8) are the series for celn+l+~(Z, q) and 8elntl+l3(z, q) with -il£ written for fJ.

4.78]

FRACTIONAL ORDER: SOLUTION

O~'

EQUATIONS

89

Symbolism for solutions with the same a but different q may be devised by the reader.

4.762. Degenerate forms of (1)-(4) § 4.761. If a remains constant as (a, q) moves towards a 2 " t in Fig. II, P, ~ 0, and in (1), (2) § 4.761 -+ A~2n), KP2r -+ Agn), and q;2r -+ 0, r ~ I. When j1 = 0, by proper choice of K, we obtain ce2n(z, q). As (a, q) moves towards b2n , !!' ~ 0, and Eo -?- 0, Ke2r ~ Bgn), P2r ~ -17T, r ~ 1, so with ~ = 0 we get ±se2n (z, q). Similarly the degenerate forms of (3), (4) §4.761 are ce 2n +1(z, q) and ±se2n +1(z,q). These forms may also be derived from (3), (4) 6° §4.752.

ss,

4.763. Functions of order m+lL, q follows: ceu 21£ +1'( ± Z, -q)

=

=

<

O. These are defined as

(_I)ne=Fl 1T1' ceu 2n +I'[ ± (!1T-Z), q]

(-I)n

Ke'F/Llr{po+

Jl

(b2n+/t )

(1, 2) (-ltPzr COS( 2rZ=ftPzr)}' (3, 4)

ceu21l+1+~(±Z, -q) =

(_l)ne T I 1Tf'

ceU2 11.+1+jL[ ± (! 1T -- Z), q]

(a 2n +1 +/.L )

(5, 6) 00

= (-l)nKI e=FJU

!

r=O

(-I)r p 2r +1 sin[(2r+ l)z=fCP'!r+l]' (7, 8)

The r.h.s, of (7), (8) may be expressed in a cosine series by altering the argument in [ ] to [(2r+ 1)Z-!1T=fcP2r+l]. The definition would then be in keeping with the notation ceu. When p, = 0 ill (3), (4), (7), (M) the functions degenerate to ce 2n (z, -q), se 21l +1 (z, -q); when I! = 0, we get =fse 2 11.(z, -q), ±ce2n +1(z, -q), as may be expected from Fig. 11, where q < O. The multiplier (-1)11. ensures that the signs of the degenerate forms are in accordance with the definitions in § 2.18, provided the upper signs are taken in the case!!: = O.

4.764. Normalization. The rules must be such that the degenerate forms of the functions are those given in §§ 4.762, 4.763. Then we must have

KZ[2p~+r~t~'] = and

Kfr~:~r+1 =

I,

I,

or

K =

or K 1

1/[~p~+r~Pt.Y,

= I/L~:~r+1Y.

t Except at a = m 2, q = 0, the direction of approach to § 4.711 the approach is with q constant. 4961

N

Q 2n

(I)

(2)

is immaterial. o.g. in

GENERAL THEORY:

90

~'UNCTIONS

OF

[Chap. IV

4.77. Convergence of solutions. Substituting (1) 20 § 4.70 into Mathieu's equation and equating the coefficient of e2r~i to zero for -00 to +00, yields the recurreqce relation (1) § 4.750. This is a linear difference equation which may be treated on the lines of § 3.21. We find that r =

1'U2rl""q/4(r+I)2~0 asr~+oo,

(I)

where u 2,. = C2r+2/c2r. A similar result is obtained for r negative. Then as in §§ 3.21-3.23 it may be shown that (1) 20 § 4.70 and its derivatives are absolutely and uniformly convergent in any closed region of the z-plane. Hence the series, differentiable term by term, represent continuous functions. They may also be proved integrable term by term. These conclusions are valid for stable and also for unstable regions of the (a, q)-plane. Similar conclusions may also be drawn respecting the solutions given in §* J.7H}, 4.76:1.

4.80. Solutionofy"+(a-2qcos2z)y The equation may he written

=-~

O,averylarge»q >

y"+ay = (2qcos2z)y.

O.

(1)

Neglecting the r.h.s., which is relatively small, by hypothesis, the formal solutions are cos zct, sinza l . Substituting cos ze! for y on the r.h.s. of (1) leads to y"+ay = q[cos(al+2)z+cos(ai-2)z], (2) of which the particular integral is [cOS(al+2)Z _ c?S(al - 2)Z] at+l al-l'

_1

(3)

4q

provided a =/:= I. Using this for y in the r.h.s. of (I) and solving again, the particular integral has terms in cos{ai±4)z, and so on. Hence the first solution of (1) may be expressed in the form 00

1/1

= !

c,cos(ai-2r)z

(a l non-integral).

(4)

r=-oo

If we use sin ctz and proceed as above, the second solution is found to be

ex>

Y2 =

! c,sin(a l r=-oo

2r)z.

(5)

Substituting either (4) or (5) in the differential equation yields the recurrence relation (6)

4.80]

FRACTIONAL ORDER: SOLUTION OF EQUATIONS

or If a l

~ T,

4r(a1- r )c, - q(C' _1+ C' +1) = O. we get the approximate relation 4rc,-(q/a l )(c' _1+C,+l) = O.

As in § 3.21, for convergence of (4), (5) we must have

r-+

91

(7) (8)

c; ~ 0 as

±oo.

Now a recurrence relation for the J-Bessel function is 4rJ;.(u)- 2U(J;.-1 +J;.+1) = 0,

(9)

and since J,. ~ 0 as r ~ ±oo, c, = constant X J,. If we take the constant to be unity (for simplicity) we get

c; = J,(q/2ai ) .

(10)

When qj2ai ~ r, the B.F. may be represented by the first term of its expansion, so c, ~ (q/4al )r/r! giving cr/cr- = q/4rai . Thus the 1

coefficients decrease with increase in r and may be neglected if r > '0' where at ~ roe Accordingly, approximate solutions of Mathieu's equation, subject to the conditions a very large js- q > 0, 'al ~ ro, are

Yl

and

~

Y2 ~

I"0

~(q/2al)cos(al-2r)z

(11)

I".

J;.(qj2a 1)sin(al - 2r)z.

(12)t

"=-"0 1'=-"0

Hence the approximate complete solution with two arbitrary constants is

y ~ AYl+ BY2 = 0

ro

I

"=-'.

J,(q/2ai)cos[(ai-2r)z-ex],

(13)

where C = (A2+B2)i, ex = tan- 1(B jA ). Since (13) is bounded in z, the point (a, q) must lie in a stable region of the plane (see Fig. 8).

4.81. Transformation of Mathieu's equation to a Riccati e

type. Write y = e Jwd:, where v r:': a}, and 1C(Z) is a differentiable function of z. Then dyjdz = vwy, d2y/dz2 = vy(dw/dz+vw2 ) . (1) V

If we put [1-(2Q/a)cos 2z]

= pi,

Mathieu's equation becomes

t If r covered the range -00 to +00, it is easy to show by aid of formulae (6), (7), p. 42, reference [202], that the respective representations of {l 1), (12) would be : :[ ai ( z -

t,sin 2Z)]. This form is obtained at (7), (8) § 4.81 by a different method.

GENERAL THEORY: FUNCTIONS OF

92

[Chap. IV

+

y" ap2y = o. Using the above substitution it is transformed to the Riccati type [133] (2)

since y ¢. o. Now if a ~ q > 0, a being very large, the first member of (2) may be neglected, and we get w

=

±ip = ±i(l- ~cos2z)1

(3)

~ ±i(1-!COS2Z).

(4)

Hence

v

j wdz ~ ±ial(z-ixsin2z),

(5)

and, therefore,

y

= e j wdz ~ e±ial[z-(q/2a)8In2z).

(6)

V

Then by the theory of linear differential equations, we may combine the two solutions in (6) as follows:

y1 ~

Hei a l [z- (q/2tI)sln 2z)+ e-ial[z-(q/2a)ll1n 2z1} = cos[a 1( z- 2~ sin 2Z)

l

(7)

and

Y2

~ 2~{eifll(z-
Combining (7), (8) with two arbitrary constants leads to

Y where y

= ~1+BY2 = ocos[al(z-2~8in2Z)-cil

G = CA2+BI)t,

and ii

= tan- 1(BlJ).

(9)

Expanding (9) gives

~ o{ COS(2~l sin 2Z)008(aIZ-ci)+Sin(~1 sin 2Z)Sin(aIZ-ci)}.

(10)

Expressing the first factor of each member of the r.h.s, in B.F. series [202, p. 42], we get, with h = (qj2a l ), which may be ~ I,

Y

~ O{[.fo(h)+2r~1 JlIr(h)eos4rz]cos(a1z-ii)+ +

[2,.~oJlIr+1(h)sin(4r+2)z]8in(alz-ii?}~

(11)

00

=

OI

r--('X)

J;.(k)cos[(ai-2r)z-a].

(12)

4.81]

FRACTIONAL ORDER: SOLUTION

O~'

EQUATIONS

93

This has the same form as (13) § 4.80, but in (12) the S1l111111ation covers a doubly infinite range of T. Owing to the rapid decrease in J,.(k) beyond r = To, a finite range of T is adequate in applications. The approximate solution of y"+(a+2qcos2z)y = 0 is obtained from (12) by writing -h for h. Since ,~( -It) :-~= (-1 y.lr(h), we get

Y~0 A~

<X>

!

(-I)rJ,.(h)cos[(al-2r)z-ci].

(13)

r--oo

in § 4.80 the point (a, q) must lie in a stable region of Fig. 8.

4.82. More accurate approximate solution of

=

y"+(a-2qcos2z)y

O.

The solution at (9) § 4.81 is a circular function with periodically varying argument, but constant amplitude factor C. We shall now derive a closer approximation in which the amplitude factor is periodic in z. We start with (2) §4.81, and assume that [133J 1

1

W = WO+-w 1+2W2+'." v v

(I)

the w(z) being differentiable functions of z. Then

W2

2

2 I( 22 ) = WO+-WOw l +-2 w 1 + W O w 2 +..., v v

ldw --d v z

1,

I

,

(2)

1 ,

= -WO +v2W1 +SU'2+···· v v

Substituting (2), (3) into (2) § 4.81 gives 2+ p 2 Wo

1 (w't + w12 + 2wow2 )+··· = O. +-v1 (W o'+ 2wowt ) +2 v

Equating the coefficients of yO, v-I,

v- 2, •••

to zero yields

W~+p2 = 0,

(5)

w~+2WOWl = 0,

(6)

w~+wj+2WOW2

=

Wo

=

±ip, so v

(7)

0,

Z

From (,5),

(4)

Z

f wodz = ±iv f pdz;

(8)

From (6), WI

1 w~ = ---,

2wo

so

e I

WI

d

Z

dw o = --1 Ie -+constant 2

Wo

= log AWol = log(Ap-l)=F!1Ti.

(9)

94

GENERAL THEORY: FUNCTIONS OF

From (7),

W2= _(W;+Wf) = ![W~_~ W~IJ 2w o 4 wft 2 w3 =

f

:r i ( " ;~ T 8 3 2pp -3p ),

p

z

!v

so

w2 dz =

Then by (1), (8)-(10), if tion v

[Chap. IV

lal

=f~ 8v

~ 1,

f (2 ~

pp"-3p p3

'2)dZ.

(10)

lal > 12ql, to a second approxima-

l fie (2 "-3 '2) pp p3 p dz. fz wdz = ±w• f~ pdz +log(Ap-l)=f!7Ti=f iaT

(11)

Hence

'

eV JW

dz

= Be

v

f wdz ~ (constant)p-le± ia' f[p-(2pp'-3p'')/8ap'J dz

0

0

,

(12)

the factor e=fl1Tl being absorbed in the constant. Combining the two solutions as in § 4.81 we get Yl(Z) ~

1 fa ~(z) dZ]' ~ constant p-1C?S[a mn

(13)

o where ~(z) == p-(2pp"-3p'2)j8ap3 . Then (13) are independent solutions of y"+ ap2y == o. A representative graph is given in Fig. lOA. If a ~ 1, a > 2q > 0, both real, the solutions (13) are bounded, so the point (a, q) must lie in a stable region of Fig. 8. If la I ~ 1, lal ~ 12ql, and the argument in [ ] is imaginary or complex, the solutions are unstable, so (a, q) then lies in an unstable region of Fig. 8. Omitting the terms in p', p" in q>(z), we get Z

al

8

I ¢J(z) dz = I (a-2qcos 2z)1 dz o

0

z

= (a+2q)1 f (1-,\2 coslu) I du o

= (a+2q)i[E(~,tn')-E(~,i1l"-z)] = (a+2q)lEl(~'z),

(14)

where E(>t, z) is an incomplete elliptic integral of the second kind

4.82]

FRACTIONAL ORDER: SOLUTION 014' EQUATIONS

with modulus ~ the form

=

Yl(Z)

YI

95

2[q/(a+2q»l < 1. Then we may write (13) in

~

constant C?s[(a+2 )lE ('\,z)]. (a-2qcos 2z)1 SIn q 1

(15)

+0'025

(A)

0

-0'025 1

C ='1

+0'025

(B)

l

0 -0,025

€=!

·0·025

(C)

0

-0-02

FlO.

€=t

10 (A). Curves showing combined amplitude and frequency modulation; see (13) § 4J~2 and (4) § 15.25.

When j2ql ~ lal the approximate solution of Mathieu's equation In8rY he derived as shown in 2° § 6.20. The argument of the circular functions in (15) is periodic: so the frequency of repetition of the function fluctuates. It is defined to be

f =

W/21T == --!.. dd [(a+2q)lE.(.\,z)] 217' Z 1

= -{a+2q)I(I-~2cos2z)1. 217

(16)

GENERAL THEORY: FUNCTIONS OF

96

[Chap. IV

The 'periodicity' or reciprocal of the frequenoy is

1// =

21T/(a+2q)1(1-~2C082Z).,

(17)

whioh differs from tha,t when q = O,~n&mely, 21ra-1. From (15) the ratio (maximum/minimum) amplitude factor is [(a+ 2q)/(a- 2q)]~.

(18)

(A)

+1

(B)

t

o~~ -1

(c) £=1 Fro. 10 (8). As at. Fig. lOA, but for the derivative, e.g, I 1(t ) in (9) § 15.25.

4.83. Equation of the type y"+2Ky'+(a-2qcos2z)y = O. ASSU1l1e that K, ii, q are real, with I( ~ 0, q ~ O.Substitutil1g y ~ e-KZu(z) into the equation, we obtain the Mathieu equation u"+(a-2qcos 2z)u = 0,

(1)

with a == (li-K'2). If the parametric point (a, q) lies within a stable region of Fig, 8, the solution of the original equation takes the form

4.83]

OI{Dl'~I~:

FRACTIONAL

}4~QUArrIONS

SOLUTION O.F

97

(see (2), (5), 10 § 4.70)

== e-

Yl(Z)

ClO

KZ

I C2r cos(2r+,B)z T=-oo

ClO

I CZr 1-1 cos(2r+ l-t-,B)z, r=-oo

or e- K Z

(2)

and OC)

I

.'Iz(z) == e- Kt

c2rsin(2r+p)z

or

e- KZ

T=-~

ex>

I c2r +1sin(2r+ l+P)z r=-ClO (0

< fJ <

1,

f3

real).

(3)

Th« solutions (2), (3) are proportional to e- =cem+fJ(z, q) and respectively. These may be regarded as damped Mathieu functions of fractional order. If (a, q) lies in an unstable region of Fig. 8, the solution of (1) takes the form at (1) or (3), 2° § 4.70. Thus the solution of the original K

e--K::sPmr~(z,q)

equation is given by Yl(Z)

==

00

I

e(P.-K)z

r=-oo

and Y2(Z)

==

CZr e 2rz i

or

e -2rzi

or

ClO

e -(p. f K)z

I

r;:;:-oo

C + e(2r+l)zi 2r 1

(4)

00

C

2r

e-(P.+K)z

I

T=-oo

T=-~

Yl is Yl is Yl is Y2 is

I

00

e(P.-K)z

C + e-(2r+l)zi. 2r l

stable and -~ 0 as z -~ +00, if K > JL > O. unstable and ~ ±oo as z ~ +00, if 0 ~ Ie < ft. neutral and periodic with period 7T or 217, if K = JL stable and ~ 0 as Z ~ +00 in each case.

(5)

> o.

Conditions of solution are assumed (see 2° § 4.70) such that JL is real. If it is complex, R(p,) is to be understood above; also z is real. 4.84. Iso-fJl-t stability chart. This is illustrated in Fig. 11. The iso-fJ curves lie in the stable regions for Mathieu's equation, while the iso-u curves lie in the unstable regions. Each iso-f3 curve is single-valued in q, hut except for the region below a o, each iso-u is double-valued in q. For constant JL, q at the turning-point of the iso-« curve increases with increase in a, e.g. if JL = 0·1, q ~ 0·21 when a ~ 1, but q ~ 1·3 when a ~ 4·22. Suppose that in (1) §4.83, a = 1, q ~ 0·21, then (a,q) lies on the iso-u curve JL = 0·1 between bl and ai- Thus the solution of the equation is unstable. Referring to the original equation, if Ie > 0-1, the solution is stable and tends to zero as z ~ +00. Since (1,0-21) lies between the characteristics for se l and eel' by 2° § 4.70 the solutions take the form at the extreme right of (4), (5) § 4.83. If K = 0·1, (p.-le) = 0, so one solution is periodic with period 211, while the other 4961

0

98

GENERAL THEORY: FUNCTIONS OF

[Chap. IV

-+ 0 as z ~ +00. The iso-fL curve fL = 0·1 is its characteristic or boundary curve separating the stable and unstable regions. Thus for

y"+0·2y'

+(a-2q-eos 2z)y =

0,

(1)

the stable regions are larger than those for Mathieu's equation, by

7 6 5

4

t:t

'S3 en ~

::s

62

-!J

.-ro---+--_ _+---=a,--f-_ _~

Z

2

4

3

5

Values of fJ

.......'., . .'..,.~ .~

-1 L.....

'., ·,0,8 .". 1'.1

FIG.

.........

--=~~~-L--_--l..-_-

----L---~

Qo

II. Iso-pI' stability chart for Mathieu functions of fractional order. The iso-f3 and iso-I-' curves are symmetrical about the a-axis.

the areas included between the iso-I-' curves p, = 0·1 and the bm , am curves to which they are asymptotic. The range of f£ is -00 < f£ < 00: if positive it is 0 ~ I-' < 00.

4.85. Solution of y"+2KY' + (ci-2q cos 2z)y = 0, when a ~ 2q > 0, a > 1(1. We assume that the point [(a-I(I), q] lies in a-stable region of Figs. 8 or II. Then by § 4.83 Y = e-lC·u(z), u(z) being a solution of

4.85]

FRACTIONAL ORDER: SOLUTION OF EQUATIONS

(1) § 4.83. Hence its value is given by § 4.81 with a ingly, with hI == q/2(a-K 2 )i , we get

y

~

=

a-1(2.

99

Accord-

co

I J;.(h1)e-KZcos[{(a-Kt)i-2r}z-£Xl]· ,.::c-oo

01

(1)

4.86~ More accurate approximate solution of equation in § 4.85. It is evident that the required solution may be derived on writing (a-K 2 ) for a in (15) § 4.82 and multiplying bye- Kz • Thus we get

KZ Yl(Z) ~ _a constant eC?s[(a_K2+2 )iE (,\ ,z)], 2 Y2 (a-K - 2q cos 2z)1 SIn q 1 1

(1)

where '\f = 4q/(a-K 2 + 2q), and the conditions respecting (a, q) in § 4.82 obtain. In the foregoing cases, when q is negative the solution is obtained by writing (!1T-Z) for z in that already given, excepting the exponential index.

4.90. Solution of y" +(a-2qcos 2z)y

=

0 in an unstable region

of the (a, q)-plane. In § 4.12 et seq. the solution was shown to have the form

= ~c/J(z}, } (I) Y2 e-~ep( -z). Now ep(z) is periodic in z with period tr or 21', so it can be expressed in a series of Mathieu functions se2p, ce2p or se2p+1 , ce2p +l • In accordance with 2° § 4.70, for (a, q) in an unstable region between the curves b2P and a 2P' we write Yl(Z)

00

¢J(z)

= I

[02pce2P(z,q)+82pse2P(z,Q)];

(2)

pmO

and for (a, q) in an unstable region between b2P +1 and a 2:P+l we write 00

¢J(z)

= p::zO I [C2P+1 ce2P +1(Z' q)+82P +1se2P +1(Z' q)],

(3)

the 0, S being determinable constants. Putting r = 2p or 2p+l, as the case may be, our proposed solution takes the form Yl(Z} = e:J:,.1I

f

[C,ce,(z,q}±S,se,(z,q}].

(4)

Yt p=O Substituting Yl(Z) from (1) into Mathieu's equation yields ep"+2fL4>'+(a+1£2-2qcos2z)ep = O. (5) From (2), (3) with r = 2p or 2p+ 1, assuming uniform convergence, 00

ep"(z) =

I

(O,ce;+S,se;)

~-o 00

=

I

p-o

[C,.(2qcos 2z-a,)ce,+S,(2qcos 2z-b,)se,],

(6)

nEN}t~RAL TH}4~ORY:

100

FUNCTIONS OF

[Chap- IV

where a" b, are the characteristic numbers for ce., ser' respectively, corresponding to the value of q used in the equation. Also CX)

I

eP'(z) =

p=o

(7)

(Cr6e;+S,se;).

Substituting from (2), (3), (6), (7) into (5) leads to 00

I

[C,(2q COR 2z-a,)cer+k~,(2q cos 2z-b,)se,]

p==o

00

+2fL ~ (C~ ce;+Aqr se;)+ l~=O

ex')

-t·(a+fL2-2qcos 2z) I «),ccr-t-A9rser) == O.

(8)

p~:"o

The series being assumed absolutely convergent, (H) may be written 00

00

00

I G,{a-a,+1-'2)cer + p=o I 8,{a-b,+JL2)sc,+2JL p=o ~ (C,ce;+S,se~) ::::: o. p==o

(9)

Multiplying (9) by ce. and integrating with respect to z from 0 to 21T, by aid of§§2.19, 2.21,14.42, we get 21T

CX)

f L Sr8c; ces dz =

1TOs(a-a8 + p.2)+ 2p.

o

1TC8 ( a - as +fL 2 )+ 2fL

or

0,

(10)

p==o 00

I s;«; == p=o

(11)

0,

217'

where

Krs

f Be; ce dz.

=

s

o

Multiplying (9) by se, and integrating as above leads to 21r'

1T8s (a - b. + p.2)+ 2p_f o

By (~) § 14.42,

2"

f ce;ses dz = o

00

I

= o.

0, ce;ses dz

(12)

p=()

2w

-

f se~ce, dz =':: -K

ST

,

and by virtue of

0

uniform convergence (12) may be written 00

1T8s (a- bs+p.2)-2JL

so

I G~ Kar = 1):::0

0,

(13)

(14)

with 8 = 2j or 2j+ 1, as the case may be. Writing m for 8 in (11) and substituting for S, from (14) yields

4.90]

FRACTIONAL ORDEI{: SOLUTI()N O}4' EQUATIONS

101

The eliminant of this set of equations is-in effect-an equation from which 1L lllay be determined (compare (:3) § 4.~O).

4.91. Approximate solution when a

> q real> o.

In this case

we see from Fig. 8 that a lies ill a narrow unstable region where bnt < a < am. It is to be expected, therefore, that the dominant terms in (2), (3) § 4.90 will be of order m. Accordingly as an approximation we shall take

ep(z) ~ O,ucem(z,q)+Smse".(z,q),

(1)

the other 0, S being assumed negligible by comparison. Then by (1) and (1) § 4.90 and

Yt(z) ~ eflZ[Cmcem(z, q)+~n sem(z, q)]

(2)

Y2(Z) ~ e-flz[ C1t~ cem(z, q) -~Il sem(z, q)].

(3)

'rhus (15) §4.90 luay be written Gm[7T2(a-am+JL2)+4JL2K~l1n/(a-b,n+JL2)] = O.

(4)

Now Om -# 0, so (4) gives

4~2K~m+(a-am+1-'2)(a-bm+1-'2)=

O.

(5)

1T

Neglecting

ft4, 011

f-L2

the assumption that IILI

~

1, we obtain from (5)

= (ltlll--a)(a-bll\)/(~2K~lIn+2a--alll-bm).

(6)

When q is small enough, by §§3.32, 14.42 and tabular values of the coefficients A, R [95 J, we find that K;,f1t ~ 1T21n 2, while (2a-a m -- bm ) <{ 4m 2 •

Hence we ruay express (6) in the approximate form J.L ~ ±[(am-a)(a.--bm)]l/~m.

(7)

In Fig, 12, (a2-a), (a--b 2 ) are the segments of the line a 2b i . Their product is a maximum when a is the mid-point. Then for an assigned q ILmax ~ (allt - bm)/4tn. (8) By virtue of this property, when q is small enough the turning-point of an iso-j, curve occurs approximately midway between the bounding curves for the region, i.e, b2n , a 2n or b2n+1 , aSn +1 as the case may be. Also, when q is small enough, the series for am' bm in § 2.151 may be used in computing p., as exemplified in § 4.92.

102

GENERAL THEORY: FUNCTIONS OF

[Chap. IV

To the degree of approximation in (1), equations (II), (13) §4.90 become (9) 1T().(a-a.+p.2)+2JL~nKm8=== 0

= o.

(10)

wG",(a-am+p,2)+2JLSmKmm = 0

(11)

o.

(12)

1TS.(a-b.+,..,,2)_'·2p.Gm K. m

and In particular when and

8

=

m, we get

n-Sm(a-bm+JL2)-2,.,.l.l"mKmm =

From (12)

2p.Om Kmm/n-(a-b m+p.2).

s'n =

o

2 ~"'lIG.

Values

3

4

Of 'I

(13)

5

12. Jllust.ruting calculation of p, p..

Substituting for 2JLKmm /1T from (5) into (13) yields S m

=

G m

(a m-a-p,2)1 a-bm +,. , 2

·

(14)

Substituting for p, from (5) and for 8 m from (13) into (9), we obtain

q, = -Om KmIJ (am-a-p,S). K

mma-as+,.,,1

(15)

Substituting for p. from (5) into (10) leads to

_

S. -

K tma [(a m-a-JL2)(a-bm +p,I)]l (a-b.+p.S) •

0",K".m

In (15), (16) it may be remarked that

8

(16)

corresponds to r in (4) §4.90.

4.92]

FRACTIONAL ORDER: SOLUTION OF EQUATIONS

103

4.92. Example. Find a first approximation to Yl(Z) for the equation yH+(1-0·32c082z)y = o. Here a = I, q = 0·16, and the point (a,q) lies ill an unstable region of the plane between the curves hI' a 1 in }"ig. 8. TheIl from (2), (3) §2.151 (1) b ~ l-q-lq2 i

a1

and Thus

~

(2)

l+q-lq2.

(a-hi) ~ Q+ i q2, } so (a1-a)(a-b1) ~ q2 __ J4q4 ~ q2. (ai-a) ~ Q-lq2, Hence from (3) above, and (7) § 4.91 with in ~ 1, \VC obtain

(3)

fL ~ ±lq = ±0·08, (4) which is the value calculated in § 5.33 using an entirely different method. Substituting for the various quantities in (14) § 4.91 with m :::::: 1, we have q2)!

s

~

1 -

~

(l-iq)!

0 (q-lq2- 1 = c 1 q+lq2+1q2 1 l+iq 0·9401 .t

(5)

Substituting for t-, 8 1 , q in (2) § 4.91 gives a first approximation to Yl(Z), namely, Yl(Z) ~ 0 1 eO' 08Z[ cet(z, 0·16)+0·94 se1(z, 0·16)], (6) 0 1 being arbitrary. Also writing --z for z, the second solution is Ys(z) ~ 0 1 e-O'08~[cel(Z' O·16)-O·94se1(z, 0·16)], (7) and the complete solution with two arbitrary constants is, therefore, Y ~ Ae-O·08~[cel-O·94sel]+Beo·08z[cel+0·94sel]' (8) the constant 01 being absorbed in A, B.

4.93. Solution of y" +(a-2qcos 2z)y = f(z). This is given by (13) § 6.22 to which section reference should be made. 4.94. Solution of y"+2KY'+(a-2qcos2z)y =/(z). The substitution y = e-ICZu(z) transforms the equation to u"+(a-2qcos2z)u = eIC1 (z), (1) 2 with a = (a-K ) . Then the solution of (1) is given by (13) § 6.22. If q < 0, or if cosh be written for cos, -00 < q < 00, in this section and in § 4.93, the procedure is first to obtain the solution with the r.h.s. zero, and then use (13) § 6.22.

t

The sign of this tenn is that of (13) § 4.91, where we assume I" is positive.

V

S()].JUTI'ON OF EQUA'l'IONH

NV~IEltI(jAL

5.10. Calculation of

/3, fL, with q

small and positive. Provided be used, AIAo (14) §4.30, or one of that type suited to the region of the (a, q)-plane where U, q lies, is valid for a integral or fractional, provided q is not too large. If an iso-fJl-' stability chart of the type in Fig. 11, but 2 or 3 feet square, and covering a greater range of a were available, {3, ,.,., could be obtained directly. If a 1 and q were plotted, the vertical spacing would be more uniform than that in Fig, II, while interpolation near the ai-axis would be almost linear. Tabular values of ai, q, and f3 would serve the same purpose and facilitate interpolation. 1°. Calculate fJfor y" +(2-0·32 cos 2z)y === o. Here a = 2, q == 0·16 and the parametric point lies in a stable region of Figs. 8, 11, between at and b2 • First we obtain an approximation, 011 the a-axis fJ = fractional part of ai, so taking a. == 2, f3 == 21- 1 ~ 0·414. This is an approximation to fJ at q = 0·16. In Fig. 11 the f3 const. curve through (2~ 0·16) meets the a-axis where a < 2. Thus the correct value of f3 < 0·414. Method 1. Using (4) §4.7 .1 and omitting the term in q6, we get a -:j.1n 2, (4) §4.74 or (2) §4.23 may

1+!J=

f3

80

J!= ,. .

17XO.164

0·162

[ 2-2-0~j62+-32X2-=-==

1'98,2·, (1)

0·4097.

As we shall see in § 5.20, this is a fairly close result. Method 2. We apply (14) §4.30, but have to calculate sin2u first. Then (12) § 4.30 may be abbreviated and written in the approximate form

I (l _11) 65 3 _ q2 cos 2a - q(1) l _ _ q2 cos2a+----q4--q2+1-a _q2

4

2

64

64

Substituting a = 2, q

=

8

== O.

(2) 0·16 gives

c08 2 2O' -

so

1536

25-1 cos 20'-158·5 = 0,

cos 20' = -5·2 or +30-3.

(3)

5.10l

NUMERICAL SOLUTION OJ." EQUATIONS

105

Referring to Fig. 8B, we see that in (3), a = -11T+i8 for the region in which the point (2, 0·16) lies. t Then cos( -1T+ 2i8) == - cosh 2fJ = - 5·2, this root of (3) being chosen since the positive one is inadmissible. Now Icos20'1 > 1, so (4) sin2a = ±i(cos2 2a - l )1 =::: ±5·1Ii, and by (14) §4.30 we obtain JL ~ ±O·4085i, so f3 = 0·4085, (5) where the positive sign is used conventionally. Thus (I), (5) differ in the third decimal place. In § 5.20 the former result is shown to be the more accurate. 2°. Example. Calculate f3 for y"+(6·25--1·6cos2z)y== o. Here a =::: 6·25, q == O·S, so the parametric point lies in a stable region of Figs. 8, 11, between the curves a 2 and ba. Then 6·25 l = 2·5, so taking the fractional part, 0·5 is a rough approximation to the value of fJ. In accordance with our remarks in 10 , fJ < 0·5. Method 1. From (4) §4.74, omitting the term in q6 (for purposes of illustration) 5·25 X 0.8 2 (31·25+ 7) X O.S'] 1 (1) 2+f3 == [ 6·25- 2-x 5.25 2= 0.64 - 32x 5·25 s-x 2.25 '

== Hence

6·1869.

=

2·4873. f3 = 0·4873.

(2)

Method 2. Using (14) § 4.30 and (2) 1°, 1L ~ (-O·4-f-O·012+0·0096-0·00·122+0·0227)sin 2a (3) ~ ±1·3S5i, so fJ ~ 0·385. (4) In this case unity is present, because (14) § 4.30 has been used outside the region stipulated in § 4.32. The value f3 == 0·385 is in marked disagreement with (2), the former being in error, since q is too large for (12), (14) §4.30 to yield accurate results. This is indicated by the fact that 0·0227 exceeds the three prior terms numerically.

3°. Example. Calculate IL for the equation y" + (1-0·64 cos 2z)y = o. The point a = 1, q = 0·32 lies in an unstable region of Fig. 8 between bl and a l . Here (2) §4.23 does not apply, since a = n~2 = 1~ so we fall back on (12), (14) § 4.30. From"the former c08 22a - 12·72 cos 2a - l ·512 = 0,

t As shown in § 4.60 this value of (I) § 4.60, a = iB. 4161

(J

applies to (2) for the region concerned. For p

106

NUl\IERICAL SOLUTION OF EQUATIONS

[Chap. V

giving 00820' = -0-115 or +12·835. (I) Referring to Fig. 8 B, we see that a is real, 80 Ic08 20'1 ~ 1 and, therefore, we choose the negative root. The reality of a entails sin 20" = ±{I-cos2 2a )l, 80 (2) sin 20' = ± 0·9934. By (14) §4.30 we find that p, = ±0·158 (see Fig. 11). (3) The above examples illustrate various formulae and their limitations. If /L is real, [cos 20'1 and Isin 20'1 ~ 1, but if imaginary they exceed unity. Hence for equal accuracy, q in (12), (14) § 4.30 may be greater when the solution is unstable than when it is stable (p. imaginary). If p, is computed using (6) §4.91, the result is 0·16, being less accurate than (3).

5.11. Calculation of f3 using continued fractions, q > o. When a, q are such that (4) §4.74 does not give sufficient accuracy, the method below may be used. Applying the procedure in §3.11 to (I) §4.17 we get the infinite continued fraction [94, 134] c2r -q/(2r+p)2 q2/(2r+p)2(2r+2+p)2 C1r- 2 = -1'::"-li/(2r+!3)2l-aJ(2r+2+!3)2 q2/(2r+ 2+p)2(2r+ 4+P)2 (1) --i--a/(2r+ 4+(J)2 The next step is to derive au alternative continued fraction for c2r/car- 2 • In (1) §4.17 we replace r by (r-l), divide throughout by qear- I , and obtain c",./ctr _ 1 = [a-(2r-2+p)2_ q(clr _,/clr _ 2 )]/q, 80

c1r-,JC ar = -q/[(2r-2+p)2-a+q(clr _ t/c 2r _ 2 ) ] _

-qj(2r-2+fJ)1

(2) (3)

q2/(2r-2+fJ)2(2r-4+fJflA

- l-a/(2r-2+,8)1-

l-a/(2r-4+,8)1

.... (4)

But (2) may be expressed in the form C,,/CI1'_1

= {[ -(2r-2+p)2+a]/q}-(C2r_./CIr_z).

(5)

Writing (r-l) for r in (4) and substituting it for the third member of (5) yields the alternative continued fraction e /e = -(2r-2+,8)I+ a q/(2r-4+,8)1 .. "'-1 q l-aJ(2r-4+p)lql/(2r- 4+fJ)I(2r- 6+fJra

+

l-a/(2r-6+f3)1

(6)

6.11]

NUMERICAL SOLUTION OF EQUATIONS

107

Then (1), (6) are equal for all r provided fJ has its correct value. Thus with r = 1, we get, respectively,

_ .,j _ -q/(fJ+2)1 C

1)0 -

Co -

ql/(fJ+2)I(fJ+4)1

l-a/(fJ+2)2-

l-a/(fJ+4)1

ql/(fJ+4)I(fJ+6)1

_

-T=-a/(,8+6)1 _ ..., (7)

_ _ J Vo - c1 Co -

a-fJ2

qJ(fJ-2)2

qIJ(fJ-2)I(fJ-4)1

-q- + l-a/(fJ-2)2- l-a/(fJ-4)1 _....

(8)

(7), (8) are based upon the form of solution (2) 1° §4.70 which progresses in even orders of r. Consequently they are suited for computing fJ when (a,q) lies in a stable region between a 2n and b2n +l • If (a,q) lies between a 2n +l , b2n +2 the appropriate solution is (5) 1° §4.70, so we use the recurrence relation

[a-(2r+l+,B)2]C2r+I-Q(C2r+3+C2r-l)

which is (1) § 4.17 with (P+ 1) for substitution for f3 in (7), (8) gives v _ 1 -

ill

=

f3

and

C2r +1

=

0,

(9)

for elr • Making this

-qJ(fJ+3)2 q2/(,8+3)2(,8+5)2 Q2/(fJ+5)2(fJ+7)2 l-a/(fJ+3)1- l-a/(fJ+5)1 _ l-a/(fJ+7)1 _ ... , (10) a-(,8+ 1)2

q

q/(fJ-I)'l.

+ I-a/{fJ -1)2-

q2/(fJ-I)2(fJ-3)2

I-a/{ fJ -3)2 - - ... ·

(11)

The correct value of p in (7), (8) or in (10), (II) is that for which Vo = vo, or VI = VI' 0 < f3 < 1. In practice we aim to make vo-vo, or VI-VI' vanish to an adequate number of decimal places. Convergence of the C.F. may be considered as in § 3.14. A numerical example is given in § 5.13 et seq.

5.12. Calculation of

j-L,

q moderate and positive, solution

unstable. When the parametric point lies in an unstable region of Figs. 8, 11 between b2n +2 and a 2n +2 , JL is real and we write fJ = -ip, in (7), (8) § 5.11. Then -Vo and vo- (a+ JL 2)/q are seen to be conjugate complex numbers. Let Vo = -(x+iy), and vo-(a+JL'J.)/q = x-iy. When (vo-vo) = 0) the imaginary part vanishes and

2R(vo) = -2x

= (a+p,2)/q.

(1)

It is possible to satisfy this equality by evaluating Vo alone such that twice its real part is equal to (a+p,I)/q, i.e.

2x+(a+JL2 )/q =

o.

(2)

108

NUMERICAL SOLUTION OF EQUATIONS

[Chap. V

When the parametric point lies between b2n+1 and a 2n +1' Pin (10), (11) §5.11 is replaced by -if'. The conditions for correct fL are R(v l )

=

R(v l )

and. l(v l )

=

l(v1 ) .

(3)

5.13. Calculate fJ for y"+(3-4cos2z)y = O. Before using the formulae in §5.11, we have to find a trial value of fJ to start the calculation. (4) §4.74 may be used for this purpose if a/q is large enough. Here a/q = 1-5 is too small, and in consequence the term in rf is too large. An iso-fJp, chart of the type in Fig. 11, or the tabular values corresponding thereto, may be used if available. We proceed now as follows: In Fig. 8 the point (3,2) lies in a stable region between at and ba. For q = 2, we find from Appendix 2 that a1 ~ 2·379, b2 ~ 3·672. Referring to Fig. 12, we divide BA = (4-1) on the a-axis in the ratio (3i - 2'3791)/(3'672. - 2'3791) = 0·508.

Thus A C = 3 X 0·508 = 1,524, so at 0, a = 2·524 and, therefore, = 2'524'-1 = 0'589. This is our trial {1, and we shall see later

fJ

that it is in error by +0·00956.... Since (a,q) lies between at, b2 we employ (10), (II) §5.11, and to demonstrate the procedure, choose a less accurate value of {1 than 0·589. We take fJ = 0·56. Neglecting components in (JO) §5.I1 beyond the third, and in (II) § 5.11 beyond the fourth-the computation being merely by way of illustration-e-we get 4/5'56 2 X 7'56 2 _ ----- ---- -- = - ------- 2. - -

q2/(f3+5)2(fJ+7)f. -

*

I-a/(fJ+7)2

•

0 0024,

1-3/7'56 2

q2/(fJ+3)2(fJ+5)2

-_

2

_. = - -4/3.56 - - - -X-5.56 ---- _ 00114,

l-a/(p+5)2-0'0024

1-3/5'562-0'0024

-q/(fJ+3)2

-2/3.562

.?

_

~~/(~~~~~(fJ__l!2~ l-a/(fJ-5)2 .92L(~_~_I)~~fJ-Yl~__

=

2

=

VI'

0.0402 '

4/0·_442X2·~~2 __ 1-3/2.442-0.0402

:c: _

l-a/(~-3)2-0'0402 .- .__ __ q/(fJ-I)2 _

4/2.44 X 4'4~ 1-3/4.442

(I)

-

1=-a/(,8+3)i=-0'0114 = 1-313,562 - 0'0114 - -0 ...09 -

=

7.64

'

2/0·44" ---- _- _ 0 .466, = ----------

l-aj(fJ-l)2-7'64

1-3/0'442-7'64

[a-(p-t- 1)2]/q = (3-1'56 2)/2 = 0,285,

so

'1;1

= 0-285-0'466

=

-0-181.

(2)

NUMERICAL SOLUTION OF EQUATIONS

6.13]

Since

109

*-

VI VI in (1), (2), we now repeat the computation with 0·58 and obtain VI = -0,207, VI == -0,208. Whether fJ should be increased or decreased to bring VI' VI closer to equality in (1) (2)

fJ =

may be ascertained by examining the continued fractions.

5.14. Interpolation. Next we use interpolation as illustrated graphically in Fig. 13 A.

In this case the crossing-point is near

fJ =0·56

v1

t:-

,8= 0'58"

0'181

(A)

,8= 0'56 (B) Flo. 13 (A), (B).

Illustrating linear interpolation for

p.

one extremity of the range of {J, which is likely to improve the accuracy of the result. '1'0 a first approximation-a higher one not being justifiable with three decimal places-,ve have

x 0·001 0-02 -x = 0-028'

so x

==

0'0007.

(J )

Hence to three decimal places

p= Using

P==

0-58-0-0007 = 0-5793 ,

(2)

0'5793, if we repeat the calculation with five component

NUMERICAL SOLUTION OF EQUATIONS

110

[Chap. V

fractions for VI' six for ill' and work to nine decimal places, a more accurate result will be obtained, Thereafter the accuracy may be improved by interpolation, and 80 on until the desired number of decimal places is reached. " To illustrate the procedure, in Table 5 we give the results of a calculation using two values of fJ different from those employed hitherto [94]. 5

TABLE

P VI

e,

0·6766 -0·20726 88 -0·20348 13

0·5794 -0-20683 52 -0·20679 23

ap =

0·0028 0·00043 36 = - 0·00331 10

av. = Af)l

VI - 1'1

= -

~Vl-~Vl =

0·00004 29 --0·0037446

Then to a closer approximation f3

=

0·5794+ (_~l-VL)dP,

dvi-av i

= 0·57943 2.

(3)

Continuing the calculation we obtain the data in Table 6 [94]. TABLE

fJ VI

61

0·5794 -0·20683 52 -0·20679 23

0·57943 2 -0·20683 049 -0·20683 017

6

~fJ -= 3·2 x 10- 6 ~Vl = 4·71 x 10-' ~VI = - 3·787 X 10-6

VI-V. =

~Vl --~Vl

-- 3·2 X

10-7

= -4·26 X 10- 6

To eight decimal places

fJ

= 0·57943

V -v 2+ ( --.?~ -~~)dfJ = 0·57943 224.

~Vl-~Vl

(4)

In interpolating, it will often be found that !d'i'\/ ~f31 }> 1~'l"1/~f31 or vice versa. It is essential, therefore, to ensure that the portions of the curves between the points of interpolation are sensibly linear. A check may be made by calculating VI' VI at an equal small interval 8fJ on each side of the crossing-point. The points so obtained should lie on the original lines.

5.15. Trial values of fJ. We shall now compare trial values of P found by the method used in §5.13, and also by (4) §4.74. The former may be crystallized in the empirical formula

NUMERICAL SOLUTION OF EQUATIONS

5.16]

III

the q in the differential equation. If [(2m+ 1}/ml]c/J is small enough, a first approximation to (1) is (2)

7. Trial values of fJ calculated from d'ifjerent [ormulae

TABLE

I a 3 (m

=

1)

6 (m = 2) 8 (m = 2) 36 (m = 5)

Formula used

j--q

I

2

2 3 16

_.~_1)_1 0·589 0·27 0·67 0·51

P (2)

' m too small 0·29

0·79 0·54

More accurate value 01 P

,B (4) § 4.74 inapplicable: see § 5.13 0·35 0·72 0·62

0·5794 0·34 0·70 0·583

In the second, third, and fourth rows of Table 7 the superiority of (4) § 4.74 is evident, although a/q is comparatively small, If in the fourth row q were 3, a/q = 12 instead of 2·25, and the accuracy would be improved appreciably. Wherever possible, the use of (4) §4.74 is preferable to obtain the best trial value. Although we have concentrated our attention on the evaluation of fJ, since it is needed in differential equations, the same principles may be used to calculate a if 13, q are assigned. First we obtain a trial a using either (6) § 2.16 or (1) § 5.15 suitably transformed, and then (if necessary) proceed with the method of continued fractions as in § 5.13 et seq. to obtain V o = vo, or VI == VI. 5.20. Solution of equations [134]. The methods adopted may be shown most readily by a series of numerical examples. We commence with the equation y"+(2-0·32cos2z)y = 0,

(1)

whose parametric point (a = 2, q = 0·16) lies in a stable region of Fig. 8 between a'i and b2 • Accordingly we choose the form of solution at (4) 10 § 4.70. Substituting it into (1), equating the coefficient of e(2r+l+#>zi to zero, r = -00 to +00, and taking 13 = 0·409 from § 5.10, we obtain the recurrence relation [2-(2r+l·409)2]C2r+l-0·16(c2r+s+C2r-l) =

o.

(2)

We have taken fJ = 0·409 instead of the more accurate value 0·4097 to illustrate a check later OD. In § 4.77 when r is large enough we showed that Ic2r +2/ C2r J ~ q/4(r+ 1)2, 80 that after a, certain r is reached the [c] decrease very rapidly. To start the computation we assume

NU~lERICAL

112

SOLUTION OJ4' EQUATIONS

LChap. V

that C2r +3 vanishes to the number of decimal places required in the solution, Taking r = 3 in (2) and neglecting c9 ' we get C2r - 1 =

so

c6

[2-(2r+I-409)2]0'21+l/ 0-16,

==

(2-7-409 2)6-25c

=

-330c ,_

(3)

7

(4)

With r = 2 in (2)

ca = -(2-5-409 2)6-25X330C 7 -

C7

= 5-62 X I04 c, _

(5)

With r == 1 in (2) c1

'rhus

c,

:::::: (~- 3-40n 2)6-25)<

5·62 )( 1()4 C7

+ 330c

7

= -3-38 X I06 c7_

(6)

=

(7)

-2·96 X IO--7C1 _

Using (7) in (4), (5) yields ca = -1-661 X IO-2C1 , c6 = 9-77 X IO-5 CI , C,

=

}

-2-96 X lO-'cI -

(8) (9)

(7)

These numerical values illustrate the rapid decrease in the c with increase in r, when a = 2, q = 0-16_ To calculate C- 2r- 1 we use the above method, and in the present instance commence by neglecting c- 9 - Then with r === - 4 and c- 9 == 0 in (2)

c- 5 With r ==

= ==

(2-6-591 2 )6-25c_7 -259c_7 -

(10)

-~3

c- 3 === -(2-4·59I2)6-25x 259c_1-c_1

= If ~::=

-2

r ::_: --I Thus

3-lxI04c_ 7 _

(II)

c- 1 == (2-2-591 2)6-25X 3-1 X I04 c_7 + 259c_7

= c1 = == c- 7

-9-12x I05 c_1 •

(12)

-(2-0-591 2)6-25X9-12XI05c_'l-3-1Xl04c_'l

-9-44X I06 C_ 7 _

(13)

== -1-06X 10-7 C1 ,

(14)

NUMERICAL SOLUTION 014' EQUATIONS

5.20)

and since Ic- 7/c- sl

<

4X 10-3 , we may anticipate that

lc- a/c-71 < SO

by (14), Ic-al

<

113

4X 10-3 ,

4X IO-IOC t • Using (14) in (10)-(12) yields

c- 1 == -9·12x I05C_ 7 == 9·67x IO-2 C1 , c- 3 = 3·1 X 104c_7 === -3·286 X IO-3cl , c- 5 = -259c_7 2·72 X IO-5 ct , c- 7 = -1·06X IO-7 Ct • Check on

fJ.

=

Writing r

(15)

(16) (17) (18)

0 in (2) gives

(2-1·409 2 )C1 = O·16(c3+c_t ) ·

(19)

Using (8), (15), the two sides of (19) are 1·472xIO-2C1

1·28XIO-2C1 ,

and

respectively,

(20)

from which it follows that the value f1 === 0·409 is a little too small. Now an error in the fourth decimal place in f3 causes a trifling error in O·16(c3+c_1 ) , so we may write 2-(1+,8)2 == 1·28 X 10-2,

(21 )

and an improved value of f3 is then found to be

f3

(22)

== 0·4097,

which is in accord with (1) § 5.10. 5.21. Solutionofy"--t-(2-0·:J2cos2z)y =--=

Y -where we have taken (1), \\"0 obtain y(z}

=

fJ ~

IX)

~

k

r=

-00

o.

By (4) 1°§4.70,

c2r+l e(2r+l·41)zi ,

(1)

..

0·41. Substituting for

C2r +1

from §5.20 in

c1[ ... -1·06X IO-7e- 6·59zi+ 2·72 X lO-5e- 4'59zi _

-3·286 >~ IO-3e- 2·s9zi+ 9·67 X 10-2e-o·59zi+el·41Zi_ -}·061,< 10--2eS·41zi-t-9·77 X lO-5 e5'41=i - 2·96 X lO-7 e7'4lz i +

...].

(2)

This form is inconvenient, 80 we refer to (5) 1 § 4.70. If the latter is substituted in the above differential equation, the recurrence relation (2) § 5.20 is obtained. Hence, apart from a constant. multiplier, 0

49tH

114

NUMERICAL SOLUTION OF EQUATIONS

[Chap. V

the coefficients C2r +1 must be the same in both cases. Accordingly a first solution of the equation is co

Yl(Z)

=

I C2r +1 cos(2r+l+,8) r=-co

= c1 [

(3)

-1-06X 10-7 cos 6-59z + 2-72 X lO-5cos4-59z-

-3-286 X 10-3 cos 2-S9z+9-67 X 10-2 c08 O-S9z+

+cos 1,41z-}-661 X 10-2 cos 3-41z+9-77 X 10-5 cos 5-41z-2-96 X 10-7 cos 7-41z+ ... J_ (4) The second solution is derived by writing sin for cos in the I terms in (3), but not in (4), since the terms corresponding to negative values of r would then have the wrong signs. Thus we get !/2(Z) = c 1[ ••• +1·06x 10- 7 sin 6-59z-2·72 X IO-5 sin 4-59z+ +3-2~6 X

10-3 sin 2-S9z-9·67 X 10- 2 sin 0-59z+sin 1·41z-

-1·661 X 10-2 sin 3-41z+9·77 X IO-5 sin 5·41z-

-2·96X 10-7 sin 7·41z+ .. _J_

(5)

If (2) is expressed in circular functions, (4) is the real and (5) the imaginary part, i.e, the real and imaginary parts are linearly independent solutions of the equation, which is to be expected. The complete solution, using (4), (5), is y =- AYl+BY2' ,6) where A, B are arbitrary constants, real or complex, and this defines a fundamental system. In an application, A, B would he obtained from the initial conditions. Check on the solution. (4) may be checked as in §§3.13, 3.16, by aid of the relationship

y"(O) = (2q-a)y(0) = -1-68y(O). '1"0 an adequate approximation we find that each side has the value -- I·HI. If greater accuracy is needed, Pshould be computed as shown in § 5.13 commencing with the value 0-4097_ The coefficients C2r +1

may then be found to the same degree of accuracy as {3, if r is increased and more significant figures used in the calculation.

+

5.22. Second method of solving y" (2- 0-32 cos 2z)y =-= o. Here we use the analysis in § 4.30 and refer to § 5_10. Then fL = 0-41'it in (5) 1° §5.10 corresponds to sin~u = -S-Ili, c0820' = -5·2, o = (-~1T+i8). Substituting in (16), (17) §4.30 we find that t In accordance with (22) § 5.20 we have taken this approximate value of fJi.

6_21]

NUMERICAL SOLUTION O}4' EQUATIONS

III

= -0·02478, Cs = -O-00817i_ Thus by (I), (2) §4.30 the first solution is

8a

Yl(Z) = eO'41zi ( sin(z-i8+11T)-0-02478 sin(3z-i8+11T)-

=

-O-00817i cos(3z--:i8+11T)+ _..] (1) eO' 41zi( cos(z-i8)-O·02478 cos(3z-i8)+

+O·00817isin(3z-i8)+ ...]. Expressing the circular functions in exponentials, (2) becomes 1Il(Z)

=

(2)

leB[(oo8 1-41z+i sin 1-41z)-1-661 X 10-2 (008 3-41z+

+i sin 3-41z)+e- 28{(cos 0-59z-i sin 0·59z)-3-295 X 10-2(00s 2-59z-i sin 2·59z)+ ...}]. (3)

Now sin2u = sin(-1T+2i8) = -isinh28 = -S-lli, 80 8 is positive and it has the value 1-166. Then e- 28 = 9·7 X 10-2, and (3) becomes Yl(Z)

= A 1[(COB1·41z+i sin 1-41z)-1·661 X 10- 2(oos 3·41z+ +i sin 3-41z)+ _.. +9·7 X 10-2(cos 0·59z-i sin 0·59z)-3-2x 10-3 (cos 2-59z- i sin 2-59z)+ ...],

(4)

with Al = le Making allowance for the reduced number of terms, (4) is seen to be a linear combination of (4), (5) § 5.21. For apart from the multipliers AI' iA 1 , the coefficients agree well enough for the purpose of illustration. Also, since the real and imaginary parts of (4) are linearly independent solutions of the equation, apart from arbitrary constants, (4) contains the complete solution given at (6) § 5.21. Proceeding to find Y2(Z), as shown in § 4.30, we write - 0 for o, Since c0820 is even but sin 2u is odd, we get 8.

fL = -O·41i,

8a

=

-0-02478,

ca = O·00817i.

Thus the second solution is y,,(z) = e- O' 41zi ( cos(z+i8)-O-02478 cos(3z+i8)-

-0·00817isin(3z+i8)+ .._],

(5)

which is merely (2) above with the sign of i changed. It follows, therefore, that the solutions obtained herein have the forms Yl(Z) = A t [! 1(Z)+ i/ 2(z)]

(6)

and Y2(Z) = A 1[! I(Z)- i! 2(Z)]. (7) In applications these forms will usually be inconvenient. Thus when fL is imaginary, the forms at (4), (5) §5.21 are preferable, They may,

NUMERICAL SOLUTION OF EQUATIONS

116

[Chap. V

of course, be derived from above, since fl(Z), !,.(z) are linearly independent solutions of the differential equation. Then the complete solution may be written y(z)

=

Afl(Z)+B/2 (z),

(8)

where A, B are arbitrary constante.

5.23. Relation between solutions of types § 4.13 and § 4.30. Although in § 4.50 we demonstrated symbolically that the solutions sre alike, as they must be for uniqueness, it is interesting to asoer.ain whether or not the two expressions for Zo in (11) §4.50 yield dentical numerical results. From § 5.20, c- 1 = 9'67 X 10- 2C1 , t and Tom §5.10, sin 20' = -S'lli, cos 20' = -5,2. Then 200s 2O' - l = -5-2, so cos o = ±1·449i. Now a = (-!1T+i8), 8 = 1·166, so ~os(-I1T+i8) = isinh8 = +I·449i. Also sin a = sin2a/2cosa = -1,763. Substituting these values in (II) §4.50, we obtain

.zo =

-(C1+c_1)/sina

= c1X 1'0967/1-763 =

0-622c t

(1)

c1 X 0'9033/1'449 = 0-623c1 . (2) l'he approximate agreement of (1), (2) is sufficient to illustrate the ioint in question. Here Zo is real, since the C are real by virtue of he parametric point of the equation lying in a stable region of the z,q)-plane of Fig. II.

.nd

Zo =

i(ct-c_1)/cos a

=

5.30. Example: Solve y"+(3-4c082z)y === O. In § 5.14 we alculated fJ for this equation to a high degree of accuracy, but for he purpose of illustration we shall take f3 = 0-58. Adopting the iheme outlined in § 5.20, we neglect CIS et seq. Then with r = 5 in re recurrence relation (1)

e get

=5

=4 =3

=2

C9

==

·1(3-II-58 2)cl 1 -6S-5cl l ;

== c7 == -!(3-9'58 2) X 65-5c l l - el l = 2-899 X I03c 11 ; Cs = !(3-7-58 2)X 2899c 11- C. = -7-893x I04CI 1 ; Cs

= =

-1(3-5-582 )

X 7893Ocl 1

- C,

1·102x I06C11 ;

r To conform with the notation in § 4.50, bars have been placed above the c.

(2) (3)

(4)

(5)

5.30]

NUMERICAL SOLUTI.QN OF

r=l

== 1(3-- 3-58 2 ) x 1-102 ><

C1

:=~

'rhus

c7

(6)

(-0-206830c l ) ,

(7)

0-OI4744c t ) ,

(8)

(-0-00054 2ct ),

(9)

(0-00001 2ct ) ,

(10)

(

-0-00054 3c1 0-00001 228cl

117

I06CII-C5

-5·331 X I06C11 •

ca = -0-2068c1 t Cs == 0-01478c1

== cg == C11 ==

J1~QUATIONS

-0-00000 01875c I

(II)

'To calculate c-2r --1' we commence by neglecting C- I 3 et seq.; so with r == -6

==

(;-9

~(3-10-422)C_ll

(12)

== -52·7c- 1l ;

r

~-5

_ C- 7 -

~_

(3-- 8.422 )

-

~------~ 2

C 11 52 7c-11--

== 1-78 X 103C_ l 1 ; r === -4

(3-6-42 2 )

C~-5 === - - -- ~ ----

( 13)

1780c_ 11 --

C_ 9

== -:l·405 X l04 C __ I1 ;

r

== -3

C 3 --- -

-==

(:~ --- 4.42 2 ) -

--2-- --

2·8p7 X lOSe

( 14)

3·405;< 104C_I1-C~7 (15)

11;

_ (:~- 2-42 ) • 5 e 5 --------2 807 X 10 c112 2

r == -2

C- 1 -

=== -3-674 X 10sC_II;

r= -1 Thus

c1

c- 1

(16)

(3-0-42 2 )

==

---2--3.674 x l05C_II-C_3

==

-8X IOsC_I 1 •

==

0·4592c l

(17) (0-45952 7cl ) ,

(18)

c- 3 === --0·3509c 1 c- 5 == 0·04256c 1

(-0-351:34 9cl ) ,

(19)

(0-04275 2c1 ) ,

(20)

c- 7 == - 0·00222 5c1

(-

0·00224 Oc 1 ) ,

(21)

(0-00006 6ct ) ,

J22)

c- 9 C- 11

=

0-00006 6c1

== -0·00000 125c 1

(23)

t The ratio C3/C 1 = - 0·2068 should be equal to VI in § 5.14. They differ in the fifth decimal place. Tho values in ( ) were obtained by machine using au elaborate method of computation [94].

NUMERICAL SOLUTION OF EQUATIONS

118

r

[Chap. V

Check on calculation. This may be effected by computing Then from (I)

= o.

c- 1 =

= =

C-1

for

(3-1.58 2 )

-"-2- C1-Ca (O-252+0·2068)c1

4·588 X 10-lc1 ,

(24)

which differs from (18) in the third decimal place. Mort) accurate c than those we have computed are given in brackets at the r.h.s. In (18), (24), c-1 is slightly too small. This defect could be rectified by using a more accurate {J, and working to six or eight significant figures.

5.310. Solutionofy"+(3-4cos2z)y = 0. Since the point a = 3, q = 2 lies in a stable region between the curves ai' b2 in Fig. 8, the preferred form of solution is ClO

I

Yl(Z) =

clr+ 1 cos(2r+ l + p)z

(1)

"a-CO

~

0·459 cos 0·42z+cos 1·58z-0·351 cos 2·42z-0·207 C08 3·58z+0·043 cos 4-42z+0·015 cos 5-58z-0·002cos6·42z-...

(2)

with c1 = 1. The second solution is (1) with sin for cos, so y,,(z) ~ -0·459 sin 0·42z+sin 1·58z+0-351 sin 2·42z-

-0·207 sin 3·58z-0·043 sin 4-42z+0·015 sin 5-58z+ +0·002 sin 6·42z-. __ .

(3)

The complete solution with two arbitrary constants is (4)

5.311. The Mathieu functions cel -58 (z, 2), sel -5S(z , 2)_ Referring we have to compute the normalizing constant

to §4.72,

(I)

Using the c in § 5.30, we find that K is unity, so by §§4.71, 4.72

and

~

O·S52!c1 • Now

C1

in § 5.310

ce t .58(z, 2)

~ 0-852Yt(z)

(2)

se 1.6S(z, 2)

~

(3)

O·852Y2(Z).

5.32]

NUMERICAL SOLUTION OF EQUATIONS

119

5.32. Alternative method of calculating {J, JL [134]. The method of checking (J in § 5.20 suggests how it may be calculated. Suppose the equation to be solved is that in § 5.30. First we obtain & trial fJ as in § 5.13, say, 0·56. Then we use (I) § 5.30 with r = 0, p for 0·58, and get [3-(I+,8)2]C1 = 2(c3+c_ t ). (I) Analysis as in §§ 5.20, 5.30 leads to the values of Ca, C-1 in terms of Ct. Neglecting c" c_, and coefficients of higher orders, we obtain

=

[3-(I+,B)2]C1

so

2(-0·2095+0·468)c1 ,

(2)

,8 = 0·576,

(3)

choosing the positive root. The process may now be repeated using more significant figures in the working, and commencing with, say, C15' c- 1S• The result will be more accurate than before. Repetition may continue until the desired accuracy has been attained. It may be preferable, however, to use interpolation. From (1) with fJ = 0·56 and by aid of (2), we have (3-1·56 2)cl - 2(- 0·2095+ 0·468)c1

givIng

L1

=

= L1cl ,

0·0494.

Taking f3 = 0·58, we get ca' c- 1 from § 5.30, and (I) we have

(5) 011

substitution in

(3-1·58 2)c l-2(-0·2068+0·4592)c1 = L 2 c1 ,

so

L2

(4)

== -0·0012.

(6)

(7)

Having obtained values of L on each side of zero, we interpolate and calculate fJ when L is zero. Thus (Fig. 13B) x = (0'58-0-56)0,0012 ~ 0-00047 0·0494+0·0012

(8)

·

{J ~ 0·58-0·00047 = 0·5795. (9) This may be improved by repetition and application of procedure similar to that in §5.14, where an accuracy to eight decimal places Hence

is achieved.

The foregoing method is applicable to the determination of IL (real) when (a, q) lies in an unstable region of Figs. 8, II, but the starting value of JL would need to be assumed, unless a rough trial value could be obtained from (6), (7) §4.91, or as in 30 §5.IO, 10 § 5.33. Also, if f3 or fL is assigned for a given q, the value of a may be calculated.

NUMEI~ICAL

120

SOLUTION OF EQUATIONS

[Chap. V

5.33. Example. Solve y"+(1-O·32 cos 2z)y = O.t 1°. From Fig. 8 \\re see that the point a = 1, q = 0-16 lies in an unstable region between bl and a l . Thus a, J.L will be real and [cos zo] < 1. Using (2) IO§5_IO"we obtain the approximate equation cos 2 2a - 25 cos 20'-1-5 == 0, cos 2a

80

Since [cos 2<11

<

= =

(1)

12-5± 12-56 --0-06 or +25·06.

(2)

1, we take

cos 2a

= -

0-06,

sin 20' == =f(I-cos 2 2a )1 = =f0-9982.

so

By Fig.

~BJ

-i1T ~ a

~ 0,

a

:=::

so

--46-72° == -=-0·815 radian.

(3)

Using (] 4) § 4.30 we obtain J.L ~ ±O·OS

(compare (4) §4.92).

(4)

Also by (l6), (17) § 4.30 8a ~

ca ~ -0·0012.

-0·02,

Substituting a, fl-, etc., into (I), namely, !ll(Z) ~

eO·08z[Hin(z+

(~)

(5)

§ 4.30 yields the first solution,

O·8J 5)-0·02 sin(3z+0·815)-

-O·OOI2cos(3z+0·S15)+ ...].

(6)

To derive the second independent solution ,ve write -0' for a, Then c3 being odd functions of a, change sign. Thus Y2(Z) ~ e- o·o&e[sin (z- O·SI 5)- O·02 sin (3z- 0-S15)+

JL,

+0-0012 cos(3z-0·S15)+ ... J.

Now Yl

~

±oo as z -~

Y2 -+ 0 as z ~~

+00,

(7)

so this part of the complete solution is unstable,

+00, so this part of the complete solution is stable,

Hence the complete solution, with arbitrary constants A, B, namely,

y

==

AYl+B,'I2'

is unstable and tends to ±oo as z ~

t

+00.

See § 4.92 for a different method of solution.

(8)

5.33]

NUMERICAL SOLUTION 014' EQUATIONS

121

2°_ We shall now obtain the solution in the form given at (3), 2° §4_70, namely, Y1 (z) -- efLZ

00

~

~

r=-oo

c2r+l e(2r+l)zi ,

(1)

using the value of JL at (4), 1°. The recurrence formula for the c is (1),2° §4.750, which may be written

{a+ [JL+ (2r+ 1)i]2}C2r+l-Q(C2r+3+C2r-l) ==

o.

(2)

Putting JL = O·08,t a == I, q === 0·16 in (2) gives {1+[O·08+(2r+ l)i]2}C2r+l-0-16(car+3+c2r-l)

=

O.

(3)

To illustrate the procedure in calculating the c, assume that the real and imaginary parts of c7 vanish to an adequate number of decimal places, i.e. in the computation we put C1 === 0. Then with r

=

2

c3

=

-5(30-'i)c6 ,

(4)

where 0-08 2/0 .16 = 0·04 has been neglected in comparison with --150_ With

r

= 1

c1 == (-50+3i)c3 - cS

and by (4) so

=== 5(1496-140i)cs, Cli

==

(1-325 X 10-4

+ 1-24 X 10-6i )c1 _

ca == -(1-994X 10-2+1-198X IO-3i)e l _ To calculate the c- 2r +V neglect c- 7 , and with From (4), (5)

r r r

= == :.::=

-3 in (3),

-2

= c- 1 == c- 3

c1

-I

-5(30+i)c_s ,

(5) (6) (7)

-(50+3i)c_a- c_5

= 5(1496+ 140i)c_5 ,

(8)

== =

(~)

(0·04-i)c_1-c_3 5(229-S4-1489-4i)c_ 6 ,

the 0-04, omitted previously, being included here. Thus

= c- 3 = C- 1

c_- 5

=

(5-95x IO-2+0·998i)c1 , -(2·38X 10-3+1·978X IO-2i)c 1 ,

(2-024x IO-5+1-313X 10-4i)c1 _

(10) (II) (12)

t When p. cannot be calculated with adequate accuracy by the method used in 10 , or by the formulae in § 4.91, the procedure outlined in either § 5.11 (continued fractious) or § 5.34 (recurrence relation) may be used. The latter is the simpler of the two. To obtain a rough indication of the order of p. to start a computation (trialp.) 88 in § 5.34, formula (7) § 4.91 may be used beyond its normal range. Having found p.. to the required accuracy, the solution is obtained as shown in the present section. It may, if desired, be expressed in the form given in § 4.751. tNI

B

122

NUMERICAL SOLUTION OF EQUATIONS

As a check we shall find C- 1

=

C- 1

by taking r

= 0 in

[Chap. V

(3). Then

[(O·08+i)I+IJa-25c1 - ca

= (5·994x 1O-2+1-00li)ct ,

(13)

which differs slightly from (10)_ Alternatively, from (2) with r

=

0

(14)

(JL+i)2+1 = 0-16(c a+c_ t)/c t •

Substituting for the c, on equating real and imaginary parts, we find in each case that f' is nearly 0-08. 3°. The solution, Substituting the values of p, and the o in Yl = eIJ-Z I C2r +1 e(2r+l)zi gives the first solution: Yl(Z) = c1 eO· 08I[( 1-06+0·998i)cos z- (2·232+ 2·1i)10- 2cos 3z+ +(1·529+ 1-44i)lO-4 cos 5z- ... +(O·998+0·94i)sinz- (1-86+ 1-752i)IO-2 sin 3z+ (1-189+ 1·123i)10-4 sin 5z- ...]

=

(1)

c1 eO·OSZ[I·06+0-998i][c08z-2·1 X 10-2 cos 3z+

+ 1-44 X 10- c 0 8 5z- ... +0·94sinz-l·75 X 10- sin 3z+ 4

2

+1-12xlO- 4sin5z-._.].

(2)

The second solution is obtained by changing the sign of z in (2). The exponential becomes e-o,08.z, while the sine terms alter sign. 4°. Odd and even eolutione. The solutions in 3° are neither odd nor even. Nevertheless such solutions may be constructed using linear combinations of those in 3°. Omitting external multipliers, the solutions in 3° may be written

eO'oSlfI cos+ I

sin.] and

e-o' o8Z[I cos- I

sin].

(1)

By addition we get the even solution iiI = 2[ cosh O·08z I coa-l-sinh 0-08z I sin], and by subtraction we get the odd solution

(2)

2[sinhO·08z I cos-l-coshu-Ose I sin].

(3)

ti2 =

The complete solution is

y

= AYl+ BYa,

(4) where A, B are arbitrary constants, determinable from the initial conditions. The utility of such solutions in practical problems is doubtful since they both tend to infinity with z. 5°. RelattOn8hip between (6) 1° and (2) 3°. This is expressed by z, in (II) §4.50. The two formulae for Zo should give the same result,

NUMERICAL SOLUTION OF EQUATIONS

5.331

123

which ought to be a multiple of (1-06+0-998i) in (2) 3°_ Then by (11) §4.50, with c1 = c1 = 1,

Zo =

-(I-0595+0-998i)/sina and

Zo =

(0·0405i+O·098)/cosa. (1) Now from 1°, a = -46·72°, so sin o = -0·728 and oos o = 0·686. Substituting these into the two fractions in (1), we find that they are in good agreement, having the value

Zo ~ 1-37(I·06+0-998i) = 1·9ge43·asoi. The angle of §4.751, Zo

(2)

z, is the same as that of Zo in 3° §4.752, and since by

= MoZ o, it follows that Mo = 1·90.

(3)

Reference should be made to § 4.752, where the conjugate properties of the coefficients in 2° are discussed and another form of solution is given,

5.34. Example. Solve y"-(I·25+2 cos 2z)y = o. From Figs. 8, 11, we see that the parametric point a = -1-25, q = I, lies in an unstable region below ao, where the iso-JL curves are asymptotic to a o as q ~ ±oo. When q = 0, p, = ia i , taking the positive root, so the curvature of the iSO-it curve through the point (-1-25, 1) is less than that of ao (q = I). On this basis we proceed to find a trial JL to start the computation. A few assumed iso-JL curves, which may be drawn in pencil on a large version of Fig, SA, show that the curve through ( -1-25, 1) will probablyintersect the a-axis between - (0-81)1 and -1, i.e. -0-9 and -1, ifiai is plotted instead of e.] Thus our trial JL lies between 0·9 and I, so we shall choose O·95.t The recurrence relation for the coefficients in the solution of type (1) 20 §4.70 is (I) 1° §4.750. Inserting a = -1·25, q = I, JL = 0-95 therein, we get -[1-25+(2r-O-95i)2]c2r-(c2r+2+C2r_2) = O. (1) Let r = 3, neglect cs, and (I) gives (using a 10-inch slide-rule as we are merely demonstrating the procedure) Cc = -[1·25+(6-0·95i)2]cs = -36-35(1-0-314i)cs . (2) i t When a > 0, a is plotted. : If (4) § 4.74: is used to calculate ,.,., 88 indicated in § 4.753, the value 1·002 is obtained. As shown later, this is a better value than 0·95. The purpose of the above procedure, however, is to illustrate what may sometimes be done when the formula

cannot be used,

NUMERICAL SOLUTION O}4"

12..

r = 2

r

=1

c2 = -[l Co

=

e25+(4-0-95i)2]c.-c.

-[1-25+(2-0-95i)2Jc2- c.

.~QUATIONS

[Chap. V

= 506(1-0-914i)c..

(3)

=

(4)

3929(-0-107+i)c,_

Then from (4),

ce = -2-52 X 10.-4(0-107 -t-i)co,

(5)

from (2), (5),

c. =

(6)

from (3), (5),

c2 = -(O-13+0-115i)co'

9-53x 10-3(0-421+0-966i)co,

(7)

To check the trial p. we put r = 0 in (I), thereby obtaining -[1-25-0-95 2]co for the l.h.s,

and

(8)

since by § 4-_750, c2 and C- 2 are conjugate, Thus the l.h.s, is -O-3rlCo' and the r.h.s, is -O-26co- Hence the trial j, is too small, and a hetter value is found by substituting JL for 0-95 and solving the resulting equation. By so doing we find that f' ~ 0-995, taking the positive root.] We 110W repeat the computation using f" = I, and obtain the following results: C2 = -(O-124+0-119i)co, } c4 = (3-20+8-93i)lO-3co, (9) Cs = -(6'23+248i)lO-8co' The remainder of the solution is left as an exercise for the reader. Since c2r and c- 2r are conjugate, c- 2 ' c- 4 ' c- 8 follow immediately from (9) by changing the sign of i. The computation of these coefficients would constitute a check. A partial check is obtained by repeating the procedure at (8). Then with f" = 1, we get -[1-25-I]co = -0'2000 for the Lh.s,

and

2R(c 2 )

=

-O-248co for the r.h.s,

(10)

Solving for I-' as before, we have (a+1-'2)c O= 2qR(c.), 80

and, therefore,

1-'2

=

-0·248+1·25 I-' ~ 1-001,

(II)

= 1·002 (12)

taking the positive root. Thus the results are accurate enough for

t This procedure should be used only when the difference between the two values of IL is a small fraction of the trial value. When this is not so, a better trial,." must be chosen.

NUMERICAL SOLUTION O}4' EQUATIONR

5.34]

125

the purpose of illustration. Greater accuracy may be obtained by increasing r at the beginning, e.g. take r = 5, neglect e12 , and work to (say) seven decimal plaees.] with J.L = 1·001. An improved value of J.L will follow, the computation is repeated, and interpolation then used as in § 5.32. The requisite technique is gradually acquired hy practice.

5.35. Solutions with q < O. III the solutions with q positive write (!1T-Z) for z. In the case of unstable solutions like (2), 3° § 5.33, the constant multiplier

e:!:hTJL

may be omitted.

5.40. Solutions of y"+(a=F2qsin2z)y =

o.

If

ill

y"+(a-2qcos2z)y = 0 we write (!7T+Z) for Z, it takes the form of the first equation. Hence if q is positive, the solutions of the equation are derived from those of y" (a- 2q cos 2z)y = 0 by writing (!7T=fZ) for z.

+

5.41. Solutions of y"+[a-2q cos(2z±8)]y === o. If we write (!8±z) for z in Mathieu's equation, the above form is reproduced. Accordingly for q positive, solutions of the first equation may be derived from those of the second by making this change of variable. When q < 0, write [1(1T+(J)±z] for z in the solution of Mathieu's equation.

5.50. Example. Solve y"-(:l-4 cosh 2z)y == o. If in § 5.:~O we write iz for z, the equation there takes this form, Hence the first solution is (2) § 5.310 with cosh written for cos, while the second solution is (3) § 5.310 with sinh written for sin, the multiplier i being omitted as it is a constant. 5.51. Example. Solve y" -(3+4 cosh 2z)y = o. The solutions derived by writing (l~i+z) for Z in those of the equation in § 5.50 are unduly complicated. The simplest procedure is that given in § 4.71 for q negative, i.e, change alternate signs in § 5.50.

5.60. Example. Solve y" +4y' +(5-0·32 C08 2z)y = o. By § 4.83 we have K = 2, a = 5, a = a - K2 = 1, q = 0·16, so the auxiliary equation is u"+(1-0e32cos2z)u = O. (1) The point a

= \, q =

0·16 lies in an unstable region of Fig. 8 between

t

Using a machine,

126

NUMERICAL SOLUTION OJ4" EQUATIONS

[Chap. V

the curves b1, at, and the first solution of (I) is given by (2), 30 §5.33. Then Yt(z) = e- 2eut(z), f40 Yt(z) === c1[ I·06+-0·998iJe--1'! 2.:(cos z- 2·1 X 10-2cos3z+

+

+0·94sinz-I·75X 10-2 sin 3z+

).

(2)

The second solution is derived from (2), 30 § 5.33 by writing -z for z and multiplying by e-2z • Thus Y2(Z) = ct[I·06+0·998i]e- 2·OSZ(cosz-2·1 X 10-2 cos 3z+

-

-O·94sinz+l·75X 10-2 sin3z- ).

(3)

In (2), (3) we may write C = ct [ I ·06+ 0·998i], a constant multiplier. Both solutions tend to zero as z ~ +00, and are, therefore, stable but non-periodic. The complete solution, with two arbitrary constants, is (4) y == AYl + BY2' and this constitutes a fundamental system.

VI HILL'S EQUATION

6.10. Form of equation. For general purposes this may be taken as d2 J+[a-2qIjJ(CLIz)]y

=

(1)

0,

where t/J(wz) is a periodic differentiable function.] such that Ic/J(wz) Imax = 1. Usually .p(wz) is even in z with min I"

f IjJ(CLIZ) dz = O. o Sometimes 14s(wz) IJnax =1= Ir/J(wz) lmin' but this ease is not considered herein. We assume a, q to be rea1. When ifJ(wz) == cos 2z, (1) becomes the standard Mathieu equation. In certain applications it is con-

venient to write (1) in the form

day

-d2 +

z

(8 + 2 ~ 8 eos 2rz)y == 0, 00

0

r=l

(2)

21

co

where 00' 82 , 8" ... are assigned parameters and ~

,.=1

/02rl

converges.

The theory in §§4.10-4.16 applies to (1), (2). Stability is discussed in§§4.14, 6.50.

6.11. Solution of (2) § 6.10. In accordance with § 4.13, we take a solution of the form [215]-see comments in §6.3100

Yl(Z)

=

e~ ~

r:s -00

c2re2rei ,

(1)

where fL may be real, imaginary, or complex. Substituting the r.h.s, of (1) into (2) §6.10, we obtain 00

~ -co

00

(f'+2ri)2c2re2rzi+8o

~ c 2re 2m

00

+ 8.

~ c2r[e2(" +l)zi

+e2(,.-l).zi] +

-00-00

co 00 +8~ ~ c2r[e2(r+2)zi+ e2(r-2)ri]+Oe ~ c2r [ e2(r t-3)zi+ e2(r-3)zi]+ ••• • -co -00

(2)

t t/J(wz) may also be periodic and finitely discontinuous (piecewise continuous). but it cannot be differentiated at its discontinuities. The saw-tooth function treated later is a case in point.

128

HILL'S EQUATION

[Chap. VI

Equating the coefficient of e2rzl to zero, r = -00 to 00, we get (JL+2ri)2c2r+OOc2r+82(C2r-2+C2r+2)+84(CIr-4+C2r+4)+

+8s(c2r-e+c2r+8)+ ... = 0,

(3)

00

or

(JL+ 2ri)2c2r +

I

m=-CX)

(4)

82mC2r +2m = 0,

with the convention that 82m = (J-2m' As in §4.20 we have a set of equations, infinite in number, giving relations between the coefficients e2r • Dividing (3) throughout by (p,+2ri)2+80 = 80-(2r-ip,)2 = ¢J2r,

(5)

we obtain c2r

82 ( c2r-2+c2r+2 ) +~ 8, (c2r-4+c2r~.4)+'" == o. +y'f'2r 'fI2r

(6)

Giving r the values ... -1, 0, 1,... in succession leads to the set of equations

The eliminant for the c is ~(ilL)

=0,

= 1

O2/1>-2 84/c/Jo

8e/c/J2 8s/cpt

O2 /1>-4 84/1>-4 88/1>-4 08/1>-4 1 82/eP-2 04/eP-2 08/eP-2 82 /cpo 84 /cpo 82/c/Jo IT] 1 82 /CP2 84/CP2 82/CP2

8e/CP4

84/CP4

82/CP4

(8)

1

provided 00 =I- (2r-iJL)2.

