watson atreatiseonthetheoryofbesselfunctions text

G.N.WATSON G.N.WATSON

PRESS

A TREATISE ONTHE THEORY OFBESSEL FUNCTIONSG. N. Watson

The late Professor G. N. Watson wrote hismonumental treatise on the theory of Besselfunctions with two objects in view. The firstwas the development of applications of thefundamental processes of the theory of com-plex variables, and the second the compila-tion of a collection of results of value tomathematicians and physicists who en-counter Bessel functions in the course oftheir researches.The completeness of the theoretical ac-

count, combined with the wide scope of thecollection of practical examples and theextensive numerical tables, have resulted ina book which is indispensable to pure mathe-maticians, to applied mathematicians, andto physicists alike.

'In Professor Watson's treatise, which is amonument of erudition and its often toorare accompaniment, clear exposition, wehave a rigorous mathematical treatment ofall types of Bessel functions, their properties,integral representations, asymptotic ex-pansions, integrals containing them, alliedfunctions, series, zeros, tabulation, togetherwith extensive numerical tables.'L. M. Milne-Thomson in Nature'A veritable mine of information.

. .indis-

pensable to all those who have occasion touse Bessel functions.'S. Chandrasekhar inTheAstrophysicalJournal

Also available as a paperback

THEORY OFBESSEL FUNCTIONS

W. B. F.

A TREATISE ON THETHEORY OF

BESSEL FUNCTIONS

BY

G. N. WATSON

SECOND EDITION

CAMBRIDGEAT THE UNIVERSITY PRESS

1966

PUBLISHED BYTHE SYNDICS OF THE CAMBRIDGE UNIVERSITY PRESS

Bentley House, 200 Euston Road, London, N.W. 1American Branch: 32 East 57th Street, New York, N.Y. 10022

West African Office: P.M.B. 5181, Ibadan, Nigeria

First Edition 1922Second Edition 1944

Reprinted 1952195819621966

First paperbackedition 1966

/~--fcU^6\\*s

First printed in Great Britain at the University Press, CambridgeReprinted by lithography in Great Britain byHazell Watson & Viney Ltd, Aylesbury, Bucks

PREFACETHIS book has been designed with two objects in view. The first is the

development of applications of the fundamental processes of the theory offunctions of complex variables. For this purpose Bessel functions are admirablyadapted; while they offer at the same time a rather wider scope for the appli-cation of parts of the theory of functions of a real variable than is provided bytrigonometrical functions in the theory of Fourier series.

The second object is the compilation of a collection of results which wouldbe of value to the increasing number of Mathematicians and Physicists whoencounter Bessel functions in the course of their researches. The existence ofsuch a collection seems to be demanded by the greater abstruseness of propertiesof Bessel functions (especially of functions of large order) which have beenrequired in recent years in various problems of Mathematical Physics.

While my endeavour has been to give an account of the theory of Besselfunctions which a Pure Mathematician would regard as fairly complete, I haveconsequently also endeavoured to include all formulae, whether general orspecial, which, although without theoretical interest, are likely to be requiredin practical applications; and such results are given, so. far as possible, in aform appropriate for these purposes. The breadth of these aims, combinedwith the necessity for keeping the size of the book within bounds, has madeit necessary to be as concise as is compatible with intelligibility.

Since the book is, for the most part, a development of the theory of func-tions as expounded in the Course of Modern Analysis by Professor Whittakerand myself, it has been convenient to regard that treatise as a standard workof reference for general theorems, rather than to refer the reader to originalsources.

It is desirable to draw attention here to the function which I have regardedas the canonical function of the second kind, namelt the function which wasdefined by Weber and used subsequently by Schlafli, by Graf and Gubler andby Nielsen. For historical and sentimental reasons it would have been pleasingto have felt justified in using Hankel's function of the second kind; but threeconsiderations prevented this. The first is the necessity for standardizing thefunction of the second kind; and, in my opinion, the authority of the groupof mathematicians who use Weber's function has greater weight than theauthority of the mathematicians who use any other one function of the secondkind. The second is the parallelism which the use of Weber's function exhibitsbetween the two kinds of Bessel functions and the two kinds (cosine and sine)

VI PREFACE

of trigonometrical functions. The third is the existence of the device by whichinterpolation is made possible in Tables I and III at the end of Chapter XX,which seems to make the use of Weber's function inevitable in numerical work.

It has been my policy to give, in connexion with each section, referencesto any memoirs or treatises in which the results of the section have beenpreviously enunciated; but it is not to be inferred that proofs given in thisbook are necessarily those given in any of the sources cited. The bibliographyat the end of the book has been made as complete as possible, though doubtlessomissions will be found in it. While I do not profess to have inserted everymemoir in which Bessel functions are mentioned, I have not consciously omittedany memoir containing an original contribution, however slight to the theoryof the functions; with regard to the related topic of Riccati's equation, I havebeen eclectic to the extent of inserting only those memoirs which seemed tobe relevant to the general scheme.

In the case of an analytical treatise such as this, it is probably useless tohope that no mistakes, clerical or other, have remained undetected; but thenumber of such mistakes has been considerably diminished by the criticismsand the vigilance of my colleagues Mr C. T. Preece and Mr T. A. Lumsden,whose labours to remove errors and obscurities have been of the greatestvalue. To these gentlemen and to the staff of the University Press, who havegiven every assistance, with unfailing patience, in a work of great typographicalcomplexity, I offer my grateful thanks.

G. N. W.

August 21, 1922.

PREFACE TO THE SECOND EDITIONTo incorporate in this work the discoveries of the last twenty years would

necessitate the rewriting of at least Chapters XIIXIX; my interest inBessel functions, however, has waned since 1922, and I am consequently notprepared to undertake such a task to the detriment of my other activities.In the preparation of this new edition I have therefore limited myself to thecorrection of minor errors and misprints and to the emendation of a fewassertions (such as those about the unproven character of Bourget's hypo-thesis) which, though they may have been true in 1922, would have beendefinitely false had they been made in 1941.

My thanks are due to many friends for their kindness in informing me oferrors which they had noticed; in particular, I cannot miss this opportunityof expressing my gratitude to Professor J. R. Wilton for the vigilance whichhe must have exercised in the compilation of his list of corrigenda.

G. N. W.March 31, 1941.

CONTENTSCHAP.

I.

II.

III.

IV.

V.

VI.

VII.

VIII.

IX.

X.

XI.

XII.

XIII.

XIV.

XV.

XVI.

XVII.

XVIII.

XIX.

XX.

BESSEL FUNCTIONS BEFORE 1826

THE BESSEL COEFFICIENTS

BESSEL FUNCTIONS

DIFFERENTIAL EQUATIONSMISCELLANEOUS PROPERTIES OF BESSEL FUNCTIONS

INTEGRAL REPRESENTATIONS OF BESSEL FUNCTIONS

ASYMPTOTIC EXPANSIONS OF BESSEL FUNCTIONS.

BESSEL FUNCTIONS OF LARGE ORDER ....

POLYNOMIALS ASSOCIATED WITH BESSEL FUNCTIONSFUNCTIONS ASSOCIATED WITH BESSEL FUNCTIONS .ADDITION THEOREMS

DEFINITE INTEGRALS

INFINITE INTEGRALS

MULTIPLE INTEGRALS

THE ZEROS OF BESSEL FUNCTIONS .

NEUMANN SERIES AND LOMMEL'S FUNCTIONSVARIABLES

KAPTEYN SERIES

SERIES OF FOURIER-BESSEL AND DINI .

SCHLOMILCH SERIES

THE TABULATION OF BESSEL FUNCTIONS .

TABLES OF BESSEL FUNCTIONS

BIBLIOGRAPHY

INDEX OF SYMBOLS

LIST OF AUTHORS QUOTED ....GENERAL INDEX

OF TWO

PAGE

1

H38

85

132

160

194

225

271

308

358

373

383

450

477

522

551

576

618

654

665

753

789

791

796

To stand upon every point, and go over things at large, and to be curious in

particulars, belongeth to the first author of the story : but to use brevity,

and avoid much labouring of the work, is to be granted to him that willmake an abridgement.

2 Maccabees ii. 30, 31.

CHAPTER I

BESSEL FUNCTIONS BEFORE 1826

1*1. Riccati's differential equation.

The theory of Bessel functions is intimately connected with the theory ofa certain type of differential equation of the first order, known as Riccati'sequation. In fact a Bessel function is usually defined as a particular solution

ofa linear differential equation of the second order (known as Bessel's equation)which is derived from Riccati's equation by an elementary transformation.

The earliest appearance in Analysis of an equation of Riccati's type occursin a paper* on curves which was published by John Bernoulli in 1694. Inthis paper Bernoulli gives, as an example, an equation of this type and statesthat he has not solved it"f\

In various letters! to Leibniz, written between 1697 and 1704, JamesBernoulli refers to the equation, which he gives in the form

dy = yydx + xxdx,

and states, more than once, his inability to solve it. Thus he writes (Jan. 27,1697): " Vellem porro ex Te scire num et hanc tentaveris dy yydx + xxdx.Ego in mille formas transmutavi, sed operam meam improbum Problema per-petuo lusit." Five years later he succeeded in reducing the equation to a linearequation of the second order and wrote to Leibniz (Nov. 15, 1702) : " Quaoccasione recordor aequationes alias memoratae dy yydx + a?dx in qua nun-quam separare potui indeterminatas a se invicem, sicut aequatio maneretsimpliciter differentialis : sed separavi illas reducendo aequationem ad hancdifferentio-differentialem|| ddy.y x& dx*."

When this discovery had been made, it was a simple step to solve the lastequation in series, and so to obtain the solution of the equation of the firstorder as the quotient of two power-series.

* Acta Eruditorum publicata Lipsiae, 1694, pp. 435437.

f "E8to proposita aequatio differentialis haec xidx+ y*dx= a?dy quae an per separationemindeterminatarum construi possit nondum tentavi " (p. 436).

t See Leibnizens gesamellte Werki, Dritte Folge (Mathematik), in. (Halle, 1855), pp. 5087. Ibid. p. 65. Bernoulli's procedure was, effectively, to take a new variable u defined by the

formula1 du

_

u dx ~*

in the equation dyldx=x'2 + y2, and then to replace by y.il The connexion between this equation and a special form of Bessel's equation will be seen

in 4-3.

2 THEORY OF BESSEL FUNCTIONS [CHAP. I

And, in fact, this form of the solution was communicated to Leibniz byJames Bernoulli within a year (Oct. 3, 1703) in the following terms*:

"Reduco autem aequationem dy= yydx+xxdx ad fractionem cujus uterqueterminus per seriem exprimitur, ita

X3 X7 X11 XVa X19

3 3.4.7 3.4.7.8.11 3. 4.7.8. 11 . 12. 15^3. 4. 7.8. 11. 12. 15. 16. 19V

,X* X* X12 xlti

3.4 T 3.4.7.8 3.4.7.8.11.12 T 3.4.7.8.11.12.15.16

quae series quidem actuali divisione in unam conflari possunt, sed in quaratio progressionis non tarn facile patescat, scil.

