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 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
(-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/ 16r23
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 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 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>
ezi+ 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
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.
(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
=
! [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- 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(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)
=
=
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-
!
(-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
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 ) 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)
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
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,.
(8)
(r ~ 2).
(9)
7.41. Behaviour of 1m' 9m as q -+ o. By § 3.32, A~)-+ 1, B2r+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)[[(!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
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
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 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, 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)8evn,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
AA(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 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 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-
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]
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 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
>
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
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.
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
(-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
~
[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·
(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
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~(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 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.,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 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) ~ - =
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.
=
- 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)
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 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", 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 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- 22)
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
= _
= _
22
[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. 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.
REFERENCES
316
41. Einaudi, R. Sulle configurazioni di equilibrio instabilo di una piaatra sollecitata da sforzi tangentiali pulsante. Atti Accad. Gioenia Catania, mem.10, I, 1936; memo 5, 1, 1937. 42. Emersleben, O. Freie Schwingungen in Kondensatorkreisen. Phya. Zeita. 22, ~93, 1921. 43. Erdelyi, A. fiber die kleinen Schwingungen eines Pendels mit oszillierendem Aufhangepunkt. Z.A.M.M. 14,435, 1934. 44. - - Uber die freien Schwingungen in Kondensatorkreisen mit periodisch veranderlieher Kapaaitat, A. der P. 19, 585, 1934. 45. - - Zur Theorie des Pendelriickkopplers. A. der P. 23, 21, 1935. 46. - - Uber die rechnerisehe Ermittlung von Schwingungsvorgangen in Kreisen mit periodisch schwankenden Parametern. A. fur E. 29, 473, 1935. 47. - - Uber die Integration der Mathieuschon Diffcrentialgleichung 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
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 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