6.12. Evaluation of (8) § 6.11. Adopting the procedure in § 4.21, we can show that the determinant is absolutely convergent, provided none of c/J2r vanishes. Then by an argument akin to that in §§ 4.22,

HILL'S EQUATION

6.12]

l:lU

4.23, we find that [215]

cos iP.1T = 1- a (0)[ 1- cos 1T8&] or

( 1) (2)

sin2Iip.1T = a(0)sin211T8A,

where

~(O)

is (8) § 6.11 with

p,

= 0, and 80

=1= 4r 2 • Thus

Ll(O) =

82/(80 - 16) 8./(8 0 - 16) 86/(80 - 16) 88 /(80 - 16) 1- -

-

~

-------

• - -

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

(}2/«(}0- 4)

1

(}2/(OO- 4)

(J.!8 0

(J2/00

L!J

...

-----

....

_---

O.!(80 - 4) : 86/(80 - 4)

8./80

82/80

8./(80 - 4) 82/(°0- 4) °e/(8 0 - 4) 8s!(8o- 16) 86!(8o- 16) 8./(8 0 - 16) 82!(8o- 16) 1

t _________________________________ 1

0,,1(80 - 4) 1

(3)

The remarks in §4.24 on the roots of (3) §4.20 apply equally to those of (8) §6.11. When 02' 84 , 8e,..• are small enough, it can be shown by aid of the expansion for cot x that

~(O) ~

1+ 1Tcotl1TO&[~+~+~+ ...]. 2-8

48&

1-80

2

0

32-00

(4)

A less accurate formula can be obtained by expanding the thirdorder determinant in (3) about the origin. Then if 82 , 8.,... are small enough Ll(O)"""; 1+~j__ +~1f)4---~L (5) 80(2 2-80 ) 80(2 2-80 )2 (22-80 )2-

6.13. Example. In the lunar perigee problem [70] which gave rise to Hill's equation, the following data were used: 80 =

1·15884 39396 0,

82 = -0·05704 40187 5, 8. = 0·00038 32380 0, 88 = -0·00000 91732 9. Calculate the value of JL. 1°. We shall get a first approximation using (5) § 6.12. The last two terms will be neglected, since they are of order ~ 10-7 • Then a(O) ~ 1+[281/80(4-80)] SO 4N'

[~(O)]i ~

± 1·00098 5; s

also

= 1·00197, SA ~ 1·07649 6.

(1)

(2)

HILL'S

130

sin 11riJL

Thus

[Chap. VI

EQU~TION

:=;:

[~(O)]I sin

~

± 1·00098 5 cos( tn X 0·07649 6)

!1r8A

= ±1·0009~ 5c080·12016 0

= ± 1·00098 5 X 0·99278 9 = ±0·99376 7.

(3)

We find that

11TiJ.L 80

= ± 1-45909

1

or

± 1·68250 1 radian,

IL ~ ±O-92888 6i

or

±1·071114i,

(4)

the sum being ±2i. Since (5) § 6.12 is not a very accurate formula, we are not justified in giving the value of J.£ to more than three decimal places. Thus we take p,~±0·929i or ±1·07Ii. (5)

2°. We shall now obtain a more accurate value of fL using (4) § 6.12. Then

8A = In8A =

~~ot*1TOA

1·07649 61400 8 1·69095 61826 5

=

-0.08809 15080 2

=

-0·02048 56419 7

40A

cotl1TOA = -0-12074 15203 9 sinl1T8i = 0·99278 94864 7

0~/(1-0o)

8~/(4-0o) =

O·

516 9

[

0:/(9-80 ) =

O·

1

]=

-0·02048 55902 7

Thus and

~(O)

:=;:

[Ll(O)]l

=

sin l1TiJL JL

=

(1)

±1·00090 18965 6.

(2)

= [Ll(O)]l sin !1TO i

= so

1·00180 46065 4

±0·99368 48798 9,

±O·92841 62006 3i

or

±1·07158 37993 7i.

(3) (4)

These values of p, are correct to six decimal places. More accurate values may be calculated by aid of (7) § 6.30 which gives p, = ±O·92841 67276i

and

± 1·07158 32724i.

(5)

6.14. Calculation of coefficients in (I) § 6.11. Having found u if 81 , 84 , ••• are small enough compared with 80' an approximate solution may be obtained by neglecting all c except, say, c-4 , c_I ' co' C2 ' Ct. In (7) § 6.11 the five equations, the central one with two immediately above and below it, each containing only C-4, •.•, c" are solved in the usual way. A rough approximation may be got by solving the three equations within the broken-line rectangle, which include only c_1 ,

HILL'S EQUATION

6.14]

131

Co' ca. In any case, unless 82 , 8., 88 , ••• are small enough compared with 80' the solution is troublesome. Additional methods are given

below.

6.20. Transformation of Hill's equation to a Riccati type. 1°. The form of (I) §6.10 is such that the analysis in §4.82 may be applied provided ifJ(wz) be written for cos 2z. Thus the Riccati type of equation is (2) §4.81, where p" = [1-(2qja).p(wz)]; and if lal ~ q > 0, lifJ(wz)lmax = 1, its solution is (13) §4.82. In terms of .p we have [133] Yl(Z)

Y2

~

C?S[a1 fZ (1- 2Q.p) 1dz +

constant

[1- (2Q/a)ifJ(wz)]1 SIn

2 fZ {"

wq

+4a1

.p

o

a

o

.p'2

5q

+ 2a [1-(2qja).p]

}

dz

]

[l-(2qja).p]I·

(1)

As in § 4.82, the argument of the circular functions is periodic, the frequency of repetition being

f

]

1 d [

= 21T

=

dz .

q

2

1 [( i

2; a l-(i.p

)1+ w4a2ql {[1-(2qja).p]1+2a(I-(2qja).p]t .p" 5Q«/J'2 )]·

Under the conditions lal (2) the approximation

f

~

-1

~

q

>

0, 1.p(wz)lmax

=

(2)

1, we derive from

[a1_qa-1.p+ lw2Qa- l{ifJ"(1+ 3qa- 1cP)+ lqa- 1.p'2( I + 5qa- 1.p)}]. (3)

211

The periodicity is the reciprocal of (3). The solution of (2) §6.10 is obtained if in (13) §4.82 we write 80 for a, and take p2 = [1+<2/80 } tion, with
=

Yl(Z) ~

con:tant

Y'I.

p we get

00

!

r=l

82, c08 2rz]. As a first approxima-

C?s[oife {1+(2jOo)!OarCOS2rz}ldZ].

[1+(2/0o)!OarCos2rz]lSm

0

r=l

(4)

r=l ClO

This solution rests on the assumption that 180 1 ~ 2 I 18trl, and in r=l

common with (1) its boundedness implies stability, i.e. (4) does not hold for an unstable region of the a, q-plane for Hill's equation.

HILL'S EQUATION

132

2°. 2q ~ a > 0, 1.p(wz)lmax form

= 1.

[Chap. VI

We now write (I) §6.10 in the (5)

yi wdz Substituting Y = e with

,,2 = 2q, we get

~~ = yy(~~ +yw2), so from (5), (6), if

,2

(6)

= [(aj2q)-.p], (5) transforms to

! dw +W~+t'2 = O.

(7)

Y dz

On comparison with (2) § 4.81, the approximate solution of (5) is obtained by writing y for v, and, for p in (13) § 4.82. Thus Yl(Z) ~

~ (constant),-lC~S[(2q)l mn

r{e-(2ef'-3f )/ I6qea} -l 2

(8)

1/·(wZ)}' dZ].

(9)

h

By omitting the terms in ,', ,", we get Yl(Z)

Y2

~ constant(2ql~C?s[ fZ {a-2 [a- 2q.p(wz)]1

BIn

q'fl

o The solution of Mathieu's equation under the above conditions is found from (9) by writing cos 2z for ,p(wz).

6.21. Solution of (2) § 6.10 involving Mathieu functions. The equation may be written y"+(80+282 cos 2z)y = -2[J/2rCOS 2rz]y.

(I)

If for a first approximation the r.h.s. of (I) may be neglected, and if ,." is imaginary, the solutions are Yl(Z)

=

cem+~(z, -q)

and

Y2(Z)

=

sem+p(z, -q),

(2)

provided 80 is not a characteristic number for cern' sem , 0 < fJ < 1 and q = +(J2.t For a second approximation to the first solution we

t If 80 is a characteristic number, then fJ = 0 or 1, and with 8, > 0, by Chapters II, VII, the solutions are Y. = cem(z, -q), Y. = fem(z, -q) (80 = am)' or

With 8.

Yl =. 8e ll,(z, -q),

YI = gem(z, - q)

(80 = b".).

< 0, -q becomes +q,and (-1)' is absent from the second ~ in (3). See § 4.71.

HILL'S EQUATION

6.21]

133

use Yl(Z) for y on the r.h.s, of (1). Then we get y" +(80+282 cos 2z)y

=

(-I)n+1 2 [ ! 92r cos 2rz] r=2

!

3= -00

(-I)HA~+Jl){cos(228++fJl+Q)z}, 23+1 8

(3)

fl.

fJ or (fJ+I) being used in accordance with §§4.70, 4.71. Taking only the important terms on the r.h.s. of (3), the equation now to be solved may be written d2y dz 2 + (80+282 cos 2z)y =I(z). (4)

6.22. Solution of (4) § 6.21. We write the equation in the form d2y dz 2+(a+2qcos2z)y =/(z).

(I)

When f(z) = 0, we take the complete solution as the sum of two Mathieu functions Yl(Z), Y2(Z) with the arbitrary constants A, B. Thus y(z) = AY1+BY2. (2) \Vhen the r.h.s, of (I) has the valuef(z), suppose that Y = AY1+ B Y2+ A l(Z)Yl + B 2 (z )Yt ,

o=

dA l dB" Yl dz + Y2 dz '

(3)

( ) 4

where A 1 (z), B 2(z) are variable parameters (functions of z) added to the respective arbitrary constants in (2). Differentiating (3), we get

dy ' B 2Y2+ ' A''lYl+ D ' D2YI· dz = A' Yl+ B' Y2+A tYl+

(

5)

Using (4) in (5) gives

so

~~ = (A+Al)y;+(B+B2)Y~'

(6)

~~ = (A+Al)y~+(B+B2)y;+A~y~+B'2Y~.

(7)

Substituting (7) into (1) and using (3) leads to (A +Al)[Y~ +(a+ 2q cos 2Z)Yl]+

+(B+B2)[y;+ (a+ 2qcos 2Z)Y2]+A~y~+B'2Y~ = 1(1,). Now the [ ] are both zero, so (8) reduces to A;y;+~y~ = f(z). Solving (4), (9) gives A~ = YI!(Z)/(Y" y~ -Yl y~).

(8)

(9)

(10)

HILL'S EQUATION

134

[Chap. VI

By § 2.191 the value of the denominator in (10) is -a2 , so e

-~ f yz!(z) dz. •

A; = -y,.!(z)/cz,

ox;.

Al =

Yd(z)/c z,

or

s, = :z

.lfa =

Also

f

yd(z) dz.

(11)

(12)

Substituting for AI' B 2 from (11), (12) into (3) yields the complete solution of (1). Hence Y

-~[Yl(Z)

j

j

=

[AYl+Byz]

=

complementary function + particular integral.

Yz(u)!(u) du -Yz(z)

Yl(U)!(U) dU] (13)

(14)

This solution is valid if -q be written for q in (1), provided Yl' Y2 are then solutions of y"+(a-2qcos2z)y = 0, and c2 is calculated therefrom. If (a, q) lies in a stable region of Fig. 8, these solutions are cem+~(z, q), sem+~(z, q). When the point (a, q) in (1) lies in an unstable region of Fig. 8, so long as Yl' Y2 can be found, (13) gives the solution of (1). The method in § 4.90 for solving Mathieu's equation in terms of Mathieu functions is applicable to Hill's equation. Reference may also be made to § 10.70 et seq., where integral equations and their solution in connexion with Hill's equation are discussed.

6.30. Solution of Y"+[8 o+2! 82rcos2rz]y r=1 § 4.30 [81-3]. We assume a solution Yl = e,a4>(z, 0'), where JL

=

= 0

by method of (1)

82eP2(a)+84aP4(a)+86aP6(a)+ ... + +81 ~2(a)+8= g4(a) + ...

+

+82 8. ~2,4(a)+82 86 ~2,6(a)+84 86 ~4,6(a)+ ... + +8=X2(a)+ ... +

+....

(2)

~(z, 0') = sin(z-a)+82f,,(z, 0')

+ 84/4(z, a)+ +

+8~g2(Z, a}+8~g4(Z,

a)+

+

+82 84 g2,4(Z, a}+ ... + +8~h2(Z, a}+ ...

+ ...

+

a periodic function,

(3)

HILL'S EQUATION

6.30]

135

a being found from the relation

80 = 1+82
+

+...+

+828./1/,,4(0')+ ... + +81i'2(a)+.... Substituting (1) into the differential equation yields

~"+2ft~' + [ft2+ 8o+ 2r~ 82JoC08 21'Z]~ =

(4)

O.

(5)

Inserting the series (2)-(4) into (5) and proceeding as in §4.30 to get a periodic solution for C/>, we derive expressions for 80' JL, 12' 92' etc. The analysis is much too extensive, however, for inclusion here, so the original memoir should be consulted. Expressions for 80 and JL are given below for making a computation. Then 8 1 1 (J2 1 2 11 828 13 8 8 8 o = - 4 2 - 60•- 192 2 4 - 192 2 4 6

+ 481 842 -

893 8! 2 21184 2

+... +

1 +C08 20'[82 + ~8284 + 1~ 848. - 648:+ 1:2 8~ 8.1 (J (J2 175 (J3 (J 1 85 -144 2 4-9612 24.+288 2

+cos

4 [ 1 82 (J'

5 82 8

13 (J 8 (J

II (J4

9181

+ ... ] + 8!

]

"8 2+ 64 2 '+576 2 4 ,- 512 2+ 176-9472 1+'" +

+C08 60'[ 5~2 8:84- 4~:6 8~ + ".J + +C0880'[ -3::68~+"'] +.... p. =

(6)

8in2a[!91+!8184+~e488-~~+~8~88+ 2 8 24 128 384 7

2

+ 288(J2 8, -

211 137 9216 ~84+ 36864 81 +...

. [1 1 3 337 ] -l-sin so 128818.+ 1152828486-1024 tJ:+ 4: 423680:+ ..·

J+

+

+8in60'[1:24~84- 8:92~+'''] + +8in8a[-6;:36~+ ...J+....

(7)

136

HILL'S EQUATION

[Chap. VI

6.31. Calculation of p,. Using the numerical values of 80' 82 , ••• given in §6.13, we find from (6) § 6.30 that [81] 1·15884 39396 0

=

1-0·00081

3~804

9-0·05704 65855 7 cos 2a+

+0·00040 66226 27 cos 4a+ 0.0 945 cos 6a9

--O·Olo368cos8a.

(1)

'rhus cos2a

= -2·79871 82492 1+0·00712 79047 3cos4a+ +0·0'165 65 cos 6a-0·0864 51 cos 8a.

(2)

Expressing the r.h.s, of (2) in terms of cos 2a leads to the equation 0·0'516 08cos42a-0·07662 60cos32a-O·01425 58603 6cos 22a+ +cos 2a+2·S0584 60209 7

= o. (3)

Using six significant figures and the last three terms of (3), we obtain the first approximation cos 2a ~ -2-70178. Then by the process of successive approximation it is found that cos 2a

=

-2·70178 48031 8.

(4)

Since [cos So] >1, by §4.60, a= -l1T+i(X, and from four-figure tables (X = 0·8255, so a = -!1T+0·8255i. Also

=

=f2·50990 86064 Oi,

(5)

sin4a = 2 sin 2acos 2a

=

± 13-56247i,

(6)

sin6a = sin2a{4cos22a-l)

= =f70·7758i,

sin2a

sin 80.

=

::=

=Fi(cos22a-l)i

4 sin 2a(2 cos320'-cos 2a} = ±368-8i.

(7) (S)

Inserting the values of 82 , 84 , 88 in (7) § 6.30 we get fL = -0·02852 03942 5sin2a-0·07212 52sin4a+

+0-093162sin6a-0·01326sin8a.

(9)

Using (5)-(8) in (9) leads to the result p, = ±O-07158 32724i.

(IO)

By aid of the formulae in [81], fa' fl.' g2' etc., in (3) § 6.30 may be found, so the first solution in the form (1) § 6_30 is obtained. The second solution is derived therefrom by changing a to -a and +,." to '--p,. Since p, is imaginary the solution is stable, and if the number of decimal places is fixed (conventionally) it is periodic, It may of course be expressed in the form (4) 10 §4.70 if desired. In this connexion we remark that the values of ,." in (10) differ by ±i from those at 2° §6.13 (up to the seventh decimal place), because in §6.11

HILL'S EQUATION

6.31]

137

we assumed the form of solution at (1) 1° §4.70. Without a stability chart for Hill's equation-like Figs. 8, II-the particular form of solution required to keep p, in the range 0 < p, < i is not always predictable. In the present instance, however, (}., 8, are so small numerically compared with (J2' that the equation is approximately a Mathieu type. From Fig. 11 the point 80,82 lies in a stable region between aI' b2 , so there would have been justification for replacing 2r by 2r+l in (1) §6.11 in accordance with (4) 1° §4.70. Moreover, an approximate value of fL = ifJ may be calculated from (4) §4.74. Using only three terms, we find that JL ~ ±0·0713i (10-inch sliderule). The iso . .fJ curve fJ = 0·0713 lying just above a 1 in Fig. II is an approximation to that for Hill's equation with the (J in § 6.13.

6.40. Example. We shall determine fL for the equation [44]

~~ + 2K~ + [a' -2q,p(wt)]y -= By § 4.83 the substitution y

O.

(1)

= e-Ktu(t) leads to the equation

d

2u

-dt2 + [a-2q¢J(wt)]u = 0,

(2)

with a = (a' - K 2 ) . For ~(wt) we choose the saw-tooth function repre.. sented graphically in Fig. 14. It has period 21T/W, and the Fourier expansion is ,p(wt) = _~ ~ (_l)n-t sin nwt. (3) '1T~

n

n=l

.,,/w

We see that

f

-~w

¢J(wt) dt = 0, and that IljJ(wt)/max

the interval -1T/W < t

~n

=

1. We consider

< 1T/W, where the function is defined to ¢J(wt) = WtJ1T.

be (4)

For this interval alone, (2) may be written

u+a(l- a1T t)u = o.

2

d dt 2

Writing T

=

2qw

(I-yot), where Yo = 2qw/a1T, (5) transforms to d2u d,.,2 + b3T U = 0,

(5)

(6)

with bl = a 37J"2J 4q2w 2 . By aid of example 46, p. 38, reference [202] we find two linearly independent solutions of (6) to be 4N1

u 1 = TlJ,(<XTt)

and T

u 2 = T1Yi(CXTI),

(7)

138

[Chap. VI

HILL'S EQUATION

where ex = 1Tat j 3wq. Although these solutions are valid for the interval -1T/W < t < 'TT/w only, they enable us to determine JL by aid of (9) § 4.152. We now procee~ to obtain the various components in that formula. dUl

dT

!! (CX'T')Ji(CX'T )+ !'T-lJ1(CX'T1) dT

=

'T1

=

!CXTJ1(CXT f )

'

+IT-lJ1(cxTI).

(8)

ij f(lIJt) max. 11; t(lAJt) min. :FIG. 14. Saw-tooth wave form.

Using the recurrence relation

J~(z)

~:l =

in (8) leads to Then Similarly

== -~Jv(Z)+JV-l(Z) Z

dU

i'CX'TJ_1(cx'T i ) .

du; dr

l

J

di1 = dT dt = -a T du 2(wt ) _ dt

- -a

l T

Y ( -I

(10)

i -f(CXT ).

aT

(9)

f)

•

(11 ) (12)

6.41. Values of the various functions at t = ~1T/W. Writing 2q/a = y in the formulae in § 6.40, we obtain the following results: Ut(=F1T) = (l±r)IJ.[ex(l±y)I],

U~(=f1T) = -a1(1±y)J_f[ex(1±y)I];

(1)

U2(=F1T) = (l±y)lY.[o:(l±y)'],

U~(=F1T) = -a l (l ± y)Y_.[o:(l ± y)t].

(2)

8.41]

HILL'S EQUATION

139

The Wronskian for the Bessel functions of argument <X(l±,,)t is [202, p. 156, ex. 61 a] Cl(l±y)I[J. Y1-Ji

¥il =

2/1T.

(3)

Using (9) § 6.40 for J 1and Y1in (3) gives cx(l±y)I[Ji Y-t-J- f

Y.J = 2/7(.

(4)

By (1) § 2.191, u 1 U~-U2U~ = the argument in this case being =F1T. Substituting from (1), (2) into this identity yields c2 ,

-al (l ± y )f[J i Y-t-J- 1 ¥iJ = c2

(5)

and from (4), (5) it follows that 2c2

=

-4a I /1TCi. .

(6)

6.42. Evaluation of JL. We have now found expressions for all the components in formula (9) §4.152. Substituting from (1), (2), (6) § 6.41 therein, and writing WI = o:(l+y)l, W? = o:{l-y)l, we obtain after a little reduction cosh 2JL1T

=

t

o:(l - y2)1{(l-y)1[J-I(W2)¥i(Wl)-J.(Wl)Y-t(W2)]+

1T

+ (1 +y)l[J-f(Wl)¥I(W2)-J,(W2)Y-I(Wl)]}.

(1)

If the values of U)l' tV 2 are large enough, (1) may be simplified by using the dominant terms in the asymptotic expansions of the Bessel

functions. Then [202, pp. 160, 161] J-f(w) --

(~,)l cOS(W-!1T+!1T),

Jl(w) --

(~)' cos(w-!1T-l1T); (2)

Y-f(w) --

(1T~)'sin(w-!1T+l1T),

l!(w) --

(~)' sin(w-!1T-l1T); (3)

the part omitted in each case being O(l/w i ). Substituting these into (1), after some reduction we find that

cosh 2JL7T = {y/{I- y2)1[(1 +1')1_ (1-1')1 ]}coS(W1-W2).

(4)

Since stability of the solutions of (1) § 6.40 depends upon the factor (see §4.83), this subject may be investigated using (4). The circuit of Fig. 23 D is examined in this respect in [44], which may be consulted, Stability conditions for various mechanical and electrical devices are discussed in u number of the technological papers cited on }>p. 373-81, and to these the technical reader is referred.

e(p.-IC)l

HILL'S EQUATION

140

[Chap. VI

6.50. Stability of solutions. In § 4.60 we discussed this question for the equation y"+(a-2qcos2z)y = 0, use being made of Fig. 8B. It can be shown that, when t/J(wz) is assigned, the (a, q)-plane for the equation y"+[a-2q,p(wz)]y = 0 may be divided into stable and unstable regions after the manner of Fig. 8. The stable (unstable) regions lie between characteristic curves for solutions of integral order having period '1'1 and 21ft ('1'1 and tr, or 21T and 21T). The stability diagram is symmetrical about the a-axis, provided .p(wz) is syms.".

metrical about the origin, and is periodic with

f ¢J(CJJz) ds = o. o

Characteristic curves for coexistent solutions of fractional order having period 2811, 8 ;?; 2, and corresponding to the iso-fJ curves of Fig. 11, may be drawn in the stable regions of the plane. Here JL is imaginary, but in the unstable regions it is either real] or complex, according to the form of solution assumed (see § 4.70). Solutions of period 11 or 277' do not coexist (unless q = 0), and the second solutions corresponding thereto are probably non-periodic (like the Mathieu case), although a proof of these statements has not been published. The asymptotic nature of the stability diagram has been discussed at Borne length in reference [171], which should be consulted for information on this topic. Amongst other matters it is shown that in the region corresponding to that below curve a o in Fig. 8 B, JL is real and the solutions unstable-as in the Mathieu case. This arises mainly from the fact that lal > 2q and a is negative. In the regions corresponding to those above the characteristic 4 0 ,

a ~ -2qI.p(wz)lmax+O(ql)

(a,q on right of a-axis)

(1)

and a ~ -2Iqllt/J(wz)lmJn+Olqll (" left " ). (2) If, as contemplated hitherto, 1t/J(wz)lmax = 1, (12) § 12.21 is repro.. min

duoed. t Provided

t/J has period n, If '" has period 2Tr, the boundaries correspond to solutions with periods 21r, 41T. Reference [82] may be consulted in connexion with 8tability for small values of the 8 in (2) § 6.10. t In this case iso-" curves may be plotted 88 in Fig. 11.

VII NON-PERIODIC SECOND SOLUTIONS CORRESPONDING TO cem , se m , Cem , Sem 7.10. Non-periodicity of solution. If the first solution has period 'IT or 2'7T, it may be shown that any second linearly independent solution cannot have either period. We now illustrate this by a particular case [84]: THEOREM: If a be assigned for ce2n(z,q), the odd second solution cannot have period 7T' when q =1= o. Proof. Suppose the second solution is the odd function 00

!

C2r + 2

r=O

sin(2r-t: 2)z,

which has period sr, By (I) §3.10 we have the recurrence formula for ce2n(z, q): (a-4)A 2 - q(A 4+2A O) = 0, (1) and if the assumed second solution is substituted in Mathieu's equation, we obtain the relation (a-4)c 2-qc4

== o.

(2)

Multiplying (1) by c2 ' (2) by A 2 , and subtracting leads to

IA z Atl =

A 2 c4 - A 4 c2 =

(3)

2A oca·

C2 C4

When r

~

2 the recurrence relations are, respectively, (a-4r 2)A Ir

and (a-4r 2)c 2r Dividing (4) by (5) yields

= =

AarCar+2-AzT+2C2r

q(A 2r +2 + A 2I-- 2)'

(4)

Q(c2r+2+Car-a)·

(5)

=

(6)

A2r-2C2r-AarC2r-z,

so [84] A 2r

I

A2r+21 =

jA

2r - 2

Air! = ... = IA At' = 2A oc I

l•

(7)

C.. C1r +1 CIr-2 c2r Ct ct For convergence of the series representing each of the two solutions, we must have Air ~ 0, car ~ 0 as r ~ +00. It follows that each determinant in (7) does likewise, 80 Aoc, = o. But A o =1= 0, save in the case q = 0, 11, > 0, so C1 = o. Hence all the c are zero, and a second solution having period tt does not exist. The same conclusion

142

NON-PERIODIC SECOND SOLUTIONS

[Chap. VII

may be reached for a solution of period 217'. When the period is 2817', 2, it may be inferred from §§2.16, 4.16 that the two solutions of Mathieu's equation coexist, and have the same period. This may be established by a method siniilar to that used above. Since the period cannot b~ a sub-multiple of 17', it follows that the second solution corresponding to ce2n (z, q) is non-periodic. The cases of ce 2n +1 , se m may be argued in like manner. Since Cem(z, q), Sem(z, q) are derived from cem(z, q), sem(z, q), respectively, by writing iz for z, the same conclusion may be reached here. The periods are, of course, 17'i and 217'i. It may be remarked that a general treatment of the periodicity of the solutions of Mathieu's equation is given in [150], to which reference should be made. A different approach to the non-periodicity aspect will be found in [71, 89, 1261. 8 ~

7.20. Relationship between first and second solutions. The following analysis is applicable when the first solution has period 17', 21', and the second is non-periodic. It does not hold when both solutions are periodic with period 2817', 8 ~ 2. Choose ce2n{z, q) as the first solution with period 17', even in z. Then we denote the second non-periodic solution, odd in z, by fe 21l(z, q). If (1T+Z) be written for z in the differential equation, it is unchanged, so fe2n(17'+z) is also a solution. Thus we may write fe 2n (1T -t- Z)

== yce 2n (z)+ 8 fe27,(z),

(1)

where /', 8 are constants, and the presence of q is presumed. Again, taking (1T+Z) for Z in (1) gives fe 2n (21T + z) = 'Y ce2n(z)+8 fC2n (1T+ Z),

(2)

since ce2n (z) has period 17'. Substituting for the last member of (2) from (1) leads to fe 2n(21T+Z) = y( 1+8)ce2n (z)+ 82 fe 2n (z).

(3)

Replacing z in (3) by (Z-1T), we get fe 2n (17'+ z)

=

1'(1 +8)ce 2n(z)-8 2 fe 2n (1T - z),

(4)

the negative sign arising from fetn, being odd in z. Putting -z for z in (1) and substituting in the third member of (4) leads to an alternative version of (1), namely, fe 2.n (1T+ Z) == 1'(1 +8-8 2)ce2n(z) +8 3 fe 2,,(z).

(5)

For the identity of (1), (5) we must have

" = ,,(1+8-8 2)

and 8

=

83 simultaneously.

(6)

7.201

C()RRESP()NDING TO ce.., se"., Ce..., Se...

143

Now fc 2n (z) is non-periodic, so by (1) neither y nor 8 can vanish. Thus 8 = 1. Inserting this in (1) yields fe 2n(1T+z)

=

yce 2n(z)+fe2n(z),

(7)

and since y is independent of z, it follows that fe 2n(1T+z)-fe 2lt(z)

=

yce 2n(z),

(8)

a periodic function having period 1T.t Thus fe 2n (z) must satisfy (8). In addition, when q == 0 it must reduce to sin 2nz, for ce2n(z) then reduces to cos 2nz, and these are two independent solutions of y"+ay = 0, a = 4n 2 • If f2tt(Z, q) is a function periodic in z, with period 1T, and we write fe2n(z, q) ===

(}2n(Q)z

ce2n(z, q)

+f2n(Z, q),

(9)

ot

(8) is satisfied provided y == 1T(/2n(q). Also, if 02n(q) ~ and f2n(Z, q) ~ sin 2nz as q ~ 0, (9) fulfils the conditions for a second

non-periodic§ solution of Mathieu's equation.

7.21. The functionf21~(z,q) and its normalization. The second member of (9) § 7.20 is odd in z, so 12"'(Z, q) must be odd in order that fe 2n(z, q) may also be odd. First normalization rule, based upon § 2.11. We take the function 00

f2n(Z, q) = sin 2nz+

!

(1)

qrSr(z),

r=1

where Sr(z) are odd continuous functions of z, the coefficient of sin 2nz is unity for all q, and f2n(z, q) ~ sin 2nz as q -+ o. Second normalization rule based upon § 2.21. Here we take 00

12n(z,q) = !J4~+~sin(2r+2)z r=O

=

00

02n(q) !f~~+~sin(2r+2)z, r=O

(2)

J4:

n ) -+ 1 where J~~~~ = 02n(q)f4~~)2. In § 7.41 it is shown that as q -+ 0, while J4~+~ ~ 0, r =1= n-l. This is true if we adopt the rule that

G~n(q)

Then, as q -+ 0,

00

00

! [f4~+~]2 === r=O ! [J4;+~]2 = r=O J4~n)

=

C2n(q)f~~n) ~ 1,

and since J~:+~ -+ 0, r =1= n-l, /2n(Z, q) ~ sin 2nz

1.

(3)

(4) (5)

t When the order of the function is (2n+ 1), celn+l has period 2". Hence 2" must then be written for " throughout. : As shown in § 7.31, it is convenient to make Oo(q) = 1. § q #= o.

[Chap. VII

NON·PERIODIC SECOND SOLUTIONS

144

as required. Now the f4~+)2 may be calculated by the method given in § 7.53. Then taking the positive root C2n(q)

=

It! {fg~~}2]l.

1

(6)

7.22. Definitions of the second solutions. By aid of §§ 7.20, 7.21 we define as follows:

= C2n(Q)z ce 2n(z, q)+ f 2n(Z, q) fe2n +1 (z, q) = C2n+1(q)z ce 2n+1(z, q)+f2n+l(z,q) ge2n+1(z, q) = S2n+l(q)z se 2n+ 1(z, Q)+g2n+l(z, q)

(a2n),

(1)

(a2n+1 ) ,

(2)

(b2n+1 ) ,

(3)

ge 2n +2(z, q) = S2n+2(q)Z se2n+2(z, Q)+g2n+2(Z, q)

(b2n+2 ) ·

(4)

fe 2n(z,q)

The series for fm(z, q), g,n(z, q) are given below. Second normalization in § 7.21

First normalization in § 7.21

Function

00

00

sin 2nz+ ~ q'S,(z)

Iln(Z, q)

,-1

0ln(q)

~ f~2'!(.,> sin(2r+ 2)z ,-0 T

ex>

11,.+I(z, q)

Nin(2n+l)z+

91n+l(%' q)

cos(2n+ l)z+

gl"+I(Z, q)

cos(2n+2)z+

I

00

(,t2U+1(Q) f~;+11) '=0

sin(2r+ l)z

(2a)

qro,(z)

San+l(Q) I g~;+11) cos(2r+ l)z

(3a)

q'O,(z)

S211+I(Q)

00

,-1

I

q'8,(z)

r-l

I

00

,-0

ex>

As

q~

~

(1 a)

..

00

.I g~;n+2) COS 2rz

(4a)

,=0

'=1

0,

Cm(q)f~)

=

J~) ~ 1,

0m(q) ~ 0,

fm(z,q) ~ sinmz;

(5)

8m(q)g~)

=

g~) ~ 1,

Sm(q) -+ 0,

gm(z,q) -+ cosmz.

(6)

The second normalization is merely (1) § 2.21 applied to fm(z, q), gm(z, q), so we get <X>

QO

Oln I

Jlr+2 = r=O

0ln+l

I flr+l = r=O

ClO

S~n+l I gL+l r=O

= Sin+2[2g~+ Jlgi,] =

1.

(7)

The f, 9 are single-valued continuous functions of q. fm(z, q), gm(z,q) are periodic, having period 1T, 21T, according as m is even or odd. The second solution comprises a non-periodic part involving the first solution, and a periodic part with the same period as the latter. By virtue of the factor z in the first part, the functions (1)-(4) tend to ±co with z. In accordance with (a) §4.14 these solutions are unstable. Thus, if (a, q) is upon a boundary line am' bm

CORRESPONDING TO ce., se.,

7.22]

~.,

Be.

143

in Figs. 8, 11, the first solution is stable, the second unstable. If (a, q) is not upon am' bm , both solutions are either (a) stable, t (b) unstable in -00 < Z < 00. When q is positive, and a is a characteristic number for a function of integral order, the complete solution of Mathieu's equation is

=

y(z)

A cem(z,q)+Bfem(z,q)

= (a =

(a

(8)

am),

or y(z) = A sem(z,q)+Bgem(z,q) bn. ) . (9) Either (8) or (9)' constitutes a fundamental system of solutions.

7.30. Determination of 0m(q), Sm(q), fm(z, q), gm(z, q) with first normalization in § 7.21. There are at least five methods of approach, of which we select one.j We exemplify this by using the function fe 1(z, q)

=

C1(q)z ce 1(z, Q)+fl(Z, q),

(1)

which corresponds to the first solution ce1(z, q) having period 21T. fl(Z,q) is the second member of (2a) §7.22 with n = o. Substituting (1) into y"+(a-2qcos2z)g = 0 (2) we get C1(q)z[ce~+(a-2q C08 2z)cel]+f~+201(q)ce~+(a-2qCOB 2Z)/1

= o. (3)

Since cel is a solution of (2) for a = ai' the [ ] vanishes leaving f~+201(q)ce~+(a-2qcos2z)fl= O.

By hypothesis 0 1 (0) this and fl(Z,Q)

=

=

(4)

ClO

0, so we assume C1(q)

=!

(X,qr. Substituting

1=1

ClO

sinz+ ~ q'S,(z) into (4), we obtain, with the aid r=l

of results in § 2.13, I~

=

2q ce~ =

-sinz+qS~+q2S;+q3S;+ ... ,

-2[SinZ-~qSin3Z+ 6~q2(-3Sin3Z+~Sin5)Z- ...]

X

X [(Xl q+tll ql+£Xaq3+ ...],

all =

[SinZ+Q81+qllS2+ ...

l[ l+Q_~qz_ ~4q3-15~6q'+

..}

-2q/l cos 2z = -.2q[I( -sinz+sin3z)+co8 2Z(qSl+q2S2+q3S~+...)]. t The period is then 28'7F, 8 ~ 2, or infinity (see § 4.71). An infinite period implies a non-periodic bounded function, t Another is given in § 10.75. 4M1

n

NON. PERIODIC SECOND SOLUTIONS

14:6

[Chap. VII

Equating the coefficient of qr to zero yields qO: -sinz+sinz = 0, identically; S~+SI-2~lsinz~2sjnz-sin 3z = 0.

q:

Now the particular integral corresponding to sin z is -}ZC08%, which is non-periodic. Since 11(z, q) is to be periodic with period 271', the coefficient of sin z must vanish, so (Xl = I. Thus

8~+81 =

sin3z,

giving 8 1 =

-~sin3Z.

S"+ S2 - 2CX2SIllZ · + 4a181n 3 · 3Z-8I. 8111Z + S1 - 2 AS1 C0 8 2Z = 2

q2:

81 2"+ 8- 2-- 2 <X2 · SIn z

80

To avoid a non-periodic term, 8;+82 = S2

giving

=

0,

· 3Z +-i SIll . 5z == O. + 85-8111 8 (X2

=

0, so

-(~sin3z+~sin5Z),

5. -SIn 3 Z+-1 -- .sIn 5z. 64 192

' al (3 · 3 5. 5)z S-,,+ 3 8-3- 2 aaSlnz+32 sin Z-3"Sln 1-

1.sln z -64

-

-

-- 8 1+-82 - 28 2 c 0 8 2z = 0, 8

80

8-3 S-,,+ 3

(2

3) 8InZ+-SIn · 35. 3Z - -sIn5z--SIn I. 1 . 7z = O. 192 8 192 As before the coefficient of sin z must vanish, so (Xa = -3/64, and ~3+32

35 81n · 3 1. 1. 8_,3, +S-3 = -192 z+-sln5z+-81n7z, 8

:i S 3 =

giving

192

35. 3 1. 5 1. z--81n Z - - - S I n 7% 1536 192 9216 '

--SIn

and so on. Thus ~

q(q)

= r~1 cxrrf' =

3 3 q- 64q3- 256

31

rt + 36864 q5+O(q6),

and

J1(z, q) = sin z _!q sin 3% + 8

3( -3 35.

1 -512 q

~q2(5 sin 3z + !sin 64 3

810

(5)

5Z)-

3 8. It 1. ) Z+381DOZ+,ISsln7z

+....

(6)

CORRESPONDING 'I'O

7.30]

00""

Be,,,, Ce..., Be".

147

Finally the series representation of the second solution, corresponding to ce1(z,q), using the first normalization rule in §7.21 is fel(z, q) = [q- :4

e- 2:6 lf+ 3638164qll+O(q8) ]zcel(z, q)+

+ [sin z- ~q sin 3z+ 6~ qa(5 sin 3z+ ~sin 5Z) -

3(

- -1q 512

35. 3Z+-Sln 8. 5Z+-Sln 1 . 7) Z 3 3 18

--SIn

+

1 n4( --sIn 17. 3Z - 343. 5 61. 7 1 . 9) +--'1 S l n z+-sln z+-sln Z 4096 3 54 108 180

]

.•••

(7)

5 4 3

2

fe1(%'O'O~

1

OIL---~---"";;~:""'--_.....,..-1~---'---I--~--

5IT

Values of FlO.

%

15. Graphs of feo(z, q), fe 1(z, q}, q = 0·08.

By virtue of the first member on the r.h.s. fe 1(z, q) is non-periodic, unless there are values of q I=- 0 for which C11(q) vanishes. When q -+ 0 in (7), tel(Z, q) --~ sin z. Also fe1(21T+z)-fe1(z)

=

(}I(q)21Tce 1(z,q), a function with period 217',

(8)

the conditions in § 7.20 are satisfied (see footnote respecting oddorder functions). The graphs of feo(z, 0'08), fe1(z, 0'08) are depicted in Fig. 15.

80

7.31. 0m(q), Sm(q) for first normalization in § 7.21. The following were obtained by procedure different from that in § 7.30 [80]:

feo

Co(q)

=

I

(by convention],']

(1)

fel

CI(q)

=

q-

:4q3-

(2)

2:6 lf

+ 3::64 q6+ O(tf),

t When a = q = 0, Mathieu's equation becomes 1/* = O. Thus y = C 1 z+c t t where = 1/-'J2 = ca' and feo(z, 0) =- Co(q)z ceo(z, 0) = 0o(q)z/v2. If we take Co(q) = I, c1 = 1/v2, and y = (l/v2)(z + I).

ceo(Zt 0)

148

NON .PERIODIC SECOND SOLU1'IONS

~qll-l:~2q4+0(q8),

fe2

Gll(q) =

fea

1 ~ Cs(q) = 192 Q3+ O(q4);

gel

f:J ( ) -

gel

811(q) =

gea

8a(q)

1

[Chap. VII

3 3

q - q- 64 q

(3)

(4)

3" 31 s O( 8) + 256 ~ + 36864 q + q,

(5)

~qa+ 6:~2q"+O(q8),

(6)

= 1~2q3+0(q4).

(7)

7.32. Gm(q), Sm(q) for second normalization in § 7.21, when o. By § 7.41, !~), g~), the respective coefficients of sin ms, cosmz in the periodic part of the series for fem(z, q), gem(z, q), tend to unity as q -+ 0, the other coefficients tending to zero. By (1 &)-(4a) § 7.22 the coefficient of sinmz, cosmz is unity for all q, and the remainder tend to zero with q. Hence when q --+ 0, Gm(q), S,n(q) for the second

q -+

normalization are the limiting forms of these functions in the first normalization. Retaining only the terms of lowest order in § 7.31 gives G1(q)

= 81(q) =

Gs(q)

q;

= 8s{q) =

02(q)

=

1 8

82(q) = _q2;

1

192 q3.

(I)

Accordingly these are the forms of the various functions as q -+ o. To illustrate this, the values of 0 1 (1) and Ca{l) were computed by the method of § 7.54 using (7) § 7.22. The results were

01 (1)

~

0·9526,

(2)

Ca{l) ~ 0·00513 ~ 1/195.

(3)

These data indicate that 01(1) -+1, and Ca{l) -+1/192.

7.40. Recurrence relations for f~m), g~m). If (I) § 7.22 is substituted in series form into (2) § 7.30 and the coefficient of sin(2r+2)z equated to zero for r = 0, 1, 2,... , we obtain the two recurrence relations (omitting superscripts on the f) fezn(z,q):

and

(tt-4)!Z-q!4 = 4A~2n) Z (a-4r )! tr- Q(! 2r+2+ f zr- ,,) = 4rA~~n)

(1)

(r ~ 2).

(2)

CORRESPONDING

7.40]

'ro

Be,.., CeM , Be,.

<'e".,

149

Similar procedure using (2)-(4) § 7.22 leads to the following:

£el "+I(: ' q):

(a-I +q)!I-q!a = 2A~2n+l),

=

[a-(2r+ 1)2J!2r+l-Q(!2r+s+flr-l)

ge 1n+l(z. q):

(3)

2(~r+l)A~~'+1l)

(r ~ 1).

(a-I-q)Ul-qga = -2Bi2n +1),

=

[a-(2r+l)2]gzr+l-q(g2r+3+g2r-l)

geSn +2(z, q):

(5)

-2(2r+l)B~~'t1l) (r ~ 1). (6)

agO-qg2 = 0,

(7)

(a-4)g2-Q(g4+ 2UO) = _4B~2n+2), (a-4r 2 )g2r- Q(g2r+2+ g2r- ?)

=

(4)

-4rB~~n+2>

(8)

(r ~ 2).

(9)

7.41. Behaviour of 1m' 9m as q -+ o. By § 3.32, A~)-+ 1, B<,:'>-+ 1, and the remaining A, B ~ 0 as q ~ O. We shall now demonstrate that 1m' Um have similar properties. We take the particular case of fes(z,q). By §§2.151, 3.32, 3.33, 7.31, we have a=

9+-kq2+61~3+ ..., A~a)

=

O(q),

Ai3 )

=

O(q),

A~3) -+

I,

when q ~ o.

Ca(q) = O(q3),

(1)

Since 12r+l = !2r+l/Ca(Q), (3), (4) § 7.40 may be written (a-I +q)!I-q!a = 2C3(q)A~S),

(2)

[a-(2r+l)2]!2r+l-q(!2r+a+!2r-l) = 2(2r+l)Ca(q)A~~~1

(r ~ 1). (3)

Substituting for a, A, from (1) into (2) gives

(8+q+-k q2+ ... )!t-q!a

(4)

= O(q4).

Now assume,that!a tends to a constant value as qlJ~

o.

11 -+ qla!(8+q) ~ lq( l-lq)/s -+ o.

Then (5)

Substituting from (I) into (3) with r = 1, 2, we get (-kq2+,~a+...}la-q(!5+!1) ~ 6Ga(q)

and

(-kq2+~a+... -16)!5-q(J1+J3)

=

O(q4).

(6) (7)

Since la -+ a constant as q ~ 0, it follows that 10 ~ o. Similarly this result may be derived for all the lof higher order than 5. Neglecting 17 in (7), we obtain :t "" _~ ,7 J5 -

HqJa·

(8)

Substituting from (5), (8) into (6) yields

(-kq2+liq3+ ...+-kql_iql+liq3)/a ~ 60s(q)·

(9)

l

(10)

Hence as q ~ 0,

Ca(q) ~ aq3/192.

NON-PERIODIC SEC()ND

150

S()LU~'IONS

[Chap. VII

This result is compatible with (4) § 7.31, provided la -+ 1 as q -+ 0, so the assumption above (5) has been justified. By similar analysis it may be shown that J~) and g~n) ~ I, while the remaining J, g ~ 0 as q -+ o. This latter result is independent of the normalization, which must be compatible with it. The two normalizations in § 7.21 fulfil this requirement. (4

7.42. Behaviour of /p, gp, as q -+ we get by aid of (4) § 7.31

o.

Since 1~~~1

= !~~~I/Ga(q)J

f2r+l -+ 192!2r+l/q3. (1) From § 7.41, when q ~ 0, it may be deduced that Jl' J6 are O(q), 17 = O(q2), J. = O{q3), and so on. Since 13 ~ 1, it follows from (1) that 11' /S, /5' /7 ~ 00, but this is inconsequential. Also by analysis of the type in § 7.41 it may be shown that Cm(q), Sm(q) are O(qm).

Hence when q

=

0, jf m

=

2n

2n+l

} all j, g of order lower than m,

and 2n-l} of order higher than m are infinite. Thus, including 2n

/~~), g~),

there are 3n } coefficients in the series which become 3n+l infinite. As before, this is inconsequential. 7.50. Theory underlying computation of /p, gpo The recurrence relation (4) § 7.40 is a complete linear difference equation of the second order. Let t/J2r+t' «/12r+l be independent solutions of

[a- (2r-+- ) )2JC2r+lr-q(C2r-t 3+c2r-t)

== 0,

(1)

and 12r+l be a particular solution of (4) § 7.40. Then the general solution of the latter is U 2r +1

= 12r+l+8eP2r+l +EeP2r+l'

(2)

where 8, E are arbitrary constants. As shown in § 3.21, one solution of (1) tends to zero, the other to infinity as r ~ +00. Now the A in the series for ce2n +1(z,q) satisfy a relation of the form (1), and they tend to zero as r ~ +00. Hence we may replace 8c/>2r+l by 8A 2r +1 , 8 being a constant to be determined. For convergence of the series containing 12r+l we must have /21"+1 ~ 0 as r -:,. +00. Then from (2) W 2r +1

=

(U 2r + 1 - f -P2r+l )

= f2r+l+8A2r+l ~ 0

as r ~

+00.

(3)

Now choose a value ofr = 8, which is such that: (1) IA 2r +1 1 decreases rapidly with increase in r, (2) far +8, say, vanishes to an adequate

CORRESPONDING TO ce..., 8e"., Ce"., Se",

7.50]

151

number of decimal places, (3) Iw2r - 11 ~ IW2r +s l. Since E is arbitrary,'] let it be such that when r = 8, W Zr +1 = o. Then for r < 8, 121'+1 = w2l"+1-8A~'t11).

(4)

The value of 8 depends for given a, q, m upon the value of 8. Since I"r+l' expressed to a limited number of decimal places, is independent of 8 (if large enough), it follows that w 2r +1 is dependent upon 8. Now W 2r +1 = u2r+l-E~2r+l' so E varies with 8, as we should expect.

7.51. Calculation of the w in § 7.50. The w satisfy (4) § 7.40, so with r = 8, we put W 2s +1 == 0, and neglect W 2s+3• Using tabular values of the A [52, 95], we calculate U'2s-1

Also with r

:=:

8

== -2(2~~-+-1)A~~'t11)q--l.

(I)

and (r-l) for '., since w 2B+1 = 0, (4) § 7.40 gives

[a- (2r-l )2]W2r-l-qw2r-3

==

2(2r-l )A~~':11).

(2)

Since w 2r - 1 is known from (I), W 2r- 3 may be calculated from (2). Again with (r-2) for r in (4) § 7.40, w 2r - S may be calculated using the values of w2r - 1' W 2r - 3• Proceeding in this way we ultimately reach r == 1. Then (a-9)w a- q(w 1+u's) == 6A~2n+l), (3) from which

WI

may be calculated, since w a' w 6 are known.

7.52. Determination of fJ in (4) § 7.50. Substituting from this equation into (3) § 7.40 gives

(a-l+q)(w l - 8A 1)-q(wa- 8A s) = 2Ai2n +l). Also by (2) § 3.10 qA == 1•

s (a-l-q)A

(1) (2)

Then from (1), (2) we obtain

8

= _I_[(a-l±~)wl_U's] _!. 2A 1

q

q

(3)

Since the values of all quantities on the r.h.s. of (3) are known from § 7.51, 8 may be calculated. Then the 12r+l may be computed using (4) § 7.50.

7.53. Calculation of the 12,+1 in fe 1(z, 8). Here q == 8, a = -0,43594 36013 20,. We divide (4) § 7.40 throughout by q, and substitute w for I, since w 2r +1 is a solution. Then q-l[a-(2r+l)2Jw2r_ll-(W2r+3+W.. _t} = 2(2r+l)A~~~lq-l.

t

Its value need not be known.

(I)

[Chap. VII

NON. PERIODIC SECOND SOLUTIONS

152

We now choose 8 to conform with the conditions specified below (3) § 7.50. Usually several trial 8 will be required at this stage, TABLE

8. Numerical data foro-evaluation of f~~~1 in fe1(z, 8) 8

r

I

(21'+ l)1-a 21'+1

2 3 4

u

6 . 7

8 9

=

0·62641 0·73885 O'24505 0·04030 · .398 · .. 26 · ... 1

0·17949 ... 1·17949... 3·17949 ... 6·17949 ... 10'17949 •.. 15·17949 ... 21·17949 ... 28·17949 ... 36·17949 ... 45·17949 ...

11 13 15 17 19

,J')

Wl5

0 (- 1)'2(2,,+ t)A~~\l/f

( -1)'A~~)+ 1

f

1 3 5 7

0 I

= 7,

.. ..... '"

•• I"

0·15660 0·55414 0·30632 0·07052 •. 896 ... 72 .... 4

79353 1 55385. 69636 1 13496 1 49422 5 33419 3 24547 0 . 4424 1 .. 122. .... 2,

'" " ... "

I

.....

44838 a 16539. 12045 1 73618 a ' 61200, 41903 0 04777, 16590• .. 520, ... 128

( -l),+l/g)+l

-

-_

r

( -l)'8A~~)+ 1

(-1)'1D2r + 1

..-

1·54597 1·04543 0·24124 0·02792 . . 187

0

I 2

3 4:

5

1·07865 1·27226 0·42197 0·06939 .. 686 ... 45 .... 2

725• 43 31 68 11 94 u 03 13

•••• 75615

6

II

1668

'"

.....

zero

7 8 9

= [fD2r+1-8A~~)+l]( -1)r+l

75u 89. 7 4735 67 17 18&3 34. 0 14u .7..

0·46731 0·22683 0·18072 0·04146 .. 499 •.• 37 •.•. 1

.....

e.g, r = 8, 9 in Table 8. With r = 8 in (I), we put w 2B+1 W 2B+S' and obtain W 2B- 1 = -2(28+ I)A~~1 s:'. Choosing r

=s= Wl a

7, A 15 = -4.4241 X

10-7,

=

(2)

and (2) gives

1-245470 X 10-6 ,

-1(169·43594... )1·6590,x,IO-8- w 1I

grvmg

'WI 1

r=

0, neglect

= 30x4·4241 x 10-7/8 = 1-6590. X 10-8 •

Taking r = 6 in (I), Ala hypothesis, we get

TAking

=

97 u 46 a• 79.. 728• 15so 78u 97 8, .7'1

5, All

=

=

80

with

(3) Wl5

=

= YX 1-245470 X 10-6,

-7'5615 X 10-5•

-2-63341 9a X 10-4, and using w l a ,

by

0, .

(4) WIt

in (1),

we find that (5)

CORRESPONDING TO

7.53]

00....

ee...,

ee., Be..

153

Proceeding in this way we obtain the values in column 6, Table 8. Using (3) §7.52 yields (} = 1·72194 54961.

(6)

Next we calculate 8A~~~1 and get column 7. Finally subtraction of the corresponding values in column 7 from those in column 6 yields t,hef~~~l·

Oheck that conditions in § 7.50 are satisfied. (I) The IA 2r +1 1 decrease rapidly with increase in r ~ 3; see column 4, Table 8. (2) With r = 7, /17 vanishes to seven decimal places, which is adequate here. (3) Writing (r+ I) for r in (I), we get [a-(2r+3)2]w2r+s-Q(W2r+5+W2r+l)

=

2(2r+3)A~~~3.

= 0, by hypothesis, while for r = 7, (2r+3)2 approximately

W l5

~

lal, so (7)

(7)

becomes

W2r+3+qw2T+5/(2r+3)2 = -2A~~~3/(2r+3). (8) For r ~ 3, the data in column 6, Table 8, show that the w decrease rapidly with increase in r.t Then IW 17 > \wual, so in (8) the term in W 2r +5 may be neglected. Hence WI? ~ -2A 17 / 17 = -2x 1·22.x 10-8 / 17 1

~

-1·44X 10-9 •

(9)

Thus /W 13 /W17 I ~ 1·659x 10-6/1·44x ~ 1·15X (10) We have shown, therefore, that the imposed conditions are satisfied. It is of interest to remark that in a second set of calculations with WID = 0, (J ~ 1·946, which exceeds the value at (6) above, while for a given r ~ 6, w exceeds that in Table 8. This is in accordance with § 7.50. The f4~~1 are in agreement to the eighth decimal place, r=Oto7. 10-9

10 3 _

7.54. Normalization. We use the second normalization rule in §7.21. Then from the values off2r+l in Table 8 00

I flr+l = r=O

0·30424 88 4 •

(1)

Substituting in (7) § 7.22, we obtain G1(q) = 1/(0-30424 88.)1 = 1·81294 87 8 •

t Since w.. :=--:; 0, it Ini~dlt be preferable to say tha.t with increase in r, the (!8HC of r = 7 being omitted, 49Rl

x

Iwt "

(2) s/u·tr+1l der-reases rapirlJy

NON-PERIODIC SECOND SOLUTIONS

154

[Chap. VII

Multiplying the figures in column 8, Table 8, by (2) yields column 3, Table 9. Then co

fe1(z,8) = (I·SI2...)zcertz,8)+ !JJ~t18in(2r+l)z.

(3)

r=O

TABLE 9 2r+l

r

1

0 1 2 3 4

3 5

7 9 11 13 15

r, 6

7

1~~1

01(q)JJ:~1

=

0·84722 0·41123 0·32765 0·07517 0·00904 0·00068 0·00003 0·00000

670 95 7 04. 80 7 93. 501 58 7 138

7.55. Formulae for calculating f2n' g2n+l' g2n+2. These may be derived by analysis similar to that in § 7.50, and we get: c. "(2n) -- w2r - 8A(2n) (I) le 2n ("N,q ).· Jlr 2r'

8= ., 2~J(4 qa)W

4] - ~ .

2+W

ge 2n +1(Z, q):

g~~~11)

=

W2r+1-8B~~'+11),

2~J(a-;-q)WI-W3] -~.

8= -

g~~'t~2) = W2r+2-8B~~'t~'a),

8=

!.-(W2-~WO). B q

(2) (3)

(4) (5)

(6)

2

The procedure in calculating the f, 9 is identical in form with that in §§7.53, 7.54, and the normalization rule at (7) §7.22 is used to determine Cm(q), Sm(q).

7.60. Second solutions corresponding to cem(z, -q), Bem(z, -q). These are derived by writing (!1T-Z) for z in (1)-(4) § 7.22. 'I'hus, with the aid of (2)-(5) § 2.18, we obtain feln(z, -q) = (_1)1&+1 fe"n(!1T-Z, q) =

(a l n )

(1)

-02n(q) [ (11l'-z)ce 2n(z, -q)+( -I)n r~ (-I)'1~~)28in(2r+2)z], (2)

7.60]

CORRESP.ONDING TO

fe 1n +1(z, -q) = (-I)" ge 2n +1(!1T-Z, q)

00.., 8e""

155

Cc"" Sc",

(b2n +1 )

(3)

= SlIn+l(q*!1T-z)cell n+1(z, -q)+( -l)n r~ (- Wg~~'t1l)sin(2r+ l)Z]; (4)

ge 1n +1(z, -q) = (-I)nfe 2n +1(! 1T- z, q)

(a 1n +1)

= Olln+1(q)[
Jo(-- Wf4~tl)

(5) cos(2r+l)z], (6)

ge 2n +l{z, '-q) = (-I)nge1n +2(! 1T- z, q)

(b2n+2 )

= San+II(q)[(!1T-z)selln+lI(z, -q)+(-l)n

Jo (_l)rg~~n)cos2rz].

(7) (8)

Then by § 7.22, when q ~ 0,

fem(z, -q) ~ +sinmz, gem(z, -q) ~ +cosmz, the multipliers ( -1)n ensuring the conventional positive signs. Since cem(z, -q) ~ +cosmz and sem(z, -q) ~ +sinmz, we obtain the two solutions of y" +m2y = 0, which is the degenerate form of Mathieu's equation when q = O. The complete solution of y"+(a+2qcos2z)y = 0 for q positive, l.e, q negative in the standard form, is obtained from (8), (9) § 7.22 by writing -q for q, the characteristic numbers being those above and in §2.18.

7.61. Second solutions corresponding to Cem(z, q), Sem(z, q). These are solutions of y"-(a-2qcosh2z)y = 0, corresponding to a = am' (L= bm respectively. They are derived from (1)-(4) §7.22 by writing iz for z. Thus for the second normalization of §7.21, with the aid of (2)-(5) § 2.30, we get:

Fe1n(z, q)

= - i fe 2n(zi, q) = G2n (q)z Ce2n(z, q)+F2n (z, q)

(1)

=

-ife 2n +1(zi,q) = G2/t +1{q)z Ce2n +1(z, q)+F2n +1 ( Z , q) Ge I 1l+1(z, q) = ge 2n +1(zi , q) Fe 2n + 1(z,q)

= -San+l(q)zSe2n+l(Z,q)+G2n+l(Z,Q)

(2)

(b1n +1 ) ,

(3)

(b1n +I ) .

(4)

Ge1n+l{z, q) = gean+a{zi, q)

=

-8sn+,(Q)zSeln+,(z,q)+Gln+S(z,q)

NON -PERIODIC 8EC()ND 8()LU'fIONS

156

[Chap. VII

By writing iz for z in the third members of (10,)-(40,) § 7.22, we obtain ao

F2 ,t(z, q)

=

02n(Q) If4~~)28inh(2r+2)z,

F2/t +1(Z, q)

=

02n+l(Q) If~~tl)sinh(2r+l)z;

00

(2a)

r=O co

G1n+1(Z,q) = 8 I R+1 (q)

I

r=O co

G2n +2(Z, q )

(La)

r=O

= 8 2n +2(q) I

g~;'t1l)cosh(2r+l)z,

(30,)

g~~n+2)cosh

(,ta)

2rz.

r=O

By §7.22, when q~O, Fem(z,q)-~+sinhmz, Gem(z,q)-++coshmz. The complete solution of the differential equation for q positive is y(z)

or

= A Cem(z,q)+BFem(z,q) (am)

y(z) = ASem(z,q)+BGem(z,q)

(5)

(6)

(bm ) ,

according 8S the characteristic number is am or bm • Either (5) or (6) constitutes a, fundamental system of solutions for the modified Mathieu functions of integral order. '

7.62. Second solutions corresponding to Cem(z, -q), Sem(z, -q). These are solutions of y"-(a+2qcosh2z)y = o. They are derived from (1)-(4) §7.61 by writing (l1Ti+z) for z, since this alters the sign of q in the equation in § 7.61. Thus from (I) §7.61 we have (-1)"'Fe 2n (!'7Ti+ z, q) = (}211(Q>[(!7Ti+z)Ce 2n(z, -q)+(-l)"

f (-I)r+1fJ;~~8inh(2r+2)z].t

"1110

(1)

Now, if z is real, the r.h.s, represents a complex function, but it is expedient that the function defined should be real. Since Ce1n(z, -q) is a first solution of the differential equation, there is no need to retain t1Ti Ce2n(z, -q). Accordingly we adopt the following definitions:

Fe.,,(z, -q) = Oln(q)[zCe1n(z, -q)+(-l)n

t

f

r-O

(_I)r+lf~~18inh(2r+2)z]

(a",),

This result is obtained also, if we take -ife.,.(iz, -q), using (1) § 7.60.

(2)

C()RRESP()NDINO TO re.,

7.621

1'4El..,

Oe., Be...

157

Fe2n +1(z, -q)

=

-&1I+1(q)[ZCe 2'1+1(Z.

_q)+(_I)n+1,.~(-ltg~~1I)sinh(2r+l)z] (b2n +1 ) ;

(3)

Ge 21& +1(z, -q)

= G~n+1(q)[z Se2n + 1(z, -q)+ (-I)n Jo (-I)':f4~~tllcosh(2r+ l)z] (at n +1 ) ,

Ge2n +2(z, -q) =

-82n+2(Q)[zSe2n+2(Z, _q)+(_I)n

By § 7.22, when q ~ 0,

(4)

i (-I)rgg,nHlcosh2rz]

'-0

(b1n +I ) .

(5)

Fem(z, -q) ~ + sinh mz, Ge,n(z, -q) ~ + cosh mz, 2 these being solutions of y" -m y = 0, the degenerate form of the modified Mathieu equation when q = o. As in § 7.60 the multipliers (-l)n ensure the conventional positive signs. The complete solution of y"-(a+2qcosh2z)y = 0 for q positive, i.e. q negative in the standard form, is obtained by writing -q for q in (5), (6) §7.61, the characteristic numbers being those above and in §2.31.

VIII tf'

SOLUTIONS IN SERIES OF BESSEL FUNCTIONS

8.10. First solution of y"-(a-2qcosh2z)y = 0, q > u = 2kcoshz, k 2 = q > 0, and the equation becomes [52] (u 2- 4k 2)y "+ u y ' + (U2_ p 2)y = 0,

with

p2

=

(a+2k 2 ) .

o.

Let (1)

Assume a solution ex>

!

y =

(2)

(-I)rc 2rJ2r(u),

r=O

and substitute it into (1); then we get ex>

!

"=0

(-1)rc2r[(u2-4k2)J;r+uJ~r+(u2-p2)J2T] =

o.

(3)

From Bessel's equation u2J;r+uJ~r+u2J2r= 4r 2J 2T,

(4)

and by recurrence relations 4J;r

==

(5)

J2r-2-2J2T+J2r+2.

Substituting (4), (5) into (3) leads to ex>

! (-I)Tc2r[(4r2-a)J2r-k2(JIr_2+J2r+2)] r:.:O

== 0,

(6)

since 2k 2 - p 2 = -a. Equating coefficients of J Ir to zero (r = 0, 1, 2,... ), we obtain Jo aco- k 2c 2 = 0, (7) 2(c J2 (a-4)c 2 - k 4+2co) = 0, (8) J2r

(a-4r2)c2r-k2(C2T+2+C2T-2)

=

°

(r ~ 2).

(9)

Now (9) is a linear difference equation of the second order, and it has two independent solutions, so the complete solution takes the form U = YC +8d , ( 10) 2T

2r

2r

where y, 8 are arbitrary constants. When r ~ +00, C2r ~ 0, while by § 3.21 Id2r 1 -+ 00 in such a way that /V2r+21 ~ 4(r+ 1)2q - l . Thus by 2° § 8.50, (2) would diverge, so the solution dar. is inadmissible, i.e, 8 = O.t Since (7)-(9) are identical in form with the recurrence relations (I) § 3.10 for the A in QO

Ce2n(z, q)

=!

r=O

t

.A~) cosh 2rz

See § 3.21.

(a 2n ) ,

(11)

8.10]

SOLUTIONS IN SERIES OF BESSEL FUNCTIONS

it follows that

CIr

US9

is a constant multiple of A~~n). Hence 00

K

I

(-1)'A~~n)J2r(2kcosh z)

(12)

r=O

is a solution of y" -(a-2qcosh 2z)y = o. Now both (11), (12) are even functions of z with period 1Ti and they satisfy this equation for the same (a, q). Thus one is a constant multiple of the other. Accordingly 00

=

Ce2n(z,q)

I

K

r=O

(-I)'A~;n)J2r(2kcoshz).

(13)

By 20 §8.50 this series is absolutely and uniformly convergent in any finite part of the a-plane. Putting z == l1Ti gives K = Ce21l(!1T,q)/A~2n), (14) since all the B.F. vanish except when r = 0, giving Jo(O) = 1. Hence from (13), (14) it follows that Ce2n (z, q) = ce2nA(!(2n) 1T,q) L ~ (-I)'A(2n),,(2kcoshz) (a2n )• (15) II" 2r o r=O By taking u = 2k sinh z in the differential equation and proceeding as shown above; we derive the solution ex>

K 1 I A~~n)J2r(2ksinhz).

(16)

r=O

This is even with period 1Ti, and is also a constant multiple of Ce2n (z,q). With z = 0, K I = ce2n(0,q)/Ab2n ), 80 2n (O, q) ~ A(2n) T (2k · h) Ce l n (Z, q) -- ce A(ln)-- L 2r tJ 2r SIn z

o

(a )· zn

(17)

reO

Thus (15), (17) are alternative forms of the first solution. The multipliers K pertaining to various functions in the sections which follow are derived in Chapter X.

8.11. The second solution. Since the recurrence relations for the Bessel functions Y2,. are identical in form with those for J I ,., and both functions satisfy the same differential equation, it follows from (15), (17) § 8.10 that the function defined by n (~1T , ~ Fey111 (z, q) = cel A(2n)

o

_ cean(O,q) -

A(tn)

o

~ L

(-I)r A (2n)1: (2k cosh e) I'

2r

(lcosh e]

~ L

"=0

A(2n )l': (2ksinhz) tr 2r

> 1) (l)t

r=O

(

t The 11 in Fey signifies the r-Bessel function.

ISinh ZI R(z)

1)

> >0

(2)

160 SOLUTIONS IN SERIES OF BESSEL FUNCTIONS

[Chap. VIII

is an independent solution for a = a 2n • The restrictions indicated on the right are necessary for absolute and uniform convergence, and are obtained in 3° § 8.50. Actually, although both series are non-uniformly convergent as Icoshzl or [sinh a] ~ I, they converge when these arguments are unity, but the rate of convergence is dead slow! Solutions in B.F. products are free from these disadvantages (see § 13.61). Using the well-known expansion for the Y functions in (1), (2), we find that FeYzlI(z,q) =-

3[Y+Iog(k~~:~;)JCezl/(Z,q)+two

double summations, (3)

i' being Euler's constant, The first member on the r.h.s. may be

written (4)

from which it is clear that neither (1) nor (2) is periodic in z. In Chapter XIII we shall obtain a relationship between ~'eY2n(z, q) and the alternative second solution Fe 2n (z, q) of Chapter VII. It is not possible to do this using the series (1), (2), since the non-uniform convergence of (1) and the divergence of (2) at the origin renders term-by-term differentiation invalid.t This applies to kindred functions in succeeding sections. The complete solution of the equation, corresponding to a = a 2n , is (5)

where A, B are arbitrary constants. This constitutes a fundamental system.

8.12. The solutions Ce2n+1(Z, q), FeY2n+l(z,Q). similar to th4t In § 8.10, we obtain (1), (3) below:

By analysis

Cean+1(z, q) -

-

_

ce~n+l(!1T,q) ~ (-I)'A(2n+Ucl (2kcoshz) kA (In+l) L.t 2r+l 21"+1 1

kA12n+l)

In+l

)

(1)

reO

- ce2n+1(O,q)cothz

-

(a

~ (2r+I)A(Sft+l)tl (2k sinh z}; L.t Ir+l 11"+1 ,

(2)

r=O

t The result of of (9) § 13.31.

80

doing to (I) gives

I4·ey~,.((), q) -=- (),

which is untrue by virtue

SOLUTIONS IN SERIES OF BESSEL FUNCTIONS

8.12]

161

FeYln+l(Z, q) -

-

ce~n+l(Vr,q) ~ leA (2n+l) ~

_

1

(-I)'A(2n+l).Y: 21"+1

2r+l

(2leCOBh~) ~

(3)

'=0

- ce 2n+!(O,q)cothz kA (2n+l) 1

~ (2r+l)A(2~+1)l': (2le sinh z) 2r+l 2r+l •

(4)

~ r=O

The restrictions at (I), (2) §8.11 apply to (3), (4) respectively. To derive (2), (4), we observe that Ce2n +1(z,q) is an even function ofz, whereas J2r +1 (2k sinh z) is an odd one. If we multiply the latter by the odd function coth z the product function is even. Assuming that y = cothzw(u), with u = 2ksinhz, and proceeding as in § 8.10 leads to the series solutions (2), (4). By virtue of the logarithmic term in the expansion of 1;r+l' these solutions are non-periodic, but (1), (2) have period 27Ti. T};1e complete solution takes the same form as (5) §8.11.

8.13. The solutions Sem(z,q), GeYm(z,q). As before we assume a Bessel series with argument u === 2k cosh z, namely, 00

u,(u) ==

!

r=--O

(1)

(-1 )'C 2r -t 1 J 2r+ 1(u ),

this being an even function of z. Rut He 2n +1 (z, q) is odd in z, so we take the odd function y == tanh z lO(U) (2) and substitute it in the differential equation. Then the transformed equation is [52] d (3) (u 2- 4k 2)w" +uw' +(U 2_p 2)W+ 8k 2du (wju) == o. Proceeding as in § 8.10, we find that C 2r +1 ::::::: K 1(2r+ 1)B~~+11), K 1 being a constant multiplier. Hence tanhzw(u)

=

(4)

00

tanh z ~ (-1)r(2r+I)B~~'+11)J2r+l(2kcoshz) r=O

(5)

is a first solution of y" -(a-2q cosh 2z)y corresponding to the characteristic number a = b2n + 1 • By analysis of the type in § 8.10 we obtain Se2n + 1 (z, q)

= S~211_:tJ(*!",q)tanhz ~ (-lY(2r+l)B~;+-P)J2r+I(2k cosh z) kBC2n +1) L." r=O

1

= S~~'!lj::l(O'9) ~ kB(21t+l) ~ un

1

B(2n+1)J. 2r+l 2r+l

(2ksinhz).

'=0

y

(b2n+I)

(6)

(7)

162 SOLUTIONS IN SEIlIES

O~'

BESSEL .[4'UNCTIONS [Chap. VIII

These representations both have period 2m. Since sinhz is odd in z, (7) was obtained by substituting (1) with u = 2ksinhz in the differential equation. The remaining solutions derived as shown above and in previous sections are GeY2n+l(z, q)

~

-- se2n +1(! 1T, q) tanh z kB(2n+l) -

(-I)r(2r+I)B(2n+l)y; 2r+1

(2kcoshz)

2r+1

(b

2n+1

se~n+l(O, q) ~ kB(2n+l) ~ 1

B(2n+l)Y; (2k· h 2r+1 2r+1 SIn

)

(8)

r=O

1

-

~

z.)

(9)

r=O

These are the second solutions corresponding to (6), (7). By virtue of the logarithmic term in the expansion of the Y function, (8), (9) are non-periodic. Attention is drawn to the restrictions and comments in §8.11. Se2n +2(z, q)

= - 8e~n+2(11T,q)tanhz ~ (-I)r(2r+2)B<2n+2}.J (2kcoshz) k2 B(2n+2) ~ 2r+2 2r+2 2

(b

2n+2

)

(10)

r=O

= se~n+2(O,q)cothz ~ (2r+2)B(2nH).l (2ksinhz) k 2 B(2n+2) ~ 2r+2 2r+2 •

(11)

r=O

2

The B.F. in (10), (11) being of even order are both even in z, but the hyperbolic multipliers are both odd. Hence the representations are odd in z, also they' have period 1Ti. The second solutions (apart from Gem(z, q) in Chapter VII) corresponding to (10), (11) are Gey 2n+2(z, q)

8e~_'0-2(!1T, q)tanh z ~ ~ (-I)r(2r+2)B(zn+2»)'; (2k cosh z) k 2 B(2n+2) 2r+2 2r+2

= -

2

,

=

(0)

(b

2n+2

)

r=O

(12)

(2r+ ll) B (2n+2> Y;2r+2(2ksinhz) • .. 2r+2

(13)

00

~~~+~--~cothz '" k 2 B(2",+2) ~ 2

r=O

.

See remarks below (9) above. The complete solution takes the form y

=

A Sem(z,q)+BGeYm(z,q).

(14)

8.14. The solutions Fekm(z,q), (jekm(z,q). These are derived from §§ 8.11-8.13 by expressing the Y functions in the series for

SOLUTIONS IN SERIES OF BESSEL FUNCTIONS

8.14]

163

FeYm(z,q), GeYm(z,q) in terms of the J- and K-Bessel functions. Thus K.,(u) = 11Tei1r(v+l)iH~I)(iu),

(1)

(2)

80

Writing

v

= 2r, and applying (2) to (1) § 8.11 yields

(t 1T, q)

Ce2n

Af2~ X

FeY2n(z,Q) =

o

Jo (-l)rA~~n)J2r(2kcoshz)-;2 L A~~n)K2r(-2ikcoshz) 00

X

i [

00

]

r=O

(k

= iCe2n (z, q)-

>

0, [oosh e]

>

1)

(a 2n )

2F ek 2n (z, q), .

(3) (4)

where we adopt the definition 2n (! 1T, Q) ~ F ek 211- (Z,q ) -- ce - 1TA(2n) k

o

h) 2r (-2·k ~ cos z

A(2n)K 2r

(a2n)•

(5)

r=O

In like manner (2) § 8.11 yields T.'

2n (O, q) ~ ( k (z,q ) -- ce ( 2·k· h ) -";'A(21l) k - 1)'A(2n)K 2r 2r ~ SIn z. o r=O

£e 21t

(6)

Since Fey2n(z, q), C1e 2n(z, q) are real if z is real, it follows from (4) that Fek 2n(z, q) is complex. The restrictions at (1), (2) § 8.11 apply here, and convergence is considered in § 8.50. By virtue of the logarithmic term in the series for the K function, (5), (6) are non-periodic in z. These remarks apply also to the remaining functions given below: . FeY2n+l(Z, q)

=

with

i[Ce 2n +1(z, q)+2 Fek 2n + 1(Z, q)],

(7)

(-2ikcoshz)

(8)

Fek 2n +1 (Z, q)

= -

ce;n+l(!1T,Q) 1TkA(2u+l)

k~

1

r=O

A(2n+l)K

= ~_2n+1(O,q)cothz 1TA·A(2,,+1) k~ 1

r=O

2r+l

2r+l

(-I)'(2r+l)A<21l+ UK 2r+l

2r+l

(a) 2n+l

(-2iksinhz). (9)

164 SOLUTIONS IN SERIES OF BESSEL FUNCTIONS

[Chap. VIII

with

Gek 2n +1(z, q)

-

-

~2"+1(~,q)tanhz ~ (2r+ I)B<2n+l)K (-2ik cosh z) 1TkBi2n+1) L.., 2r+l 2,+1 r=O

_

-

~~n+l(O,_~) ~ (-I)'B<22r.,-1 'TTk i2n +1) L..,

fJ;+1)K

B

II ~1

(-2iksinhz)

(b2n +1 ) (II)

(12)

•

r=O

(13)

GeY2n+2(Z, q) = i Se2n +2(z, q)-2 Gek 2n +2(z, q), with Gek 2n +2(z, q)

= ~~1I:+2
(b

2n+2

=

8e~n+2(0, q)

- :"1i£Jj(2'/t-2) co 2

)

(14)

r=O

tl 1 Z L.., ~ ( - 1)r(~)eiJr + 9) B(2n+2) h ) 2r+2 K 2r12 ( - 2·k· l SIll Z. r=O (15) J

If desired the K functions Inay be expressed in terms of Hankel functions by aid of (1). The complete solutions given at (5) §8.11, (14) §8.13 may sometimes be unsuitable. However, in any complete solution of that type it is permissible to substitute lfek m for FeYm' and Gek., for GeYm' by virtue of (4), (7), (10), (13).

8.20. First solution of y1i+(a-2qcos2z)y = o. By writing -iz for z in the series for Cern' Se m in §§8.10, 8.12, 8.13 we obtain n ( 11T , _91 ~ ce I n (z' q) -- ce iA(211) L.., o r=O

_

-

ceS1l+1( z,q)

~~~_n_(~~ ~ (-I)rA2(2n)12r(2ksinz) A (2n) L.., I'

o

t 1T , q)- L.., ~ (= - ce~1l+1( ---kA(2n+l) =

•

(a2n )

(1) (2)

,

r=O

1 )'A(2n+l) T (2k cosz ) (a + ) (3) 2r+l "'2r+l 2n 1

r=O

1

~e.2n+1(~~q)cotz ~ (-It(2r+I)A(27l+1lL (2ksinz) , kA (In +1) L.., 21'+1 21'+1 1

se 2n +1 (Z, q)

(-- I )r A 21' (2n)J, (2k cos z) 2,

~ (= Se2n+l(!1T, - kB(2n+l)q) t an z L.., 1

1)'(9eiJr

+ 1)B(2n+l) (2k 2r+l tl2r+l T

1'=0

= ~e~,,+!~~~_~_~ ~ l.B(2" +1) L..t 1

(4)

'=0

'=0

(-I)'.m2n+Ul 2,.+ 1

(2~"sinz),

21'+1

COS Z

(b2n +1 )

)

(5)

(6)

8.20]

SOLUTIONS IN SERIES OF BESSEL FUNCTIONS

se1n + 2( Z, ) q

=

Se~n+2(!1T, ~ (k B (l n + 2)q)tan Z L.,

-

2

2

-

165

1)'(2 r +2)B(2n+2) T (2'· ) 21'+2 tJ2r+1 I( COS Z (b2n+2 )

r=O

se~n+2(0,q)cotz ~ (-I)'(2r+2)BCln+I)L (2ksinz) k2B(2n+2) L., 2r+2 21"+1 •

(7)

(8)

r=O

2

8.30. Solutions of y"-(a+2qcosh2z)y = 0, q > O. If we write (!1Ti+z) for z in y"-(a-2qcosh2z)y = 0, it takes the above form, i.e, the sign of q is changed. Making this change of variable in series derived in § 8.10 at seq. leads to the following representations:'] Ce2n(z, -q) = (-1)nCe 2n(!1Ti+z,q) (a2n) (1) = (_1)n c e 2n (! 7T, q) A~2n)

~

L.,

A(2n)L (2ksinhz) 2r

21

(2)

r-=O

i (-I)rA~~n)I2r(2kcoshz).

