A TREATISE ON THE
G. N". WATSON, Sc.D., F.RS. .

hirst Edition 1 U'22 Second Edition KMi IN OW3.VT UIUTAIN
PREFACE book has heen designed with two objects in view. The first is the development of applications of the fundamental processes of the theory of functions of complex variables. For this purpose Bessel functions are admirably adapted; while they offer at the same time a rather wider scope for the appli- application of parts of the theory of functions of a real variable than is provided by trigonometrical functions in the theory of Fourier series. The second object is the compilation of a collection of results which would be of value to the increasing number of Mathematicians and Physicists who encounter Bessel functions in the course of their researches. The existence of such a collection seems to be; demanded by the greater abstruseness of properties of Bessel functions (especially of functions of large order) which have been required in recent years in various problems of Mathematical Physics. While my endeavour has been to give an account of the theory of Bessel functions which a Pure Mathematician would regard as fairly complete, I have consequently also endeavoured to include all formulae, whether general or special, which, although without theoretical interest, are likely to be required in practical applications; and such results are given, so. far as possible, in a form appropriate lor these purposes. The breadth of these aims, combined with the necessity for keeping the size of the book within bounds, has made it necessary to be as concise as is compatible with intelligibility. Since, the hook is, for the most part, a development of the theory of func- functions as expounded in the Course of Modern Analysis by Professor Whittaker and myself, it has been convenient to regard that treatise as a standard work of reference for general theorems, rather than to refer the reader to original .sources. It is desirable to draw attention here to the function which I have regarded as the canonical function of the second kind, namely the function which was defined by Weber and used subsequently by Schltifli, by Graf and Gubler and by Nielsen. F\>r historical and sentimental reasons it would have been pleasing to have felt justified in using Hank el's function of the second kind; but three considerations prevented this. The; first is the; necessity for standardizing the function of the second kind; and, in my opinion, the authority of the group of mathematicians who use Weber's function has greater weight than the authority of the mathematicians who use any other one function of the second kind. The second is the parallelism which the use of Weber's function exhibits between the two kin<Is of Bessel functions and the two kinds (cosine and sine)
VI PREFACE of trigonometrical functions. The third is the existence of the device, by which interpolation 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, references to any memoirs or treatises in which the results of the section have been previously enunciated; but it is not to be inferred that proofs given in this book are necessarily those given in any of the sources cited.* The bibliography at the end of the book haw been made as complete as possible, though doubtless omissions will be found in it. While I do not profess to have inserted every memoir in which Bessel functions are mentioned, I have not consciously omitted any memoir containing an original contribution, however slight, to the theory of the functions; with regard to the related topic of Riccati's equation,! have been eclectic to the extent of inserting only those memoirs which seemed to be relevant to the general scheme. In the case of an analytical treatise such as this, it is probably useless to hope that no mistakes, clerical or other, have remained undetected; but the number of such mistakes has been considerably diminished by the criticisms and the vigilance of my colleagues Mr C. T. Preece and Mr T. A. Luaisden, whose labours to remove errors and obscurities have been of the greatest value. To these gentlemen and to the staff of the University Press, who have given every assistance, with unfailing patience, in a work of great typographical complexity, I otter my grateful thanks. O. N. W. Auguaf&\, 102-2. PREFACE TO THE SECOND EDITION To incorporate in this work the discoveries of the last twenty years would necessitate the rewriting of at least Chapters XII—XIX; my interest in Bessel functions, however, has waned since 1922, and I am consequently not prepared to undertake such a task to the detriment of tny other activities. In the preparation of this new edition 1 have therefore limited myself to the correction of minor errors and misprints and to the emendation of a few assertions (such as those about the unproven character of Bourget's hypo- hypothesis) which, though they may have been true in 1922, would have been definitely false had they been, made in 1941. My thanks are due to many friends for their kindness in informing me of errors which they had noticed; in.particular, I cannot miss this opportunity of expressing my gratitude to Professor J. R. Wilton for the vigilance which he must have exercised in the compilation of his list of corrigenda. G. N. W. March 31, 1941.