_a? x1 2x11 13-r15y ~ 3

+3.3. 7 + 373. 3. 7. 11

+ 3.3.3.3.5. 7777II +

Of course, at that time, mathematicians concentrated their energy, so faras differential equations were concerned, on obtaining solutions in finite terms,and consequently James Bernoulli seems to have received hardly the full creditto which his discovery entitled him. Thus, twenty-two years later, the paperf

,

in which Count Riccati first referred to an equation of the type which nowbears his name, was followed by a note:,: by Daniel Bernoulli in which it wasstated that the solution of the equation

ax11 dx + uudx = bduwas a hitherto unsolved problem. The note ended with an announcement inan anagram of the solution : " Solutio problematis ab 111. Riccato propositocharacteribus occultis involuta 24a, 6b, 6c, Sd, 33e, hf, 2g, 4>h, 33i, 61, 21m,

26n, 16o, Sp, oq, I7r, 16s, 25t, 32m, 5#, By, +, -, , , =, 4, 2, 1."

The anagram appears never to have been solved ; but Bernoulli publishedhis solution || of the problem about a year after the publication of the anagram.

The solution consists of the determination of a set of values of n, namely 4m/(27/i 1), where m is any integer, for any one of which the equation issoluble in finite terms; the details of this solution will be given in 4*1, 4*11.

The prominence given to the work of Riccati by Daniel Bernoulli, combinedwith the fact that Riccati's equation was of a slightly more general type than

* See Leibnizens gesamellte Werke, Dritte Folge (Mathematik), m. (Halle, 1855), p. 75.

t Acta Eruditorum, Suppl. vm. (1724), pp. 6673. The form in which Riccati took theequation was

xmdq = du + uu dx:q,where q = xn .

% Ibid. pp. 7375. Daniel Bernoulli mentioned that solutions had been obtained by threeother members of his familyJohn, Nicholas and the younger Nicholas.

The reader should observe that the substitution

bdzu=

z dx

gives rise to an equation which is easily soluble in series.

|j Exercitationes quaedam matheviaticae (Venice, 1724), pp. 7780; Acta Eruditorum, 1725,

pp. 465473.

1*2] BESSEL FUNCTIONS BEFORE 1826 3

John Bernoulli's equation* has resulted in the name of Riccati being associatednot only with the equation which he discussed without solving, but also witha still more general type of equation.

It is now customary to give the namef Riccati's generalised equation toany equation of the form

where P, Q, R are given functions of x.It is supposed that neither P nor R is identically zero. If U=0, the equation is linear;

if P=0, the equation is reducible to the linear form by taking 1/y as a new variable.

The last equation was studied by Euler J ; it is reducible to the generallinear equation of the second order, and this equation is sometimes reducibleto Bessel's equation by an elementary transformation (cf. 3*1, 4*3, 4'31).

Mention should be made here of two memoirs by Euler. In the first itis proved that, when a particular integral yx of Riccati's generalised equationis known, the equation is reducible to a linear equation of the first order byreplacing y by yx + 1/u, and so the general solution can be effected by twoquadratures. It is also shewn (ibid. p. 59) that, if two particular solutions areknown, the equation can be integrated completely by a single quadrature; andthis result is also to be found in the second || of the two papers. A brief dis-cussion of these theorems will be given in ChaDter iv.

1*2. Daniel Bernoulli s mechanical problem.

In 1738 Daniel Bernoulli published a memoir! containing enunciations ofa number of theorems on the oscillations of heavy chains. The eighth ** ofthese is as follows : " Defigura catenae uniformiter oscillantis. Sit catena AGuniformiter gravis et perfecte flexilis suspensa de puncto A, eaque oscillationesfacere uniformes intelligatur: pervenerit catena in siturn AMF; fueritquelongitudo catenae = 1: longitudo cujuscunque partis FM x, sumatur n ejusvalorisff ut fit

/ Jl P_ l^_ Is

n+4ww 4 . 9n

+4 . 9 . 16n4 4.9.16. 25ns

'

* See James Bernoulli, Opera Omnia, n. (Geneva, 1744), pp. 10541057 ; it is stated that thepoint of Riccati's problem is the determination of a solution in finite terms, and a solution whichresembles the solution by Daniel Bernoulli is given.

t The term * Hiccati's equation ' was used by D'Alembert, Hist, de I' Acad. R. de Set. de Berlin,xix. (1763), [published 1770], p. 242.

X Iiutitutiones Calculi Integralis, n. (Petersburg, 1769), 831, pp. 8889. In connexion withthe reduction, see James Bernoulli's letter to Leibniz already quoted.

Novi Coram. Acad. Petrop. vni. (17601761), [published 1763], p. 32.|| Ibid. ix. (17621763), [published 1764], pp. 163164.If " Theoremata de osoillationibus corporum filo flexili connexorum et catenae verticaliter

suspensae," Comm. Acad. Set. Imp. Petrop. vi. (17323), [published 1738], pp. 108122.** Loc. cit: p. 116.

ft The length of the simple equivalent pendulum is n.


Ponatur porro distantia extremi puncti F ah linea'verticali = 1, dico foredistantiam puncti ubicunque assumpti M ab eadem linea verticali aequalem

1x

i

xx nn 4.9w3 4.9.16n4 4.9.16.25w8

He goes on to says "Invenitur brevissimo calculo n = proxime 0691 I....Habet autem littera n infinitos valores alios."

The last series is now described as a Bessel function * of order zero andargument 2 V(#/n); and the last quotation states that this function has aninfinite number of zeros.

Bernoulli publishedf proofs of his theorems soon afterwards; in theoremviii, he obtained the equation of motion by considering the forces acting onthe portion FM of length x. The equation of motion was also obtained byEuler\ many years later from a consideration of the forces acting on an elementof the chain.

The following is the substance of Euler's investigation :

Let p be the line density of the chain (supposed uniform) and let T be the tension atheight x above the lowest point of the chain in its undisturbed position. The motion beingtransversal, we obtain the equation bT=gpbx by resolving vertically for an element ofchain of length bx. The integral of the equation is Tgpx.

The horizontal component of the tension is, effectively, T{dyjdx) where y is the (hori-zontal) displacement of the element ; and so the equation of motion is

'*-(r2)-If we substitute for T and proceed to the limit, we find that

dt* y dx\ dxj '

If / is the length of the simple equivalent pendulum for any one normal vibration, wewrite

where A and f are constants ; and then n (x/f) is a solution of the equation

A ( d-\ +- =dx\ dx) f

If x/f= ii, we obtain the solution in the form of Bernoulli's series, namelyu ul v? u*

v= l H +1 1.4 1.4.9 1.4.9.16

* On the Continent, the functions are usually called cylinder functions, or, occasionally, func-tions of Fourier-Bessel, after Heine, Journal fur Math. lxix. (1868), p. 128; see also Math. Ann.

m. (1871), pp. 609610.

t Comm. Acad. Petrop. vn. (17345), [published 1740], pp. 162179.X Acta Acad. Petrop. v. pars 1 (Mathematics), (1781), [published 1784], pp. 157177. Euler

took the weight of length e of the chain to be E, and he denned g to be the measure of thedistance (not twice the distance) fallen by a particle from rest under gravity in a second. Euler's

notation has been followed in the text apart from the significance of g and the introduction ofp and 5 (for d).


.

, where C andD are constants. Since y is finite when #=0, C must be zero.

Ifa is the whole length of the chain, y=0 when .= a, and so the equation to determine/* is1 j. a% a3 _"

1./+TT4/2 ~ 1 . 4. 9/3

""'

~

a

By an extremely ingenious analysis, which will be given fully in Chapter xv, Eulerproceeded to shew that the three smallest roots of the equation in a/f are 1-445795, 7*6658and 18-63. [More accurate values are 1*4457965, 7*6178156 and 18*7217517.]

In 'the memoir'"' immediately following this investigation Euler obtained the general

solution (in the form of series) of the equation -j- (u-r-\+v0, but his statement of the

law of formation of successive coefficients is rather incomplete. The law of formation had,however, been stated in his Institutiones Calculi Integrality, n. (Petersburg, 1769), 977,pp. 233-235.

1*3. Euler 's mechanical problem.The vibrations of a stretched membrane were investigated by Euler}: in

1764. He arrived at the equationld?z_d?z Idz 1 d*ze* dt2 ~ dr*

+rdr* r'cty3 '

where z is the transverse displacement at time t at the point whose polarcoordinates are (r,

); and e is a constant depending on the density andtension of the membrane.

To obtain a normal solution he wrote

z = u sin (at + A) sin (/9


circular membrane of radius a with a fixed boundary* are to be determinedfrom the consideration that u vanishes when r = a.

This investigation by Euler contains the earliest appearance in Analysis ofa Bessel coefficient of general integral order.

1*4. The researches of Lagrange, Carlini and Laplace.

Only a few year* after Euler had arrived at the general Bessel coefficientin his researches on vibrating membranes, the functions reappeared, in anastronomical problem. It was shewn by Lagrangef in 1770 that, in the ellipticmotion of a planet about the sun at the focus attracting according to the lawof the inverse square, the relations between the radius vector r, the meananomaly M and the eccentric anomaly E, which assume the forms

M = E - e sin E, r = a(l-e cos E),give rise to the expansions

j, 00E=M+ Z A n sinnM, -=l + he2 + 2 BH cosnM,n=l & n=\

in which a and e are the semi-major axis and the eccentricity of the orbit, and._ 9

(-)" M ti-n-i 6n+m^ ^

(_)m( H + m) , ron-t-m-i en+2m

n ~"m=o 2+"*"ro ! (n + w)'! '

n ~~ -

mt 2'l+2t w!(w + m)!Lagrange gave these expressions for n = 1, 2, 3. The object of the expansionsis to obtain expressions for the eccentric anomaly and the radius vector interms of the time.

In modern notation these formulae are written

A n -2Jn {tu)/n, Bn = -2(e[n)Jn'(m).It was noted by Poisson, Connaissance des Terns, 1836 [published 1833], p. 6 that

j? f dA nn at

a memoir by Lefort, Journal de Math. xi. (1846), pp. 142 152, in which an error made byPoisson is corrected, should also be consulted.

A remarkable investigation of the approximate value of A n when n is largeand < e < 1 is due to Carlini J: though the analysis is nob rigorous (and itwould be difficult to make it rigorous) it is of sufficient interest for a briefaccount of it to be given here.

* Cf. Bourget, Ann. Sci. de VEcole norm. sup. in. (1866), pp. 5595, and Chree, QuarterlyJournal, xxi. (1886), p. 298.

t Hist, de VAcad. R. des Sci. de Berlin, xxv. (1769), [published 17711, pp. 204233. [Oeuvres,in. (1869), pp. 113138.]

X Ricerche sulla convergenza delta serie che serva alia soluzione del problema di Keplero(Milan, 1817). This work was translated into German by Jacobi, Astr. Nach. xxx. (1850),col. 197254 [Werke, yii. (1891), pp. 189245]. See also two papers by Scheibner dated 1856,reprinted in Math. Ann. xvn. (1880), pp. 531544, 545560.