(_I)nce~~~:._?J

=

o

The multiplier (-1 )'1. is needed by virtue of that in § 2.31. ~""eY21l(z, -q) =: (-I)nFeY2n(!1Ti+z,q) (a 2n)

= (_I)nce2!~!~q~ o

i (-IYA~~n)~r(2iksinhz)

-

o

= (_1)n~2n(1~'22 ~ AC2n)K (2ksinhz) 7TA (2n) L., 2r 2r o

=

(6)

(7) (8)

r=O

(-I)nc~;5i(~-'l) o

FeY2n(z, -q)

(5)

r=O

Fek 2n(z, -q) = (-1)nFek 2n(!1Ti+z,q)

=

(4)

r=O

(_1)nce2!!~~,q~ ~ A(2n)y' (2ikcoshz) • A(2n) L., 2r 2r

-

(3)

reO

i

(-1)rA~~n)K2r(2kcoshz).

(9)

r==o

i Ce2n (z, -q)-2Fek 2U{z, -q).

(10)

Ce2/1, +I {Z , -q)

=

(-I)ni- l i Se2n+1(! 1Ti+ z,q) (b2n ..l )

= (_I)n~2!~,+,,(!1T,q)cothz ~ (2r+I)B(2nl-lll (2ksinhz) kB<2n+l) L., 2r+l 21-+1 1

= (-l)n~i]ii~~~~q) 1

(II) (12)

r=O

i (-lyB<~~1l)I2r+1(2kcoshz).

(13)

r-O

t Formulae 2, 3, 9, 12, 13, 19, 22, 23, 29, 32, 33, 39 are given in reference r6] without the Inultipliers external to the E. The method ofdeeivation differs from that used here.

166 SOLUTIONS IN SERIES OF BESSEL FUNCTIONS [Chap. VIII

FeY2n+l(Z, -q) = (_I)n+li GeY2n+l(17Ti+z, q)

(b2n +1 )

(14)

(_I)n+lise2n+l(tn",q~cothz ~ (-I)"(2r+I)B(2n+l)¥: (2iksinhz) kB(2n+l) ~ 2r+l 2r+l

=

1

(15)

"=0

(_I)n+li8e~n+1(O,q) ~ B(2n+1)~ (2ikcoshz) • kBi2",+I ) ~ 2,.+1 2,.+1

=

(16)

r=O

Fek 1n +1(Z, -q) = (-I}RGek 2n +1 (I 1Ti + z, q)

= (-l)n~;fBf!.:i-Pcothz 1

(17)

i

(2r+l)B~;~11)K2r+l(2ksinhz)

(18)

"=0

= (_I)lIse~II+1(Q'91 ~ (-IYB(2n+1)K (2kcoshz) 7TkB~2n+l) ~ 2r+l 2r+l •

(19)

r=O

}~eY2rt+t(z, -q) ===

i Ce2n tt(z, -q)+2Fek 2n +1 (z, -q).

Se2n+1(z, -q) = (-l)tl+li Ce2n +1(! 7Ti + z,q)

(a2n +1 )

(21)

= (_I)n+l~~!!..+l(!~,q) ~ A(2n+l)1 (2ksinhz) kA (2n+l) ~ 2r+l 2r+l 1

=

(20)

(22)

r=O

(-I}n~~~~l_(O,q]tanhz ~ kA (2u+l) ~ 1

(-I)r(2r+I)A(2n+l)L 2r+l

2r+l

,.=0

(2kcoshz) • (23)

GeY2n+l(z, -q) = (-1)1l+1i ~"eY21l+1(!7Ti+z,q) (a 2n + 1 ) =

(24)

(_I)lliC~~~+l(~1T,q) ~ (-lYA~2~:tl)~ 1(2ihinhz) "A (2" +1) ~ r r+ ~

1

(25)

"=0

= (_I)n+li CC2l'A!!..+(2"l(O:Q! ta nh z ~ (')r+l)A(2n+l)~ (2ik cosh z) +1) ~ 2r+l 2r+l • 1

(26)

r=O

Gek 2n +1 (Z, -q)

=

(-l)n Fek 2,&+1(! 7Ti + z, q)

= (-I )11+1 C~~n-±lJ_!1T,q~ ~ A~21+l'i 7T!'A(211+1) ~ r 1

(27) 1)

K2

r+

1 (2k sinh

z)

(28)

r=O

= (-I)II~~2.-n+1(~,-q~tanhz ~ (-I)r(2r+I)A~2~1l)K2r+1(2kcoshz). 1T~·.4 (2" +1) L.r 1 r=O GeY2,,+1(Z' -q) = i Se2'~+1(Z' -q)+2 Gek21l+1(Z, -q).

(29)

(30)

8.30]

SOLUTIONS IN SERIES OF BESSEL FUNCTIONS

167

Se2n+2(Z' -q) = (_l)n+l Se2n+2(! 1Ti + z , q)

(b2n+2)

(31)

= (-I)n+l~~n+_~J!1T,q)cothz ~ (2r+2)B(2n+2)L (2ksinhz) k2B<2n+2) L.t 2r+2 2r+2 2

=

(-1)n

~~J;(~~~~)tanh z 2

i:

(32)

r=O

(-1 t(2r+

r=O

2)B<2~,+~2)12r +1I(2k cosh e). (33)

GeY2n+2(Z' -q)

=

(_I)n+1GeY2n+2(!1Ti+z,q)

=

(_1)n~e~n+2(!n",q)cothz ~ (-1)r(2r+2)B(2n+2)Y; (2iksinhz) k2B(2n+2) L.t 2r+2 2r+2 2

(34)

(b2n+2 )

(35)

T:lIIO

= (-1 )n+l se~n+2(O, q) tanh z k2 B(2n+2) 2

~ (2r+ 2)B(lIn+2)y; (2ik cosh z) (36) L.t 2r+2 2r+2 •

r=O

Gek 2n +2(Z, -q) =

(-I)n+lGek 2n+2{! 1Ti + z,q)

=

(_1)n+l~e~1l+2H_1T,q)cothz ~ (2r+2)B(2n+2)K (2k sinh z) 1Tk2B(2n+2) L.t 2r+2 2r+2 2

(37)

(38)

rcO

, (0) 00 == (_l)nse~~+~-.-2.9_ tanh z ~ (--1)T(2r+2)B(2n+2)K

L.t

1Tk2 B(2n+2) 2

2r+2

2r+2

(2kcoshz)

•

(39)

r=O

GeY21L+2(z, -q) == iSe 21l+2 (z, -q)-2Gek 2n+ 2{Z, -q).

(40)

The functions involving the Y-Bessel function are complex, but those involving the K -Bessel function are real, if z is real.

8.40. First solution of y"+(a+2qcos2z)y = O,q > be derived by writing -iz for z in the first solution of

o.

This may

y" - (a+ 2q cosh 2z)y == 0 given in § 8.30, or by putting (!1T-Z) for z in the first solution of y"+(a-2qcos2z)y == 0 given in §8.20. Thus we find that ce2n (z, -q) = (-1)"ce 2n (! 1T - z, q)

=

(-1)fl

=

(-I)nCe 2n ( - i z,q)

(a2n ) (I)

lJ) (-1)rA~~n)J2r(2ksinz) A\t;:;_ o r=O

(2)

i

cell

= (-1)11~~2.'l(~-,ql ~ (-1)'A~lIn)I2r(2kco8z). A (21l) L.t r o

r-o

(3)

168 SOLUTIONS IN SERIES OF BESSEL FUNCTIONS [Chap. VIII

cean+1(Z, -q) = (-I)nSe 2n +1( ! 1T - Z, q)

(4)

(b2n +1)

= (_I)n Besn±! (!1T, Q) Cot z kB(2n+l) 1

~

~

(_I)r(2r+I)BC2n+l).l 2r+l

2r+l

(2ksinz)

(5)

r=O

= (_l)lt~e~!t-!l(O,q) ~ kB<21l -n ~ 1

(_I)rB!2n.fl)l

2r+l

2r+l

(2kcosz).

(6)

".:0

se21l+ 1(z, -q)

=

(-l)n ce 2n +1(-j7T-z, q) (a 2n + 1 ) , (1 ) co = (-I )1l+1 ~~~t7"l ~7!,-ll_ " (-I)rA~2~1l)J2 tl(2k sin z) kA (2n+l) ~ r r 1

=

kA(2n+l) 1

==:

(8)

1=0

(_1)n celln_±l(O, V tan z

Se 2n-f2(Z'

(7)

1l / 1I +t(2kcosz). ~ (-I)'(2r+I)A&21t r r

~

(9),

r=O

-q)

(-I) ' t se21t ~2(j1T-Z,q) (h2/1 t2) , (I) ( I)nl.l~e2n~2 2 1T, Q. cot z ~ (-I)r("2r-t-·»)B(2nt2)J.

(10)

(y)

::::::=

~~2B!2"-f2)

~ r~-O

2

:-. : :

(_l)I,,~e~!, ~ L~ 1c B(f2(O,Q)t,.1l11 2" ~ 2) 2

UI

2

*J

*J

4J

2r 1- 2

(lo)ksinz) (11)

21"-12'"

(_I),(oJI'_f-"»JJ(2tlI2JI

,.,

r

2r-+2

~r +2

l

(-)I·1\ COQZ ) o. /oJ

(1"») *J

0

8.50. Convergence of solutions. 1°. Formulae asumptotic in m, '1"0 discuss convergence we require formulae for the various Bessel functions, valid when m, the order of the function, is very large and positive, and far exceeds the argu-

merit 1'ltJ. Then the first terms in the ordinary expansions of Jm(u), /,,.(u) predominate, and we nuty write J,n(u) ,...., !,Il(u) ,....,

Similarly we have

~(~)1l~.

(I)

m.2

2 r 1 (2)1n Y;n(u) ~ ---Rm(u) ~ --(m-l)! - . 1T

1T

(2)

U

From (I), (2) we infer that under the above condition 1J:,,(u) [ ~ IJm(u)1

IIm (u)l. 2°. Series int'ol1'ing J and J [unctions. If 2r p 12kcoshz), and

IKm(u)l}>

(3)

(1) 10

gives, with r extremely large, 1 ~r(2k cosh z) ,...., (2r)! (k cosh z)ZI'.

(I)

SOLUTIOSS IN SERIES

8.50]

O~' BESS}4~L

FUNCTIONS

169

When r is large enough, by (4) § 3.21

IA~~~2/A~~n) I ~ Ik 2 //4(r+ 1)2.

(2)

Thus, taking the ratio of the moduli of the (r+ l)th and rth terms in (15) § 8.10, we get

1

~~!,j-2 J~_+_2 ~ 2_~_~?~~_~) 1 ~ I~AJI~?sh~l~~ 0 as A2rJ2r(2kcoshz)

r

16(r+ 1)4

~

+00

(3) '

if z is finite but unrestricted. Hence the series is absolutely convergent. By applying the' M' test, it may be proved to converge uniformly in any closed region of the z-plane, or in any closed interval, z real. These conclusions are applicable to all series in this chapter which involve the J and I functions. 3°. Series involving Y and K functions. We consider (1) §8.II. Then (2) 1° gives Y2r(2k cosh z)

t'.I

-

(2r-l) !/1T(k cosh z)2r

(1)

and, therefore, by aid of (2) 2°,

1

~2r~ 2 Y~r+2( 2k ~~~~_:)_ 1 ~ . _1~2J(2r±1)2r __ ~ 1/Icoahzl2 A 2r Y2r(2k cosh z)

4(r+ 1 )21 k 2 1 1 cosh z 12

(2)

as r ~ +00. If [cosh z] > 1, the series is absolutely convergent, and the'M' test shows that it is uniformly convergent. As z ~ 0 the convergence becomes non-uniform. The series converges at the origin, but the rate of convergence is very slow. If [cosh z] < 1, z is imaginary, [cosz] < I, and the series diverges. When the argument of the Y function is 2k sinh z, the third member of (2) is 1/lsinhzI 2 • Hence the series is absolutely and uniformly con .. vergent for [sinh z] > 1; for [sinh z] = 1 it converges non-uniformly, but when Isinh zl < 1 it diverges. Thus (1) § 8.11 represents FeY2n(z, q) up to and including the origin, whereas (2) § 8.11 holds up to [sinh s] = 1 only, so z = loge{I+2 1) . Owing to the logarithmic term in the expansions of the Y and K functions of integral order, the additional restriction that R(z) > 0 must be imposed when the argument is 2k sinh z. Despite non -uniformity of convergence, as shown in § 13.60, the function FeY2n(z, q) is continuous, but its derivative at z = 0 cannot be obtained from either (1), (2) § 8.11 by term-byterm differentiation. These conclusions apply to all series in this chapter involving Y or K functions.

4061

z

IX WAVE EQUATION IN ELLIPTICAL COORDINATES: ORTHOGONALITY THEOREM

9.10. The two-dimensional wave equation. In two-dimensional problems associated with sinusoidal "rave motion, the displacement " or its equivalent, at any point (x, y), must satisfy the wave equation, namely, 2

2

+ oy21 8

8 ' ox2

' -L

2

k1 '

=

0,

(1)

where k1 is a constant dependent upon the properties of the medium, and the pulsatance of the disturbance therein. In problems per" taining to a rectangular boundary, e.g. a rectangular membrane, (1) is solved in the form given, subject to the particular boundary conditions imposed. If the boundary is circular or elliptical in shape, it is expedient, prior to solution, to transform (1) to either polar or elliptical coordinates, as the case may be.

9.11. Elliptical coordinates. We write x+iy == hcosh(g+i1J}, so that, equating real and imaginary parts, x == h sinh gsin 1]. 'rhus

== h cosh ~ cos 1],

y

X

2

2

Y

inl 12/;s h- 2 cos I12~~ + h--2 SIn

and

x2

_

--

cos

2

•

2

-1

1]+8111 7] - -

~2

•

- - - - - h 2 '". 2 == cosh 2, - Slnh 2, h 2 cos 2 'YJ SIn 'YJ

(1)

,

=:.:

1.

(2)

Then (1) represents a family of confocal ellipses with major axes 2h cosh g, minor axes 2h sinh g, the common foci being the points x = ±h, y = O. Also (2) represents a family of confocal hyperbolas with the same foci, as illustrated in Fig. 16A. The two families of conics intersect orthogonally, and each intersection corresponds to a point defined by the coordinates x = b cosh ~ cos 'YJ, Y = h sinh gsin 'YJ. The angle ~ varies from 0 to 217' in passing once round an ellipse, while if we consider a stretched elastic membrane clamped between two similar elliptical rings, g varies from zero along the line of foci to at the rings. At the extremities of the major axis of any confocal ellipse (Fig. 16 A), 'YJ = 0, TT, Y == 0, and x == ±h cosh == ±a. At the ends of the minor axis, '1J == !1T, 117', X == 0, and y == ±hsinhg :::-- ±b. If e is

eo

e

9.11]

\VAVE

}1~QlTATION

IN ELLIPTICAL COORDINATES

171

the eccentricity of the ellipse, h == ae, so cosh~ = e- 1 , and when e. ~ 1, ~ ~ 0, while a ~ It. Thus a long elliptical cylinder degenerates to a ribbon of equal length, whose width is 211., i.e, the interfocal

y

1J= 30° ~ =15

0

X TJ =0°

(c)

y

y e=O

e=1

x yf

y'

Flo. 16. (A) Orthogonally intersecting confocal ellipses and hyperbolas for elliptical coordinates, P has elliptical coordinates , =- 2, T'J = !1T (60°), and cartesian coordinates x = It cosh 2 cos j1T, Y = hsinh 2 sin j1T. (B) Degenerate form of ellipse when e = 1. As e -+ 1, a -~ h, b - ~ () and the ellipse-+ the intorfocal line of length 21t. (c) Degenerate form of ellipse, when eccentricity e = o. The foci coalesce at the centre of a circle radius r, equal to the semi-major axis. The original hyperbola through P (see A) is now radii making c/> = ±i1T with X'OX.

distance (see Fig. 16B). With a constant, if e -* 0, ~ -* 00, and the ellipse tends to a circle of radius a. Since k = ae, It ~ 0, the foci

~"AVE

172

EQUATION IN ELLIPTICAL

[Chap. IX

tend to coalesce at the origin, and h COS}l g ~ It sinh g -+ a. If h be constant, as a ~ 00, ~ 00, and e ~ 0, 80 that the confocal ellipses tend to become concentric circles. Now (2) may be written x2 y2 - - - - =k2 (3)

e

COS2~

8Un2~

•

As k ~ 0, yJx ~ ±tan T}, 80 T} ~ 4>, cos T} ~ cos 4>, the confocal hyperbolas ultimately become radii of the circle and make angles tP with the X-axis, as in Fig. 16c. y

(B)

y

o y'

y'

Flo. 17. (A) Hyperbolic (ds l ) , elliptic (ds s ) arc length, and radius vector r. (B) Area (d8 1 d8,) enclosed by two contiguous pairs of orthogonally intersecting confocal ellipses and hyperbolas.

9.12. Arc lengths ds!, ds 2 , and radius vector r, Referring to Fig. 17, the hyperbolic and elliptic arc lengths are, respectively,

= d82 =

de, and

Now

[(8xjog)2+{oy/8e)2]ldg

(1)

[{oxjOYJ)2+ (oyjOYJ)2]l dTJ.

(2)

ox/O, = h sinh, cos YJ, oX/OT}

oy/o,

= -hcoshesinT},

=

/1, cosh, sin TJ,

ay/aT} = hsinhecos1],

so each bracketed member in (1), (2) is II = k[cosh 2 sin21]+sinh2~ cos21]]l =

e

h(cosh2~-COS21])'

h

= 2i (cosh 2g-C08 2TJ)l.

(3) (4)

Hence from (1)-(3) we obtain

dS1 = l1d

e

and dsf,

=

l1d 1].

(5)

Since cUt is along the direction of the normal to the ellipse, we may write

dn

= ' 1 tIl.

(6)

9.12]

COORDIN' ..- \TES: ORTHOGONALITY THEOREl\f

173

The distance of RIl)T point (z, y) front the origin, expressed in elliptical coordinates, is r == (X2+ y2)! == h[cosh 2e cos 27J + sinh 2gsin2 1]]1 = h(cosh 2e-sin2 1])1 (7)

h

== 2 (cosh 2~+cos 21])i.

(8)

t

e

When is large enough, cosh ~ 1'-1 sinh ~ 1'-1 let, so by (3), (7) with h constant, 11 1'-1 It cosh 1'-1 h sinh 1'-1 r the radius vector. Hence we may write

e

dSI

I-....J

rde

I-....J

e

dr;

dS2

I-....J

rdTJ;

dS1 dS2

rdrdTJ.

I-....J

(9)

9.20. Transformation of (1) § 9.10 to elliptical coordinates. z == x-iy == hcosh(e-iT]), then

Write z = x+iy == hcosh(e+iTJ), %Z = X2+y2, and

2

2z2

40

ozoz =

Putting' ,~

==

== ,+i1], ~ =

e+1J 2

8' 0%

2

•

(0

(

2)

-

.... (

8x2 + oy2 zz.

,-iT], we get

%

==

1)

kcosh', Z == kcosh{, and

Thus a~

1

1 hsinh~'

ai =

= hsinh"

80

1 0 -o == ----,

oz

Hence or

-0 ==

k sinh' 8'

82

482

82

1

-----~--

48 2

0 --, . and

--------

k sinh ~ o~

82

o,e~

=

81

4

2

0

2

1

2

(0

8

2

2

2

(2)

ozoz = ox2+ oy2 = hisinh-'sinii-~ a,o~' 8

2

0 0 --+-. ee OTJ2

)

ox2+ay2 = k2(cosh2~-cos27}j o~ll+a7}2 ·

(3)

Applying (3) to (1) §9.10 leads to the equation 2

t2 a' o~

2

2 + -01] 8 2 + 2k ( cosh 2,'

cos ~1])' = 0,

(4)

with 2k = k1h. Then (4) is the two-dimensional wave equation (I) § 9.10 expressed in elliptical coordinates.

9.21. Solution of (4) § 9.20. Let the desired form of solution be ,(e,1]) = ,p(')c!>(T]), where ,p is a function of

of 1] alone. Then we obtain d2,p d2e/>

ealone, and c/J a function

if> d~2 +!fld7}2 + 2k2(cosh 2~-C08 27})!flif> = O.

(1)

174

[Chap. IX

WAV'E EQlTATION IX ELLIPTICAL

Dividing throughout by

if1~

and rearranging leads to

~ ~~ +:!k2 cosh 2g =. - ~ ~~ +2k2 cos 2,/.

(2)

e,

Since the l.h.s, is independent of 1J, and the r.h.s. of each side must be a constant, say a. Accordingly we obtain the two ordinary equations

d 2q, ---+(a-2k 2cos21J}eP

d1J2

= 0

:;~-(a-2k2cOsh2g)if1=

and

(3)

0,

(4)

where a is the separation constant. t Then (3), (4) are the canonical forms with which we have dealt hitherto, and the above analysis illustrates the genesis of the equations which bear Mathieu's name, [f in (3) we write ±i~ for Tj, it is transformed into (4), while the latter is transformed into (:l) if ±i1] be written for ~.

9.30. Integral order solutions of (1) § 9.21. A solution comprises the product of any two functions which arc solutions of (3), (4) § U.21, respectively, for the same values of a and q. Since a may have any value, the number of solutions is unlimited, In practical applications the appropriate solutions are usually given by ordinary and modified Mathieu functions of integral order, i.e. solutions of (3), (4) § 9.21 corresponding to a == am' bm • The solutions 10. Soiuiiotu: of (1) § 9.21 when a has values for Mathieu functions of integral order. q > 0

TABLE

,

:=

CCm (

, =

tPm(~)c/>m(TI)

for a =

e, q)CC

f ll

(.1m (

TI, q)

Fe1ll (f , q)cem("1, q) Cem(~' q)fcm(TJ, q)

Fem(e, q)fem ( ,,/, q) m = 0,1,2,...

t

}'l'operty

_

Period tr or 21f' in 1], m even or odd Period 1f'i or 21Ti in ~, In even or odd Period 1f' or 21f' in "1, ,n even or odd Non-periodic in ~ Non-periodic in TJ Period 1f'i or 21f'i in m even or odd Non-periodic in "/'

e, e

*_M

tPm( ~)c/>m( TJ) bm

for a

:-7.:

•

])roperty

_

Hpm(~' q)~m(TJ' q)

As in column 2.

Gem(e, q)gem("/' q) m = 1,2,3,...

This must not be confused with the semi-major axis of the ellipse.

9.30]

COORDINATES: ORTHOGONALITY

THEOR~~~I

175

= cem(1J,q) or selll(T/,q) with a, q real are important by virtue of their periodicity in 1T, 21T, and single-valuedness. Various product pairs, including second solutions of (3), (4) §9.21, i.e. ordinary and modified Mathieu functions of the second kind, are set out in Table 10. When q < 0, the solutions in Table 10 are valid provided the definitions of the various functions in Chapters II. VII are used. Alternative solutions. As shown in Chapter VIII, there are alternative second solutions of equation (4) §9.21. These are set out in 'fable 11. (See Chapter XIII also.)

q,(1J)

TABLE

Solution Fem(e, q) Ge1n(~' q)

11. Alternative second solutions of (1) §9.21

Alternative solution.

Defined at

FeYm(e, q) Fekm(~' q) GeYm(e, q) Gekm(~' q)

§§ 8.11, 8.12 § 8.14 § 8.13 § 8.14

I

Alternative

~olutiot~ __ ~ution . _

Fem(e, -q)

Feym(e, Fekm(e, Gem(e, -q) GeYm({' I Gek m ( { ,

I

Defined at_ _

-q) -q) -q) -q)

§ 8.30 " " "

--------------------9.31. Solutions of (1) § 9.21 of real fractional order. When (a, q) lies on an iso-fJ curve in Fig. 11 the order of the solution is == m+f3, 0 < f3 < l , and the solutions of (3), (4) § 9.2] coexist for a == a m 1 fJ- Then the solutions of (1) § 9.21 are as shown in Table 12. II

Definitions of the various functions are given in §§ 4.71, 4.76. 12. Solutions of (1) § 9.21 when a == am+fJ for Mathieu functions of real fractional order v = m+:S, and q is positive

'fABLE

,=

«P1I(f)ef>v(TJ) for a = a v

-

Ce v(

Se v( Ce v(

Propertq of solutions

------- - - - - - - - - - - -

e, q)cc ( TJ, q) e, q)ce v TJ, q) e, q)se TJ, q) ll

l

(

.(

SCv ( f, q)8ev<11, q)

If

fJ

is a rational fraction pis in its lowest terms, 0 < p]« < I, all solutions in TJ have period 2S7T, and in have period 2S7Ti. If fJ is irrational, all solutions are non-periodic in TJ'

e

e.

When q is negative Table 12 is applicable provided appropriate definitions of the functions are used, e.g. in § 4.71. We omit consideration of solutions when (a, q) lies in an unstable region of Figs, 8, 11.

9.40. Orthogonality theorem [135]. Let qn,m be such that for n a positive integer and ~ = ~o, l/Jn(~o, qn,m) = 0, t where -Pn is a solution of (4) §9.21. Then if a, q are real, the point (an,m,qn,nl) lies on t qn.m is a parametric zero of cPn: see §§ 12.40, 12.41.

[Chap. I

,V AVE EQUATION IN ELLIIlTICAL

176

the characteristic curve for the function 4> n ( 7], q), the latter being an integral-order solution of (3) § 9.21. For instance, the curve a 2 in Fig. 8 corresponds to ce2(7] , q) and..also to Ce2(' , q). It follows that we may write (1)

(2)

and Put 'n,m = -Pn c/>n' then for (a ll ,m' qn,m) we have

a2oft ' n m+a1]ia2' nm+ 2qn,m (cos}1 2i:.~-cos 21] )r':>n,m = 0 ·

(3)

Also for (ap,r' qp,r) 2

2

a ' p,r a ' p,r 2 ( h 21: 2)r -a~ + 01]2 + qp" cos ~-cos 1] ':>p"

- 0 ·

(4)

Multiplying (3) by 'p,r' (4) by 'n,m' and subtracting the second from the first we obtain

a [r a'n,m r a'p,r] + 01]a [r':>p" a'n,m r O'p,r] ·o-.;} - ':>n,m --a~- +

of ':>p,'-ije - ':>n,m - fJg

+2(qn,m-qp,r)(cosh 2,-cos 21J)'n,m'p,r =

Integrating (5) with respect to ~ from 0 to 21T, we get

'0' and to

7]

o.

(5)

from 0 to

[r':>P,' eea'n,m ':>n,m r -ar a'p,r]~· d 1] + If [r':>P" a'- 61]n,m - ':>n,m r a'p,r] 211 dl: 01] ~ + 0

1 111

o

0

0

0

fo

+2(qn,m-qp,,)

II

211

(cosh 2f-cos 21]){n,m {p" dfd1] = O.

(6)

o 0

If {n,m = Gan,m cen,m or Sen,m sen,m' {p,r = Cep,r cep,r or Sep,rsep,r' the first integrand in (6) vanishes at , = 0, while the second vanishes by virtue of its periodicity in 7]. Hence it follows that the last integral is zero if p =1= n, i.e.

'0'

t.l11

fo f (cosh 2f-cos 21])'n,m 'p"dfd1] = O. 0

(7)

COORDIN.\TES: ORT1{()QONALITY THEOREM

9.40]

This holds also if }) == n, r =1= m. If p = nand r integral does not vanish. Then we have f.

21f

f f ce:.m(~)ce:.m(7J)}

o

0

Sen m(,)se n m(7]) '

=

177

m, the double

(cosh 2g-cos 27]) d~d1J ¥= O.

(8)

,

It may be remarked that if ePn = cen,m or sen,m' these functions do not satisfy the usual orthogonal relations (§2.19) since cen,m, cep ,,. have different q values, as also have sen,",' se p ,,., n =1= p, m i= r. The 11 integrals in (S) may be evaluated by aid of the following formulae: 1 f211

-

n

ce22n ('\'l)cos 2-n., dn-, --,

00

A<2n>A(2n)+ ~ A(2n)A(2n) 0 2 ~ 2r 2r+2 -

r=O

o

0 2n'

(9)

Those parts of the g integrals involving cosh 2~ may be expressed in terms of derivatives by aid of (7) § 14.21.

4961

Aa

X INTEGRAL EQUATIONS AND RELATIONS

10.10. Definition. The equation b

y(z)

= -'f X(u, z)y(u) du,

(1)

a

in which X(u, z) is a known function of the variables u, Z, and" a particular constant, is called a homogeneous linear integral equation for the unknown function y(u). X(u, z) is termed the nucleus or kernel, and it is symmetrical if X(u, z) == X(z, u), e.g. e- UZ = e- ZU • The equation has a continuous solution only for a discrete set of values of A. These are called the characteristic values of the nucleus. The nuclei with which we shall deal herein are continuous and symmetrical in u, z, except in § 10.30 et seq. THEOREM.

Let

1°. ep(u), ep'(u), ep"(u) be continuous with period 1T or 21T, such that = 0;

eP"+(a-2k 2cos2u)ep

2°. X(u, z), X~,Z' X~,Z be continuous in u, z, such that

.J..' ]11,=211 0, (a ) [.J..' 'rXU-'rU X 11,=0 = (b) 82X_ 82X-2k2(cos2u-cos2z)x =

ou2

OZ2

o,t

211

then

y(z)

= -'

f x(u, z)c/>(u) du

(1)

o

satisfies Mathieu's equation.

Proof. By (1) 211

211

::~+(a-2k2coS2Z)Y = -' f [~:~-(2k2COS2Z)x].pdU +-'a f Xc/>du o

0

2w

= -' f ::~c/> du +-' f (a-2k2cos2u)Xc/> du o

t

This is the wave equation

(3)

0

::~ + ::~ + 2kl ( cosh 2u -

the modified elliptical coordinates x for u. .

(2)

211

=

h COH

U

cos 2%)X

=

0 expressed in

cos z, y = ill sin u sin z, i.e. iu is written

10.10]

INTEGRAL EQUATIONS AND RELATIONS

179

by 2° (b). Now

f

2"

f

'2"

t/J ::~ du

=

t/J d(X~) =

[t/JX~]:" -

f

2"

t/J' dX

(4)

0 0 0

= [t/Jx~-t/J~x]:"+

217'

Io t/J"x duo

(5)

By hypothesis [ ] vanishes, so 217'

f o

f

2"

02 ePJ:. du = Ou 2

-~2 X du. Ou 02J.

(6)

0

Substituting from (6) into the r.h.s, of (3) leads to

f [~~+(a-2k2C082U)t/J]xdU 211

~~+(a-2k2C082Z)Y

=,\

o = 0,

by hypothesis.

(7)

Hence y(z) satisfies Mathieu's equation. If X(u, z) is periodic in u, z, with period 7T, 27T, y{'u) has the same period as eP(u), so eP(u) = AoY(U). Accordingly we have the homogeneous linear integral equation of the first kind for the periodic functions cem(z, q), sem(z, q), namely, '-

217'

y(z)

= .\0

Ix(u, z)y(u) duo

(8)

o

If [ePX~-eP~ X): == 0, the uPIler limit in (8) lnay be 7T. Tho theorem is valid for C~cm(z, q), Scm(z, q) if for z we write iz in (1),80 that y satisfies y"-(a-2k 2cosh 2z)y == o.

10.11. The eight primary nuclei for cC,n(z, q), sem(z, q). We commence with the wave equation 82X+ 82x+k2

ox2

oy2

1

X

=

0

,

(1)

where k1 h = 2k. By substitution we confirm that simple solutions are: X = ei k1X , ei k l 1l, yei k 1x , and xei k 1'l1. The real and imaginary parts of these functions are separate solutions, so we obtain the eight given in Table 13.

INTEGRAL EQUATIONS AND

180

TABLE

REI~ATIONS

[Chap. X

13

Solutions of (1) eosk1x sink1x sink1y ysink1x

cos e, Y xcosk1y yeos k1x xsink1y

In modified elliptical coordinates, x = h cos u cos Z, y and by §9.20, (1) may be transformed to 02

===

ih sin u sin z,

02X 8z2 - 2k2(cos 2u-cos 2z)X = 0,

ouX2 -

(2)

which is (b) 20 § 10.10. Substituting for x, y from above in Table 13 and omitting the multiplier i, we obtain eight primary solutions of (2). These are the eight primary nuclei X(u, z), and they may be used in (8) § 10.10 with the upper limit 'IT. They are set out in Table 14. TABLE

14. The eight primary nuclei for cem(z, q), sem(z, q), q

x(u, z)

=

nucleu« of (8) § 10.10, and a solution of (2)

---_...

>

0

y(z)

~

1/

2

cos( 2k cos u cos z) sin( 2k cos u cos z)

"i"~al

3

sinh( 2k sin u sin z)

#l2a+l

3'

11-2" t

4'

1

4 18inU8inzx x sin(2k cos u cos z)

I

itl

2

2'

I

cosh(2k sin u sin z) "~II cos U cos z X ,,~" i-l X cosh(2k sin u sin z) sin u sin e X IA-~,,+ 1 X cos(2k cos u cos z) , casu cos z X #1-2" +2 x sinh(2k Hin u sin z) I

ce2"(Z, q) ce2" t l(Z, q) Be2"+l(Z, q)

se2" f- 2(Z, q)

Each nucleus in Table 14 is symmetrical and periodic in u, z. (1), (4) have period 1T, while (2), (3) have period 21T. The functions

y(z) are allocated to their respective nuclei by considering periodicity, evenness, and oddness. Since cem(z, q), sem(z, q) satisfy 8, homo-

geneous linear integral equation with a symmetrical nucleus, it follows that they are orthogonal [215] (see also § 2.19). We have now to derive the characteristic values ~m' ILm corresponding to various nuclei. This may be effected by aid of Bessel series for the Mathieu functions, as shown hereafter.

10.12. Bessel function expansions. Inserting nucleus 1 Table 14, § 10.11 and ce2n(u,q) in (8) § 10.10, with limits 0, 1T, we get

I cos(2kcosucosz)cez,,(u) 'IF'

ce21l(z,q) = "Ill

o

du

(alII)·

(1)

10.12]

lSI

INTEGRAL EQUATIONR AND ItELATIONS

Substituting the expansions of the circular [202, p. ~3] and Mathieu functions, (1) becomes

f [J (2k cos z)+ 2Jl(- -I tJ2r(2k cos z)cos 2rlt] o w,

= '\211

ce 2tt(z, q)

00

o

X

00

X

I

A~~'l) cos 2su

du,

(2)

8:::.0

Consider any r in the first~. By virtue of orthogonality of the circular functions, all integrals vanish save when r == 8. Thus we get (-1 Y2'\2tt J2r(2k cos z)A~~n)

w

f cos 2rlt du 2

o

= (--I Y"2n 1TA~;.n).J2r(2k cose).

Hence the expansion of the Mathieu function in

B.~".

(3)

is

00

ce2n(z, q)

=

"2n 11 ~

r;; 0

(-1 t A~~1L)J2T(2k cos z).

(4)

By 2° § 8.50, the r.h.s, of (4) is absolutely and uniformly convergent in any closed rectangle of the z-plane, k 2 real> 0. Since the r.h.s, represents ce2n(z, q), it and also (I) is a solution of y"+(a-2qcos2z)y ~ 0, q > 0, a = a 2n • This remark applies as well to the integral equations and their expansions for ce2n +1(z,q), sem(z, q), with q > 0, a = a 2n +1J bm , respectively, given in later sections. Determination of "2n' In (4) put z == j1T, then cosz := 0, and all the J vanish except Jo(O) == 1. Therefore "211.

== ce2n ( ~7T, q)!1TA&2n) ==

00

~ (-I)rA~~n)/1TA~2n), r=O

(5)

so "2n is a function of q. Using the tabular values of the A, we have for n = 2, q == 8, 8

ce4 (! 1T, 8) ~ ~ (-I)T A~~) = 0·24703 39-0-59450 H8+ r=O

+0·63941 57+0·33453 OO+O'{){)(JH9 34+0'00607 17+ +0-00039 05+0-00001 77+0,00000 06 == 0·{)9384 47.t

(6)

0-69384 47/1TXO'24703 39 = 0-89275...

(7)

Hence by (5), (6) we get

"4 = when q == 8.

f 'rhis may

be obtained from (95 J to five decimal places.

INTEGRAL EQUATIONS AND RELATIONS

182

[Chap. X

10.13. Exponential nucleus. We shall now demonstrate that

f o

."

~lln

celln(z,q) =

e2ikCOBUCOUOOlln(U)

,"

du

(alln ) ·

(1)

To do so, we merely need to prove that its imaginary part vanishes, since by (1) § 10.12 its real part represents ce2n(z,q). Then .

Imag. part = ~lln

00

'II'

f2! o

(_l)mJ2m+1(2kcosz)cos(2m+l)u oolln(U) du,

m==O

(2)

and this vanishes by virtue of the orthogonality of the circular functions. In the same way it may be shown that integral equations in the sections below can have exponential nuclei. Either the real or the imaginary part of the integral vanishes, as the case may be.

10.14. Bessel series for ce2n +1(z, q). Using this function and nucleus 2, 'fable 14, § 10.11 in (8) § 10.10, a repetition of the analysis ill § 10.12 yields 00

ce211 +1(z,q) = "2n+1'TT

! (-l)rA~;'t11)J2r+1(2kcosz) r=O

(1)

(a2n +1 ).

By 2° § 8.50 this series is absolutely and uniformly convergent in any closed rectangle of the z-plane, k 2 real > 0, as also is its first derivative. Consequently term-by-term differentiation is permissible, so ce~n+l(z,q) = -"2n+121Tksinzx 00

X

!

r==O

(-I)r A~~~11)[d J2r+1(2k cos z)jd(2k cos z)].

Now J~,(u.) =-= !['~Il -l(u)-Jm~l(lt)l, and since Jp(O) = 0 if p when z == ~7T, the only non-zero term is iJo(O) == -1. 'rhus

ce;n+l( !'TT, q) = -1TkA~2n+l)"2n+l'

(2)

=P

0, (3)

so \ tl -"2n

-co.' (J 1T' Q)j1T1\" A 1(211 -f l) -~ 2n+l 2

00

~

~

r=O

(-I)"(')r+l)A(211.+1)j1TkA<2n+l) .. 17"+1 1 • I

(4)

10.15. Additional expansions. Using nuclei 1', 3 from Table 14 § 10.11 and proceeding as in §§ 10.12, 10.14, we find that 00

ce2n(z, q) = "~n 'TT ! (-1)'A~;n)12r(2k sin z) r-O

(aan)'

(1)

10.15]

183

INTEGR,AL EQUATIONS AND RELATIONS

_ _1_ ~ A (2n) _

'"

with

A (2n>L.,

2n-

1T

o

2r

-

ce2~O, q),

r==O

1TA(2n) 0

(2)

,

(4)

with

10.16. Determination of JL21L+2' ";"+1' P.~1l+1' P.~II t-2· If we select f'2n+2'

the integral equation, using nucleus 4, Table 14, is

= p'2n+ll f sinusinzsin(2kcosucosz)x .".

se 2n + ll(Z, q )

o

X se2n +2(U, q) du .".

=

(b2n +2)

(1)

00

f sinusinz ~ (-I)rJ2r+1(2kcosz)cos(2r+l)ux

2p,2n+2

o

rzzO

00

X ~ B~~'+-t2) sin(2s+ 2)'u du

(2)

8:::110

f sinzr~o (-IYJ .".

=

p'2n+ll

00

2r+1(2kcosz)cos(2r+l)ux

o 00

X ~ B~~~~2)[cos(28+1)-u-cos(2s+3)u]du

(3)

8=0

=

JL2n+2 ! 7T sin z X CX)

X ~ (-1)rB~~~~t2)[J2r+l(2kcosz}+J2r+3(2kcosz)]. ,.... 0

(4)

2r+2)J2T +2(U) = ![J2r+l(U)+J2r+3(U)], so (4) takes the form Now (-use 2n +s(z, q)

= P,2n+21T~nz

L (-IY(2r+2)B~~~~2)J2r+2(2kcosz). 00

(5)

r=O

This series and its first derivative is absolutely and uniformly convergent for all finite values of z, k 2 real > o. Thus term-by-term differentiation with respect to z is valid and we get se;n+2(Z, q)

=

p'2n+2 ; [

i (-IY(2r+2)B~~~~2){sec2z

.f:ar+2(2k cos z)-

r==O

-k tan z sin z[J2r+1(2k cos z)-J2r+ 8(2k cos z)]}].

(6)

IXT~~GRAL

184

EQUATIONS AND

As z ~ 11T all terms vanish except -ktanzsinzJ1(2kcosz) ~ -k2 • Hence

1J-2n-+2

sec2zJ2(~kcosz}~

[Chap. X

!k2

'TT (2n +2) -..fL21l+22 kB 2 ,

, (1 ) se21t +2 }1T, q SO

R}~LATIOXS

(7)

-2se;"+2(!'TT,q)/1TkB~2n+2)

=

'J

00

... , ( I )r(2 - 1TkB(2n-t 2) L., r 2

and

(~)

+ ?)B(2n+2) 2r+2·

(9)

aJ

r=O

For the remaining nuclei in 'fable 14 § 10.11, we obtain , 7T cotz ce 21l+1(z, q) = '\2n+l----VC- X ex>

X ~ ~ (-1)r(~r+I)A(2n+l)l 2r+l 2r+1 (2ksinz) '

(10)

r=O

00

I

X

r=O

'u1

\\'1

'

fL2'1+1

=

(-I)r(2r-t-l)B~;'+1l)J2r-+l(2kcosz)

(b21l ..1)'

2 se2t1+1(11T, q) 2 ~(I)rB(2n+l) - B(2n+l) :::::.:: -B<2n+i1 L.., 2r+l· 7T

1

1

11'

(12) (13)

r-O

, 1T cotz se21t+2(z, q) = !J-2U+22k X 00

X

,h

WIt

'

:2 (-1)r(2r+2)~~,+~2)I2r+2(2ksinz) r=O

- 2se;tI+2(O,q) _

JL2n+2 -

-;kBC2-n+2) -. 2

(b2n t2)' (14)

2 ~ (2 +2)B(2n+2) ;kB(2n+2) L.., r 2r·f 2 • 2

(15)

r=O

By 20 § 8.50 all series in § 10.15 and the present section are absolutely and uniformly convergent in any closed rectangle of the a-plane, k 2 = q real> o. The characteristic values in Table 15 are expressed in terms of the coefficients in the series for the corresponding Mathieu functions. By § 3.21 the sigma terms are absolutely convergent in 0 ~ q ~ qo. By §3.25 the A, B are continuous functions of q, and by § 3.30 A~2n), A~~U+l), B~2,,+l), IJ<22n +2) have no zeros in q > 0, but tend to zero monotonically as q ~ +00. ,\O~,\~ ~ I/1T and ~~, JL~ ~ 2/1T as q ~ 0, and are continuo~s in 0 ~ q ~ qo. For n > 0, "211' "~n' "~n+l' JL~n+l'

INTEGRAL EQUATIONS AND RELATIONS

10.16]

15. Oharacterietic values for the nuclei in Table 14

TABLE

~ = ce 2 ,.(I" , q)

~ = ce 2n (O, q)

"A~2")

II

I

"2ntl

'=0

ex>

(-l)t'A~;"

IA~~")

,=0

---;A~2n)-

=

ce~,. + 1 (I-w, q)

~

= - -7TkA\2,.-t - - 1)-

"A~2t&)

_ ~ce2"-!_1(O,_ q) 1TA~2,,+1)

n •1 -

00

IX)

~ (-1)'(2r-t-I)A~;~1l)

2

,.:c0

=

1l-2" +1 =

- - - 1r'kA~2n+l) - ----

,

2 Be2,. + 1( i1T , q) 1TB~2,.+1)

fL2"+1 =

1)-

00

~ (2r+ l)B~;~~l)

#J.2,.i ~ =

.2 ~ (-l)'B~;~~l)

,=0

reO

--frkli~2n-+l)--

-

1TB~2"+ 1)

I

2 se~n +~(i"', q) --';;kB~2" +

2'-

'

-

1'-2,. I 2

I

2

-

se~,,~l?~ ~~ 1TkB~2n + 2)

00

2

A~~~\l)

1TA?ii""+r>

=

se~n +1(0, q)

-;;kiJti,,-j

I

,c:0

ex>

=

ce 2n(z, q)

"A~~II)

"

ex>

=

185

I

00

,=0

(-lY(2r+2)B~~~~2)

2

I

r:::O

(2r-t-2)B~+~2)

- --;'kiJ~i£t+2) -- -

and for n ~ U, A2n + 1 , fL2n+l' fL2,,,+2' fL~1t+2 are continuous, 0 < q ~ qo. The discontinuity in each of the first set at q = 0 is due to vanishing of the coefficient in the denominator. That in each of the second set arises from the factor k-1 = s:', and for n > 0 to vanishing of the coefficient in the denominator. The form of the latter may be derived from the general result (9) § 3.33. Thus as q ~ 0, we obtain

Ab2n ) ~ qllj2 2n - 1(2n )! ;

B~2n+2) ~ (n+l)qnj2 2n(2n+l)!;

Ai2 1t+1) ~ Bi2 1t+1) ~ qnj2

2 l1(2n )!

(17)

The reciprocal of each coefficient lias an infinity of order n at q TABLE

16. Illustrating behaviour of characteristic values as

q-+O Characteri8tic t1Q.ltu

~"

~,.tl

11-2"+1 ~,,+2

I Behaviour as q

~

~oo,n

0

Oharacteristic value

> 0

~"

~"Tl ~"+-1 ~"~2

-+ 00 ~oo

-+ 00

Db

(16)

Behaviour

Q8

q~O

-+ 00, n -+ 00, n -+ 00, n ~oo

> > >

0 ' 0 0

==

O.

186

INTEGRAL EQUATIONS AND RELATIONS

[Chap. X

10.17. Theorem on integral relations. If y(u) is a Mathieu function of period", or 2",; X(u, z) the corresponding nucleus for the integral equation; ,~(u,z) = X(u,z), so that ,(u,z) =

u

f X(u,z) du and

[y(u),(u,z)]::; = 0, then

y(z)

= -,\

j[ j

X(u,z) du ]Y'(U) duo o Proof. Substituting for X into (8) § 10.10, we get

f ~~(u, z)y(-u) du f y(u) d[~(u, z))u o '" = [y(u)~(u, z))::;- f ~(u, z)y'(u) du, o y(z) = -,\ j[j x(u, z) du ] y'(u) du, ~

y(z)/'\

(I)

~

=

=

(2)

0

80

(3) (4)

o

since the first member on the r.h.s. of (3) vanishes, by hypothesis.

10.18. Example. To illustrate the application of (4) § 10.17, we take nucleus 4 in Table 14 § 10.11, namely, X('u,z) = sinusinzsin(2kcosucosz) with y(u) = se2n +2(u, q). Then

f u

~(u,z) =

f u

X(u,z) du = -

t~~

sin(2kcosucosz) d(2kcosucosz) (1)

=

tanz -k-cos(2kcosucosz).

(2)

2

z)]::; == 0, so

But [y(u),(u,

cot z se2n+2(z, q) 1r

= - ~~ ;

=

f cos( 2k cos u cos z)se~n

+2(u.)

du

(3)

o

-i:f [Jo(2kCOSz)+~,.~o(-I)'+lJ2r-t2(2~'COSZ)COS(2r+2)u]x o

00

X

!

(28+2)B~~~~2)cos(28+2)u du

(4)

8-0

= ;~~(-I)r(2r+2)BW~_~2)J2r-t2(2kCOSZ). r-o

(5)

187

INTEGRAL EQUATIONS AND RELATIONS

10.18]

Comparison with (5) § 10.16 shows that A =

f'2n+2'

so from (3) and

(1) § 10.16 we get

f sin u cos 1T

2k

Z sin(2k cos u

cos z}sezn+z(u, q} du

o

f cos(2kcosucosz}se~n+Z(u,q) duo 1T

=-

(6)

o

The integral relations given below may be derived by analysis akin to that above. 1T

f sinh(2ksinusinz)ce~n+1(u,q) du,

= - A~t

tanzce:on+l(z,q)

(7)

o

f sin(2kcosucosz)se~n+1(u,q) _iLtn1~ f cosh(2ksinusinz)se~ntz(u,q) 'IT

~~t'

cotzse zn+1(z,q) =

du,

(8)

o

'IT

tanzsezn+z(z,q) =

duo

(9)

o

10.19. Formulae for cem(z, -q), sem(z, -q). These functions are solutions of the equation y" +(a+2qcos 2z)y ~~ n. Thus the required formulae may be derived from those in preceding sections by aid of § 2.18. The values of q and k remain as before, i.e, q > O. 10.20. Integral relations for Cem(z, q), Sem(z, q). These are derived by applying the relationships in (2)-(5) § 2.30 to the integral e9uations for cem(z,q), sem(z,q) in § 10.12 et seq. Thus Ce2n(z, q)

f

1T

== ce2n(-~1T, - -;'A~2n) q)

f

cosi(?k z cos U cos I1 Z) ce2n (U, q) dU

(I)

o

'IT

~

2n (O, q) ce --;'A&2-n) -

inh z) ( ) d U,. cos (')1.' ~h:, SIll U SIll 1 Z ce 21l U, q

(2)

o

Ce2n +1(z, q) "IT

--

-

Ce~n+l(t1T, -:;kA1 + l)-q)' j~ in

f

· (2k cos

SIll

U

) dU cos I1 Z) ce 2n +1 ( U, q

(3)

o

1T

=

2--;A~2n+l) ce 2n +1(0, q)

o

I (2k· COSUCOSIZCOS csm

· h z)ce 2n +1( ) d u; (4) u sm u,q

188

INTEGRAL EQUATIC)NS AND RELATIONS

[Chap. X

Sezn+1(Z, q) 'If

=

-- f·

Zn +1 (! 1T, Q) 2se 1TB~Il,,+iY

·h

S1DUS1D

) ZcO~(2k cosucosh z )selln+l ( u,q

du () 5

o (6)

1T

f'

2se~n+2(i7T, q) = - -------- --7TkB(2n+2) 2

.

SIn U SIn h Z

0

. k cos u cosh z)se 2( ) dU sIn(2 2 n+ u , q (7)

'If

2 se~n+2(0, q) f cos ~ cosh zZ SIn si (2k' inh1 z) ( ) d u, = -:;;icB~2n+-"),. SIn U SIll Z se 2n +2 U, q

(8)

o The evaluations of (1)-(8) are given at (15), (17) § 8.10; (1), (2) § 8.12; (6), (7), (10), (II) § 8.13, respectively. From § 10.18 and Table 15 § 10.16 additional relations in which ce~n+l' se~n+l' se~n+2 appear under the integral sign may be written down. All the above representations of the modified Mathieu functions are first solutions of (1) § 2.30, provided a has its proper value.

10.21. Integral relations for Cem(z, -q), Sem(z, -q). These may be derived from (1)-(8) § 10.20 by applying the definitions (1), (11), (21), (31), § 8.30. Thus Ce2n(z, -q)

f f

'If

= c_!l:~tll~\l1T~V

cosh(2k cos u sinhz}celln(u, q} du

o

·

'If

--

( - l )nce (0 q) . A (2n) ~_n_'_ 1T 0

cosh(2kslnucoshz)ce 2n (u , q) du·,

(I) (2)

o

f· 'If

=

(1)n 2 se 2n +1( !1T,_.q)- Slnucosh zcosh (2k COSUSln · h z)se 2n +1( ) dU - ------u,q 1TB(2n+l) 1

0

(3)

."

. h(2k sIn · u cos h z )se 2n +I (q)du· = (-l)nSe~n+l(O,q)f --- --- S1l1 u, , 1TkB~"n+l) o

(4)

10.21]

INTEGRAL EQUATIONS AND RELATIONS

189

Se2n +1(z, -q)

=

(_l)n+lce~n+l(!1T,q) 7TkA~2n+t}

f sin· h(2k cos ."

· I)

U SIll 1 Z

() d

ce 2n +1

U,

q

(5)

U

o 1r

(-1)1l2Ce211.-t(o,Q)f COSUSln · h zeosh(2k· . ism te cos h z)ce2n +1(u ,q) du ;

= -----;;'A<2n+l-)- 1

(6)

0

Se 2 n +z(z, -q) ."

· h ssi · h z)se + d = (-1)n+12se~n+2(!1T,q)f --~-iJ(2n+2)--- smucos asm h(2k COSUSln 2tz 2(u,q) u, 2

(7)

0

2 se~n+2(O, = (-1)'" -- -;;'kB(2-n+2) - q) --

f

·h

1r

cos U BIn

· h(2k sm " u cosh Z )SC + ( ) d u, 2n 2 U, q

Z SIn

0

2

(8)

The evaluations of (1)-(8) are given at (2), (:l), (12), (13), (22), (23), (32), (33) § 8.30. All the representations of the modified Mathieu functions are first solutions of (1) § 2.31, provided a has its proper value.

10.30. Bessel-circular function nuclei. Hitherto our nuclei have been limited to circular and hyperbolic functions. But the wave equation admits of a plurality of solutions, and other nuclei ulay, therefore, be derived. Any solution expressed in 1nod~fied elliptical coordinates, which satisfies conditions (a), (b), 2° 10.10, may be used as a nucleus for the integral (1) § IO.IO.t We commence by transforming 02X 02X -(I) ox2+ -oy2+ ki X = 0,

*

to cylindrical polar coordinates, where established procedure we obtain

.'l:

===

r cos (~, y =

02 X+! oX+~ 02 X+ = o. or2 r or r 2 O(l.2 l.21 X

1"sin

n.

By (2)

Writing X = [J(r)J'(~) and using analysis akin to that in § U.21 yields the two ordinary equations

d2~+~ dU +(k~-~)[T = '0, 2 2 dr

and

r dr

r

d2V - 2 -l- v 2V dcx I

=

0,

t Provided the integral converges.

(3) (4)

190

INTEGRAL EQUATIONS AND RELATIONS

[Chap. X

v2 being the separation constant. The formal solutions of (3), (4) are,

respectively, Jv(k1 r), J:(k 1 r); and cos Vet, sin Vet. Thus if Jv(k1 r)cos Vet, Jv(k1 r)sin vcx, J:,(k1 r)cos vc.:, J:(k1 r)sjn vc.: are expressed in modified elliptical coordinates, we get the sets of B.F. nuclei in Table 17. Each nucleus satisfies conditions (a), (b), 2° § 10.10 and the integral (1) § 10.10 converges if due regard is paid to the phase and modulus of z. 'fABLE 17.

Bessel-circular function nuclei: k2 real > 0

-jNUCltU8x(r.ex) = U(r)V(ex)! 1_",_ (m = 0, 1, 2,... ) ~_.

I ~ _

I Rema,!_8_ _

lI(iz)

+

r = h[ 1(008 2u c082z)Jl 1 J 2m (k l r ) C?" (2ma) co (u,q) Sc e (z,q) ~ee2,,(Z,q) for y(z); X in 1,2, are e 2" SU1 802" v then periodic and symc II metrical in tt, z: period 2 I J 2m +1 (k1 r ) C?s {2m + l )a: co (tt,q) co (z,q):Se (z,q) 'T'f, 2'T'f, respectively. sm se211-+ 1 118e2 " t 1 j °2" I 1 In (3), (4) replace C08 2z }4' by cosh 2z, R(z) > o. X 3 Y 2m(k1 r) C?8 (2ma:) co iu, q) GOY (z.q) in 3,4, are non-periodic, 1 sm s{'2,. I eY2" as also are Fey, Gey cos co F (see§S.II). The nuclei 4 Y2"11 1(k1r) . (2m+ 1)0: (u,q) ey (z,q) for y(iz) are skewBin S02,. t" 1 eY2n -1 1 I symmetrical in u, z.

I I

10

I

II

Formula for r. r == (X 2+ y 2)l with x = hcosucosz, y == ihsinusinz, the modified elliptical coordinates. Thus r

==

h[l(cos 2u+cos 2z)]l

(5)

k1r

=

k[2(cos2u+cos2z)]l.

(6)

and, since 1·1 h = 2k,

Yp has a singularity at the origin, and since k1 r has zeros when cos 2u == -cos 2z, (6) cannot be used as the argument of the Y function if z is real. \Vriting iz for z, R(z) > 0, there are no zeros (u real), so we obtain the argument for the J, Y functions used in the nuclei for the modified Mathieu functions in column 5, Table 17, i.e, kIf

=

k[2(cos2u+cosh2z)]l.

(7)

Formulae for eoep«, sin p«, From above, and

cos ex == x/r = cosucosz/[!(cos2u+cos2z)]i

(8)

sin « = ylr = isinusinz/[!(cos2u+cos2z)]'.

(9)

Then by de Moivre's theorem

+

(cos pex i sin pa)

=

(cos ex + i sin ex)1',

( 10)

10.30]

INTEGRAL EQUATIONS AND RELATIONS

191

so COs Pc/" sinpc/' are obtained by equating real and imaginary parts in (10), and substituting from (8), (9). The remarks in column 6, Table 17, may now be confirmed. 10.31. Hankel-circular function nuclei: k 2 real> O. By (4)

§ 8.14 Fek 2n(z,q) = ![i Ce2n (z, q)- }"eY2n(z,Q)]

(a2n ) ,

(1)

so by Table 17, § 10.30 we see that the nucleus for Fek 2n(z, q) is

![iJ2m (k 1 r)-J;m(k 1 r)]cos 2ma:

= !iH~~(kl r)cos 2ma:,

in which k 1 r is given by (7) § 10.30, with R(z) set out in Table 18. TABLE

O. Other nuclei are

18. Hankel-circular function nuclei.' k 2 real> 0 Nucleus x(r, ()) (m

= 0, 1, 2, ... )

1l~(klr)C?s(2r.n()) Sin

2

>

(2)

ll~~ -.. 1 (k 1 r) C?S (2m sm

-+ l)/l:

ep(u)

y(z}

in (1) § 10.10

in (1) § 10.10

ce (u, q) 8e2n

Fek

ce

Fek

(u, q)

se2n+l

Ge~,,(Z, q) (z, q)

Gek 211 +1

The Hankel function may be expressed in terms of the K-Bessel function by the relationship (1) § 8.14.

10.32. Nuclei for k 2 real < O. Here we write -ki for kf in (3) § 10.30: (4) § 10.30 is unchanged. l'he formal solutions of (3) § 10.30 are now the modified B.F. Iv(k 1 r) and Kv(k 1 r). Hence we have the nuclei Iv(k 1 r)cos 1Ic/', Iy(k 1 r)sin vcx, Ky(k 1 r)cos II~, and K.,(k1 r)sin v<X. Y.,(ik1r) is also a solution of (3) § 10.30, so Yv(ik1r)cosv<x and Y.,(ik1 r)sin va: may be used as alternative nuclei. The nuclei are set out in Table 19. TABLE

~ 1

3 4

Nucleu8 x(r, ex) (rn = 0, 1,2,... ) C08

I I

4>(u)

ce

I Icese

!I(z)

I . I y_('_z)_ _ Ce

i-

<

0

Remark8

_

Those in Table 17 1 12". {k 1 r ) ° (2m<x) (u,-q) (z,-q) SA.... (z,-q) sin I S~II 2n -"II apply here, tnutatis mutandi8. Ce (z,-q ) 12m + 1(k.r) cos ° (2m+l)() I ce (u,-q) ce A.... (z,-q) SA.. sin se211+1 S-4511+1 -..,,,+1 If Y(ik 1 r) is used cos ce Fek ( ) in (3), (4), Fey, K 2".(k1 r ) sin (2m() se2" (u,-q) Gek 2n z,-q Gcy are obtained for y(iz). • cos eft ; Fek ( ) E 2 ,lt t l (!ol r ) sin (2m + l )a.: se2n+l(U,-Q) Uek2ntlZ,-q I

I

2

19. Bessel-circular function nuclei.' k 2 real

192

INTEGRAL EQUATIONS AND RELATIONS

[Chap. X

10.33. Integral equation for ce2n(z, q) with Bessel nucleus. In I, Table 17 put m = 0, ep(u) = ce2n (u , q), and insert X' t/J in (8) § 10.10 with upper limit 1T. Then

f Jo{k[2(cos 2u+cos 2z)]I}ce 2"(U, .".

cei/~(z, q)

= 'Pi"

q) duo

(I)

o ~2/L'

To determine

=

let z

0, then from (I)

I Jo(2kc08U) I o '"

ex>

A~~n)cos2rudu.

(2)

I [Jg(k) +2 m~1(-I)mJ~{k)cOs2mu]cos2rudu

(3)

=

ce 2n (O, q )

'P2n

r-O

Now by (2) § 10.34, with z

=

0,

Io'" Jo

(2k cos u)cos 2ru du '"

=

ex>

o

=

(-I)r1T J : (k).

(4)

Hence from (2), (4) we obtain q>21t

== ce2n (O, q)/1T

i: (-I t A~;n)J~(k).

(5)

r=O

An alternative expression for Gl2n may be derived as follows: Insert the value of ~Il from (5) § 10.12 into (4) § 10.12 and integrate both sides of the latter with respect to z from 0 to 1T. Then

JL A~;n) '"

o

00

cos 2rz dz

= 1TA~2n)

~e2'~_( t 1T ,-til A (2n)

JL..t~

r-:aO

--

o

'"

0

ex>

(-1)"A 2r (2n)el (2k cos z) dz 2r

== ce2,_~(-l1T, q~ [1T L..t ~ (-I)'A (2n)J2(k)] A (2n) 2r r , o

(6)

r=-O

(7)

,.=0

."

since

JJ r(2k cos z) dz =

o

2

11J:(k). Thus from (7) we get

00

I

r=-O

(-I)rA~~n)J~(k) = [Ab2A >]2j ce2n (! 1T, Q).

(8)

Substituting the r.h.s, of (8) in the denominator of (5) leads to crilL

==

ce~h~(OJ q)cea 1t ( '~1T, q)/1T[ A~2n)]2.

(9)

INTEGJ~AL j.~Ql" ATI()NS

10.34]

AN D RELATI()NS

193

10.34. Expansions of Mathieu functions in Bessel function products. We apply the addition theorem <JJ

JO[(V~2+V~2+ 2v~ V~ cos 2U)I] =

where gives

EO

=

1,

Em

=

I (-I)m EmJm(V~)Jm(V~)COS 2mu, m=O

2 for m ~ 1. Let t·~

= keiz ,

t'~

= ke:",

(1)

and (1)

ex>

Jo{k[2(cos2u+cos2z)]6} == I (-I)mEmJm(ke-iz)Jm(keiz)cos2rnu. (2) 1'n-=O

Substituting from (2) into (1) § 10.33 we obtain ce2n(z, q) ~

= fP2n

00

I

I

r==O m==O

(-I)m A~~n)Em Jm(ke-U)Jm(keiZ)

~

J cos 2ru cos 2mu du, (3)

o

term-by-term integration being permissible, since the series before and after integration are absolutely and uniformly convergent. Now ?T

1T

Jcos2rucos2mudu=o

I 1T

o Hence (3) becomes

(m = r

=

0), }

(m=l=r>O), (m

=

r

>

(4)

0).

OC)

ce2n(z, q)

==

fP2'l1T

I

(-I)r A~~n)~(ke-iz)~(keiZ),

(5)

r:.::O

so this series satisfies y"+(a-2k 2 cos 2z )y = 0, k 2 > o. When z == 0, (5) gives (6) q>2n = ce 2n (O, q)/ 1Tr~o (-I)TA~~n)J~(k), as at (5) § 10.33. If in (5) we write iz for z, this being permissible by § 13.60, then OC)

Ce 2n(z, q) 80

=

fP2n

?T

I

(-I)r A~;n)J;.(ke-Z)Jr(keZ),

(7)

r=O

the r.h.s. satisfies y"-(a-2k 2cosh2z)y = 0, k 2 Substituting (!1T-Z) for z in (5) leads to ex>

ce2n(z, -q) = (-1 )1lq>2n 17'

!

> o.

(-I)r A~~n)Ir(ke-iZ)Ir(keiZ) ,

(8)

r=O

80

the r.h.s. satisfies y"+(a+2k 2cos 2z)y = 0, k 2 > o. Putting iz for z in (8), or (!1Ti+z) for z in (7), yields Ce2n(z, -q)

=

00

(-1 )"q>2n 1T I (-I)r A~~n)lr(ke-Z)Ir(ke·),

(9)

r=O

so the r.h.s. satisfies y"-(a+2k2 cosh 2z)y = 0, k 2 > o. The multiplier (-I)n is used in accordance with the definitions in §§ 2.18, 2.31. 4861

oC

INTEGRAL EQUATIONS AND RELATIONS

194

[Chap. X

10.35. Integral relation for Se21t+1(z, q) with Bessel nucleus. In (2) Table 17, if we put m = 0, iz for z, take ~(u) = sesn+l(u, q), (8) § 10.10 gives, with the aid of (4) § 2.30, ."

where k l , = k[2(cos2u+cosh2z)]. = k(e2z+e- 2z +2cos 2u)•. Now if (£ is a, cylinder function, then [214, (4), p. 365]

~

VVVV

maIO

I

2

with 1U2=vf+vl+2vIV2C082u, Ivil < Iv 21, while C~(cos2u) is the coefficient of ~r in the expansion of (1-2~1 cos 2u+o:¥)-V in ascending powers of (Xl- Thus ~,m

OV (cos2u) = "

f:t.

m

(-1)8

2m - 2s r ( + ) m-2s 2 .~ m-8 CO~__ ~ (m- 2s)! s! rep}

(3)

and in particular sinuO:n(cos2u) = sin(2m+2)uj2cosu = (-I)m

m

~

(-I)Psin(2p+l)u,

p==O

(4)

the third member being obtained by expressing the second in exponentials and expanding. Also from (5), (9) § 10.30 with -iz for z

= hsinhzsinu/r = 2ksinhzsinu/klr. Writing J for (£, VI = ke:", V 2 = ke", II = 1 in (2), and using (4), sin ex

=

(5)

yjr

obtain

~(klr)sin(X = ~sinhz

i

(m+l)wm+1

m-O

i

(5), we

(-1)Psin(2p+l)u, (6)

p::::aO

with Hence from (1), (6) 4 .

Se2n+1(z,q)

= lJl2n+1k smh z

2: wm+1(m+ l ) 00

X

m-O

f ~(-1)Psin(2p+l)u~B~ilP)sin(28+1)udu. o p-o 2" m

X

Q()

8-0

(7)

INTEGRAL EQUATIONS AND RELATIONS

10.35]

f

195

211

Since

sin(2m+ 1)usin(2s+ l)u du

=

0 «m -:1= 8)

m=8~O

7T

)}, the value of

o (m+ 1) times the integral in (7) is found to be

'TTB2m+1 = 7T(m+I)[BI-Ba+Bs-···+(-I)mB.m+l]·

(8)

Hence by (7), (8) Se2n +1(z,q) =

CP2n+l

4'TT sinh z ~ k . : B 2m +1 W m+l·

(9)

m-O

10.36. Alternative form of (9) § 10.35. By applying the formula 2(v+ I) J.,(v) = --J"+1(V)-J"+2(V), V ~

to

=

J.,(Vl)J.,+1(V2)-Jv+l(Vl)Jv(VS)'

(1)

(2)

(3)

Applying (3) to its r.h.s. repeatedly and writing m for v yields Wm =

~sinhz[(m+l)Wm+1+(m+2)Wm+2+(m+3)Wm+8+''']'

(4)

Accordingly we have 00

k"" m=O

(-I)m B<2n+l>Jt: 2m+l m =

~sinhz[Bl(Wl+2W2+3W8+''')-

= ~SinhZ

i

-Ba(2w2+3wa+ ... )+Bs(3wa+4w4+ ... )- ...]

(5)

(m+l)[Bl-Ba+B5-· ..+(-1)mBlIm+l]wm+1

(6)

m-O

(7)

Hence from (9) § 10.35 and (7) above, we obtain

196

INTEGRAL EQUATIONS AND RELATIONS

10.37. Determination of

~211 +1.

When z -)-

[Chap. X

+00, the asymptotic

form of ~e2'l+1(z, q) is, by (8) § 10.30, He 21l +1(z, q) "-I

-q')2n+l

n.B1(2/1Tl'2)1 cos(v2+11T).

(1)

Comparing this with (5) § 11.10, we see that <'P21t+1

(2)

== 8 21l+1/ 1TBJ •

With this value of CP2n+l in (8) § 10.36, (5) § 13.11 is reproduced.

10.38. Integral relations for FeY2n(z, q), GeY2n+l(z, q). From Table 17 we get 217'

FeY2n(Z,q) = Cf}2n

f Yo{k[2(cos 2u+cosh 2z)p}ce211(u, q) duo

(1) o By § 10.34 and (2) § 10.35 we see that the value of the integral may be obtained from the r.h.s. of (7) § 10.:34, if y;'(!·e Z ) be written for .l,(keZ ) . Th us w

FeY2U(z, q) = ({)2n 277 ! (-I)rA~~n)J;.(ke--Z)y;'(keZ).

(2)

r=O

If we let z ~ +00 and equate the asymptotic form of the r.h.s. to (9) § 11.10, we find that

=

CP2n

ce2n(0, q)ce 2n(iTT, q)/2TTA~.

(3)

With this value ofq52n' (2) gives the B.F. product series for FeY2n(z, q) at (1) §13.20. Again, from Table 17 we get 217'

GeY2tl+1(z, q)

f

= Cf}2n+1 Yl{k[2(cos 2u+cosh 2z)]1}sin se2n+1(u,q) du; (X

(4)

o

and from (2) § 10.35 we see that the analysis for the present case is the same as that in §§ 10.35, 10.36, provided Ym(ke') be written for J,n(ke'). Then by (8) § 10.36 00

GeY2n+l(Z, q)

=

fP2n+1 1T ! (-1)' B~~~11)[J;.(Vl)Y;.+l(V2)-J;.+l(Vl)y;'(V2)]. r=O

(5)

If we determine

~2n+l

as indicated above, the expansion (8) § 13.20 is reproduced. Moreover, we have here a method of obtaining the B.F. product series representations of the modified Mathieu functions of the first and second kinds, which is an alternative to that given in Chapter XIII.

10.40]

q

INTEGRAL EQlrATIONS AND RELATIONS

197

10.40. Integral relations for Fek., Ciem, Fey"., (~ekm. Se"a. GeY"I' 0, with infinite upper limit. We commence "pith the formula

>

f e-,:coSh1tcosh 2ru duo 00

K 2r(z)

=

(1)

o

This integral converges uniformly with respect to z if R(z)

~:r,

x > o. Under this condition, if -2ik cosh z be written for z, we get 00

~ A~nlK2r( - 2ik cosh z)

f ~ e2ikcOSbll:cosb A~~n) cosh 2ru du = f e2ikc08bll:cosh Ce2n(u , q) duo 00

=

r=O

00

11

0

(2)

r~O

00

11

(3)

o If z and k are real, the above restriction may be removed, since the real and imaginary parts of (3) are ahsolutely and uniformly convergent by § 10.42. Then by (5) § S.14 and (3) above we have the integral relationship 00

}'ek211, (z, q) = c~~n( A!~,~) (2n) 7T 0

f ~2ik

cosh s eosh U

Ce2 (u q) du • n'

(4)

o

Applying (4) § 8.14 to (4), on equating real and imaginary parts, we obtain 00

Ce2n(z, q) =

q) -2 ce2n('~7T, - 71'A~27i)--

f sin. (2k cosh z cosh)C u e2,.(u, q) d·u

(5)

o

and

00

FeY2n(Z,q)

=-

2C;::At:'-lfl f cos(2k cosh z cosh u)Ce

2n(u, q)

duo (6)

o

Similar analysis leads to the following:

f ~~;:;~~~t~q) f

00

Fek

(z q) =- 2n+l ,

ce~n+l(L7T,q2 k7TA~2n+l)

e2ikcoshzcoshuCe ~

(u q) au

2n+l"

(7)

o

co

Ce2n+1(z,q) =

cos(2k cosh z cosh u)Ce 2n+1(u,q) du, (8)

o

and F eY2n+1 ( z,q)

00

=

11 2 ce~n+ k7l'A11 ( 1+1), q) 2n

f sin· (2k cosh zcosh)u Celln+1(u,q) d

U.

o

(9)

INTEGRAL EQUATIONS AND RELATIONS

198

[Chap. X

To derive integral relations for Gek m, Sem, Geym we commence with the integral relationship [132]

J rLJ

vKv(z)

=

z

"sinh u sinh ve du

e-zcosh

=

t7TJleI1l'(v+lliH~1)(zi),

(10)

o

R(z)

>

o. Then by aid of formulae

Gek2n +1(Z, q) - -

f

in

§ 8.14 we find that

00

2ise 2n +1(Vr, q)

TT B i2n +1)

e2ikC08bzco8husinh Z sinh uSe

(u q) du

2n+l'

,

o

(11)

Se2n +1(z, q) ClO

=

4se 2n +1 (! TT,Q) J -----:;;Jj£'in+l-)-

· (2k"C08h ZCOS h) · h z sm · h U Se 2n +1 (u,q ) du, SIn U sin

o

(12)

GeY2n+l(z, q) 00

=

#

J cos (2k cos h zcosh)· · h U Se2n +1(U,q )dU, 1TBi2n(!+TT,q) U SInh z sm 1)

4se 2n +1

o

(13)

Gek 2n +2(Z, q)

-

J 00

_ 2i se~n+2( t q) k1T B~2n+2) TT ,

e2i k cosh z cosh U sinh z sinh uSe

(u q) du

2n+2'

o Se2n+2(Z, q)

,

(14)

co

· h u Se2n +2(U, q )du, = - 48e~n+2(!1T,q)J k1T B~2n+2) cos (2k cos h Z cosh)· U SIll h Z sin o

(15)

Gey 2n+2(Z' q) 00

· (2k cos h z cos h)· · h u Se2n +2(u, q )du. = -.48e~n+2(17T,q)J -11T B~2n+2Y-- sm U SID h Z SIn o Relations (11)-(16) may also be expressed as follows:

Gek 2n +1(Z, q)

co

- se2n+1 (111, q) tanh z -

k7TB12n+l)

(16)

Je

2i k cosh z cosh 11 Se'

(u q) du

2n+l'

,

(17)

o Se2n+l(Z, q)

J 00

- 2 k7TBi2n+l)

se 2n +1(11T, q)

tan h z

cos(2k cosh Z C08 h)8' U e2n +l (u, q ) du, (18)

o

INTEGRAL EQUATIONS AND RELATIONS

10.40]

GeYan+l (Z, q)

199

f· ee

2 se +1 (! 1r, q) tanh z k7ranBit1l.+1)

=-

sin (2k cosh z cosh)8' u et1l.+1 (U, q) dU, (19)

o Gek 2n+a(z, q)

- se~n+a(trr,q)tanhz - k21rB<2n+a) a

J 00

o

Sean+2(Z, q) =

e2ikcoshzcoshuSe' (u q) du 2n+2' , .

2se~n+2(11T,q)t -p.1TB&2n +2)- anh z

(20)

f· ClO

( ) dU, sm (Ie 2 cosh z cosh)8' U e2n H U, q

(21)

o

GeY2n+a(z,Q) ClO

= - 2se~n+2(l1r,q) k 21TB~2nH) tanh z

Jcos(2k cosh z cosh)8'e

( ) 2n H U, q

U

du. (22)

o

On integrating by parts and using the result that Sem(O)

=

Sem(oo)

=

0,

(17)-(22) reduce to (11)-(16), respectively. (4), (7), (11), (14), (17), (20) converge if k cosh z is complex, pro-

vided that its imaginary part is greater than zero. The arguments of the circular functions in the integrals obtained by separation into real and imaginary parts must be real. The,nuclei in the integral equations (5), (8), (12), (15) are symmetrical in u and z, §§ 10.4210.45 are based on k > O,Z real: z ~ 0 in (4)-(9), z > 0 in (11)-(22).

10.41. Integral relations for Fek,n(z, -q), Gekm(z, -q). These are obtained by applying (7), etc., § 8.40 to (4), (7), (11), (14) § 10.40. For example, if R(z) > 0, 00

Fek

(z -q) - (-I)nce 2n (! 1T, q) 2n" A(2n) 1r 0

f

e-2ksinhscoshuCe

(u q)

an'

au·

(I)

o

The remaining relations may be obtained by the reader. There are, however, alternative relations for these functions, which we shall now give. Applying (I) § 10.40 to (9), (19) §8.30, we obtain

J 00

Fek 2 n (z -q) - ~~2n(O,q) ' A(2n) 1T 0

e-2kcoshzcoshuCe

o

(u -q) du

211.'·

(2)

200

INTgORAL EQtTATI()NS AND RELATIONS

It should be noticed that for Ce 2n , q

>

<

0 in (1) and

J

[Chap. X

0 in (2).

ex>

Fek 2n+l (z, -q) = se~n-f:~~!-~l k B(2n+l) 1T

1

e-2kcosbzcoshuCe

(u -q) du

2n+l'·

"

o

(3)

Similarly from (10) § 10.40 and (29), (39) § 8.30 we find that if z > 0 Gek 2n +1 (Z, -q)

f

00

- ~~e~'~l(~-,_~ 1TA~2n+l)

e-2kcolihzcoshusinhzsillh'uSe2

+l(U n'

-q) du

,

(4)

o

Gek 2n +2(Z, -q)

f

00

= 2se~n+~J!?_,q) kTT B~2n+2)

e-2kcoshzcoBhusinhzsinhuSe

2n +2

(u -q) du ,

•

(5)

o

If (!1Ti+z) be written for z in the integrals in § 10.40 corresponding to Cern' Feym» Se m , GeYl!l' they diverge.

10.42. Convergence of integrals (4)-(9) § 10.40. In that section functions are defined by infinite integrals whose properties we shall now investigate. Constant multipliers will be omitted for brevity. We exemplify using (6) § 10.40. Both members of the integrand are continuous in 0 u ~ U o, however large U o may be, the continuity of the first member holding in any closed interval of

<

f exists, so we consider f with 1#0

z real. Thus

00

Uo

extremely large.

o Uo Then by (2) § 11.10 the asymptotic form of Ce2n(u)

o: x-tsin(x+!TT), where x = 2kcoshu, du ~ de]», With these substitutions the integral for consideration is 00

I

= f cos{xcoshz)sin(x+!1T)it-1 da;

(1)

Xo

Since Icos(xcosh z)sin{x+l1T) I ~ 1 if z is real, we get 00

I ~

f x-I dx = 2xol ,

(2)

Xo

so the original integral converges absolutely. Further, the convergence is independent of z in any closed interval 0 ~ z ~ Z2' so (6) § 10.40 is absolutely and uniformly convergent therein. Consequently it represents a continuous function in the interval. In § 13.60

INTEGRAL EQUATIONS AND RELATIONS

10.42]

201

it is demonstrated that the series for FeY2n(z, q)-the function represented by (6) § 10.4o-is uniformly convergent in ZI ~ Z ~ ZI' and so represents a continuous function therein. The convergence of (4), (7) § 10.40 follows from that of their real and imaginary parts.

10.43. Differentiation under the integral sign. This is valid provided the resulting integral is uniformly convergent. To exemplify, we differentiate (6) § 10.40 with respect to z, and obtain (omitting the external multiplier) eo

sinh z

J2kcoshusin(2kcoshz coshu)Ce

2n(u, q)

duo

(1)

o

Thus we consider the integral

J = lsinhz J{-cos[x(coshz+l)+!1T]+ CD

I

= sinhz sin(xcoshz)sin(x+17T)x-1 dx

(2)

x.