To stand upon every point, and go over things at largo, and to bo cu particulars, belongeth to tho first author of tin* story : but to ti.se* and avoid much labouring of tho work, is to bo granted to him t make an abridgement. •2 Maccaiikks ii. :io,
CHAPTER I BESS EL FUNCTIONS BEFORE 1820 l'l. Riccati's differential equation. The theory of Bessel functions is intimately connected with the theory of a certain type of differential equation of the first order, known as Riccati's equation. In fact a Bessel function is usually defined as a particular solution of a linear differential equation of the .second order (known aw Bcssel'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 occurs in a paper* on curves which was published by John Bernoulli in 1004-. Tn this paper Bernoulli gives, aw an example, an equation of this type and states that he has not solved it"f\ In various letters']; to Leibniz, written between 101O and 1704, James Bernoulli refers to the equation, which he gives in the form dy — yydx + tnvdx, and states, more than once, his inability to solve it. Thus he writes (Jan. 27, 1097): " Vellum porro ex To so ire mini ot hanc. tent.averis dy = yydx-\- ,r.vd,r. Ego in mille formas transmutavi,sed ope ram meam improhuin Problema per- pettio lusit." Five years later he succeeded in reducing the equation to a linear equation of the second order and wrote§ to Leibniz (Nov. 15, 1702): " (a)ua occasione recordor aequationes alias memoral.ae dy~yydx-\-x"du: in qua nun- quarn separare potui indeterrninatas a se invieem, sicut aequatio maneivt simpliciter differential is: sed separavi i 1 las reducendo aequationem ad haue differentio-differentialem|| ddy: y = — ,ra dx"." When this discovery had been made, it was a simple step to solve the last equation in series, and so to obtain the solution of the equation of the first order as the quotient of two power-series. * Ada F. nidi toy inn publicatu Lipxian, 1(>94, ]ip. 4;{.r)—4H7. t " K«to proposita aequatio differential in luicc xulx-]- y-d.r ~a"ilij (inac an pur Hcparationcin indctormiiiataruni conHtrui poKnit nondum tcntavi" ([). 4,!J()). X Sec faibnizem getiamellte Werkt, J^iitto FoIrb (Mathonmtik), in. (Hallo, lH,r>r>), pp. /i0—H7. § Ibid. p. 615. Bernoulli's procedure wuh, eiTcctivuly, to take a new variable a defined by tho formula 1 du u dx =y in the equation di/[(lx = x"-\-y1i, and then to replaee it by y. || The connexion between this equation and a special form of Bessel'n equation will bo seou in §4-3. W. B. F. 1
2 THEORY OF BESSEL FUNCTIONS [CHAP. I And, in fact, this form of the solution was communicated to Leibniz by James Bernoulli within a year (Oct. 3, 1703) in the following terms*: "Reduco autem aequationem dy^yydx+ocxdx ad fracfcionem c-ujus u torque, terminus per seriem exprimitur, ita , , 347811 8478111215  3.4.7 3.4. 7.8. 11 . 12. 10. JU. ll> y" , ^ , * *" , ^"^ ~ 3.4 + T 3.4. 7. 8. U . 12. 15 . Hi quae series quidem actuali divisione in unam conflari possunt, hc<1 in qua ratio progressionis non tam facile patescat, scil. _ a* x> 2a;11 _ 13a?16 „ y ~ 3 + 3.3.7 + + 3.373.' 3. 5. 7 . 7 .31 °" Of course, at that time, mathematicians concentrated their energy, so tar as differential equations were concerned, on obtaining solutions in 'finite terms, and consequently James Bernoulli seems to have received hardly the full credit to which his discovery entitled him. Thus, twenty-two years later, the pa per f, in which Count Riccati first referred to an equation of the type which now bears his name, was followed by a notej hy Daniel 'Bernoulli in which it, was stated that the solution of the equation^ axn dx + uudx — bdu was a hitherto unsolved problem. The note ended with an announcement, in an anagram of the solution: "Solutio problematis ab 111. Kiecato propu.sito characteribus occultis involuta 24a, %, 6c, 8d, 33e, 5/* 2//, 47>, ',YM, HI, 21 ni, 26n, 16o, 8p, oq, 17r, 