14] BESSEL FUNCTIONS BEFORE 1826 7

It is easy to shew that An is a solution of the differential equation

c2^ +


The earlier portion of Laplace's investigation is based on the principlethat, in the case of a series of positive terms in which the terms steadily in-crease up to a certain point and then steadily decrease, the order of magnitudeof the sum of the series may frequently be obtained from a consideration ofthe order of magnitude of the greatest term of the series.

For other and more recent applications of this principle, see Stokes, Proc. Camb. Phil.Soc. vi. (1889), pp. 362366 [Math, and Phys. Papers, v. (1905), pp. 221225], and Hardy,Proc. London Math. Soc. (2) n. (1905), pp. 332339 ; Messenger, xxxiv. (1905)., pp. 97101.A statement of the principle was given by Borel, Acta Mathematica, xx. (1897), pp. 393

394.

The following exposition of the principle applied to the example consideredby Laplace may not be without interest

:

The series considered is(n + 2m) nn + 2m~ 2 tn + *>"

iL


integration (cf. 8*31). Laplace seems to have been dubious as to the validityof his inference because, immediately after his statement about real andimaginary variables, he mentioned, by way of confirmation, that he hadanother proof; but the latter proof does not appear to be extant.

1"5. The researches of Fourier.

In 1822 appeared the classical treatise by Fourier*, La TMorie analytiquede la Chaleur; in this work Bessel functions of order zero occur in the dis-cussion of the symmetrical motion of heat in a solid circular cylinder. It isshewn by Fourier ( 118120) that the temperature v, at time t, at distancex from the axis of the cylinder, satisfies the equation

dv_K_/d?v lcfoAJt~lW\dtf + xdx)'

where K, G, D denote respectively the Thermal Conductivity, Specific Heatand Density of the material of the cylinder; and he obtained the solution

v e~ fgx* tf


The expansion of an arbitrary function into a series of Bessel functions oforder zero was also examined by Fourier ( 314320); he gave the formulafor the general coefficient in the expansion as a definite integral.

The validity of Fourier's expansion was examined much more recently by Hankel,Math. Ann. vm. (1875), pp. 471494; Schlafli, Math. Ann. x. (1876), pp. 137142; Diui,Serie di Fourier, i. (Pisa, 1880), pp. 246269 ; Hobson, Proc. London Math. Soc. (2) vn.(1909), pp. 359388; and Young, Proc. London Math. Soc. (2) xvni. (1920), pp. 163200.This expansion will be dealt with in Chapter xvni.

1/6. The researches of Poisson.

The unsymmetrical motions of heat in a solid sphere and also in a solidcylinder were investigated by Poisson* in a lengthy memoir published in 1823.

In the problem of the sphere f, he obtained the equation

where r denotes the distance from the centre, p is a constant, n is a positiveinteger (zero included), and R is that factor of the temperature, in a normalmode, which is a function of the radius vector. It was shewn by Poisson thata solution of the equation is

Jocos (rp cos a>) sin2n+1 coda

and he discussed the cases ?i = 0, 1, 2 in detail. It will appear subsequently( 3*3) that the definite integral is (save for a factor) a Bessel function oforder n + ^.

In the problem of the cylinder (ibid. p. 340 et seq.) the analogous integral is

V cos (h\ cos co) sin2n G>cifo>,.'o

where w= 0, 1, 2, ... and \ is the distance from the axis of the cylinder. Theintegral is now known as Poisson's integral ( 2'3).

In the case n 0, an important approximate formula for the last integraland its derivate was obtained by Poisson (ibid., pp. 350352) when the variableis large; the following is the substance of his investigation:

Let J J (k)= - I cos (i cos ) da, J ' (k)= / cos a> sin (k cos ) dm.it J ir J

Then J (k) is a solution of the equation

*&ffl + (1+i),_a* Journal de I'ficole JR. Polytechnique, xn. (cahier 19), (1823), pp. 249403.+ Ibid. p. 300 etseq. The equation was also studied by Plana, Mem. della R. Accad. delle Sei.

di Torino, xxv. (1821), pp. 532534, and has since been studied by numerous writers, some ofwhom are mentioned in 4*3. See also Poisson, La TMorie Mathimatique de la Chaleur (Paris,1835), pp. 366, 369.

t See also Rohrs, Proc. London Math. Soc. v. (1874), pp. 136187. The notation J (k) wasnot used by Poisson.


When k is large, l/(42) may be neglected in comparison with unity and so we may expectthat Jo{k)sJk is approximately of the form A cos k+Bsm k where A and B are constants.

To determine A and B observe that

cos k.J (k) - sin k . / ' ()= - I {cos2 ba> cos (2k sin2 a>) + sin2 ; and therefore

i

J^

=

J(*k) K1 + ffc^ CS * + ^ +^ sin ^'

^o' (*) =77^m t - (!+*) sin *+(! + ?*) cos ^'3-

It was then assumed by Poisson that t/ () is expressible in the form

where A =.5=1. The series are, however, not convergent but asymptotic, and the validityof this expansion was not established, until nearly forty years later, when it was investi-gated by Lipschitz, Journalfur Math. lvi. (1859), pp. 189196.

The result of formally operating on the expansion assumed by Poisson for the functioneft iJ (k) *J(irk) with the operator-jt2+ 1 + rn is

.rz.l.B -jA 2. 2B"- (I. 2+ 1) A' 2. 3B'"-(2. 3+ j) A" 1-*[ & & * + - J

+8m*[_ p + Ji + +- J'

* Cf. Watson, Complex Integration and Cauchy's Theorem (Camb. Math. Tracts, no. 15, 1914),p. 71, for a proof of these results by using contour integrals.


and so, by equating to zero the various coefficients, we find that

A'- 1 A" A A'"*= 9' 25 RA---H, A -"

2 g2 *,A

- 2.3.83 ' '

and hence the expansion of Poisson's integral is

/"" /V\*IY 1 9 9-25 \ ,j o

OM(*ooe.)A,-^JLV

1 -8I--2T8^ + 2V3."83l3+-" >)

C08 *

/.,

1 9 9.25 , \ . ,1V1 + 8l"2T8^~2.3.88ife + -; 8m *J-

But, since the series on the right are not convergent, the researches of Lipschitz andsubsequent writers are a necessary preliminary to the investigation of the significance of

the latter portion of Poisson's investigation.

It should be mentioned that an explicit formula for the general term in the expansionwas first given by W. B. Hamilton, Trans. B. Irish Acad. xix. (1843), p. 313; his resultwas expressed thus

:

-

"["cos (20 sin a) da=-^ 1 [0] ([ - *])2 (43) cos (20 - $n,r - }n),

and he described the expansion as semi-convergent; the expressions [0] -B and [-$]" areto be interpreted as 1/n ! and (-|)( f) ... (-+)

A result of some importance, which was obtained by Poisson in a subsequentmemoir*, is that the general solution of the equation

is y = Ax% f e- hxcoSu> da> + Bx$ \ e~hxcoaa log (x sin2 m) da>,

Jo Jo

where A and B are constants.It follows at once that the general solution of the equation

dxr x dx

is y = A\ e~ hxcoau da) + B I e -hxcoa '\og (a? sin2 w) cZo>..'o Jo

This result was quoted by Stokesf as a known theorem in 1850, and it islikely that he derived his knowledge of it from the integral given in Poisson'smemoir; but the fact that the integral is substantially due to Poisson hasbeen sometimes overlooked J.

* Journal de VEcole R. Polytechnique, xn. (cahier 19), (1823), p. 476. The correspondinggeneral integral of an associated partial differential equation was given in an earlier memoir,ibid, p. 227.

t Camb. Phil. Trans, ix. (1856), p. [38], [Math, and Phys. Papers, hi. (1901), p. 42].J See Encyclopidie des Set. Math. n. 28 ( 53), p. 213.

1'7] BESSEL FUNCTIONS BEFORE 1826 13

17. The researches of Bessel.

The memoir* in which Bessel examined in detail the functions which nowbear his name was written in 1824, but in an earlier memoirf he had shewnthat the expansion of the radius vector in planetary motion is

- = 1 + ^2 + 2 Bn cosnM,

where Bn = / sin u sin (nu ne sin u) du ;

this expression for Bn should be compared with the series given in 1'4.In the memoir of 1824 Bessel investigated systematically the function Ikh

defined by the integral JIf 2"

Ikh = k I cos (hu k sin u) du.zvrJo '

He took h to be an integer and obtained many of the results which will begiven in detail in Chapter n. Bessel's integral is not adapted for defining thefunction which is most worth study when h is not an integer (see 10-1) ; thefunction which is of most interest for non-integral values of h is not Ikh butthe function defined by Lommel which will be studied in Chapter in.

After the time of Bessel investigations on the functions became so numerousthat it seems convenient at this stage to abandon the chronological accountand to develop the theory in a systematic and logical order.

An historical account of researches from the time of Fourier to 1858 has been compiledby Wagner, Bern MittheUungen, 1894, pp. 204266 ; a briefer account of the early historywas given by Maggi, Atti delta It. Accad. dei Lincei, (Tra/isunti), (3) iv. (1880), pp. 259263.

* Berliner Abh. 1824 [published 1826], pp. 152. The date of this memoir, " Untersuchungdes Theils der planetarischen Storungen, welcher aus der Bewegung der Sonne entsteht," isJan. 29, 1824.

t Berliner Abh. 181617 [published 1819], pp. 4'J55.J This integral occurs in the expansion of the eccentric anomaly ; with the notation of 1-4,

nA n = 21\t ,a formula given by Poisson, Connaissance des Terns, 1825 [published 1822], p. 383.

CHAPTEE II

THE BESSEL COEFFICIENTS

21. The definition of the Bessel coefficients.

The object of this chapter is the discussion of the fundamental properties

of a set of functions known as Bessel coefficients. There are several ways ofdenning these functions ; the method which will be adopted in this work is to

define them as the coefficients in a certain expansion. This procedure is due

to Schlomilch*, who derived many properties of the functions from his defi-nition, and proved incidentally that the functions thus defined are equal to the

definite integrals by which they had previously been defined by Bessel f. Itshould, however, be mentioned that the converse theorem that Bessel's inte-

grals are equal to the coefficients in the expansion, was discovered by Hansen:}:fourteen years before the publication of Schlomilch's memoir. Some similarresults had been published in 1836 by Jacobi ( 222).

The generating function of the Bessel coefficients is

It will be shewn that this function can be developed into a Laurent series,

qua function of t; the coefficient of tn in the expansion is called the Bessel

coefficient of argument z and order n, and it is denoted by the symbol Jn (z),so that

(1)*('"?)- I t"Jn (z).

=-oo

To establish this development, observe that e^zt can be expanded into an

absolutely convergent series of ascending powers of t ; and for all values of t,

with the exception of zero, e~ ielt can be expanded into an absolutely conver-

gent series of descending powers of t. When these series are multipliedtogether, their product is an absolutely convergent series, and so it may be

arranged according to powers of t ; that is to say, we have an expansion of the

form (1), which is valid for all values of z and t, t = excepted.