1

1-+

+00

Xo

+cos[x(cosh Z-I)-11T ]}x- l dx. (3)

Now -cos[ ] oscillates between finite limits for any z in ZI ~ Z ~ ZI as l ~ +00, while x- 1 ~ 0 monotonically. Hence by a known theorem'] the integral converges uniformly with respect to z in the interval. It vanishes at z = 0, owing to the factor sinh z. In the second integral of (3) write x(coshz-l) = y,

dx

and we get

= dyj(coshz-l),

J 'II

-

12 -

lsinhz (coshz-l)l

coS(y-l7T)Y

-1

dy,

(4)

'Uo

the upper limit being y

=

x( cosh z-I), x ~

Yo

=

+00, and the lower limit

xo(coshz-l).

From what precedes it is clear that, excluding the region of z = 0, (4) is uniformly convergent in any closed interval ZI ~ Z ~ %2. When z ~ 0, the external factor' tends to I,/Y2, the upper limit y ~ !xz2 , and Yo ~ ixoz2 • Thus as z ~ 0, Yo ~ 0, but y ~ 0 or 00 according to the way z ~ 0, i.e. it is not unique. Hence (1) is uniformly convergent in 0 < ZI ~ Z ~ ZI. Accordingly FeY~n(z, q), as defined t Bee Bromwich, Theory o/Infinite Senu. 4961 ' Dd

202

INTEGRAL EQUATIONS AND RELATIONS

[Chap. X

by the integral, is continuous in z > o. A second differentiation under the integral sign is not permissible, since the resulting integral is oscillatory and diverges owing to presence of the factor xl in the integrand. The preceding conclusions apply equally to (5), (8), (9) § 10.40.

10.44. Convergence of integrals (11)-(16) § 10.40. The inves-tigation follows on the lines of that in the preceding section. In all cases we get an integral akin to (3) § 10.43, so the integrals are uniformly convergent in %1 ~ Z ~ %2. Thus they represent continuous functions. These integrals cannot be differentiated under the sign, since the resulting integrals diverge. As in § 10.43, ZI real> o. 10.45. Convergence of integrals in § 10.41. The analytical procedure is similar to that in §§ 10.42, 10.43. Thus for (2) § 10.41 we consider

f

00

f

00

e-XCOSh8(x-lex)

dx]«

Xo

=

e-x(coshll-l)x- t

dx .

(1)

Xo

with x = 2k cosh u. It is obvious that (1) is absolutely and uniformly convergent in 0 ~ Z ~ %2' so Fek 2n(z, -q), as defined by (2) § 10.41, is a continuous function of z. Differentiating (2) § 10.41 under the integral sign with respect to z leads to a consideration of 00

f e-x(coshz-l)x- 1 dx.

(2)

Xo

This is seen to be absolutely and uniformly convergent in Z1 ~ Z ~ Z2' 80 long as the neighbourhood of z = 0 is excluded. It diverges if z = o. These conclusions apply equally to (3) § 10.41. In the case of (4), (5) § 10.41, integrals of type (2) above are involved, 80 the preceding conclusion applies. These latter integrals can be differentiated under the sign, except in the neighbourhood of z = o.

10.50. Integral equations for cem(z, q), sem(z, q) with nucleus It is readily confirmed by substitu-

e2ik(cosh f cos "1 cos 8+slnh f sin TJ sin 8>. tion that [183]

,= f

217' eik1(x cos 8+11 sin

8>j(8)

eo.

(1)

o 1(8) being an arbitrary differentiable function of 8, is a solution of 2

1

-+-+k!1' = o. 8ox 8Oy' '2

'

(2)

INTEGRAL EQUATIONS AND RELATIONS

10.50]

203

Introducing elliptical coordinates (Chap. IX), and taking 4k 2

=

kfh 2

>

0,

(1), (2) become, respectively,

f e2ikwf(0) se, 211

, =

(3)

o 2

2

t2+-2+2k2(cosh2~-cos21])' = 0, 8f)~ 8' ' 87]

and

e

(4)

e

with w = cosh cos TJ cos 8+sinh sin TJ sine. The physical interpretation of (3), (4) deserves to be mentioned. (4) is the equation for propagation of 'elliptical' waves, and (3) is a solution thereof. Now e2i kw represents a system of omnidirectional plane waves, since 8 varies from 0 to 217'. Hence we may visualize 'elliptical' waves as being synthesized from 'plane' waves moving in all directions (8) and properly coordinated in amplitude and phase [f(8)]. As a suitable solution of (4) we take' === 11(')/2(TJ), where!I'!1 are solutions of

~¥: and

-

(a- 2k2 cosh 2~)fl = 0

(5)

d 2f d'Y}:+ (a-2k 2cos 2'Y})f2 = 0,

(6)

respectively. The next step is to define f(O) , so that (3) is a solution of (6) with , written for f2. From (3)

d 2' (j2 TJ

= -

f [4k 211

2(sinh

~ cos 'Y} sinO-cosh ~ sin 'Y} cos 0)2+

0

+ 2ik(cosh, cos TJ cos8+sinh, sin l] sin8)]e2i k wj(8) dO.

(7)

Now

e

e

(sinh cos 7] sin 8- cosh sin 1] cos 8)2

e

e

= (sinh sin 7] cosO-cosh eos 1] sin8)2-l(eos 21]-eos 28).

(8)

Substituting from (8) into (7) gives

::~ =

2"

f f(O) [ 2k2(cos 2'Y}-cos 20) +=2]e2ikw se. o

(9)

2M

INTEGRAL EQUATIONS AND RELATIONS

[Chap. X

Then by (3), (9)

f

2".

;:~+(a-2lc2COS2f1" = f(8~[a-2lc2COS28+:~]e2ikl.OdO.

(10)

o

But

f(8)~(e2ikW) = !.-[f(8)!... (e2ikw)] _ of(8) ~(e2ikw) a82 ee a(l a8 a8 =

2ikw• ~ [f(8)!... (e2ikW) _ of(8) e2ikw] + 02fe 2

88 Substituting from (11) into (10) leads to 00

2

80

d ' + (a-2lc2cos 21]){ = d2 ~

a0

(11)

f2"[f"(8) + (a-2lc 2cos 28)f(8)]e2ikWdO + 0

+ [f(8) ~(e2ikW)_ of(8)e2ikw]ll1l' 88 ee 0

(12)

If f(8) satisfies (6), the first [ ] in (12) vanishes, while if 1(8) has period 11', 211', the second [ ] vanishes at the limits. Under these conditions' satisfies (6) as required, so f{O) must be a multiple of cem(O, q), se,n{8, q). Also with the same a, q in (5), (6) it follows that Il("q)OC Cem{"q) or Sem("q), and f2(Tj,Q)OC cem(Tj,q) or sem(Tj,q). Thus (3) yields the integral equations 2".

f e2ikWcem(8) dO

Cem(e)cem("1) = Pm

(13)

o

Sem(~)sem(f1)

and

f e2ikwse 2'"

=

Um

m(8)

se.

(14)

o

Pm' u". are the characteristic values of the nucleus e2i k w •

10.51. Evaluation of (13), (14) § 10.50. We commence by writing 2kw = 2k(x 1 cos8+ylsin8) =

%1

cos(8-cx),

(1)

where Xl ZI

=

= x/h =

cosh,coSTj,

Yl

and ex =

tan- 1(Yl/X1) ,

= y/h = sinhtsin1],

=

2k(cosh2t cos21J+sinh2esin21])t

2k(cosh2'-sin21]).

= k1 T,

or tan « = tanhttan1]. Next we use the ex-

pansion [202, p. 43] CX)

ei[Z'l cos(B-tX)]

= JO(Zl)

+2 I

·ip cosp(8-~)Jp(Zl)'

1'-1

(2)

10.51]

205

INTEGRAL EQUATIONS AND RELATIONS

together with that of ce2n(8) , in the integrand of (13) § 10.50, thereby obtaining 211

Cean(f')cean(l1) = Pan

co

f [JO(Zl) +2p~tcoSP(8-a:)Jp(Zl)] X

o

co

X

Now 211

I

~~n) cos 21'0 r-O

se.

(3)

co

f [JO(Zl) +2 ~ ip cosp(8-a:)Jp(Zl)]COS 2r8 dO = 21Tillr cos 2ra:JIlr(Zl)'

o

P

(4)

1

the other integrals vanishing by virtue of orthogonality of the circular functions. Also, the series concerned are absolutely and uniformly convergent. Hence by (3), (4) we obtain the result Ce2n(e)ce2n (7J)

eo

21TP2n ~ (-1)'A~~n) cos 2ra:J2,(zl)'

=

(5)

r=O

this series being absolutely and uniformly convergent. Since the r.h.s. of (3) when evaluated is real, it follows that the imaginary part vanishes, so 211'

f sin[zl cos(8-a:)]cesn(8) dO =

O.

(6)

o

To evaluate P2n in (5), put 7J = 0, then ~ri~

Ce2n (e)ce2n (O)

=

(X

= 0 and Zl = 2kcoshe,

~

21TP2n I (-I)rA~~n)J2r(2kco8he). r=O

(7)

Thus from (7) above and (15) § 8.10, P2n = ce2n {0, q)Ce 2n (t 1T, q)/21TA~ln) = P2n/21T.

(8)

Using (8) in (13) § 10.50, and remembering that the imaginary part vanishes by (6), we get the integral equation for ce2n (7J , q), namely,

Cean(~)Cean( 11). = Pan

211'

f cos[Zlcos(8-a:)]cean(8) dO

(9)

o co

= P2n I

r=O

(-I)rA~~n)cos2rQ:J2r(zl)'

(10)

10.52. The remaining integral equations. Analysis similar to that in § 10.51, using the multipliers Pm' 8 m defined in Appendix I, leads to the following: P2n+l = Pln+l/ 2n',

t

When used in (13), (14) § 10.50, 1.h.s, are real.

PII&+l' al R+l

(l)t

must be multiplied by -i, since the

206

INTEGRAL EQUATIONS AND RELATIONS

[Chap. X

2".

Celln+l(f)celln+l(1J) =

Plln+1

f Sin[ZI COS(O-lX)]Celln+l(O) dO o

(2)

00

= -P2n+l

I (-I)'Ag.tt1 1)cos(2r + l )cx J2r+1(Zl )' 1'-0

(3)

2".

f C08[Z1 C08(O-lX)]celln+1(O) dO = O.

(4)

o

0'2n+l

=

(5)

8 2n +1/21T,

2".

=

Se an+1(€)selln+1(1J)

O'lIn+1

f sin[ZICOS(O-lX)]se2n+1(O) dO

(a)

o eo

= 8 2n +1 I

1'-0

(-1)1' B~~~11) sin(2r+ 1)a:J2r +1 (Zl)'

(7)

211

f C08[ZI COS(O-lX)]se2n+l(O) dO = O. o U2n+2

=

8 2n +2

/21T,

(8) (9)

21T

Se2n+2(f)se2n+2(1J)

= 0'2n+2 f COS[ZI COS(O-lX)]se2n+2(O) dO

(10)

o

=

eo

-82n+2

I

1"-0

(-1)'Bg.~~2)si.n(2r+2)(XJ2r+2(Zl)' (11)

2".

f sin[ZtCos(O-o.:)]selln+2(O) dO = o.

(12)

o

By analysis similar to that in § 8.50 it may be shown that the series herein and in § 10.51 are absolutely and uniformly convergent in any finite region of the z-plane.

10.53. Expansions of C?s{2k(cosh SIll

k2

>

e

C081]

cos 8+sinh~8in1J sin 8)},

O. Assume that

cos[2 1 cos(8-cx)]

Multiply both sides by ce2n(8), integrate with respect to 8 from 0 to 211, and we get 2"

f C08[ZI COS(O-lX)]cean(O) dO = 1TC2n(1J )Cetn (f ).

o

(2)

INTEGRAL EQUATIONS AND RELATIONS

10.53]

207

the other integrals vanishing by virtue of orthogonality (§ 2.19). From (2) above, (8), (9) § 10.51, it follows that C2n {"1 )

=

2 ce2n("1 )/P2n.

(3)

Multiplying both sides of (I) by Se2n+2(O) and proceeding as before,

8 2n+2 ( 1]) = 2 se2n+2 ( "1)/8 2n+2 •

we get

(4)

Substituting (3), (4) into (1) yields the expansion cos{2k(cosh ~ cos YJ cosO+sinh ~ sin YJ sinO)}

=

i

2

[Ce1n.<~)ce~n (TJ )cel'~ (0)

+ Se2n+2(~)Se;n+2( TJ )sean+a(8)] . .(5) 2n+2

P2n

ft-O

By similar analysis we find that sin{2k( cosh, C08"1 C08 8+sinh, sin "1 sin8)}

=2

'5' [Ce2n_+1(~)Celn+1(TJ)Cean+1(0) + Sean+1(~)sean+1( TJ)sean+1(O)]. (6)

i,

~

8 2n +1

P2n+l

In (5), (6) write for" then, referring to § 10.60, the r.h.s, of (5) above reduces to the form (7) forO = 0, and to (8) for 0 = !1T, while (6) above reduces to the form (9) for 0 = 0, and to (10) for 0 = 11T. Accordingly we obtain the expansion ei~l C08(8-tk)

=

eikl(XCos8+lIsinB)

=

2

L CX>

[

1

=

cos{k1(x cosO+y sin 8) }+i sin{k 1(x C08 8+y sin 8)}

-Ce2n(e)ce2n{7J)ce2n(O) n-o P2n

1 +-8-Se2n+2(,)se2n+2(YJ)se2n+2(8) + In+2

+i{_l_ Ce2n+l{,)ce2n+l(7J)ce2n+ + P2n+l 1{8)

+ _1-Se2n+l(e)Se2n+l('Y})Se2n+l{8)}].

(7)

8 2n+1

When the fundamental ellipse tends to a circle, it may be shown, by aid of Appendix I, that with (J = 0, well-known expansions in B.F.

t1T,

(5), (6) degenerate to

10.60. Expansions of nuclei in characteristic functions, q >

o.

If in (2) § 10.11 we put X = R(u)8{z), and proceed as in §9.21, we obtain the two ordinary equations

dlR du a + (a- 2qcos 2)R u =

°

.

(1)

INTEGRAL EQUATIONS AND RELATIONS

208

d 2S dz2 + (a-2qcos 2z)S

and

=

[Chap. X

(2)

O.

For a giren value of q, if a = am' .the products of the first solutions of (1), (2) are Xmc == cn l cem(u)cem(z), while if a == bn& we have Xma

em and

=

8,n se m(u )se m(z),

being arbitrary constants, m taking positive integral values. Hence we may write 8m

co

ex>

X

= I

m-O

Cm cem(u)cem(z)+

I

m=l

8 m sem(u)sem(z).

(3)

Now X = e2ikcosUC08~ and e-2kslnusinz are also solutions of (2) § 10.11. It appears, then, that these composite integral function solutions may be expressed in infinite series of periodic Mathieu functions. The real part of the first, being even in u, Z, is the nucleus for ce2n(z, q). Now the second series on the r.h.s. of (3) is odd in u, Z, while ce 2n(z, q) has period 17'. Let us assume, then, that ex>

I

cos(2kcos'ucosz) =

8=0

c2s ce 2B(u )ce 28(z),

(4)

the r.h.s, of which is even in u, z and admits the period 17' in both variables. To determine Cis multiply both sides of (4) by ce 2n(u) and integrate with respect to u from 0 to 17'. Then w

f cos(2kcosucosz)celln(u) du = cllncelln(z) f ce~n(u) du, ~

o

(5)

0

all other terms on the r.h.s. vanishing by virtue of orthogonality (§ 2.19). Hence by (5) above and (1) § 10.12, ce 2n (z)/A2n =

!1TC 2n

ce2n(z),

so, using (5) § 10.12, c2n = 2/7TA 2n

=

2A&2n>/ce2n(11T,Q)·

(6)

Consequently from (4), (6) we obtain ex>

cos(2kcosucosz) = 2

L

A(2n)

__J} ce2" (u)ce 21l(z), n=O ce2n(iTT, q)

(7)

which is the expansion of the nucleus for ce2n(z, q) in terms of the functions themselves. The seven remaining primary nuclei in

INTEGRAL EQCATIO:SS AND RELATIONS

10.60J

209

Table 14 Inay be expanded in a similar way, and the results are given below [97]: A(2n)

co

cosh(2ksin'llsinz) = 2 ~ ---~()ce 2n(u)ce 2n(z), ~ cC 2n 0 1'10=0 A(2n+l) 2 ce'-! (-1-1' ~ .; 2 --,B~2"'+1) 2k

(8)

00

sin(2kcosu cosz) = -2k

2.l+1

n=O

·

..

~ slnh(2ksln zz sin z)

ce2n+l(u)ce2n+l(z),

(9)

)

00

-----(-.:-)Se21l+1(u)se2n+l(Z)' (10)

:=::

1'10==0

se 2n + 1 0

A(2n+l)

(1:)

coa a cos e coshrzc sin e sin s) = ~ . - 1. - ce2n+l(u)ce2n+l(z), ~ CC 2n 'il (O) 1'10=0

(11)

B(2n-tl)

CX>

.

sin u sin z cos(2k cos 'U, cos z) = ""'~-_.J. r, sC2n+l(u)se2n+l(z), . : se 2n +1( 2 17) n=O .

(12)

B(21l,+2)

00

sin u sin z sin(2k cos u cos z) == -k "'" ., ~·--l. se2n+2(u)se2n+2(Z), ~ se (17)

(13)

2n+2 B(2n+2)

n=O

00

cos u cos z Sillh(2k sin u sin z) = k ,

_,2 -_. -- -

~ se 2 n +2(O)

se2n+2(u)se2n+2(z).

(14)

n=O

By writing iz for z in the above relationships, series are obtained in terms of Mathieu and modified Mathieu functions. For z real, convergence of the series follows from the fact that the A, B ~ 0 as n -+ +00 (see § 3.35). 10.61. Results deducible from § 10.60. Writing u (7) § 10.60 yields

= 11T

in

ex)

1= 2

I

n=O

A~211)ce2n(z).

(1)

Assuming the r.h.s. of (9) § 10.60 and its first derivative are uniformly convergent with respect. to u, by differentiating and substituting 'it

= !1T we

get. oc.

cos z =

I

n=O

A~2n +1)ce 2l1 -i I(Z).

(2)

In a similar ,,·ay we find that ex:>

sin z --

sin ')." -

C'

--

--

~ B(2n -tl)se

~

nICO

1

• 21l+1

(z) ,

(3)

("",)

(4)

co

~ B(2n t2)\olo

~

'1 -0

2

u" 21l ;.2 "., •

INTEGRAL EQUATIONS AND RELATIONS

210

[Chap. X

Inserting the expansion of ce2n(z) in (1) gives 00

1= 2

00

I

I

A~2n)

A~~n)cos2rz.

(5)

vr-O

11,==0

Since the l.h.s. is independent of z, equating constant terms, i.e. r = O~ yields

ctJ

I

= 2 I [Ab21t)] 2.

(6)

11,=0

Equating the coefficient of cos 2rz to zero leads to co

I

=

Ab2n)A~~n)

(r =1= 0).

0

(7)

11,=0

Similarly we deduce that 00

I

[Ai 21t+1)]2 =

11,=0

00

I

11,=0

[Bi 2n +l)]2 =

00

I

11,==0

[B~2n+2)]2

=

1.

(8)

Using the orthogonal properties of the functions the following expanSiOIlR nlay be obtained: 00

cos 2rz ==

2 A~;n) ce2n(z),

(9)

11,-=0 00

A 2r+ (211,+1) 1 ce 2n+l (z) ,

( 10)

sin(2r+ l)z ==

I B~~'l11) se 2n + l (z), 11,=0

(11 )

sin(2r+ 2)z ==

I B~~~~2), se2n + 11,=0

c08(2r+ l)z ==

~ £.

11,=0 00

00

00

I

[A~;n)]2 ==

11,=0

00

I

[A~~~iI)J2 ==

11,=0

(12)

2(z),

00

I

11,=0

[B~~~11)]2

00

== I

[B~~~~2)]2 == 1, (13)

n=O

T>O 00

I

ft'-O

rA~~n)A~~n)]

==

00

~ [A~~~11)A~~~1l)]

11,=0

== -

00

2 [B&~~11)B~~ril)] n =0 ~

k

n=O

[B(2n+2)B(2n+2)] -

2r+2

28+2

-

0

(14)

provided r =1= 8. Additional relationships between, the functions of integral order will be found in reference l H7].

10.62. Additional expansions. Assuming uniform convergence of the series in § !O.6(), numerous results may be deduced by differentiating or by integrating term by term. Two examples involving B.F. will suffice to illustrate this point. Integrate both sides of (7) § 10.60 with respect to u from 0 to 11', and we get Cf.J

,~.(~k cos c) ::-...:

2

~ [A~2n)]2 c~2n(Z, q)/ce2,,( 111', q). n .... O

() )

10.62]

If z

INTEGRAL EQUATIONS AND RELATIONS

=

0,

while for z =

Jo(2k)

=

211

00

2

I

11.==0

!17

[A~2n)]2 ce 2n (O, q)/ce2n(!1T, q);

=

1

(2)

00

2

I

[A~2n)]2,

(3)

11.=0

as at (6) § 10.61.

2°. Multiplying both sides of (7) § 10.60 by cos 2mu and integrating as before leads to 00

J2m(2k cosz) =

I' A~21t)A~~)ce2n(z,q)/ce2n(!7T,q) 11.=0

(m

>

0).

(4)

Rigorous proofs of the validity of results in §§ 10.53, 10.60, 10.61, and a demonstration that, under conditions similar to those for the Fourier case, a function of period 7T, 21r may be expanded in a series of Mathieu functions of integral order, are beyond the scope of the text. For functio~s having period 287T, 8 ~ 2, the Mathieu functions of order 2n+~, 2n+ 1+,8, ,8 = pis, 0 < ,8 < 1, would be needed (see §§ 2.20, 4.71).

10.70. Integral equations of the second kind. We commence with the equation

d2y du 2 + [a-2q,p(wu)]y

=

(I)

0,

where .p(wu) has the same properties as in (1) §6.10. Multiplying throughout by sin vU, v 2 = a, gives

sinvuy"+asinvuy z

z

so

= 2q¢(wu)sinvuy,

f sinvu dy' +a f sinvuy du =

(2)

z

2q f ifJ(wu)sinvuy duo

(3)

0 0 0

Thus

[y'sinvu]~-v

z

z

z

f cosvudy +a f sinvuydu = o

2q f ifJ(wu)sinvuydu, (4)

0

0

2

and ultimately, since a = v , we get z

y'(z)sinvz-vy(z)cosvz

f ifJ(wu)sinvuy duo

-vy(O)+2q

(5) o If in (2) sin vu is replaced by cos vu, the equation corresponding to (5) is =

z

y'(z)co8vu+vy(z)sinvz

=

y'(O)+2q

JifJ(wu)cosvuydu. o

(6)

212

INTEGRAL EQUATIONS AND RELATIONS

[Chap. X

Multiplying (5) by cos lIZ/V, (6) by sin vzjv, and subtracting the first from the second, yields the integral equation ~

y(z)

Jsin v(z-u).p(wu)y(u) du, , (7)

= y(O)cos vz + y'(O)sin vz:+ 2q v

v

o

Hence we are led to two solutions of (1), in the form of the integral equations of the second kind with variable upper limits, namely, z

Yl(Z)

= Yl(O)cosvz+Y~(0!.sinvz+ 2q J sinv{z-u),p(WU)Yl(U) du, v v o

and Y2(Z)

= Y2(0)cosvz+

~

----+- J·

y~(O)sinvZ

2q

v

v

(8)

slnv(z-u)«/J(wu)Y2(u) du,

(9)

o

If we specify the initial conditions, as in §4.10, to be Yl(O) Y2{O)

(8), (9) give

= =

1, 0,

y~(O)

=

0,

y;{O)

=

1,

z

Yl(Z)

=

cosvz+ -;

Jsinv(z-u)ifs(wu)Yl(U) du

(10)

o

J t:

and

Y2(Z)

= ~nvz + 2q v

v

sinv(z-u)¢J(WU}Y2(U) du,

( 11)

o

respectively. For Mathieu's equation «/J(wu)

==

cos 2u.

10.71. Integral equations for y"-(a-2qcosh 2z)y = o. Using the procedure in § 10.70 but with sinhvu, coshvu for sinvu, and COSVU, respectively, we get y(z)

= Y(O)coshvz+Y'(O)~-~~ v

J ~

2q v

sinhv(z-u}cosh 2uy(u) du. (I)

o

For the initial conditions in § 10.70, (I) yields z

Yl(Z)

=

coshvz-;

sinhv(z-u)cosh 2UYl(U) du

(z finite)

(2)

(z finite),

(3)

o z

and Y2(Z)

J

=

sinh vz- 2q v

v

J

sinh v(z-u)cosh 2u Y2{U) d'u

o

where Yl(Z), Y2(Z) are, respectively, even and odd solutions of the above differential equation.

213

INTEGRAL EQUATIONS AND RELATIONS

10.72]

10.72. Solution of (10), (11) § 10.70. Write p

= 2q/v, and assume

that Yt(z) = cOSVZ+pCl(Z)+p2C2(Z)+p3C3(Z)+ ... , (1) the c being continuous functions of z. Substituting (1) into (10) § 10.70 we get pc t(Z)+p2C2(Z)+ ... z

= p f sinv(z-u).p(wu)[cosvu+pc1(U)+p2C2(U)+ ...] duo

(2)

o

Equating the coefficients of p on each side of (2), we have z

c1(z)

= f sin v(z-u)cos vu.p(wu) du o z

= 1 f [sinv(z-2u)+sinvz].p(wu) du o

=

z

lsinvz

z

f .p(wu)du +1 f sinv(z-2u).p(wu) duo o

(3)

0

This is now substituted for 0l(U) into the r.h.s. of (2) and c2(z) obtained by equating the coefficients of p2 on each side. If p and pJ4J(wu)J are small enough, a first approximation is given by Yl(Z)

~ cosvz+(q/v{sinvz j .p(wu) du + j sinv(z-2u).p(wu) o

the omitted part being

y,(z)

~ Si:VZ-

O(I/v2 ) .

(q/v2{ cosvz

-!

0

(4)

Similarly for (11) § 10.70, we find that

j .p(wu)du - j cosv(z-2u).p(wu) dul

o

0

(5)

the omitted part being O(I/v3 ) .

10.73. Alternative forms of (7) § 10.70. This may be written 'y(z)

=

q,cosvz+S"sinvz+ ~q

•

f sinv(z-u).p(wu)y(u) du,

(1)

o

where 0" = y(O), 8" = y'(O)Jv. Now let a integer, then (1) § 10.70 becomes either

= m 2 + ,\2, m being an

d2y2 + [m2_{_,\2+2q.p(wz)}]y

dz

or

d"g dz 2+[,\1I-{2q.p(wz)-m'}]Y

=0

(2)

=

(3)

O.

214

INTEGRAL EQUATIONS AND RELATIONS

[Chap. X

Since (1) is an integral equation for (1) § 10.70, it follows that the integral equations of the second kind for (2), (3) are, respectively,

f s

= Om cos mz+Sm sin mz + ~

y(z)

sinm(z-u)[2qifs(wu)_,\2]y(u) du,

(4)

o z

y(z)

= O.\cos.\z+S.\sin.\z+~

f sin.\(z-u)[2qifs(wu)-m2]y(u)du. o

(5)

Alternatively, if we take a = ~f+cx=, by varying either <Xl or 0:2' keeping a constant, we can write down an infinite number of integral equations. The 0, S are functions of q.

10.74. Integro-differentlal equations for y"+2Ky'+[a-2q.p(wz)]y

= o.

The integro-differential equations corresponding to (1), (4), (5) §lO.73 are, respectively, z

y(z) =

If sinv(z-u) [2q.p(wu)-2I
.

0

(1)

y(z) = G,ncosmz+Smsinmz+ z

+~ f Sinm(Z-U)[ 2qrp(wU)-.\2_2te d~]Y(U) du,

(2)

o

and y(z)

=

[Acos,\z+8,\sin,u+

+~

f•

sin.\(z-u)[ 2qifs(wu)-m 2-2te ;u]Y(U) duo (3)

o

The 0, S are functions of q.

10.75. Second solution of Mathieu '8 equation using (4) § 10.73. The integral equation may be split up into the two parts, namely,

f = Sm8inmz+~ f ~

Yl(Z)

= Om cosmz +~

sin m(z-u)(2qC08 2U-~2)Yl(U) du

(1)

sinm(z-u)(2qcos2u-A')y,(U) duo

(2)

o

and

z

Y2(Z)

o

INTEGRAL EQUATIONS AND RELATIONS

10.70]

215

Let a be the characteristic number for ces(z,q), then we have to determine Y2(Z) from (2). For ce2 (z, q), by (5) § 2.151

a=4 ~2

so with m = 2,

2_~ 4 +~ 12q 13824 q +...,

5

_

2

- 12 q

763

4i-

-13824 q

(3)

(4)

....

Assume that Y2(U) = 8o(U)+ qsl('U) +q282{ u)-t- ... ,

(5)

8 2(q) = 1+fJlq+fJ2q2+fJ3q3+ ... .

(6)

Substituting (4), (5), (6) into (2) gives 80(Z)+qSl(Z)+q2S 2(Z)+q3S S(Z) z

+

f

+... = (I +f11 q+/32q2+ /3sQ3+ ... )sin 2z+

5 sin 2(Z-U)[qcos 2u- 24 q2+ 2~::8q4_... ] X

o

[SO(U)+qSl(U)+q2 8 2(U) +...] duo

X

(7)

Equating like powers of q on both sides of (7) leads to qO q

t~o(z) =

sin 2z,

81 (Z) =

.81 sin 2z+

(8) z

f sin 2(z-u)cos 2u

8 0(U)

du

o z

=

.81 sin 2z+1

.

f sin 2(z-u)sin 4u du o

= (~1+1)sin

2z-h sin 4z.

(9)

If for the sake of illustration we adopt the normalization of § 2.1 ] , the coefficient of sin 2z is unity for all q. Thus in 81(Z), 82(Z), ... the coefficient of sin 2z must vanish, so ( 10)

f f

z

q2 82(Z)

= .82 sin 2z+

sin 2(Z-U>[ cos 2/.t 8.(/l)

o

-i:

1

80(1/) du

z

=

.82sin2z-

Sin2(Z-U)[}2cos2usin4u+:4sin2u] du

o =-=

9).

(132 - ----l~~

SIn

I

I ..

'J .z+ --zeos 4~~z+- SIn bz 8 384'

(11)

216

INTEGRAL EQUATIONS AND RELATIONS

Thus

fl./,

8 3(Z)

q3

=

9

=

128

and

82(Z)

=

[Chap. X

1

!zcos 2z+-- sin 6z. 8

384

(12)

(f3a- 619 )Sin2Z- ., 34560

53. 4 I.] I I 4 - [ -32z+96zCOS z-13824 s1n z+23040s1n8z ,

giving

Q

619 34560'

---

fJ3 -

and 8 3 (Z )

Y2(Z)

1 I 4z----sln4z+--s1n8z 53. 1.] = - [ --Z+-ZCOS 3~ 96 13824 ~3040 '

(13)

= r=O i q"s,(z) = Sin2z-1~)qSin43+q2(-81ZC082Z+-J-8in6Z)~ 384 53 1 _q3(_~1)Z+~ZCOS4Z3 sin4z+- Rinsz)+ .... 3~ 96 1824 2'040 3

(14)

Thus

Y2(Z)

=

fcz(z,q)

= !q2(1_ 1474q2- ... )zce2(Z,q)+Sin2Z-~qSin4Z+ 8 I 12

'. 53. 4z -1---8111 1 .Sz )+ 1 q2 SIll 6z- -1q3( ---SIn + -384 012 27 , 45 ...

(15)

I

=

(

9 q2+ _ 619 . l __1 q+ __ _ q3_ ...) sln2z-t6

12M

34560

~

+

f Sin2(Z-1t)[qCOS2U-;4q2+2~:~8q4- ...]fe2(U,q)dU.

(16)

o

A similar treatment of equation (1) would yield the series for ce2(z, q), the first solution of Mathieu's equation with a == a2 , as given at (3) § 2.150. In general, if Yl(Z) ~- cem(z, q) and the q series for a is unknown, write a = 1n2+Cl:lq+a2q2+l~3q3+ ..., (17) make the coefficient of cos mz unity, and determine (1:1' (l:2"" so that all terms in the expansion are periodic in 'T1' or 21T, according as m is even or odd (see § 2.13 et seq.). 1'0 obtain sem(z, q), treat (2) in this ,\.a,~". ge".{z. q) can be obtained by analysis similar to that from (3)-( 15).

INTEGRAL EQUATIONS AND RELATIONS

10.76]

217

10.76. Integral equations for Yl(Z)Y2(Z). Let Yl(Z), Y2(Z) be independent solutions of y"+(a-2qcos2u)y = 0, for the same a, q. Then writing e(u) = (2qcos2u-a), we have

(1)

y~ -~(U)Yl

= 0 (2) y;-,(U)Y2 == O. (3) Multiplying (2) by y~, (3) by y~, adding and integrating with respect to u from 0 to z, gives and

~

~

Io (Y~Y~+Y~Y;) du = I (Y~Y2+Y~Yl)'(U) du, Io d(y~y~) = I ,(u) d(YIY2)'

(4)

0

~

so

e

(5)

0

Hence by partial integration of the r.h.s. of (5) we find that y~(z)y~(z)

=

y~(O)y~(O)-Yl(O)Y2(O),(O)+ z

IYl(U)Y2(u)sin

+Yl(Z)Y2(Z),(Z)+4q

2u duo

(6)

o

If the initial conditions are Yl(O) == y~(O) = 1, y~(O) = Y2(0) = 0, the first two members on the r.h.s. of (6) vanish, and we obtain

y~(z)y~(z)

z

=

Yl(Z)Y2(Z),(Z)+4q

I

Yl(U)Y2(u)sin 2u duo

(7)

o

Results similar to (6), (7) may be obtained in connexion with Hill's equation. For the equation y"-(a-2qcosh2u)y == 0, with X(u) = (2qcosh 2u-a) we get y~(z)y~(z)

=

y~(O)y~(O)+Yl(O)Y2(O)X(O)~

-Yl(Z)Y2(Z)X(Z)+4q

I Yl(U)Y2(u)sinh 2u duo

(8)

o

10.77. Differential equation satisfied by w(z).= Yl(Z)Y2(Z). Mathieu's equation may be written

where ~ = 80

y" -fy =0, (2q cos 2z-a). Differentiating (1) gives y'"-'Y' -,'y =

°

Y;' -~Y~-"Y2 =

o.

yl' -~y; -~'Yl

and 4961

Ff

(1) = 0, (2) (3)

218

INTEGRAL EQUATIONS AND RELATIONS

[Chap. X

Multiplying (2) b)" y,., (3) by YI' and adding leads to the equation Y;"Y2+Y;'Yl-2~'YIY?-~(YIY~+Y2Y~)= O.

(4)

From (1) we have

(Y;-'Yl)Y~ = 0

(5)

and so by addition

(Y;-'Y2)Y~

==

0,

" '+ YaYI ", = 5i:.(YIY2, + Y2YI') = sW /:' • YIY2 Al80 '" +3(" " ') W = YIY?+Y2YI YIY?'+Y?YI· Hence from (4), (7), (8) the equation sought is w'" -4,w' -2g'w = o. Iff

Iff

(6) (7)

(8) (9)

This result applies to Hill's equation (1) § 6.10, provided

,=

[2q~(wz)-a].

The complete solution of (9) with three arbitrary constants is w

= Ayf+BYIy?+Cyl.

(10)

XI ASYMPTOTIC

FORMUT~AE

11.10. Approximations. The modified Mathieu functions lend themselves readily to the derivation of approximate asymptotic formulae when R(z) is large and positive. If k and z are such that 2kcoshz ~ kef!, = v, this being appreciably greater than the order of the function, the dominant term in the asymptotic expansion of J:ar(2k cosh z) is [202, p. 158) (2/1TV)lcos(v-!1T-r1T)

=

(-1)r(2/7Tv)1 sin(v+!w).

Substituting (1) in (15) §8.10 and using the multipliers P,n' Appendix I, leads to the approximate result Ce2n(z, q)

Writing

"-J

P2n{2/11'v)l sin(v+ 111')

(1) 8m

(a2nJ.

in (2)

J2r +1 (2k cosh z) ,......, (2/1rV)l cos {v-l1T- !(2r+ 1)7T} = (_1)r+l(2/1TV)1

cos(v+l1T)

(3)

in (1) § 8.12 leads to ·Ce2n+1(z, q) ~ -P2n+l(2/1TV)' cos(v+i'7T)

(a 2tt+1 ) .

(4)

The same result follows from (2) § 8.12, while (6), (7) § 8.13 give with tanhz

~

cothz

Se 2n +1(z, q) ,......,

.From (4), (5) when R(z)

~

2ksinhz

I,

~

2kcoshz

-S2n+l(2/TTV)1: cos(v+l'7T)

>

~

kes,

(b2n + 1 ).

(5)

0 is large enough, it follows that (6)

Ce2n +1(z, q) ,......, (Pzn+l!s2n+l)Sezn+1(z, q).

}'rom (10), (II) §8.13 we derive Se2n + 2 ( Z , q) "-J

and

80

by (2), (7), when R(z)

8 2n + 2

>

{2/7TV)I: sin(v+ !1T)

(b21£ +2 ) ,

(7)

0 is large enough, we may write

Ce2n +2(z, q) ,....., (Pan+2/sZn+2)SeZn+2(z, q).

(8)

As z (large and real) increases, all the above functions, and those at (9)-( 12), alternate'] with ever-increasing rapidity, but decreasing amplitudes. When z -+ +00 they all tend to zero exponentially. The second solution« corresponding to the first given above are defined in §§ 8.11-8.13. Using the multipliers in Appendix I and t An alternating function is one which oscillates and has real zeros.

ASYMPTOTIC FORMULAE

220

[Chap. XI

proceeding as at (1)-(4), we obtain the following approximate asymptotic representations:

FeYt,t(z,q),....,

-Ptn(:k)tCOs(v+l7T)

(TTV2)1sin(V+!1T)

(a 2n +1 ) ,

(10)

-82n t 1 TTV

(2)1sin{v+ }17')

(b21t+-1)'

(II)

-82n+2(~-)t COS(V+ 17T) 11'1'

(bt n +t ).

(12)

FeY2n+l(Z, q)

1'.1

-P2n ..l

GeY21l+1(Z' q)

1'.1

GeY2n+2{Z, q) 1'.1

(9)

The dominant term in the asymptotic formula for FeY2Tl+2(z,Q) is a constant multiple of that of Gey 2n +2' the constant being P2n+2/82n+". A similar remark applies to (10), (11). These formulae are valid if R(v) ~ +00 with v, so -in < phase z < in. Me~),(2)(z, q), Neg),(2)(z, q), Gekm{z, q). Using the dominant term in the asymptotic expansions of the Hankel functions in § 13.40,t if Ivi n we get:

11.11. Approximate formulae for

~"ekm(z, q),

>

Me~~·(t)(z,q) ,....,

Ptn(:v)t e±i(~-hr)

Me(l),(2)(z q) 2n+l'

f""ttt,J

p 2n+l 1rV

(l ), (2)(Z q) Ne 2n+l'

1'.1

8

Ne(l),(2)(z q) "" 2n+2'

8

tn),

(1)

e±i(v-I1T)

(a 2n+l ) ,

(2)

e±i(v-hr)

(b2n+l ) ,

(3)

e±i(v-hr)

(b2n+2 )•

(4)

(-!)l (.!.)! (.!.)!

2n+l 1TV

2n +2 TTV

(a

Note that (1) is a constant multiple of (4), and (2) of (3). Applying the relationships in § 13.41 to (1)-(4), formulae for Fek ,ll(z, q), (~ekm{z, q) are derived immediately, As z ~ +00 the first function tends to a constant multiple of the second, for any m.

11.12. Formulae for q < o. Using the formulae in § 8.30 and proceeding as in § 11.10, we obtain the following: (1) Ce2n (z, -q) P~n e"/(21Tv)1 (al n ) , 1'.1

t

Ce1n +1(Z, -q) ,....., 8~n+l e"/(21rV)l

(ban+1 ) ,

(2)

Se2n +1(z, -q) ,....., P~n+l e"/(21rV)1 Se21l +S{z, -q) "'" 8~n+2 e"/{21TV) 1

(aan+l)'

(3)

(b2n +2) .

(4)

Or the formulae in § 11.10, substituted into (1), (2) § 13.40.

AS YM PT()T rc FO Rl\llT l.A~;

11.12]

221

These functions are constant multiples of one another. and each tends to +00 monotonically as z ~ +00. The formulae for Fek 2,,(z, -q), ~'ek2"~1(Z, -q). (:ek 211 f 1 (Z, _.q). Gek 21l f-2(Z, -q) are obtained from (J )-(4). respectively. on replacing eV by e -v. Each function tends to zero monotonically ItS Z . > t oc. All formulae given hitherto in this chapter depend upon c ht'ing large enough for the Ressel functions in §§ S.IO-S.:~O to he represented (approximately) by the dominant terms in their asymptotic PXIUU1sions. We shall now remove this restriction on z, and establish what may be regarded as accurate asymptotic expansions.

11.20. Accurate asymptotic expansions in z, We commence with the modified Mathieu equation y" -- (a- 2k 2 cosh 2z)!J

===

o.

(I)

If in (I) we write x == -- ikp:: === --- io, it becomes [;l7] 4

d211 J dll ( a k ) dX2+xdJ:~- 1+;;2+ X4lJ:~

o.

(:!)

A second transformation is effected hy putting 1I .; -: u-e ", and we now get

~~lll_ 2 dx

( 4 (2 _x~)~W _ 1 + ~ + k )U' dx x x X

_=-

o.

with

Co

4

2

(:3)

(X)

To solve (3) assume that to

==

~ (-l)'crx- r --l ,

r=O

==

I. Insert.ing

this in (3) and equating coefficients of like powers of x to zero, we obtain the recurrence relations

+(4a - 1) -.:-= 0, 16c2 +(4a -- n)r 1 == u, ~r 1

(4 ) ( f»)

2(r+ I)Cr~1+[a,---(1"-t-A)2]rr+k·I(·r-2~- 0

(r ~~ 2).

(U)

From (4)-(6) we find the expressions for the c to be as follows t 10;3J: (4a - 12)

c1 = -'--u'---, o

c2

=

(4a- 12)(4tl --- :~2) ""~2' ... t'"

(4a-1 2)(4a-32)(4a-52 )

Ca

= ----------------3183-

- ---- --- --

1

(4a-1 2 ) ( 4a- 32 ) ( 4a- 52 ) ( 4a-7 2 )

c" = .-.. -- _.... - --- 4' -~,f-.0

4

-:fik',

(7) (8)

k4

--- -- ----- +~4;(4a-13),

(0)

.tiJ • •

and so on. It should be noted that, (I) the dominant term and the

two which follow are independent of k, (2) the first member of each

[Chap. XI

ASYl\IPTOTIC FORl\IULAE

222

c is identical with that in the asymptotic series for Jv(v) when a = vI, i.e. the degenerate form of the c when k ~ O. If x remains constant meanwhile, equation (2) reduces tc) the standard form for the B.F. Jv(v) and Yv(v). Frequently the terms involving k in (8), (9) are small numerically, com pared with the earlier terms in the series, and may be neglected, but this point should always be checked. The solution of (2), with x == <-i», is Y

== ue:» ==

ei tJ

----I-

(-tv)l

La:>

ei(r f I1T) La:>

(-I)rc (-i'v)---r = -'- r

T-O

e -r [(I-c i (v +l 1T)

=

2 V-

vi

c ('iv)-r

T-O

r

2+C v- 4 - ••• )-i(c v-1-C v- 3 4 a 1

+...)] (11 )

e1(v +11T)

=

(10)

T[P-iQ],

(12)

== Q ==

l-c 2 1) - 2 -f- c 4 v- 4 - •••

(13)

C1 v-l_C3V~-3+C5 1)-5----....

(14)

P

where and

The real and imaginary parts of (12) arc linearly independent solutions of (1), so we have the two solutions and When

Z~

Yl(Z)

== l,-i[Pcos(v+l 11 )+ (I Hill (I.' -f-! 1T )J

( 15)

Y2(Z)

== v-i[Psin(v+l 11 )-

(1 t»

+00,

we may write Yt{z) ~

and

Q cos(o+ l 7T ) )·

Y2{Z)

~

cos(v+t1T)

(17)

v- l sin(v+ l 1T),

(18)

v-I

these being the dominant terms in the asymptotic expansions (15), ( 16) respectively.

11.21. Asymptotic expansions for Cern' Se m, Feym' Geym' q > 0, If (18) § 11.20 is multiplied by P2n(2/11)l, we obtain (2) § 11.10, 'the dominant term in the asymptotic formula for Ce2n(z, q). Hence using (16) § 11.20 we infer that Ce2n(z, q) ~ P2n(2l7TV)i[P~~ sin(v+ 111 ) - Q~r:l cos(v+ 17T) ]

(a2n ) . (I)

The subscript 2n and the superscript (a) signify that a 2n is to be used for a in the formulae for the c in (7)-(9) § 11.20. If (17) § 11.20 is multiplied by -P2n{2/1T)I, we obtain (9) § 11.10, the dominant term in the asymptotic formula for FeY2n(z,Q). Hence using (15) § 11.20 we infer that FeY2n(Z,Q)""'" -P2n(2/1TV)t[P~~cos(v+11T)+Q~~sin(v+t1T)]

(azn). (2)

ASYMPTOTIC FORl\IULAE

11.21]

223

Writing

~~ = (:v)l[p~~Sin(v+ I 1T ) -Q~~ cos(v+l1T)]

(a 211),

(3)

S~~ = (:v)![P~~C08(V+i1T)+Q~:lsin(v+!1T)]

(a211) ,

(4)

and proceeding as shown above, we arrive at the following:

Ce2n +1( Z , q) Se2n +1(z, q)

f'J

-P2n+l

f'J

- 82n +1

S~c:1+1

(a2n +1 ) ,

(5)

S~t;l+l

(b2n +1 ) ,

(6)

(b2n +2 ) ,

(7)

R~bJ+2

Se2n +2(z, q)

f'J

8 2n +2

FeY2n+l(Z, q)

f'J

-P2n+l R~c:l+l

(a 2n +1 ) ,

(8)

- 82n+1 R&bJ+l

(b2n +1 ) ,

(9)

GeY2n+l(z, q) ""

-82n+28&~+2 (b2n +2). (10) In the series for R~),(b), S~),(b) the c in § 11.20 are obtained using the GeY2n+2(Z,Q)

I'-'

characteristic numbers given at the r.h.s, The asymptotic expansions for Meg>,(2)(z, q), Ne~\),(2)(z, q) may be written down immediately by aid of (1), (2) § 13.40 and the appropriate formulae above. Then by means of § 13.41 the asymptotic expansions for Fekm(z, q), Gekm(z,q) may be derived. The phase range in the above formulae is -!1T < phase z < !7T.

11.22. Degeneration of expansions in § 11.21 to those for Jm(k 1 r), Ym(k 1 r). B)T Appendix I when Z ~ +00, l~ -+ 0, a ~ 4n 2, then kez ~ k1 rand Ce2n (z, q) -+ P~n J2n (k 1 r), (1) As k -+ 0, the series p~~, Q~~ degenerate to those in the asymptotic expansion of J2n (k 1 r), excluding the factors

(2/1Tk1 r)l :f~(kl r- !1T-n1T) [see reference 202, p. 158]. In a similar way, under the above conditions, the asymptotic expansion of FeY2n(z, q) degenerates to P~n times that of Y2n{k 1 r), and so on.

11.23. Accurate asymptotic expansions for q < o. These are derived by applying the definitions at (1), (4),... § 8.30 to the expansions for q obtain

>

0 in §11.21. Thus from (1) §8.30 and (1) §11.21 we ,

00

0:>

Cean(z,-q)'1-' PIn [evIc,v-r-ie-VI(-I)rCr,v-r]

(27TV)l

,.-0

r-O

(u2n ) ,

(1)

[Chap. XI

ASYMPTOTIC FORMULAE

224

the c being those in § 11.20 with a 00

=I

Ce2 ,,(z, -q)

=

a 211 , Co = 1. Since

(-l)rA~~n) cosh 2rz,

r=O "

it is a real function if z is real. The imaginary part of (I) must then be omitted. This remark applies also to (3)-(5) below. The phaserange for formulae in this section is -117 < phase z < tlT. Under the conditions stated in § 11.22 Ce2 ,t(z, -q) ~ P~n 12n(k1 r), (2) and the part in [ ] in (1) degenerates to the series which occur in the asymptotic formula for 12n(k1 r). For the other functions we obtain Cell n +1(z, -q) "" the

C

~~Ml_ [ev.f crv-r+ie-

V

(21711)1

r=O

being those in § 11.20 with a Se 2n +1(z, -q)

t'tJ

Sellll+ll(z, -q) ""

_(P2~n+11 [as (,l7V)1"

=

! (-I)r r

b2n +1 •

at (3) but a

8~n+llJas at (1)

(21TV)lr

(b2n + 1 ) , (3)

C V-r]

r=O

but a

=

a lllt+1]

(a2n+1)'

(4)

=

blln+ll]

(blln+lI ) .

(5)

Each of the four functions above tends to

+00 monotonically as

Z-)- +00.

~"'ey 2n{z, -q) Ji"eY2n+l(Z, -q)

1"..1

1"..1

-2P~n i [as at (3) but a = ( lTV)lr

a lln]

(alln),

-(~2~n+\i [as at (1) , lTV)·

b2n +1]

(b2n +1 )' (7)

1

but a

=

(6)

GeY2n+l(z, -q) ""

&~:)~i [as at (1) but a =

a 2n +1]

(a2n +1)' (8)

GeY2n+2(Z, -q) ""

(~;;)l i [as at (3) but a =

blln+lI]

(b2n +2 ) . (9)

Each of the four functions in (6)-(~) tends to +ioo monotonically as z ~ -t-oo. Applying the relationship (10) §8.30 to the expansions (1), (6) above, yields , 00 Fek 2n (z, -q)

t'tJ

Pin

(21TV)i

e-1' L., ~ (-I)rcr v- r r=O

(aIn ) ,

(10)

the c being those in § 11.20 with a = a 2,,, Co = 1. The formulae for Fek 2n +1(z, -q), Gek 2n +1(Z, -q), Gek l n +2(Z, -q) are obtained from

225

ASYMPTOTIC FORMULAE

11.23]

(10) on replacing P~n by 8~n+l' P~n+l' 8~lt+2' respectively, and using the c corresponding to the characteristic numbers b2n +1 , a 2n +1 , b2n +2 • These functions tend to zero monotonically as z ~ +00. When the conditions in § 11.22 are satisfied,

_1_. e-ll ~ (-l)rcrv-r ~ !K2n(k1 r), (21TV)

6,

(11)

'IT

i.e. l/Tr times the asymptotic expansion of the K -Bessel function. The degenerate forms of the four asymptotic expansions above are then constant multiples of each other. Formulae for Meg>,(2)(z, -q), Ne(1),(2)(z, -q) may be derived from (10) and kindred formulae, by aid of the relationships in § 13.41.

11.30. Alternative asymptotic expansions in z, with argument 2k cosh z. Previously the argument has been v == ke', but it may sometimes be expedient to use u == 2k cosh z, so we shall develop the appropriate asymptotic series. The/unctions Cem(z,q), FeYm(z,q), q (1) § 11.20 and we get

> o.

Let x = -2ikcoshz in

(x 2 + 4k 2)y " -t-xy' _(X 2+ p 2)y == 0,

where

p2

=

(a+2k 2 ) . (x

with

pi ==

Writing y == toe:" in (I) transforms it to

2

2+

4k )w" - (2x 2 - x + 8k 2 )w ' - (x-pf}w

(2k2 -

(1)

=

0,

(2)

a ). As in § 11.20 we now assume that 00

w ==

!

(-I)rdrx- r- l ,

(3)

r=O

with do == 1. Inserting (3) in {2} and equating coefficients of like powers of x to zero, we obtain the relations [52] (4) d1 === !(p¥+!),

d2 =

2(r+ l)d r ~.l

1(pf+!)(pf+i):-k2,

=== [p~+1(2r+ 1 )2]dr-

(5)

+

4k 2(2r -

l )dr_1 +k2 (2r - l )(2r - 3)d' _2 (r ~ 2). (6) By expressing (4), (5) in a slightly different way, they are readily compared with (7) §11.20. Thus (4a-~ 12-Rk9 ) (4a-1 2 ) (7) d1 = == +k2 , 8

d 4161

2

=

(4a-1

8

2)(4a-32

) _1k 2(a +

~

2! 8 2

og

a- k 2) . 4

(8)

[Chap. XI

ASYMPTOTIC FORMULAE

226

T II

Writing

=

=

I-d2 u - 2+d. u- 4 - ••• , dlU-l-dau-3+d6U-s- ...,

(9)

(10)

by aid of§§11.10, 11.21 we obtaifi

Cean(z, g) ""'" Pan (:U)![T~':lsin(U+!1T)-

U~':l cOS(U+!1T)] (a2n ) ,

Cean+1(z, g) !"'oJ

(11)

-Pan+1 (:U)l [T~':l+1 cos(u+ !1T)+ U~':l+1sin(u+l1T)] (a2n +1 ) ,

(12)

FeYan(z, g) ""'" -pan(:U)![aS at (12) but with 2n for 2n+l] (a2n ) ,

Feyan +1 (z,g)

""'" -Pan+1(:U)![as at (11) but with

(13)

2n+l for 2n] (a2n +1 ) .

(14)

The phase range of (11)-(14) is -17T < phase z < 17T. When k ~ 0, the parts (2/7TU)1[ ] in (11)-(14) degenerate to the asymptotic expansions for the Bessel functions J2n (u ), J2n +1(U ) , Y2n (u ), l'in+l(u), respectively (see § 11.22).

11.31. Formulae for q < o. To obtain these use definitions (1), (11), etc. §8.30 in (11)-(14) § 11.30. Then u = 2kcoshz becomes 2iksinhz, and the required expansions are found if in (1), (3), (6), (7) § 11.23 we write 2k sinh Z for v and d, for Cr. 11.32. The functions Sem(z, q), GeYm(z, q), q > the differential equation for x(z) y(z)

=

=

o.

First we derive

y(z)/sinhz by writing

X(z)sinhz

in (1) §11.20. It transforms to d2X dX dZI+2cothz dz + (4k2cosh2z+ l - p l)X = 0,

(I)

where pI = a+2kl • The solution of (1) is, therefore, y/sinhz, where y may be Sem(z,q). GeYm(z,q). Now write x = -2ikcoshz and (1) becomes [52] (xa 4ka)dax.2 3x dx. _ (p 2+ x a- l )x. = O. (2) dx dx If in (2) we substitute X = we-x, we obtain , (x 2 + 4k l)w"-(2x2-3x+8k2)w'-(3x-pf-l)w = 0, (3) l with pI = (2k - a).

+

+

11.32]

ASYMPTOTIC FORMULAE

227

To solve (3) assume that ClO

W

=

I

(-l)"e,x-,.-t.

(4)

r-O

Inserting (4) into (3) and equating coefficients of like powers of x to zero, we obtain [52]

e1 = !(pf+i),

(5) l

e2 = !(pf+i)(pf+!)-3k 2(r+ l)e'+l

(6)

,

= [(pf+ 1)+!(2r+3)(2r-l)Je,.-

-4k2 (2r + l )e"_1+ k2(4r 2 - 1)e,_s (r ~ 2). Then using the procedure in § 11.20 with u = 2kcoshz,

V = l-e 2 u - 2+e4 u - 4 -

(8)

...

W = etu-l-esu-3+e5u-5- ..., we obtain the asymptotic expansions and

Se211+1(z, q) '"

- 8211 +1

tanhz(~)l

(7)

(9)

X

X [V~~+l COS(U+!1T)+ W~';l+l

sin(u+!1T)] (b2n +1), (10)

Se211 +2(z, q) '" 8 211 +2tanh z (~) I X X [V~~+2 sin(u+t1T) - W~~+2 cos(u+t1T) ]

GeY211+1(z, q) ,...., -8211+1 tanhz(~)l

(11)

(b2n +1),

(12)

(b2n +2 ) .

(13)

X

X [as at (II) but 2n+1 for 2n+2]

GeY2n+2(z,q) '"

(b2n +s),

-82n+2tanhz(~)1X X [as at (10) but 2n+2 for 2n+l]

The phase range of (10)-(13) is -111 < phase z < 111. When k ~ 0, the parts (2/11U)l[ ] in (10)-(13) degenerate to the asymptotic expansions for the Bessel functions J2n +1(U ), J2n +2(U ) , ~n+l(U), ~n+2(u), respectively (see § 11.22).

11.33. Formulae for q < o. Write 2k sinh z for v, e; for c, in (4), (5), (8), (9) § 11.23 and multiply by cothz. The expansions 80 obtained are identical with those found by applying the definitions (21), (31), (24), (34) § 8.30, respectively, to (10)-(13) § 11.32.

228

ASYMPTOTIC FORMULAE

[Chap. Xl

11.34. Formulae for Fekm(z, -q), Gekm(z, -q). These may be obtained from the asymptotic expansions for Cem(z, -q), FeYm(z, -q), Sem(z, -q), GeYm(z, -q) and application of formulae (10), (20), (30), (40) § 8.30. 11.40. Asymptotic formulae for cem(z, q), sem(z, q) when q is large and positive [52]. From (12) § 12.21 we see that when q is very large and positive, a ~ -2q+O(ql), so we shall assume that a = -2q+~ql+~o+~1 q-l+~2q-l+~3q-t+ ..., (1) where ex, ~o' ~1'''' are constants dependent upon the function and its order. Putting this series for a in the standard Mathieu equation for cem , sem gives y"--(4qcos2z)y+(exql+Cto+a:lQ-l+CX2q-l+ ... )y == O. (2) To solve this equation we assume that [73, 101] Y == eq lx, (z)[1+q-1fl (z)+q-- ~2(Z) +...]. (3) Substituting (3) into (2) and equating coefficients of q, qi, qO, q-l, s" to zero yields the equations [X']2 == 4 cos2z, (4) 2X"'+(cx+X")'

==

0,

(5)

'''+(cxo+2X'!{)'

=

0,

(6)

= f2'''+2f~''+(f2''+2x'f~+cxof2+Cilfl+(X2)' ==

0,

(7)

O.

(8)

fl'''+2!{''+(!;+2x'f~+cxofl+Cil)'

Solving (4), (5) we obtain X = ±2sinz+a constant } and , = a constant/(cosz)1{tan(!z+11T)}±~/4. Inserting (9) into (3) leads to the first approximations: Yl = e2k8inz/(cosz)l{tan(!z+11T)}«/4 and

Y" = e-2k8inz{tan(lz+11T)}ct/4/(cosz)1.

(9)

(10) (11)

These are formal approximate solutions of y"+(a-2qcos2z)y = 0, when q is large and positive. A first approximation to (1) is a+2q ~ cxqi, (12) and from § 12.30 a+2q ~ 2(2p+l)ql, (13) where p = 0 for ceo, se l ; P

== m for

cern, sena+1 • Thus we take ex = 2(2p+l).

(14)

11.40]

ASYMPTOTIC FORMUI"AE

229

If in addition we use the formulae

= [tan(lz+!1T)]l =

[tan(!z+!1T)]-l

21cos(lz+11T)/cos1z,

(15)

21 sin(~z+!1T)/coslz,

(16)

(10), (11) may be expressed in the form

and Writing

Yl

=

2P+le2kslnz[oOS(!Z+!1T)]2P+l/(COSZ)P+l

(17)

Y2

=

2P+le-2kSlnZ[sin(!z+!1T)]2P+l/(cosz)P+l.

(18)

-z for z in (18) gives (17), and vice versa.

11.41. Construction of asymptotic formulae for ce m, sem' q large and positive [52]. First we remark that (17) § 11.40 has poles at z = ...-~1T, -11T, !1T, ~1T, ..., and in general at z = (4r+3)!1T, r == -00 to +00. Also (18) § 11.40 has poles at z == l1T(2r+ 1). Since cen p sem are bounded functions, the variable z JlIUSt be restricted to exclude the singularities of (17), (18) § 11.40. Secondly, using the two latter formulae, the solutions of Mathieu's equation may be expressed by Y = EYl±8Y2' where E, 8 are appropriate constants: for ceo, se l , P = 0, so bearing in mind the interchangeability of (17), (18) § 11.40, we choose € = 8 == 0 0 2- 1. Thus we obtain

0o[e2k sinz cos(1z +11T)± e-2k sin z sin(}z + I 1T)]/ cos z,

(1)

these being valid in -11T < Z < t1T and !1T < z < j1T, thereby avoiding the singularities mentioned above. We have now to discriminate between these solutions to ascertain which represents ceo and which se1 • With the upper/lower sign (1) is even/odd ill -111' < z < 1-11', and with the lower/upper sign it is even/odd in 111' < Z < in. Hence we have deduced that when q > 0 is large enough, ceo(z, q) "" Core2k slne COS(!Z

se l

the open interval to se l . Also

+11T) +e-2k sinz sin( lz+11T)]/cos z,

-111' < z < !1T pertaining to ceo and in < z < I'"

ceo(z, q) "" Oo[ e2ksinz 008(lz+ llT)-e-2kslna sin(lz+ 11T)]/cos Z, se l the open interval

(2)

in < Z < 111' for

ceo and

(3)

-t1T < Z < t'" for sel .

ASYMPTOTIC FORMULAE

230

[Chap. XI

An argument similar to that above enables the following to be derived: cem (z, q) in sem +1

-l1T < Z < I1T}

'" ~m {e2k Sln e[ cos(lz+ t 1T)]2m+l±e-2kslnz[sin(lz+ t1T)]2m+1}fcosm+1z.

m+l

(4)

For the interval 11T < Z < i1T, alter the centre signs to =F. The constants 0, S are determined in § 11.42. q large and negative. The formulae for this case may be derived from (2)-(4) by applying the relationships in §2.18 and altering the range of z accordingly.

11.42. Formulae for Cem(z, q), Sem +1 (z, q), q large and positive. If the method given in §§ 11.40, 11.41 is applied to (1) § 2.30, use being made of the fact that Ce is even and Se odd, the results obtained are those in § 11.41 with zi written for z. Thus Cem(z, q) 210m [ e2iksinhZ{cosh !z-i sinh Iz)2m+l+ +e-2ikslnhz(cosh !z+i sinh iz)2m+l]/(2 coshz)m+l. (1) Putting a = cosh !z, b = sinh lz, 8 = tan- 1(bja), (1) becomes f'.I

2

+2

Ce (z, q) '" [l(a b )]m+iOm {ei l2k slnhe-(2m+ll8J+ e-i l2k sinh e-(2m +1)8)}, (2) m (coshz)m+l

0m hi cos[2ksinhz-(2m+l)tan-1(tanhlz)]. cos z In the same way we find that = 21 m-

Sem+1(z,q) '" 2

~m+1 hi sin[2ksinhz-(2m+l)tan-1(tanh!z)]. cos z

m-

(3)

(4)

The phase range of (3), (4) is -}1T < phase z < t1T. For z imaginary, formulae in §§ 11.41, 11.43 may be used. When z ~ +00, cosh z ~ v/2k, 2k sinh z ~ v, tanh lz ~ 1, so that (3) degenerates to

;:!: (~)l

cos(v-t1T-lm1T).

(5)

Ce2n (z, q) '" Olln(-l)n~::~:(~)!Sin(v+Vr)'

(6)

Cem(z, q) '" Om

Thus with m = 2n, 2n+ 1 we get

Ce2n +1(z, q)

,

I-....J

(1Tk)l ( 2

02n+l( _I)n+l 22n+

l

)! cos(v+l1T)·

1TV

(7)

ASYMPTOTIC FORMULAE

11.42]

231

From (4) we have Sem+1(z,q) -- 8m+1 ~:!: (:V)lsin(V-i'r-Im'll')'

(8)

so

Sesn+1(z,q) --8Iln+1(-1)n+1~::~:(:v)lcos(v+i'r)

(9)

and

Selln+ll(z,q) -- 81ln+1l(-1)n+l ~::l:(:V)lsin(v+l'11').

(10)

The expressions for the 0 and S may now be written down on comparison with (2), (4), (5), (7) § 11.10, since as z ~ +00 the corresponding formulae tend to equality, Thus we get [52]

=

(-I)n22n-1ce2n(0)ce2n(!1T)/A~2n)(1Tk)1,

(11)

C2n+1 = {_I)n+122n+1 ce2n+l(0)ce~n+l(11T)/kAi2n+I)(1Tk)1,

(12)

0211,

S2n+l

=

8 2n+2 =

(-1)11,2211,-1 Se~n+l(0)Se2n+l(t1T)/kBi2n+I){1Tk)·,

t

(-1 )n+122n+lse~n+2(O)Se~n+2( 1T )/k 2B~n+2)(1Tk)1.

(13) (14)

q large and negative. The formulae for this case may be derived by applying (1), (11); (21), (31) § 8.30 to (3); (4) above, and taking the real part, when z is real.

11.43. Higher approximations [52]. (6) § 11.40 may be expressed in the form /; =

-[cxo+'"/'J/2X'.

(1)

Substituting from (9) § 11.40 into (I) and taking m1 = (2m+l) gives II(z)

=

=F(al!)[{[(mf+3)sinz=f4ml]/cos2z}+

+(mf+ 1+ 8cxo)logetan(lz+i'r)].

(2) The additive constant of integration has been omitted, since it does not affect the ultimate result. Then the second approximations are cem (s, q) in sem +1

-11T < Z < !1T} ~

K m[Yl(I + _2/ 1Q- 1)± Ya(1+ 2/ I Q- I)]/2fn +1• (3)

For the interval t1T < Z < f1T, alter the centre signs in (3) to =f. YI' Y2 are, respectively, (17), (18) § 11.40, while -2/1' 211 represent /1 in (2) with two negative or two positive signs. Since cem, Bem+ l are periodic, (3) must not alter, save in sign, when -z is written for z. Hence the logarithmic term must vanish, so CXo

=

-(mf+1)J8

=

-(2m2+2m+l)J4.

(4)

232

ASYMPT01'IC

~"OR~lULAE

[Chap. XI

Thus the second approximations for q large and positive are cem (z,q) in se m +1 "'"

0.

m

«:

-}1T < Z < ~1T}

..,

{e2kSlnZ[cos(!z+ 11T)]2m+1 X X [1 +{(2m+ 1)-(m2 + m + l)sin z}/8k

cos2z]±

±e-2kslnz[sin(!z+ !1T)]2m+l X X

[1 +{(21n+ 1)+ (m2 + m + 1)sin z}/8k cOs 2z]} /cos 'U+1z.

(5)

For the interval j-r < Z < !1T, alter the centre signs to =F. Remembering that the log term in (2) is zero, we see thatfl(z) has singularities at z = t1T(2r+ I), r any integer, and these are common to (5). Thus the formula is invalid in the neighbourhood of any of these points, so (5) must be restricted to the condition that k cos2z is large enough.

11.44. Asymptotic expansion for a [52, 92, 93]. If in § 11.43 the analysis is extended up to 16' and at each stage the logarithmic term is made to vanish by equating its coefficient to zero, the values of the ~ in (1) § 11.40 are obtained in terms of m1 and q. Then we get the following asymptotic expansion: a "-' -2q+2m l ql-(mf+ 1)2-3_(mf+3ml)q-12-7- (5mf+ 34m¥+ 9)q- 12-12 - (33mt+ 410mf+ 405m 1)q- 12-17 - (63mt+ 126Omf+ 2943mf+486)q-22-20-

- (527mI

+ 15617mf+ 69001mf+41607ml)q-12-2&- ... ,

(I)

with m l == (2p+l). P == 0 gives a o, bl ; p . = 1 gives aI' b2; and in general p = m gives am' bm +1 • Since all the characteristic curves am+fJ lying between am' bm +1 are mutually asymptotic when q is large enough, (I) gives am+p for p == m, This will be clear from Figs. 8, II. In references [92, 93] formula (1) is derived by a procedure different from that outlined above. Both methods involve some very heavy algebra. When q ~ +00, (I) may be written

a 2q+ 2(2p+ 1)ql, (2) so the form in § 12.30 is reproduced. For p moderate, (1) gives accurate results for comparatively small values of q. For an assigned q, the accuracy decreases with increase 'in p. Thus to maintain accuracy to a definite number of decimal "-I

-

ASYMPTOTIC FORMULAE

11.44]

233

places (or units in the sth significant figure, whichever is preferred) q must increase with increase in p. We may say that (1) gives adequate accuracy, provided pl/q is not too large. To illustrate the order of accuracy attained for various q, Table 20 is appended. When q > 40, formula (1) with p = 0, 1 gives a correct to the fifth decimal place at least. The accurate values were obtained from reference [52]. Additional information regarding the accuracy of (1) is given on p. 299 reference [93]. TABLE

Values oj

--

q 8 16 40

20. Data illu8trating accuracy of formula (1)

-ao, -b l

Accurate values

from (1)

-at

10-60604: 24-25868 67·64216

10-60672 24-25868 67-64216

-b l

-a 1 , - b l from (1)

0-41144: 10-60536 24-25864 9-33456 67-64216 I 43·35228

Accurate ooluu

--

-al

-b l

0-43596 9-35268 43-35228

0-38936 9-33412 43-35228

In reference [56] asymptotic formulae are deduced for the difference between two characteristic numbers in a stable region of the (a, q) plane. There are also asymptotic formulae for a when the curves anI' bm are approached from an unstable region.

-1961

ub

XII ZEROS OF THE MATHIEU AND MODIFIED MATHIEU FUNCTIONS OF INTEGRAL ORDER 12.10. Real zeros of cem(z, q), sem(z, q). If q is fixed and m ~ I, these functions vanish for certain real values of z, Consider the case q = 0 when the functions reduce to cos mz, sin mz, respectively. In the interval 0 ~ z < 1T, the graphs of cosmz, sinmz cross the z-axis m times, 80 each function has m real zeros in this interval. The case m = 3 is illustrated for cos3z in Fig. 18A, there being three zeros in 0 ~ z < tt, We shall now demonstrate that the number of zeros of cea(z, q) in q ~ 0 remains constant [87]. Let q be increased from zero, and suppose that a fourth zero appeared. Its genesis would entail the curve of Fig. 18 A bending towards the z-axis as shown by the broken line. From the theory of equations, the existence of a minimum value above the s-axis would entail the occurrence of two conjugate complex zeros of the type Zl = ex+ifJ, Z2 = ex-ifJ. Further increase in q would be accompanied by the minimum approaching and ultimately being tangential to the z-axis, thereby introducing a double zero, i.e. z = ex bis (Fig. 18B). For a greater q the minimum would occur below the axis, thereby entailing two different simple real zeros (Fig. 18 c). If z = a were a double zero, we could write cea(z, q) Then

=

=

say.

(J )

y'(z) = 2(z-a)f(z)+(z-a)2f'(z),

(2)

(z-ex)2f(z)

y(z),

= O. (3) Now y"+(a-2qcos 2z)y = 0, and y(cx) = 0, by hypothesis, so y"(cx) = o. Hence y(a) = y'(a) = y"(cx) = 0, so y(z) must be a null so

y'(ex)

=

o. But ce3(z, q) is not a null function, so a double function, i.e. y(z) zero cannot occur. A similar argument may be used to show that the number of zeros cannot decrease. It follows that the number of zeros of ooa(z, q) in 0 ~ z < TT is independent of q in q ~ o. In general a discussion on the above basis leads to the conclusion that cem , sem have m simple zeros in 0 ~ Z < TT. senl has a simple zero at the origin, 80 it has (m-l) zeros in 0 < z < 1T. When q = 0 the zeros are equally spaced, Le, for the circular functions, but, as shown later, they tend to cluster about z = iTT as q ~ +00, excepting that

12.10] ZEROS OF MATHIEU FUNCTIONS OF INTEGRAL ORDER

23Li

of sem at the origin. This discussion shows that as the number of zeros in a given interval is independent of q, the graphs of cem' sem are distorted versions of those of cos me, sin mz (see Figs. 1-4).

(A)

(c)

FIG.

18. Illustrating that the number of zeros of a Mathieu function (ce, sa) in a given interval is independent of q.

12.20. An inequality for a, q. We obtain this by aid of Sturm's first comparison theorem which is used for comparing Mathieu's equation with two forms derived therefrom. First we remark that integral-order solutions have rm zeros in the interval 0 ~ z < r'IT, r being a positive integer> 0, e.g. ce2 , seahave 6, 9 zeros, respectively, in 0 ~ z < 31T. Secondly, the first maximum of (a- 2q cos 2z) occurs when z =- t1T and has the value (a+2q), so we consider the equation y"+(a+2q)y = O. (1) Its formal solutions are cosz(a+2q)1, sinz(a+2q)i, and in the interval o ~ z < r1T both of these functions will have at least rm zeros, provided that (a+2q)l

>

(m- ;r)'

(2)

By Sturm's first comparison theorem the solutions of Mathieu's equation cannot have more zeros in 0 ~ Z < 111' than those in each solution of (1). Hence by (2), if cem, sem each have rm zeros, it follows that

a>

(m-;rr

-2q.

(3)

236

ZEROS OF THE MATHIEU AND MODIFIED

[Chap. XII

But r is arbitrary, so we can make it tend to +00, thereby obtaining the inequality (4) a ~ m 2-2q, q ~ 0, the equality sign being for~ q = o. Thirdly, the first minimum of (a-2qcos2z) occurs when z = 0 and has the value (a-2q). Proceeding as before, we consider the comparison equation y"+(a-2q)y = 0, (5) whose formal solutions are cosz(a-2q)1, sinz(a-2q)1. In the interval o ~ z < r'TT, the number of zeros will not exceed rm, provided that

(a-2q)l < (m+ :r)'

(6)

By the comparison theorem the solutions cem, sem each have as many zeros as (but no more than) the solutions of (5) in 0 ~ z < r'TT. Hence

(m+:rf a < (m+ :rf+2q.

(4-2q)

or As before, let r

~

<

(7) (8)

+00, and we obtain the inequality

a ~ m 2 + 2q, (9) q ~ o. Combining the inequalities (4), (9) we find that the characteristic numbers for cem, sem satisfy m 2-2q ~ a ~ m 2+2q, (10) for q ~ 0, the equality signs pertaining to q = 0. If q < 0, we have m 2 - 2q ~ a ~ m 2 +2q. The analysis herein is based upon reference [87].

(11)

12.21. Form of a when q is large. Here and in the succeeding section it is expedient to use the equation y"+(a+2qcos2z)y = 0, where q

(1)

> o. From the expansion of cos 2z we get the inequality

1-2z2 ~ cos 2z ~ 1-2z2+2z4 /3. (2) If in (1) we replace cos 2z by the first and third members of (2), we obtain the two comparison equations [91] and

y"+(a+2q-4qz2)y = 0 y"+(a+2q-4qz2+kz4)y = O.

(3) (4)

12.21]

MATHIEU FUNCTIONS OF INTEGRAL ORDER

237

Then by Sturm's first comparison theorem the odd or even solution of (1) has at least as many zeros, in a finite interval, as that of (3), but no more than that of (4). Writing z = xJ2ql in (3), (4) yields

d2y+[(a+2q)_~X2]y=0 2 dx

and

4ql

(5)

4

:~+[(a~~)_~X2+4~.]Y = O.

(6)

Now the interval of z to be considered is 0 ~ z < sr, and since the corresponding range of x is 0 ~ x < 2q1'1r, x can be made as large as we please by suitable choice of q. Substituting l+! for (a+2q)/4ql in (5) leads to the equation for the parabolic cylinder functions Dz(x), i.e.

:~+[(l+!)-lx2]y =

O.

Using the notation of (5), it is known [140] that with integer, if

(7) 8

a positive

a+2q

28-1 < 4ql ::::;; 28+1, the odd solutions of (7) have 8 zeros in 0

(8)

< x < +00, while if

28-1 < a+2q 4ql ::::;; 28+1,

(9)

its even solutions have 8 zeros in the same interval. Thus for any solution of (1) having a finite number of zeros in 0 ~ Z < 'Ir-and this includes periodic and non-periodic classes of solution-it follows that, as q ~ +00,

a~~ =

0(1),

(10)

i.e. it is bounded. It can also be shown that as q ~ -00, a-2q = 0(1) (11) 41qll · Finally, therefore, we obtain the form of a for Iqllarge, namely [91], a = -2\ql+Olqll.

(12)

12.22. Distribution of zeros as q ~ 00 [91]. When q ~ x in (5), (6) § 12.21 the term in x4 in the latter may be ignored. Then the two equations are the same, so their respective solutions have an equal number of zeros in a given interval of x. Now z = xf2ql, 80 when

238

ZEROS OF THE MATHIEU AND MODIFIED

[Chap. XII

X is finite and q large enough, the solutions of (3), (4) § 12.21 have an equal number of zeros in a given interval. Also the solutions of (3) § 12.21 are non-alterneting'[ if ...

(a+2q-4qz2)

i.e. if

Z2

But from (12) § 12.21, with

<

0,

> (a+2q)/4q. q > 0 in (1) § 12.21,

(1)

a = -2q+O(ql), so (1) takes the form Z2

> O(ql)j4q = A2q- l,

(2)

A being bounded and positive. Thus in the small interval -Aq-l

Aq-t,

a+2q-4qz2

>

0,

both solutions of (3) § 12.21 are alternating functions.j Moreover, they have the same number of zeros in the above interval. Hence, 8S q ~ +00, any solution of (1) § 12.21 has a finite number of zeros in the interval (-Aq-l,Aq-l). Since a = -2q+O(ql), we get 80

a+2qcos2z

=

2q(cos2z-1)+O(ql),

(3)

and in the two intervals -1T+Aq-t < z < -Aq-t, Aq-l < z < 7T-Aq-l, neither of which includes the origin, (3) is negative. It follows that the solutions in these intervals do not alternate, and no more than one zero can occur in each. The above argument rests upon the hypothesis that q is large. Accordingly as q ~ +00 the interval (-,\q-l,Aq-l) ~ 0, and the zeros of (1) § 12.21 in the interval -111' ~ z < 111' congregate or condense on each side of the origin. To obtain the result of letting q ~ +00 for cem(z, q), sem(z, q), we remark that yH + (a+ 2q cos 2z}y = 0 takes the canonical form lIH+(a-2qcos 2z)y = 0 if (}IT+Z) be written for z. Doing this with the above interval, we conclude that as q -+ +00 the zeros of cem(z, q), sem(z,q) in 0 ~ z < 11' condense about 111, with the exception of one, namely, the zero of sem(z, q) at the origin. The interval of condensation is (llT-Aq-l,llT+Aq-l). The condensation phenomenon is illustrated by the curves in Figs. 19 A, B, plotted from data in reference [96]. They are asymptotic to the line 8 = 90° = t~. The function ceo(z,q) has double zeros at z = 0,11',211', .•• as q ~ +00, i.e. the graph is tangential to the s-axia (see Fig. 3, ceo(z, 8». It may be remarked that in 0 < z < 111', ce2n(z, q), ce2n +1(z, q), se2n +1(z, q), se 2n +2(z, q) each

t

Oscillatory but devoid of real zeros, i.e. J(z) i:- o. : Oscillatory but having real zeros.

12.22]

MATHIEtJ FUNCTIONS OF INTEGRAL ORDER

239

have n simple zeros. For this reason sesn+s(z, q) is used in preference to sesn(z,q) which has only (n-I) zeros in the interval.

(A)

e ~

N

t.s 65° ~

go> <{

..

~ 4So~------II..o.-_---o 20 40 Values

off

(D) «n L. :J U

g GO° ot; ~

N

...c

i

u

40°

..L.)

~ ~

~

-c tOO~-----,"---II

~

FIG.

ZO Values

off

--

+0

19. Illustrating condensation of zeros at

i1T:

(A) for ce,(z, q),

(B) C6s(Z, q), se.(z, q) 88 q ~ +00. The curves are asymptotic to

8=

!1T.

ce l , see each have two zeros in 0

< z < 1'77' (see Figs. 1,4).