16s, 2St, 32u, 5x, 3y, +, -, ——, +, =, -I-, 2, 1." The* anagram appears never to have been solved; but Bernoulli published his 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 — 4w/Bm ± 1), where m is any integer, for any one of which the equation in soluble in finite terms; the details of this solution will be given in $ 4f 1, \-1 1. The prominence given to the. work of Riccati by Daniel Bernoulli, combined with the fact that Riccati's equation was of a slightly more general type than • See Leibnizem gesavidlic Werke, Dritte Folge (Miithematik), lit. (Hallo, lH5.r>), p. 7,">. ¦\ Acta Eruditorum, Suppl. vin. A724), pp. 06—73. The form in which lUcoali look tho equation was xmdq — du + uu dxiq, where q=xn. t Ibid. pp. 73—75. Daniel Bernoulli mentioned that solutions had been obtained by thn-o other members of his family—John, Nicholas and the younger Nicholas. § The reader should observe that the substitution hdz u= — - z dx gives rise to an equation which is easily soluble in series. || Exercitationes quaedam mathematicae (Venice, 1724), pp. 77—80; Ada Eruditorum, 172C pp. 465—473.
1*2] BESSEL FUNCTIONS BEFORE 1826 3 John Bernoulli's equation* has resulted in the name of Riccati being associated not only with the equation which he discussed without solving, but also with a still more general type of equation. It is now customary to give the namef Riccati's generalised equation to any equation of the form where P, Q, R are given functions of x. It is supposed that neither P nor 11 is identically zero. If 22=0, the equation is linear; if P=O, the equation is reducible to the linear form by taking \\y as a new variable. The last equation was studied by Euler J; it is reducible to the general linear equation of the second order, and this equation is sometimes reducible to 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§ it is proved that, when a particular integral yx of Riccati's generalised equation is known, the. equation is reducible to a linear equation of the first order by replacing y by yx + l/u, and so the general solution can be effected by two quadratures. It is also shewn (ibid. p. 59) that, if two particular solutions are known, the equation can be integrated completely by a single quadrature; and this result is also to be found in the second|| of the two papers. A brief dis- discussion of these theorems will be given in Chapter iv. 12. Daniel Bernoulli's mechanical problem. In 1738 .Daniel Bernoulli published a memoiriT containing enunciations of a number of theorems on the oscillations of heavy chains. Thu eighth ** of these is as follows: " De fiyura catenae uniformiter oscillantis. Sit catena AG uniformiter gravis ot perfects flexilis suspensa de puncto A, eaque oscillationes facere uniformes intelligatur: porvenorit catena in situm AMF; fuoritque longitudo catenae = 1: longitudu cujuscunque partis FM — x, sumatur n ejus valorisff ut fit _ L JL _ l* il _ —A" + 4~4.9n3+479.16ft4 4".~9.16.25ns + * See James Bernoulli, Opera Omnia, u. (Geneva, 1744), pp. 1054—1057; it is stated that the point of liiccati'ti problem is the determination of a solution in finite terms, and a solution which rt'Rerubles the solution by Daniel Bernoulli is given. t Thu term ¦ lliceati's equation ' was used by D'Alembert, ItiBt. de VAcad. R. dea Sci. de Berlin, xix. A7C8), [published 1770], p. 242. X Institutiones Calculi Intefjraliii, n. (Petersburg, 1769), §881, pp. 88—89. In connexion with the reduction, see James Bernoulli's letter to Leibniz already quoted. § Novi Comm. Acad. Petrop. vin. A760—1761), [published 1763], p. 82. || Ibid. rx. A7E2—1763), [published 1764], pp. 163—164. IF " Theoremata de osoillutionibus oorporurn filo flexili oonnexorum et catenae verticaliter BUspenBae," Comm. Acad. Sci. Imp. Petrop. vi. A732—3), [published 1738], pp. 108—122. ** Loc. cit: p. 116. ft The length of the simple equivalent pendulum is n.