* ZeiUchriftfiir Math, und Phys. n. (1857), pp. 137165. For a somewhat similar expansion,

namely that of e SCO80 , see Frullani, Mem. Soc. Ital. (Modena), xvm. (1820), p. 503. It must be

pointed out that Schlomilch, following Hansen, denoted by Jftn what we now write as Jn {2z)

;

but the definition given in the text is now universally adopted. Traces of Hansen's notation

are to be found elsewhere, e.g. Schlafli, Math. Ann. m. (1871), p. 148.

t Berliner Abh. 1824 [published 1826], p. 22.+ Ermittelung der Absoluten Storungen in Ellipsen von beliebiger Excentricitdt und Neigung,

i. theil, [Schriften der Sternwarte Seeburg : Gotha, 1843], p. 106. See also the French transla-

tion, Memoire sur la determination des perturbations ab$olues (Paris, 1845), p. 100, and Leipziger

Abh. n. (1855>, pp. 250251.

2*1, 2 '11] THE BESSEL COEFFICIENTS 15

If in (1) we write \jt for t, we get

n -x

= 5 i-tyj_n {Z),n x

on replacing n by - w. Since the Laurent expansion of a function is unique*,a comparison of this formula with (1) shews that

(2 ) J-n{z) = (-YJn (*),where n is any integer - a formula derived by Bessel from his definition ofJn (z) as an integral.

From (2) it is evident that (1) may be written in the form

(3) eiz(t-llt) = Jq {z) + J {tn + { _ )n t-n] Jn {2)w-1

A summary of elementary results concerning Jn (z) has been given by Hall, The Analyst,i. (1874), pp. 8184, and an account of elementary applications of these functions toproblems of Mathematical Physics has been compiled by Harris, American Journal ofMath, xxxiv. (1912), pp. 391420.

The function of order unity has been encountered by Turriere, Nouv. Ann. de Math. (4)ix. (1909), pp. 433441, in connexion with the steepest curves on the surface z=y (5.r2 -y).

2*11. The ascending series for Jn (z).An explicit expression for Jn (z) in the form of an ascending series of powers

of z is obtainable by considering the series for exp (h zt) and exp ( - kz/t), thus

exp (**(-- 1/0)= I flT v L- **>'"'-r=o rl l=t> ml

When n is a positive integer or zero, the only term of the first series on theright which, when associated with the general term of the second series givesrise to a term involving tn is the term for which r = n + m ; and, since n > 0,there is always one term for which r has this value. On associating theseterms for all the values of m, we see that the coefficient of tn in the product is

m=o(n-rm)l mlWe therefore have the result

(1) Jn(z)= S l ^ K* Z)

* For, if not, zero could be expanded into a Laurent series in t, in which some of thecoefficients (say, in particular, that of

16 THEORY OF BESSEL FUNCTIONS [CHAP. II

where n is a positive integer or zero. The first few terms of the series are

given by the formulazn ( z* * 1

(2) Jn () = 2^1 { 1 ~ 22 . 1 . (n + 1

)

+2-

. 1 . 2 . ( + 1) (n + 2) '"]'

In particular

(3) Jo (?) 1 22 ^ 22 . 42 22 . 42 . 62 "

'

To obtain the Bessel coefficients of negative order, we select the terms in-

volving irn in the product of the series representing exp tyzt) and exp(- \z\t\where n is still a positive integer. The term of the second series which, when

associated with the general term of the first series gives rise to a term in t~n

is the term for which ra = n + r ; and so we have

J-n{Z)~

rZo r\" (n + r)l'

whence we evidently obtain anew the formula 2'1 (2), namely

J_^) = (-)n ^(*)-It is to be observed that, in the series (1), the ratio of the (w + l)th term

to the rath term is - s2/{ra (n + ra)}, and this tends to zero as ra oo , for all

values of z and n. By D'Alembert's ratio test for convergence, it follows that

the series representing Jn (z) is convergent for all values of z and n, and so it

is an integral function of z when n = 0, + 1, 2, + 3, . . .

.

It will appear later (4"73) that Jn (z) is not an algebraic function of z

and so it is a transcendental function ; moreover, it is not an elementary

transcendent, that is to say it is not expressible as a finite combination of

exponential, logarithmic and algebraic functions operated ou by signs of

indefinite integration.

From (1) we can obtain two useful inequalities, which are of some import-

ance (cf. Chapter xvi) in the discussion of series whose general term is a

multiple of a Bessel coefficient.

Whether z be real or complex, we haveao i l z i2W

* n! mZom\(n + l)m 'and so, when n ^ 0, we have

(4) ! J. I ).This result was given in substance by Cauchy, Comptes Rendus, xm. (1841), pp. 687,

854 ; a similar but weaker inequality, namely

lJr

.l< l-^P(l*l ,l.was given by Neumann, Theorie der BesseVschen Functionen (Leipzig, 1867), p. 27.

2*12] THE BESSEL COEFFICIENTS 17

By considering all the terms of the series for Jn (z) except the first, it isfound that

() j.()-^(i+),

where ||


Again, differentiate the fundamental expansion with respect to z ; and then

^(t-l/t)eiz

2-13, 2'2] THE BESSEL COEFFICIENTS 19

The expansion thus obtained,

(11) *./,(*) = 4 I {-)n-l nJ2n (z),n-l

is useful in the developments of Neumann's theory of Bessel functions ( 3'o7).

2*13. The differential equation satisfied by Jn {?)>When the formulae 2'12 (5) and (6) are written in the forms

the result of eliminating J,t_, (z) is seen to be

d f-^Wni*)} = -z*-nJn (z),dzthat is to say

and so we have Bessel's differential equation*

(1) ziW)

+ zdJAz)

+ (z2 _ wl) Jh {z) ^

The analysis is simplified by using the operator ^ defined as z (dldz).

Thus the recurrence formulae are

(S + n) Jn (z) = zJn_x {z), (^ - n + 1) Jn-i (z) = - zJn (z),and so

(^

-

n + 1 ) [z-i (* + n) Jn (z)} = -zJn (z),that is

z-> (^ - n) (* + n) Jn (z) =-zJn (z),and the equation

C&-n*)Ju {z) = -z*Jn (z)reduces at once to Bessel's equation.

Corollary. The same differential equation is obtained if Jn + l (z) is eliminated from theformulae

(S+ n+ 1) Jn + l (z)= zJn (z), (-n)Jn (z)=-zJn ^ l (z).

2*2. Bessel's integral for the Bessel coefficients.

We shall now prove that

(1) Jn (z) = J- I "' cos {n$ - z sin B) d6.Lit

'o

This equation was taken by Bessel f as the definition of Jn (z), and hederived the other properties of the functions from this definition.

* Berliner Abh. 1824 [published 1826], p. 34 ; see also Frullani, Mem. Soc. ItaL.(Modena), xvm.(1820), p. 504.

t Ibid. pp. 22 and 35.

20 THEORY OP BESSEL FUNCTIONS [CHAP. II

It is frequently convenient to modify (1) by bisecting the range of in-tegration and writing 2tt for in the latter part. This procedure gives

1 f"(2) Jn (z)=- cos (n0 - z sin 0) dO.

IT JO

Since the integrand has period 27r, the first equation may be transformedinto

1 /*2ir+a

(3) Jn (z) = 2~ I cos(n0-zsm0)d0,

where a is any angle.

To prove (1), multiply the fundamental expansion of 2 - l (1) by t~n~

x andintegrate* round a contour which encircles the origin once counterclockwise.

We thus get

2771 J =- Z7rt J

The integrals on the right all vanish except the one for which m = n; andso we obtain the formula

Take the contour to be a circle of unit radius and write t = e~i0t so thatmay be taken to decrease from 2tt + a to a. It is thus found that

I rtor+a

(5) Jn(*)^-/ *


1 '**Jn {z)- I cos n$ cos (z sin 6) ddK)

2 ,^ f("even).

= - I cos w cos (z sin &) d0ir Jo

If be replaced by tt - 77 in the latter parts of (6) and (7), it is found that

(8 ) /n (*) - - (-)*

22 THEORY OJF BESSEL FUNCTIONS [CHAP. II

Catalan's integral may be established independently by using the formula

i'(0+)'(0+)

no that

./ (2i\'*)= 2 -i_-J-. 2 -, t-^-Wdt

= J_. /"(0 + )

exp^ + !N

)^ = ;l [* exp {*+*-}

2-22] THE BESSEL COEFFICIENTS 23

will be of frequent occurrence in the sequel, enables us to write (1) and (2) inthe compact forms:

oo

(5) cos (^ sin 0) = X e.m J2n (z) cos 2nd,=o

ao

(6) sin (z sin 0) = X e2w+1 Jm+1 (z) sin (2n + 1) 6.=o

If we pub in (5), we find

0) 1=2 emJm (z).If we differentiate (5) and (6) any number of times before putting 6 = 0, weobtain expressions for various polynomials as series of Bessel coefficients. Weshall, however, use a slightly different method subsequently ( 27) to provethat zm is expansible into a series of.Bessel coefficients when m is any positiveinteger. It is then obvious that any polynomial is thus expansible. This is aspecial case of an expansion theorem, due to Neumann, which will be investi-gated in Chapter xvi.

For the present, we will merely notice that, if (6) be differentiated oncebefore 6 is put equal to 0, there results

(8) *= 2 e2n+1 (2n + l)J2n+1 (z),

while, if 6 be put equal to %-ir after two differentiations of (5) and (6), then

(9) zsinz = 2{2*J2 (z)-4?Ji (z)+6*J6 (z) -...},(10) zcosz = 2 jl./;(*)-3s J3 (z) + 5 2 J!i (z)- ...}.

These results are due to Lommel*.

Note. The expression exp {$z{t- 1/0} introduced in $ 2 1 is not a generating functionin the strict sense. The generating function t associated with enJn (?) is 2 **"/ (z).

If this expression be called S, by using the recurrence formula 2-12 (2), we have

If we solve this differential equation we get

(11) S=ei*-W+l(t + tyj*V-wJZ

e-*V-V')j(t)di:

A result equivalent to this was given by Brenke, Bull. American Math. Soc. xvi. (1910),pp. 225230.

* Studien ilber die Bessel'schen Functionen (Leipzig, 1868), p. 41.t It will be seen in Chapter xvi. that this is a form of " Lommel's function of two variables."


2 3. Poissons integral for the Bessel coefficients.

Shortly before the appearance of Bessel's memoir on planetary perturbations,Poisson had published an important work on the Conduction of Heat*, in the

course of which he investigated integrals of the typesf

^ cos (z cos 6) sin2"-1

"1 Odd, (' cos (z cos 6) sin*1 0d0,Jo Jo

where n is a positive integer or zero. He proved that these integrals aresolutions of certain differential equations^ and gave the investigation, whichhas already been reproduced in 1*6, to determine an approximation to the

latter integral when z is large and positive, in the special case n = 0.

We shall now prove that

and, in view of the importance of Poisson 's researches, it seems appropriate to

describe the expressions on the right as Poissons integrals for Jn (z). In thecase n = 0, Poisson's integral reduces to Parseval's integral ( 2'2).