12.30. Approximate asymptotic formulae for a [91]. To obtain these we have to determine a value for O(ql) in (12) § 12.21. Consider ce2U(z, -q), ce2n +1(z, -q), the even solutions of integral order of y"+(a+2qcos2z)y = o. If q > 0 is large enough, by (9) § 12.21 we may write a+2q 4ql 28+}, (1) /"OW

the asymptotic sign referring to q. Transposing (1) and writing n for 8 gives the desired asymptotic formulae, namely, a2U( -q) "" -2q+(8n+2)ql, (2) a2'~+1(-q) '" -2q+(8n+2)ql.

(3)

240

ZEROS OF THE MATHIEU AND MODIFIED [Chap. XII

If q > 0 is large enough, se2n +1(z, -q), se2n +2(z, -q) each have (2n+ 1) zeros in the neighbourhood of the origin, the additional zero of se 2n -HI occurring at z = !11. Since sem(z, -q) is odd in z, by (8) § 12.21 we get "

a+2q /"OJ 4ql

28+1.

(4)

This leads to the asymptotic formulae

b2n +1(- q) I'-....J -2q+(8n+6)ql,

(5)

-2q+(8n+6)ql. Formulae (2), (3), (5), (6) apply to the equation y"+(a-2qcos2z)y = 0 when q < o. For q positive it can be deduced that a 2n (q) ~ -2q+(81t+2)q~, b2n +2(- q)

(6)

f'J

(7)

a2n +1(q) ~ -2q+(8n+6)ql,

(8)

b2n +1(q) ~ -2q+(8n+2)ql, b2n +2(q) ro-J -2q+(8n+6)ql.

(9) (10)

From the preceding formulae we have the relationships, asymptotic in q (see Fig. 8),

a2n ( -q)

f'"I

a2n +1 ( -q),

a1n(q)

I'-...J

b2n +1(q),

b2n +1( -q)

~

b2n +2( -q);

a 2n +1(q) "-J b2n +2(q),

am(q) ~ bm +1(q);

or a 2n In+l

(-q)

I'-....J

(II)

a 2n (q),

b2n +1

(12)

b2n +1 ( -q)

"-I

a2n +1(q).

(13)

b2n +2

2n+2

In the above formulae n is assumed to be finite.

12.31. Accuracy of formulae in § 12.30. This is illustrated for several cases by the data in Table 21. TABLE -ao(q)

"""-J

-b1(q)

21 -- -

-a 2(q) """-J -ba(q)

Valuea oj q

Value Jr01tl (7) § 12.30

Accurate t'alue

Value from (7) § 12.30

Accurate value

80 160 1600

142·12... 294·71. .. 3120·00

142·3671..• 294·0453... 3120·2508...

70·60... 193·51... 2ROO'OO

73-9437 ... 196·8521... 2803·2779...

-----

For n = 0 in (7), (9) § 12.30, i.e. for ceo(z,q), se1(z,q), the discrepancy occurs in the first decimal place, whereas "pith n = 1, i.e, cea(z,q),

12.31]

MATHIEU FUNCTIONS OF INTEGRAL ORDER

241

sea(z, q), there is a difference of 3 in each of the digits. Thus for an assigned n the accuracy increases with increase in q, but for a given q it decreases with increase in n. The' accurate' values were obtained from ref. [52], p. 325.

12.40. Zeros of Cem(z, q), Sem(z, q), FeYm(z, q), GeYm(z, q), z large and positive. Reference to the approximate asymptotic formulae in § 11.10 shows that for any k > 0, each of these functions has an infinity of zeros. As z increases, the graphs of the functions oscillate about the z-axis with steadily decreasing amplitude. The zeros of Ce2n(z, q), Se 2n+2 (z, q), FeY2n+l(Z, q), GeY2n+l(Z, q) occur approximately when sin(v+!'7T) vanishes. Thus we have v+l7T ~ r'7T

z,

so

~

or

ke Z ~ (r-l)1T,

loge[(4r-I)1T/4kJ.

(1)

(2)

»

1 but integral. From (2) it follows that, Now v ~ I, so by (1) r to the order of approximation contemplated, the zeros in z are independent of the order of the function. By § 11.10 the zeros of Ce2n +1 (z, q), Se 2n +1 (z, q), FeY2n(z,q), GeY2n+2(Z, q) occur approximately when cos(v+i'7T) vanishes. Thus

z,

~

loge[(4r+ 1)1T/4k].

(3)

Parametric zeros, z large and positive. These are the values of q for which, with z assigned, the function vanishes. From (1) we get

qr =

k~ ~

(4r-l)2'7T 2e- 2zf 16,

(4)

which gives the parametric zeros of the four functions prior to (1). From (3) qr = k~ ~ (4r+ 1)21T2e-2z/16, (5) which gives the parametric zeros of the four functions just above (3). These formulae are independent of the order of the function.

As z ~ +00, the interval between successive zeros tends to vanish, so they condense in the neighbourhood of q = +0.

12.41. More accurate formulae for parametric zeros of Cem(z,q), Senl(z,q). These may be derived from (3), (4) § 11.42. From the former we get for Ce2n (z, q)

2k sinh z

~

(2r+ 1)1T/2+(4n+ l)tan-1 (tanh lz),

(1)

provided Ie is large enough. In problems on elliptical membranes and wave guides, cosh %0 = e- 1 at the bounding ellipse. Thus 4961 I i

242 ZEROS OF MATHIEU FUNCTIONS OF INTEGRAL ORDER [Chap. XII

ainhzo = e- 1(I - el )l, tanhizo = [(l-e)/(l+e)]l. Substituting these in (1) leads to Cean(z,q) ~ ~

{(r+l)1T+(4n+l)tan-1[(1- e)/ (1+ e)]1}2e2/4(I - e2). (2)

For Ce1n+1(Z,q) write (4n+3) for (4n+ 1) in (2). Similarly (4) § 11.42 leads to Sean+l(z, q) k~ ~ {(r+l)1T+(4n+l)tan- I[(1-e)/(1+e)]1}2 e2/4(1-e 2 ) .

(3)

For Se 2n +2(z,q) write (4n+3) for (4n+l) in (3). In these formulae r takes all integral values ~ I. The accuracy improves with increase in r, also as e ~ 1, i.e. as the ellipse becomes long and narrow, thereby increasing ki;

12.42. Large zeros of Ce:n(z, q), Se:n(z, q). By differentiating the formulae in § 11.42 we find that the large zeros of are in close agreement with those of Sem +l , and a similar remark applies to Se:n and Cern-I. Accordingly the large zeros of Ce~n(z, q) are given by (3) § 12.41, while those of Se~n+l(z,q) are given by (2) § 12.41, and 80 on.

ee:n

12.43. Large zeros of Jm(z). By Appendix I as e ~ 0, the ellipse tends to a circle, Cem(z,q)~P:nJm(kla), Sem(z,q)-+s:nJm(k1a): also klk = 2k, h = ae, so kla = 2ke-1• Thus (2), (3) §12.41 degenerate to the well-known formulae for the large zeros of the Bessel functions, namely, J1n :

k1a ~ (r+l)17'+(4n+I)1T/4 = (r+n+i)1T,

(I)

Jan+1 :

k1a ~ (r+l)1T+(4n+I)1TJ4 = (r+n+!)1T.

(2)

XIII

SOLUTIONS IN BESSEL FUNCTION PRODUCT SERIES 13.10. Method of solution [36]. The equation whose solutions we require is y"-(a-2qcosh2z)y = 0, (1) i.e. the modified Mathieu equation, which may be written in the form y"+[k2(e28+e- 2s )- a]y = 0,

=

with k 2

q =F WI

while

o.

=

WI

(2)

Now

J,.(ke-Z ) satisfies

= ~(kea) satisfies

w~

+ (k 2e- 2s -

w;+(k2ek _

= 0, p2)w s = 0, r 2 )w l

(3) (4)

r integral, p integral or fractional. Multiplying (3) by w 2 , (4) by and adding gives

wi wI+w; WI + [k2(e2z + e- 2z )-

r 2 _ p2]w l WI

=

WI'

0,

(5)

2w:w~.

(6)

which is equivalent to

::2(W1w2)+(2k2cosh2z-r2_p2)wl Writing y

=

wl

W z and

W2=

adding -ay to both sides, (6) becomes

:~+(2k2Cosh2Z-a)y= -2k2J~J~;+-(r2+p2-a)J,.~.

(7)

By aid of recurrence relations we obtain

2J~J; = -

2;: J,.~+J,.-1~-1+J,.+1~+1"

(8)

Substituting from (8) into the r.h.s. of (7) yields the equation

::~ +

(2kllcosh 2z-a)y =

[(r+p)2-a]J,.~-kll(J,.-1~-1+J,.+1~+1).

(9)

Writing p = r+v, assume that co

y =

I

(-I)rd2r~(ke-Z)J,.+.,(ke·)

(10)

1"=--00

is a solution of (1). If VI = ke:", co

I

r--QO

VI

=

ke', then we must have, by (9),

(-1)rcl~{(2r+v)S-a}J,.(vl)J,.+.,(vl)-

244

BESSEl.. FUNCTION PRODUCT SERIES

[Chap. XIII

In (11) the coefficient of J,.( Vl)~+V( v2 ) will vanish provided (12) ClO

Ce 2n +v {z,q) =

Now

I

A~~n+v)cosh(2r+v)z,

(13)

1'=-00

satisfies (1), and the recurrence formula for the A is identical in form with (12). Hence for the same a, q, if d2r = K2n+vA~~n-fv), K2n +v being a constant, it follows that when k ::f. 0, CX)

y

= K 2n+v

I

(-I)rA~~n-t-Jl)J;.(vl)~+v(v2)

(14)

1'=-00

is a solution of (1). Since (I) is unaltered if we change z to -z, then CX)

=

y

K2n+v

I

(-I)rA~~n t-Jl)~+v(Vl)~(V2)

(15)

1'=-CX)

is also a solution. Adding and subtracting (14), (15) yields the two series, even and odd in z, 00

K 2n+v

I

(-I)T A~~n+v)[~(Vl)Jr+v(V2)±J;.+v(Vl)J;.(V2)]'

(16)

r--CX)

both of which satisfy (1). By analysis similar to that in § 13.60, we find that these series are absolutely and uniformly convergent in any closed region of the z-plane, including z = o.

13.11. B.F. product series for Cem(z, q), Sem(z, q). Write p = v, and apply (2) §4.711 to (16) §13.10 as f3~0. Then the first series becomes (I)

while the second series vanishes. Now (1) represents an even function of z with period 'TTi, which is a solution of (I) § 13.10 for a = aln. Hence we infer that (1) is a constant multiple of Ce1n(z, q), to whioh (13) § 13.10 also reduces as v ~ O. We now determine KIn. As Z -)0 +00, all terms of (1) are negligible in comparison with the first, and this has the value (2)

when the dominant term of the asymptotic expansion for JO(v2 ) is used [202, p. 158, no. 13]. By (2) § 11.10, if K2n = P2,JA~2n), (2) is

245

BESSEL FUNCTION PRODUCT SERIES

13.11]

the dominant term in the asymptotic expansion of we obtain'] co

Ce2n(z, q) = (P2nI A &2n») I

r==O

Ce2t~(Z, ']).

'rhus

(-I)'A~~n)~(Vl)~(V2)'

(3)

When fJ ~ 1, by (4) §4.711, the first member of (16) § 13.10 vanishes, while the second becomes 00

K 2n+1 I (-1)'B&~ttill~(Vl)~+1(V2)-J;.+1(t'1)J;.(V2)] (b2n+1 ) . r-O

(4)

This series, odd in z with period 21Ti, is a solution of (1) § 13.10 with a = b2 1l,+1' so we infer it to be a constant multiple of 8e211o+1(Z, q). Then procedure akin to that above yields Se 2n +1 (e, q)

=

(S2n+l1 Bi2n+1 »

00

I

r=O

(-1)' B~~'+11)[J;.(Vl)J;.+1(V2)-J;.+1(Vl)~(V2)]. (5)

Writing v = fJ+I in (16) § 13.10, letting fJ ~ 0 and 1 in turn, using (6), (8) §4.711, and following the procedure already set out, leads to Ce2n +1(z, q)

=

i

(P2n+l/ A 2n+1»

and

00

I (-I)'A&~tt11)[J;.(Vl)J;.+I(V2)+J;.+l(Vl)~(V2)] (a2'~+I) r=O (6)

Se21t+2(z, q)

en

= -(S2n+2/ B&2n+2»

I (-1)'B&~+~2)[J;.(Vl)~+2(V2)-J;.+2(Vl)J;.(V2)] r=O

(b2n +2 ) . (7) Solutions of the type herein may be derived by means of integral relations, as shown in § 10.34 et seq.

13.12. Series for Ce2n+~(Z, q), Se2n+~(z, q). Ifin (16) § 13.10, v

= 13,

o < 13 < 1,

comparison of the first solution therein with the series for Ce2n+p(z, q) at (13) § 13.10 shows: (a) both are even in z; (b) any A in (16) § 13.10 is the same as the corresponding coefficient in (13) § 13.10; (c) iffJ = pis, a rational fraction in its lowest terms, both solutions have period 2S1Ti. (b), (c) apply also to the odd solution in (16) § 13.10 and Se2n+p(z,q) at (12) §4.71. Further, the solutions at (16) § 13.10 constitute a fundamental system, as also do Ce2n +p, Se2n+~. Hence we infer that they are constant multiples of each other, so we write

~2n

(z.g) = e .,. p

~Bn+p ~ (-l)rA~~n+p>[J,.(Vl)J,.;p(V.)±J,+p'(Vl)J,(V.)] 2n+fJ r=a -00

t See Appendix I for 1?,,'

(

•

')

azn+~

·

(1)

BESSEL FUNCTION PRODUCT SERIES

246

Let z = 0, and we get K 2n+fJ

[Chap. XIII

/2

= celln+p(O, q) r-~ ee(-1 r A~~n+fJ)J,(k)Jr+lJ(k).

(2)

v

Differentiating and putting z = 0 gives

12k _~ (-1 t A~~n+IJ>[J,(k )J;+p(k) -J,+p(k)J;(k)].

KlIn+p = se~n+p(O, q)

r-

(3)

00

The series for Ce2n+l+~' Se2n+l+~ are obtained fro III above by writing (1+,8) for fJ. Analysis similar to that in § 13.60 shows that the series (1) are absolutely and uniformly convergent for z finite.

13.20. Second solutions of (1) § 13.10: functions of integral order. Yp and Jp satisfy the same D.E. and recurrence relations of the same type. Since it is by virtue of these that the B.F. product series satisfy (1) § 13.10, it follows that a second linearly independent solution may be obtained on replacing J,n(v 2 ) by Ym(v 2 ) in the first solution in § 13.11. Thus, corresponding to (3) § 13.11, we get FeY:an(z,q)

=

(P2n/A~2n»

00

I

r-O

(-I)rA~~n)~(Vl)y;'(V2)'

(1)

This function is non-periodic owing to the term loge(keZ ) in the expansion of y;'{V2) [202, p. 161, no. 61]; for loge(kel:) = logek +z. Using the said expansion gives J,(ke--)y;'(ke-)

= ~ (,,+ log lk )J,(ke",

II)J,(

+

keel'

+~zJ,.(ke-4)J,(keZ)+I cosh terms-]- I sinh terms. (2)

'"

The arguments of the cosh, sinh terms are even multiples of z, and result from ,the relation e±2m11 = cosh 2mz±sinh 2mz. Applying (2) to (1) leads to the representation

FeYlIn(z,q)

= [~(,,+loglk)Ce2n(Z,q)+(P2n/A~lIn» I

+ [~z Cell..(z, q)+ (Pan/A~lIn» I =

cosh terms] + sinh terms]

(3)

even function of z, with period 1Ti+odd non-periodic (4) function of z. In Chapter VII it is shown that if one solution of (I) § 13.10 is even and periodio in z with period wi, the other is odd in z but non-periodic. A combination with two arbitrary constants constitutes a fundamental system. Aooordinglyon comparing (3) with the solutions in §§ 2.30,

BESSEL FUNCTION PRODUCT SERIES

13.20]

247

7.61 we infer that the first I is a multiple of Ce2n(z,q), ·while the odd function is a multiple of Fe2n(z, q). Thus we may write

FeYan(z,q) = A Ce 2n (z, q)+BFe2n(z, q), (5) where A, B are particular constants. A similar relation pertains to functions of order (2n+ I): also GeYm(z,q) = ASem(z,q)+BGem(z,q). (6) It appears, therefore, that Fey m(z, q), Gey m(z, q) include the first and second solutions, but none of these functions constitutes a fundamental system in itself. The A, B, are determined in § 13.21. Proceeding as heretofore we obtain: Fey2n+l (z, q)

=

(P2n+l/ Ai2n +1» X 00

X

I (-1)'A~~~11)[~(Vl)~+1(V2)+J;.+l(Vl)~{V2)]' (7) r=O

GeY2tl+l(Z, q) == (S2n+l! B~2n+l) X

00

X

I

r=-O

(-l)r B~~~~2)[J;.(Vl)~+2(V2)-J;.+2(vl)~(V2)]. (9)

Series for Fekm(z,q), Gek,n(z,q) may be derived from (1), (7)-(9) by aid of (4), (7), (10), (13) § 8.14.

z

13.21. Relationships between FeYm' Fe,n; GeYm' Gem. Writing 0 in (5) § 13.20, with m for 2n, we get (1) A = Fey 7n(O, q)/Cem(O, q)

==

since }'em(O, q) =

so when z

==

o.

0, Ce~J,

Also

Fey:n = A Ce:n+BFe:n; = 0 and B = }-'eY:n(O, q)/Fe;"'(O, q).

(2) (3)

Hence by (I), {3} above, and (5) § 13.20, we find that

FeYm(z,q) = Fey,~(O,qlcem(z,q)+ Fe~~(o,q)Fem(z,q). Cem(O, q) Fem(O, q) Similarly we obtain ( ) GeYm(O,q)G ( ) GeYm(z,q) = Gey:n(O,q)S Se;"(O,q) em z,q + Gem(O,q) em z,q · See § 13.31 respecting Fey:n(O,q), Gey".(O,q).

(4)

(5)

248

BESSEL FUNCTION PRODUCT SERIES

[Chap. XIII

By (4) § 8.14 we have FeY2n

= i Ce2n-2 Fek 2n = A Ce2n + B Fe2n •

(6) (7)

Hence

Fek 2n(z,q) = !(i-A)Ce2n(z,q)-!BFe2n(z,q). (8) A, B being given by (1), (3). It may be remarked that by §oS.I4, Fek 2n (z, q) is complex for z real. Relations of the type (S) may be found for Fek 2n + 1' Gek m•

13.30. Integral order solutions of y"-(a+2qcosh2z)y = o. Applying (I), (II), (21), (31) § 8.30 to (3), (5)-(7) § 13.11, each to each, we obtain the first solutions: Ce2n(z, -q)

= (P~n/A~2n»

00

~ (-I)'A&~n).z;.(Vl)J;.(V2)'

(I)

r~O

Se2n +l(z, -q)

Se2n+2(z, -q)

=

co

(8~n+2/B~n+2» ~ (-1)'B~~+t2)[1,(v 1)I,+2( v 2 ) - I,+2(vl)I,(v 2 ) ] . r==O

( 4)

The second solutions involving the K-Bessel function may be obtained from the series in §§ 13.20, 13.30 by applying (4), (10); (14), (20); (24), (30); (34), (40) §8.30 and (2) §S.14. They are as follow: Fek 2n(z, -q)

= (P~n/1TA~2n»

Gek 2n +2(Z, -q)

00

~ A~~n)I,(vl)K,(V2)'

(5)

r==O

co

= (8~n+2/1TB~2n+2» ~ B~~'+-t2)[ I,(VI)K,+2(V2)-Ir+2(Vl)Kr(V2)]. .. 1'-0 Absence of (-1)' in these series should be noted.

(8)

13.30]

BESSEJ.. FUNCTION PRODUCT SERIES

249

Series for FeYm(z, -q), GeYm(z, -q) may be derived from (1)-(8) using (10), (20), (30), (40) § 8.30. Relationships of the type in § 13.21 may be obtained also, e.g. Fek 2n(0, -q) C Fek~n(O, -q) F ( ) --1-------eZ,t(z, -q)+ , e Zn z, -q · Ce 2u (O, -q) Fe 2n (O, -q) (9) Fek:n(O, -q), Gekm(O, -q) are given in § 13.31: note that Fekzn(z, -q) =

Ce2 ,t(0, -q) === cez,,( 111', q). 13.31. Evaluation of FeY;n(O, q), GeYm(O, q), Fek~(O, -q), Gekm(O, -q). These are obtained from the series in §§ 13.20, 13.30, by aid of the following B.F. relations, with argument k:

J,. Y;-

J;y;. =

2/1Tlc

(the Wronskian relation), JrY;+l-J~Y;.+l+Jr+lY~-J~+ll:= 2(2r+l)j'TTk2, JrY;.+l-~+lY;.

(I) (2)

= -2/",k,

(3)

Jr Y;.+2- J;. +2y;. = -4(r+ 1)/rrk

2

,

(4)

IrK;-I;Kr = -Ilk (Wronskian for modified B.F.), IrK;+l+I;+lKr-Ir+lK~-I;Kr+l

= -(2r+I)/k

2

IrKr+l+Ir+1Kr = 11k, IrKr+2-Ir+2Kr = (2r+2)jk 2 •

,

(5)

(6) (7)

(8)

It will be seen that the r.h.s, of (5)-(8) are given by -111 times (1)-(4), each to each. For (I), (5) see [202, p. 156, ex. 60, 61J. The remainder may be derived therefrom by aid of recurrence relations. The results obtained are as follow:

FeY~n(O,q) = 2AP(~:)ce2n(!1T,q). Fey'

,

GeY2 +l(O,q) n

Gay

0

_

2P21t+lce~n+l(!'TT,q) 1TkA12n +1 ) '

(10)

= - ~~~~-t!_!~~_'.!--tl(l~~~l 7TkBi2n+l) ,

(11)

(0 q) =

2n+l

2n+2

(0 q) = _ 28~n+28e~!!-.t2(lrr,q) 1Tk 2lJ<;.2n.+ 2)

,

Fek2' (0 _q) = _1?~~_c~_~~(O,q) n

4961

(9)

11

1TA~2n)

,

Kk

,

,

(12) (13)

BESSEL FUNCTION PRODUCT SERIES

250

l?ek 2' +1(0 -q) = _ n'

Gek Gek

2n+l

~~.!'+1 S~i!t.±!~O, q) 1TkB~2n+l)

(0 _q) = ~n~1 Ce~n+l(~!.-~

1TkAi2n +1 )

,

[Chap. XIII

(14)

,

(15)

'

(0 _q) = ~;n+2 se~!'_~i~, q) 2n+2 , 1Tk2B~2n+2>·

(16)

13.40. Combination solutions. In problems on wave motion using elliptical coordinates, it is expedient to have solutions akin to the Hankel functions Hg),(2)(Z). The latter are used to represent incoming and outgoing waves in problems pertaining to circular cylinders. Accordingly for wave-motion problems involving elliptical cylinders we shall adopt the following definitions: Meg>,(2)(z,q) = Cem(z,q)±iFeYm(z,q} and

(am)'

(1)

Neg),(2)(z,q) = Sem(z,q)±iGeYm(z,q) (bm ) .

(2)

Then by aid of§§8.lo-S.13, 13.11, 13.20 we get

Me~U,(2l{z,q) = ~~2A\t'-~) o

i (-1)rA~~nlH~V,(2l{2kcoshz)

= ce~n(O,~2 6"" A(22rnlH(ll,(2l(2ksinhz) ~(2n) 2r o

Ne(1),(I)(z q) 2n+l

,

=

(3)

r-O

(4)

r-

se 2n +1 (! 1T, q)t anh z x kB(2n+l) 1

QO

X

I

(-1)r(2r+l)B~~'+11)H~~~f)(2kcoshz) r-O

= se;n+l (O, q) . kB(2n+l)

1

~ n(ln+tlH(ll,(2l(2k sinh z)

~

r-O

.LIir+l

2r+l

(9) (10)

13.40]

BESSEL FUNCTION PRODUCT SERIES

2ft}

(X)

X

I (-I)r(2r+2)B~~,+~2)H~Vf~)(2kcoshz) r:::cO

(12)

(X)

X

I (2r+2)B~~,+~2)H~~4<:)(2ksinhz) r==O

(13)

(X)

= - (S2n+2/ B~2nt 2»

I (-I)'B~~'t~2)[J;.(vl)H~~~2)(V2)raaO -J;.+2(Vl)H~l),(2)(V2)].

Series for q definitions:

< o.

Me~~,(2){z,

-q)

(14)

These may be obtained by aid of the following

=

(-I)nMe~~,(2)(!7Ti+z,q)

== Ce2n(z, -q)±i FeY2n(z, -q), Ne~~~~){z, -q) ==

(15)

(_l)n+l Ne~~1~)(~1Ti+z,q)

=

Se2n +2(z, -q)±iGeY2n+2(Z, -q),

(16)

Me&~i~)(z, -q) = (_I)n+liNe&~i~)(!7Ti+z,q)

== Ce2n +1(z, -q)±i FeY2n+l(z, -q), Ne~~~i)(z, -q)

==

(17)

(-I)n+liMe&~~~)(i1Ti+z,q)

== Se 2n+1 (z, -q)±iGeY2n+l(z, -q).

(18)

13.41. Relationship between Me~), Fekm ; Neg), Gek.. Applying (4), (7), (10), (13) § 8.14, and (10), (20), (30), (40), § 8.30 to (1), (2) § 13.40 leads to (1) Me~~(z, ±q) = -2iFek 2n(z, ±q), Me&~+l(z, q)

=

-2 Fek 2n+1 (Z, q),

(2)

Me&~+l(z, -q) = 2i Fek 2n +1 (Z, -q),

(3)

Ne~~+l(z, q)

= -2 Gek2n+1(z, q), Ne~~+l{z, -q) = 2iGek 2n +1{z, -q), Ne~J+2(z, ±q) = -2i Gek 2n +2{Z, ±q).

(4) (5)

(6)

252

BESSEL FUNCTION PRODUCT SERIES

13.50. Series for ce (z, q), ce se m

sem+p

[Chap. XIII

(z, q). Writing -·iz for z in

(3) § 13.11 gives

ce 2n (z, q) = (P2n/A~2n»

..,00

!

r==O

(-l)rA~~n)J;.(v;)J;.(v~),

(1)

where v~ = ke1.z, v~ = ke:". The r.h.s, of (I) is even in z. If z is real, v~ and v~ are conjugate, 80 the r.h.s, is real, as we should expect. If z is complex, v~, v~ are not conjugate and (1) is, therefore, complex. Writing -iz for z in (6) § 13.11 yields ce2n+1(z, q)

=

(P2n+l/Ai2ntl» X

the r.h.s. being even in z. If z is real the two product pairs are conjugate, since one is derived from the other by changing the sign of i. Thus the part in [ ] takes the form

(x+iy)+(x-iy) == 2x,

(3)

a real function expressible in a cosine series, since e±iz == cos z±i sin z. By virtue of (3), if z is real, 00

ce21l +1(z, q) == (2P21l+1/Ai2n+1»

!

r==O

(-I)r A~~~_il) }{eal[~(v~)J;.+l(v~)].

Similarly we obtain se2n+1(Z, q) --

(8 2n+11l·B<2n+l» 1

(4)

X

and, if z is real, se 2n+ 1(z, q) == (28 2n+11B~2n+l» X 00

X

se2n+ 2(z, q)

== -

! (-I)rB~;~1l)Imag[Jr(V~)~+1(V~)]; r=O

(6)

(82"'+2/iB~2n+2» X

and, if z is real, se2n+2(z,q) = -(282n+2IB~2"'+2» X ex>

X

! (-I)r B~~'t~2) Imag[~(v~)J;.+2(v~)J. r=O

(8)

Series for cem+p(z,q), sem+p(z,q) may be derived from § 13.12 by the above substitutions.

253

BESSEL FUNCTION PRODUCT SERIES

13.51]

13.51. Series for cem(z, -q), sem(z, -q). Applying the relationships (2)-(5) § 2.18 to the formulae in § 13.50 yields the results given below. ce2n(z, -q)

=

ex>

(P~n/Ab2n»! (-I)rA~~n)Ir(v;)Ir(v~).

(1)

r=O

This function, as represented by the r.h.s., is even in z, real or complex with z. ce2n +I(z, -q) = - (lJ;n+l/ Bi2'l+1» X

and, if z is real, ce2n+1(z, -q) == - (28~n+l/B~2n 1-1» X <X)

X

se2n +1 (Z, -q ) and, if z is real, se2n +l(z, -q)

! (-1)'B~~'+11) Real[Ir(v;)Ir+l(v~)]; r=O

' / ~'A<2n+l» == - (P2nAI I

=

(3)

X

-(2P~n+l/Ai21l+1» X <X)

X

! (-1)'A~;'+11) Imag[ Ir(v;)I'+l(v~)]; r=O

(5)

se2n +2(z, -q) = (8~n+2/iB~2n+2» X

and, if z is real, se2n+2(z, -q)

=

(28~n+2/ B~2n+2» X 00

X

! (-1)' B~~~~2) Imag[I,(v;)I'+2(v~)]. 7=0

(7)

13.52. Series for fe 2ll(z, q). Substituting -iz for z in (1) § 13.20 gives. <X)

FeY2,t(-iz,q) = (P2n/A~2n»! (-l)'A~~)J,(v;)~(v~).

(1)

r-O

Consider the product J,(v~)~(v~); substitute the expansion of the Y function [ref. 202, p. 161, no. 61], and we get

J,.(v;>~(v~) = 'TT~['Y+logelk]J,.(v;>J,.(v,:>+ !

cosines-

-i[~~J,.(V;)J,.(V~)+ !

sines}

(2)

BESSEL FUNCTION PRODUCT SERIES

[Chap. XIII

Using (2) in (1) leads to FeYan(-iz,q) 2 ~ = :-(y+logelk)celn(z,q)+(PlnTA~ln» I (-I)rA~~n) I cosinesw r-O

-i[~zcelln(z,q)+(Plln/A~lIn» i W

r-O

(-I)rAgn) I Sines].

(3)

Now, if z is real, the real and imaginary parts (even and odd functions, respectively) of (3) are linearly independent solutions of y"+(aln-2qcosz)y = o. Thus the second member ofther.h.s. must be a constant multiple of ce1n(z, q), while the imaginary member must be a constant multiple of fe 2n(z, q). Hence for z real we define fe 2n(z, q) = multiple of Imag{FeYln(-iz,q)}. (4)

Now by (1) § 7.22 fesn(z,q) = 02n(q)[zce2n(z,q)+

! fJ~!V2sin(2r+2)z],

r-O

(5)

so by (1) we obtain fe 2n(z,q) = G~n(P2n/A<;n»

00

I

r-O

(-l)rA~~n)Imag[J;.(keilS)~(ke-U)].

(6)

In this and the sections which follow, the series in B.F. products are valid only if Ie i= o.

13.53. Determination of O~n. Differentiation term by term in (6) § 13.52 is valid since by § 13.60 the series is uniformly convergent, as also is the resulting series. Thus fe~n(O, q)

=

O~n(P2n/A~2n»

co

I

r-O

(-I)r A~~n)Imag[ikeuJ;(keie)~(ke-iz)-ike-uJ;.(keU)Y;(ke-iz)]z=o,

(1)

where J;(v) = dJ,.(v)/dv. Now J;(k)~(Ie)-J;.(k)Y;(k) = -2/wk, by (1) § 13.31, so (1) gives

f~n(O,q) = -C~n(P2n/A~2n»~ '1T

i

r-O

(-ltA~~n),

(2)

and, therefore, O~n

=

-iwA~n)fe~n(O,q)/P2nce2n(lw,q).

(3)

13.54. Relationship between FeYI"(-iz, q), ce 2n(z, q), and fetn(z, q). (3) § 13.52 may be written

FeYln(-iz,q)

=

Ace2n(z,q)+iBfel,,(z,q).

(1)

13.54]

BESSEL FUNCTION PRODUCT SERIES

2M

Since fe 2n (O, q) = 0, we get

A = FeY2n(0, q)/ce 2n(0, q). Differentiating (1), and putting z = 0, yields

(2)

B = -FeY~n(O, q)/fe~n(O, q).

(3)

. ) FeY2n(0,q) .FeY;n(O,q)~ ( ) F eY2n ( -~z,q = ce (0 , q )-ce 2n (z,q ) -~~'(-O -) le 2n z,q · .le ,q

(4)

Hence by (1)-(3) 2n

2n

This will be clear from (4) § 13.21.

13.55. Series for fe 2n+1(z, q), gem(z, q). Analysis similar to that in § 13.52 leads to the following results for real values of z: 00

X

!

r-O

with

O~n+l(P2n+l/A~2n+l» X

=

fe 2n+1(z, q)

(-l)rA~~'+11) Imag[J;.(v;)Y;.+l(V~)+J;.+l(V;)y;'(v~)], (1)

C~n+l = 17TkAi2n+lJ fe~n+l(O, q)/P2n+lce;n+l(!1T, q).

(2)

The relationship corresponding to (4) § 13.54 is obtained by writing 2n+l for 2n. ge2n+1(z, q)

=

S;n+l(s2n+l/ Bi2n + l) X

00

X

! (-l)rB~~'+11)ReaI[~(v~)Y;.+l(V;)-~+l(V~)y;'(v~)], (3) r .... O

8 2n +1 = . ) GeY2n+l ( -~z,Q

-!7TkB~2n+l) ge 2n +1 (0, Q)/S2n+l se2n+1(! 7T, q),

(4)

q)e + ( ) = GeY;n+l(O,q) -,-------- -. ---~e21l+1 (z,q) +GeY2n+l(O, ----g 2n 1 z,g ,

se2n +1(O, q)

ge 2n +1 (0, q)

(5) 00

X

! (-I)r B~;~~2) Real[J;.(v;)Y;.+2(v~)-J;.+2(v~)y;'(v~)], (6) r=aO

S~n+2

= -trrk2 B~2n+2) ge 2n +2(O, Q)/S2n+2 se~n+2(!1T, q).

(7)

The relationship corresponding to (5) is found therefrom if 2n+2 is written for 2n+ 1.

13.56. Series for fe2n (z, -q). By (1) § 7.60 fe 2n (z, -q) = (-I)n+lfe 2n(11T-z,Q). Applying the r.h.s. of (1) to (6) '§ 13.52 gives

(1)

fesn(z, -q) = (_I)n+lG~n(P2n/A~ln» X 00

X

!

r-O

(-I)rA&~n)Imag[~(ike-i·rY;'(-ikei.e)].

(2)

256

BESSEL FUNCTION PRODUCT SERIES

[Chap. XIII

The part in [ ] has the form x+iy, so if z is real, (2) is unaltered if we write - i for i and (_I)'t for (_1)11.+1. Thus for z real fe2n (z, -q)

=

(-1)nC~n(P2n/A.~2n»x -o

X

!

(-I)rA&~n)Imag[~(-ikeiZ)~(ike-iZ)].

(3)

r-O

J,(-ikeiz )

Now

~(ike-iz) =

and

= i- r Ir(kei Z)

(4)

i- r[ -~Kr(ke-iZ)+(-lYiIr(ke-iz)l

(5)

Using (4), (5) in (3) leads to fe 2u(z, -q) X

=

C~n(P~n/Ab2n» X

{f (-l)rA~~Il)Ir(keiZ)Ir(ke-iZ)-~

~ A~~Il)Imag[Ir(keiZ)Kr(ke-i.e)]}

n ~ r-O

r-O

=

O~n[ce2n(Z,

(6)

-q)-2Imag{Fek 2n(-iz, -q)}]

= C~Il[ce2n(z, -q)-2fek 2n(z, -q)],

(7)

(8)

where the definition of fek 2n(z, -q) is obvious.

Replacing the K functions in (6) by their expansions [ref. 202, p. 165] leads to fetn(z, -q)

=

const.[(ln-z)ce 2n(z, -q)-+a sine series],

(9)

and this has the same form as (2) § 7.60.

13.57. Series for fe 2n + 1(z, -q), gem(z, -q). Applying (3) § 7.60 to (3) § 13.55 gives ~ ( ) S'211.+1 (' te 8 211.+1 /B<2n+l» 1 X 2n +1 Z, -q = IX)

X

! (-1)rB~~'+1l)Real[~(iv~)~+1(-iv~)-~+1(iv~)~(-iv;)]. r=:O

(I)

If z is real, (1) is unaltered if we write - i for i, since the real part in [ ] is unaffected. Doing this and following on the lines of § 13.56 leads to fe 2n +1(z, -q)

= =

S~n+l[cean+l(Z, -q)-2Imag{Fekan+1(-iz, -q)}]

(2)

8~"'+l[ce2n+l(z, -q)-2fek 2n+1(z, -q)]

(3)

(z real).

Similarly we obtain gean+1(z, -q) = O~lt+l[sean+l(Z, -q)-2Real{Gekan+1(-iz, -q)}]

(4)

= G~n+l[seln+l(Z, -q)-2gek an+1(z, -q)] (z real),

(5)

BESSEL FUNCTION PRODUCT SERIES

13.07]

257

ge2n+ 2(z, -q) = S~n+2[se2n+ 2(z, -q)- 2 Real{Gek 2n +z( -iz, -q)}]

=

S~n+2[se2n+2(z, -q)-2gek 2n+2(Z, -q)]

(Z

real).

(6)

(7)

In each case by using the expansions for the K -Bessel functions, as at (9) § 13.56, the forms in § 7.60 may be obtained.

13.60. Convergence of solutions. By analysis similar to that given below, all the B.F. product series in this chapter may be proved absolutely and uniformly convergent in any finite region of the z-plane. We shall exemplify this point, using the series 00

I

r==O

(-I)r A~~n)~(VI)y;'(V2)'

(1)

Then by (1), (2) § 8.50 we get ~(VI)y;'(V2) ~ r- 1e- 2rz

(r ~

Ivll or

Iv2!).

(2)

Using (2), and (4) § 3.21, the ratio of the (r+ l)th to the rth term is

1

~_-tlQJ1-)~_f:l(V2)A~~~21 J;.(VI)~(v2)A~~n)

,-..J

I-

Ik2e- 2.z 1 ~ 0

4(r+ 1 ) 2 '

as r ~

+00

(3) ,

for all real or complex z provided R(z) is finite. Hence under this condition series (1) is absolutely convergent. By applying the 'M' test the convergence may be proved uniform. It follows that 00

I (- I)r A~~n)J;.(keiz)y;'(ke-iz)

(4)

r=O

is absolutely and uniformly convergent, provided R(iz) is finite, i.e. Imag(z) is finite. By virtue of uniform convergence, it follows that the functions represented by the R.F. product series are continuous in any finite region of the z-plane.

13.61. Advantages of B.F. product series over those in Chapter VIII. These are as follows: 1. The product series converge more rapidly than those with argument 2k cosh z, 2k sinh z. The rate of convergence increases with

increase in z (real). 2. The product series for FeYm' Fek m , GeYm' Gek., converge uni-

formly in any finite region of the z-plane, including the origin. The other series do not converge uniformly in a restricted region, including the origin. 4961 L I

BESSEL FUNCTION PRODUCT SERIES

~58

[Chap. XIII

3. By virtue of (1), (2), the product series are preferable for calculating the values of the functions. This feature is illustrated vividly in Table 22. TABLE

I

22. Oomputation of Feko(z, -4); k z

C6 0 (0,

= 0, 2k cosh z = 4

4) (-1)' x

Ao

r

--

°I 2 3 4 5 6

-Sum-

X

K t ,.(2k cosh z) 0·00078 0079 130 1687 93 6560 67 359 . 5025 .. 3990 ..

+0,0046 .....

z

ceo(O, 4)ceo(j-1r, 4)

A o~

X

x

I,.(ke-Z)K,.(ke Z)

=

2, q = 4

= log,2, 2k cosh z = 5

A8 in column 2

A8 in column 3

15 39 35 80 27 07

0·00025 801 39 706 22965 12 127 6306 3 30. 17 ..

+0·00227 -0· 121 +0, 8 -0'

+0·00909 51

0·00111 905

+0·00114 137

+0·04189 -0,03841 +0'00607 47 --0' +0, 2 -0'

970 818 215 230

When z = 0 the K 2r series, being non-uniformly convergent, converges so slowly that six terms contribute much less than three of the IrKr series. In columns 3, 5 the rapidity of convergence of the IrKr series is strikingly displayed, the value of the function being obtained to at least seven decimal places using only four terms, See also [M].

XIV MISCELLANEOUS INTEGRALS INVOLVING MATHIEU FUNCTIONS e

14.10. Evaluation of

f y cosh 2u du, when y satisfies

o

y"-(a-2qcosh2u)y

= o.

Integrating the D.E. with respect to u from 0 to z

[y'J:-a z

I

we obtain

z

f y du +2q f yeosh 2u du = 0,

o 80

Z,

yeosh2udu

(1)

0

=~

o

z

I YdU-2~[Y'(Z)-Y'(0)],

(2)

0

where y is any solution. In particular, if y z

f

=

Ce2n(u, q),

~

Ce2 n (u' q) du -- A (2n )z+ ~ ~ 2r ~ A (2n) sinh 2rz , 0 2r

o

(3)

r=l

d IU:"I: - Ce2n(u, q) dU u=o

=!eo 2rA~~n) sinh 2rz.

(4)

r-l

Substituting (3), (4) into (2) gives z

f Ce2,,(u,q)eosh 2u du = !A~2n)z+lAf.2n)sinh 2z+ o

+-4 L: - (A<2n) +A(2n) ) · h rz, 00

1

1

r-l

r

2r+2

by (1) §3.10. In a similar way for y"+(a-2qcos2u)y

f o

f

z

z

yeos 2u du = :q

=

2r-2 SIn

2

(5)

0, we find that

y du+ 2~'[y'(Z)-Y'(0)],

(6)

0

and in particular, when y === ce2n(u,q), the r.h.s, is

l~ ~n)z+ lA~n)sin 2z + ~

i r-l

~ (A~~'r2+A~~~2)sin 2rz.

(7)

260

MISCELLANEOUS INTEGRALS INVOLVING [Chap. XIV

14.20. Product integrals. We commence with the equations

= y;-(a-2Q2QOsh 2U)Y2 = y~-(a-2ql cosh 2U)Yl

0

(1)

and 0, (2) the a being the same but ql =1= q2. Multiplying (1) by Y2' (2) by Yl' subtracting and integrating with respect to u from ZI to Z2' we get e.

2(qa-ql)

f Yl(u)Ya(u)cosh 2u du = [y;(u)Ya(u)-Y~(U)Yl(U)]::.

(3)

%1

When the fundamental ellipse becomes a circle, (3) degenerates to the well-known form for the Bessel functions [202, chap. vi]. If in (1), (2) ql = q2 and Yl' Y2 are independent solutions of the same equation, with %1 constant and Z2 = z variable, we get a constant. t (4) For given functions Yl' Y2' c2 is independent of z, but not of q. By (4)

Y;(Z)Y2(Z)-Y~(Z)Yl(Z)= Y;(Zl)Y2(Zl)-Y~(Zl)Yl(Zl) = c2 ,

;u[log(Yl!Ya)] z

f

80

dU!YIYa =

Taking Yl = Ce2n+ 1 (u , q), Y2

f•

=

Y~Y2-Y~Yl =

Ca!YIYa,

(5)

~log(Yl!Ya)'

(6)

FeY2n+l(U, q), (6) asserts that

du![Cean+1(u) FeYal1+1(U)]

Again, from (4),

=

= ~log[Cean+1(Z)/FeYan+1(Z)].

(7)

y~

gives

c2 , so dividing throughout by

(Y;/Y2)-(Y~Yl/Y~)

d(Yl!Ya) du

or

=

c2/y:,

= ca!y~.

(8)

(9)

Integrating (9) leads to %

f du!YI+A, Yl = ctya f du!yl+ AYa·

Yl!Ya

= ca

(10)

z

or

(11)

This gives one solution in terms of the other, but it is a troublesome form to use in practice. The lower limit is an arbitrary constant, related to A. t For eolutions of Bessel's equation the equivalent of (4) takes the form [202, p. 156] JliYI-Y;Yl = a constant!z. The absence of

Z-1 on the r.h.s. of (4) is by virtue of Mathieu's equation having no differential of the first order.

MATHIEU FUNCTIONS

14.20]

261

The results herein are valid for solutions of y" +(a-2qcos 2u)y = 0, provided that cosh 2u is replaced by - cos 2u.

14.21. Theorem. IfYI(u), Y2(U) are solutions of y"-(a-2qcosh2u)y = 0 for the same a, corresponding respectively to ql' q2' such that y~(O)Y2(O)-y~{0)YI(0)= 0, and either YI(z,qt) = Y2(Z,QI) = 0 or y~ (z, ql) = Y~(z, q2) = 0, then z

f Yl(U)Y2(U.)cosh 2u du = O. o This follows from (3) § 14.20. If YI(U)

=

(1)

Y2(U),

z

f yf(u)cosh 2u du ¥= O.

(2)

o

We now proceed to evaluate this integral. In (3) § 14.20 take 6.q = (q2-ql) to be small.] Zt = 0, Z2 = Z, then we have z [Y;(U)Y2(U)-Y~(U)Yl(U)]:-2~q Yl(U)Y2(U)cosh2udu = O. (3)

f o

By Taylor's theorem Y2

= Yl+~q~+ (~q~ oil1_+... .

2! oq2 Substituting into (3) and neglecting terms in (6.q)2, ... leads to [

oq

I) ('

1)] -

A By A 0 oy Yl, ( YI+uq-- --:'II YI+uq--

aq

(4)

au oq

z

0

I

-2~q Yl(Yl+~qtl)C08h 2u du = O.

(5)

o

Dividing throughout by Sq and letting ~q tend to zero yields, with y for YI' Z Oy By_ Oy]Z _ 2 y 2(u }cosh 2u du = O. (6) [CU oq au aq 0

y!-

J

o

Hence

JZ y 2(u)cosh 2u du = ![Oy Oy _y!-(Oy)]fl. o

2ouoq

ouoq

(7)

0

8y/oq is not found by differentiating the coefficients in the series for y, since both q, a vary on a characteristic curve. t Y. -+ Yl as ~q -. o.

262

MISCELLANEOUS INTEGRALS INVOLVING [Chap. XIV

14.22. Example. Evaluate (7) § 14.21 when y = Ce2n(u ), given tkat Ce2n(eo,q) = 0, the upper limit being z = ~o. Here oy/Bulu=o = 0, since all terms in the series vanish at the origin. Then

t.

f

~

y2cosh 2u du

=

![Oy Oy] +![y.! (8Y)lJ 2 e« oq u=e. 2 OU oq

(1)

11=0

o

=

!Ce~n(eO,q)(~Y)

uq u=f.

+lce2,,(o,q)[: (~Y)]

,

ou uq u=o

(2)

where, with a constant,

oy/oq =

lim{[Ce2n+p(u,q+~q)-Ce2n(u,q)]/dq}. 4~::g

(3)

The only way of evaluating (2) seems to be by a numerical process. ~

14.23. Evaluation of

f (y'2+ ay2) du, when y satisfies °

y" -(a-2qcosh 2u)y

= o.

The D.E. may be written

-y"+ay

=

(2qcosh2u)y.

(1)

Multiplying both sides of (1) by y, and integrating with respect to 1£ from to z, yields

°

•

c

c

- I y dy' +a I y2 du = 2qI y2cosh 2u duo

(2)

0 0 0

Thus

a

f (y'2+ ay2)du = [yy'~+2q f y2cosh 2u du o ~

(3)

0

(4)

by (7) § 14.21.

14.24. Evaluate (4) § 14.23 if y = Ce 2n(u, q), given that

Since Ce2n (eO' q) = 0, and Ce~n(O, q) = 0, the first member on the r.h.s. of (4) § 14.23 vanishes. Hence by (7) § 14.21, (2) § 14.22 and (4) § 14.23 Celn(~o, q)

= 0, z = ~o.

e. f [Ce~(u,q)+a ee:n(u,q)) du = 2q times (2) § 14.22. o

(1)

MATHIEU FUNCTIONS

263

14.30. Integrals involving sinh 2u. Let y be a solution of the differential equation in § 14.23. Differentiating . ~(u) = y'l(u)-(a-2qcosh2u)yl(u) (I) with respect to u, we obtain

:: = =

2y'y"+4q(yy' cosh 2U+y2 sinh 2u)-2ayy' 2y'[y"-(a-2qcosh2u)y]+4qy2sinh2u

Hence

=

(2)

4qy 2 sinh 2u.

(3)

a

4q

f y2sinh 2u du = =

y'2(z)-(a-2qcosh 2Z)y2(Z)

(4)

y'2(Z)_Y(Z)Y"(Z).

Integrating by parts gives ~

Ie

f y2sinh2udu = 2 f y2coshudcoshu =

(5)

l

j

2 [y2 cosh 2z- 1 y2sinh 2u du -2 fyy' cosh2u du

(6)

f y2sinh 2u du = y2(z)cosh2z-2 f yy' coshSuduo

(7)

Therefore S

Ie

Hence by (4), (7) we find that s

f yy' coshSu du = (q+ia)y2(Z)-lY'2(Z).

4q

Since cosh 2u

=

,(cosh 2u+I), (8) may be written

f yy' cosh 2u du = =

f yy' du Z

Ie

4!

(8)

(2q+a)y2(z)_y'2(z)_4q ay2(Z)_y'2(Z).

(9) (10)

14.40. Integrals for the ordinary Mathieu functions. If t' is a solution of v"+(a-2qcos 2u)v = 0, the integrals corresponding to (4), (8), (10) § 14.30 are

f

Ie

4q v 2sin2udu = v'2(z)+(a-2qcos2z)v2(z)

=

V'2(Z)-V(z)v"(z),

(1)

(2)

~

4q

f w' cos2u.du = (1+ ia)v

2(z)

+ !V'2(Z),

(3)

z

4q

f vv' cos 2u du = av 2(z)+v'2(z).

(4)

264

MISCELLANEOUS INTEGRALS INVOLVING [Chap. XIV

=

Using (~) with v

cem(u,q) or sem(u,q), we find that

f ce~(U}8in 2u du = J8e~(u}8in 2u du = O. 2"

2"

o

(5)

0

Using (4) leads to

f cem(u)ce:n(u)c08 2u du f 8em(U)se~(u)cos 2u du = O. 2"

2"

=

o From § 9.40 we have

(6)

0

f ce~(u)c082udu = 1T0

fo se~(u)cos 2u du = 2"

2"

m;

o 2"

1T'Ym'

(7)

2"

f

f

14.41. Evaluation of ce~(z, q) dz, se~(z, q) dz. We have o

0 2~

2"

2n

f y'2(z) dz = f y' dy = [yy,]:1T- f yyH dy.

(I)

0 0 0

If y = cem(z,q) or sem(z,q), the third member of (1) vanishes at both limits. Since y is a solution ofy"+(a-2qcos2z)y = 0, the first and last members of (1) give 2"

2n

Jy'2(z) dz = f (a- 2qcos 2Z)y2 dy. o

(2)

0

The evaluation of the r.h.s. is obtained from (1) § 2.21, and (7) § 14.40, so we get

f

2n

if y

=

y'2(Z) dz =

1T[a-2q{:

: }]

= 1T(~),

(3)

o ce m or sem" 2~

14.42. Evaluation of

2n

f cem(z, q)se~(z, q) dz, f sen(z, q)ce:n(z, q) dz.

o

0

First we have 2n

f cem 8e~ dz =

[ce msen]~1T -

o

2n

f sence:n dz,

(1)

0

by partial integration. The member in [ ] vanishes at the limits. Hence 2n

f cemse~ dz =

o

2~

-

f sen ce~, dz. 0

(2)

l\IATHIEU }4"UNCTIONS

14.42]

265

Now 2",

2'7T

f ce2mse~ndz= f

o

0

00

2'7T

2r

r=l

(3)

r~l

r=O

00

I

=

00

[IA~~m)COS2rzI2rB~~n)COS2rZ]dz •

f A~~m)B~~n) cos22rz dz,

(4)

0

since the other integrals vanish, I being moved outside the integral sign by virtue of uniform convergence. Therefore by (2), (4)

In a similar way we find that 2#

2",

f ce2m+1Se~n+1 dz = - f se o

ce~m+1 dz

2n+1

0

2",

2'7T

f cemce~ dz = f semse~ dz = O.

and

o

(7)

0

These results are valid for m =1= n or m == n,

14.50. An integral identity. In § 14.20 we showed that if Yl' Y2 were solutions of y"-(a-2qcosh2u)y = 0 for the same a, q, then for all u a constant. (1) Squaring both sides of (1) gives y;2y~+y~2yf-2Y;Y~YIY2== c4 •

(2)

Now, by § 10.76,

Y~Y~

z

= ~1.11(O)-Yt(Z)Y2(Z)X(Z)+4q f Yt(u)Y2(u)sinh 2u du,

(3)

o

where gl.2(O) = y;(O)y~(0)+Yl(O)Y2(O)X(O). Using (3) in (2) leads'[ to

c4 == yl(Z)el.1(O)+yf(z)~S.2(O)- 2Yl(Z)Y2(Z)~1.2(O)+ e

+4q

t 4861

f [Y2(Z)Yt(u)-Yt(Z)Y2(u)]2 sinh2u duo o

With the aid of (4) § 14.30.

Mm

(4)

MISCELLANEOUS INTEGRALS

266

[Chap. XIV

Taking the initial conditions Yl(O) = y~(O) = 1, Y~(O) = Y2(O) = 0, we find that ~l,l(O) = (2q-a), ~2,2(O) = 1, ~1,2(0) = 0, c4 = 1. Hence (4) becomes z

Yf(z)+(2q-a)y~(z)+4q f [YlI(Z)Yl(U)-Yl(Z)YlI(U)]llsinh 2u du

= I, (5)

o

or

f•

4q [YlI(Z)Yl(U)-Yl(Z)YlI(U) ]1 sinh 2u du

= I-yf(z) +(a- 2q)y=(z). (6)

o

For the equation y" + (a- 2q cos 2z)y = 0, the identity corresponding to (6) is z

4q

f [YlI(Z)Yl(U)-Yl(Z)Yll(u)]lI sin2u du = l-yf(z)-(a-2q)y~(z). (7)

o

If a, q are fixed, (6), (7) are valid for all real finite values of z. When q = 0, (7) degenerates to y~(z)+ay~(z) =

1,

(8)

while the equation becomes

y" +ay = 0, and the appropriate solutions which conform with (8) are y (z) = cosaiz, sina1zfa 1 respectively.

(9)

PART II APPLICATIONS OF MATHIEU FUNCTIONS XV

APPLICATIONS OF yH+(a-2qcos2z)y = 0 15.10. Amplitude distortion in moving-coil loud-speaker. A conventional diagram of this apparatus is shown in Fig. 20 A. A sinusoidal current, of low audio frequency, in the coil causes it to vibrate axially about a central or equilibrium position. For absence of distortion arising from amplitude of motion, the amplitude must be proportional to that of the coil current. This condition is fulfilled if the total magnetic flux passing radially through the coil remains constant throughout its travel.'] When the axial length of the coil is equal to that of the outer pole face (Fig. 20B), and the amplitude is relatively great, i.e, the sound output is large, an appreciable length of the coil moves into the leakage field at each side of the outer pole. The amplitude cannot then be proportional to the current, since the field decreases rapidly with increase in axial distance from either side of the outer pole. In practice the coil usually tends to move into the weaker part of the field, but its excursion is limited by the axial restoring force exerted by the centring spider (not shown), and the outer surround' which supports the diaphragm. By virtue of this superimposed unidirectional motion, speech and music are reproduced with perceptible distortion because the axial constraint is non-linear for large coil displacement.j If, however, the axial length of the coil is adequately greater than that of the outer pole (Fig. 20n), the magnetic flux passing radially through the coil is almost constant throughout the travel. Then the effect described above is negligible, being counteracted easily by the axial restoring force. A well-designed loud-speaker satisfies this requirement [201], provided the coil amplitude does not exceed the linearity limits of the axial control. t It has been tacitly assumed that the axial constraint is proportional to the displacement of the coil. : Whon the coil current arises from speech or music, intermodulation occurs between the various component frequencies, owing to the constraint-displacement relationship being non-linear. The distortion is unpleasant.

APPLICA1'I()NS O}4' y4'+(a-2q(,():i 2z).lJ ---= 0

268

[Chap. XV

(A)

~irecbon

~offlux

Energising coil

(8) .

(D) ~

l.E!J Inside

magnet

Leakage

leakag

~

.:i .

Cell)

::Sa)

au

.- U

'5~

CS~

:.§&.-

.:::;&..

~

~'IO.

20

~

c:-

Q).Qt

A-]).

leakage flux

Sehematio diagrams for moving-coil Ioud-speaker,

15.11. The equation of motion. To study the phenomenon, described in § 15.10, analytically, we consider a relatively narrow coil situated in 8, radial magnetic field which decreases linearly'] from the outer side of the centre pole (Fig. 21) towards the left, i.e. away from the magnet. The axial force on the coil due to a current locos 2wt is (1) f = 21Trnc~Iocos 2wt, where r = mean coil radius, n = number of coil turns, uniformly spaced, t In practice the relationship between flux and axial distance takes the form shown in Fig. 20 c.

A II II LIe AT I () N S o F yIP +(a - 2q cos 2z)y ~:: ()

15.11]

269

ce

B == = magnetic flux density when coil centre is distant, from the origin, c = slope of line in Fig. 21.

21. Assumed relation between flux density B and distance from outer face of magnet in Fig. 20.

FIG.

For simplicity we assume that: (a) the axial restoring force on the diaphragm is negligible; (b) the mechanical loss is negligible;

(c) the electrical impedance of the circuit external to the coil is large enough to preserve constant current whatever i.e, the back e.m.f. induced in the coil by virtue of motion in the magnetic field is small compared with the potential difference applied to the circuit. If me is the effective mass of the coil and diaphragm, the equation of motion is d2 (2) m, dt,,-21Trnc,Iocos2wt = o.

e,

e

Writing z

=

wt, q

=

1Trnclolw2me, (2) becomes

die

dz2 - (2q cos 2z)e

= o.

(3)

If a linear axial restoring force 8 per unit axial displacement is present, the term Be must be added to the l.h.s, of (2). Then we have

~:!+(a-2qc082Z)e=

0

(4)

270

APPLICATIONS

()}4'

y·+(a-2qco82z)y

=0

[Chap. XV

with a = a/w"me- In addition, if linear damping r e per unit axial velocity is present, the equation of motion is d"~ d~ medtl+redt+8~ ...... 21Trncelocos2wt = 0,

d"e de dz" + 2IC + (a-2qcos2z)e =

or

dz

0,

(5) (6)

where 21C = re/wme. Under the simplifying assumptions given above, the motion of the coil in a linearly decreasing radial magnetic field is defined by (3). As a practical value of q we take 0·16, so the equation to be solved is

e

d2 dz2 - (0-32 cos 2z)e

=

O.

(7)

15.12. Solutionof(7)§ 15.11. From Fig. 8we see that the point a = 0, q = 0·16 lies in a stable region of the plane between ao and bl . By (1) §4.74 with v = {J, a = 0, we obtain {J2 ~ q2/2(1-fJ2), from which we find that fJ ~ 0·1138. (1) 0 For the first solution, in accordance with 1 § 4.70 we take ~1

co

= I

r=-oo

(2)

02rcos(2r+0·114)z.

By (1) § 4.17 the recurrence formula for the c is (2r+0-114)2c2r+O·16(c2r+2+c2r_2) =

o.

(3)

Neglecting cs , with r = 2, we get C2

=

-4-114 2 x 6-25c4

=-1-057xl0 2c4 • r

=1

Co

=

2-95 X 103c4.

(5)

Thus c2 = -3-58 X co . and C4 = 3-39 X 10-4 co Neglecting c-s, with r = -3 c- 4 = -5-886 2 X 6-25c_ e 10- 2

= -2·165X 10 2c_e.

r = -2

c_1

(4)

= 2-114 2X 6-25X 1·057 X 10 2c4 - c4

=

3-886 2 X 6-25

(6) (7)

(8)

X 2-165 X 10"c_8 - c- e

= 2-04 X 104c_8 _

(9)

=0

271

-1.8862 X 6·25 X 2·04 X 10·C_8 -c_, = -4·55 X I05c_8.

(10)

APPLICATIONS OF y#+(a-2qcos2z)y

15.12]

Co

r =-1

Thus

=

c- 2 = -4·5X 10-2co, c- 4 = 4·77 X IO-4co, c- 8 = -2·2 X 10-6CO. If in (3) we put r = 0, f1 for 0·114, and insert the values of c2 '

we obtain

cof1 2

=

-0·I6(c 2+c_2 )

=

(11) (12) (13) c_1 ,

0·16x8·08x IO-2co,

f1 = 0·1137, (14) which checks the value at (1) and also the computation in generalup to a point! Substituting for f1 and the c in (2), the first solution is

so'

'1 =

co[cosO·114z-.:....4·5XI0-

2cosl·886z-

-3·58x 10-2cos2·114z+ ...]. By 10 § 4.70 the second solution is

(15)

-3·58X 10-2sin2·114z- ...].

(16)

'2 = co[sinO·114z+4·5X 10-2sin 1·886z-

By §4.71, (15), (16) are multiples of ceO' 114(z, 0·16) and se O'11' (Z, 0·16), respectively. Using (3) § 4.72, we have K

Hence

~

l/co[I + 0·0452 + O·0362]1 ~ 1/1·002co.

(17)

'2

(18)

'1 =

1·002coce O' 11' ( Z , 0·16) and = 1·002coseO.11,(z,O-I6), and the complete solution with two arbitrary constants is

g = Ag1+Bg2•

(19)

t

15.13. Initial conditions. We take, = '0' = 0 at t = 0, i.e, when z = o. Then by (15), (16), (19) § 15.12 the first condition gives go = Ac o(I-0·045-0·0358), so A = I·088'0/c o. (I) The second condition gives B = o. Hence the appropriate solution of the problem is

, =

1·088'0[cosO·114z-0-045cos I·SS6z-0·036cos2·114z+ ...] (2)

= 1·09fo ceO' 114(z, 0·16),

where z = wt. Since (2), (3) is IOOO7r.

f1 =

(3) 0·114

=

57/500, by §4.71 the period of

APPLICATIONS OF 1I--+(a-2qcos2z)y

272

=0

[Chap. XV

15.14. Interpretation of (2) § 15.13. Cos O·114z is the dominant term, and as the coefficients of the other terms are comparatively small, these terms may be considered to represent ripples of pulsatance > O·114w. The motion of 11he coil is illustrated by the heavy curve of Fig. 22. The coil reaches the position = 0 after a time

e

%

FlO.

ex: bime

-0· 5cos 1·88Gx. -0,036 cos Z'114%

22. Curve showing displacement of L.S. coil outside magnet.

lapse of approximately the first quarter period of cos O·114wt, i.e, T = 1T/O·228w. If the pulsatance of the coil current were w = 351T, T ~ 1/8 second. Since we assumed absence of constraint and loss, if we make the additional assumption that the magnetic field is zero when ~ < 0, the coil having reached ~ = 0 will move onwards by virtue of its momentum. The time taken for tho coil to travel the distance f = fo may be increased by reducing q [201].

15.15. Electro-mechanical rectification. The effect described above has been called by this name because a sinusoidal current in the coil causes a unidirectional motion. If, however, the magnetic field were negative behind the origin-antisymmetrical about 0the coil would oscillate about 0, so the question of rectification would not arise. Attention is drawn to tile following features: (a) In (2) § 15.13 there is no term of pulsatanoe equal to that of the current, 2w.

15.15]

(b)

APPLICATIONS

O~..

y·+(a-2qcos2z)y

=0

273

011 the dominant component cos 0'114wt is superimposed an infinite number of ripples of different pulsatanoes, which are non-integral multiples of 0·114w.

15.16. Parametric point in unstable region of Fig. 8. The solution in § 15.12 pertains to a stable region. If (a, q) lies in an unstable region the form of solution is given at (1) or (3), 2° § 4.70. This entails the rate of oscillation of the coil being the same as the frequency of the current, while ~nlax increases with increase in time. The reader may find it of interest to investigate the unstable case with and without damping. 15.17. Pulsatance multiplication. In Fig. 11 each iso-f3 curve (if continued far enough) intersects the q-axis. Between the intersections of a o, bl with this axis, 0 < f3 < 1, a = 0, and each f3 corresponds to a different pulsatance of the dominant term in (2) § 15.13. Hence in this range, by varying q we Inay obtain any pulsatance in the range 0 < pw < w. When the point is in any other stable region (q > 0) the dominant term will not always have the lowest pulsa.. tance, e.g. in (2) § 5.310 the dominant term has a pulsatance about four times that of the first tern}. Thus in a system satisfying (4) § 15.11, by varying the driving force (proportional to q) and/or the constraint (proportional to a), the pulsatance of the dominant component nlay be any multiple or sub-multiple of 2w, the pulsatanee of the driving force.

15.20. Frequency modulation. In radio telephony, e.g. broadcasting, a wave train of constant frequency and amplitude is radiated from the aerial at the transmitting station during intervals. This is called the 'carrier' wave. The process at the transmitter whereby this wave is made to 'carry' signals, which become audible at the receiver, is known as modulation, i.e. the signal characters modulate the carrier wave, In the receiver there is a circuital arrangement for demodulating the radio frequency, thereby reproducing the original signals, usually rendered audible by a loud-speaker, There are three types of modulation, (a) frequency, (b) amplitude, (c) phase. We shall deal with (a), where it is implied that signals cause a variation in the frequency of the radio oscillations at the transmitter. The same principle is used in the case of the audio warble tone employed in acoustical test rooms to reduce standing wave effect. Instead of using a constant test frequency, the latter undergoes cyclic 4~

Nn

APPLICATIONS OF y·+(a-2qcos2z)y = 0

274

[Chap. XV

variation covering a band of 50 cycles per second or more, t at a rate which may be altered.

15.21. Classification. Frequency modulation may be classified under three heads: 1°. Direct capacitance modulation; 2°. Exact frequency modulation; 3°. Inverse capacitance modulation [2]. 10 • Direct capacitance modulation. Consider the loss-free circuit shown schematically in Fig. 23 A. Inductance L is in series with

~

(c)

(COCOS 2CtJ1 t

Co

Back plate

~

R

(D)

23. (A) Schematic of loss-free electrical circuit with periodically variable capacitance. (B) Illustrating frequency modulation without amplitude modulation. (c) Illustrating capacitance type microphone. (n) As at (A) but with resistance R in circuit.

FIG.

capacitance C, which varies with time, so we write C = C(t), a function of t. We assume that in practice the loss in L, C, and the connecting wires would be neutralized by the negative resistance effect of a thermionic valve. If Q denotes the quantity of electricity in the capacitance, the circuital differential equation is d 2Q Q dt 2 + LO(/,) = O. (1) We shall assume that C(t) = Go( 1+E cos 2wl t), Go being constant and E ~ 1. If E = 0, the central or carrier pulsatance of the circuit is Wo = 1/(Wo)l. As G(t) alters from Oo{I-E) to 0 0 (1+E) and back again periodically, the pulsatance varies over the range l/[Wo(l-E)]l to l/[LCo( I + E)] l , and the central or carrier wave is said to be frequency modulated. A frequency modulated wave is illustrated in Fig. 23 B. t Usually a percentage of the central or test frequencr.

AI>PLICATIONS OF y"+(a-2qcos2%)y

15.21].

==

0

275

Using the binomial expansion of (1+Ecos2w1t)- 1, (1) takes the form d2 Q -dt2 +W~(I-Ecos2wlt+E2COS22wlt-E3cos32wlt+ ... )Q = o. (2)

If we write z = WI t, expand the circular functions, and put 80 = (WO/Wl)2[1+1E2+ ...], 282 = -(WO/Wl)2[E+fE3+...], etc., (2) becomes (3)

which is Hill's equation, whose solution is treated in § 6.11 et seq., and § 15.25.

Since the extreme values of C(t) are w~ax

=

Co~l±E), we

have

I/LOo( I - E),

so Wmax ~ wo(I+1E): also £Omin ~ £00(1-1£). Thus the pulsatance variation on either side of £00 is approximately !(£Omax-£Omln)

=

~£O = l£OoE,

so £ = 2~w/wo' Substituting this in (2) and omitting terms in £2, E3 , ••• , as a first approximation we obtain

d2 Q

-dTi+(w~-2wo~wcos2wlt)Q

Writing z

=

WI

t, Q

=

=

O.

(4)

y in (4) leads to the canonical form

d'y dz 2 + (a- 2q cos 2z)y = 0,

(5)

with a = (WO/wl)2, q = wo~w/wf. The solution is in § 15.26. If 0 were fixed and L varied cyclically, an equation similar to (5) would be obtained. 2°. Exact frequency modulation. This is a mathematical concept, and so far as the analysis is concerned, we need not consider the method by which the desired practical result is obtained. The differential equation takes the form d 2Q dt 2 +t/J(t)Q = 0,

(1)

where .p(t) is a periodic continuous or piecewise continuous function of t. If [.p(t)J. = £Oo-~w cos 2Wl t, the pulsetanoe varies from

APPLICATIONS OF y#+(a-2qcos2z)y = 0

276

[Chap. XV

(wo+~w) to (wo-~w) exactly, and not approximately as in 1°. Substituting this value of .p(t) in (1) leads to

d"Q dt2 +(w:+!~2w-2wo~wc
=

0,

(2)

which is a form of extended Mathieu equation, with 80

=

(w:+!~2w)jwf,

8" = -2wo~wjwf,

84

=

1~2wjwf,

lJe = 8s = ...

= 0, Z = wI t. If Jiw is small enough, the terms in Ji2w may be neglected and (2) is then identical with (4) § 1°. Otherwise the equation must be solved without this reduction. 3°. Inverse capacitance modulation. An example of this class is found in a certain type of condenser microphone shown schematically in Fig. 23 o. Assuming that a sound wave of pulsatance WI impinges on the diaphragm, the capacitance of the microphone varies inversely as the distance d = do(I-Ecos2w1t) from the back-plate. Then the capacitance at any instant is given by C(t) = Kjdo(I-Ecos2w1t), where do is the separation between the diaphragm and back-plate in quiescent intervals, and K is a constant for the microphone. By (1) § 1° the differential equation for the system is, with Co = Kjd o'

or

d2 Q Q -+---(1-Ecos2w1 t) = 0 dt" LCo '

(1)

~t~ +w~(l-Ecos2wlt)Q=

(2)

0,

w:

where = l1LOo• The nomenclature 'inverse capacitance modulation' is due to (I-Ecos2wlt) being in the numerator in (1), whereas in (1) § 1° it is in the denominator.

15.22. Modulation zones. The differential equation for the three classes of modulation considered in § 15.21 may for present purposes be expressed in the form d 2y dz2+a(I-2ycos2z)y where z

=

=

0,

(1)

WI t, 2w1 being the modulating or signal pulsatance;

a = (wolcvl)'A, Wo being the central or carrier pulsatance which is modulated; " = ~w/wo, dw being the deviation on either side of Woe 2wl is the rate at which the deviation occurs. Equation (1) may be used for In, 2°, § 15.21 provided the requisite approximation is permissible.

15.22]

APPLICATIONS OF yN+(a-2qcos2z)y

=0

277

Referring to the stability chart Fig. 8, it is expedient for present purposes to use that half where q ~ 0, and to interchange the a, q ?'= 0·5

12

y= 0·4

[:¥J~;;l:b===:::I====:::::tt::=:::J" =0·025 1

9

4

:FIG. 24. Stability chart for frequency modulation [2].

II

= 0·025

(W FIG.

O/W

1)~= a

25. Modulation zones in Fig. 24- [2].

axes. Then we have the diagram Fig. 24, and by giving y a series of values, the lines q = ay may be inserted as shown. It is convenient to divide Fig. 24 into three zones as illustrated in Fig, 25. Zone 1, covering the range 0·1 ~ y < 0·5, is of theoretical interest only. 2~w approaches to wo, and 2wl to wo0 The line y == 0·5

278

APPLICATIONS OF y'+(G-2qC082z)y

=

0

[Chap. XV

may be regarded as a boundary, since in practice y < 0-5, in fact it is usually much less than 0-5. The second zone includes the audio warble tone and covers the range 2 X 10 2 ~ a ~ 2·4 X 108 , with o < y ~ 0·1, ~w < WO, wI ~ Wo0 Zone 3 covers the range of frequency modulation in radio transmission, where 9X 10 5 ~ a ~ 1·2x 10 13 , o < Y ~ 0·025, ~w ~ WO' WI ~ WOo For a given value of wo, since y = ~wjwo, any of the straight lines in Fig. 24 represents a condition of constant pulsatance deviation. 2y may be regarded as the modulation coefficient; the signal strength is proportional thereto.

15.23. Stability of oscillations. Consider the circuit of Fig. 23A in conjunction with the line" y = 0·025 in Figs. 24, 25. Imagine the circuital parameters to be continuously variable so that as Wo = Ij(LCo)t increases, the ratio (WO/Wl)2 = a does likewise, but y, the fractional deviation, remains constant at 0·025. Commencing at a = 0, the line y = 0·025 intersects the (a, q) curves where they are close to each other. and as a increases, the intercepts in the stable regions steadily increase, while those in the unstable regions decrease. As a rough approximation the intersections occur when a = m 2 , m a positive integer. With a > 106 , the probability of instability in a radio circuit, whose parameters are chosen at random, is small. Moreover, if a is large enough, the operating point (a, q) will probably lie in a stable region between two characteristics for Mathieu functions of integral order. This statement is based on the assumption that changes in a = (wO/w1 )2 are likely to be small. The curves a l OOO ' bl oo l cross the a-axis at a = 106 and a = (10 3+ 1)2 = 106+2001, respectively. Thus when q is small and positive, if we select a point midway between a l OOO ' b100l ' the permissible variation in pulsatance ratio a l = wO/wl' to avoid encroaching upon an unstable region would be less than 1 part in 2,000. When q > 0 exceeds a certain value the two characteristics begin to approach each other rapidly, and the permissible variation in (WO/Wl) to avoid instability then decreases with increase in q. In precision work, where thermostatic control and temperaturecompensated components are used, the pulsatance variation of an oscillator would be very small. The greater y, e.g. 0·5, the wider the intercept in an unstable region and the higher the probability of unstable operation. It is clear, therefore, that to ensure stable operation over a wide range

15.23]

APPLICATIONS OF y'+(G-2qcos2z)y

=0

279

of a, and consequently of the signal pulsatance ZW1,Y = ~w/wo should be small compared with 1/2.

15.24. The line spectrum in radio frequency modulation [21]. We have mentioned that in this case a is very large. If y~ 1, then a ~ 2a'Y = 2q > 0, and the condition for validity of (13) §4.80 is fulfilled. Apart from a constant multiplier, this series represents Q, the quantity of electricity in the electrical circuit Fig. 23 A. Thus writing Q for y, and Wtt for z in (1), (13) §4.80, we obtain the current dQ

I = dt

=

ro

-Wl,.~~,.. C,.Jr(q/2a l)sin[(al-2r)w1 t- cx],

(1)

where Or = O(a l - 2r ) ~ Cal, since in §4.80, a l ~ roo Now q = ay, Q = (wO/w 1)2, 'Y = Aw/wo, so (1) may be written ro

I ~ -woO ~ J;.(Aw/2wt)sin[(wo-2rwt)t-£x].

(2)

r=--r.

Taking ~w/2Wl = h, J- r

=

(-I)rJ;., (2) may be set out in the form Pulsatances

Discrete components -1/woO = Jo(h)sin(wot-cx> +Jl(h){sin[(wo-2wt)t-ex]-sin[(wo+2wl)t-£X]}+ + Js(h){sin[(wo- 4Wl)t- ex] +sin[(wo+ 4Wl)t- £x]} +

+

+ .

wo' wO-2wl' Wn- 4WI'

the carrier WO+ 2Wl wo+ 4w t

. +

+JT,(h){sin[(wo-2rowt)t-ex]±sin[(wo+ 2rOwl)t- (X]} wo-2rowt, WO+2rOwl

(3)

It follows from (3) that when the carrier of pulsatanee Wo is modulated by an oscillation of pulsatance 2wt ~ Wo, there results a line spectrum or series of discrete pulsatance components. These are spaced at equal intervals 2w1 on each side of the carrier or central oscillation of pulsatanoe wo, as illustrated in Fig. 26. The amplitude of the component of pulsatance (wO±2rwl) is proportional to the J-Bessel function of order r, namely, J;.(Aw/2wt}. In Fig. 26, Aw/2wl = 0·1. When> 1, the amplitudes of some components may exceed that of the carrier, as will be seen from tables of ~(u).

15.25. Approximate stable solution of (1) 1° § 15.21. The equation may be written

Q"+ ap2Q = 0, with a

=

w~

= I/Wo, P =

(I)

(1+£c08 2wlt)-l. If the conditions a ~ 1,

y~+(a-2qcos2%)y =

APPLICATIONS OF

280

0

[Chap. XV

0<£ < I are satisfied, then by (13) §4.82 an approximate stable solution is t

= A(I+ECOS2w 1t)lcos[wo

Q1(t)

j dt/(l+ECOS2w1t)i]

(2)

t

=

f dt!(I-,VSin 1t)iJ. ... 2w

A(I+Ecos2w1t) cos[fwo!(I+E)i}

(3)

0

~

-e, ~ 0

~

....

~

~

0 <:»

~

'-f

0

...~

~~ 11

~

~

t.)

::::s

-0

...L.)

~

Ci.. E ro

='

Q...

E

-s u

C'1t

0:

c:

u

c: cv ..u tV .J!!

""'0

"Vi

-0

c:

-c

0 U

L.

~

~

c/)

-6~1

-4(1)1

FlO.

Q..

E

ra

-s v c:

~

6,)

~

..L)

::J

Q..

-c

:::J

..J...)

to ~

·Vi

'"'t:s

"'

-A..)

::::J 0-

~

21

-0

~

'::::'

"'

"':::s

c~

Ow

-0

~ .~ ~

I~

.~

""'0

Ii E

'<

I~ "-

.~

0i:

c:

"'

L:

- 2lU 1

L.

OJ

u

0

r--..

C'

e ~ ~

8

-s ~

-0

::s ......,

E

"'u-s

c:

"' en

...L.,)

C'O

~

o.. ~

-0

~

~ II

w -a

,

Lt')

....N

21

~

-0

Ci...

~

~

:::s

e

0...

tJ (.)

-s

os c:

"' t'O

..J..)

~

:::J

a~

-0

0c;;

-c

c:

0

"'

~

2GJ1

~

0 0

::s

-en

L

~

c:

U

V)

4tAJ1

E

~

U

c:::

os

-LJ

C'U

~ ::J

Cl~

-0

0c;;

-0 .... :c

I-

6ltJ1

26. Illustrating relative amplitudes of side bands in frequency modulat.ion due to a single modulat.ing wave.

with A2 = 2£/(1+£), A being an arbitrary constant, the second one not being needed here. Thus, with WO/~I == wo~ QI(t}:= A(1+£cos2w1t)lcos[{wo/(I+£)i}F(A,wIt)],

(4)

where F is an incomplete elliptic integral of the first kind with modulus A, i.e, the integral part in (3). The type of function represented by (4) is depicted in Fig. lOA_ Both amplitude and frequency modulation are exhibited. The former is associated with the factor (1+£cos2wIt)1, and the latter with F()', WI t). These expressions are periodic in t, with period 'IT/WI' The ratio max.jmin, amplitude for £ = 0·75 is [(1+0-75}/{1-0-75}]1 = 1-63/1. This is an extreme case and corresponds to i' = 0-375 (see

APPLICATIONS OF y'+(a-2qco82z)y

15.25]

=

0

281

Fig. 25). If € were 0-02 for radio frequency modulation, the amplitude ratio would be 1·01, i.e. 1 per cent. amplitude modulation. By (2) above and (16) §4.82 the 'instantaneous frequency' is

,

f(t)

wo

= d-t!r t 21T L

f

dt/(I+E:COS2w1t)l]

=

(WO/21T)/(I+E:cos2w1 t)l. (5)

o

If E = 0-5 the max. and min. frequencies are approximately 1.41/0 and 0.82/0 , with 10 = WO/21T. Thus the variation is asymmetrical about/o_ For E: = 0·02 the values are nearly (1±0·01)/0, 80 iffo were 10 megacycles per second, the frequency would vary 10 5 cycles per second on each side of the carrier, at a rate WI/1T c.p.s. The number of oscillations in one period of the modulating frequency WI/'IT is the integral of the' instantaneous frequency' over the period. Thus by (5)

f

~/wl

N

=

Wo 271'

Writing

-~o..