4 THEOBY OF BESSEL FUNCTIONS [CIIAL\ I Ponatur porro distantia extrerai puncti Fab linea'vertical! = 1, (lieu fore distantiam puncti ubicunque assumpti M ab eadem linea verticali aequalem 1'" n + 4^ ~ 479^ + OTfen4 "* 479.1G. 25/in He goes on to say: "Invenitur brevissimo calculo n = proxime IHiiW /.... Habet autem littera n infinitos valores alios." The last series is now described as a Bessel function* of order zero and argument 2*J(xjv); and the last quotation states that this function has an infinite number of zeros. Bernoulli published^ proofs of his theorems.soon afterwards; in theorem vin, he obtained the equation of motion by considering the forces acting on the portion FM of length x. The equation of motion was also obtained \>y EulerJ many years later from a consideration of the forces acting on an element of 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 lie. the ten.sinn u\ height x above the lowest point of the chain in its undisturbed position. The unit inn lieini; transversal, we obtain the equation 8T=gp8;c by resolving vertintllv l«»r an element <<f chain of length 8x. The integral of the equation is T=g/).v. The horizontal component of the tension is, effectively, T(dijjdx) where >/ i.s the Imrt zontal) displacement of the element; and so the equation of motion \» 9 If we substitute for T and proceed to the limit, we find thai, (Fy_ d ( dy\ d~>~(Jd\Vd) If / is the length of the simple equivalent pendulum for any one normul viUrat inn, \vi- write where A and ? are constants ; and then II (x/f) is a solution of tlio ('(juatiDn d ( do\ o . dx \ ax) f If a?//=u5 we obtain the solution in the form of Bernoulli's .series, namely 1 1.4 1.4.9+ ¦"¦ * On the Continent, the functions are usually called cylinder fuuriioiiH, or, oowumnuilly, func- functions of Fourier-Bessel, after Heine, Journal fur Math. lxix. A808), p. V2.H- hcc iiIho Muth Ann in. A871), pp. 609—610. + Comm. Acad. Petrop. vn. A734-5), [published 1740], pp. 162—17D. t Ada Acad. Petrop. v. pars 1 (Matliematica), A781), [published 1784], pp. 1.-57 177. Kuler took the weight of length e of the chain to be E, and he defined ,j to he the iiuuihiuo of tlu- distance (not twice the distance) fallen by a particle from rest under gravity in a mioond. KuIit'h notation lias been followed in the text apart from the significance of g and tho introduction of p and 8 (for d).
1*3] BESSEL FUNCTIONS BEFORE 1826 5 The general solution of the equation is then shown to be Di' + Ov I - 2, where C and I) are constants. Since y is finite when A'=0, 0 must l>e zero. If a is the whole length of the chain, y=0 when x=a, and so the equation to dofcermine / is By an extremely ingenious analysis, which will be given fully in Chapter xv, Eulot proceeded to shew that the three smallest roots of the equation in a/faVQ l*44-79.r>, 716Gf>K and 18-63. [More accurate values are l-44f»79fl."), 7'017Hlf)fi and 18-7217517.] In the memoir* immediately following this investigation Euler obtained the general solution (in the form of series) of the equation ,- («¦,-) + r~0, but his statement of the v ' l du \ dwj ' law of formation of .successive coefficients is rather incomplete. The law of formation had, however, been stated in his Institutions* Calct'li [nttufwlix^, it. (Petersburg, 17C!)), ? iO7, pp. 233-235. 