It is easy to prove that the expressions under consideration are equal to

Jn (z); for, if we expand the integrand in powers of z and then integrateterm-by-term ||, we have

I fir I oo / \m sm fir- cos (* cos 0)8^0(20 = - t V cos2'0sin=00r ' 2.4.6... (2n + 2m)

-1.3.5...(2-1) 2oli

'

~

2H+m m!(n + m)rand the result is obvious.

* Journal de VEcole R. Polyteckuique, xn. (cahier 19), (1823), pp. 249403.

t Ibid. p. 293, et seq. ; p. 340, et seq. Integrals equivalent to them had previously been

examined by Euler, hut. Calc. Int. u. (Petersburg, 1769), Ch. x. 1036, but Poisson's forms are

more elegant, and his study of them is more systematic. See also 3'3.

J E.g. on p. 300, be proved that, if

R=rn+1 I cos (rp cos ) sin2"1"1 udu,Jo

then R satisfies the differential equation

dr1 r-

Nielsen, Handbuch der Tfieorie der Cylinderfunktionen (Leipzig, 1904), p. 51, calls themBessel's second integral, but the above nomenclature seems preferable.

|| The series to be integrated is obviously uniformly convergent ; the procedure adopted is due

to Poisson, ibid. pp. 314, 340.

2-3, 2*31] THE BESSEL COEFFICIENTS 25

Poisson also observed* that

(' e^* sin8'* Odd = f*cos (z cos 0) sin 0d0\

Jo Jo

this is evident when we consider the arithmetic mean of the integral on theleft and the integral derived from it by replacing by it 0.

We thus get

A slight modification of this formula, namely

(3) /(*) =r(n4|^V, (1 _ PY-iM>

has suggested important developments (cf. 6*1) in the theory of Besselfunctions.

It should also be noticed that

(4)!' cos (z cos 0) sin2" 0d0 = 2 \

'

cos {z cos 0) sin1" 0d0Jo Jo

f*= 2 cosCssin^cos*1^,

Jo

and each of these expressions gives rise to a modified form of Poisson's integral.

An interesting application of Bessel's and Poisson's integrals was obtainedby Lommelf who multiplied the formula

, * / ,m 4w8 {4n8 -28}...{4n2 -(2ro-2)2} . .

cos2n0= 2 (-)m s J /a v. -sin*"^ -o (zm)\

by cos (z cos 0) and integrated. It thus follows that

(5) J (z)-(-Y I ( r 48 f4n-2r... (W-(2iii -*Y) Jm (s)

2'31. Bessel's investigation of Poisson's integral.

The proof, that Jn (z) is equal to Poisson's integral, which was given byBessel J, is somewhat elaborate; it is substantially as follows

:

It is seen on differentiation that

-^ I cos sin2"-1 cos (z cos 0) 9,- sin2*1"1

"1 sin (z cos 0)

=|(2w - 1) sin8"-8 - 2n sin8" +

**

sin2"*8 cos (* cos 0),

* Poisson actually made the statement (p. 293) concerning the integral which containssinSH+1 6 ; but, as he points out on p. 340, odd powers may be replaced by even powers throughouthis aualysis.

f Studien Uber die BesseVscken Functional (Leipzig, 1868), p. 30.+ Berliner Abh. 1824 [published 1826], pp. 3637. Jacobi, Journal filr Math. xv. (1836), p. 13,

[Get. Math. Werke, vi. (1891), p. 102], when giving his proof ( 2*32) of Poisson's integral formula,objected to the artificial character of Bessel's demonstration.

w. B. jr. 2

26 THEORY OF BESSEL FUNCTIONS [CHAP. Hand hence, on integration, when n > 1,

(2ra -1)1 cos (z cos 0) sin8""2 6d0 - %n j cos (z cos 0) sin3" 0d0^2 r*

+ 2n + i Jcos (* cos 0) sin2"-*"8 0d0 = 0.

If now we write

rin + ^ra)/^ 008

where fi = cos 0. We shall assume this formula for the moment, and, since nosimple direct proof of it seems to have been previously published, we shallgive an account of various proofs in 2*3212323.

If we observe that the first n 1 derivates of (1 /i2)M-i , with respect tofi, vanish when fi = 1, it is evident that, by n partial integrations, we have

zn j cos (z cos 0) sin-'1 Odd = zn I cos (zfi) . (1 - /a2)"-* dfi

= (-)nJ_ i

COS foa - \ mr) -^di } d.

* Journal fiir Math. xv. (1830), pp. 1213. [Ges. Math. Werke, vi. (1801), pp. 101102.] Seealso Journal de Math. i. (1836), pp. 195196.

2*32, 2*321] THE BESSEL COEFFICIENTS 27

If we now use Jacobi's formula, this becomes

1.3.5...(2n-l)f 1 dsinndIf 1 . , .dsinnd ,cos {Zfi \n-jr)

j

dfi

= 1 . 3 . 5 . . . (2n - 1) cos (z cos - ^nir) cos n0d0Jo

= 1.3.5...(2n-l)wJn (5),by Jacobi's modification of 2'2 (8) and (9), since cos^cos 6 \nir) is equalto ()*** cos (z cos 6) or (-)^n-1) sin(^cos 9) according as n is even or odd; andthis establishes the transformation.

2*321. Proofs ofJacobus transformation.

Jacobi's proof of the transformation formula used in 2-32 consisted in deriving it

as a special case of a formula due to Lacroix*; but the proof which Lacroix gave ofhis formula is open to objection in that it involves the use of infinite series to obtaina result of an elementary character. A proof, based on the theory of linear differentialequations, was discovered by Liouville, Journal de Math. vi. (1841), pp. 6973; thisproof will be given in 2*322. Two years after Liouville, an interesting symbolic proofwas published by Boole, Camb. Math. Journal, in. (1843), pp. 216224. An elementaryproof by induction was given by Grunert, Archio der Math, und Phys. iv. (1844), pp. 104

109. This proof consists in shewing that, if


2*322. LiouvMds proof of Jacob?* transformation.The proof given by Liouville of Jacobi's formula is as follows

:

Let y=(lfi2)

n~k and let D be written for d/dp ; then obviously

(l-V)Z)y+(2n-l) My=0.Differentiate this equation n times ; and then

(1 - fi2) Dtt+l i/-

(iD^+^D^-^^O :but (l"^-^-^^^-"^.so that (^ +

JZ)- y= 0.

Hence D*~ 1y= A sin nd+ BcoB n6,where A and B are constants ; since Dn~ ly is obviously an odd function of 6, B is zero.To determine A compare the coefficients of 6 in the expansions of D*- 1y and A sinnd inascending powers of 6. The term involving 6 in D*~ ly is easily seen to be

(-^)n" 1^" 1=(" )"~ 1(2w " 1)(2w~ 3) - 3 - 1 -

2-322-2*33] THE BESSEL COEFFICIENTS 29

233. An application of Jacobi s transformation.

The formal expansion

/'(cos x) cos nxdx = 2 (-)mamfin+2m>(cosx)dx,

Jo Jo t-0

in which am is the coefficient of tn+2m in the expansion of Jn (t)/J (t) in as-cending powers of t, has been studied by Jacobi*. To establish it, integratethe expression on the left n times by parts ; it transforms ( 2*32) into

1.3.5...(2n-l) Slf(n) (CS *} 8inm*dx>

and, when sin2n# is replaced by a series of cosines of multiples of x, this becomes

1 fw XM , J. 2n 2n(n-l) A 1 ,

We now integrate / (n,.(cosa?)cos2#, /

gives


2*4. The addition formula for the Bessel coefficients.

The Bessel coefficients possess an addition formula by which Jn (y + z)may be expressed in terms of Bessel coefficients of y and z. This formula,which was first given by Neumann* and Lommelf, is

00

(1) J(y + z) = 2 Jm (y)Jn-r,i(z).m- -oo

The simplest way of proving this result is from the formula 2*2 (4), which

1 f

2'4-2'6] THE BESSEL COEFFICIENTS 31

From the general formula we find that

(2) Jn (2z)-S/f (#)Jn-r (*) + 2 (-YJt (*) Jn+r (z),r=0 r=l

when the Bessel coefficients of negative order are removed by using 2*1 (2).Similarly, since

j I rjr (s)\\ 5 {-rv*Jni {z)\= exp [z (t - 1/t)} exp [\z (- t + 1/t)}= 1

>

it follows that

(3) J >(z) + 2% Jr*(z) = l,

(4) 2 (-yJr (z) Jm_r (z) + 2X Jr (z) Jm+r (z) = 0.r-0 r-1

Equation (4) is derived by considering the coefficient of tm in the Laurentexpansion ; the result of considering the coefficient of tm+1 is nugatory.

A very important consequence of (3), namely that, when x is real,(5) \J,{*)\*1, |^,(*)j+*> e

-(nfl+sin +, dOdj).

To reduce this double integral to a single integral take new variables definedby the equations

8- = 2X , 0+ = 2yJr,so that

d(0,)9 (x Vr)

It follows that

= 2.

J2 (*) = 2^*11*"""* ""** 8in * cos

* d*;cty,

where the field of integration is the square for which

tt^x yk^'7r> "" < X + ^ ^ 7r-Since the integrand is unaffected if both % and yfr are increased by tt, or if

^is increased by ir while yfr is simultaneously decreased by ir, the field of inte-gration may evidently be taken to be the rectangle for which

0

32 THEOEY OF BESSEL FUNCTIONS [CHAP, n

Hence

Jn (*) - 2^-aff* e2"**-2** sin * * ctydy1 C"

= -\J** (2* cos x)

2*61, 2'7] THE BESSEL COEFFICIENTS

This result was written by Neumann in the form

W Jn{Z) ~ (n\f I l(2n + l) + 1.2.(2n + l)(2n + 2) J'2w + l

33

where

(2)

1\ = 2m- 2'

(2n+l)(2n + 3)4~(2n + 2)(2w + 4)'

(2,. + l)(2tt + 3)(2n + 5)"(2n + 2)(2n + 4)(2n + 6) ,

This expansion is a special case of a more general expansion (due toSchlafli) for the product of any two Bessel functions as a series of powers with

comparatively simple coefficients ( 5*41).

2*7. Schlomilch's expansion of zm in a series of Bessel coefficients.

We shall now obtain the result which was foreshadowed in 2*22 con-cerning the expansibility of zm in a series of Bessel coefficients, where m is anypositive integer. The result for m = has already been given in 2*22 (7).

In the results 2*22(1) and (2) substitute for cos 2nd and sin(2n+ 1)0their expansions in powers of sins 6. These expansions are*

n.(n + s -1)!cos2n#= 2 (-)'

, wo\t,=o (n-s)!(2s)!

(2 sin ey,

The results of substitution are

( cos (x sin 6) - J, (x) + 2 ^1 J (x)j(-) ij&"^i! (2 sin *)}

,

| sin (x sin 9) = 1 Jm+, (X) { I (-) gj$| (*- )""( If we rearrange the series on the right as power series in sin 6 (assuming

that it is permissible to do so), we have

( / -/i\ irMLPvrnL? (-)"(2sin0)f 3 2n.(n+ s-l)\ T \1 , n\ 5 (~)*(2sin0)+1 f " (2n + l).(n + s)! r , Jsin (z sin6)= X L-LroTT\i

\2 r^-^r J+i(*)

* Cf. Hobson, PZajie Trigonometry (1918), 80, 82.