=

T,

= (WO/WI~

7T(l+E:)l

f

0

(6)

i~

dT/(I-A2sin2T)i

=

o

Taking Wo = period W 1/1T are

N =

wl

=

(WO/Wl) F(A, 171'). 1T(I+E:).

(7)

21TX 1()3, by (7) the oscillations per

10' 1TX 1·01

X 1·5874 ~ 5000.

Current. corresponding to (4). Since 11(t ) of (4) leads to

= -

dtj(I-A 2 sin 2w 1 t )l .

(6) becomes

21TX 107 ,

11 (t)

f

~/wl

271'(1 +e)i

0

wit

N

dt/(1+E:cos2w1t)1 =

(8)

= ;t Ql(t), differentiation

fl+EC~2Wlt)t[(l~oe)lsin{(I~oe)iF(A'WJ.t)}+ E:Wl sin 2wI t {wo F(~ )}] + ~i(i+-~oos-2~ltjlCOS (T+€)l ,Wit ·

(9)

The type of function represented by (9) is illustrated in Fig. 10 B. When Wo ~ WI' and e is such that the second term in [ ] may be neglected in comparison with the first, I l (t)max occurs approximately at the same t as Ql(t)m!D (see Figs, lOA, B). ~1

00

282

APPLICATIONS OF y'+(a-2qcos2z)y = 0

[Chap. XV

15.26. Approximate solution of (4) 1°§ 15.21. Referring to (I) §15.25, we now take pi = [1-(2~w/wO)C082wlt]. Writing e

the solution is

=

2aw/wo, v

t

Ql(t) = [B/(I-E cos 2wl t)l]C08[ Wo

Now

f (I-E cos 2wl t)l dt °

l

(I)

E (). I (I-Ecos2w1t)1 dt = (I+E)1 I (1-~2COS2wlt)l dt = ~1(1+:)-1' t

t

o

°

t)

where E 1 is an incomplete elliptic integral defined at (14) § 4.82, with ).1 = 2€/(I+E). Thus Q1(t) = [B/(l-E cos 2wI t)l]cos[ WO(1+E)lE1 ( ). , WI t)],

(2)

B being an arbitrary constant, the second one not being required here. The 'instantaneous frequency' is given by f(t)

=

(wo/21T)(I-t" cos 2WI t)l,

(3)

1

and

N = -(wo/wI)(l+t")IE()', 11T). 1T

(4)

15.30. Circuit having resistance. Hitherto we have discussed resistaneelesa circuits, the differential equation having the form at (I), 1° § 15.21. Suppose that a constant resistance R is included in circuit, as shown schematically in Fig. 23 D. The p.d. across R is RI = RdQ/dt, 80 the equation for this case is d 2Q

dt 2

If

€

RdQ

+L

dt

Q

+ LO(t) =

(1)

O.

is small enough, we may take l/C(t) ~ (1-€cos2w1t)/CO' and

(1) becomes 2Q

d dtl

+~ dQ +JL(l-ECOS 2w1 t ) = L dt

Leo

O.

(2)

Writing Q=y, IfWo = w3, z=w1t, Rjw1L=2K, a=(wO/wl)2, 2q = ii«, (2) takes the form y"+2I
= o.

(3)

The solution of equations of this type is treated in §§ 4.83-4.86, 5.60, and stability conditions are given in § 4.83. An unstable oscillatory

APPLICATIONS OF yW+(a-2qcos2z)y

15.30]

=

0

283

cirouitj may be stabilized by adjusting R to some suitable value depending upon the other parameters. Moreover, &S shown in § 4.84, the stable regions for (3) cover a greater area than those if IC = o. They increase in extent with increase in R and, therefore, IC. When the point (ii, q) is on a characteristic curve for equation (3), Ie = JL in §4.84, and the motion is periodic, i.e. a neutral condition. Once started, the oscillation would (in theory) continue unaltered. In practice, however, it would usually either decay to zero or increase until its growth was arrested by the natural limitations of the system.

If (a,q) lies in a. oscillation} ~ 0

~±oo

stabbile} region of Fig. 11,t I( unsta e JL as t ~

> fL} and the > IC

+00.

15.31. Approximate stable solution of (1) § 15.30. Substitute

Q = ve- K' , with RfL = 21<, and we get

d2v

dt 2

+(ap2_ 1(2)V =

=

where

a

(ap2_1(2)

=

w~ = I/LCo, p2 = 1/(I+£cos 2w1 t ). Now a
d2v

dt 2 +acp2v =

1, 0 < E solution of (2) is

If a

0,

~

<

1, 1(2Ja

~

o.

(1)

write

(2)

1, by (13) §4.82 an approximate stable t

v1(t)

=

Btp-l cos

B being an arbitrary constant. But

[w o f tp dt],

(3)

°

= (1+ £cos 2wl t) -6[ 1- (1(2/a) ( 1+ Ecos 2Wl t) ]1

( 4)

~ (1+£cos2w1t)-I_(1(2/2a)(I+£cos2w1t)i,

(5)

(6)

and by (4)

also

(I+Ecos2w1t)i =

(I+E)I(I-~2sin2wlt).

(7)

t It is/ tacitly implied in such circuits that a thermionic valve is incorporated, so that a negative resistance effeet may be obtained. t Reference is made here to the stability diagram for (3). The boundary lines are I'

=

K.

APPLICATIONS OF y*+(a-2qcos2z)y

284

=0

with,\2 = 2EJ(1+E). Hence from (3)-(7), on substituting Q1 we obtain

[Chap. XV

=

VI

e- Id ,

Ql(t) ~ - !J~_~/(I(1+f co~_~Cl!!-~~: cos{w [!~'\,_~!!l_ 1(2 E (>.,w 1t)]}. o (l+E)l [1-(1(2/4a)(I+Ecos 2w 1 t)] 2a(1+E)-i (8)

The type of function represented by (8) is that of Fig. lOA but having an exponential decay, i.e. the influence of R is to cause extinction of the oscillation, unless a negative resistance effect were introduced to counteract it.

15.32. Circuit with periodically varying resistance. By (I) § 15.30, the equation for the free oscillations is

d2 Q RdQ Q dt2 + L dt+ LC = 0, where L, G are constant, but R 21Tfwl' mean value zero, Icpl ~ 1.

Write Q = e-l«l)u(t), with K(t)

(1)

= R o+R 1 ~(Wl t),

= 2~

f

tp having period

t

R(T) dr, then (1) becomes

U"-t-(_lLO__ 4L2 R2 _ R~.)u = 2£

0

·

(2)

Now R2 = R~+2RoRl~+Rfcp2, R' == dR/dt = w 1 R 1 r.p ' , and if we put 1(0 = R o/2L, 1(1 = R 1/2L, w~ = (1/La}-(R~/4L2) = a, we obtain

R') _

R2 1 ( LC- 4L2-2L

[21<'011 +Klll2 +wl({l ']

2 -,WO-K1

= a-2q.p(w 1t).

Thus from (2), (4)

(3)

(4) (5)

which is a standard form of Hill's equation, whose solution may be obtained by the procedure in Chapter VI. If we commence with the circuital equation

L ~: + RI + ~ and write I

=

f I dt = 0

(6)

e-K(l)w(t) , we obtain U'

"+ (1L(J- 4L2+ R2 R') 2£ w = 0,

(7)

APPLICATIONS OF y'+(a-2qcos2z)y

15.32]

=

0

285

which is identical in form with (2) except for the sign of R' /2L. Then (7) may be written

w"+[a-2q.pl(W 1t)]w

=

0,

(8)

where 2q.pl(Wl t) = 1(1{2I
L~~ +RI

+b II dt

=

E(t).

(9)

Then with the above substitution we get

w"+[a-2q,pl(W1t)]w = (eKCJ>/L) (dE/dt) ,

(10)

which may be solved by aid of Chapter VI. These equations occur in the theory of the super-regenerative radio receiver, and reference [45] should be consulted for further details which are too extensive for inclusion here.

15.40. Dynamical system satisfying Mathieu's equation. A simple link-mass-spring mechanism is shown schematically in Fig. 5 A. The mass m slides over a frictionless horizontal plane, along the straight line CA. The links are massless and the pin-joints at A, 0, B are frictionless. Also the spring is massless and unstressed when m is at D, while AB, OA are long enough for motion parallel to BJJ to be negligible in comparison with that along the axis of the spring. The driving force Fo == lfocos 2wt is applied to the cross-head B. It may be resolved into two components, one along AB, the other along DA. The latter is nearly (focos2wt)y, and it causes m to slide along the line CDA. Three forces are to be considered in connexion with m, namely, (1) the inertia md 2y/dt2 , (2) the constraint sy due to the spring, (3) the driving force (Jocos 2wt)y. Then the equation of motion is (1)

Substituting z form

= wt, a = s/w2m, 2q = folw2m, we obtain the standard d 2y dz 2 + (a-2qcos 2z)y =

If we write

q

==

o.

(2)

aI', where I' =-= 10/ 28, (2) becomes

y"+a(I-2ycos2z)y = O.

(3)

286

APPLICATIONS OF yH+(a-2qcos2z)y

=0

[Chap. XV

15.41. Behaviour of system in § 15.40. Draw a line a = q/y in Fig. 27. Since a = s/w 2m, w will be large at points on the line near the origin, but will decrease gradually in passing along the positive direction. When w is such that the line intersects one of the curves bm, am' e.g. the points 1'-8, the motion of m is periodic and neutral.

FIG.

27. Stability diagram for mechanism in Fig. 5

A.

Otherwise it is either stable or unstable according to the region in which the point (a,q) lies. The remarks in § 15.17 on variation of pulsatance apply in the present case when (a, q) lies within or on a border-line of a stable region. For a component of given pulsatance to have an amplitude greater than that of any other, (a,q) must be suitably selected. For instance, suppose the motion includes the first sub-harmonic of the driving force. Since all Mathieu functions of odd integral order, namely, ce2n +1(wt , q) and se2n +1(wt, q), have a subharmonic of pulsatance w, the values of a, q at the intersections of the respective curves a 2n +1 , b2n +1 , and the line a = slv fulfil the required condition. For predominance of the sub-harmonic we consider the relative magnitudes of the coefficients A, B in the series

15.41]

APPLICATIONS OF y·+(a-2qcos2z)y

=0

287

for ce1n +1 , se2n +1 , n = 0, 1, 2,.... From [95], with q small and positive, A~~n+il) predominates in one series and B~~n+\t) in the other. For example, if q = 1, Ail) ~ 0·99, the remaining A being negligible: Bil ) ~ 0·994, the remaining B being negligible. In these cases the sub-harmonic predominates. If q = 24, At ~ 0-398, As ~ -0-744, A 5 ~ 0·499, A 1 ~ -0·190, so the amplitudes of the 3/2 and 5/2 harmonics exceed that of the sub-harmonic: also B 1 ~ 0-818, B s ~ -0·527, B" ~ 0·224, 80 the sub-harmonic predominates, but to a much smaller extent than for q = 1. Tabular values show that the amplitude of the sub-harmonic in se, exceeds that in eel' ces, ceo' se s , ses for q in 0 < q ~ 40. (See § 29, p. 394.)

15.42. Demonstration of sub-harmonic. For this purpose the device of Fig. 28 is preferable to that of Fig. 5 A. The link-messspring mechanism in the latter is replaced by a relatively heavy mass m Spring steel strip

.t

.

~~""":WWjJbaM.

~.

- - -:-_- - ~ ;'-"

IIr--

hcos2~t.

. . . . ----r-l--.. . . . . . . 8 E%Vt:~~ (driving force) '---'

Flo. 28. Dynamical device for illustrating sub-harmonic.

mass at the centre of a light flat strip of spring steel fixed at one end A, but free to move along the direction AB at the other end. So long as the pulsatance of the driving force at B is small enough for absence of wave motion on the spring, this mechanism may be used to demonstrate the effect of varying the parameters m, 8, ui, /0 in (1) § 15.40. In § 15.41 we showed that the best condition for a predominant sub-harmonic was with (a, q) on b1 , q being small or moderate. Thus a = 8/w 2m and y = /0/28 must be small or moderate. To render the motion of m visible, w should be low, so 1n must be relatively large and fo small, since we postulate s to be small. If it is Bot feasible to make w low enough, the motion of m could be rendered visible stroboscopically. By using suitable values of m, 8, the mechanisms of Figs. 5 A, 28 may be used to demonstrate stability and instability.

15.43. System of Fi~. 5 A with Iinear damping. If r is a constant representing the resistance per unit velocity, (1) § 15.40 takes the form d 2y dy mdti+r dt +sy = (focos 2wt )y . (1)

APPLICATIONS OF !l'+(a-2qcos2z)y

288

Then with z

=

0

(Chap. XV

= wt

we get y"+2Ky'+(a-2qcos2z)y = 0, (2) where K = r/2wm, a = s/w2m, 2q = fo/w2m. (2) is identical in form with (3) § 15.30, and the remarks in that section apply here also.

15.50. Vibration of stretched strtng, The classical case of a stretched string with immovable ends is a simple one. But when one end is held and the other is given a periodic motion along the length, the analysis is more complicated and leads to a Mathieu equation. We commence with constant tension and assume the string tZ'=o

rr---~

l------:::-----.....~ d~df f'/,~

~..J(L

2H. Illustrating a vibrating string with periodically varying tension superimposed upon constant tension.

to be elastic, uniform, and loss-free. Referring to Fig. 29, consider an elemental length ds. T is the horizontal component of the tension at ds, and PI its vertical component. By similar triangles o,/8x = 1i/T, so T1 = To~jox. (1) The rate of change of 1i with respect to x is by (1) o'l.~/ox = TCJ2gjox 2 , (2) so the difference in T1 at the ends of the element ds is 01i = Tdxo 2ejox2• (3) When the amplitude of vibration is small, we may write dx for ds, 80 the vertical force to accelerate the element ds of mass m dx is aT! = (mdx)8 2ejot 2• (4) Equating the r.h.s, of (3), (4) leads to the equation of motion, namely, mdx0 2,/ot2 = Tdx0 2f/ox s,

or

2 0, 2

ot

_!.m

2 0,

ox2

= O.

(5)

(6)

15.51. Solution of (6) § 15.50. During vibration there are stationary waves on the string, the boundary conditions being, = 0 at ,:r == 0, 1. OUf solution must satisfy these. Now eib(z+cU), with a 2 = Tim, satisfies the equation, so its real and imaginary parts must

15.51]

APPLICATIONS OF y"+(a-2qcos2z)y

= 0

289

both be solutions. Then eih(z+~) = cosb(x+cxt)+isinb(x+cxt), and since = 0 at x = 0, the solution cannot contain cos bx, Thus of the four product pairs in the expansion of the real and imaginary parts we must choose sin b» sin bOtt, sin b» cos bOtt. Hence the appropriate formal solution of (6) § 15.50 is ~ = (A sin bcxt+B cos bcxt)sin bx = f(t)sinbx, (1)

e

e

A, B being arbitrary constants. Evanescence of at x = 0, l, entails b = 1T1l,/l, n a positive integer =1= o. The pulsatances of the natural modes of vibration are given by W n = ba = (7Tn/l)(T/m)i, n = 1, 2, 3,.... The number of nodes between the ends of the string is (n-l), and there are n antinodes or points of maximum displacement, corresponding to the nth mode of vibration.

15.52. Periodically varying tension. Suppose that, instead of being constant, the tension is expressed by T = To(I-2ycos2wt), then (6) § 15.50 becomes

02{_~(1_2YC082wt)02~ =0 2

ot

ox2

m

(I)

with 'Y < 1. By virtue of similarity we assume (1) § 15.51 as a solution and obtain the equation d2J dt 2

2

+ bmT:0(1-2ycos 2wt)f =

0,

(2)

where I(t} is different from its former value. Writing z = wt, a = b2To/w2m , q =: yb 2To/w2m = ya, (2) transforms to the standard form d2J (3) dzi+(a-2qcos2z)! = o. Since a = slr and 'Y < 1, we get the case depicted in Fig. 25, where the straight line having this equation intersects all the characteristic curves. Unless the parametera a, q are specially selected, the point (a, q) will not in general lie on a characteristic for a function of integral order. Moreover, if (a, q) lies inside a stable region the motion will be represented by a Mathieu function of real fractional order, e.g. ce2n+~(z, q). In practice owing to damping, the motion once started would die away with increase in time.

15.53. Melde '8 experiment. This was described briefly in Chapter I. If an electrically maintained steel reed, having a frequency of the order 50 cycles per second, is used to vary the tension ~1

pp

290

APPLICATIONS OF y"'+(a-2qcos2z)y

=0

[Chap. XV

of the string, the rate of vibration will be sensibly constant. The tension To or the length 1, or both, may be adjusted so that the string vibrates without a node. Its pulsatance is then half that of the reed, i.e. a subharmonic occurs. If now"'the string is stopped and released, its amplitude increases with increase in time, and attains an ultimate value. By reducing the reed amplitude such that q < some value qo, the vibration of the string dies away. To explain these observations, terms representing both damping and non-linearity must be incorporated in the differential equation.

15.54. The modified equation. Adding viscous damping proportional to the velocity and a non-linear control term (odd in y, in virtue of symmetry of motion about the central position), we obtain y"+2Ky'+(a+py2-2qcos2z)y == 0 (p > 0). (I) For a subharmonic to occur as in § 15.53, a ~ I and (u,q) lies in an unstable region between hI and a l in Fig. II. A first approximation solution may be obtained by taking y == Atcosz+Btsinz. Then, with A = (Af+Bf)l, the ultimate amplitude is IAI ~ [(4/3p){(I-a)+(q2-4K 2)1]]I. (2) The condition for maintenance of the vibration, i.e. the reality of

IA I, is

q

> qo ==

{4K2 + (I - a )2}1.

(3)

Amplitude limitation is caused by a mis-tuning effect due to the nonlinear tension control py3. t

15.55. String whose mass per unit length is periodic. If in (6) § 15.50 we write m = mo(I-2ycoS<Xlx) for m, the equation becomes 02, 02, ot 2 - {T /[m o(I - 2y coS<Xl X)]}ox2 = O. (I)

e

Assume that = j(t)g(x), where f is a function of t alone, and gone of x alone. Then d~

d~

gliii-{T/[m o(I-2ycoscx1x)]}!dx 2

=

O.

(2)

Dividing (2) throughout by fg leads to 1 d2f 1 d2g jdt 2 = {T/[m o(l-2 y c OS lX1 X)]} g dx 2 '

(3)

t A detailed analysis is given in McLachlan, Ordinary Non-linear Differential EquationB in Engineering and Physical Sciences (Oxford, 1950).

15.55]

.4.PPLICATIONS OF y·+(a-2qcos2%)y

==

0

291

Since the l.h.s. is independent of x and the r.h.s, of t, each must be a constant, say, ,\t. Then using the l.h.s. and the separation constant ,\2, we get d1f+>..,/= 0

dtl

I=

80

J

sin>J or

(4)

,

cos >J.

(5)

From the r.h.s. of (3) we get

dig

dx 2

,\2

+ Tmo(I-2ycoS<XIX)g =

0,

d2g

dz2 -t-a(1-2y cos 2z)g = 0,

or

(6) (7)

where z = tal X, a = 4,\2m o/af T, q = aI', Y < 0·5. The solution of (7) should be stable and satisfy the boundary condition 9 = 0 at x = 0, l. The stable solutions for q > 0 are given at (1)-(4) §4.71. The boundary condition excludes (1), (3) §4.71, 80 we consider (2), (4) §4.71. Now 9 in (6) mayor may not be periodio in x. If periodic, fJ = pis, a rational fraction in its lowest terms, o < fJ < 1. Write z = 87TX/l, then

sin(2r+p/s)(S1TX/l)

=

sin(2r+ 1+p/S)(s1Tx/l) = 0

for all r when x = 0, l. Hence we may take se m+p /s (s1Txll, q) as those solutions of (6) which satisfy the boundary condition. Before the complete solution can be written down we have to determine A. As in §§ 15.22, 15.41 we have the relationship a = str. y being a known constant. Then the straight line 80 defined may be drawn as shown in Fig. 27. Its intersections with the iso-p]« curves for sem+p{s (z,q)-see the iso-p curves of Fig. II-give the values of (a, q) for which (6) satisfies the boundary condition. Then we have ~+P/s = am+p/s<xf T/4m o' If there is an integral number of periods of mass-distribution on the string, we may take <Xl = 2n1T/l, so the pulsatance of the vibration is (8)

Then the complete solution of (1), with two arbitrary constants, appropriate to the boundary condition, is

f = m~o[A"ICOS>.m+PI8t+BtnsmAm+PI8t]sem+PI8 (81TX) -l-,am+PlsY • 00

•

(9)

APPLICATIONS OF yW+(a-2qcos2z)y = 0

292

[Chap. XV

15.60. Column subjected to axial pull with periodic component [121]. Referring to Fig. 30, the pin-jointed column sustains a steady axial pull ~, but owing to out-of-balance machinery an alternating or ripple component -2y~cos 2wt is superimposed thereon. Thus the total instantaneous axial load is P = Po(I-2ycos2wt),

-------------~~-----~----I

---------t1.-t ~=o

Flo. 30. Illustrating

long column with pinned ends: 8 periodic force is superimposed upon a constant axial pull.

8

where usually 'Y < 0-5. To render the analysis tractable we make the following assumptions: 1. The column is a uniform solid or hollow homogeneous cylinder, its outer diameter being small compared with the length l. 2. The lateral displacement due to P is small compared with l. 3. The maximum stress is within the elastic limit of the material. 4. Shear stress and rotatory inertia are negligible. 5. The ripple pulsatance is well below that of the first longitudinal mode of the column, i.e, there is no longitudinal wave motion, and the column moves substantially as a whole in an axial direction. Let E = Young's modulus of elasticity; I = second moment of cross-section about a diameter; m = mass of unit length of the column; = lateral displacement at any point distant x from the right end (Fig, 30). Then it may be shown that the equation of motion is

e

04e

or

a2e

a2e

EI axc-~(1-2'Ycos2wt)ax2+mBt2 = 0,

(1)

f) 2e _ ~(1-2yco8 2wt)f) 2e + EI f)te = o.

(2)

att

m

e

ax2

m axe

The displacement and the bending moment El o2e/ax! must vanish

15.60]

APPLICATIONS OF y'+(a-2qcos2z)y

=

0

293

at the ends of a pinned column. Thus the boundary conditions are ~ = 0, 02e/ox 2 = 0 at x = 0, l. From §§ 15.51, 15.52 we see that a, solution of the form, = f(t)sinbx, with b = n'lr/l, satisfies these conditions. Substituting this solution in (2) leads to the equation 2

:n

2

2

d f + [b dt m (Elb 2+Po)-2')1 E b cos 2wt ] f 2 Writing z = wt, a = b2(Elb2+Po)/w2m , q the standard form

ddz2f + (a-2qcos 2z)1 = 2

If now we write

')11

=

= o.

(3)

YPob2/w2m , (3) takes

o.

(4)

=

yPo/(Po+ E l b2 )', (4) becomes d2f dz 2 + a( 1- 2')11 cos 2z)f = 0,

(5)

so the question of stability for a given n in b = 'TTn/l may be considered in the same way as in § 15.23. The stability in practice exceeds that predicted from theory owing to internal loss in the material, and an idealized equation incorporating this loss takes the form at (2) § 15.43. The complete solution of (2), with constants All' may be written

LAllfll(t)sin~~X, 00

g=

(6)

n=l

fn(t) being that solution of (4) corresponding to the particular n. Proof of the convergence of a solution of type (6) will be found in reference [121]. When the inherent stiffness of the column vanishes, it may be regarded as a flexible string. 'I'o obtain this metamorphosis mathematically, we write EI = 0 in (3), thereby reducing it to

d 2f

b2P,

dt 2 + m 0 (1- 2y cos 2wt)f = 0,

(7)

which is identical with (2) § 15.52, but with Po for To. The stability of both column and string is discussed at some length in [121], to which reference may be made.

XVI APPLICATION OF TH.E WAVE EQUATION TO VIBRATIONAL SYSTEMS 16.10. Elliptical membrane [130]. Let m be the mass per unit area, T the uniform tension per unit arc length, t , the displacement normal to the equilibrium plane of the membrane (horizontal). Then in vacuo the equation of motion of a homogeneous loss-free membrane in the x, y plane is 2

2

2

0 ' +iJy20' nn aX"i T 0ot'2 =

O.

(1)

This is (6) § 15.50 with the addition of the term -(T/nn)(o2'/oy2), thereby including the coordinate y. If' varies sinusoidally with time t, we may take' = eiwlj(x,y), so that (1) becomes 02{

ox2

+ 02{ + m w 2{ = O.

(2)

oy2 T Comparison with (1) §9.10 shows that kf = nnw"/T, so 2q = nnw2k 2/2T . Hence with this value of 2q, (3), (4) § 9.21 are the ordinary differential equations into which (2), expressed in elliptic coordinates, may be divided. The appropriate solutions, in product form, may be selected from Tables 10-12 to satisfy the physical conditions of the problem. 16.11. Physical conditions. Referring to Fig. 31, if we start at 1] = 0 and move counter-clockwise round a confocal ellipse = 1 < to the displacement, at any instant alters continuously. It '1fI1Jy be

e e

repeated between TJ = '" and TJ = 21T, but it is repeated after TJ = 21T. Hence' is single-valued and periodic in the coordinate TJ. The period is either 1T or 21T, so that 1)) = 1]+17') or '(f, 1]+217') as the case may be. Thus of the product functions in , = t/J(f)(1J) = cem (1J) or sem(1J ) or a constant multiple thereof. In addition to periodicity in 7], at any point (0,7]) and the corresponding point (0, -7]) on the interfooal line, we must have: (a) continuity of displacement, i.e,

,(e,

'(0,1])

t

,(e,

=

'(0, -1]);

(3)

To obtain uniform radial tension per unit arc length, the membrane is stretched uniformly and clamped between circular rings. Elliptica.l rings are then clamped on the membrane within the circular ones.

16.11]

WAVE EQU..~TION TO VIBRATIONAL SYSTEMS

29&

(b) continuity of gradient, i.e,

o

a~['(E, 71)]~ ....o =

-

0

()~mE, -71)]~-+o'

(4)

These two conditions imply continuity of displacement and gradient in crossing the interfoeal line orthogonally.

}"IG.

For ce m ( 1])

,=

=

31. Diagram for membranal problem, from which the Mathieu functions originated.

Cem(e)cem (1]), Ce"&(O) is a non-zero constant, while cem ( -1] ). Hence (a) is satisfied. Also (}

cem(±71)()E Cem(~)~-+o

so (b) is satisfied. For' = Sem(e)sem( 1]), '(0)

= 0,

8 sem(71) ()~Sem(E)€-+o

=

=

0,

(5)

so (a) is satisfied, while 0 -sem( -71) ()~Sem(~)€-+o

(6)

thereby satisfying (b). For' = Fe m (, }cem (1J), Fem(O) = 0, so (a) is satisfied, but 8

cem(71) ()~Fem(~)€-+o

=

8

cem( -71) ()~Fem(~)€-+o i= 0,

(7)

except for special values of 11, so in general (b) is not satisfied. Proceeding in this way, we find that the only permissible forms of solution are

'(',11, t) = and

Om Cem(~' q)cem(1J, q)cos(wmt+£m) (am) ~(',7J,t) =- S,nSem("q)sem(7],q)cos(wmt+im) (bm )·

(8) (9)

296

APPLICATION OF THE

[Chap. XVI

0l1P s'n are arbitrary constants determinable from the conditions specified for the configuration of and velocity distribution over the membrane at t = O. W m is the pulsatance of the mth free mode of vibration, and Em its relative phase angle. Since each integral value gives a, separate solution of the differential equation, the complete , solution of (1) § 16.10 when expressed in elliptic coordinates is co

,(g,Tj,t)

=

IOmCem(e,q)cem('f},q)cos(wmt+£m)+

m=O

co

+I

SmSem(g,q)sem(1J,q)COS(Wmt+Em)·

(lO)

1n=1

16.12. Symmetry. In any product pair solutions, Ce or Se is constant on any confocal ellipse. Hence the symmetry is governed by ce or se, Since ce2n(1J, q) = ce2n(1T± 1J, q) = ce 2n(21T- 1J , q), it follows that the displacement expressed by Ce2n(" q)ce2n(1J, q) is symmetrical about both the major and minor axes of the ellipse. ceSn+1 (1J, q) = -ce 2n+1(1T± 1J ,q) = ce2n+l (21T- 1J, q), so that the displacement expressed by Ce2n+l(g,q)ce2n+l(1J,q) is symmetrical about the major axis but anti-symmetrical about the minor axis. se2n+1(1J ,q) = =Fse2n+1(1T± 1J, Q) = -se2n+1(21T- 1], q), 80 Se2n+l(f,q)sean+l(1J,q) corresponds to a displacement anti-symmetrical about the major axis but symmetrical about the minor axis. Finally se2n+2( 1], q) = ±se2n+2(1T± 1], q) = -se 2n +2(21T- 1J , q), so Se2n+2(~' q)se2n +2( 1], q) corresponds to a displacement anti-symmetrical about both axes. These deductions. will be understood more readily by reference to Figs. 1-4. As a guide to memorization, it may be remarked that the symmetry or anti-symmetry about the major and minor axes is the same as that for the degenerate forms cosmz, sinmz about z = 0, t1T, respectively (see remark at end of § 12.10).

16.13. Vibrational modes of membrane. Each individual solution in (10) § 16.11 for m = 0, I, 2,... corresponds to a different mode of vibration. When q has its appropriate value, the dynamic deformation surface of the membrane and the pulsatanoe differ for each m. Any mode may exist separately-at least in theory-or they may all be present. The maximum displacement of the surface for a particular mode depends upon the value of Om or 8m. These in turn are governed by the configuration of and the normal velocity

WAVE EQUATION TO VIBRATIONAL SYSTEMS

16.13]

297

distribution over the membrane at time t = o. For instance, it might be pulled from the centre into a conoidal shape and released at t = o. The boundary condition for all TJ is that' = 0 at the clamping rings where = ~o. Since cem(1J, q), sem(TJ,q) are independent of " we must have

e

Cem(~o, q)

and

Sem('o,q)

= =

0

for m = 0, 1, 2, ...

(1)

0

for m = 1, 2, ...

(2)

eo

Now is fixed, so we need those positive values of q, say qm,r' qm,r for which the respective functions in (1), (2) vanish. These may be regarded as the positive parametric zeros of the functions. Tiley define a series of confocal nodal ellipses. There is an infinity of zeros for each m, i.e, one for each r = 1,2,3,.... Hence for Cem , ifr = p, there are (p-I) nodal ellipses within the clamping rings, and likewise for Se m , but their locations differ. (I), (2) are known as the period or pulsatance equations, since their roots are used to calculate the pulsatanoes of the modes of the membrane. For q > 0, m ~ 1, the functions ce m ( TJ, q), se m ( TJ, q) have zeros in TJ (see Chap. XII). Hence' in (8), (9) § 16.11 vanishes also if '1 satisfies the respective equations

and

ce m ( '1, qm,r) = 0

(3)

sem(TJ,iim,r) = O.

(4)

The roots define a series of confocal nodal hyperbolas. Now cem(TJ, qm,r) , sem(TJ,iim,r) each have m zeros in 0 ~ TJ < 17' (see § 12.10), so for a given m each function gives rise to m nodal hyperbolas.

16.14. Expansion of function. From physical considerations it is clear that' in (10) § 16.11 must be a continuous function of ~, 7], within the ellipse, which vanishes at its periphery. This suggests the Theorem: That any function of ~, '1, continuous and single-valued within the ellipse, which vanishes on its boundary, may be expanded at any point of the interior in the form ofa double series, namely [52] {(" '1) =

! [! Cm.rCem(~,qm.r)cem(1J,qm.r)]·+ + f [! 8m.rSem(~,qm.r)sem(1J,qm.r)]' (1)

m==O r=l

1u=1

r=:l

A formal proof of the theorem is outside our present purview, -&961

Qq

APPLICATION OF THE

298

[Chap. XVI

16.15. Determination of Om,,., 8m,,.. Suppose that '(~, 1]) has the properties stated above. Multiply both sides of (1) § 16.14 by (cosh 2e-cos 21J)Ce.(e, qn,p)oon( 1], qn,p) , (1) and integrate with respect to '7 from 0 to 277', and with respect to f from 0 to eo. Then by (7) § 9.40 all terms vanish except when 11, = m, r = p. Hence OI "

Jf

o

Cem(e, qm.r)cem( n, qm,r)(cosh 2e-cos 2"/),(,, ,,/) ded,,/

0

l.2ff'

= Om,r

f f Ce~U, qm.r)ce:n("/,qm.r)(oosh 2e-cos 2,,/) d~d,,/. o

(2)

0

Now by (7) § 14.40 a" f ce:n(,,/, qm,r)cos 2"/ d"/ = 110m•r

(3)

o

so we obtain ~o 2ff'

Om,r --

f f Cem(~' qm,r)cem( "/,qm.r)'(~' ,,/)(cosh 2~-cos 2,,/) d~d,,/ eo f Ce~("qm,r)[cosh 2,-0m,rJ d,

0 0

(4)

11

o

tr

In like manner we can show that

Sem(" iim,r)sem(,,/, iim,r){(" ,,/)(cosh 2,-cos 2,,/) d,d,,/

8.m,,.-

0 0

·

f.

(5)

f Se~(',iim.r)[cosh2'-'fm.r] d~

11

o The denominators of (4), (5) may be evaluated numerically (see§ 9.40).

16.16. Transition to circular membrane. Using the results in Appendix I, and omitting the time factor, (10) § 16.11 degenerates to the well-known form [202, p. 27] ClO

{(r,8)

where

c; =

=I

00

O~Jm(klr)cosm8+

m-~

I 8~Jm(klr)sinm8J m-l

(1)

P:nOm' S;,. = s:nSm. The pulsatanee equations (1), (2)

§ 16.13 become

Jm(k1 a) = 0, (2) whose roots define the nodal circles, while the nodal diameters are defined by the roots of c08{m8-tan-l(S~/O;")}=

o.

(3)

16.17]

WAVE EQUATION TO VIBRATIONAL SYSTEMS

299

16.17. Example. We shall consider the mode in which the displacement is proportional to Ce1 (e, ql,l )ce 1(1], ql,t), with a = 5 em., b = 3 em., e = 0-8. The first step is to calculate qI,t' the lowest positive parametric zero, for which Ce1(' 0, qI,t) = 0_ (1) I Then e- = cosh = 1-25, and from tables = 0-6931: also eft = 2, e-fo = !, e in these latter cases being Napier's base. Writing ieo for z in (16) § 2.13, we obtain to order 3 in q-this being adequate for illustration-

'0

'0

Ce1 (, 0, q) ~ -ex3 (I\ cosh 7'0-3 cosh 5'0+1 cosh 3'0)-1-

2(!cosh 5eo-cosh 3'o)-cxcosh 3'0+ cosh '0 =

+cx

0,

(2)

where ex = lq. Using the preceding numerical values in (2) leads to the approximate equation ex3+O·578ex2-1-85ex+O-569 = 0,

whose lowest root is found to be ex of the mode in question is Wt

= ae ==

Now k1h = 2ql , h and (4) gives

WI

~

(3)

0·393. By § 16.10 the pulsatanee

= k1 (T jm )l .

4, so with ql,t == 8ex

(4)

=

= 0·S86(Tjrn)l.

3·144, k1 = 0·886,

(5)

Omitting the time factor, the normal displacement of the membrane at any point (" 'YI), is expressed by 't(~, 1]) =

C1 Ce(" ql,t)ce(1],qt,t),

(6)

0 1 being an arbitrary constant dependent upon the greatest amplitude. The corresponding cartesian coordinates are x = h cosh, cos 1], y = hsinh,sin1], with h = ae = 4. Since ce1(1],Ql,l) = 0 when TJ = tn, in, the minor axis is a nodal line. Some idea of the configuration of the membrane at any instant. may be gleaned from the graph of ce1{7J, q) in Fig. 1. The displacement is symmetrical about the major, but anti-symmetrical about the minor axis.

16.18. Transition to circular membrane. If we apply the formulae in Appendix I when e ~ 0, a remaining constant, (6) § 16.17 degenerates to '1 (r ,8)

When {}

= 11T, f1T,

=== O~ J 1(k1 r )C08 8.

(I )

cos8 = 0, so there is a nodal diameter. The

APPLICATION OF THE

300

first positive zero of J1 (k l a) is k1 a = 3·832, and since a k1 = 3·832/5 = 0·766. Then WI = k1 (T j m )1 -:;- O·766(T/m)l.

[Chap. XVI

=

5 om., (2)

Comparing this with (5) § 16.17, we find that with a constant, an increase in e from 0 to 0·8 raises' the pulsatance of the mode by about 16 per cent.

16.20. Free oscillations of water in elliptical lake [104]. In Fig. 32 A, , is the vertical displacement of the water surface from its

y' FIG.

32. (A), (B) Illustrating problem of elliptical lake.

equilibrium position, and d the uniform depth. We assume that (1) the lake is stationary in space, (2) , varies as ei wt with respect to time. If the displacement is small enough for its square to be neglected, it can be shown that the differential equation of motion in rectangular coordinates is 2

2

2

ox" +0oy" + wc2

0

'

'

,

= 0,

(1)

where c" = gd, c being the velocity of a free wave in a very large expanse of water of uniform depth d, and 9 the acceleration due to gravity. Applying the transformation in § 9.20 to (1) leads to

16.20]

WAVE EQUATION TO VIBRATIONAL SYSTEMS

301

as' as, a~+a'I'J2+2k2(C08h2~-C082'I'J)' = 0,

(2)

with 2k = wh/c. Then, by an argument akin to that in § 16.11, it can be shown that the required solution takes the form 00

,(e,7/,t)

= I

m-O

GmCem(e, q)cem(1], q)COs(wmt+ £m)+ 00

+ m-l I8mSem(e,q)sem(Tj,q)coS(Wmt+Em).

(3)

When the appropriate value of q is assigned, each term of the series in (3) corresponds to a normal mode of oscillation. There are two types of mode, namely, Cemcem and Semsem. An infinity of modes of either type corresponds to each m. The instantaneous displacements of the water for these modes are proportional, respectively, to Cem ( " q)cem(7], q)cos(wmt+E'm) and

Sem("q)sem(7],q)cos(wmt+im ) ,

(4)

q having an infinity of values for each m, Each corresponds to a distinctive dynamic deformation surface of the water. When the appropriate values Om' 8m are used, , in (3) represents the configuration of the surface of the water at any instant t ~ o. As usual Om' 8m, are obtained by aid of the displacement and velocity distribution over the surface at t = o.

16.21. Boundary condition. Consider any elemental arc length ds, and another one dn normal thereto, as depicted in Fig. 32 B. Then it may be deduced that the water-particle velocity in the direction of dn is ig

Un

By (6) §9.12, dn

a,

= --. w on

= ds! = ltd" so

(1)

(1) may be written

a,

ig un = wi! a~'

'0'

(2)

At the boundary of the lake, , = and the velocity of the water normal thereto is zero. Hence at e = Un = 0, so the boundary condition is

'0'

[8ae']t=fo -_ · 0

(3)

302

APPLICATION OF THE

[Chap. XVI

In the pairs of product functions Cemcem, Semse". in (3) § 16.20, cem, are independent of~. Thus (3) entails

&em

= Se:n(fo, q) =

Ce:n(~,q)

and

0

(4)

0,

(5)

ee:n

80 we require qm,p, qm,p, the positive parametric zeros of and Se:n respectively. (4), (5) are designated the pulsatance or period equations.

16.22. Expansion of function. From purely physical considerations it is evident that, omitting the time factor, , in (3) § 16.20 must be a function continuous within the ellipse, but having zero normal gradient at its boundary. This suggests the Theorem: That any function of (" 7]) continuous and single-valued within the ellipse, having zero normal gradient at its boundary, may be expanded at any point of the interior in the form of a double series like (1) § 16.14 [52]. In the present case the values of q are designated qm,p, qm,p, these being the positive parametric roots of Ce:n(~o, q) = 0 Se:n(~o, q)

and

=

(1)

O.

(2)

The coefficients Gm , l~m in the expansion are determined in a way similar to that in § 16.15.

16.23. Example. To illustrate the analysis in § 16.20 et seq., we shall consider the mode defined by m = I, P = I: the tide height being proportional to Ce1(e,ql,1)ce1(TJ,ql,1). If the eccentricity of the ellipse is e = 0·8, then the ratio of the axes is bla = (l-e 2)1 = 0·6. To determine the pulsatance of the water mainly parallel to the direction of the major axis, we have to calculate qt,l' such that Ce~(eo, qt,t) =

o.

(1)

Differentiating (2) § 16.17 with respect to ~ and putting ~ we get

= lq,

Ce~(fo,q) ~ -a3 ({g sinh 7eo-fsinh5fo+sinhSfo)+ +~2(lsinh5eo-3sinh3eo)-3~sinh3eo+sinheo =

o.

(2)

Now coshfo = = 1·25, fo = 0·6931, ell = 2, and e-t • = 0·5. Expressing the hyperbolic functions in exponentials and using these numerical values leads to the approximate equatior e-1

13·4a3+29·7a2-2S·6cx+l·5 = O.

(3)

16.23]

WAVE EQUATION TO VIBRATIONAL SYSTEMS

The smallest positive root of (3) is 4q = w 2h 2/ C2 , with h = ae, we have

wI =

{X

~

303

0·07, so ql,l ~ 0·56. Since

4lJl,tcl/alel.

(4)

With the above numerical values (4) yields [104] WI ~ 1·87c/a. (5) For the corresponding mode of a circular lake of radius a the

FIG.

33. Diagrams for lowest asymmetrical mode of water in elliptical lake. (A) Velocity u in x direction. (B) Tide height (.

boundary condition is given by the degenerate form of (4), (5) § 16.21 (see Appendix I), so J~(wl alc) = 0, (6) and from Bessel-function tables we find that the lowest root is WI

ale

~

1·84

or

WI ~

1·84c/a.

(7)

Thus the pulsatance of the mode for a lake having eccentricity e = 0·8 is only 1·6 per cent. higher than that for e = o. Moreover, in a practical sense the decrease in width of the lake from 2a to 1·2a = 2b has a negligible influence on the pulsatance of the (mainly) longitudinal oscillation. For this mode, since '1 cc Cel(e, qt,l)ce t( 11, qt,l) , (8) the vertical displacement of the water surface is symmetrical about the major axis, but anti-symmetrical with respect to the minor axis, as illustrated in Fig. 33 B~ and demonstrated analytically in § 16.12.

APPLICATION OF THE

304

[Chap. XVI

The minor axis is defined by x = 0, so x = b cosh, cos 7J = 0, giving 1] = !17' or i17'. Thus ce t (l1 , qi,t ) = 0, so , = 0, and the minor axis is a nodal line. ., 16.24. Example. We now pass on to study the mode m = 1, P = 1 in which the tide height is proportional to Se1("ql.1)se1(11,ql,1)' the oscillation being mainly across the lake. t Since se, = 0 when 7J = 0,11, the major axis is a nodal line. The pulsatance equation is Se~(,o, qt,l)

=

0,

(1)

and we have to calculate the smallest q > 0 to satisfy this. Writing iz for z in (2) § 2.14, and differentiating with respect to z, gives, with ii = lq, Se~('o,q) ~ -
+
= o. (2)

Proceeding as in § 16.23 we arrive at the approximate equation 129·1ci3-77·7
(3)

of which the smallest positive root is Ci ~ 0·175, giving iit,l ~ 1·4. Substituting this in the formulae wi = 4Qt,lc2/a2e2 (see (4) § 16.23), we obtain [104] £01 = 2·96cJa = I·7Sc/b. (4) Thus WI/WI = 2·96/1·87 = 1·58, so

wI =

I·SSw!_

(5)

Accordingly the pulsatance of the first mode transverse to the major axis is 58 per cent. greater than that of the longitudinal mode. This is due to the inmor axis being 40 per cent. shorter than the major axis, 80 the time taken for the disturbance to travel across the lake is less than that to travel along it. From (4) and Table 23 we see that WI is 3·3 per cent. less than that of the corresponding mode of a circular lake of radius b. It follows that if we commence with a circular lake and keep the length in the direction of the x-axis equal to 2a, the effect of reducing the breadth is to raise the pulsatanee of the longitudinal oscillation, whereas an increase in breadth causes a decrease in pulsatance. The order of the changes is small, however, even for large alterations in breadth. For the modes of oscillation corresponding to Ceo ceo, Cel eel' Sel se2,. .. the above method of calculating the values of qm,p, qm,p

t

See § 16.12 regarding symmetry about the axes.

16.24]

WAVE EQUATION TO VIBRATIONAL SYSTEMS

305

fails owing to slowness of convergence of the series in powers of q. Under appropriate conditions formulae in § 12.41 may be used.

16.30. Lake whose eccentricity e -+ 1. The plan of the lake is now a long narrow ellipse. Since cosh = e> 0, so in (2) § 16.23, in order to obtain an approximate value of ql,l for the gravest longitudinal mode, we may replace sinh by The tide height is given by , = 0 1 Ce1(' , QI,I)ce1("1 , Ql,I). (1) Then, with an accuracy adequate for illustration, we find that the equation for <X = q/8 is

eo

5·39cx3_0·667
1,'0 ,-+ '0 '0.

1 = 0,

(2)

whose smallest positive root is
=

(3)

}·SS6c/a.

16.31. Comparison of numerical results in §§ 16.23, 16.30. The three sets of results are set out in Table 23. It is obvious therefrom that the breadth of the ellipse has little influence on the frequency of the slowest longitudinal mode. When the sides of the canal are straight and parallel, however, WI = t'ITcja = 1·57Ic/a, which is 17 per cent. less than the pulsatance when the eccentricity of the ellipse tends to unity (see third row of Table 23). TABLE

23

Plan of lake

Puleatanc« oj 8WwUt longitudinal mode (Wt)

Circle of radius a Ellipse major axis 2a, minor axis 1·2a Ellipse major axis 2a, eccentricity e -+- 1

I·S4c/a I·S7c/a I·S86c/a

I

The comparatively large difference may be attributed to curvature of the sides of the ellipse, which entails a diminution in crosssectional area towards each end of the lake. This feature introduces reflection which is progressive and continuous from the centre towards the extremities of the major axis. Consequently the relative energy of the water per unit mass near the central part of the lake exceeds that in the straight-sided canal of uniform breadth.tHence the' effective length' of the elliptical lake is less than that of a uniform canal, and the lowest pulsatance in the first case exceeds that

t 4i61

Assuming the total energy to be equal in both ceses.

Rr

306

APPLICATION OF THE

[Chap. XVI

in the second. It is of interest to remark that at any distance x from the centre, there is little difference in either the particle velocity in the x direction, or in the tide height in the two preceding cases.

16.32. Surface elevation of water (tide height) in § 16.30. This is expressed symbolically by

't =

0 1 Ce1(f , QI,1)ce1(71, ql,l)' (1) 01 being an arbitrary constant, cos Wt t being omitted but tacitly assumed present. Since e~ 0 and ql,1 is fixed, Cel is constant and, therefore, oc cel (1], qt,t)· (2) Thus the elevation of the water surface is dependent upon the angular position "I, but independent of ,. Moreover, at any point distant x from the centre, but not too near to it, the displacement is substantially constant over the breadth of the lake. From (2) § 2.17 cet (1J, qt,t) = A~l) cOS1J+A&l) cos 371+A~t) cos 571+ ... . (3) Using tabular values [52, 95], we find the A corresponding to qt,t = 0-888 by interpolation. Substituting these in (3) yields C~I(1J, 0·888) ~ 0-990cos 7]-0·123 cos 37]+0-005 cos 57]. (4) In a, canal of uniform breadth and length 2a, the tide height for the slowest longitudinal mode at a distance x from the centre is proportional to sin7Tx/2a. Taking cos TJ = xla, we get , oc sin !n"(cos 7]) = 2[~(!7T)COS 7]-J3(! 7T )COS 37]+J6(! 1T)C08 57]- ...] (5) OC 0-990 cos 7]-0·121 cos 31]+0·0039c0851]. (6) Comparing (4), (6), the difference lies in the slightly larger coefficients of cos 37], cos fYrJ in the case of the elliptical lake where e ~ 1. Consequently in passing from the right-hand ends of the lakes towards their centres, the tide heights calculated from (4), (6) are almost equal for all 1] in 0-!1T. A rough idea of the shape of the water surface may be gleaned from the graph of ce1(z, 2) in Fig. I. When frictional 10s8 is taken into account there is no significant difference in the tide heights of the water at corresponding points in the two cases considered.

'1

16.33. Pulsatances of modes corresponding to Ce2 , Ce3 , ••• when e ~ 1. In [53] it is shown that when fo ~ 0, i.e, e ~ 1, there is only one value of q > 0 for which Ce;,.(e,q) = 0, m ~ I. Thus there

16.33]

WAVE EQUATION

r.ro VIBRATIONAL SYSTEMS

307

is only one mode of oscillation corresponding to each m. If q > qm.l satisfied the pulsatanee equation, we should have ee:n<e,qm,p) = 0 thereby entailing the existence of nodal ellipses. Since for < there is only one parametric zero, namely, gm,l' there are no nodal ellipses associated with the modes of oscillation of an elliptical lake whose eccentricity e ~ I. To calculate the roots of q) = 0, we use a trarrseendental equation like (8) § 3.11, write 2q for a on the Lh.s. and in the continued fraction,'] and proceed to calculate q and the A as in § 3.12 et seq. The following results were obtained [54]: qt,l = 0·88982 0 giving cel ( 71, ql,l) = 0·99241 cos 71-0.12288 cos 371+°-00471 cos 51]-0·00009 cos 71]+ ... ; (1) q2,1 = 3·03907 36 giving oe2(1], q2,t) = 0-39516+0-79032 cos 21]-0-24989 cos 41]+

e eo,

ee:n(eo,

+0·02552 cos 61]-0·00134 cos 81]+0·00004 cos 101]- ... ; qa,l

=

(2)

6-42586 16 giving

ce3(1], qa,l) = 0·71339 cos "1+0·60237 cos 31]-0·35232 cos 51]+ +0·06370 cos 71]-0-00604 C08 91]+0-00036 cos 111]-0·00001 cos 137)+ ... ; Q4,1 = 11·04799 28 giving oe4(1], qt,l) = 0·32544+0·65107 cos 211+°.41535 cos 41]-

(3)

-0-42190 cos 671+0·11561 cos 81]-0·01661 cos 101]+ +0·00151 cos 121]-0·00010 cos 147)+0·00001 c08167)- ... ;

°

(4)

Q5,1 = 16·90474 giving ce6 ( YJ, qo,t) = 0-61238 cos "1+0·57615 cos 31]+0·23319 cos 51]-

-0·45463 cos 71]+0-17534 cos 9YJ-0·03485 cos 111]+ +0·00441 cos 131]-0·00039c081511+0-0000300s 171J-....

(5)

The various series (1)-(5) exemplify (a) the rapid decrease in the numerical values of the coefficients beyond a certain term, (b) the remarks at the end of § 3.33_ Using the above values of qm,l' the pulsatances of the first five longitudinal modes of a lake whose eccentricity approaches unity were calculated, and are given in Table 24_ t This substitution is based on the result that as e ~ 1, a -+ 2q [53].

[Chap. XVI

APPLICATION OF THE

308

TABLE

Mode oj oacillation I 2

Function

Value oJ~.t

ce l ce l ce.

0·88982 0 3·03907 4 6·42586 2 11·04799 3 16·904740

----

3 4: 5

004

ee.

t

24 Puleaumc«

(CIt

(CII for 8traight uniJorm canal

1·5708 •• c!at 3·1416.. cla 4·7124 .. cIa 6·2832 .. cla 7·8540 .• c!a

1·8866c/a 3·4866c/a 5·0699c!a 6·6477c!a 8·2231c!a

The numbers are multiples of .".

The tide ranges for several modes calculated by aid of (2)-(5) are portrayed in Figs. 34-7 [54, 104].

~---.~ Equilibrium

or free level

X Zl FIG. 34. Tide height , in lowest symmetrical mode of water in elliptical lake. The broken line is for a canal of uniform breadth.

c

FlO.

Second asymmetrical mode

36. Tide height , in second asymmetrical mode of water in elliptical lake. The broken line is for a canal of uniform breadth.

16.40. Transverse vibrational modes of ~as in 10nl1 hollow elliptical cylinder. If ep is the velocity potential at any point x, Y in a cross-section, the differential equation of wave motion is 02ep

02ep

ox2 + (}y2 +kf4> =

0,

(1)

16.40]

WAVE EQUATION TO VIBRATIONAL SYSTEMS

,

309

Second symmetrical mode

nd

FlO.

36. Tide height Cin second symmetrical mode of water in elliptical lake. The broken line is for a canal of uniform breadth.

\

FIG.

,,_/

/

/

I

/

I

I

I

37. Tide height' in third asymmetrical mode of water in elliptical lake. The broken line is for a canal of uniform breadth.

where k1 = w/c, w = 21T X frequency, and c is the velocity of sound waves in an unconfined atmosphere of the gas, of uniform density. If the inner surface of the cylinder on which the gas particles impinge is rigid, the velocity normal thereto is zero. Hence the particle velocity v = -(oep!on) = 0, n indicating the direction of the norma] to the inner surface. Accordingly the problem is analytically similar to that of the elliptical lake studied in § 16.20 et seq., so the pulsatance equations are those at (4), (5) § 16.21. When the ellipse degenerates to a circle the vibrational modes of the gas in a hollow circular cylinder are obtained. The pulsatance equation is now (6) § 16.23. It is assumed that the cylinder is long enough for end effect to be neglected, or that practical conditions ensure absence of interference due to the ends of a short cylinder. 16.50. Vibrational modes of elliptical plate. Let p be the density, t the uniform thickness, a Poisson's ratio < 1, E Young's

[Chap. XVI

APPLICATION OF 'fHE

310

modulus, and c2 = Et 2/ 12p(I - a 2 ) . Assume that " the displacement normal to the plane of the plate, varies as ei wl • Then with let = w 2/e" it can be shown that the differential equation of motion of a homogeneous loss-free plate vibrating with small amplitude in vacuo is

04'

4

8' 204' k4' - 0, ox4+ 8y4+ ox2oy21 2

2

2

82 8 ) ( 8 -+ -8 - k f) ( -+-+kf 8x2 oy2 ox2 oy2

or

2

Hence

0 '1

and

CJ2'2

ox2

(1)

,= o.

+ D2't + k2' = 0, oy2 1 1

2 + 0 ' 2_ ox! oy!

k2' 1 2

(2) (3)

= o.

(4)

By § 9.20, (3), (4) expressed in elliptical coordinates are, respec-

tively,

2

8 i:21+--l+2k2(cosh2,-cos27J)'I 02{ ' Os> 2

(5)

2k2(cosh 2f-cos 21J)'2 = 0, + 001J:'

(6)

2

8Of22 '

and

=

0,

OYJ

where 2k = kIlt = (w/c)lh, giving q = wk2/4c. This analysis is valid for a variety of boundary conditions amongst which we may mention: 1°. Free everywhere. 2°. Clamped edge, centre free. 3°. Clamped edge, confocal elliptical hole at centre. 4°. Clamped edge, clamped at a central confocal ellipse. 5°. Clamped at a central confocal ellipse, edge free. The solutions for all five cases must be periodic in '1], with period 1r or 217'. For reasons akin to those given in § 16.11 a formal solution of (5) in cases 1°, 2° is (omitting the time factor)

'im ) =

Om Cem(~,q)cem(1J,q)+8mSem(~,q)8em(1J,q).

(7)

For (6) we have ,~m)

Om Cem(~' -q)cem('1], -q)+8mSem(~, -q)sem('1] , -q). Then ,em) = 'im)+,~m), and the complete formal solution is

,= I

=

(8)

co

m-o

{Om Cem(f, q)cem('1], q)+Om Cem(e, -q)cem(1J, -q)}+ IX)

+ m-I I {8mSem("

q)sem('1] ,q)+ SmSem(f, -q)sem(1J , -q)}.

(9)

18.50]

WAVE EQUATION TO VIBRATIONAL SYSTEMS

311

When the centre of the plate is removed or clamped, the solutions Feym' GeYtn' Fek m, Ge~ must be included, 80 we have for cases 30-5°: ,~m) = omCem(e,q)cem('Y},q)+FmFeYm(~,q)cem('Y},q)+

+Bm Sem(e, q)sem ( 7], q)+ Gm GeYm(" q)se,n('1, q) (10) and ,~m) =

Om Cem(e, -q)cem('1, -q)+FmFekm(e, -q)cem (1], -q)+ +SmSe(e, -q)sem(1J , -q)+GmGekm(e, -q)se m(1J, -q).

(11)

The complete solution takes a form similar to (9).

16.51. Clamped edge, centre free. Consider those modes where the displacement is either symmetrical about both axes or about the minor axis only. The first necessitates a solution with ce2 1P and the second with ce 2n +1 (see § 16.12), the sen~ functions being inadmissible. The appropriate formal solution for the mth mode is, by (9) § 16.50,

= em Cem(e, q)cem (1], q)+Cm Cem(" -q)cem (1J' -q). (l)t The boundary conditions are , == d'jd, = 0 at the clamped edge where, = eo- Using these in (1) leads to the two conditional equat<m)

tions

em Cem(,o,q)ce 'Y}, q)+ e,n Cem('o, -q)cem( '1, -q) = 0 em ee:n(,o, q)cem( 71, q)+ am Ce;n(go~ -q)cem( lJ' -q) == o. m(

and

(2) (3)

From (2), (3) we deduce that

[Cem(eo, q)Ce:n(eo, -q)- Ce:n(~o,q)Cem(~o, -q)]cem { 71, q)cem ( 1], -q) = O. (4)

Hence we have the pulsatanoe equation Cem(~o, q)Ce;,.(eo, -q)- Ce:n(~o, q)Cem(,o, -q)

=

0,

(5)

which may be written

d

d1[Cem(~,q)/Ce"M, -q)]~=f.

=

O.

(6)

This equation is satisfied by q = qm,8' 8 = 1, 2,.... and defines a system of confocal nodal .ellipses 0 < ~ ~ eo' From (4) we have the equations defining a system of nodal hyperbolae, namely,

cem ( 7J, q) t "Then m C~,

ee for Se,

= 2n+ 1, and q 86.

See [229J.

= <

0

and

ce".( 7], -q)

=

0, in (1)-(7) write Se, ae for

O.

ee, ce:

(7) in (8), (9) write

APPLICATION OF THE

312

[Chap. XVI

Nodal ellipses and nodal hyperbolae coexist, being analogous to nodal circles and radii on a freely vibrating circular plate. For modes in which the displacement is anti-symmetrical about the major axis or about both a·xes, the pulsatance equation is Sem(eo,q)Se:n(eo, -q)-Se:n(eo, q)Sem(eo, -q) = O. (8) This defines a second system of confocal nodal ellipses. The equations for the accompanying system of nodal hyperbolae are

sem (7J,q) = 0 and sem (7J, -g) = O. (9) Accordingly there is a double infinity of both classes of nodal pattern.

16.52. Determination of g2n,8 in (6) § 16.51. If q is small, the procedure in § 16.17 may be followed. For large values of q = k 2 the asymptotic formula (3) § 11.42 may be used. Applying the relationship Ce2n (e, -q) = (-I)nCe2n (t ".i + ' , q) thereto, and selecting the real part, we obtain with, large enough Ce 2n (e, -q) ~ C2n [ cosh WI cos 111+sinh WI sin 1D"]/22n sinhl "

where WI = 2kcoshe, and (3) § 11.42

111 ~ (4n+l)tan-1(tanh!e).

(1)

Thus by (1) above

.Ce211(~' q) ......, tanhl~ cos X Ce 2n (, , - q) 21 cosh Xl ,

with

X = 2ksinh,-(4n+l)tan-l(tanh!,)

and

cosh Xl = cosh WI cos 111+sinh WI sin 111.

(2)

Differentiating (2), as indicated in (6) § 16.51, leads to the pulsatanee equation [X' tan X+ X~ tanh Xl]t=fo = (sinh 2'0)-1, or

[tan X+ (X; /X')tanh Xl]e=f. = (X'8inh2e)f~fo.

'0

(3)

Now k is large by hypothesis, so when is large enough the r.h.s, of (3) is very small, (X;/X')tanh Xl ~ I, and (3) approximates to 8

tan X = - l o r X = 81T-l1T = (8-1)1T, being an integer. This entails 2ksinheo = (s-!)1T+(4n+l)tan-1(tanhieo).

Then for 2k sinh

(4)

(5)

eo -+ keel and tanh leo -+ 1, we have keto ~ (8+n)1T,

or Equation

kl"., = (~)

Q2n••

~ (8+n)2,,2e - 2t·.

§ 16.51 may be treated in a similar way,

(6)

16.53]

WAVE EQUATION TO VIBRATIONAL SYSTEMS

313

16.53. Transition to circular plate. Using the degenerate forms of the modified Mathieu functions in Appendix I, both of the pulsatance equations (5), (8) § 16.51 become Jm(kla)I:n(kla)-J:n(kla)Im(kla) = 0

(ki

= wlc),

(1)

which is the known result for the modes of a clamped circular plate of radius a. Using the dominant terms in the asymptotic formulae for the Bessel functions, (1) gives cos(k1a-11T-lm1T)+sin(k1a-l1T-!m1T) = 0

or

tan(k1a-!1T-lm1T)

so, if 8 is an integer,

k1a

==

=

-1,

(2)

(3) (4)

(s+!m)1T.

By Appendix I, as the hounding ellipse tends to a circle of radius a, keto ~ k1 a, so if m = 2n, (4) above and (6) § 16.52 agree. Since kl = wlc, (4) gives for the higher modes of vibration of a clamped circular plate of radius a and thickness t w

= (8+!m)1T~ [_.._~]!. 2 2a

3p(l-a

(5)

)

Also by Appendix I, we see that (7), (8) § 16.50 degenerate to

'lm) =

(G~cosm8+S~sinm8)Jm(klr)

(6)

=

(O~cosm8+S~sinm8)Im(klr),

(7)

,~m)

and

the known forms for a circular plate.

16.60. Elliptical cylinder in viscous fluid [28, 116, 137, 154]. A long cylinder of elliptical cross-section'] has its axis perpendicular to the plane of the paper. A viscous fluid, e.g. air, moves past the cylinder with undisturbed velocity U in a direction 8 to the major axis, as illustrated in Fig. 38. In the absence of viscosity the motion of the fluid would be irrotationaI. Owing, however, to friction between the surface of the cylinder and the fluid, by virtue of viscosity, the flow is accompanied by vortex motion, i.e. a perturbation is superimposed upon the irrotational motion. The problem is to determine the vorticity' when the motion is steady. It is given by the expression (1)

t

The cylinder is many times longer than its major axis. 88

314

APPLICATION OF THE

(Chap. XVI

where .p is the stream function. For steady motion the vorticity is known to be expressed by the differential equation 2'+02 o 0 ox2 oy2' _~/uo' v\ ox + V oy' ) = 0 ,

(2)

U= Undisturbed

velocit~

of fluid

IY' FIG.

38. Illustrating cylinder in viscous fluid.

being the velocity components of the fluid parallel to the x and y axes, respectively, tv == fLip, fL the kinematic coefficient of'viscosity, p the density of the undisturbed fluid. Then, if c/> is the velocity potential, we have U, V

u

= _ o,p =

_

oy

oc/J

(3)

ox'

v=o.p=_oc/J. ox

(4)

By

For the undisturbed motion, t the velocity potential and stream function expressed in rectangular coordinates are q,o = -U(xcos8+ysinfJ), (5) tPo= -U(gcosiJ-xsin9), (6) these satisfying (3), (4). Substituting from the two latter into (2) yields 2 2

88x2+ t 8 0" 8'Oy) = 0• ay2+ ~(~ ax 0, ox + oy '

v

(7)

Applying formulae in § 9.20, (7) is transformed to elliptical coordinates and becomes

02{ + a2{ +~(Oc/J o{ + oc/J 0') =

fJ,2

07J2

II

0, o~

O.

071 071

t In tho absence of friction between the fluid and the surface of the cylinder.

(8)

16.61]

WAVE EQUATION TO VIBRATIONAL SYSTEMS

315

16.61. Solution of (8) § 16.60. Assume that the solution takes the form'

= e-4/"f(f, 7J).

81~

81~

Then since of' + o7J1 = 0, we obtain

e-"" {Olf + olf _-!.[(0~)1+(&/J)1]/} = 8~s OTJ2 v2 8~ oT}

O.

(1)

Expressing (5), (6) § 16.60 in elliptical coordinates by aid of the substitutions x = kcO&h,c087]; y = ksinh,sin7], leads to

4>0 = -Ult(coshfcos7Jcos8+sinhfsin'1sin8) t/Jo = -Uh(sinhtsin7]cos8-coshecos7]sin8).

and

From these we find that

~o

and

(2) (3)

tPo satisfy

Ocp)2 (&P)S U2A? 0' + o7J = --2(

(cosh 2,-cos 2'1).

'4)

Now (2), (3) are for undisturbed motion, so that if (4) is substituted into (1).& first approximation, based on Oseen's analysis. is

0"1

02f

U ik 2

of'+87J2- 2v2 (cosh 2f-c0827J)f = O. As in § 9.21 assume that f(f, '1) = f1(,)f2('1), and we get all d2j U2k2 f2df:+f1d7J:- 2v. (cosh2f-cos2TJ)fd. = O.

(5)

(6)

Dividing throughout by fIla yields 2

-!. d / 1 _ 11 de 2

-~ d / 2 _ 2k2 cos 2'Yl = a a constant, (7) 5 /2 d7J2 "/' 2k2 , or k2 = TJ 2h2/4v2 • Then we get the two Mathieu

2k2cosh 2i: =

with U 2h2/2v2 = equations

2

~r:-(a+2k2c08h2f)fl= 0, d2f

dTJ:+ (a+ 2k· cos 2TJ)J. = O.

(8) (9)

16.62. Conditions to be satisfied. (a) As in § 16.11 the solution must be periodic in 7] with period 'TT' or 21T; (b) the amplitude of the motion must tend to zero as ~ ~ +00; (c) the circulation of the fluid at ~ = 00 must be bounded, otherwise the force on a finite length of cylinder would be infinite. (a) is satisfied if we choose cen(z, -q), sen(z, -q) as the solutions of (9) § 16.61. Now , = e-t/>/"!I(e)!2(TJ),

(1)

APPLICATION 0}4' THE

316

[Chap. XVI

and to fulfil (b), e-.'1I/1 (' ) must ~ 0 as '~+oo. Suppose we select the solutions Fekn(e, -q), Gekn(e, -q) for (8) § 16.61. Then, using their asymptotic forms from §ll.12 with cPo from (2) § 16.61, we must have

e-J,J~fl(e)oc e2k(cosbf cos 'I cos8+slnbfsln 'I sin 8)-keE From §9.12, keosh, ~ r, hsinhe ~ r, as Appendix I, since thee ~ r, kef ~ k 1 r as ~ ~ and substituting in (2) we obtain

J(2;ee)

~ o.

(2)

,~+oo;

+00.

also by §2 Writing 2k = lei b

ek 1r{cos(1J- 8) - l}

e-cMV!I(f) oc ~- ---. -v(2kl r)

(3)

Now {cos(7]-8)-I} < 0 except for (7]-(}) = 0 when it vanishes. Hence (3) tends to zero as r ~ +00, and, therefore, as ~ ~ +00. Accordingly a solution of the form

, = e-~J~L~o c; Fek,,(e, -q)ce,,(1], -q)+

+n~lDn Gek'l(e, -q)se n(1], -q)}

(4)

complies with conditions (a), (b). We shall now demonstrate that a modification of (4) is necessary, in order that condition (c) may be satisfied.

16.63. Circulation of fluid as ~ ~ +00 [116]. We shall evaluate the difference in circulation of fluid over the ring defined by the conand ~ -= ~2 as ~ +00, which arises from one focal ellipses, = term in (4) § 16.62. We take the term Cn e-c/>/v Fekll,(e, -q)ce,~(1J' -q), and the circulation is proportional to

'I

'2

ff e-~J~ FeknU, -q)ce

n(1],

-q) ds! ds 2 ,

(1)

where dsl , dS2 are elemental arcs, as defined in §9.12, the integration extending over the ring. From §9.12, when, is large enough, ds1 d82 ~ rdrd7], so using (3) § 16.62 and the series for een(1J, -q), (1) may be written

= const.

~

f e-kl'rlXtl dr f ekl'COs('1-8>[m~o (-I)mA~) cosm1]] d1], r.

In

Tl

2"

0

(2)t

16.63]

WAVE EQUATION TO VIBRATI()NAL SYSTEMS

317

where r l corresponds to '1' r2 to '2' and XIl

= 1 + (4an-l) + (4an-l)(4an~9) + (4a n8k I r

2! (8k l r)2

l )···(4a n -25~ 3! (8k l , ) 3

+ •.. ,

(3)

which is the series part of the asymptotic expansion of Fekn{z, -q) in (10) § 11.23, if the term in k4 in (8) § 11.20 is neglected. Now ektrCOfl(Tj-(J)cosm7J

=

{lo(k1r)+2 p~/p(k1 r)cosP(7J-lJ)}cosm7J

=

lo(k i r)cosm1J+

00

I

p=l

Ip(k i r)[cos{1J(p+m)-pO}+ +cos{1J(p-m)-p8}].

(4)

Using (4) in (2), all the integrals vanish except when p = m, so

f

217' ektr<X's(Tj-(J) cos m7J

d7J

=

2171m (k 1 r)cos mO.

(5)

o

But when kir is very large [ref. 202, p. 163]

1 (k r) __ ~{1_~~2-1) + (4m m 1 v'(27Tk l r) SkI r 2! (SkI r)2

2-1)(4m2-9)

_

}

(6)

... ·

Substituting from (5), (6) into (2) yields r,

In =

const.

00

f Xn[ L (-l)mA~)cosmOx r.

m=o

X { 1-

(4m 2 - I ) -Sk r 1

2-1)(4m2-9)

+.(4m 2! (Sk1 r)2- -

}] •.. dr.

(7)

00

Since

I

(-1)8+mA~)m28cosm8 =

m=-O

(d28/d028)[cen{O, -q)], the integrand

of (7) may be written

Xn[ oe; +

S:1 r (cen+4ce;) + 21

(S~l r)2 (16ce~ +40ce;+9cen)+ ...]. (8)

Designating the coefficients of l/r, 1/,2, ... in (3), (8) by ai' a2' ... , PI' P2'···' respectively, (8) yields (1 +al r- 1+0:2r- 2+... )(fJO+fJl r- 1+P2r- 2+...)

= ,80+((XI ,80+ ,81),-1+ ((X2 ,80+(Xl,8l + ,82),-2+... .

(9)

[Chap. XVI

31H

Inserting this ill (7) and integrating with respect to r leads to

In =

const.(fJo(r2-rl)+(a:lf30+Pl)log~+ •

r1

+ «()2,80+()t,8t +,82) (~-~j + ...) (10)

= A n ,8o{(r2 - rt)+ «()t+ ,8t/,8o)log(rs/rt)+ + «()2+ ()t ,8t/,80+,82/,80) (~- ~j + ...). (11) Since (11) ~ 00 with f 2, it follows that the presence of a term of the form e-;/vCnFekn (" -q)ce,,(T], -q), as in (4) § 16.62, introduces an infinite circulation as r 2 ~ 00, which is physically impossible. It is necessary, therefore, to modify the term in question to give a tillite solution. Only the first two terms in (II) yield infinities, so we consider these and disregard the remainder of the series, because it is con.. vergent and finite. Then

(Xl+fJl/{30

1 = -k {(4an -

8

l I )+ I + 4ce;} = 2k

cen

1

1

(an+ce~/cen)'

(12)

Now ce;+(a n + 2k 2 cos 28)cen = 0, so that an+ce~/cen

== -2k 2 cos 28,

(13)

which gives

(a:l+f31/f30)

=

k2 Uk 2 --cos29 == ---cos28. k1 4v

(14)

Using (14) in (11), if 8 is fixed-as it would be in practice-with Po = cell. or ceu +1 ' we get

Ill/An cen (15)

:- {('2- r t)-

Uk2

.}

4v COS 28Iogr s/rt+Ln +1 (finite as T2-->- OO) ·

Although both (15), (16)

~ 00

(16)

with r 2 , their difference

(In/A", ce,~)- (In+1/A n+1 cen +1) = L,,,- L n+1,

(17)

16.63]

WAVE EQUATION TO VIBRATIONAL SYSTEMS

319

is finite. Hence the first part of the solution (4) § 16.62 must be taken in the form F'k".t~+l

e-;/V[C", Fek n ce; -0"'+1 ~'ek/~+1 cell +1], (18) are constants whose ratio is chosen to annul the first

=

where On' On +-1 two terms at (I5), (16). ~ The results for the second part of the solution at (4) § 16.62, namely, e-t/>/v Dn Gek n sen are identical in form with those above, except that sin is written for cos, and se for ce. Thus to preserve finite circulation we take Gk n ,U+l == e-cfJlv[Dn(jek/~~e/l,-JJ,t+-lGekn+lse'''+I]. (19) Finally the complete solution which satisfies conditions (a), (b), (c) in § 16.62 is given by

, =

e-cfJ/v {

!

n=O

Fk n ,n+1+

!

Gkn.n+t}.

(20)

1&=1

The arbitrary constants C, D in (18), (19) are found from the boundary conditions as usual.

XVII ELECTRICAL AND THERMAL -e

DI}"~"USION

11.10. Eddy current loss in core of solenoid. When a sinusoidal current flows in a solenoid having a metal core of elliptical crosssection, the varying magnetic field induces eddy currents therein, the core behaving like the secondary winding of a transformer. To calculate the loss arising from these currents we assume that: 1. The current in the winding is everywhere in phase. 2. The core is a uniform metal bar of elliptical cross-section, having

resistivity p and permeability J-L, the latter being independent of H the magnetizing force. 3. The uniformly wound solenoid of n turns is long compared with its cross-sectional dimensions. 4. H is uniform at the curved surface of the bar. Differential equation for H. It can be shown that, in rectangular coordinates, the equation for H at any point (x, y) of a cross-section of the bar is [167] 2H 2H a + a _ 4'Trp. aH = O. (1) ox2 By 2 p at

If H = HI ei wl , oH/ot = iwH, and (1) becomes

alH + a 11 _i(41TJ-LW)H ox2 oy2 p

= o.

(2)

Introducing elliptical coordinates as in § 9.20, this equation is transformed to 02B a~2

02B

+ aTJ2 + 2k 2(cosh 2~-cos 2TJ)H =

0,

(3)

where 4k2 = k~h2 = -ik2(4TTp,w/p}. Thus k 2 is negative imaginary and 2k = i1km, with m = (41TJ.LW/p) 1. Then by § 9.21 a suitable solution of (3) is H = X(e)(TJ), X being a solution of

d2X

~2-(a-2kloosh2E)x = 0,

(4)

and ep a solution of (5)

ELECTRICAL AND

17.11]

THER~I.AL

DIFFUSION

321

17.11. Physical conditions. The distribution of H round any confocal ellipse in a. cross-section of the core is symmetrical about the major and minor axes. Also H is single-valued and periodic in TJ with period 'TT. Hence ep must be a multiple of ce 2n(1], q). Further, to satisfy (b) § 16.11 the solution X

=

Fe 2n (" q)

must be excluded. Thus Ce2n(e, q)ce2n(7] , q) is the only admissible type of solution of (3) § 17.10. Accordingly at any point (e, TJ) of the cross-section we have 00

H

= !

11.==0

(1)

G2n Ce2n (" q)ce2n( 7], q),

the constants G2n being determinable from the boundary conditions. 17.12. Determination of G2n in (1) § 17.11. At the surface of the core, H = Ho, a constant given by 47Tnlr.m .s.!l, I r .m .s. being the root mean square value of the current'] and l the axial length of the solenoid. Thus at , = ~o co

tt, = n::=lO ! G2n Ce2n (eo, q)ce 2n ( 1], q).

(1)

Multiplying both sides of (1) by ce2n(lJ,q) and integrating with respect to 'YJ from 0 to 211, we get 2"

u, f

ce2n('I],q) dTJ

=

2n

(e

G2n Ce2n o, q)

° where

f ce~n(TJ,q) d'l] °

£!2n

217'

f cel

=

n ( 1],

= G2n Ce2n(, 0, Q)£!2n'

(2)

q) d1], the other integrals vanishing by virtue

° The value of the first integral in (2) is 27THoAb2n), of orthogonality.j it being understood that since k 2 is negative imaginary, the A in the series for ce2n (TJ, q) are complex (see § 3.40 et seq.), Thus we obtain G2n

=

21THoA b2n )/Ce 2n (eo, q)£2n.

(3)

Hence by (1) § 17.11, (3) the magnetizing force at any point (',1]) of a core-section is given by Hf.'fJ

=

co

(e

21THo ! A~2n) Ce2n(',q)ce2n(1],Q)/.22n Ce2n o,Q).

"-0

Since all quantities under the! sign are complex, so also is in general. t The r.m.s, value is selected merely for convenience, since it is used in : When q 4961

=

(4) H~,rl'

§ 17.20. k l is real the value of the integral is n, In (2), q is negative imaginary.

T

t

322

ELECTRICAL AND THERMAL DIFFUSION [Chap. XVII

17.13. Transition to circular cross-section. By Appendix I Ce2n (g, q) ~ P~n J2n (k1 r), Ce2n(go, q) ~ P~n J2n {k 1 a), a being the radius of the core, while ce 2n ( 1], q) ~ cos .., 2n8, n ~ 1, ceo(1), q) ~ A~O) = (2)-1, A~2n) ~ 0, n ~ I. Also

i?2n

~

217'

f cos

22n8

o

(4) § 17.12 vanish except when n

=

dO

=

Hence all terms of

'IT.

0, and for a core of circular

section we get

(I)

Now k 1 = i'm, so (1) becomes

H

= HoJo(ilmr)jJo(iama).

(2)

Expressing the J function in polar form, with Jv(i1z)

=

bervz+ibeivz

=

M v(z)ei 9v(Z),

leads to H = R Mo(mr) ei[9o(mr)-9o(m a)]. o Mo(ma)

Mo(z), 80 (z) are tabulated

ill

(3)

reference [202].

17.14. Total flux in elliptical core. This is found by integrating ~H over a cross-section. Thus the total flux is, with ki = i'm,

cI> = ~

I

H H) II (02 -+ki

~ HdA = - -

2

8

ox2

oy2

(I)

dxdy,

by (2) § 17.1 O. Now if n is the outward normal and ds an elemental surface arc, (I) may be written

= -

JL

i.. 8H de =

kf j' on

-

I kf og 211

~

fJH d'1l

o

(2)

""

by virtue of the relationships ds = II d7], on = lIe, in § 9.12. Substituting in (2) for H from (4) § 17.12 leads to = - 21TILHo ~

kf

6

A~2n)Ce~n("q)

~lln Celln(go, q)

I () 211

o

celln '1]

d

(3) '1],

= _ 41T P.H o ~ [A~lln)]ll Ce~n(g,q) kf ~ £2n Cesn('o,q) • ll

This represents the total flux within an ellipse defined by , The total core flux is (4) with for, in the numerator.

'0

(4)

<

eo.