1. Eiders mechanical problem.. The vibrations of a stretched membrane were investigated by Euler} in L764. He arrived at the equation 1 d-z tf-s 1 dz 1 d'-z e'1 dt~ dr" r dr r'2d<fr' where z is the transverse displacement, at time t at the point whose polar coordinates are (r, <f>); and e is a constant depending on the density and .tension of the membrane. To obtain a normal solution he wrote z «= it sin (at. + A ) sin (fi<j> + li), where a, A, ft, H are constants and u is a function of r; and the result of substitution of this value of ^ is the differential equation dru , J dit fa" ft-\ ., dr- r dr ¦ e- r-J The solution of this equation which is finite at the origin is given on p. 2a(i of Euler's memoir; it is ,. _ rp j i a 7 i. ... a f . . ._ i. where n has been written^ in place of 2/0+ T. This differential equation is now known as Bessel's equation lor functions of order ft; and ft may have || any of the values 0, 1,2, .... Save for an omitted constant factor the series is now called a liessel coefficient of order ft and argument ar/e. The periods of vibration, "IttJol, of a * Artn Actul. J'etnrp. v. pars 1 (Mnlhi'iiuitica), A7H1), [published 17H-1], pp. 17H—11H. + See also §§085, 91W (p. 1H7 ft. ssq.) ibr tho nolutitm of an utmocialiul onuation which will Ir- dificusHed in jj 3-52. X Novi Comm. Actul. Petvop. x. A70i), [publlshod 1700], pp. 218—42<j(). ^ Thu reason why Euler made this change of notation in not obvioua. || If p were not an integer, the displacement would not be a one-valued function of position, in viuw of the faotor sin ((Sij> + fi).
6 THEOBY OF BESSBL FUNCTIONS [CHAP. T circular membrane of radius a with a fixed boundary* are to be determined from the consideration that u vanishes when r = a. This investigation by Euler contains the earliest appearance in Analysis of a Bessel coefficient of general integral order. 14. The researches of Lagrange, Carlini and Laplace. Only a few years after Euler had arrived at the general Bessel coefficient. in his researches on vibrating membranes, the functions reappeared, in an astronomical problem. It was shewn by Lagrangef in 1770 that, in the elliptic, motion of a planet about the sun at the focus attracting according to the law of the inverse square, the relations between the radius vector r, the mean anomaly M and the eccentric anomaly E, which assume the forms M - E «- e sin E, r - a A - e cos E), give rise to the expansions E = M + 2 An sinnM, - * 1 + ie2 + S Bn cos vM, «=i a " n~i in which a and e are the semi-major axis and the eccentricity of the orbit and 2'l+2'rt m! (</t + m)\ Lagrange gave these expressions for n = 1, 2, 3. The object of the expansions is to obtain expressions for the eccentric anomaly and the radius vector in terms of the bime. In modern notation these formulae are written J.n = 2/n(ne)/n, Bn = - 2 («/») Jn' (tw). It was noted by Poisson, Connaissance des Terns, 1836 [published 1833], p. <i that, n _ 6 (lAn a memoir by Lefort, Journal de Math. XI. A846), pp. 142—152, in which an orror nmdc ly Poisson is corrected, should also be consulted. A remarkable investigation of the approximate value of A n when n is large and 0< e< 1 is due to Carlini^; though the analysis is not rigorous (and it would be difficult to make it rigorous) it is of sufficient interest for a brief account of it to be given here. * Cf. Bourget, Am. Sci. de VEcole norm. tup. ra. A866), pp. 55-95, and Clin-o, Quarterly Jovrnat. xxi. A886), p. 298. t Hilt, de VAcad. R. des Sci. de Berlin, xxv. A769), [publiahed 17711, pp. 20-1—231*. Weuvrei in. A869), pp. 113—138.] % JRicerche sulla convergenza delta Krie che servo, alia soluzione del problema di KfiUem (MUan, 1817). This work was translated into German by Jaoobi, A*tr. Nach xxx. AH50) col. 197-254 [Werke, vn. A891), pp. 189-245]. See also two papers by ScheiW datod ' reprinted in Math. Amu xvu. A880), pp. 531—544, 545—560.