If we expand the left-hand sides in powers of sin $ and equate coefficients,we find that

l=Jo(s) + 2 5 J2n (4n=l

I(* ' "-. Jn^7)\ Jm+1

(z>- (* *

*'

2>

>

The first of these is the result already obtained ; the others may be com-bined into the single formula

(1) (i.r = I C + *.)( + . -Dl Jm>w (m _ 1( 2| 3...)w=0

The particular cases of (1) for which m = 1, 2, 3, were given by Schlomilch*He also shewed how to obtain the general formula which was given explicitlysome years later by Neumannf and LommelJ.

The rearrangement of the double series now needs justification ; the rearrangement ispermissible if we can establish the absolute convergence of the double series.

If we make use of the inequalities

IA +1 (*)|


[which has been proved in 2'22 (8) in the special case m = 1], we have

/i \,+, S (m + 2n) . (m + n 1) ! , r , vn=0 n -

i r / x. 5 ((wi+ n)! , (ra + w 1)!) . r , x= \z

.

m \ Jm (z) + 2^ r^- + rn _ ly I* J~+ (*)

2 (wi -Mi)! f , r , , r / ,,=0 n\

_

" (m+l + 2n).Q + n) ir ,

.

=0 w!

Since (m + m)' ^i+sn (z)/n\-*- as w -* oo , the rearrangement in the thirdline of the analysis is permissible. It is obvious from this result that the in-

duction holds for m = 2, 3, 4,

An extremely elegant proof of the expansion, due to A. C. Dixon*, is as follows :

Let t be a complex variable and let u be denned by the equation u= -

-

2 ,so that when

t describes a small circuit round the origin (inside the circle11 \ = 1), u does the same.

We then have

wi! /"expi-ia^-l/O}---^*

^/lw -p


were obtained by Schlomilch, Zeitschrift fur Math, und Phys. II. (1857), p. 141, and hegave, as the value of P^p\

m

w>JLo (2^Ti J-o" JL (a*)l".iJoLM*l (20""* iJ-oP fAz^2m 2m

m=0 v*wl7 ! Jfc-0

=2(-)l jP-Zff,pS=o

since terms equidistant from the beginning and the end of the summation with respect toI- are equal. The truth of equation (1) is now evident, and equation (2) is proved in asimilar manner from 2*22 (2).

The reader will easily establish the following special cases, which were stated bySchlomilch

:

P Jx (*)+ 33Ja (i)+ 53 Jb (*)+ ... = (+**),

(4) '2V2 (?)+4V4 W+eV6 ()+...=^,2.3.473 (*)+ 4.5.6./6 (*)+6.7.8.f7 (*)+...=$*3.

2*72. Neumanns expansion ofz as a seizes of squares of Bessel coefficients.

From Schlomilch's expansion ( 2'7) of z1 as a series of Bessel coefficientsof even order, it is easy to derive an expansion of z* as a series of squares ofBessel coefficients, by using Neumann's integral given in 2*6.

Thus, if we take the expansion

, am v (2m + 2n).(2m + n-l)! , /a . -.(2sin0)= 2 v '-A- JWi+m (2zsm0),

and integrate with respect to 0, we find that

I?.?&>*- I (^ + 2)-(2m + n-l),ir Jo =o m

so that (when m > 0)

(I) aM*.S2 v (2m + 2n).(2m + r-l) !(1) (**>"(2m)!*,

.*!/-*

2-72] THE BESSEL COEFFICIENTS 37

This result was given by Neumann*. An alternative form is

(2)

and this is true when m = 0, for it then reduces to Hansen's formula of 25

CHAPTER IIIBESSEL FUNCTIONS

3*1. The generalisation of Bessel's differential equation.

The Bessel coefficients, which were discussed in Chapter II, are functionsof two variables, z and n, of which z is unrestricted but n has hitherto beenrequired to be an integer. We shall now generalise these functions so as tohave functions of two unrestricted (complex) variables.

This generalisation was effected by Lommel*, whose definition of a Besselfunction was effected by a generalisation of Poisson's integral ; in the courseof his analysis he shewed that the function, so defined, is a solution of thelinear differential equation which is to be discussed in this section. Lommel'sdefinition of the Bessel function Jv {z) of argument z and order v wasf

Jv (*) =rfr+ffirft) C

cos {z cos 6) sin2" 0d0 '

and the integral on the right is convergent for general complex values .of vfor which R(v) exceeds |. Lommel apparently contemplated only realvalues of v, the extension to complex values being effected by HankelJ;functions of order less than - \ were defined by Lommel by means of an ex-tension of the recurrence formulae of 212.

The reader will observe, on comparing 3*3 with 1*6 that Plana andPoisson had investigated Bessel functions whose order is half of an odd integernearly half a century before the publication of Lommel's treatise.

We shall now replace the integer n which occurs in Bessel's differentialequation by an unrestricted (real or complex) number v, and then define aBessel function of order v to be a certain solution of this equation ; it is ofcourse desirable to select such a solution as reduces to Jn (z) when v assumesthe integral value n.

We shall therefore discuss solutions of the differential equation

(1) z>*l + z^ + (z>- v>) y = 0,

which will be called Bessel's equation for functions of order v.

* Studieji iiber die BesseVschen Functionen (Leipzig, 1868), p. 1.+ Integrals resembling this (with v not necessarily an integer) were studied by Duhamel, Cours

d Analyse, n. (Paris, 1840), pp. 118121.% Math. Ann. i. (1869), p. 469.

Following Lommel, we use the symbols v, ft. to denote unrestricted numbers, the symbols, m being reserved for integers. This distinction is customary on the Continent, though it hasnot yet come into general use in this country. It has the obvious advantage of shewing at aglance whether a result is true for unrestricted functions or for functions of integral order only.

3-1] BESSEL FUNCTIONS 39

Let us now construct a solution of (1) which is valid near the oiigin; theform assumed for such a solution is a series of ascending powers of z, say

y= 2 cm z*+m,m-0

where the index o and the coefficients cm are to be determined, with the pro-viso that c is not zero.

For brevity the differential operator which occurs in (1) will be called V,so that

w+s+*-*It is easy to see that*

V, 2 cm z*+m = 2 cw {(a + m)2 -i/2}s"+m + 2 cmza+m+a.

The expression on the right reduces to the first term of the first series,namely c (a8 Vs) za, if we choose the coefficients cm so that the coefficients ofcorresponding powers of z in the two series on the right cancel.

This choice gives the system of equations

/(^{(a + l)*-*2 } =0ca {(a + 2)2 -*,2j+c =0c8 {(a + 3)2 -i/2} + Cl =0

(3)

c,{(a + ra)2 -v2}+cM_2 =

If, then, these equations are satisfied, we have

(4) V 2 cTO^+'tt=c (aa -i/2)2a.m-0

From this result, it is evident that the postulated series can be a solutionof (1) only if a = + v; for c is not zero, and za vanishes only for exceptionalvalues of z.

Now consider the mth. equation in the system (3) when w > 1. It can bewritten in the form

cm (o- v + m) (a + v + m) + Cm-* = 0,and so it determines cm in terms of Cm^ for all values of m greater than 1unless a-i>ora + i/isa negative integer, that is, unless 2v is a negativeinteger (when a= v) or unless 2v is a negative integer (when a v).

We disregard these exceptional values of v for the moment (see 3" 11,3*5), and then (o + m)2 p* does not vanish when m 1, 2, 3, .... It now

* When the constants a and cm have been determined by the following analysis, the seriesobtained by formal processes is easily seen to be convergent and differentiable, so that the formalprocedure actually produces a solution of the differential equation.

40 THEORY OF BESSEL FUNCTIONS [CHAP. Ill

follows from the equations (3) that d = c3 = c5 = ... = 0, and that c2Ml is ex-pressible in terms of c by the equation

c =(-r*

.

"* (a-v + 2)(a-v + 4,)...(a-v + 2m)(a+v + 2)(a + v + 4>)...(a + v+ 2myThe system of equations (3) is now satisfied; and, if we take a = v, we see

from (4) that

(-)" (**)""(5) c zv

a formal soli

formal solution

1+ 2 -i m ! (v + 1) {v + 2) . . . (v + m)

is a formal solution of equation (1). If we take a = v, we obtain a second

(6) c 'z-"\i+ s(-)m (^y

,-i m\(- v + 1) (- v + 2) ... (-v + m)]'In the latter, c ' has been written in place of c , because the procedure of

obtaining (6) can evidently be carried out without reference to the existenceof (5), so that the constants c and c ' are independent.

Any values independent of z may be assigned to the constants c and c '

;

but, in view of the desirability of obtaining solutions reducible to Jn (z) whenv -* n, we define them by the formulae *

(') co = oirv.. i i \

>

c=

2T(v+l)' 2-T{-p + l)'The series (5) and (6) may now be written

| (-)m (bz)v+2m * (-)w (i-g)~"+m,=o m\r(v +m + l)' (. m!r(-i/ + m + l)'

In the circumstances considered, namely when 2v is not an integer, these seriesof powers converge for all values of z, (z = excepted) and so term-by-termdifferentiations are permissible. The operations involved in the analysisf bywhich they were obtained are consequently legitimate, and so we have obtainedtwo solutions of equation (1).

The first of the two series defines a function called a Bessel function oforder v and argument z, of the first kindX', and the function is denoted bythe symbol J (z). Since v is unrestricted (apart from the condition that, forthe present, 2v is not an integer), the second series is evidently J_ (z).

Accordingly, the function Jv (z) is defined by the equation

(8) MZ) =m? m!I> + m+l)-

It is evident from 2*11 that this definition continues to hold when v is apositive integer (zero included), a Bessel function of integral order beingidentical with a Bessel coefficient.

* For properties of the Gamma-ftmction, see Modern Analysis, ch. xn.t Which, up to the present, has been purely formal.X Functions of the second and third kinds are denned in 3*5, 3 54, 3-57, 3-6.

3*11] BBSSBL FUNCTIONS 41

An interesting symbolic solution of Bessel's equation has been given by Cotter* in theform

[l+zv D- 1 z-2"- 1 D- 1 zv+1]- 1 (Azv+Bz- v),where Dsd/dz while A and B are constants. This may be derived by writing successively

[D(zD~2v)+z]zvy=0,[zD-2p+D~ 1z]z"y= -2vB,

zD (z~ vy) + z~2v D~V+V= - 1vBz-%\

z-vy+D- xz-^- xD- x zv+ly=A +Bz~ tv

,

which gives Cotter's result.

3'11. Functions whose order is half of an odd integer.

In 3*1, two cases of Bessel's generalised equation were temporarily omittedfrom consideration, namely (i) when v is half of an odd integer, (ii) when v isan integerf. It will now be shewn that case (i) may be included in the generaltheory for unrestricted values of v.

When v is half of an odd integer, letS = (r + $r,

where r is a positive integer or zero.