17.15]

~~L~~CTRICAL

AND THERMAL Dllt(."USION

323

17.15. Transition to clrcularcross-sectlon. By aid of Appendix I, (4) § 17.14 degenerates to the form <J)

= _ 2'TTap,Ho J~(kl a) = k1

JO(k1 a)

2'TTap,Ho ~(ifma) mit Jo(itma)

(1)

= 'TTa 2/-LH. [~ M 1 (ma )] ei [81(ma)- 8o(m a )- t 1T] .

(2)

o rna Mo(rna)

Mo, 80 , M1 , 81 are tabulated in reference [202]. 17.20. Effective resistance R, and effective inductance L, due to core. If I r.m.s, is the root mean square value of the current in the solenoid, the p.d. across its ends is E r •m .8 • = (Re+ iwL, )4 .m.s., so

z:

e) Er .m .s . (R t·w L) 1r.m.s, -- Er.m.s, (Re-iwL R:+w2L~ = et'

(1)

Accordingly we assume that I r .m .s. has two components in phase quadrature. One of these causes a flux e (equal but of opposite phase to that arising from eddy currents induced in the core) such that nde/dt = R e / r .m .s.• Thus the core loss is given by d e RI2 4> l r.m.s, . e r.m.s, =n dt

(2)

Substituting Ho == 4'TTnlr .m .s.!l in (4) § 17.14 the total r.m.s. core flux is 4> -

161T3/Ln4 .m.s. ~ [Afrn)]2Ce~n(eo,q)

(3)

- - ----ffl-- ~ - £2n Ce2n(t~~qj- · n=O

e is the real part of (3), so if the flux varies as ei wt , the core loss is, by (2), (3), P

=

n d e 1 dt r.m.s,

[i 60 kf i!2n Ce

= _ ~~!!:~~~~2_!~.~~8_. Real ~ ~~2~2r ~!l({!I~~)] . l

2n(,o, q)

(4)

Substituting k1 = itm in (4), the power dissipated in the core is represented by

p = ~_~'TT3JL~!1'~]2 m2l

r.m.s.

Re911[~ [A~~~~]~_C~~n({~!.q2].

c:

n==O

£ 2n Ce2n (t.so' q)

(5)

Since P = Be l~.m.s.' the effective resistance due to the metal core is given by (5), if l~.m.s. is omitted. The so-called 'imaginary' power or energy per unit time stored in

}~LECTRICAL

324

.AND THERMAL DIFF"USION [Chap. XVII

the solenoid and restored to the power source periodically is found by similar analysis, using the imaginary part of (3), to be

q)]

= 16~~wn2]2

vIrna [~[Ab2n)]2Ce~n('0, (6) g ~ ~ Ce it: ) • n-O 2n 2n'~O' q Hence the effective inductance of the solenoid due to the metal core alone is given by (6), provided wI~.m.8. is omitted. wL ]2

m2l

e r.m.s.

r.m.a,

17.21. Transltlon to circular cross-section. By § 2 Appendix I we find that

~ [A~2~!J~ Ce~!t('o,_q) ~ _ k1 aJ1(k l a)

~ n-O 80

with kf

= -im2 ,

(1)

21T,zO( k1 a) ,

i!2n Ce2n ('0' q)

(5) § 17.20 degenerates to

[J

p = 81T JLwan 12 Real 1 (i f ma ) e- l 1Ti ] ml r.m.s, Jo(i1ma) 2

2

2

-- 81T fLwan ml

2

M1(ma) cos(8 - tJ r.m.s·Mo(ma) 1 0

J2

1) 1T

(2)

•

(3)

The effective resistance is given by (3), devoid of l:.m.s.. In a similar way (6) § 17.20 degenerates to 2

2

L - 81T p.an M1(ma) · (8 -{} _.1-) e ml Mo(ma)sln 1 0 4"·

(4)

17.30. Resistance of long straight conductors of elliptical and rectangular section to high-frequency alternating currents. At sufficiently high frequencies the current is confined mainly to a surface layer of the conductor, and the current density is negligible within the conductor. Thus the component of magnetic force normal to the surface tends to evanescence, and the magnetic force may be assumed wholly tangential thereto. It follows that, at the surface, the vector potential is constant and satisfies the same conditions as the scalar potential. Consequently the surface distribution of current density is identical with that of an electrostatic charge. The total current flowing axially through the conductor corresponds to the total surface charge [169].

17.31. Formula for charge and current density. We have now to find a formula for the charge at any point on the cylinder, bearing in mind that the surface potential eP is constant. Using

325

ELECTRICi\L AND THERMAL DIFFUSI()N

17.31]

elliptical coordinates (" 1J}-see Fig. 39A-tlle potential function constant on the surface is, apart from a constant multiplier, cP

x

=

~

at

~

=

~o.

f

1J=1T

1J=0

(1)

x

Surface of conductor where E = lo

FIG.

39. (A) Illustrating conductor of elliptical cross-section foci Fit Fl. The hyperbola determines the coordinate 1] at the surface where ~ =

eo.

FIG.

39. (B) Ellipse having same area and axes ratio as rectangular section.

Now the charge at any point P whose coordinates are ('0' 1J) is al

=

«,dep/dn,

(2)

where K1 is a constant and n denotes the direction of the normal to the surface. By § 9.12, dn = h(cosh 2'0-cos2 7J)i de, so from (1), (2) 0'1

= K1(~): = K1/h(cosh2~O-COS2'YJ)1.

(3)

(3) is proportional to the charge at any surface point (eo, 71)· Since al is a maximum when 71 = 0, 1T at the ends of the major axis. It is a minimum ~t the ends of the minor axis, where

eo is constant,

326

TJ

= 111, j1T.

ELECTRICAL AND THERMAL

[Chap. XVII

The values of 0'1 at these respective points are O'l(max) = Kl/h(cosh2~O-I)i

and

DI~"'FUSION

= X I /(a 2-h2)i

O'l(mfn) = K1/hcosh"eo = KI/a.

(4)

(5)

Using 0' for current density, the magnetic force tangential to the surface at (eo,.TJ) is, with an appropriate multiplier K [169],

II,

=

41T0' =

41TK/h(cosh2~O-COS2'YJ)i.

(6)

17.32. Power loss. When II, varies sinusoidally at high-frequency, it is shown in § 18.53 that the power loss per unit surface area of a conductor of unit permeability is (I/81T)Hf(pf)l, where p = resistivity, f =.: frequency, II, is the r.m.s. value, and the radius of curvature does not vanish at any point on the surface. The length of an elemental arc on the surface of the conductor (Fig. 17) is by§9.12 ds = h(cosh 2eo-cos2'YJ)id'YJ. (1) Thus the power loss over the periphery, per unit axial length, is, by (1) above and (6) § 17.31, i1r P = (~l Hf ds = 817K2(pj)1 dTJ/h(cosh 2eo-cos2TJ)I. (2)

f

f

o

Now kcosh~o = a, the semi-major axis, while (coSh~O)-l = e, the eccentricity of the bounding ellipse. Accordingly (2) may be written (3)

= ~17~~(1!j)1 F(e, 117) a

(4)

[169], where F is a complete elliptic integral of the first kind with modulus e. The total current in the conductor corresponds to the total charge on unit length, i.e, 2"

fads = K

f dTJ = 217K. o

(5)

Hence the resistance per unit length of conductor to high-frequency alternating current is, by (4), (5),

Rae

.·

=

P/4172K2 = 2(pj)1 F(e, !17). 11a

(6)

17.33]

ELECTRICAL AND THERMAL DIFFUSION

327

17.33. Ratio Ra.c.!Rd .c.. The resistance to a constant unidirectional current is R d•c. = pJTTab (1) per unit length of conductor. Thus by (1) and (6) § 17.32 we have Ra.c'! R d •c.

= 2b(fJp)lF(e, 117).

(2)

When the eccentricity e ~ 1, it can be shown that [215] F(e, 117)

Hence for an ellipse with alb

~ 10ge4aJb.

(3)

> 1, (2) may be written

Ra.c.!Rd .c. ~ 2b(f/p)1Ioge(4a/b). When e -)- 0 the ellipse degenerates to a circle, F(e, !1T)

(4) ~

!1T, and

(2) becomes

Ra.c.!Rd .c. ~ 7Ta~f/p)l, (5) a being the radius. By (6) § 17.32 the a.c, resistance per unit length is R a .c. ~ (pf)i/a.

(6)

17.34. Resistance of rectangular strip. Referring to Fig. 39 B we suppose that al/b l ~ 1 and assume that the strip may be replaced by an elliptical conductor of equal cross-sectional area with aJb = at/bt" Then it follows that b = 2b t 7T- 1• (1) Substituting this in (4) § 17.33 leads to the result that for a rectangular strip, at an adequately high frequency Ra.c.!Rd .c. ~ 4b t (/ /7Tp)i loge(4a 1Jb1 ) .

(2)

Numerical example. Let

2a1 = 2·5 em.} 4a 1 /b1 = 50. 2b1 = 0·2 em. f = 40 X 106 cycles per second. p = 1600 absolute units, for copper.

Then

loge(4a 1 /bl )

=

2·3026 loglO 50

~

3·92.

(3)

Hence by (2)

Ra.c.lRd .c. :;0

~

2 X O·2( 40 X 106/17 X 1600)1 X 3·92

~

140,

Ra.~. ~ 140Ru.c.'

(4)

(5)

328

ELECTRICAL AND THERMAL DIFFUSION [Chap. XVII

17.40. Heat conduction in elliptical cylinder [135]. We treat the case where the cylinder is long enough in comparison with its major axis for end effect to be neglected. If 8 is temperature, I( diffusivity, and t time, the two-dimensional equation for heat transmission, expressed in rectangular coordinates, is 2 fJ 8 fJx 2

2 fJ 8

1 fJ8

+ fJy2-~ ot =

0

(1)

·

Let (8-80 ) = 81 = f(x, y)e- Ky21, wheref(x, y) is a function independent of t, and the purpose of v will appear later. 'I'hen 08 =

ot

_ KV2fJ

(2)

'

and on substituting into (1) we obtain 2

2

+ 0 / +v2j = o. ox2 oy2 0/

(3)

Transforming (3) to elliptical coordinates, by § 9.20 we get

a"!

o2j

fJ~: + fJ1]: + 2k2(cosh 2g-cos 21])/1

where Ie?

= q = !v2h?, and 11 is a

function of

=

0,

(4)

~,'1.

17.41. Particular form of solution of (4) § 17.40. If X(,) • .p(TJ) are functions of g, TJ alone, respectively, the appropriate solution takes the form X(e)«/J(TJ). Herein we consider a case where the temperature distribution is symmetrical about both axes of the ellipse. Thus ¢J(1]) must be even and periodic in 1], with period 17. The only solution of (4) § 17.40 having these properties involves ce2n (1], q) as a factor. For reasons similar to those given in § 17.11, the other factor is Ce2n ( " q). Hence a formal solution of (1) §17.40, expressed in elliptical coordinates is 81

=

00

(8-fJo)

= fl(e, TJ)e- lCvl l = I

n-O

G2n Ce2n(~,q)ce2n(1J,q)e-Ky21. (1)

17.42. Boundary conditions. We take the simple conditions that (a) (J = 0, t < 0, throughout the cylinder; (b) 8 = 80' t ~ 0, at the surface where = '0; (c) 8 ~ 80' t ~ +00, 0 ~ , ~ The problem is to determine the temperature at any internal point

'0.

e

ELECTRICAL AND THERMAL DIFFUSION

17.42]

329

e

when t > o. At the surface = '0' and when t ~ 0, (8-80) = 81 = 0, so by (1) § 17.41 we must have Ce2n(~0' q) = O. (1) Thus q has those values Q2n,m which make Ce2n(~0' q) vanish, i.e, the parametric zeros of the function. Now as t ~ -0, 8 = 0, 81 = -80, and (1) § 17.41 becomes co

-80

ex>

=! m==l ! 0211, Ce 2n (" n==O

(2)

Q2n,m)ce 2n( Tj, Q2n,m) ,

so we have to determine the C2n •

17.43. Determination of C2n • We employ the orthogonality theorem in § 9.40. Accordingly we multiply each side of (2) § 17.42 by Ce 2P (" Q2P,r)ce 2P( 1], Q2P,r) (cosh 2,-C08 21)} and integrate with respect to 7J from 0 to 21T, and with respect to from 0 to Then the r.h.s. vanishes except when p = n, r ::-:: m, so

e

eo.

~o 271

f f Cean(g, qan.m)cea,~(1], qan.m)(cosh 2g-cos 21]) dgd1] fo = o.; f f ee:n(g, qan.m)c~n(1], qan.m)(cosh 2e-cos 21])dgd1].

-()o

o

27T

0

(1)

o 0

Taking individual integrals, we have (see (9) § 9.40 for 0 2n ) 2"

f cean(1], qan.m)cos 21] d1] = 1TA~an); f c~n( 1], qan.m) d1] = 1T; o o f cean(1],qall,m) d1] = 21TA~an); f c~n( qan.m)cos 21] d1] = ,,0an· 2"

27T

27T

1],

o

o

(2)

Hence (1) may be written

eo

-nlJo

f Cean(g, qan.m)[2A~an) cosh 2g-A~an)J de fo = 1T02n f C~n(e, q2n.m)[cosh 2e-0 2nJ de, o to f Cean(e, q2n.m)[2A~2n) cosh 2g-A~an)J de c - ----- to Jo Ce~n(e,q2n.m)[cosh2e-02nlde

o

(3)

-()o

so

2n -

Q-.

-----

------ - ---- - -- -

--- ----- ---------

(4)

The first integral in the numerator iR evaluated in § 14.10; for that ill

the denominator see § 14.~2. 4961

uu

330

ELECTRICAL AND THERMAL DIFFUSION [Chap. XVII

Using (4) in (1) § 17.41 we obtain 8

= 80{1- n-O ! ma:::l ! e-ItJ'I...ICe21nct,Q2n,m)ce2n(1],Q2n,m) X

ffo Ce2n(g, Q2n,m)[2A~n) cosh 2g-A~2n)] dg X (t

.-

-~~---

---

(5)

--------~----

f Ce~n(g' Q21n,m)[ cosh 2g-E>21n] dg o

where ~n,mk2 = 4q2n,m. This gives the temperature at any point (',1]) of the cross-section when t ~ o. Since the factor in (5) involving the integrals is independent of ',1], (5) is a solution of (1) §17.40 expressed in elliptical coordinates. Also (1) 8 = 80 for t ~ 0 at the surface where g = go, since Ce2n (go, Q2n,m) = 0; (2) 8 ~ 80 everywhere when t ~ +00, since e-Kvl",m l ~ o. Hence the boundary conditions (b), (e) § 17.42 are satisfied. 17.44. Transition to circular cylinder. We commence with (1) § 17.41 and by aid of Appendix I find that it degenerates to ex>

! C2nP~n e-KkfIJ2n(kl r)cos 2n
81 =

(1)

with k 1 = v2n,m. Since 81 is now independent of~, by virtue of condition (b) § 17.42, all terms in (1) must vanish except n = O.t Thus 81 = Cop~ Jo(kl r)e- Kk: l • (2) At the surface r = a, 8 = 80 , so k1a = ~O,m' m = 1, 2, 3,..., these being the roots of JO(k l a) = o. Alsoase~O,g~+oo,sinh~~coshf, r ~ It cosh ~ (h ~ 0), dr ~ hsinh ,d~ (see § 9.12), cosh 2,d~ ~ 2 cosh , sinh,d, ~ 2rd'l'/k2 • Using these and Ceo("Qo,m) ~ p~Jo(~o,mr/a), ceo(1],qo,m) ~ 2- 1, A~O) ~ 2- 1, A~O) -)0 0, 8 2ft ~ 0, ~,m ~ cXf.,m/a2 (since by Appendix I, q ~ as e ~ 0), (5) § 17.43 degenerates to

°

a

f rJo(Otmria) dr

ClO

8 = 80 1-

!

e-K(X:'l/a'Jo((Xmr/a)~

,

(3)

f rJ~(Otm ria) dr

m-l

o

where

am

has been written for a

Now

(XO,m'

f rJo(Otmrja) dr =

o

2

~Jl(Otm)' ~m

t A mathematical proof is given in l135J.

(4)

17.44J

331

and a

fo rJ~(otmrla) dr = !a2[Jf(otm)+J~(otm)] = !a2Jf(otm),

(5)

since Jo(am ) = o. Substituting from (4), (5) into (3) leads to the well-known result for a circular cylinder, namely, 8 = 80

[1-2~ Jo(otf!lr/a) e-K~tla']. J ~

m=l

(l:m

(6)

l(a:m)

17.50. An electromagnetic problem analogous to heat conduction [135]. The problems in § 17.10 and § 17.40 are analogous, provided the conditions are analogous. For the eleotromagnetic problem they are as follow: (a) H = 0, t < 0, throughout the metal core; (b) H = Ho = 4tm10/l, t ~ 0, at the surface where ~ = ~o; (c) H ~Ho, t~ +00, 0 ~ ~ ~ ~o. Interpreted physically these conditions imply that-at t = 0 a battery is connected to the solenoid through a high resistance R, such that the time taken for the current to attain a substantially steady value 10 is small enough to be ignored. Thus for analytical purposes the current-time relationship is represented by Heaviside's unit function. The problem is to find H at any point of the cross-section of the metal core, and thence to calculate the inductance (variable) of the solenoid due to the core alone, when t > o. The differential equation for H is (1) § 17.10. If in (1) § 17.40 we write H for e and 4rrJ.tlp for 11K, the two equations are alike. Hence mutatis mutandis, (I) §17.41 is a formal solution of (I) §17.10, so we have eo H == ! C2n Ce2n (g, q)ce2n (1] , q)e-pvtt/41T1L. (1) <

u,

n=O

The solution which satisfies the above boundary conditions is (5) § 17.43, with H written for (}. Thus we get H

= 4rrnL __0 [ ] -! ! 00

l

<X>

e-PV~,.,natI41TJ.L

Ce2n (g, Q2n,m)ce 2n ('T} , Q21l,m) X

11-::0 111.=1

~o

J Ce X

2 11

O~-r

fo

([ ) [

A~2n) cosh 2[-A~n)] d[ _

JCe~n([)[cosh2[-021l]d~ o

(2)

332

J4~LB(;'fRICAL

AND 'fHERMAL

})Il"'~'USION

[Chap. XVII

When the ellipse tends to a circle, this expression degenerates to a known form for a metal core of circular cross-section [203; (4) p. 242]. .., 17.51. Inductance due to core alone. This is defined to be the total flux interlinkage arising from the magnetization of the core, when unit current (unidirectional) flows in the n turn winding. If cJ) is the total core flux, then

L = tt>nJ1o

=

i: ff

H ds1dsa•

(1)

where dB! ds. is an elemental area of cross-section. Now by § 9.12 k dB! = 2' (cosh 2,-cos 21])1 (2)

de

k dss = 2' (cosh 2,-cos 21J)l d7],

and

(3)

"2.,,.

L= h;i; ff H(cosh 2~-cos 27]) d~d7].

so

(4)

00'

Substituting for H from (2) § 17.50 into (4) and using the analysis in § 17.43 leads to

L

= 41r"n;oof' [ 1- ~ ~

x

{l' 0

i

e-pvl...,t/4f1'p. X

n.-:O m==l

CeanU,

qan.m)[2A~2n) cosh 2~-A~an)] d~r

10 f Ce=n(~' qan.m)[cosh 2~- Sanl ~

-.

(5)

o

For evaluation of some of the integrals, see §§ 14.10, 14.22. When the ellipse tends to a circle, (5) degenerates to a known form [203; (14) p. 240].

XVIII ELECTROMAGNETIC WAVE GUIDES 18.10. Introduction. From 1888, when electromagnetic waves were first demonstrated, till about 1936, attention was concentrated chiefly on their propagation in the atmosphere surrounding the earth. It is possible, however, to confine electromagnetic waves to the air space within a metal tube. Propagation of this nature was treated experimentally and theoretically more than half a century ago. The subject has been retrieved from the academical archives, owing to developments in ultra high-frequency technique. There are now ultra high-frequency electronic devices for generating frequencies exceeding 3 X 109 cycles per second, i.e. wave-lengths less than 10 em. Analysis has revealed that if the electromagnetic power is transmitted to an aerial by a transmission line, e.g. a parallel wire or a coaxial cylinder system, the Ioss] exceeds appreciably that incurred when the power is transmitted as electromagnetic waves through the air in a uniform highly conducting tube. Since the power is confined within the tube, it is guided, so the tube is called a wave guide. The subject has now attained considerable importance, and deserves detailed treatment here. Moreover, we shall not confine our analysis to the case where Mathieu functions are involved, but lead up to it from two simpler cases, thereby introducing a difficult subject gradually. References to numerous papers dealing with wave guides will be found in [199]. 18.11. Rectangular wave guide. Analytically this is the simplest case. The convention adopted for cartesian coordinates is illustrated in Fig. 40, the e-axis being parallel to the axis of the guide and constituting the inner left lower corner. At the near end, containing the origin 0, we suppose that there is a device for exciting electromagnetic waves, which are propagated parallel to the z-axis, in the direction of the arrow. We assume that the guide is so long that the influence of reflection from the far end at an intermediate point may be disregarded. Alternatively we may consider the guide to be relatively short, but terminated at its far end by an appropriate t Expressed as a percentage of the input power at the tra.nsmitting end of the line.

334

ELECTROMAGNE'TIC \VAVE GlJIDES

[Chap. XVIII

impedance, such that reflection is negligible, as is the case in practice. Moreover, the transmission is unidirectional. Power is transmitted down the guide in one or more of the' modes' to which the system responds, t To investigate these we have to solve Maxwell's equations for the electromagnetic field, subject to the particular condition which obtains at the inner surface of the guide. Unless we postulate absence of loss due to eddy currents in

o :FIG. 40. Coordinate system in rectangular wave guide.

the wall of the guide, it is difficult to formulate a precise boundary oondition.j Assuming zero resistivity for the metal of the guide, the condition is that the tangential component of the electric field in the air must vanish at the inner surface.

18.20. Maxwell's equations. For an isotropic non-conducting dielectric of permeability f" and dielectric constant e, the 'curl' equations in vector form (heavy type) are

BE = curlH,

€-

at

8H

at =

fL-

-curlE,

(1)

(2)

where E designates an electric, and H a magnetic force space vector. The vectors E and H can each be resolved into components parallel to the axes X. Y, Z of Fig. 40. Let the scalar components be denoted t For power to be propagated down the guide, it may be shown by § 18.24 that the width must be > l~, where A is the wavelength in an unbounded dielectric. : Loss in the dielectric (air) is also assumed to be zero, i.e, it is a perfect insulator.

18.20]

ELECTROMAGNETIC \VAV'g GlJIDES

335

s; n;

by s; and Hz, H,P Hz, respectively. Then in terms of cartesian coordinates, (1) yields the equations: EoEx

at

=

_?J!.II.+ o~ oz

By ,

(3)

(4)

(5)

To aid memorization it should be observed that ill (1), (2) E and H are interchanged, E is on the l.h.s. with E, and JL on the l.h.s. with H, but with a negative curl. Equations (3)-(5) come from (1), and their subscripts are in cyclic order. The subscripts and coordinates (in denominators) in the last two members of each equation are interchanged. Thus by memorizing (1)-(3) all formulae of the set (3)-(8) nlay be written down, For the components of magnetic force we have from (2)-using (3) as a pattern-

en,

aE1/

es,

j.L-=--at OZ oy'

(6)

= oEz _ oEx ox oz '

(7)

oH1J ft

at

a~ It at

=

BEx _ BEy By

ox ·

(8)

18.21. Cosinusoidal variation of E and H. It is expedient to make this choice so that various transmission modes may be segregated. At any section distant z from 0 in Fig. 40 the electric and magnetic force vectors are in phase and vary in accordance with the real part of eiwt . If the phase at any z is referred to 0, account must be taken of the velocity of phase propagation down the guide. The instantaneous distribution of E and H along the guide depends upon the real part of eiwle-iPz, i.e. cos(wt-f3z), where f3 is a coefficient such that pz defines the phase at z with respect to its value at O. For transmission of power down the guide, it can be shown that f3 must be real [see 210 J.

336

ELECTROMAGNETIC WAVE GUIDES

[Chap. XVIII

18.22. Simplification of equations in § 18.20. This is effected by writing iw for a/at, and -ifJ for %z, in accordance with § 18.21. Then (3)-(8) § 18.20 become

· E

oHz 1/+-' ay

"oRR

~w£ x = ~fJ

· E'II =

---'tfJ

·QHx'

(2)

= _ oHz + oHu , ax

(3)

oEz

(4)

~W£

iw€E

z

. H uau. r: x

aHz

(I)

ox

oy

=

·pE

-1,

11

----

By'

z oE 'fJEx' --+t

(5)

iWJLH = oEz _ oEu •

(6)

· HII = uap.

z

ox

oy

ox

Substituting for iElI from (2) into (4), and for iEx from (1) into (5), gives

oQoHz +'tWE. es,

oy

(7)

= _iQoHz_iw£oE!

(8)

k12 H -_ x

and

l.2H "1. '1/

-tfJ-

OX

fJ

ox '

8y

where kf = JLEW 2- fJ2 = (w/C)2_fJ2 and c = I/(JLE)i, the velocity of light. Substituting for iHlI from (5) into (1), and for iHx from (4) into (2), gives . k12 Ez = -1,.p8E (9) oxz- 'l,WJL-ay

eu,

and

k12 E -_ U

+ 'tWfL-· · oHz

·a oEz

-'tfJ-

oy

ox

(10)

Using (7), (8) in (3) for Hz, HII leads to the two-dimensional wave equation o2Ez+02Ez+k2E = o. (II) ox2 ay2 1 Z Similarly from (9), (10), and (6), we obtain

a2B.. + ""I12 H.. '~ _ox 2. +--oy2 ..

8 2B..

O.

( 12)

18.23]

ELECTROMAGNETIC WAVE GUIDES

337

18.23. Solution of (11) § 18.22. Write Ea = .p(x)~(y), where t/J is a function of x alone, and is a function of y alone. Then d2ifJ d2 f dx z +t/J dyz = -kf,pe, (1)

e

e

where kf = (WfC)2_f32. Dividing throughout by

«/1" (1) becomes

I d2«/J 2 _ I d Z, ~dxZ+kl - -~ dyz·

(2)

Introducing a separation constant l2, as in §9.21, we obtain two ordinary differential equations, namely, dZrp2 + (kf-lZ),p dx

:~+lZe =

and

=0

(3)

O.

(4)

The formal solutions of (3), (4) are: (a) cos[x(kf-l2)i], sin[x(kf-l2)i]; (b) cosly, sinly, respectively. From these we have to construct a solution which satisfies the boundary condition Ez == 0, at the inner plane faces of the guide, i.e. at x = 0, a; '!J == 0, b. Since oos(O) = 1, the cosine solutions are inadmissible, so we take Ez

= cPe =

A sin[x(k~-l2)i]sinly,

(5)

A being an arbitrary constant. Now Ez vanishes when x = 0 or y = o. For it to vanish at x = a, y == b, we must have, respectively, a(k~-l2)l = m7T, bl = n1T, m and n being any positive integers. Then eliminating l Dy squaring and adding, we get (6)

Selecting the real part of the phase factor ei (wl - (3z), an appropriate solution of (II) § 18.22 is

s, = A sin ~x sin ~y cos(wt-f3z).

(7)

This defines a wave system in the rectangular guide. By assigning integral values to m, n the modes of transmission may be specified. There is, however, a limitation associated with {J, which we shall now investigate.

18.24. Cut-oft or critical pulsatance. Since fJ2 = (w/c)2-kf, from (6) § 18.23 we get

p2 = (w/c)2- 1T2[(m/a)2+
xx

(1)

338

ELECTROMAGNETIC WAVE GUIDES

For power to be transmitted down the guide, so it follows that

>

w

[Chap. XVIII

fJ must be real [210],

1Tc[(m/a)2+(n/b)2Jt.

(2)

Hence, if m, n be assigned, there is a pulsatance Wc

= 1Tc[{mJa)2+{n/b)2]l,

(3)

below which transmission of power ceases. This is termed the cut-off or critical pulsatance, and it corresponds to {3 = O. Its value depends for a given guide upon m, n. If w is fixed but exceeds wc' there is only a, finite number of m, n combinations (modes of transmission) for which the inequality can be satisfied. 18.25. E or transverse magnetic wave. (11), (12) § 18.22 are two linearly independent equations, so the solution of (11) holds if that of (12) is Hz 0, which implies oHz/ox - 0 - o~/oy. Then from (9), (10) § 18.22 _ ~o~. _ ~o~ Ex - -k2 - ' and E II - -k2 - ' (1)

=

lOX

lOy

Applying these to (7) § 18.23 yields, using - i = e- i 1Ti , and taking the real part, E = A -f3

7T'm

ei (wI - {3z)

1T'mX. 1Tny · (

for cos(wt-{1z),

R)

kf -aC O S -a- S l n -bS l n wt-fJz ; fJ Tfn . 7T1nX 'TTny. E = A- -sln-cos-sln(wt-Rz) 11 kf b a b fJ· • :e

(2)

Also from (1), (2) § 18.22

Ha; = -(w£j/3)EJ/;

H1I

=

(W £//3) E:x;.

(3)

Collecting the six results, we have

Ex = A(fJ/kf)(mnja)cos(1Tmx/a)sin(1Tnyjb)sin(wt-{3z), Ell

=

A(P/kf){1Tnjb)sin('TTmx/a)cos('TTnyjb)sin(wt-pz),

E. = A sin('TTmxja)sin(1Tnyjb)cos(wt-{Jz); Hz = -A{wEjk~){1Tn/b)sin{'TTmxja)cos(1Tnyjb)sin(wt-fJz), B 1I = A(WEjkf)('TTmja)cos('TTmx/a)sin('TTnyjb)sin(wt-{1z), Hz = O.

(4) (5) (6) (7) (8)

(9) In the type of wave to which these results refer, the electric force at any non-boundary point has components in the x, y, z directions. The magnetic force, however, has components in the x, y directions only, so it is wholly transverse to the direction of propagation. For this reason the "rave is called an E or transverse magnetic type. It may conveniently be denoted by the symbol Em,n' m, n indicating

ELECTl{'():\IAGKJ:4~TIC 'VAVJ:4~

18.25]

GUIDES

339

the particular mode of transmission, When m == 0 for any n, or vice versa, (4)-(8) vanish. Hence transverse magnetic waves of the t~ypes Eo,n, Em,o do not exist. The lowest DIode possible is with m == n == 1, i.e. E l fl , which has the lowest cut-off frequency.

18.26. H or transverse electric wave. The linear independence of (11), (12) § 18.22 permits us to take E; == 0, which implies 8Ez /ox = 0 oEz/oy. Then from (9), (10) § 18.22

=

aHz = ik¥Ex; aHz = _ ikf E . (1) oy WJL ox WI-' II Since Ex == 0 at y == 0, b, whilst E1I == 0 at x == 0, a, it follows from (1) that the appropriate conditions to be satisfied by Hz are ollz/oy == 0 at y == O,b; oHz/ox == 0 at x = O,a. (2) Referring back to § 18.23 we now use the cosine solutions of (12) § 18.22 and obtain, with an arbitrary constant B, mnx 7Tny (3) Hz = Bcos--a-cosT cos(wt-f3z), which satisfies the conditions (2). TIlen from (7), (8) § 18.22 II = _ x

ifJ oHz •

= _ i{3 oH

11

kf ox '

2

k~

11

oy ,

(4)

while from (4), (5) §18.22

s, = w; HlI ;

' "'

Ell

= -{: Hz.

(5)

Applying (4), (5) to (3), we obtain 7Tn 7Tmx. 7Tny · f1z) E = - B -WJL -cos-sln-sln(wtx kfb a b '

=

E

BI-'w 7Tm sin 7Tmx cos TTny sin(wt-f3z)

kfa

II

a

b

'

s;« 0; H

:x;

~

fJ 7Tm . mn» '1Tny. -B- - S l n - cos-sln(wt-pz), k~a

a

b

TTn 'TT'mx. 7Tny · ( p) - C O S - - S l n - S l n wt- z k¥ b a b ' mnx 7Tny fJz). Hz = B cos-acosTcos(wt-

H

11

= - B -{3

(6)

(7) (8) (9)

(10)

(11)

In the type of wave to which these results pertain, the magnetic force at any non-boundary point has components in the x, y, z

J.~LECTROMAGNETIC WAVE GUIDES

340

[Chap. XVIII

directions. The electric force, however, has components in the z, y directions only, so it is wholly transverse to the direction of propagation. For this reason the wave is called an H or transverse electric

- - Electric

;ntens;~~

--- Magnetic intensity (A)

Side view

T0 1

- l,./

L-' ~

-~

-~-

-

I-

--- - -

-i-

--

--

-p..-

- -_.

I-

- ... -.

Z ZZZT~g :j~fz zzZzzzzZa ZZZZZZZ :::::-..,\ ,.. :::-.:::::_--=.:.. .'\ ,.. . ::::~-

22

2 22 222

--.. . , '\',': /" . . -----.. . . . . ,'\'f't,'"~,----.... ,--, ,,\ \'ll,:" ,',---.... ,~ l ,

-

Elecbric inbensity

--- Magnetic intensity

2

't

I

I"

I

',\ , ....- - -...., , " ,::, /

',1111 (" ( J "" \

,

/

,--

, \ 'd : ,' ( " ' I 'Iii, _..,1 , ) <:> ,1 " ---,,',' ':,\" ' . . . . ----", ~' / It \" " _---_.... "'~ I \ \, ' ....-------............' " ' \._~:: :,:,:,:,:,-,::,--,' ' .... -::::: I

,n\" ,

:'1\" " " . . .

--_-::::-,;

2222?2222222212222222212222222

(B)

FlO.

41. Lines of electric (--) and magnetic force (- - - -) in rectangular wave guide: (A) E 1• t wave; (B) H I • o wave.

type. It may conveniently be denoted by the symbol Hm,n, m, n indicating the particular mode of transmission. When m = n = 0, all components in (6)-(11) vanish except Hz. Since the electric and magnetic components cannot exist independently, there is no wave of type Ho,o. But ifm = 0, n = I, or m = 1, n = 0, the waves exist. If b > a, the HO,l wave has the lowest cut-off frequency, but if b < a, H1,o possesses this attribute, as will be evident from (3) § 18.24.

18.26]

ELECTROMAGNETIC WAVE GUIDES

341

The lines of electric and magnetic force in a rectangular wave guide are illustrated in Fig. 41. 18.30. Orthogonal curvilinear coordinates. In solving Maxwell's equations for the electromagnetic field, the equations for the components of electric and magnetic force must be expressed in terms of the appropriate coordinates. For a rectangular wave guide, the equations may be written down immediately in cartesian coordinates. When the guide has a curved cross-section it is expedient

y'

FIG.

42. Orthogonal curvilinear coordinates.

to employ orthogonal curvilinear coordinates. These are determined by the common point of three surfaces which intersect mutually at right angles. '~his may be illustrated by aid of Fig. 42 which refers to the cylindrical polar coordinates r, 0, z. The point P is the common intersection of (1) the curved surface of the cylinder of radius r, (2) a plane through P perpendicular to the axis Oz, (3) a plane containing Oz and passing through P. These three surfaces intersect orthogonally. Then from Fig. 42A,B we see that x = rcosB, y = r sinO, z = z itself. The lengths of the edges of the elemental solid in Fig. 42 B are, respectively, dr, rdB, and dz. The first and third are straight, but the second is curved. Now consider the coordinates from a general viewpoint (each of the three curves at the intersection having finite curvature), and let them be ({)1' tp",
342

ELECTROMAGNETIC WAVE GUIDES

[Chap. XVIII

cylindrical polars CPl = r, CP2 = fJ, CPa = Z, dSI = dr, d8,. = rdfJ, dss = dz; so II = 1, l2 r, and ls 1. For elliptical coordinates, Fig. 16, CPI CP2 = "J, CPs = z; so by (4) § 9.12, 1} = 12 = h[(C08h2,-C08 27])/2]-, and since z = z, 13 = 1. ..,

=

=

="

18.31. Vector differential formulae. We now introduce general formulae by means of which the scalar components AI' A 2 , As of a vector quantity A may be expressed in terms of orthogonal curvilinear coordinates: [_ o(la A2)+o(lsA a)] (1) (curIA) ! = 1_1 1 ~n ~, 2

a

vrs

v'r2

(curl A), = _1 [_ 8(la Aa) -I- a(lIA I ) ] ~}

lal} and

[_ O(l141)+O(l2 (curIA) 3 = 1_1 1 ~ ,. 1 2

V't'2

,

(2)

·

(3)

(Xpa

A2)]

(Xpl

As an aid to memorization, note that, (a) the A occur in cyclic order, (b) the subscript on the l.h.s. does not occur on the r.h.s., (c) the subscripts on the r.h.s, are interchanged in denominator and numerator of the two differentials. In the wave guides treated herein la = 1, o/B<pa = o/8z may be replaced (as in § 18.22) by -ifJ, whilst As = A z . Thus (1)-(3) become (curl A), = i/3A 2

and

+!. 12

8A.,

(4)

&P2

(curIA)z

= _!. 8AIJ_ifJAI'

(curl A),

=

II &PI

(5)

_1 {_~(llAI!+8(lzA z)}.

(6) l1 l2 (Xp2 oCPl These three formulae enable us to write down the components of the electric and magnetic fields for any type of uniform cylindrical wave guide.

18.32. Form of general differential equation. From (1), (2) § 18.20 and (4), (5), (6) § 18.31 we obtain · E1

'lWf

=

·QU

1 8H~ 2 Ocp2

'tJJLl2+l- - ,

(1)

eu.

(2)

. E2 = - 1 -~-lr' ·QHl' II Orpi

IWf

ELECTROMAGNETIC WAVE GUIDES

18.32]

iw£~ = _1_

l: 0(1

1 HI) 0
l1l 2

II

+ ~12 H2)}. O
• U 'tJ-Lw.uJ = -'t·,BE2 -1-oE -'

z

l2 0CP2

•

U

'tJ-LW.u2

=

1 oEz

-Z - ; 1 u<'P1

+ 'tfJ'QE

z

=

kf HI = i{~ oEz_p' ol/z}, II 0<'Pl

l2 Oq;>2

(3)

(4) (5)

l'

_1_{O(ll E I ) _ 8(l2E 2 ) } . lI l 2 (Xp2 Oq;>1 Substituting from (4) into (2) for iE2 gives

iJ-LwH

343

(6)

(7)

with kf = (w/C)2_f32. Substituting from (5) into (I) for iEI gives kfH2

= _i{w£ oEz+f!. o~}.

(8)

i; oq;>2 Using the values of HI' H 2 from (7), (8) in (3) leads to the general differential equation .!. 02 Ez+.!. 02Ez + _1 {o(12/11 ) oEz+ 0(11/12 ) OEIIJ}+kfE == 0 (9)

If &PI

II

O({)I

l~ o~~

l1l 2 Oq;>l (j(pi 0<'P2 0
18.33. General formulae for E waves. In accordance with WE:' put n, o. Then from (1), (2) § 18.32 we get

§ 18.25

=

E1

-P -H -

W€

(1)

2

and

(2)

=

iw£ oEz

By (7), (8) §18.32

HI

and

H 2 == _icoe -_oE. _.

(3)

k~ l2 0ff2 'J

kflt O~l'

(4)

From (1), (4) and (2), (3) we have

E = _ ifJ oEz l

and

kill ~I

= _ if3

E 2

eE:I

k~ 12 C'P2 •

(5)

(6)

344

ELECTROMAGNETIC WAVE GUIDES

[Chap. XVIII

Hence if E. is determined from (9) § 18.32, the remaining components follow immediately from (3)-(6).

18.34. General formulae for H waves. In accordance with § 18.26 we put Ell

=o.

Then from (5), (4) § 18.32

= pmH.

(1)

E2 = _/LpHl •

(2)

= _ ifJ fJlIa

(3)

an.

El

and

fJ

By (7), (8) § 18.32

H

and

H s = -k2l ~.

(4)

E = _ ipm aHz

(5)

l

k¥ II &PI ifJ

1 2 "'''&'2

From (1), (4) and (2), (3) l

kfll ~z

E _ ip,w alIa I - k~ II Ot.pl •

and

(6)

Hence, if H. is determined from (9) § 18.32, the remaining components follow immediately from (3)-(6). Having derived general formulae from E and H waves, we are now in a position to obtain the scalar components of the electric and magnetic forces in any uniform cylindrical wave guide whose cross-section is expressible in terms of two orthogonal curvilinear coordinates.

18.40. Application of formulae in preceding sections to circular wave guide. Referring to § 18.30 we have q>1 = r,
= 1,12 =

r so o(ll/la)/Ocpa = 8(l/r)/oO = 0, and (9) § 18.32 becomes

fJ2E.c+! fJE.c+.!. fJlEz+ kl E Ora r or r 2 082 1 II

=0

(1) •

Assume that E. = '(r)X(8), where' is a function of r alone, and X one of 8 alone. Proceeding as in § 9.21 we get

~ dl'2 +~ ~ +,Ikf =

_! dlXI = ml , sa.y 'dr , dr X dB ' 2 m being the separation constant. Then the equation for' is

d +! ~ +(k dr r dr 2 {

2

2) 2 _ 1n , 1 r2

=

0,

(2)

(3)

18.40]

ELECTROMAGNETIC WAVE GUIDES

345

which is the standard differential equation for the Bessel functions of the two kinds of order m, namely, Jm (k1 r ), Ym(k1 r ). Now E, must be finite on the axis of the guide, but Ym(O) = -00, so the Y solution is inadmissible, and we retain only Jm(kl r). The equation for X is (4)

whose formal solutions cosmO, sin mO are appropriate here. Accordingly as a composite solution of (1), with two arbitrary constants, we take 'mXm = Jm(k1r)[Amcosm8+Bmsinm8]

= AmJm(k1r)cos(m8-c¥m)' (5)

where Am = (.A~+B~)l,
Ez

= I

m-O

AmJm(k1r)cos(m8-cxm)J

(7)

the time factor being omitted. The boundary condition is Ez = 0 when r = a at the inner surface of the guide. Hence Jm(k1 a) = 0, so that kl a must be a positive real zero of the B.F. of order m. For each m the B.F. has an infinite number of zeros, i.e, writing km,r for kl , Jm(km,ra) = 0 for r = 1, 2, 3,.... Also there are an infinite number of B.F. corresponding to m = 0, 1, 2,.... Hence a double infinity of zeros is obtained. For power to be transmitted down the guide, by [210] f3 must be real. But p2 = (WJC)2_ k :n.,., so for the reality of f3 we must have wJc > km,r. Hence for each m, power is transmitted only in those modes, finite in number, for which w satisfies this inequality. If m = 0, the first zero [202, p. 9] occurs when k O,l a = 2·405, so k O, l === 2.4000-1 • Then for an Eo,l wave E~O,l 4861

= A oJo(ko,l r)cos(wt-Pz). yy

(8)

ELECTROMAGNETIC WAVE GUIDES

346

[Chap. XVIII

Now fJ2 = (w/c)2- ki,I' so the cut-off or lowest possible frequency for an EO,1 wave in an air dielectric is (fJ = 0), w c/2'TT = kO,1 C/21T = 2·40liX 3 X 1010/21Ta = 1·15 X 10 10ja cycles per second if a is in centimetres. This is equivalent to a wave-length of 2·62a em. in free space. E waves in circular guide. Applying the formulae in § 18.33 to (6), and including the time factor, leads to the following results:

s; = EO

Am{ J:n(k1r)cosmOsin(wt-pz),

(9)

= - Ak~pm Jm(k1r)sinmOsin(wt-pz), r

(10)

I

s, = Am Jm(k1 r)cosm8cos(wt-{Jz); H,

= Am wk~m Jm(klr)sihmOsin(wt-pz),

I'

Ho = Am ~E J:n(k1r)cosmlJsin(wt-l3z),

(II) (12)

(13)

1

Hz - 0,

by hypothesis. (14) From a purely mathematical viewpoint the complete solution in each of the preceding cases involves a double summation like (1) § 16.14, in m and in r (not to be confused with r the radius). In practice, as we have seen, for any m, power is transmitted down the guide in a finite number of modes, i.e. the range of r is finite. In (9)-(13), kl = k m" is such that J,n(km"a) = o.

18.41. H waves in circular guide, Using (1) § 18.40 for Hz, the formal solution is identical in type with (7) § 18.40, so Hz". where m

=

= B m J,n(kl r)cos m8 cos(wt-fJz),

(1)

0, 1, 2, .... From (6) § 18.34

E - iJLw a~ 8k21 or'

(2)

and this must vanish when r :.- a, since Eo is then tangential to the inner surface of the guide. Hence oHzlor = 0, and from (1) with k:n" for k1 we get J:n(k:n"a) = o. (3) If m = 0, the first zero occurs when k~tl a ~ 3·832, so k~,l ~ 3·S32a-1 • Substituting this in (1) gives for an HO•1 "rave Hzo.l

=

BoJo(A·~,lr)cos(wt-fJz).

(4)

ELECTROMAGNETIC WAVE GUIDES

18.41]

347

The cut-off frequency (fJ = 0) for this mode is (air-dielectric) W

c/27T = k~,l Cj21T

=

3·832 X 3 X 1010 j27Ta

=

1·83 X 1010 ja

cycles per second if a is in centimetres. This is equivalent to a wavelength of 1·64a em. in free space. Applying the formulae in § 18.34 to (1) yields the results: E, = -BmP,k~m Jm (k 1 r )sin m8 sin(wt - {3z), lr

(5)

E(J = -Bm~w J:n(k1r)cosmfJsin(wt-fJz),

(6)

1

Ez

=

u, =

0,

by hypothesis;

(7)

Bmf J:n(k1r)cosmfJsin(wt-fJz),

(8)

1

H(J = ~ =

-BmfJk~ Jm(k1r)sin m8 sin(wt-f3z),

lr BmJm(k1r)cosm8cos(wt-{3z).

(9) (10)

The remarks in the last paragraph of § 18.40 are applicable here, except that now k 1 = k:n,r, such that J:n(km,ra) = o. The lines of electric and magnetic force in a circular wave guide are depicted in Fig. 43.

18.50. Elliptical wave guide [22]. The system of coordinates used is illustrated in Fig. 16. From § 18.30 we get CPI = ~, <'P2 = "1, epa = Z, 11 = l2 = h[!(cosh 2~-cos 21])]l. Using these in (9) § 18.32 leads to the equation 82E 82E ag2z+ a1/+!k~h2(cosh2g-cOS2'7)Ez= 0,

(1)

which has the same form as (2) § 16.20. For reasons similar to those given in § 16.11, the appropriate formal solution of (1) is 00

Ez

=

fY:)

!OmUem("q)cem(Tj,q)+ !SmSem(~,q)sem(Tj,q),

m=O

(2)

m=1

the time factor being omitted, but see (3), (4) below. For any m there are two types of solution, namely, and

Ez me

= Om Cem ( "

E.".,

=

where kf h 2 == 4q and

q)cem("1 , q)cos(wt-!3z)

(3)

SmSem("q)sem(Tj,q)cos(wt-{3z),

(4)

G~n'

8 m are arbitrary constants determinable in

348

[Chap. XVIII

ELECTROMAGNETIC WAVE UUIDES

the usual way. For E waves the boundary condition is that E. = 0 at the inner surface of the guide where = ~o. Hence we must have

e

Cem(fo,q) and

=

0

(5)

Sem(eo,q) = O.

(6)

c

c d I

-->.-'

';:_i·:·-_-:~~ ~:.. ~'~ ~ .l.~ - - - .... - ... - ~ .,; I"

I .{: 4

.\".::~ ••.. ;:.• :

-.

(> ; ' , , ;

I

-:-,"".:~--~.'~

I,. 'C

- - ,

-

-. ' : :

:

- -

•

-

<;"~': -.~ • , . I -

_':._()~·:.~'o OI,o~O ~_-_q.-:~··:-_-_-c:.-;:l

1

IOf 'r'~j~,,:-~~:_~~~:~:.;,~::~ ·1~~}~~;/:~:3~~~~ HO,l'Nl ve

Sections through c-d

d

43. Approximate configuration of lines of electric (--) and magnetic force (- - - -) in circular wave guide: small solid circles represent lines of force directed towards observer. Propagation is towards the r.h.s, of page.

FIG.

Let qm,r' iim.r be the respective rth roots of these equations for a given m. Then the appropriate solutions of (1) corresponding thereto are and

EZ rrte = 0m.r Cem(" qm,r)cem{'Y}, qm,r)cos(wt-Pz)

(7)

Ezrru = 8m.,Sem(',qm,,)sem(1J,qm,r)cos(wt-!3z),

(8)

where m

349

ELECTROMAGNETIC WAVE GUIDES

18.50] ~

0 in (7), m ~ 1 in (8), and r ~ 1. For each m, in (7)

Cem(~' qm,r) = 0, with r = 1, 2,.... Also for each r, m has an infinity of values. Hence there is a double infinity of zeros. Power is transmitted down the guide only if f3 is real, so wlc > km,r' where k~,,.h2 = 4qm.,., or k m•r = 2Qtn,r/ae. Thus for each m, power is transmitted in those modes, finite in number, for which the above inequality is satisfied. Similar remarks apply in connexion with Sem ( " qm,r) in (8). For a Cern cem mode of transmission the lowest pulsatance occurs when m = 0, r = 1. Thus k~,l h2 = 4qo,l' 80 ko,l = 2lJt"l/k = 2Q&,l/ae (e being the eccentrioity of the ellipse), and We = ko,1 C = 2cq&,t/ae. (9) Applying the formulae in§ 18.33 to (7), (8),the results/or E waves are:

=

E f

E 11

-.L{Om,rCe:n(e, qm,r)cem('Y}, qm,r)}sin(wt-pz)

kf II 8m,r Se:n(" qm,r)sem ( 'Y}, qm,r)

(10)

,

= -.L{Om,rCem(e,Qm,r)Ce:n(7],Qm,r)}sin(wt_PZ) kf II 8m" Sem(e, qm,r)se:n( 7], ijm,r)

(11)

,

E = {Om.r Cem(~' qm,r)ce m ( 7], qm,r)}cos(wt-PZ)' z 8 m,r Sem(~' qm.r)se m(1], gm,,.) ,

(12)

He = - apE E1/'

(13)

H." = apE E~,

(14)

Hz = 0, by hypothesis. (15) The respective arbitrary constants in (10)-(14) are the same. Convenient symbols for E-wave modes are Ecm" , E,m,r, m, r having the values stated below (8), so that wlc > k m,,. or km". 18.51. H waves in elliptical guide. Here E z = 0 by hypothesis, and ~ satisfies an equation like (1) § 18.50. Thus the formal solution for the mth mode is

u, =

{~mcem(e,q)cem('YJ,q)}COS(wt_fJZ).

s;s-,«. q)se

m ( 7],

(1)

q)

The tangential component of E must vanish at the inner surface of the guide, so E71 = 0 when e= eo' Then by (6) § 18.34, since q>1 = it follows that

[OR] -rl _ = 0, s

e·-eo

'

= 0, Se~(,o, q) = O.

so Cem(eo,q)

and

e,

(2)

(3)

.f~LECTROMAGNETIC \VAV}4~ GUID}4~S

350

[Chap. XVIII

Let qm.'P' qm.p be the respective pth roots of these equations for a given

m. Then the appropriate solutions of the differential equation corresponding to the pth modes are

=

/1

with m

FIG.

~

{Qm.

1J

Sm,p

!!J

v

Cem(~,qm,p)cem('IJ,qm,p)}COS(wt_f3Z)

Se m (" qm.p)sern ( 'fJ' qm.p)

0 for Cem cem , m

~

1 for Semsem , and p

~

(4)

,

1, such that

44. Lines of electric ( - ) and magnetic force (- - - -) in elliptical wave guide: eccentricity of ellipse is 0·75. (A) HIl,l wave ; (D) E n. l wave.

wlc > km,p or km•p • The lowest pulsatance is obtained by the procedure given in § 16.23. Applying the formulae in § 18.34 to (4), the results for H waves are: Ef

· (W t == -tu» ({)mpcem("qmp)ce~t('Y/,qmp)} - ' , • SIn k~ II Sm,p Sem ( "

iim,p)se:n('Y/, qm,p)

Q)

fJZ

,

(5)

{~m,p ce:n{"qm,p)cem(TJ,qm,p)} · ( t- Q ) E:11", == _ kJ-tW e - ) ( - ) SIn W fJZ, 2l S 11 m,p S' em(s-,qm,pSem'IJ,qm,p

(6)

Es

(7)

=0;

Ht =

_1.. E7/' J-tw

(8)

H7/ =

Ji. Et • f'W

(9)

H, as at (4) above.

(10)

The constants Gm,p, Sm,p are the same throughout. Convenient symbols for H waves are Hcm,p, Hsm,p, m, p having the values given below (4). The lines of electric and magnetic force in an elliptical wave guide are portrayed in Fig. 44.

18.52]

ELECTROMAGNETIC WAVE GUIDES

351

18.52. Power transmitted down elliptical wave guide. The power passing through each unit area of cross-section in the direction of the z-axis, i.e. the time rate of energy flow per unit area, may be calculated by aid of the Poynting vector p. If E, H are the electric and magnetic force vectors at any point (e, TJ) in the cross-section, then p

=

EHj47T,

(I)

where the vector product is to be taken. Expressed in terms of scalar quantities 1

pz = 47T[EeH'fJ-E1JH,].

(2)

Substituting from (10), (II), (I3), (14) § 18.50 for the various quantities, (2) becomes

v, = 4~~~~ IGmI2[Ce~(e,qm.r)ce~J11,qm.r)+

+ Ce~(" qm.r)ce~( 1], qrn,r)]sin 2(wt-

,8z) (3)

for an Ecm,r tYlJe of wave, Grn being a constant. The mean power transmitted through unit area, taken over a cycle of sin wt at any z, is one-half the maximum value of (3). Hence the total mean power for the m mode in question is found by omitting sin 2(wt-pz) and halving the integral of (3) over the cross-section. Thus, omitting qm,r for brevity,

pII:

=

ICmI2,8w~ Sl7kt

II

[Cem 12(r: ) 2 ( )+c 2 (t.) 12( )]dsl d82 seem TJ em s cem TJ l~'

dsl , ds2 , and II being defined in § 9.12. But dSI d82jl~ (4) may be written

=

(4)

d,d7}, so

fo 271

~ = IG~l~WE f f [Ce~(e)ce~(TJ)+Ce~(e)ce~(TJ)] dedTJ· o

0

21T

By (1) § 2.21,

(5)

I ce~(TJ) dTJ = o

211

17,

and by (3) § 14.41,

I ce~(1J) d1J = {)ml7 . 0

~o

Thus

p.~ --

J[C

IGm I2PW -Skf .-€

'2(i:r,;,qm.r ) +V _0. C 2 /: d em m em(s,qm.r)] ,.

(6)

o

For an E,m" type of wave, ~ is given by (6) on replacing Cem by Sem' qm" by qm", and {}m by m'm· Formulae for H waves are obtained from (6) if € is replaced by p" and q""" qm., by q""P' qm,p, respectively.

352

ELECTRO~fAGNETIC

WAVE GUIDES

[Chap. XVIII

18.53. Attenuation of waves due to imperfect conductor [199]. Hitherto the metal of the guide has been assumed a perfect conductor, so that waves travel from end to end without loss. In practice, however, loss must be t;ken into account. We confine our attention to metallic loss and assume absence of loss in the air dielectric. If the resistivity of the metal is included in Maxwell's

FIG. 45. Illustrating penetration of current into very large flat slab, induced by sinusoidally varying magnetic field tangential to free surface.

equations the loss may be determined, but not without undue complication. It is adequate for practical purposes to assume that the magnetic field at the inner surface of the guide is the same as it would be if the metal had zero resistivity. Then introducing PI > 0 into the analysis, we calculate the eddy-current loss as in the case of a metal slab. Consider a very large metal slab of resistivity PI whose upper face lies in the XZ plane, Fig. 45. Let a cosinusoidal electromagnetic field exist above the slab, the maximum value of the magnetic component tangential thereto being H,. This induces a surface current of density a in a non-resistive slab, such that

H, = 41TU, or U= H,/41T.

(1)

Since Pl > 0 in practice, there is a volume distribution of current whose density at a depth 'U below the surface is al. This depends upon Pl' the magnetic permeability ILl' and f the frequency of the electromagnetic field. The propagation of power into the slab perpendicular to its surface is similar to that along a uniform trans-

ELECTROMAGNETIC WAVE GUIDES

18.53]

353

mission line. If 0'0 cos wt is the surface-current density, it IDay be shown that at any depth y below the surface 0'1 = aoe-!3111cos{wt-{JIY)' (2) where PI = 21T{JL1//Pl)i. We have now to determine 0'o, 80 that the volume distribution of current in the slab is equal to a, i.e. co

maximum value of

fo

0'1

dy

= H,/4Tr.

(3)

Substituting from (2) for 0'1 into the integral in (3) and evaluating yields] aocos(wt-11T)j2 1f31. The angle 11T is due to the current not being in phase throughout the depth of the slab. The loss is unaffected by omitting 17'1', so equating the maximum values in (3) leads to (10 = 1311ft/2 1'TT . Thus the equivalent root mean square value of (11 in (2) is Q U -{3IY/41T• (4) a r.rn.s, = 1-11 fll e Then the power

108S

per unit surface area of the slab is given by

00

P1

f

f e-2~111 00

0'2

r.m.s.

dy

=

PIPl!!1 161T 2

o

dy '

(5)

0

so ~m.2 = fJIP 1 Hr/321T 2 = (l-'lPl!)IH1/161T. (6) Since the depth of penetration into the metal of a guide at ultra high frequency is very small, the preceding analysis is applicable to curved surfaces, provided the curvature is not excessive. This condition obtains in practical wave guides where the ratio (depth of penetration/radius of curvature) is of the order 2 X 10-5 or less, except In the region of a corner. The proportionate surface area affected is then quite small.

18.54. Application of analysis in § 18.53 to wave guide. If ds is an elemental arc-length of a cross-section at the inner surface of the guide, the area of a surface element of unit axial length is dA = ds X 1. The magnetic field component tangential to the element is 1ft. Then by (6) § 18.53 the power loss in the element is dP = {fLIPl!)lHfds/161T, so the total loss in unit length of the guide is p

= 1~1T (fL1P1!)l

f Hr

ds,

(1)

the integration being taken round the inner surface;

t

f eo 00

fla ll co8(!lt

Y- tl:l ) dy

=

1 2 R (C08<Jt+ 8in tl:t ) fJl

zz

=

1 2t R cos(tl:l-11T) · fJl

ELECTROMAGNETIC WAVE GUIDES

354

[Chap. XVIII

As the waves travel down the guide, the power absorbed by the metal per unit length is given by (1). Consequently it represents -d~/dz,t the distance rate of power loss sustained by the waves. If we define this loss to be -d~/dz = 2ex~,

P=

(2)

then ex is known as the attenuation constant. The multiplier 2 arises as follows: If Hm is the maximum magnetic field at the transmitter where z = 0, its value at a point z > 0 is Hme- CU • The power, however, is proportional to (Hm e- CU ) 2 = H:n e- 2
giving

=

2aPm e- 2cu

ex =

=

2ex~,

P/2~.

(3) (4)

For purposes of calculation the value of z is immaterial, since ex is constant.

18.55. Attenuation in elliptical wa"e guide. For an Ecm,,. type of wave the tangential component of magnetic force at the inner surface of the guide, where e= eo, is

H" =

o;wl Ce:n(~o,qm,,.)cem(7J,qm,,.), £

(1)

1 1

Om being a constant. Also by §9.12, ds = lld'Y}, so by aid of (I) §18.54 we get the loss per unit length 8S P

= IOmlll(JLlii;~WlIElI[Ce;:(go,qm.r)]

f

211

ce:n(7],qm,r) d7]/ll'

(2)

o

Now

II = k(cosh2f-cos21J)' = kcoshfo[l-(cOS1J/coshfo)2]'

= a(l-e1cos27J)l, e being the eccentricity of the bounding ellipse. Substituting this value of II into (2) leads to

P

= IOmI1l(JLli~~~lIElI[Ce;:(go,q7ll.r)]

f

111

o

ce:n(7],qm,r)d7]/(I-e1cosl7])I. (3)

t The negative sign indicates decrease in ~ with increase in z. P, is the total power passing a section distant z from the transmitter.

ELECTROMAGNETIC WAVE GUIDES

18.65]

355

Since e < 1, the denominator of the integrand may be expanded binomially, and we get for the integral I"

f ce~(1],qm,r) X

I =

o X [

.1 I cos21] 1+ "Ie

, 1.3.5 6 6 + Jd1]. + 21.3, ••• 1 • 2! e cos 1] + 23 • 3! e cos 1]

(4)

On substituting the series for cean(1], qan,,.) in (4) and using (9) § 9.40, we obtain (5)

Inserting (5) into (3), with m = 217, the expression for the power loss per unit length of guide is

I02nlll(lLli~~WllE2[~~(~O,qlln.r)][ 1+:ll e2{1+0 lln}+ ..

p =

J

(6)

From (6) § 18.52 and (6) above, the attenuation constant is

_ (X

-

1 (JLIPl/€)! Ce~~(eo,q2n,,.)

[1-Uc/f)2]l

4a -JL-

[1

+ :2ell{I+02n}+ ...]

r[~(~,q2n,r)+8'271 Celn(~,q2n.r)] ~

•

o

In (7) t; is the cut-off frequency. For Hem waves the attenuation constant may be shown to be <X

(7)

= (ILIPlf)l Ce~(~o,q ) X 4TTafJp,w m,p

f

2"

fit [ X

0

2"

06;:(1], qm,p) d1] + kfall feel {7J, q )( 1- e2 C0827J)1 d 7J ] (l-e2 COS2 7J)1 1 m m,p

h

0

f [Ce;:(~, qm,p)+8'm Ce~(~, qm,p)] ~

'

(8)

o

which may be evaluated by procedure akin to that above (7), a numerical process being used for the denominator.

18.56. Transition to circular cross-section. The results in Appendix I may be used to reproduce the formulae for a circular section given in §§ 18.40, 18.41 from those in §§ 18.50, 18.51.

ELECTROl\IAGNETIC

356

WAV~~

GUIDES

[Chap. XVIII

18.57. Influence of deviation of cross-section from circularity [22, 199]. Using the formulae in (12) § 18.50, when the ellipse tends towards a circle, by Appendix I-omitting the time factorwe have Eze".

= o,n Cem("qm,r)cem(Yj,qm,r) -~ Cmp;",Jm(k1 r )cos m8,

(1)

for an Ecm type wave, and Ez,,,. = SmSem(e,qm,r)sem(Yj,qm,r) ~ Sm 8

:n Jm(k r )sin m8, 1

(2)

for an Earn type wave. Consider both solutions in (5) § 18.40 for

FIG.

46. (A, B) Illustrating deformation of circular wave guide to elliptical form.

waves in a circular guide for the modes m = 1, r = 1, 2,... , and suppose that the deformation occurs in the position shown in Fig. 46A. If the orientation of the wave-looking down the guide-is such that 81 = 0, then

s, =

01P~ ~(kl r)c088 +- 01 Cel ( " Q1,r)ce1(1J, Q1,r)

(3)

and at the deformation the wave will tend to an ECI type. In the same way, if the deformation occurs as shown in Fig. 46 B, i.e, 1'" away from the position indicated in Fig. 46A, and the orientation of the wave is such that 01 = 0,

s, = 818~ ~(kl r)sinO +- 81 Se

1( "

Ql,r)se1(Yj , q1,,.),

(4)

80 the wave will tend to an E S I type. Generally the deformation will occur in a position such that neither 0 1 nor 8 1 is zero, so that both Ec l , ESI waves may be created. Now these corresponding types

18.57]

ELECTROMAGNETIC WAVE GUIDES

357

have different phase velocities. t Consequently in travelling down the guide the two waves will fall out of step progressively. It follows, therefore, that unless C1 =-= 8 1 = 0, i.e. the deformation occurs along an axis of symmetry of the wave pattern (see Fig. 44), non-circularity introduces a degree of instability. Since there is no E,m wave of order m = 0, slight deformation will not cause splitting of an Eel wave. For m = I, 2,... splitting of the wave into two components with unequal phase velocities will occur. Similar conclusions apply for H waves. It may be remarked that with an E1 or an HI wave, after splitting occurs, one component has a higher attenuation than the other, and after travelling an adequate distance down the guide, the component with the greater attenuation will be of negligible comparative amplitude,

t From § 18.24, fJ2 = (w/c)2-(wclc)2, so fJ = (wlc)[l -(wc/w)2Jl, where wc/2.". is the cut-off frequency for the transmission mode concerned. The phase velocity is vj) = w/fJ = c[I-(wc/w)2]- t. As w ... C.cJc+O, "» ~ -+ co, which is a well-known phenomenon in geometrical optics. By the principle of relativity, no actual velocity can exceed that of light, so that phase velocity must be regarded as a useful fiction!

XIX DIFFRACTION OF SOUND AND ELECTROMAGNETIC WAVES

19.10. Scattering of sound by elliptical cylinder. Refening to Fig. 47, a plane sound wave in air travels towards the origin from the lower side and impinges at an angle 8 on a right elliptical cylinder with axis Z'OZ, normal to the plane of the paper. The cylinder is long compared with its major axis, being immobile and non-absorbent. Its presence causes the sound to be scattered. Analytically we may for convenience consider the wave to have two components. There is (1) a plane wave identical with that in absence of the cylinder, (2) a scattered wave, diverging away from 0, for which we have to derive a symbolical representation. 19.11. Equation of sound propagation. The two-dimensional equation is (1)

where q, is the velocity potential, k 1 = w/c, and c is the velocity of sound in the medium [see reference 201, chap. ii]. Since we are dealing with an elliptical cylinder, we transform (1) to elliptical coordinates. Thus by § 9.20 we obtain

a2q, a2q,

og2 + 07J2

+ 2k2(cosh 2g-cos 27J)q, = 0,

(2)

with 2k = k1 h.

19.12. Velocity potential of plane wave. For a plane wave travelling as shown in Fig. 47 the root mean square value] of the velocity potential at any point on the wave front at a perpendicular distance p from the origin 0, in absence of the cylinder, is represented by (1)

From Fig. 47 p

=

xcosO+ysinO,

t Sinusoidal variation implied.

(2)

359

ELECTROMAGNETIC WAVES

19.12]

and by (7) §10.53, (I) expressed in elliptical coordinates is

4>1 = 2c/J oe- i wl x X ' co

~ n==O

~I 1 -Ce2n(~)Ce2n(1})Ce2n(8)+--Se2n+2(e)Se2n+2(TJ)Se2n+2(8)+ 2n

8 2n +2

+if.-!- Ce1n +1(~)cesn+l( 1J)ce1n+l(8)+-l-Selln+1(~)selln+1( "1)selln+1(8))]. \Psn+l

8 1n +1

(3)

Iyf FIG. 4:7. Illustrating plane wave impinging upon very long rigid cylinder. Q has coordinates ~, '" defined by orthogonal intersection of confocal ellipse and hyperbola. ON = e, NQ = y, OM = x cos 8, MP = ysinS, p = OP = xcos8+ysin8 = h(coshfc-os'7}cos8+sinhfsin'1sin8). ~, F I are foci and PI Fa = 2h. MN and PQ are perpendicular to OPe

This is a solution of (2) § 19.11 and represents the velocity potential of the plane wave in absence of the cylinder.

19.13. Boundary conditions. Since the surface of the cylinder is immobile, the velocity of the air particles normal thereto must be zero. Thus the first boundary condition is that the sum of the normal velocities of the incident and scattered waves shall vanish at f = If 'Un l is the normal velocity of the incident wave and tit" that of the scattered wave, whose velocity potential is we

eo.

+

must have 'Un I

_ 'Un. -

-

(84)1 8~1) an + ~

v,.

'2'

'-E. = 0,

(1)

360

DIFFltACTION OF SOUND AND

n indicating the direction of the normal, From § 9.12 the first boundary condition may be written (

8c/Jl a~

+ i4>2)

a~ t=to

=

[Chap. XIX

on =

II o~, so

O.

(2)

+00,

(3)

The second boundary condition is

cP2 ~ 0 as ,

~

since the influence of the cylinder is nil at infinity.

19.14. Representation of scattered wave. Being different from the incident plane wave, the scattered wave cannot be represented by (3) § 19.12. Accordingly we have to choose an expression which (a) represents propagation of the scattered wave in the direction indicated in Fig. 47, (b) is a solution of (2) § 19.11, (c) permits the first boundary condition to be satisfied, (d) satisfies the second boundary condition. The solution selected comprises two sets of I terms, each set having an infinite number of component solutions. For the mth term of the first set, if we take

,pc". = em e-iwlMeg>(" q)cem(7] , q)cem«(},q),

(1)

Om being a-complex constant, all the above conditions are satisfied. Since Meg> = Cem+iFeYm' (1) represents a wave moving as shown if.. Fig. 47. We shall see later that the first boundary condition may be satisfied if Om is correctly chosen. From the asymptotic formulae for Meg)(e) in § 11.11 it is seen that tPc". ~ 0 as e ~ +00. In the second set of I terms in the proposed solution we incorporate the function Neg> = Sem+iGeYm. The required form of solution which represents wave propagation away from the origin as in Fig. 47, is

,ps". =

s;e-iwlNeg,>(e, q)sem(7], q)sem«(J, q),

(2)

which has the properties (a)-(d) given above. Thus for the scattered wave we take the solution

c/Ja = e-iculx

+Bsn+l Ne~~+l(~)seln+l(11)se2n+l (8)+

+Ban +1 Ne~~+I(f)sel"+I( 1J )se2n +I(8)].

(3)

ELECTROMAGNETIC WAVES

19.14:]

361

Using condition (2) § 19.13 and eqv .ting the sum of the derivatives of functions of the same kind and order to zero, gives for = ~o:

e

= _ 2c/>o Ce~n(fol _ _ 2c/>oC08()'lne~i«••

0 1

"

where

#I,,,

Me~~t(eo~ -

P2n

2i<po Ce~~+l (eo) M (I)' (i:) e2n+1 50

= _

= _

2
P2n+l

=

where coscx2n+l

8.

(4)

'

= Ce~n(eo)/[Ce~~(eo)+Fey~~(eo)]I;

C080: 2n

CONt+l

P2n

P2n+l

(5)

Ce~n+l(eO)/[Ce~~+l(eO)+FeY~~+I(~O)]I;

__ 2ic/> oSe;n+l(eO) _ 82n+lNe~~~1(eO) -

2n+l -

,

-

2c/>oCos!32n+le-iCP.,,+1-i1t) 82n+1 '

(6)

where cos{32n+l = Se~n+l(eO)/[Se~~+l(gO)+GeY~~+l(eO)]I; 'S2 +2

= _

11,

2c/>o Se~n+IU'o) _ _ ~cb_co8 f!2~+2 e- i fl••l ~ 8211,+2 Ne~U~2('O) -8 211,+2 '

(7)

where cos{32n+2 = Se;n+2(gO)/[Se;~+2(eO)+GeY;~+2('0)]l, and Pm' 8 m are defined in Appendix I. The velocity potential at any point on or outside the cylinder is

ep =
4>oet(kl p-wl)+4>2

[as at (3) above].

(8)

This may be regarded as a general solution of the problem. If 8 = 0, and the incident wave travels parallel to the X-axis, the terms in (3) involving§2n+l' 8 211,+2 vanish, leaving 00

cP2 = e- i wl I

m=O

When 8 = we get

OmMe~)(e)cem(1J)cem(O).

1'", the incident wave travels parallel to eo

ep2 = e- i wt I

n=O

(9)

the Y-axis, and

[02nMe~~(~)ce2n(1J)ce2n(11T)+

+8211, +1 Ne~U+1(e)Se2n+l( 1J)Se2n+l( tn-)].

'0

(10)

19.20. Long rigid ribbon. This case is obtained when ~ 0, the width of the ellipse of evanescent minor axis then tending to 2h, the interfocal distance. When eo = 0, Ce~(eo) = 0, and in (3) § 19.14 G2n = C2n +1 = 0, while

= 811&+1 = 82n+1

and 4961

-2ic/>okB~2n+l)/se2n+I(I1T)Ne~~~1(0)

(1)

-2c/>okIB~2n+2)/8e~n+l(r)Ne~~!~2(O).

(2)

3A

362

DIFFRACTION OF SOUND AND

[Chap. XIX

Inserting these in (3) § 19.14 we obtain the velocity potential of the scattered wave at any point, whose coordinates are ~, 1], on or outside the ribbon. Thus

At the surface of the ribbon g = 0, so by (3) § 19.12 the velocity potential due to the plane wave is

t/Jl = 2e- i OJlt/Jo~ [At2n)ce2n(1J)C~2~_ ikAi2n+l)?e2n+1(1J)OO2n+1(8)]. L..,

n=O

cesn(! 1T, q)

ce2n +1 (tn", q)

(4)

Since Sem(O) = 0, by (3) above the velocity potential due to the scattered wave is .J.. 'f'1 -

2 -csu. k L:Q() e 'YO n==O

[Bi2n +1) GeYan+l(O)se2n+l(TJ)se2n+l(8) Ne2n (I)' (0) 1q) + 1 se2n+I ()"1T,

_ ~kB~2n+2) GeYan+2(O)SeSft+s(TJ)Seaft+2(8)] Ne~~~2(O)se;n+2(tn",q) ·

(5)

The resultant velocity potential at the ribbon surface due to both the incident and scattered waves is the sum of (4) and (5). If Po is the root mean square pressure of the incident wave, then, since p = APeP/Ot = -ipwt/J,t the pressure is obtained from (4), (5) by writing Po for c/Jo' PI' P2 for cPI' ePa' respectively. Polar diagrams for plane sound waves scattered by a ribbon of width 2h for various wave-lengths (as a function of 2h) and angles of incidence are shown in Fig. 48. The solution for the diffraction of sound waves at a slit of width 2k in an infinite rigid plane may be obtained by analysis akin to that above, but the boundary condition corresponding to (2) § 19.13 is now that the velocity of the sound waves normal to the plane is zero.

t pJs the density of the undisturbed medium, while p is not to be confused with the multipliers in (4), (5), § 19.14.

ELECTROMAGNETIC W AVES

19.21]

363

A.~:h .7Th -21Th ----..~--"""'r'---

jfDirection 0 ;nciden~ beam~

/

FIG.

48. Polar diagrams for sound scattered by a long ribbon [144].

19.21. Formulae for large distance from ribbon. The confocal ellipses are now sensibly concentric circles, and we may use the asymptotic formulae for the Ne functions, with kl , = v (see §9.12 and Appendix I). Thus from (3), (4) § 11.11, if Ivl ~ n,

and

Ne(l) (t.) "" _ se;n+I(0)se2n+l~!~(~)1 ei (v + l 7T) 2n+l 5 kBi2n +l ) 'lTV

(1)

Ne(l) (t.) '" - i se~n+2(0)8e~n+2(!1T)(~)tei(11+11T) 2n+2 s k2B~2n+2) 'lTV •

(2)

Substituting "(1), (2) into (3) § 19.20 leads to the formula

2

ePa "-' ePo(_2_)t ei(k,r-wl+f1T) ~ se ('TJ q)se (0 q) se:n(O, q) 'TI'k r ~ m' m' Ne(l)'{O q). 1 m-l m ,

(3)

This represents the velocity potential of the scattered wave at a great distance' from the ribbon. The pressure is Po0eP,JOt, so

r« = -iptftJepa "" 2pwepo(~)! ei(klf"-wl+l1T) X 'lTk l r

~

(

se:n(O, q) sam(8) ,q Ne(l)'(O ). m ,q

)

X ~ sem 1], q m-l

(4)

19.30. Scattering of electromagnetic waves by long elliptical metal cylinder. Consider the incident wave to be plane polarized, the electric force vector being parallel to the axis Z'OZ, and let the

DIJ4"FRACTION OF SOUND AND

364

[Chap. XIX

direction of propagation be that indicated in Fig. 47. t Assuming that the cylinder has zero resistivity, che boundary condition is that the sum of the incident and scattered waves is zero at the surface where = ~o. For the incident plane polarized wave the electric force is represented by

e

E1 =

(1)

Eoei(klP-wl>,

so by § 19.12, in terms of elliptical coordinates we have ex>

E1 = 2e-i wIEo !

[as at (3) § 19.12].

(2)

n-=O

For the scattered wave we take E 2 = the r.h.s. of (3) § 19.14.

The boundary condition is that

=

E1+E2

0

when

~

=

(3)

'0'

(4)

so from (2), (3) we obtain

2Eo C2n (eO)

a 2n -

S.

P2n

-

8 2n +1

2iEo '3e2n+l(~0)

-

2n+l -

Writing

(5)

Me~~(go) ,

-

(6)

Ne~~+l(gO)'

Cetn('o)/Meg>(,o) = cos am e- icxna , COSCXm

=

Cem(eo)/[Ce:n('o)+FeY~('o)]l;

Sem('o)/Neg>(,o)

=

(7)

cos Pm e- ifJ".,

cosfJm = Sem(eo)/[Se~(eo)+GeY~('o)]i,

(8)

(3) may be expressed in the form

E.

=

-2e-iaJlEo ~ [A~lln)e-i<xtNcosallnMe~U(e)Celln(1])Celln~8) L, ce2n(O)ce 2n (in) n=-O

+

+k l BLsn+ll)e-iPJN+o oos fJlln+ll N e~U+ll(e)selln+s(1])selln+s(8!+ se~n+l( 0)se~n+2( i1T)

+ .{-kA~Sn+l)e-iCXa.+lCOSasn+1MeL~+1(e)celln+1(1])cesn+1(8) + ce2n+l(O)ce~n+l(in)

,

+kBiIn+l)e-ip..

+l

coslsn+! Ne~~+1(e)seln+1(1])sean+1(8)}]. se2ft +1 (0)se2n +1 (in)

(9)

t The equation of propagation is (11) § 18.22 in rectangular coordinates, and (I) § 18.lS0 in elliptical coordinates.

ELE~TROl\IAaNETIC

19.30]

When (J = 11T, ce2n +1 ((J)

= se2n+2((J ) =

WAVES

365

0, and (3), (5), (6) yield

2n(, o) M (l>(i:) ~ [ CeMe(l)(g) E2 = -e -iwl2E0 k e 2n s cein () "1 Cain(1)+ "f'7T n=O P2n

2n

0

i Se2n +1(, O) N e2n+l~· (1) (i:) +s---N-nf-(/;-)8e1n +1(TJ )8e."+1 (.1-)] I"" • 1,.,+1

(10)

e2n +1 se

If , is large enough, v = kcoshg ~ k1r, and the asymptotic forms of the Me, Ne functions may be used. Then with the dominant term only (see § 11.11)

2l\!~~~(e) ~ 2iNe_~U+!(~~ ~ _2(_2_)lei<1I+t1T>. P2n 8 2n +1 1Tk1 r

(11)

Substituting from (11) into (10) leads to

E

t--J

2

e-iwl2(_2_)leiE X 1Tk r 1

~

k X ",=-o

0

[CeII1M0)C6

2n ( 1])ce2n (

Me~~(,o)

!1T) + ~e2n+1 (g0)8e2,, +1 ( 1]) 8e2n +1( !1T)] (12) Ne~~+I('O)·

This represents the electric force due to the scattered wave at a large distance r from the axis of the cylinder. The resultant electric force at the point is E = E1+E2 •

19.31. Long rigid ribbon. 8,,. = 0, while-from (5) § 19.30 C

2EoCe2n (O) _ - P2nMe~~(O) -

-

2n -

0.

and

-

In+l -

Here

-

~o =

0, so in (6) § 19.30

2EoA b2n >

Me~~(O)ce2n(!1T)

2iEo Celln +1(O) _ 2ikA~2n+1)Eo P2n+1Me~U+l(0) - Me~~+l(O)ce~n+l(i1T)·

(1)

Substituting from (1) into (3) § 19.30 yields

E.