1'4] BESSEL FUNCTIONS BEFORE 1826 7 It is easy to shew that An is a solution of the differential equation ••'*»-" + .'•?•—0-1 A-a Define u by the formula All = 2nn~'i f/Ulk/nl and then e2 (~ + iCl \ + tu- w2 A - e2) = 0. Hence when ii is large either u or «2 or dujde must bo largo. If u = 0(na) we should expect ul and rfrt/^e to bo 0(?i!Jo) and O(«a) respectively; and on considering the highest powers of n in the various terms of the last differential equation, we find that a = 1. It is consequently assumed that u admits of an expansion in descending powers of n in the form where ii0, Uy, u.^, ... are independent of n. On substituting thi.s .series in the differential equation of the first order and equating to zero the coefficients of the various powers of ?i, wo find that wliere u^dujde ; so that v()— ± , ?<i -- "--„, and therefore f Udf = U JHr __ ; ^ _ ^ + v/( I - C2) + 1 j _ J log ( 1 _ ,2) _,_ ... } and, since the value of An shews that \udi -«log he whefi « i.s small, tho upper .sign must be taken and no constant of integration is k> bo added. From Stirling's formula it now follows at onco that and this is tho result obtained by Carlini. This method of approximation has been carried much further by Meissel (see § 8*11), while Uauchy* has also discussed approximate formulae for An in the ease of comets moving in nearly purabolu: orbits (soo § 8-42), for which Carlini's approximation is obviously inadequate. The investigation of which an account has ju.st been given is much more plausible than tho arguments employed by Laplacef to establish the corre- corresponding approximation for Bn. The investigation given by Laplace is quite rigorous and the method which he uses is of considerable importance when the value of Bn is modified by taking all the coefficients in the series to be positive—or, alternatively, by supposing that e is a pure imaginary. But Laplace goes on to argue that an approximation established in the case of purely imaginary variables may be used ' sans crainte ' in the case of real variables. To anyone who is acquainted with the modern theory of asymptotic series, the fallacious character of such reasoning will be evident. * Comptes Rendus, xxxviri. A854), pp. 900—993. f Mtcanique Cileste, supplement, t. v. [first published 1827]. Oeuvres, v. (Paris, 1882), pp. 486—489.
8 THEORY OF BBSSEL FUNCTIONS [CHAP. I The earlier portior of Laplace's investigation is based on the principle that, in the case of a series of positive terms in which the terms steadily in- increase up to a certain point and then steadily decrease, the order of magnitude of the sum of the series may frequently be obtained from a consideration of the order of magnitude of the greatest term of the series. For other and more recent applications of this principle, see Stokes, Proc. Camb. Phil. Soo. vi. A889), pp. 362—366 [Math, and Phys. Papers, v. A905), pp. 221—225], and Hardy, Proc. London Math. Soo. B) n. A905), pp. 332-^-339 ; Messenger, xxxiv. A905), pp. 97—101. A statement of the principle was given by Borel, Ada Mathematica, xx. A897), pp. 393— 394. The following exposition of the principle applied to the example considered by Laplace may not be without interest: *- The series considered is ~ «=o 2n + 2m m! (ra + m)! ' in whioh n is large and e has a, fixed positive value. The greatest tonn i« that for which «is=/i, where /z is the greatest integer such that V(n + p) (n + 2/x - 2) < (n + 2/z) n*e8, and so p is approximately equal to Now, if um denotes the general term in Bn^\ it is easy to verify by Stirling's theorem that, to a first approximation, -4±±.' ~ qt\ where U Hence BnW^u^ {I + 2q + since* q is nearly equal to 1. Now, by Stirling's theorem, ««-' exp { and so {1+^A+6*)}" ¦ The inference which Laplace drew from this result is that B - f2 This approximate formula happens to be valid when e<l (though the reason for this restriction is not apparent, apart from the fact that it is obviously necessary), but it is difficult to prove it without using the methods of contour * The formula l + 2S(/{ ~\/['irl{l-q)\ may be inferred from general theorems on series; cf. Bromwioh, Theory of Infinite Series, § 51. It is also a consequence of Jacobi's transformation formula in the theory of elliptic functions, M0h) = (-'>)-Hn@|-T-i); see Modern Analysis, § 21-51.