If we take a = r + in the analysis of 3*1, we find that

(c1 .l(2r + 2) =0, . ,,(1) 1 / . ci ,v r! (m>l)v ' (cTO .m(m + 2r+l) + cm_2 = 0, v '

and so(y q

(2) Ctm = 2.4...(2m).(2r + 3)(2r + 5)...(2r + 27/i + l")

'

which is the value of c^ given by 3'1 when a and v are replaced by r + \.If we take

1c-2~r"+4r(r+ f)'

we obtain the solution

,= m!r(r + m + f)'which is naturally denoted by the symbol Jr+^(z), so that the definition of 31 (8) is still valid.

If, however, we take a= r , the equations which determine cm become

(S)foi.l(-2r) =0,

(3) \cm .m(m-l -2r) + Crn-* = 0. (m>1)As before, clt c3 , ... } cv-i are all zero, but the equation to determine c^+x is

Cgr+i + Czrl 0,

and this equation is satisfied by an arbitrary value of Car+r> when ra > r, C2m+iis defined by the equation

. =(-)w-r c2r+i

27,1+1 (2r + 3) (2r + 5) ... (2m + 1) . 2 . 4 ... (2m- 2r)

'

* Proc. R. Irish. Acad. xxvn. (A), (1909), pp. 157161.

f The cases combine to form the case in which 2v is an integer.


If Jv (z) be defined by 3*1 (8) when v r \, the solution now con-structed is*

Co 2-"-* T(\-r) /_,_! (z) + Czr+1 2*+* r (r + f)Jr+i (z).It follows that no modification in the definition of Jv (z) is necessary whenv=(r + ); the real peculiarity of the solution in this case is that thenegative root of the indicial equation gives rise to a series containing twoarbitrary constants, c and c^+i, i.o. to the general solution of the differentialequation.

3*12. A fundamental system of solutions of Bessel's equation.It is well known that, if yx and ya are two solutions of a linear differential

equation of the second order, and if y( and y2' denote their derivates withrespect to the independent variable, then the solutions are linearly inde-pendent if the Wronskian determinant^

y* y*

yl y*

does not vanish identically; and if the Wronskian does vanish identically,then, either one of the two solutions vanishes identically, or else the ratio ofthe two solutions is a constant.

If the Wronskian does not vanish identically, then any solution of thedifferential equation is expressible in the form cx yx + d y2 where cx and ca areconstants depending on the particular solution under consideration; thesolutions yx and y2 are then said to form a fundamental system.

For brevity the Wronskian of yx and y% will be written in the forms

W,{y,,y}, 8K{yi,y.},the former being used when it is necessary to specify the independent variable.

We now proceed to evaluate

If we "multiply the equations

V v J_v {z) = 0, Vv J,(z) = Qby J (z), J_v (z) respectively and subtract the results, we obtain an equationwhich may be written in the form

jz[zm{jv {z), j-^yi-q,* In connexion with series representing this solution, see Plana, Mem. della R. Aeead. delle

Sci. di Torino, xxvi. (1821), pp. 519538.

f For references to theorems concerning Wronskians, see Encyr.lqpidie des Sci. Math. n. lfr

( 23), p. 109. Proofs of the theorems quoted in the text are given by Forsyth, Treatise on

Differential Equations (1914), 7274.

3*12] BESSEL FUNCTIONS 43

and hence, on integration,

a) m[jv {z), J-Az))=-,z

where G is a determinate constant.To evaluate C, we observe that, when v is not an integer, and \z\ is small,

we have

J"{z) = l>TT) {l + {z% J"' {z) = m>5 ^ 1 + W>

with similar expressions for J-V (z) and /'_ (z); and hence

J.W^W-J-W:W-i{r(^T(^)-rW r(-, + 1)} +2sini/7r -. .

TTZ V '

If we compare this result with (1), it is evident that the expression on theright which is 0(z) must vanish, and so*

(2) m [jv {z\ j_ v (z)} = _ *!E^r.TTZ

Since sini^r is not zero (because v is not an integer), the functions Jv (z),/_ {z) form a fundamental system of solutions of equation 31 (1).

When v is an integer, n, we have seen that, with the definition of 21 (2),

and when v is made equal to n in 3*1 (8), we find that

nK)mZ m\r(-n + m + Ty

Since the first n terms of the last series vanish, the series is easily reduced to()n Jn(z)y so that the two definitions of J-n(z) are equivalent, and thefunctions Jn (z), J_n (z) do not form a fundamental system of solutions ofBessel's equation for functions of order n. The determination of a fundamentalsystem jn this case will be investigated in 363.

To sum up, the function Jv (z) is defined, for all values of v, by theexpansion of 31 (8); and J (z), so defined, is always a solution of the equationVt/ = 0. When v is not an integer, a fundamental system of solutions of thisequation is formed by the functions Jy (z) and /_ (z).A generalisation of the Bessel function has been effected by F. H. Jackson in his

vn 1researches on "basic numbers." Briefly, a basic number [] is defined as * , where pis

the base, and the basic Gamma function rp (v) is defined to satisfy the recurrence formulaTp (u+ l)= [u].rp (v).

The basic Bessel function is then defined by replacing the numbers which occur in theseries for the Bessel function by basic numbers. It has been shewn that very many theorems

* This result is due to Lommel, Math. Ann. iv. (1871), p. 104. He derived the value of C bymaking z -* and using the approximate formulae which will be investigated in Chapter vn.


concerning Bessel functions have their analogues in the theory of basic Bessel functions,

but the discussion of these analogues is outside the scope of this work. Jackson's mainresults are to be found in a series of papers, Proc. Edinburgh Math. Soc. xxi. (1903), pp.6572 ; xxn. (1904), pp. 8085; Proc. Royal Soc. Edinburgh, xxv. (1904), pp. 273276;Trans. Royal Soc. Edinburgh, xli. (1905), pp. 128, 105118, 399-408; Proc. LondonMath. Soc. (2) I. (1904), pp. 361366; (2) II. (1905), pp. 192220; (2) ill. (1905), pp. 123.

The more obvious generalisation of the Bessel function, obtained by increasing thenumber of sets of factors in the denominators of the terms of the series, will be dealt within 4 #4. In connexion with this generalisation see Cailler, Mem. de la Soc. de Phys. de

Geneve, xxxiv. (1905), p. 354; another generalisation, in the shape of Bessel functions of two

variables, has been dealt with by Whittaker, Math. Ann. lvil (1903), p. 351, and Peres,Comptes Rendus, clxi. (1915), pp. 168170.

3'13. General properties ofJv (z).

The series which defines Jv (z) converges absolutely and uniformly* in anyclosed domain of values of z [the origin not being a point of the domain whenR (y) < 0], and in any bounded domain of values of v.

For, when \v\ % N and \z\ % A, the test ratio for this series is-I*

m(v + m) m (m Ar)whenever m is taken to be greater than the positive root of the equation

m'-miV-|A8 = 0.This choice of to being independent of v and z, the result stated follows from

the test of Weierstrass.

Hencef Jv (z) is an analytic function of z for all values ofz(z = possiblybeing excepted) and it is an analytic function of v for all values of v.

An important consequence of this theorem is that term-by-term differen-tiations and integrations (with respect to z or v) of the series for Jv (z) arepermissible.

An inequality due to Nielsen J should be noticed here, namely

(i) W=tvt)(1+>where | 6\

3-13, 3-2] BESSEL FUNCTIONS 45

tion. We define it to be exp (v log z) where the phase (or argument) of z isgiven its principal value so that

ir < arg z % ir.

When it is necessary to "continue" the function Jv {z) outside this range ofvalues of arg z, explicit mention will be made of the process to be carried out.

3*2. The recurrenceformulae for Jv (z).Lommel's generalisations* of the recurrence formulae for the Bessel co-

efficients ( 2'12) are as follows:o

(1) ^(fJ +^W-v/rWz

(2) Jv_l (z)-Jv+1 (z)=2J,'(z),

(3) zJv'(z) + vJv (z) = *J*-^)>

(4) zJ: (z) -vJv {z) = - zJv+1 (z).

These are of precisely the same fonn as the results of 2*12, the only differencebeing the substitution of the unrestricted number v for the integer n.

To prove them, we observe first that

i , it V - A v (-)j*h

/_\m ^2vi +ai~ ** 7\.

Zo2"-1+im .m\r{v + m)- z"Jv^ (z).

When we differentiate out the product on the left, we at once obtain (3).In like manner,

i -k T \\-A 5 (-)m z*mdz \

z rJ'WJ-rfj mt 2".i!r(v + + l)

/_\r^2wi

l

=1 2K+2m-i . (W _ 1)! T (l/ + W + 1)oc / \m+i n+i

=

w= 2'+m+1

. ml r (v + w + 2)

whence (4) is obvious; and (2) and (1) may be obtained by adding and sub-tracting (3) and (4).

* StUdien Uber die BesseVschen Functionen (Leipzig, 1868), pp. 2, 6, 7. Formula (3) was givenwhen v is half of an odd integer by Plana, Mem. delta R. Accad. delie Sci. di Torino, xxvi. (1821),p. 533.


We can now obtain the generalised formulae

(5) (AJ [zvj> {z)] = *-~j- '(6) (^)

Wir-J.{*)} =(-)' z>"Jv+m (z)

by repeated differentiations, when m is any positive integer.

Lommel obtained all these results from his generalisation of Poisson'sintegral which has been described in 3*1.

The formula (1) has been extensively used* in the construction of Tablesof Bessel functions.

By expressing /_! 0) and Jx_ v (z) in terms of J (z) and J'v (z) by (3)and (4), we can derive Lominel's formulaf

T /vtv T-/\r/\2 sin vtr(7) J. (z) J^v (z) + JL (z) Jv., (z) = -

7TZ

from formula (2) of 312.

An interesting consequence of (1) aud (2) is that, if Qy () = J"2 (2), then

(8)

3-21, 3-3] BESSEL FUNCTIONS 47

with numerous other formulae of like character. These results seem to be ofno great importance, and consequently we merely refer the reader to thememoirs in which they were published.

In the special case in which v = 0, Bessel's equation becomes

solutions of this equation in the form of series were given by Boole* manyyears ago.

33. Lommel's expression ofJv (z) by an integral o/Poisson's type.We shall now shew that, when R(v) > - , then

(1) Jw (z) =_^g.'_JJcos (* cos 6) sm*>0d0.It was proved by Poisson-f- that, when 2v is a positive integer (zero in-

cluded), the expression on the right is a solution of Bessel's equation ; andthis expression was adopted by Lommelj as the definition of Jv (z) for positivevalues of v + \.

Lommel subsequently proved that the function, so denned, is a solution of Bessel'sgeneralised equation and that it satisfies the recurrence formulae of 32 ; and he thendefined Jv (z) for values of v in the intervals (-, -#),(_#,

_), (J| t -J), ... by suc-cessive applications of 3 -2 (1).