=

-2e-iwl.E. ~ [Ab2n~e~~(e)Ce2n('YJ)Ce2n(8) Me(1)(O)ce (1. ) ,,-0 211. 211. I 1T

10k

_ ikA~2n+l) Me~~+l (f)ce1n +1 (TJ)ce l n +l (8)]

Me~U+l(O)ce~"+l(P.)

(2) ·

This represents the scattered wave at any point (,,11) on or outside the ribbon. Polar diagrams for electromagnetic waves scattered by a ribbon

DIFFRACTION OF SOUND

366

[Chap. XIX

of zero resistivity and width 2h for various wave-lengths (as a funotion of 2k) and angles of inoidence are shown in Fig. 49. The solution for the diffraction of electromagnetic waves at a slit of width 2k in an infinite perfectly conducting flat sheet may be obtained by analysis akin to that which precedes. The boundary

~irection of incident ~

t

FIG. 49. Polar diagrams for electromagnetic waves scattered by a long ribbon [144].

condition is that the electric force vanishes at the surface of the sheet. In the incident wave the electric force vector is parallel to the axis of the slit.

19.32. Formula for large distance from ribbon. We use the asymptotic approximations

M (1)(/:.) ean 5 Me(l) (i:)

and

2n+l 5

I"J

I"J

_

(_2_)

i ce2n(O}ce2n (tn-) 1 i(k 1 ' +hr) A (211.) 1Tk r e o 1

ce2n+l(O)Ce~n+l(tn-)(_2_)iei(k kA (211.+1) 1

1Tlc1 r

1

, +!1J').

(1)

(2)

Inserting these in (2) § 19.31 leads to

E.

i:

~ 2(1T;l r)l ei, 1T

(3)

m-O

This represents the scattered wave at a great distance r from the ribbon. As in (12) § 19.30 and (2) § 19.31, e-i wl may be a separate factor.

APPENDIX I

Degenerate forms of Mathieu and modified Mathieu function8 1. Mathieu functions. When the fundamental ellipse tends to a circle, by § 9.11, k ~ 0 and, since qi = k = lle1 k, q ~ o. By § 3.32 all A1m ) in the series for cem('Y],q) tend to zero, except thatA~)-+-I. Hence as the eccentricity e ~ 0, q ~ 0 and cosm1] = cos~ep (m ~ 1), (1) the confocal hyperbolae in Fig. 16 becoming radii of the circle, with 7J = ep. For n = 0, ceo(7J, 0) ~ 2- 1 = AbO). In the same way it can be shown that as e ~ 0, cem('Y],q)

4-

sinm1J = sinmcP (m ~ 1). FractiO'tULZ orders. As e ~ 0, q ~ 0, A~m+1h ~ except for p when A~+P> -+ 1. Thus sem('Y],q)

4-

°

cem+~(z, q) ~

cos(m+{J)tP sem+~(z, q) -+ sin(m+fJ)ep.

and

(2)

=

m, (3)

(4)

2. Modified Mathieu functions. The equation for these functions, q > 0, is

y"-(a-2k2cosh2e)y = 0 (q = k 2 ) . (I) By §§9.11, 9.12 when a confocal ellipse of semi-major axis r tends to a circle with this radius, ,~ +00, h ~ 0 such that hcosh, ~ r, i.e. thee ~ r. Then 2k2 cosh 2g -+ k2e2f , while if this remains finite, k ~ 0 and, therefore, a ~ m 2 for a function of integral order m. Consequently (1) degenerates to y" (k 2ete -m2 )y = o. (2) Putting r = theE, 2k = k1h, (2) transforms to the standard Bessel

+

(I

d2y 1 dy m 2) dr2 +;; dr + k1 - fi Y =

equation

o.

(3)

With lei r = kef, the formal solutions of (3) are Jm(k1 r) and Ym(k1 r). Then as -+ +00, k ~ 0, coshg ~ sinh" lei' is finite and the solutions of (I) are constant multiples of the corresponding solutions of (3), each to each. Now as f and r -+ +00, the dominant terms in the asymptotic expansions of Cem(f, q), Sem(f, q), and Jm(lc1 ,.) are identical save for constant multipliers P:n, 8~. A similar remark

e

368

APPENDIX I

applies in the case of FeYm(~,q), GeYm(~,q), and Ym(k1 r ). Thus we have the following degenerate cases as ~ +00:

e

Cem("q)

~ p:nJm (k 1 fJ

Sem(e,q) ~ 8~Jm(klf)

~Cem(~' q) :~Sem(f, q)

-)0

p:nr :r Jm(k1 r)

-)0

8:nr:r Jm(k1 r)

= p:nk1 rJ:n(k l r)

= 8~kl rJ;"(kl r)

FeYm(~,q) ~ P~Ym(klf)

8:n Ym(kl r) :~FeYm(f,q) p:nklrY~(klr) GeYm(g, q) -+

-)0

(4)

(m ~ 1);

(5)

(m

~ 0),

(6)

(m

~ I);

(7)

(m ~ 0),

(8)

(m ~ 1);

(9)

(m

-)0

:~GeYm(f,q)

(m ~ 0),

~

0),

(10)

. . . . . . 1)• (1:) _ (m::;::::;

8:nk1rY;"(k1r)

Applying the foregoing to (1), (2) § 13.40 leads to Me~~(z,q)

=

-2iFek 2n(z,q) -~ P;nH~~(klr),

(12)

Me~U+l(z,q) = -2Fek2n+l(Z,q)~P~n+lH~U+l(klf);

(13)

Ne~U+l(z,q) = -2 Gek 2n+1 (Z,q) ~ 8~n+lH~~+l(klr),

(14)

Ne~U+2(z,q)

(15)

=

-2iGek zn+2(.z,q) ~ 8~n+2H~U+2(klr).

The degenerate forms of ]"em(z, q), Gem(z, q) may be derived by applying (4), (5), (8), (9) to (4), (5) § 13.21, while those of the functions of order (m+fJ) are identical in form with (4), (5), but m is replaced by (m+p).

3. Determination of P~, 8:n. These are obtained by comparing the dominant terms in the asymptotic expansions of the corresponding functions. Thus by (1) § 11.10 and (15) § 8.10, when

e~ +00

Ce:an(f,q),....,

while

ce:an(O,~~~nJ~'~}(;;ief)lsin(kt;~+Vr)'

(I)

(-I)n('IT~ r)lsin(k1 r+l'IT).

(2)

J:an(k1 r) ,....,

Remembering that as (4) § 2, that p~"

e~ +00, kef ~ k r, it follows from (1), (2) and 1

= (-I)"cel,,(O,q)celn(!1T,q)/Ab2n) =

(-l)"PI".

(8)

APPENDIX I

369

Similarly we find that

= 8~n+l =

P~n+l

(-I)7&+lcel7&+1(O,q)ce;n+l(11T,q)/kAi2n+1) = (-I)np2n+1J

(-I)n se~n+l(O,q)se2n+1(! 7T , q)/kBi2n+l)

=

(-I)n8sn+1 ,

(4) (5)

and 8~n+2 = (_I)n+lBe~n"'2(O,q)se;n+2(tn-,q)/k2B~2n+2)

=

(_1)n+1 8 2n +2. (6)

4. Results for q < o. We write -kf for kf in (3) § 2 thereby obtaining the equation for the modified Bessel functions Im(k1 r), Km(k1 r), The degenerate cases are: Ce2n(" -q) ~ P~n12n(klr), CC 2rt +1( " -q) --> 8~n+l I Sn +1(kl r);

(1)

Se2n +1( " -q) --> P~n+l 12n+1(k1 r), Se 2n +2(g, -q) ~ 8;11.+2 12n+2(k1 r);

(3)

:e Cean(e, -q) -+ P~nkirI~n(kir),

(2) (4) (5)

~ Cean+1(e, -q) -+ 8~n+1 ki r I~n+1(ki r);

(6)

:e sezn+1(e, -q) -+ P~n+1kirI~n+1(kir),

(7)

:eSezn+z(e, -q) -+ 8~n+z leir I~n+z( leir) ;

(8)

Fek2n (" -q) ~ P;nK2n(kl r),

(9)

Fek1n +1 (" -q) --> 8~n+l K 2n +1(k1 r); 'IT Gek 2n +1 (" -q) ~ P;n+l K 2n+l (k 1 r ),

(10) (11)

'TTGek 2n +2(f , -q) ~ 8~n+2K2n+2(klr).

(12)

'IT

11

The results for Meg,>(" -q), Neg>(" -q) may be derived from (9)-(12) and § 13.41.

5. Degeneration of cem(z, -q), sem(z, -q) to parabolic cylinder

functions. When z is small, cos 2z ~ 1- 2z2 and Mathieu's equation, for q negative, may be written y"+(a+2q-4qz2)y = O. (1) Let z = ixq-l, and suppose that 8S q ~ +00, Z -+ 0 but qz2 remains finite [91]. Then by (2) § 12.30, (a2n + 2q) r--J (8n+2)qi) 4961

3B

(2)

APPENDIX I

370

and with these changes (1) is transformed to

~+[(2~+1)-lx2]y = 0,

(3)

which is the differential equation for the parabolic cylinder function D2n (x ). Hence under the above conditions

a constant X D2n (x ) = KDsn (2zql). Similarly it may be shown that ce2n (z, -q)

~

se2n +1(z, -q) ~ a constant X Dzn+1(x ).

(4)

(5)

APPENDIX II 25. Oharacteristic numbers for cem(z,q), m = 0-5; sem(z,q), m = 1-6 TABLE

q

-b l

-at

-at

0·0000000 -1'00000 00

-1,0‫סס‬oo

1 2 3

0·45513 86 1·51395 69 2·83439 19 4·28051 88 5·8000460

-1,85910 81 -2,37919 99 -2·lH90391 -2·31800 82 -1·85818 75

4:

5

0·11024 1·39067 2·78537 4·25918 5·79008

88 65 97 29 06

+a.

-bl

00 -4·0000000

0

+b.

4·00000 00

9·0000000

-3'91702 48 -3,67223 27 -3,27692 20 -2·74688 10 -2,0994604

4·37130 5·17266 6·04519 6·82907 7·44910

10 51 69 48 97

9·04773 9·14062 9·22313 9·26144 9·23632

93 77 28 61 77 58 55 44 92 91

6 7 8 9 10

7·36883 8·97374 10·60672 12·26241 13·93698

08 25 92 42 00

7·36391 8·97120 10·60536 12·26166 13·93655

10 -1·21427 82 -1·35138 12 24 -0·4383491 -0·51754 54 0·43594 36 81 0·38936 18 17 1·38670 16 1·35881 01 2·39914 24 25 2·38215 82

7·87006 8·08662 8·11523 7·98284 7·71736

45 31 88 32 98

9·13790 8·96238 8·70991 8·38311 7·98606

12 14 16 18 20

17·33206 20·77605 24·25867 27·77284 31·31339

60 53 95 22 01

17·33191 20·77600 24·25865 27·77283 31·31338

84 04 78 32 62

4·57013 6·89340 9·33526 11·87324 14·49130

29 05 71 25 14

4·56353 6·89070 9·33410 11·87272 14·49106

99 07 97 65 33

6·87873 5·73631 4·37123 2·83305 1·15428

69 23 26 67 29

7·00056 68 0·79262 95 4·39789 62 2·84599 17 1·1607057

24 -38·45897 32 28 45·67336 96 32 52·94222 30 36 60·25556 79 4:0 67'60615 22

38·45897 45·67336 52·94222 60·25556 67·60615

24 94 29 79 22

19·92259 56 25·56174 71 31·36tH544 37·30263 91 43·35227 53

19·92254 25·56173 31·36515 37·30263 43·35227

03 -2·53976 57 -2·53807 29 -6·58806 30 -6·58758 05 -10·91435 34 -10,91420 80 -15·46677 03 -15,46672 49 -20·20794 08 - 20·20792

b.

a.

b6

at

b.

16·00000 00

16·00000 00

25·0000000

25·0000000

36·0000000

q

--

a.

79 50 90 43 54

0

9·0000000

1

9'078368t9·37032 25 9·91550 63 10·67102 71 11·54883 20

16·03297 16·12768 16·27270 16·45203 16·64821

01 80 12 53 99

16·03383 16·14120 16·33872 16·64981 17·09658

23 38 07 89 17

25·02084 25·08334 25·18707 25'33054 25·51081

08 90 98 49 60

25·02085 25·08377 25·19028 25·34375 25·54997

43 78 55 76 17

36·01428 36·05720 36·12887 36·22941 36·35886

99 70 12 14 68

12·4656007 13·35842 13 14·18188 04 14·90367 97 15·50278 44

16·84460 17·02666 17·18252 17·30301 17·38138

16 08 78 10 07

17·68878 18·41660 19·25270 20·16092 21·10463

30 87 51 64 37

25·72341 25·96244 26·22099 26·49154 26·76642

07 72 95 72 64

25·81727 26·15612 26·57775 27·09186 27·70376

20 02 33 61 87

36·51706 36·70350 36·91721 37·15669 37·41985

67 27 31 50 88

16·30153 49 16·5985405 16·48688 43 16·06197 64 15·39681 09

17·39524 17·20711 16·81868 16·24208 15·49397

97 53 37 04 76

22·97212 24·65059 26·00867 26·98776 27·59467

75 51 83 64 82

27·30001 27·76976 28·13635 28·37385 28·46822

24 67 59 82 13

29·2080550 31·0000508 32·93089 51 34·85305 87 36·64498 97

38·00600 38·64847 39·31501 39·97235 40·58966

87 19 08 11 41

13·62284 11·11107 8·29149 6·14563 1·72964

13·66279 11·12062 8·29467 6·14673 1·73004

66 27 21 75 56

27·8854408 27·28330 82 26·06244 82 24·37850 94 22·32527 63

2

3 4: 5

6 7 8 9 10 12 14 16 18

20 24 28

32 36 40

27 98 62 63 91

28·21l>3594 27·40574 88 26·10835 26 24·39606 65 22·33214 85

39·51255 41·23495 41·95351 41·92666 41·34975

19 03 12 46 44

I

41-6057099 42·22484 15 42·39394 28 42·11835 61 41·43300 52

APPENDIX III

Classification of Mathieu functions for a, q real

~

0

In Tables 26-8, m = 2n or 2n+l, n = 0,1,2, ... ; 0 < P < 1; JL real and positive. TABLE 26. Soluti0n8 o!y"+(a-2qcos2z)y = 0: ordinary functions First kind

Order

Integral cem(z, ± q) Integral 8em (z, ±q) Fractional I cem+~(z, ±q) TABLE

Alternative second kind

Second kind fem(z, ±q) gem(z, ±q) 8em+lJ(z, ~q)

fekm(z, .- q), z real gekm(z, -q), z real

27. Solutions of y" -(a-2qcosh 2z)y = 0: modified functions

Alternative second kind Third kind - -- -~--~- -- ---- - - - - 1 - - -----~------~----- - - - - - 1 - - - - Integral FeYm(z, ±q); Fekm(z, ±q) ~'em(z, ±q) Me~!·(2)(z, ±q) Cem(Z, ±q) Integral GeYm(z, ±q); Gekm(z, ±q) Gem(z, ±q) Ne~·(~n(z, ±q) Sem(z, ±q) Fractional Cem+~(z, ±q) Sem+~(z, ±q) Order

First kind

TABLE

Second kind

28. Soluti01.UJ ojy"+(a-2qcos2z)y functions of order (m-+JL) Fi,'at kind

I

=

0:

Second kind

ceu m t 14(z, ±q) ceu m +14 ( --z, ±q)

The cer, cei functions have been omitted since q is negative imaginary. Fey, Gey, Fek, Gek take priority over Fern, Gem as modified functions of the second kind because: 1. They are better suited for applications when z is large; 2. Their representations in B.F. product series converge rapidly, thereby facilitating computation; 3. They degenerate to the Y- and K-Bessel functions under the conditions in Appendix I. Excepting }"em , Gem' alternative representations of each function are given herein. For instance, four different series and five integral relations are given for Cem(z,q). FeYm(z,q) is a Y-type modified Mathieu function of the second kind of order m, q positive, whereas Fekm{z, -q) is a K-type modified Mathieu function of the second kind of order m, q negative.

REFERENCES Abbreviations used in Section A Annalen der Physik. A rchiv fur Elektrotechnik, Inaugural Dissertation. Mathematische Annalen. 4. M.A. 5. M.N.R.A.S. Monthly Notices of RouatAstronomical Society. Mathematical Tables and other Aids to Computation: 6. M.T.A.C. M athematische Zeitschrift. 7. M.Z. Proceedings Cambridqe Philosophical Society. 8. P.C.P.S. Proceedings Edinburgh Mathematical Society. 9. P.E.1'v.l.S. Proceedings International Oonqress oj Mathematicians, 10. P.I.C.M. Proceedings Institute Radio Engineers (America). 11. P.I.R.E. Proceedings London Mathematical Society. 12. P.L.M.S. Philosophical J.11 agazine. 13. P.M. !:ro.c.ILe:dings Royal Society, London. 14. P.R.S. -~ Proceedtng8 Royal Society, Edinburgh. 15.~. Quarterly Journal oj Mcuhematics (Oxford). 16. Q.J.M. Transactions Cambridqe Philosophical ~Society. 17. T.O.P.S. 18. T~M~:J. -'1."8ho1cu '~l athematical Journal. ' Zeitschrijt fur angewandte Mathematik Mechanik. 19. Z.A.M.M. Zeitachrijt fur H oehfrequenztechnik. 20. Z.jur H. Zeitschrift fur Physik. 21. Z.P. 1. A. der P. 2. A.fur E. 3. D.

una

A. SCIEN1'IFIC PAPERS 1. Ataka, H. "Super-regeneration in ultra-short wave receiver. P.I.R.E.23, 841, 1935. 2. Barrow, W. L. Frequency modulation. P.I.R.E. 20, 1626, 1932. 3. - - P.I.R.E. 22, 201, 1934. 4. - - Smith, D. B., and Baumann, :F'. W. Oscillatory circuits having periodically varying parameters. Jour. Frank. Inet, 221, 403 and 509, 1936. 5. Bateman, H. Solution of linear D.E. by definite integrals. T.O.P.S.21, 171, 1909. 6. Bickley, W. G. A class of hyperbolic Mathieu functions. P.M. 30, 312, 1940. 7. - - Tabulation of Mathieu functions. M.T.A.C. 1, 409, 1945. 8. - - and McLachlan, N. W. Mathieu functions of integral order and their tabulation. M.T.A.C. 2, 1, 1946. 9. Booher, M. P.I.O.M., Cambridge, 1,' 163, 1912. 10. Bock, Ph. D., Prague, 1932. 11. Brainerd, J. G. (Note on modulation. P.l.R.E.28, 136, 1940. 12. - - and Weygandt, C. N. Solutions of Mathieu's equation. P.M. 30, 458,1940.

374

REFERENOES

13. Brainerd, J. G. Stability of oscillations in systems obeying Mathieu's equation. Jour. Frank. Imt. 233, 135, 1942. 14. Bremekamp, H. Over de periodieke oplossinger der vergelyking van Mathieu. Nieuw. Arch. VfJor WiBlcunde, 15, 138, 1925. 15. - - Over de voortplanting van een golfbeweging in een medium van periodieke structuur. Physica, 6, 136, 1926. 16. - - On the solution of Mathieu's equation. Nieuw. Arch. voor Wiskunde, 15, 252, 1927. 17. Bruns, H. Astron. Nachr., No. 2533, 193, 1883; No. 2553, 129, 1884. 18. Burgess, A. G. Determinants for periodic solution of Mathieu's equation. P.E.M.S. 33, 25, 1915. 19. Butts, W. H. Elliptic cylinder function of class K. D., Zurich, 1908. 20. Callandreau, O. Astron. Noohr., No. 2547, 1884. 21. Carson, J. R. Notes on theory of modulation. P.l.B.E. 10, 62, 1922. 22. Chu, L. J. Electromagnetic waves in elliptic metal pipes. Jour. App, Phys. 9, 583, 1938. 23. - - and Stratton, J. A. Elliptic and spheroidal wave functions. Jour. Math. and Phys. Ma88. Inet, Techn. 20, 259, 1941. 24. Couwenhoven, A. Uber die Schiitterlerscheinungen elektrischer Loko .. motiven mit Kurbelantrieb. Forschungsarbeiten, V.D.l., Heft 218, 1919. (This is a special report listed on p. xxviii, vol. 63.) 25. Curtis, M. F. Existence of the elliptic cylinder functions. Ann. oj Math. 20, 213, 1917. 26. Dannaeher, S. Zur Theorie der Funktionen des elliptischen Zylinde1'8. D., Zurich, 1906. 27. David, S. P. L'Onde Electrique, 7, 217,1928. 28. Davies, T. V. Flow of viscous fluid past a flat plate. P.M. 31, 283, 1941. 29. Dhar, S. C. Convergence of second solutions of Mathieu's equation. Bull. Oalcutta Math. Soc. 10, 1921. 30. - - Solutions ofMathieu's equation, ofsecond kind. T.M.J.19, 175, 1921. 31. - - Integral equations for elliptic cylinder functions. Jour. Dept. Se. Oalcutta Univ. 3, 251, 1922. 32. - - Elliptic cylinder functions of second kind. Am..er. Jour. MaJ:h. 45, 217, 1923. 33. - - Integral equations and expansions of elliptic cylinder functions in Bessel functions. T.M.J. 24, 40, 1924. 34. - - JO'Ur. Indian Math. Soc. 16, 227, 1926. 35. - - Elliptic oylinder funotions of second kind. Bull. Oalcutta Math. Soc. 18, II, 1927. 36. Dougall, J. Solution of Mathieu's equation. P.E.M.S. 34-, 4, 1916. 37. - - Solution of Mathieu's equation and asymptotic expansions. P.E.M.S. 41, 26, 1923. 38. - - Solutions of Mathieu's equation, representation by contour integrals, and asymptotic expansions. P.E.M.S. 44, 57, 1926. 39. Drefus, L. Eigenschwingungen von Systemen mit periodisch verander.. lieher ElastiziUit. A. fur E. 11, 207, 1920. 40. Duffing, G. Erzwungene Schwingungen bei veranderlicher Eigenfrequenz und ihre technische Bedeutung. Sammlung View~g. Heft 41/42.

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123. Maginniss, F. J. Sinusoidal variation of parameter in single series circuit. P.I.R.E. 29, 25, 1941. 124. MaImborg, M. Om integrationen sf en klass sf lineare differentialekvationer med dubbel periotliska koefficienter, analog med de S. K. Hennite'ska differentialekvationerne. Uppsala Univ, Arss1cr. Math. Naturw. 1897. 125. Mandelstam, L., and Papalexi, N. Uber Resonanzerscheinungen bei Frequenzteilung. Z.P. 73, 223, 1931. 126. Markovic, Z. Sur 1& non-existence sirnultanee de deux fonctions de Mathieu. P.G.P.S. 23, 203, 1926. 127. - - Sur les fonetions de Mathieu de periode 11'. Acad. des Sc, des Slaves du 8'U.d de Zagreb (Gro(J,tie), 21, 34, 1925. 128. Marshall, W. Asymptotic representation of elliptic cylinder functions. D., Zurich, 1909. 129. - - Arbitrary constants in 128. P.E.M.I-'}. 40, 2, 1921.

130. Mathieu, E. Memoire sur Ie mouvement vibratoire d 'une membrane de forme elliptlque. Jour. de Math. Pures et Appliquees (Jour. de Liouville) 13, 137, 1868. McDonald, J. H. TraM. Amer. Math. Soc. 29, 647, 1927. McLachlan, N. W. P.M. 26, 695, 1938. - - Hill's differential equation. Math. Gaz. 29, 68, 1945. - - Computation of solution of Mathieu's equation. P.M. 36,403, 1945. - - Heat conduction in elliptical cylinder, and an analogous electromagnetic problem. P.M. 36,600,1945; 37,216, 1946. 136. Mei88ner, E. Ober Schiittelschwingungen in Systemen mit periodisoh veranderlicher Elastizitat, Schweizer Bauzeitunq, 72, 95, 1918. 137. Veksyn, D. Solution of Oseen's equation for an inclined elliptical cylinder in a viscous fluid. P.R.S. (A), 162, 232, 1937. 138. Mettler, E. Mitt. Forecli.-Anst. Gulehoffnungsflutte-Konzem, 8, 1, 1940. 139. - - Eorecbunqshefte am dem Gebiet des Stahlbauee, Heft, 4, 1, 1941. 140. Milne, A. RoOtH of the confluent hypergeometric functions. P.E.M.S. 33, 48, 1915. 141. Maglich, H. Beugungsersoheinungen an Korpern von ellipsoidischer Gestalt. A. der P. 83, 609, 1927. 142. Morse, P. M. Quantum mechanics of electrons in crystals. PhY8. Rev. 35,

131. 132. 133. 134. 135.

-J

1310, 1930. 143. - - Addition formulae for spheroidal functions. Proal Nat. A cad. So, Wash. 21, 56, 1935. 144. - - and Rubenstein, P. J. Diffraction of waves by ribbons and slits. PhY8. Rev. 54, 895, 1938. 145. Muller, K. E. Uber die Schtittelschwingungen des Kuppelst&ngenantriebes. D., Zurich, 1919. 146. Nisino, K. Jour. Aero. Rell. Inet., Tokyo Imp. o-u«, No. 176, May 1939. 147. Noether, F. Anwendung der Hillsehen Differentialgleichung auf die Wellenfortpflanzung in elektrischen oder akustischen Kettenleitem. Internal. Oong. Appl. Mech. Stockholm, 3, 143, 1931. 148. Pellam, J. R. Sound diffraction by 8 strip of absorbing material. Jour. ACOUB.

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ADDITIONAL RESULTS '#

Tmc following were obtained after the manuscript went to press. The analysis underlying the derivation of the formulae is usually omitted for brevity (see [227]).

1. Evaluation of Fey~(O, q)/Fe'm(O, q) and GeYm(O, q)/Gem(O, q) in § 13.21. 'By comparison it follows from (1) § 7.61, (3)-(5) § 13.20, and (3), (4) § 13.21, that for q real> B = FeY~n(O, q)/Fe~n(O, q) = 2/7TG2n(q). (1)

°

Similarly

FeY~n+l(O,q)/Fe~n+l(O, q) =

and

Gey m(O, q)/Gem(O, q)

2/7TG2n+1 (q),

(2)

== - 2/1TSm(Q).

(3)

2. Relations between C~, Cm(q); S'm, Sm(q) in §§ 13.52-13.55. By aid of (3)-(6) § 13.52, we deduce that (1)

C;n == -17TC2n(q). C~n+l

We also find that

==

-!1T02n+l(q),

(2)

S~ == -!1TSm(q).

and

(3)

From (1) § 7.22, (3) § 13.53, and (1) above, we get ClO

P2n ce 2n (! 7T, q)/Ab2 n ) == ce2n(O, q}+ ~ (2r+2)f~~~2 == fe~n(O, q)/G2n(q)·

Also

(4)

r=O ex>

-P2,,+1 ce~n+l(!1T, q)/kAftu+l) == ce 2n +1 (O, q)+ ~ (2r+ l)f~~'+il) r=O

= fe~n+l(O, q)/C2n +1(q), 82n +1 se2n +1 ( 11', q)/kBi2n+1) == ~ g~~~il) == ge2n+l(0,q)/82n+l(q),

(5)

ClO

r=O

(6)

00

82n+2se~"+2(!1T,q)/k2B~2n+2) == ~ g~~n+2) r=O

=

ge2n+2(0,q)/82n+2(Q). (7)

3. Limiting forms of FeYm' GeYm' Fekm, Gek m, Cm(q), S,n(q) as q -40 +0. (z,q) 21 (1, 2) ~ ±-(z+!logeq), 2Fek 0 (Z, -q ) tr

Fey}

Fey} (z,q) Gey 1»

Fek}

Gek m (t, -q)

~

-22m-l(m-l)!m!1T-lq-me-mZ (m

~

1), (3,4)

,

-+

Fey}

-IGey m (z,q)

(m ~ 1), (5,6)

ADDITIONAL RESULTS

383

provided q and ke±z are small enough. The forms (1)-(6) show that the functions of zero order have a branch point, while those of higher integral order have a pole of equal order at q = o. These formulae were derived from the B.F. product series, with the aid of (8), (9) § 3.33. Approximate representations of the J-, Y-, 1-, K-Bessel functions were used, and only the dominant term in the expansion retained. More accurate formulae may be derived by retaining appropriate additional terms before and after the dominant terms. Thus Fey} (z, q) 2l k ~ ±-[(y-Ioge 2)+!logeq+z][I=f!qcosh2z), (7,8) 2F e 0 (z, -q ) 'IT where (y-Ioge2) == -0·1159... , and ifn

~

2,

}'ey } (z, q) __ . 24n- 1(2n-I)! (2n)! e- 2nz ,.~ =F -----.------------ X 2Fek 2n (z, -q) 'lT q2n X

{1±lq[(2::1}-(2::1}]}. (9,IO)

When the order of the function m ~ 1, the multipliers P2n/A~2n), P2n+l/Ai2n+1), etc. (see Appendix I), external to the B.F. product series for these functions (see Chap. XIII), have singularities at q = o. For example FeY2n+l(Z, q) 00

=

[P2n+l/Ai~+l)] ~ (-I)rA~;~11)[J;.(Vl)~+1(V2)+~+1(Vl)~(V2)]' (11)

where

r=O

VI

= ke:", and v2 = kez• As q ~ 0, the multiplier

i

P2n+l/A 2n+1) ~ (-I)n2 4n(2n ) ! (2n+l)! q-2n-l,

(12)

while 00

~ (-I)r A~~'+11)[~(Vl)~+l(v2) +J;.+l (Vl)~( v s)] ~ (-1 )n+12'IT- 1q- 1li--(2n+1)z. r=O (13)

Hence, considered as independent functions of q, both (12), (13) have branch points at q = o. This is true for GeY2n+l(z, q) also. The ~ part in FeY2n{z, q) has no singularity in q, but that of GeY2n+2{z, q) has a pole of unit order. As q ~ 0,

Fem(z, q) ~ sinh mz, so

Fe~(O, q) ~

m.

(14)

Thus from (3), (14), Fey;,t(O, q)/F'e~.(O, q) -+ 22m --1(m - I)! m! 1T- 1q- m,

(15)

ADDITIONAL RESULTS

384

so from (1), (2) § 1, and (15), it follows that as q --)l- 0 Om(q)

~

qmj22m - 2(m-I)! m!

(16)

8m(q) ~ Cm(q).

Similarly, we find that

(17)

4. FeYm' GeYm' Fekm , Gek., with argument (Z+ip1T).

Fey}.

Fey)

. ee}

±q) = G (z, ±q) + 2~PS (z, ±q), (1,2) ey m e m p being any integer if m is even. When m is odd, p is even. Gey m

(Z+~P1T,

Fek}. (Z+~P1T, ±q) Gek III

=

Fek)k

. Ce}

(1

_Je

m

(z, ±q) - ~p s ~ e

(z, ±q),

(3,4)

In

p being any integer if m is even. When m is odd, p being even, write +p for -ip on the r.h.s, of (3), (4). 'I'hese formulae illustrate the non-periodic nature of the functions. By applying (1)-(6) § 13.41 to (3), (4) above, relations for Meg),(2)(z, ±q), and Neg),(2)(z, ±q) may be derived.

5. Integrals of the form at (9)-(12) § 9.40. 217'

f cein(u, q)cos 2pu du

0~~) = ~

o i(P-l)

= I

r=O

Ag.n)A~~n].2r+

= i[A12n)]2+

e~~)+1 = ~

f

i(P-2)

I r=O

00

I A~~n)A~~~2r r=O

A~~n)A~~n].2r+

(1)

00

I A~~n)A~~n~2r, r=O

(2)

217'

cein+1(u,q)cos2pudu

o

=

I[A~1t-tl)]2+. 1(1'-2)

=

'Y~~)+1

I r=O

i(p-3)

I r=O

A~~'til)A~~n_+2~)_I+

00

I A~~'+1.1)A~~n++2V+l r=O

(3)

00

A~~'+1.1)A~~~+2~)_I+ ~ A~~+11)A~~n++2~)+I' r=O

(4)

f

217'

=

~ se~n+1(u, q)cos 2pu du

-

_1[B(ln+l)]2_

o -

=

f

_

P

.(1)-2) ~

k

r~O

i(P-3)' .~

k r=O

B(2n+l)B(2n+l) 2r+l 2p-2r-l

+k 2p-2r-l

+k

ec

~

B(2n+l)B(2n+l) 2r+l 2p+2r+l

r=O

(5)

GO

B(2n+l)B(2n+l)

2r+l

~ B(2n+l)B(2n+l)

r~O

2r+l

2p+2r+l'

(6)

ADDITIONAL RESULTS

386

I"

'Y~f:~1 = ~ f seln+l(u, q)C08 2pu du o t~-~

= - r=O I

B~~,+~I) B~~"-7I~-2+

= _![B~n+2)]I_

t(J)-C)

I

r~O

~

I Bg.'t-tl ) B~~":I~+I r=O

B~'t-t2)B~~"_+I~)_I+ '

~

I

r-O

(7)

B~~I)B~++2~)+I·

(8)

In (1), (3), (5), (7), P is an odd positive integer, while in (2), (4), (6), (8) it is even ~ 2. The results (1), (2) will enable additional terms in (5)-(7) § 18.55 to be obtained.

6. Improved asymptotic formulae, q large and positive. These were derived from (5) § 11.43 by writing iz for z. Ce

} (z,q)

F ~m

where

f't.J

2Gmi'm[(1+ Xm)C?SXm±zm sin Xm], ~

(1,2)

C~

se} (z, q) "" ± 2S + I'm [( 1+X )sin Xm =Fz ~S Xm] , (3, 4) m1 m m G ey m+l cos SID 1(tanh Xm = 2ksinhz-(2m+l)tantZ) , ')1m = 2-m-i(coshz)-t,

(2m+l)j8koosh 2z, Zm = (m2+m+l)tanhzj8kcoshz. The formulae in § 11.42 may be derived by omitting the terms in x m ' z", In (1)-(4). Applying (4), (7), (10), (13) § 8.14 to (1)-(4) we obtain

Xm

=

Fek C . 2n (z. q) ,......., o2n I'an[(1 +xan)ei<xltl+l")+z2n e'XIII], G e k 2n+l 0ln+l

(5, 6)

Feklln +1 (z q) '" -CI 1& +1 ", [( 1 +x )ei XI1I +1 +za,,+1 ei<x.. +l-I1T)] • Gek ' 8 r an+l In+l . 2n+2

(7, 8)

2n+2

The real and imaginary parts in (5)-(8) are required if z is real, since Fekm(z, q), Gekm(z, q) are complex. Formulae for Me(1),(2)(z, q), Ne(1),(2)(z, q) may be derived by applying (1), (2), (4), (6) § 13.41 to (5)-(8). See comment in § 7 on accuracy. The phase angle range for all asymptotic formulae in the book is -11T < phase z < q real.

r,

7. Improved asymptotic formulae, q large and negative. These were derived from § 6 by applying the appropriate relations in § 8.30.

F~Ln(e, -q) '" CIn8In[(1-XI1&){~:;..-i:~:}+Zlln{~:; ..~:~:}]. (1, 2)

3D

ADDITIONAL RESULTS

388

ee}

Fey

In+! (e,

[{etfJ.-+ie- tfJs-}

{et/J.--ie-tfJ..}]

-q) ,..,., 811n +1 811n (I-X lIn ) ier/J'-+e-r/J.. +Z2n ier/J"-e-r/J.. •

(3, 4)

Formulae for Se 2n + 1(z, -q), GeY2n+l(Z, -q) are given by (3), (4), respectively, if (2n+ 1) is written on the r.h.s. for 2n and C2n +1 for 8 2n +1 • Those for the same functions of order (2n+ 2) are derived from (1), (2) by the first change, and writing 8 2n+ 2 for C2n . Fek 2n

Also

(z,-q)"-'

2,.,+1

C

82ne-t/J",(1-X21l-Z2n),

o2n

(5,6)

02n+l

and In (1)-(8),

cPm

= 2kcoshz-(2m+l)y,

Zm = (m2+m+l)cothz/8ksinhz,

X m = (2m+l)j8ksinh 2z,

=

811l

tanh2y = (coeh e):",

2-m - l (sinh z)-i.

If z is real, Ce, Se are real, so the imaginary part is omitted; Fey, Gey are complex, so both real and imaginary parts are used; Fek, Gek are real. Formulae for Me~),(2)(z, -q), Neg>,(2)(z, -q) nlay be derived by applying (I), (3), (5), (6) § 13.41 to (5)-(8). See § 11.44 for remarks on accuracy. For a given order m, and an assigned (;), the accuracy increases with increase in

(~), due to reduction in

the omitted terms

akin to X m, Zm.

8. Approximate solution of yH-(a-2qcosh 2z)y = 0, q large and positive, 1/1+aj2q/ < I. Writing the equation in the form y" +2q(cosh 2z-aj2q)y = 0, the procedure in §§ 4.82, 6.20 carried as far as U'2' leads to the second approximation

C?Sn,

Yl(Z) ~ a constant Y2 (co.ih 2z-!Jj2q)l sm

where

n=

2ksinhz-k[h 2gd(z)+

(1, 2)

i: I_3~;~~;~3)h2rD2r_2] + r==2

+ 1~k[gd(Z)+1(13h2-6)D2+1h2(II5h2-76)D4+ +f,h4 (77k 2 - 58)De+...],

(3)

ADDITIONAL RESULTS

hi

=

i(l+a/2q), gd(z) = 2tan-1e8 -

o

= 1.3... (2r-l)[ d(

11T, the gudermannian of z, and

)+tanhz+ +2.4 (2r-2) tanhZ] cosh z ... 3.5 (2r-l) coshl"-lZ ·

g z

2.4...2r

2r

387

(1), (2) may be used as asymptotic formulae for functions of any order (m+!3), provided the constant multiplier can be found. Consider Cem(z,q), and Sem+1{z,q). When q~ +00, by (7) § 12.30, a "" -2q+(4m+2)k, so (cosh 2z-a/2q)1 "-I 21cosh1z, and Q

"-I

2ksinhz-(2m+l)tn'.

Thus (1), (2) give Yl(Z)

Y2

~ 2- 1(Coshz)-lKKl C?8[2ksinhz-(2m+ 1)11T].

(4,5)

2 SIn

These are the forms at (3), (4) § 11.42, if tanh iz ~ 1. Hence by comparison, K1 = 0m/2m-f, K 2 = Sm+l/2m-f. With the aid of (1)-(4) § 6, we infer that

Ce}

Fey and g

(z ) m

'

q

Se} ( ) Gey m+l Z, q

f'-I

f'-I

. Om cOS n 2m - t ( cosh 2z-a/2q)1 sin ~~

(

±Sm+l sinn 2m- f( cosh 2z-a/2q)1 cos ~1.

a

(6 7)

)

= am ,

,

( b ) (8 9) a = m+l· ,

9. Parametric zeros of FeYm(z,q), GeYm+l(z,q). Applying the procedure in § 12.41 to the dominant terms in (2), (4) § 6, it is found that the formulae for Feym» Geym+l are those for Sem+1 , Cem, each to each. 10~

Pulsatance equations for elliptical ring membrane.

Referring to § 16.11 et seq., the membrane is clamped at two confocal ellipses where , = ~o at the outer, and ~ = ~l at the inner. The two forms of component solution are

'm =

[CmCem(~,q)+FmFeYm(e,q}]cem(1J,q)coS(Wmt+Em)

(am)' (I)

[SmSem(~,q)+GmGeYm(~,q)]se"l(7J,q)cos(wmt+im)

(b

and

'm =

m)· (2)

Inserting the boundary conditions in (1), (2) leads to the pulsatanee equations

=

Cem(eo,q)FeYm('l,q)-Cem(~l,q)FeYm('o,q) 0

and

(m ~ 0),

(3)

ADDITIONAL RESULTS

388

If q is a parametric zero of (3), (4), the corresponding nodal hyper.. bolae are determined, respectively, by .,

and semC'q,q) = o. (m ~ 1) (5) For an elliptical ring lake (see § 16.20 et seq.), the pulsatanoe equations are obtained from (3), (4) by using the first derivatives of Qem , FeYm' etc., with respect to Nodal hyperbolae are determined by (5), using the parametric zeros of the new pulsatance equations. Putting the leading terms of (1), (2) § 6 in (3), the parametric zeros are cem (1J,q) = 0,

e.

qm,p

=

k~p ~ [p1T+(J~)-(Jg)J2/4(sinh'0-sinh'1)2,

with 8~) = may be derived for (4).

(2m+l)tan-1(tanh

l'r), P integral ~

(6)

I. A similar formula

ce2n(!1T,q)/A~2n), ce2n(O,q)/A~2n). From (17)

11. Formulae for § 8.10 and (3) § 13.11 cesn(! 7T, q)/A~S")

= r~oA~~n)Jsr(2k sinh e)(~o (-I)rA~~n)J,.(ke-z)J;.(keZ),

and from (15) § 8.10, (3) § 13.11

(1)

cezn(O, q)/A~2n)

=r~o (-I)rA~n)~r(2kcosh z)/r~o (-ItAg.n)J,.(ke-Z)J,.(ke finite and > 0, these formulae are valid for all values ll

) .

(2)

If k is of z except those when the numerator and denominator vanish (zeros of Ce2n(z, Similar formulae may be derived for other multipliers in Chap. VIII. They may be used to calculate A2n , etc., on p. 185.

q».

12. Bessel function series for Cetn+fl(z, ±q), Senl+~(z, ±q). Using analysis of the type in §§ 8.10, 8.13, we get ClO

I r==

Ce2n +p(z, q) = G2n +p

(-I)rA~~n+P>J2r+,9(2kcoshz),

(1)

-00

and

00

I (-I)r(2r+,8)A~~n+P>J2r+p(2kco8hz), r==-oo

Se2n+p(z,q) = 8:an+ptanhz

with

(2)

CSn+{J

= Ce.n+{J(zo, q)/ r=~ ee( -1)rA~f&+fJ>Jtr+{J(2kcosh zo);

S.n+{J

=

(3)

Sesn+{J(zo, q)/ tanh ZOr_~«J( -1 Y(2r+ {J)A:'+fJ>J2r+{J(2k cosh zo)·

In these formulae [oosha] k is finite > O.

>

1,

Zo

(4) is neither imaginary nor zero, and

ADDITIONAL RESULTS

389

The r.h.s, of (1) has the following properties: (a) It is even in z; (b) it has a branch point in the u-plane (u = 2kcoshz) at the origin, and in the z-plane at z = (2r+ 1 )l1Ti, r = -00 to +00, i.e. on the imaginary axis; (c) it is multivalued, and if P= p18, a rational fraction in its lowest terms, it may be shown to have period 2tm-i in s ; (d) it is absolutelyand uniformly convergent if Icosh z I > 1, but non-uniformly convergent as [eosh e] -+ I. On the imaginary axis it diverges, i.e, when [cosh z] ~ 1. Hence there are no representations of the type (1), (2) for cem+~(z, q), sem+~(z, q). co

Ce2n+p(z, q) =

I

A~~n+p>cosh(2r+p)z

r-= -00

is analytic except at infinity where it has an essential singularity. This series and that at (1) § 13.12 in B.F. products are analytical continuations of (1) when [cosh z] ~ 1. Series (2) has properties (b)-(d), but is odd in z. For q < 0, co

I (-1)rA~~n+,8>I2r+,9(2kcoshz), r==-oo

Ce2n+,9(z, -q) and

=

C2n+~

Se2n+p(z, -q)

=

S2n+ptanhz

(5)

00

I (-I)r{2r+p)A~~n+P>I2r+,9{2kcoshz), r;;;:a-oo

with

(6)

(j 2n+fJ = Ce2n+fJ(zo, - q)/ r=~CIJ (_l)rA~~n+fJ>12r+fJ( 2k cosh zo) ; S2n+fJ

(7)

= Se 2n+fJ(zo, -q) /tanh ZOr=~CIJ(-l)r(2r+f3) X X A~~n+/3>I2r+,9(2k cosh zo),

(8)

Zo as above. (5), (6) have the same properties as (1), (2) respectively. Here and in §§ 13-16, the representations for functions of order (2n+l+p) are obtained by writing (I+P) for p, and using A~~'+11+P> for A~~n+p>. (5), (6) were derived from (1), (2) by putting ik for k (by virtue of the altered sign of q = k2 in the D.E.) and writing (-1)rA~~n+p> for A~~"+/3> in accordance with § 21.

13. Bessel function product series for cem+p(z, ±q), sem+p(z, ±q), Cem +,9(z, -q), Sem+,9(z, -q). ce} (z, q) se Z1t+,9

= .K1!.ln+p} t

2n+p

i

r==-oo

(-l)rA:'+fJ>[J,.(v~)J,.+fJ(v~)±J,.+p(v~)J,.(v~)],

(1, 2)

with v~

= ke», v~ = Ice-iz, the K being defined at (2), (3) § 13.12.

ADDITIONAL RESULTS

390

ee}

(z, _q)

Be In+~

~

= (_l)n~.n+~) In+fJ

with

f

(-I)rA~~n+~)(Ir(v~)Ir+.8(v~)±Ir+.8(v~)Ir(vm,

(3,4)

r=-co

Lan+~ = ce.n+~(O, -q)/2r=~a>(-I)rA~~n+.8>Ir(k)Ir+~(k);

(5)

Lan+.8 = se~n+.8(O, -q)/2kr=~a> (-l)rA~~+.8>[Ir(k)I~+~(k)-Ir+.8(k)I~(k)].

Ce}

Se 2n+fJ =

(6)

(Z, -q)

(-1)nL~2n+fJ) 2n+/J

f

(-IYA~~n+.8>[Ir(vl)Ir+.8(v.)±Ir+~(vl)Ir(v.)],

(7, 8)

r=-co

with VI = ke:", V" = kez• If k is finite and positive, (1)-(4), (7), (8) are absolutely and uniformly convergent in any finite region of the a-plane. (7), (8) are analytical continuations of (5), (6) § 12 when Icosh a] ~ 1.

14. Integral equations for Cem+p(z, q), Sem+p(z, q). These were derived by analysis akin to that in § 10.40,

f

co

Ce.n+.8(z,q)

= ~C.,,+.8

and

sin(2k cosh z cosh u-!!lll')Ce.n+.8(u,q) du, (1)

0

Se1n+p(z, q) 00

= -~8zn+.8ksinhz f cos(2kcoshzcoshu-!!l1l')sinhuSe.n+~(u,q)du, o

.

(2)

k, z real> 0, while C2n + p, .921l +p are given at (3), (4) § 12.

15. Asymptotic series for

o < fJ <

Cem+~(z, ±q), Sem+p(z, ±q), z

large,

1.

se IIl+fJ (z,q) ~ P2n+JJ(2/nv)I[P~~+JJsinx-Q.':l+JJcosx],

Ce}

(1,2)

where Pln+fJ = (-I)np~n+fJ = C2n+fJce21l+/3(O,q), and P, Q are given at (13), (14) § 11.20, a = aSn +/3 being used to calculate the c in (7)-(9) § 11.20, x = v+!1T-I,8n, V = kes ; also

CeJ (z, -q) ~ P~n+fJ(21TV)-i[etlf cr v- -ie-i(,,+~,,) I (-l)rcr vSe In+/J r==O r=-O r

r] .

(3,4)

ADDITIONAL RESULTS

391

The functions are real if z is real, so the second member in [ ] is omitted then. If R(z) > 0 is large, cosh(2r+,8)z "" sinh(2r+,8)z, and since the coefficients Ai~n+p) are common to the series representations of both functions (see (12) § 4.71), under the above condition they have the same asymptotic expansion.

16. Asymptotic formulae for Ce 2n +p(z, q), Se1n+p(z, q), q large and positive, 0 < fJ < 1.

ce} () z,q Se 2n+f3

~

h) 1[ (1+x ) cos tp2n • P2n+f3 (k 'IT cos Z ±z2n sin CPln ] , 2n sin tp2n-l cos ({)In-l (1, 2)

where CPm = 2ksinhz-i,8'IT-(2m+I)tan- 1(tanhlz), X 1n, Zln are defined in § 6 and P2n+fJ in § 15. See last two sentences of § 7 regarding accuracy.

17. Degenerate forms. Procedure akin to thatin§ 2 Appendix I, yields

ese}

e m+f3

(z, q) ~ p'm+f3Jm+p(kl r);

Ce} ( .) S e

Z, -~q ~

m

p~ T ('Ik , dm ~ I r ).,

(1, 2) (3,4)

8m

Cer} , ber} (k1r); · (Z)-+Pmb'

(5,6)

Ser} , ber} . (Z) -+ 8 m b' (kl r),

(7, 8)

Cei Sei

m

m

el m

ei

m

COMMENTS ON CERTAIN PARTS OF THE TEXT 18. cem(z, q), sem(z, q) as functions of q real > O. This is displayed in Figs. 167-77, pp. 288-93 in the revised American edition (1943) of Funktionentafeln by E. Jahnke and F. Emde. Except Figs. 167, 169, the normalization in the graphs and those of Figs. 1-4 herein is that of § 2.21. In Figs. 167, 169 the values of ceo(z, q) have been multiplied by 21. The reader should plot similar graphs using the tabular data in references 52, 95.

19. Computation of a, {J, /-" coefficients, etc. The procedure in Chaps. III-VII, XIII, etc., lacks the detail usually needed by the professional computer, because the text is for the general reader. The former is able to supply the missing material required in tabulation. The general reader will find that in computing fJ or JL (especially

39S

ADDITIONAL RESULTS

the latter), the recurrence formula method in § 5.32 is easier to manipulate than the continued fractions in § 5.11. For convenience in § 5.20 et seq., we have taken Co = 1, or c1 = 1, as the case may be. When the order of the function is (m+!J), CO/cm' c1/c m, etc., may be small if m is large enough with q moderate. It might then be expedient to put. Cm = 1. A similar remark applies with regard to computation of A~m+p> in §4.71, since it is proportional to cp. For instance, if m = 12, the computation could be started at, say, c20 on one side of Cl 2 and at c. on the other. To reduce errors it is expedient to work towards C11 from each side. The whole range C20 to c. or vice versa should not be covered in one direction only. An overlap to cm+4 on one side of em' and to cm - 4 on the other, provides a check. 20. Stability of solutions. In (b) § 4.14, a solution which tends to zero as z ~ +00 is defined to be stable, although (a, q) lies in an unstable region of the plane. But the same solution tends to ±oo as z ~ -00, 80 when the whole range of z is considered, it is unstable. In applications where z is a multiple of the time t, which is always real and positive, it is preferable that the definition (b) § 4.14 is adopted. 21. Changtng the sign of q. It is not always expedient to do this by writing (111-Z) for z in solutions of (I) § 2.10, e.g. (5), (6) § 4.71. Now the recurrence relations for -q are obtained from those for +q by writing (-I)'A(2n+!J> or (_I)'A(S1I.+l+P> forA(2ft+P> A(2n+l+p> as 21' J 2r+l J 11" "'+1 , the case may be. Thus, apart from the multiplier (-1)11, (7), (~) § 4.71 follow immediately from (1), (2) § 4.71 if this change is made. The same artifice may be used in §§ 5.20, 5.21, 5.310 to obtain solutions for a changed sign of q. When the argument in the series is 2k cosh z, it is also necessary to write ik for k, since (ik)2 = -q. If the external multiplier is a function of q, it should be re-determined (see § 12, where (5), (6) were derived from (1), (2) in this way). 22. Solution of (8) § 4.80. This linear difference equation is satisfied by the J- and the Y-Bessel functions. Referring to § 3.21, if we put vr = cr +1/C,., we obtain 1',.+1+1);1 = 4(a1/q)(r+I). (I) Then if a, q are finite, it follows from (1) that v, ~ 0 or to ±oo 8S r ~ ±oo. Hence one solution tends to zero, the other to infinity as r ~ ±oo. Now J,. ~ 0, while by aid of (2) lO§ 8.50 it may be shown that ~ ~ =+=00. But for convergence of (0) § 4.80, c; ~ 0, 80 the Y -Bessel function is an inadmissible solution.

ADDITIONAL RESULTS

393

23. Second solutions. Herein we have taken the first solutions of the various differential equations to be either odd or even, according to the value of a. If the first solution is even, there is one (and only one) linearly independent odd solution, and vice versa. But by virtue of the theory of linear D.E.., we may construct from these two solutions as many other second solutions as we please, whioh are neither odd nor even. Thus if Cem(z, q) is the first and even solution, the second and odd linearly independent solution is Fe",(z,q) in § 7.61. If" and 8 are any non-zero constants, y.(z)

= " Cem(z, q)+

8 F em(z, q)

(1)

represents an infinite family of linearly independent second solutions. which are neither odd nor even. Now it may happen that the most suitable second solution is obtained when y and 8 have special values. It is demonstrated in §§ 8.11,8.12, 13.20 that FeYm(z,q} is a linearly independent solution of the same D.E. as Ce,n(z, q), and by §§ 13.20, 13.21 it has the form at (1). In this instance the values of y, 8 derived in § 13.21, are fixed automatically, i.e. they are not arbitrary. Feym(z, q) is more suitable as a standard second solution than Fem(z, q}-see analysis in Chaps. VIII, X, XI, XIII, and Appendix III. Referring to § 9.30, FeYm(z,q), Fekm(z, -q) are preferable to Fem(z, ±q} in applications. They both tend to zero as Z ~ +00. Fundamental system. Any two linearly independent solutions of a Mathieu equation constitute a fundamental system, since the Wronskian relation is non-zero (see (1) § 2.191). In the text it is stated sometimes that if Yl(Z), Y2(Z) are solutions satisfying this condition, while A, B are arbitrary constants, (2)

constitutes a fundamental system. This may be regarded as an extension of the usual definition. 24. Formula (1) § 5.15 for p. If in Fig. 11, a l is plotted instead of a, the iso-fJ curves fJ = 0·1, 0·2, 0·3, ... will intersect the a l axis at equal intervals. Provided I is not too large, these curves are almost parallel to the q-axis. The Iq I range of parallelism increases with increase in ai, as will be evident from the last sentence off 2.151. Then interpolation for fJ is almost linear in ai, and the accuracy of (1) § 5.15 and (4) § 4.74 will be of the same order. This is the basis

q'

394

ADDITIONAL RESULTS

of (1) § 5.15, which is easily established by aid of paragraph one in § 5.13. In Fig. 11 for values of q > 0 beyond the maxima of the iso-fJ curves, where they begin to approach each other, the accuracy of (1) § 5.15 is likely to be low. ~ 25. Remark on (4) § 7.40 in relation to § 7.50 et seq. By § 3.21, if q is finite, as r ~ +00, IA2r+3/A2r+lf ~ q/(2r+3)2. Thus the r.h.s, of (4) § 7.40 is very small when r is large enough, and we get (approximately) (I) § 7.50 but with f for c. One solution tends to zero, the other to infinity as r ~ +00. Consequently, if we assume that /21'-1 > /21'+1 > flr+3' etc., in (4) § 7.40, and calculate the f~m) on this basis, we obtain a particular solution which tends to zero as r~ +00. The series involving the/1m ) may now be shown to have the convergence properties deduced in §§ 3.21,3.22. The coefficients g~m) may be treated similarly. 26. Convergence. Throughout the book, except perhaps at (2) 30 § 8.50, q is assumed to be finite. In fact q > 0 may usually be interpreted to mean that q is greater than zero, but finite. When q ~ +00 an explicit statement is made to this effect.

27. Relationships between FeYm' Cem, ~ekm' etc. in §§ 8.14, 8.30. These were deduced using series for Feym' Fek m , convergent only if [ooshs] > 1. But by virtue of the B.F. product representations these functions are continuous when Icosh z I ~ 1, so byanalytical continuation the said relations are valid in any finite region of the a-plane. They may, of course, be derived directly from the B.F. product series. In view of the frequent occurrence of the foregoing restriction on z, it is apposite to remark that the' forbidden neighbourhood' is the imaginary axis! 28. Remark on § 10.40. The functions represented by the r.h.s, of (4), (6), (7), (9) appear to be even in z, while those in (11), (13), (14), (16)' appear to be odd. They are, however, neither odd nor even, and this point is covered by the restrictions on z at the end of the section.

INDEX a, asymptotio formula, 140, 232, 239, 240. - oharacteristio number in Mathieu's equation, 13, 16, 17. - computation, 28-35. - tabular values, 371. a .., graph of, 40, 41, 98. amplitude distortion in loud-speaker, 267. - modulation, 273, 280, 281. .A~), behaviour when n~ +00, 48. - - when q~ 0,46. - - when q-+ +00, 47. - coefficients in series for ce"" Be"" 21. - computation, 28-37. - recurrence relations, 28. - tabular values, [95]. approximate solutions ot Mathieu's equation, 6, 90-5, 101-3. (a, q).plane, stable and unstable regions (chart), 39-41, 97, 98. asymptotes of characteristio curves a"" b",,39. asymptotic formula for ce., Be"" 230, 232. - . - for Oe"" Se"" 219-27, 230. - - for Fek"" 220-4, 226, 228. - - for Fey"" 220, 222-4. - - for Gek"., 220, 222-4, 228. - - for Gay"" 220, 223, 224, 227•• - - for Me(1), (2) Ne(1), (2) 220 225 attenuation ~f ~a;e8 ~ elli~tic81 wave guide, 354, 355. audio warble tone, 273.

Bessel function series, expansion of Fek., Gek,.., 163-7, 248 (P). - - - expansion of Fey.. , Gay., 1A966, 246 (P), 247 (P). - - - expansion of Se"" 161, 162, 166, 167, 245 (P), 248 (P). Bee under A~).

B~M),

boundary condition, diffraction of e.m. waves, 364. - - - of sound, 359. - - heat conduction, 328. - -- oscillation of water in lake, 301. -- - problems, 9. - - solenoid with metal core, 321, 331. - - vibration of membrane, 297. - - - of plate, 310. - -- wave guide, 334, 337, 339, 345-9.

canonical form of Mathieu's equation, 10. 'carrier' wave in radio telephone transmission, 6, 273. ce"" asymptotic formula in q, 230, 232. - characteristic number, 15-18, 21. - classification, 372. -- degenerate form, 367. - expansion in Bessel function series, 164, 167, 252 (P). - - in cosine series, 21. - integral equation for, 178-85. -limiting form as q~ 0, 13, I(S, 22. - normalization, 10, 24. - period, 18. - recurrence formulae for coefficients in /3, so, 59, 60, 64, 65. series, 28. - calculation of, 104-11. - solution of Mathieu's equation, 11-15. - curves of constant, 97-8. - tabular values, [95]. - formula in a, q, 82, 83. ce.+IJ' characteristic number, 79. Bessel-circular function nuclei, 189-92. - classification, 372. Bessel's differential equation, 189, 344, - degenerate form, 367. 367. - expansion in Bessel function product Bessel function product series, advantages series, 252. of, 257, 258. - - in cosine series, 79. - - series, expansion of C8"" Be"" 164, - limiting form as q -+ 0, 82. 252 (P), 253 (P). t - normalization, 82. - - - convergence of, 169. 257 (P). - period, 81. - - - expansion of ce"'+Ii' &e"'+Ii' - recurrence formula for coefficients in 252 (P). series, 65. - - - expansion of Oe., 159, 160, 165, - solution of Mathieu's equation, 64. 245 (P), 248 (P). Oe.. , asymptotic formula in q, 230. - -- - expansion of fe., ge., 254, 255, , - - - in %, 219-26. 256 (P), 257 (P). - characteristio number, 16-18, 26, 27. t (P) signifies' Beesel function product series'.

i:

4"1

3E2

396

INDEX

Oe., classification, 372. - degenerate form, 368, 369. - expansion in Bessel function series, 169, 160. - - - - product series, 245, 248. - - in cosh series, 26, 27. - infinite integral for, 197. - integral equation for, 187-90. - limiting form as q -+ 0, 27. - normalization, 82. - period, 26. - recurrence formulae for coefficients in series, 28. - solution of modified Mathieu equation, 26. Ce-+Il , classification, 372. - expansion in Bessel function product series, 245. - - in cosh series, 81. ce~ p' classification, 372. - definition, 88, 89. - normalization, 89. changing sign of q, 21, 26, 66, 165. characteristic curves a"" b"" 39. - - asymptotes, 39. - exponent pe, 7, 58, 61-3, 67-70, 101, 103. - - - calculation of, 69, 103, 105-7, 130, 136-9. - functions, expansions in, 207. ---- number, asymptotic formula, 140, 232, 239, 240. - - computation, q real, 28-37. - - - q imaginary, 48-55. - - continuity 88 a function of q, 40. - - expansion in powers of q, 16, 17, 20. - - tabular values, 371. - - transcendental equation (continued fraction) for, 29. - values of nuclei in integral equations for cellI' se..., 185. circular cylinder, heat conduction in, 330. circulation of fluid at infinity, 315-19. 0;", definition, 254-5. O",(q), 143-8.

- computation, 153. - limiting form 88 q~ 0, 148. coefficients A <;'), B<;'), behaviour of 88 +00, 48. - - - - of as q-+ 0,45-7. - - - - of as q~ +00,47. - - - calculation of, q real, 28-37. - - - - of, q imaginary, 48-54. - - - continued fractions for, 29, 33-6.

,,-+

coefficients A~"), 1J<;), in series for ce., Be"" 21, 22. - - - recurrence relations for, 28. - A~"'+tJ), behaviourofas 13-+ oand 1,81. column, vibration of long, 292. combination solutions Me~)'(~)' Ne:':)·(2), 250, 251. condensation of zeros of ce m, Bem, 238. - - - of Ce"., Se"" 241. conduction of heat in circular cylinder, 330. - - - in elliptical cylinder, 328. conductor, resistance of circular to a,c., 327. - - of elliptical to a.c., 324-6. - - of rectangular to a.c., 324-7. conjugate properties of coefflcients for unstable regions of (a, q)-plane, 83-6. continued fraction for a, 29-34. - - - - convergence of, 29, 33. -- -_. -

A<;'), B~"'), 28-34.

- - -- p, -- -- -

106.

#-'-, 107.

convergence of Bessel funot.ion product series for cem , Be"" Cem , Se"., 'Fey"" Oey"., Fekm , Oek"., 257. - - - series for ce"" sem , Ce"" Se"" Fey"" GeY,n' Fek"., Oek"" 169. - of circular function series for cem' sem' 37. -- - -- -- for ce~), se~), 38. -- of general solution of Mathieu's equation, 90. - of hyperbolic function series for Ce..., So",1 38.

-- - - - for Ce~), Se~), 38. - of infinite determinant for #-'-, 67. -- -- - integrals, 200-2. coordinates, elliptical, 170. -- --- diagram illustrating, 171. - orthogonal curvilinear, 341. - polar, 170, 189. -- rectangular (cartesian), 170. curl of vector, 342. current density in elliptical conductor carrying a.c., 324. curvilinear orthogonal coordinates, 341. cut-oft or critical pulsatanoe in circular wave guide, 345. - - - in elliptical wave guide, 349. - - - in rectangular wave guide, 337. deformation of circular wave guide, effect of, 356. degenerate form of 00., Be., as fundamental ellipse tends to circle, 367.

INDEX degenerate form of ce"+II' se,,+~, 88 fundamental ellipse tends to circle, 367. - - of Oe..., Se"" 88 fundamental ellipse tends to circle, 368. - - of Fey.. , Gey ..., }4'ek..., Gek., as fundamental ellipse tends to circle, 368. - - ofMe~)·(~), Ne~)·(2),88 fundamental ellipse tends to circle, 368. depth of penetration of current induced in metal slab by e.m. wave, 353. determinant, infinite, for 11-, 67, 128. deviation, frequency, 276. difference equation, 37, 150, 158. differential equation, heat conduction, 328. - - Hill's, canonical form, 58, 127. -- - - solution of, 127-9, 131-5. - - Mathieu's, canonical form, 10, 58. - - -- solution of, 11-16, 21, 58, 90-6, 99-103, 111-26. - - - modified, 1, 26. - - - - solution of, 26, 27, 158,243. - - Maxwell's for e.m, field, 334-6. - - oscillation of water in lake, 301. - - satisfied by Yl(Z)Y2(Z), 217. - - sound propagation, 358. - - vibration of elliptical plate, 310. - - - of membrane, 294. --- -- vortex motion, 313-15. - - wave guide (circular), 344. - - - -- (elliptical), 347. - -- - - (rectangular), 336. differentiation under integral sign, 201. diffraction, diagrams illustrating, 363, "366. '- e.m. waves at elliptical cylinder, 363. - - .- at long ribbon, 365. - - - at long slit, 366. - sound waves at elliptical cylinder, 358. - - - at long ribbon, 361. - - - at long slit, 362. discontinuous nucleus of integral equation,3. distortion, amplitude, in loud-speaker, 267. division of (a, q)-plane into stable and unstable regions, 76. dominant terms in asymptotic series for Ce,., Sa., Fey.., Gey",, Fek"., Gek.., Me(1)· (~) Ne(l). (2) 219-21 dyna;;'ic~l de~ice'illustra;ing Mathieu's equation, 18, 285. - - - -- - stability diagram for, 286. - - - sub-harmonic, 287.

397

eccentricity of ellipse, 171. eddy currents in elliptical cylinder, 7, 320-3, 331-2. effective inductance and resietance of solenoid with elliptical core: 323. electrical diffusion, 320. eloctromagnetic waves, differential equations for, 334-6. - - diffraction of, 363-6. -- --- in circular wave guide, 344-7. - -- in elliptical wave guide, 347-50. - - in rectangular wave guide, 337-41. electromechanical rectification, 272. elliptic cylinder functions, 2. elliptical coordinates, 1, 9, 170. - cylinder in viscous fluid, 313. - lake, modes of oscillation, 300. equation, difference, 37, 150, 158. - differential, Bee under differential equations. - pulsatance, for elliptical lake, 302. - - - - membrane, 297. -- - - - plate, 311. evaluation of Fey.:a(O, q), Gey ...(O.q), Fek,..(O, q), Gek".(O,q), 249, 250. - - miscellaneous integrals, 259-66. E-waves in circular wave guide, 346. - in elliptical wave guide, 349. - in rectangular wave guide, 338. expansion of function in terms of ce., Cam' 86m, Sem , 297, 302. - - nucleus in characteristic functions, 207. fem , characteristic number, 144. - claasiflcation, 372. - definition, 144. -- expansion in Bessel function product series, 254-6. - limiting form as q-+ 0, 144. - non-periodic nature of, 141-3. - normalization, 144, 153. - recurrence formulae for coefficients in series, 148. - solution of Mathieu's equation, 145. Fe"., characteristic number, 155-7. - claesiflcation, 372. - definition, 155-7. - limiting form 88 q ~ 0, 156. - normalization; 144, 153. - recurrence formulae for coefficients in series, 148. - relation between Oem' Fey"" and, 247. fek"" classification, 372. - definition, 256. Fek"" aaymptotio formula, 220-4.

398

INDEX

Fek"" characteristic number, 163-6. - cl888ification, 372. - degenerate form, 368, 369. - expansion in Bessel function product series, 247, 248. ~ ---- --- - - series, 163. --.- non-periodic nature, 163. -- relation between Cem, FeYm' and, 163-6. Fey"" asymptotic formula, 220-6. -- characteristic number, 160, 165, 166. - classification, 372. - degenerate form, 368. - expansion in Bessel function product series, 246, 247. - - - - series, 159, 161. - non-periodic nature, 160. -- relations between Cam' Fe"" and, 247. - - - - Fek"., and, 163. fluid, vortex motion in, 313-19. flux, magnetic in elliptical metal core of solenoid, 322. !",(z, q), definition, 143. - limiting form when q ~ 0, 143, 144. -- normalization, 144. - period, 144. formula, asymptotic, see under asynlptotic. - for fJ in terms of a, q, 83. f~"'), behaviour of when q-+ 0, 149, 150. - coefficients in series for fe"" ge"" Fe"" Gem' 144, 154-7. - computation of, 150-4. -- recurrence relations for, 148. fractional order, functions of, 19, 20, 79, 80. - - - of, classification, 372. frequency modulation, 6, 9, 273-82. ge"" see under feme Ge., see under Fe.... gek,.., see under fek".. Gelt"., see under Felt".. general relationships between solutions of Mathieu's equation, 66. - solutions of Mathieu's equation, 59-64. -- - - - altemative form, 70-6. - theory of Mathieu's equation, 57. Gey"" see under Fey",. 9.(z, q), 8U under j".(z, q). ~".), eee under J~"'). guide, wave, circular, 344-7. - - elliptical, 347-51. - - rectangular, 337-41. heat conduction, He under conduction.

I I

I

Hill's equation, 2, 4, 127. - - calculation of p., 136. - - solution of, 127, 131-5. -- - stability of solutions, 140. - infinite determinant, 128. H-waves in circular wave guide, 346. -- in elliptical wave guide, 349. -- in rectangular wave guide, 339. identity, a fundamental, 23. - integral, 265. , imaginary' power in electrical circuit, 323. inductance of solenoid with elliptical metal core, 323, 332. infinite continued fraction for a, 29. - - - convergence of, 29-, 33. - - - for Vir' 33. - determinant, 67, 128. initial conditions, 57. - value problems. 9. integral equation, characteristic value, 185. -- - definition, 178. - - for ce m, 180-4. --- - for Cem, 187-92, 197. - - for Fek"" Gek"., 191. -- --- for Fey"" GaYm' 190, 196. -- -- for se m , 180, 183-7. - -- for Sem , 188, 191, 194, 198. -- -- for Yl(Z)y,(z), - - nuclei for, 180, 190, 191. --- - of first kind, 179. -- -- of second kind, 211. - - solution of, 213-16. - order functions, classification, 372. - relations, 186, 197-200. integrals, miscellaneous, involving Mathieu functions, 259-66. integro-differential equation, 214. interpolation for a, 35. - for p, 109. inverse capacitance modulation, 276. irrotational fluid motion, 313. iso-fJ curves, 79. - -p. curves, 87. lake, circular, modes of oscillation, 303. - elliptical, eccentricity e -+ 1, 305. - - modes of oscillation, 301. limiting form of A<;'), B~"') as q -+ 0,45-7. - - of A~"+IJ) as P-+ 0, 81. - - - as q-+ 0,82. - - of oe., 86", as q -+ 0, 10. - - of Oe.., Be. as q-+ 0, 27. - -- of C8",+" se.. +~ as q -+ 0, 82.

INDEX limiting form of felli' ge. as q -+ 0, 144. -- - of Fe"" Ge", 88 q~ O. 156. -- -- of 11"'), g~Ht) as q --+ 0, 149. -- --- of Pm as P. -+ 0, 89. linear damping, dynamical system having, 287. lines of electric and magnetic force in wave guides, 340, 348, 350. line spectrum in frequency modulation, 279. long rigid ribbon, scattering of e.m. waves by, 363. - - - scattering of sound waves by, 362. Ioud-speaker, amplitude distortion in, 267. - electromechanical rectific-ation, 272. - pulsatance multiplicatlon, 273. lunar perigee, 2, 129, 130, 136. p., curves of constant, 87, 98.

Maclaurin series solution of Mathieu's equation, 74. Mathieu functions of first, second, third kinds, 372. Maxwell's equations, 334-6. - - application to wave guides, 338-9, 342-4. (m-f-~), functions of order, 19, 20, 79, 80. - - - classification, 372. Melde's experiment, 2, 289. Me~)·(2), application in wave propagation, 360-6. -- asymptotic formula, 220, 223, 225. -- definition, 250. - degenerate form, 368, 369. --.: expansions i~ Bessel function series, 250. -- relationship between Fek". and, 251. membrane, circular, 298, 299. - elliptical, 294-8. microphone, condenser type, 276. (m+p.), functions of order, 88-9. - - - classification, 372. modified Mathieu equation, 1, 26. - - functions, definition, 1, 26-7. modulation, frequency, classification, 274-6. - -- stability chart, 277. - - zones, 276. Ne~)·(2), Bee under Me~)·(2). nodal ellipses and hyperbolae, 5, 297, 304, 307, 311-12. non-linear D.E. for vibrating string, 290. non-periodic solutions of Mathieu's equation, 9, 141.

399

normalization of ce.. , se.., Oe"" Be..., 24. - - ce"'+I)' se"'+IJ' Ce"'+/J' Be.+/J' 82. - ~-- ceu"-t-#-&' 89. - - fe..., ge..., Fe"" Ge.. , 144, 153. nuclei for integral equations, 180, 190-1. - skew-symmetrical, 190, Table 17. numerical solution of equations, 111-26. orthogonality of ce..., Be"" 22. - - - application, 99, 206, 208, 321. ---- of cO"'+PI" 86",+p", 24. -- theorem for co"" Ce",; sa"" Sa"" 176. -- - for ce..., Ce",; &e.., 8e"., application, 298, 329. oscillation of membrane, etc., see under vibrational modes. oscillatory circuit, constant resistance, variable capacitance, 282. - - resistancelesa, variable capacitance, 274. -- - variable resistance, constant capacitance, 284. parametric point (at q), 63. -- zeros of Ca"., Se., 241. periodic coefficients, differential equation having, 57. - mass distribution on string, 290. - solutions of first kind, 11-15, 372. - -- of fractional order, 20, 24, 81. periodically varying capacitance, resistance, see under oscillatory circuit. -- - tension of vibrating string, 289. phase factor, 337. - velocity, 357. plane, the (a, q), iso-f3, iso-" curves, stable and unstable regions, 39, 76, 98. plate, vibration of circular, 313. - -- of elliptical, 310-12. p"., p;", the multipliers, 368-9. P~)' (6), definition, 222. polar coordinates, 170, 189, 342, 344. - diagrams for diffraction of e.m, and sound waves, 363, 366. power, condition for transmission in circular wave guide, 34lt - - - in elliptical wave guide, 349. - - - in rectangular wave guide, 338. - - 1088 in elliptical conductor carrying a.c.,326. - transmitted down elliptical wave guide, calculation, 351. Poynting vector, 351. problem, electromagnetic, analogous to heat conduction, 331.

400

INDEX

pulsatance equation, see under equation. - multiplication, 273.

slit, diffraction of HOWld waves at, 362. 8.,., 8;", the multipliers, 368-9. S;", definition, 255.

S::), (6), definition, 223. q, aM' b", 88 a function of, 39-41. --- A~'"), u".) as a function of, 41-8. ~ S".(q), 144-8. computation, 153. - p,P. 88 a function of, 83, 97, US. - negative imaginary in Mathieu's - value as q -+ 0, 148. solenoid, effective resistance and inducequation, 48-56, 320. tance with metal core, 323-4. - real parameter in Mathieu's equation, solution of Mathieu's equation, a large 1, 10. ~ q > 0,90-6. - seros in, Ca,.., Be,.., Fey"" Gey"" 241. Q::)'(6), definition, 222. - -- - circular function series, 64. - - - convergence, 37, 38, 90, 169, recurrence formula for A~'"), B~·), 28. 257. - -- for A~"'+~), {1, 65. - - -- exponential series, 58. - - - form of, in different regions of - -- for coefficients in asymptotic series, 221, 225. (a, q)-plane, 77, 78. -- -- - -- _.- in unstable solutions, 83. --- -- - form of, q = 0, 10, ] 1, 82. --- - for 11"'), g1"'), 148-9. --- -- -- in BCRSeI function product relationship between Ce"., Fe"., Fey"., series, 243. 247. - - - in Bessel function series, 158. ._-- - - numerical, 111-26. - -- -- J4"ek,.., Fey"" 163, 165-6. -- -- Sa,.., Ge"., Gey"., 247. -- - - period Tr, 2Tr, 11-16, 21. -- - - period 28Tr, 20, 81. -- - -- Gek"" Gey"" 163-4. resistance, rectangular metal strip to a.c., -- -- --- q series, 11-16. 327. - - - relations between, 66. - effective of solenoid with circular -- - - unstable region of (a, q)-plane, metal core, 324. 99-103. - -- - with elliptical metal core, 323. spectrum, line, in frequency modulation, e:), (6), definition, 223. 279. stability chart, 4, 39-41, 97-8. roots(large)ofCe...(z,q) = O,Se".(z,q) = 0, in q, 241. - of electrical circuit having constant - - - - in z, 241. resistance, 283. - - of Fey",(z, q) = 0, Gey",(z, q) =~ 0, -- of electrical oscillations, 278. in %, 241. --- of long column, 293. roots of infinite determinant for p., 69. _.- of solutions of Hill's equation, 137-40. - of solutions of Mathieu's equation, 59, scalar potential at surface of conductor, 60. 324. stream function, 314. scattering of e.m. and sound waves by string, vibrating, constant tension, 288. elliptical cylinder, 358-66. --- - periodic mass distribution, 290. second solution of Mathieu's equation, 5, -- - -- variation of tension, 289. 141. sub-harmonic, in dynamical system, -- - - modified equation, 155-7. 286-7. &e"" see under ce•. - in vibrating string, 3, 290. super-regenerative radio receiver, 285. se..+~, see under ce,..+~. Be.., see under Ce",. Se",+~, see under Ce",+~. theorem, expansions in terms of ce"" separation constant, 1, 174, 337, 344. Ca.,., se"" Se.,., 297, 302. side bands in frequency modulation, 6, - integral equation, 178. 280. - - relation, 186. skew-symmetrical nucleus in integral - non-periodicity of second solution of equation, 190, Table 17. Mathieu's equation, 141. slab, current distribution due to e.m. - orthogonality, 175. field, 352. thermal diffusion in elliptical cylinder, slit, diffraction of e.m, waves at, 366. 328.

INDEX tide height in elliptical lake, 302. T~), definition, 226. transformation, Mathieu's equatien, 25, 26. - - - to Riccati type, 91. - wave equation to elliptical <.'0ordinates, 173. - - - to polar coordinates, 189, 344. transition, elliptical to circular section, core of solenoid, 323-4, 332. - - - - heat diffusion, 330. - - - - . membrane, 298-9. - - - - plate, 313. -- - - - wave guide, 355. transverse electric and magnetic wa vee in circular wave guide, 346. - - - - in elliptical wave guide, 349. - - - -- in rectangular wave guide, 338-9. - vibrational modes of gas in elliptical cylinder, 308. ultra high-frequency transmission in wave guide, 333. U~), definition, 226. uniform canal, oscillation of water, 306. unstable regions of (a,q)-plane, 40, 41, 97,98. -- solutions of Mathieu's equation, 59, 60, 84-U. - --.- --. - --- classification, 372. -.- -- conjugate properties of coefficients, 83. vector, curl of, 342. - differential ~rmulae, 342. - potential at surface of conductor, 324. - P-oynting's, 351. velocity, phase, in wave guide, 357. - potential, sound, 358. -- - viscous fluid flow, 314. vibrating system with linear damping, 287. vibrational modes of gas in elliptical cylinder, 308. -- - of membrane, 296.

401

vibrational modes of plate, 309. - - of string with constant tension, 288. \ - - -- with periodically varying mass per unit length, 200. - - - with periodically varying tension,289. - - of water in lake, 301. viscosity, kinematic coefficient of, 314. V~), definition, 227. volume distribution of eddy currents in slab, 352. vorticity of fluid round elliptical cylinder, 8,313. warble tone, audio, 273. water, oscillation in elliptical lake, 300. - - in uniform canal, 306. wave equation, 1,9. .- " -- in elliptical coordinates, 173, 301, 310, 320, 328, 347, 358. --. - in orthogonal curvilinear coordinates, 343. - - in polar coordinates, 189, 344. ---. -- in rectangular (cartesian) coordinates, 1, 170, 294, 300 7 308, 320, 328, 336, 337, 358. -- guide, circular, 344. -- --- - cut-off pulsatance, 345. -- - ---- elliptical, 347. -- -- --- attenuation in, 354. --_. -- --- cut-off pulsatance, 349. --.. - -- powor transmitted down, 351. -- -- rectangular, 337. -- - -- cut-off pulsatance, 337. W::), definition, 227. zeros of A1"'), B1"'), 43-5. - of ce"" se"" 234, [96].

-- - Ce"" Se"" in q, z, 241. - - oondensatien of as q~ +00, 238. Zo, multiplier for unstable solutions, 84. Zo, multiplier connecting two forms of general solution, 75. zones, frequency modulation, 276-7. -

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