1*5] BESSEL FUNCTIONS BEFORE 1826 9 integration (cf. § 8-31). Laplace seems to have been dubious as to the validity of his inference because, immediately after his statement about real and imaginary variables, he mentioned, by way of confirmation, that he had another proof; but the latter proof does not appear to be extant. 1. The researches of Fourier. In 1822 appeared the classical treatise by Fourier*, La Theorie analytique de la Chal&iir; in this work Bessel functions of order zero occur in the dis- discussion of the symmetrical motion of heat in a solid circular cylinder. It is shewn by Fourier (§§ 118—120) that the temperature v, at time t, at distance x from the axis of the cylinder, satisfies the equation dv _ K /d2v 1 dvy It ~ Ul) [d:rn- + x t where K, G, J) denote respectively the Thermal Conductivity, Specific Heat and Density of the material of the cylinder; and he obtained the solution where (j = mC])/K and m has to be so chosen that hv + K (dv/dx) = 0 at the boundary of the cylinder, where h is the External Conductivity. Fourier proceeded to give a proof (§§ 307—309) by Rolle's theorem that the equation to determine this values of ni hasf an infinity of real roots and no complex roots. His proof is slightly incomplete because he assumes that certain theorems which have been proved for polynomials are true of integral functions; the defect is not difficult to remedy, and a memoir by HurwitzJ has the object of making Fourier's demonstration quite rigorous. It should also be mentioned that Fourier discovered the continued fraction formula (§313) for the quotient of a Bessel function of order zero and its derivatc; generalisations of this formula will bo discussed in §§ 5'6, 965. Another formula given by Fourier, namely a.'2 a* ft" I 1 + + had been proved some years earlier by Parseval§; it is a special case of what are now known as Hessel's and Poisson's integrals (§§ 2*2, 2'3). " This urcal.tii- pint of Vouimpi-'h veHcaroliiin wuh contained in a memoir dopoeifcod in the archives of the Freud) InntiUite on HcpL liH, 1H11, and crowned on Jan. G, 1812. This memoir ia to be found in the Mvm. de VAcud. tics ScL, iv. A819), [published 1824], pp. 185—56C; v. A820), | published lHUH], pp. lM—'24E. I Thin i« n ^'ciU'raliHation of Bernoulli's Htatoment quoted in § 1. :'. Math. Ann. xxxin. A889), pp. '240—2E0. 8 Mvm. den xttvanx Gtrtintjerts, t. A805), pp. E39—048. .This paper also contains the formal Htali'iuntt df thn the.oroni on Fourier conntantB which is sometimes called Pareeval's theorem; another j»a])(!i' l»y thm little known writer, Mf.m. den savaiu strangers, i. A805), pp. 379—398, con- tainH a Kenoral Holution of Laphice'H equation in a form involving arbitrary functions.