To deduce (1) from the, definition of Jv {z) adopted in this work, we trans-form the general term of the series for Jv (z) in the following manner:

(-)rw (i*),+swt=

(-y*Q*Y ** r(y+j)r(m + i)m!I> + m + l) I> + )r(*)-(2ro)f I> +m + l)

provided that R (v) > |.Now when R (v) > , the series

l= i (2wz)!

converges uniformly with respect to t throughout the interval (0, 1 ), and so itmay be integrated term-by-term ; on adding to the result the term for which

* Phil. Tram, of the Royal Soc. 1844, p. 239. See also a question set in the MathematicalTripos, 1894.

t Journal de VEcole R. Polytechnique, xn. (cahier 19), (1823), pp. 300 et teq., 340 et $eq.Strictly speaking, Poiason shewed that, when 2v is an odd integer, the expression on the rightmultiplied by Jz is a solution of the equation derived from Bessel's equation by the appropriatechange of dependent variable.

t Studien liber die BetseVschen Functionen (Leipzig, 1868), pp. 1 et seq.

48 THEORY OF BESSEL FUNCTIONS [CHAP. Iir

m = 0, namely / ^f)Ta)J.>0,,sin8

"^

The formula obtained by a partial integration of (5), namely

(7) jv (*> = ^TlVffe^1

f^ ( cos ^) sin2""^ cos^'

is sometimes usefuiy^t ^s valid, only when R{v)> \.


An expansion involving Bernoullian polynomials has been obtained from (4) by Nielsen*with the help of the expansion

f-*-(i-.-)i rfl' 1*" +,tf+1)

'= (n+ 1)!in which

n () denotes the nth Bernoullian polynomial and aizt.

[Note. Integrals of the type (3) were studied before Poissou by Plana, Mem. delta R.Accad. delle Sci. di Torino, xxvi. (1821), pp. 519538, and subsequently by Kummer,Journal filr Math, xn. (1834), pp. 14-4147; Lobatto, Journal fiir Math. xvn. (1837), pp.363371; and Duhamel, Cours d'Analyse, II. (Paris, 1840), pp. 118121.A function, substantially equivalent to Jv (2), denned by the equation

J{fjL,x)=\ (1 - v2)* cos vx . dv,J o

was investigated by Lommel, Archiv der Math, und Phys. xxxvu. (1861), pp. 349360.The converse problem of obtaining the differential equation satisfied by

f*

ezv {v-af- 1 (v-P)"- 1 dv

was also discussed by Lommel, Archiv der Math, und Phys. XL. (1863), pp. 101126. Inconnexion with this integral see also Euler, Inst. Cole. Int. II. (Petersburg, 1769), 1036,and Petzval, Integration der linearen Differentialgleichungen (Vienna, 1851), p. 48.]

3*31. Inequalities derivedfrom Poissoris integral.

From 33 (6) it follows that, if v be real and greater than \, then

(i) !-/.

50 THEORY OF BESSEL FUNCTIONS [CHAP. HI

3*32. Gegenbauer's generalisation of Poisson's integral.

The integral formula

(1 > J~ -fRPp|^/>--- 9.CJ (cos *) rfftin which Gnv (t) is the coefficient of a" in the expansion of (1 2at + o2)

-

" in

ascending powers of a, is due to Gegenbauer* ; the formula is valid whenR (v) > \ and n is any of the integers 0, 1, 2, .... When n = 0, it obviouslyreduces to Poisson's integral.

In the special case in which v = \, , the integral assumes the fonn

(2) Jn+h (z) = (_i)(ff/z cos 9 Pn (cos 0) sin dd ;this equation has been the subject of detailed study by Whittakerf.

To prove Gegenbauer's formula, we take Poisson's integral in the form

' '*W- r(, +

(

i'5*)r(t)i> (1 - (=)'+""'

*

and integrate times by parts ; the result is

^,,() (-2i)nr(, + ,,4-^ra)J_ 1e 1

3*32, 3*33] BESSEL FUNCTIONS 51

A formula which is a kind of converse of (4), namely*

(5) *** (7(^)) -r(^i^+V^ (P ) *"'in which i*"*1 denotes a generalised Legendre function, is due to Filon, Phil. Mag. (6) vi.

(1903), p. 198 ; the proof of this formula is left to the reader.

3*33. Gegenbauer's double integral of Poisson's type.

It has been shewn by Gegenbauerf that, when R (v) > 0,

(1 ) Jv (r) = v* I I exp [iZ cos tV (cos cos0 + sin o

_ (W [(=-^A / e%v sm 9 cos * cos2" _1 e sin Ofyd0-

* It is supposed tbat9*y

_

Y{y + l)z v->x

dz* ~ r("-M+i)

"

t Wiener Sitzungsberichte, ucxiv. (2), (1877), pp. 128129.t This method is effective in proving numerous formulae of which analytical proofs were

given by Gegenbauer ; and it seems not unlikely that he discovered these formulae by the methodin question ; cf. 12*12, 12-14. The device is used by Beltrami, Lombardo Reiidiconti, (2) xiii.(1880), p. 328, for a father different purpose.

The symbol jjm^ means that the integration extends over the surface of the hemisphere onwhich i is positive.


Now the integrand is an integral periodic function of yjr. and so the limits ofintegration with respect to yfr may be taken to be a and a + Itt, where a is anarbitrary (complex) number. This follows from Cauchy's theorem.

We thus get(ktffY /'i"'/"a+2wJv (w) = -, /. eiv sin e cos * cos2"-1 sin Bdslrd$TTl (v)Jo J*

=-teK I

'

*l

W

eiw8i,l *C08 (*+> cos2""1 sin ddylrdd.

irl Wio .'oWe now define a by the pair of equations

r cos o = Z z cos,

r sin a z sin,

so that(Aw)" /*" f 2"Jv (a) = - L, 1 I exp [i (Z z cos ) sin cos yfr iz sin sin t/t sin 0]

con2'-1 6 sin Bdyfrde.

The only difference between this formula and the formula

Jv {m) = v l, / exp [iw sin cos -f] cos2"-1 sin 0ety>d07Tl (v) Jo .'is in the form of the exponential factor ; and we now retrace the steps of theanalysis with the modified form of the exponential factor. When the steps areretraced the successive exponents are

i(Z z cos )l iz sin (f> . m,i(Z z cos )n iz sin

. I,

%{Zz cos) cos

iz sin s cos + sin sin cos yfr)]

sin2""1 ^ sin2" 0d\frd0,and consequently Gegenbauer's formula is established.

[Note. The device of using transformations of polar coordinates, after the manner ofthis section, to evaluate definite integrals seems to be due to Legendre, M4m. de I'Acad, desSci., 1789, p. 372, and Poisson, Mem. de FA cad. des Sci. in. (1818), p. 126.]

3*4. The expression of


[Note. Solutions in finite terms of differential equations associated with Jn+ i (z) were ob-tained by various early writers ; it was observed by Euler, Misc. Taurinensia, in. (17621765), p. 76 that a solution of the equation for ei2J

)t + ,(z) is expressible in finite terms ; while

the equation satisfied by 2* Jn+i (.s) was solved in finite terms by Laplace, Conn, des Terns.

1823 [1820], pp. 245257 and Mecanique Celeste, v. (Paris, 1825), pp. 8284 ; by Plana, Mem.della R. Accad. delle Sci. di Torino, xxvi. (1821), pp. 533534; by Paoli, Mem. di Mat. edi Fis. (Modena), xx. (1828), pp. 183188; and also by Stokes in 1850, Trans. Camh. Phil.Soc. ix. (1856), p. 187 [Math, and Phys. Papers, n. (1883), p. 356]. The investigationwhich will now be given is based on the work of Lommel, Studien liber die BesseVsehenFimctionen (Leipzig, 1868), pp. 5156.]

It is convenient to restrict n to be a positive integer (zero included), andthen, by 3-3 (4),

^(^ST.^d-^n ! \Zv

.

'_!dt

nly/ir \_e

r=0 zr+x dtr J_i'

when we integrate by parts 2n + 1 times ; since (1 - t*)n is a polynomial ofdegree 2n, the process then terminates.

To simplify the last expression we observe that if dr (l -t2)n/dtr be cal-culated from Leibniz' theorem by writing (1 -t2)n = (1 - t)n (l + t)n, the onlyterm which does not vanish at the upper limit arises from differentiating ntimes the factor (1 - t)n, and therefore from differentiating the other factorr n times; so that wo need consider only the terms for which r^n.

Hence [tOfff] =(-), 0. -5^and similarly [*&=!*j _(_>-., Q, .J^ .

It follows that

Um* KZ)Jir L

( }rrn ^+1.(r-n)!(2n-r)!

2 /"_ V+l Oan-r -, t ~\+ (_yi+i e-i2 v v V L - 1 V '

,.-s' +1 .(r-w)!(2n-r)! 'and hence

(1) i *~"


In particular we have

(3) Jj (*) = () sin m, J% (z) = (-) (S-^ - cos z) ;

the former of these results is also obvious from the power series for J (z).

Again, from the recurrence formula we have

/*

3*41] BBSSEL FUNCTIONS 55

and hence

(5) j^w -(i)*r00.(, +j,? 77yr^L!^_ v* r-o (2r+l)!(>i-2r-l)!(2)2' +1

_

In particular, we have

(6) mo- )'.* ^w-(i,y(-sf'-.)We have now expressed in finite terras any Bessel function, whose order is

half of an odd integer, by means of algebraic and trigonometrical functions.

The explicit expression of a number of these functions can be written down fromnumerical results contained in a letter from Hermite to Gordan, Journal fur Math. lxvi.(1873), pp. 303311.

3*41. Notations for functions whose order is half of an odd integer.

Functions of the types /(n+j)(s) occur with such frequency in variousbranches of Mathematical Physics that various writers have found it desirableto denote them by a special functional symbol. Unfortunately no commonnotation has been agreed upon and none of the many existing notations canbe said to predominate over the others. Consequently, apart from the summarywhich will now be given, the notations in question will not be used in this work.

In his researches on vibrating spheres surrounded by a gas, Stokes, Phil. Trans, of theRoyal Soc. clviii. (1868), p. 451 {Math, and Phys. Papers, iv. (1904), p. 306], made use ofthe series

n(+ l ) (-!)(+ !) (+ 2)"r 2.imr 2. 4. fair)2

"K "'

which is annihilated by the operator

d* . d n(n+ l)dr2 dr r2

This series Stokes denoted by the symbol /(') and he wrote

>*

=

Sm e ~ *


In order to obtain a solution finite at the origin, Rayleigh found it necessary to take

Sn'= ( - )n + 1 Sn in the course of his analysis, and then

t>

-

ir i (ir\ / r \ n e~ irIt follows from 3-4 that -

Q?-' = ( % -4- ) ,rn+1 \ rcrj r

and that Jn+i {r)=-j

)^

[e- ir in + ifn (ir) + e-i--ifn {-ir)].

In order to have a simple notation for the combinations of the types e* ir/(tr) whichare required for solutions finite at the origin, Lamb found it convenient to write

**~ l ~ 2(2n + 3) + 2 . 4. (jSh+ 3) (2/i+5)~ "*'in his earlier papers, Proc. Londo

watson atreatiseonthetheoryofbesselfunctions text

Documents

theory of functions

purpose bessel functions

bytrigonometrical functions

theory of besselfunctions

encounter bessel functions

theory of com

functions of large order

theory of fourier series