10 THEORY OP BESSEL FUNCTIONS [CHAP. I The expansion of an arbitrary function into a series of Bessel functions of order zero was also examined by Fourier (§§ 314—320); he gave the formula for 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. A875), pp. 471—494; Schlafli, Math. Ann. x. A876), pp. 137—142; Dini, Serie di Fourier, i. (Pisa, 1880), pp. 246—269; Hobson, Proc. London Math. Sue. B) vn. A909), pp. 359—388; and Young, Proc London Math. Soc. B) xvin. A920), pp. 163-200. This expansion will be dealt with in Chapter xvin. 1*6. The researches of Poisson. The unaymmetrical motions of heat in a solid sphere and also in a solid cylinder were investigated by Poisson* in a lengthy memoir published in 1823. In the problem of the. sphere f, he obtained the equation d?R n(n + l) where r denotes the distance from the centre, p is a constant, n is a positive integer (zero included), and R is that factor of the temperature, in a normal mode, which is a function of the radius vector. It was sheAvn by Poisson that a solution of the equation is rn+1 cos (rp cos to) sin2U+1 todco Jo and he discussed the cases n — 0, 1, 2 in detail. It will appear subsequently (| 3*3) that the definite integral is (save for a factor) a Bessel function of order n + ^. In the problem of the cylinder (ibid. p. 340 et seq.) the analogous integral is Jo cos (hX cos &)) sin2n co dco, where n= 0,1, 2,... and X is the distance from the axis of the cylinder. The integral is now known as Poisson's integral (§ 2'3). In the case n = 0, an important approximate formula for the last integral and its derivate was obtained by Poisson (ibid., pp. 350—352) when the variable is large; the following is the substance of his investigation: LetJ t70(i)=- I cos (? cos w) da>, Jo' (k)= I cos a sin (k cos «) i«. "¦ J o it J o Then Jo (k) is a solution of the equation * Journal de Vficolc R. Poly technique, m, (cahier 19), A823), pp. 249—403. + Ibid. p. 300 et seq. The equation was also studied by Plana, Mem. della Ji. Accad. delle Sci. di Torino, xxv. A821), pp. 532—534, and has since been Btudied by numerous writers, some of whom are mentioned in § 4-3. See abo Poisson, La ThSorie Mathtmatique de la Chaleur (Paris, ' 1835), pp. 366, 369. $ See also Rdhrs, Proc London Math. Soc. v. A874), pp. 136—137. The notation Ja(k) was not used by Poisaon.
1] BESSEL FUNCTIONS BEFORE 1826 11 When k is large, 1/DP) may be neglected in comparison with unity and so we may expect that Jo (k) s/k is approximately of the form A cos ? + 2? sin k where A and B are constants. To determine A and B observe that cos k.J0 (X-)-sin k.J6' (k)=- I * {cosH<w cos BAsin2?co) + sinHw cos B& cos2 ?co)} da. "¦ J o Write 7T - <u for <w in the latter half of the integral and then cos k. J{) (k) - sin k. J{{ (k) = — I cos2 \<a cos Bk sin2 %a) da> f j o and similarly sink../<, (k)+cosk. J({ (k) = —~r \ A -~ )" si [ VBfc) / ;?2\2 COS /*°°COS But hm I (i_ __ ) . a>2.^=| . .r2. ota?=-i J(i7r), fc-^oo Jo \ 2k J sm Jo sm - ¦* by a well known formula*. [Note. It is not easy to prove rigorously that the passage to the limit is permissible; the simplest procedure is to appeal to Bromwich's integral form of Tannery's theorem, Bromwich, Theory of Infinite Series, § 174.] It follows that cos k ../,, (k) - sin /-. Jo' (k) = . A + tk), <in k. JK) (k) + cos k. •/,,' (k)= -rr—r-: A i-yk), where <*-*-<) and rjk-»~O as k-»-tx>; and therefore Jo (k) = -j—.v [A + «*) cos /t + A + r)k) sin -{.•], It was then assumed by Poisson that Ju(k) is expressible in the form + + % + ... k kl where ;i = /i=l. The series are, however, not convergent but asymptotic, and the validity of thin expansion was not established, until nearly forty years later, when it was investi- investigated by Lipschitz, Journal filr MatL i-vi. A859), pp. 180—196. The result of formally operating on the expansion assumed by Poisson for the function tfl 1 J0(k) -JGrk) with the operator -¦ + 1 +/m 'a ,[<¦!. ].ff-;M , 2.2^'-A.2 + jr)yl' . 2 . SB'"-B. 3 + j-) A" , 1 * Cf. WatBon, Complex Integration and Cauchy's Theorem (Camb. Math. Tracts, no. 15,1914), p. 71, for a proof of these results by using contour integrals.
12 THEORY OF BESSEL FUNCTIONS [CHAP. I and so, by equating to zero the various coefficients, we find that «-g5, A"=-^-^A, A'"- ^—i?, ... 1 a »« 9 a T>>" 9'25 A and hence the expansion of Poisson's integral is ^ * 9 . 9-25 / 1 9 9.25 \ . . ~| v +m ~ 2: 8*#" otpz* + • • •;sm J ¦ But, since the series on the right are not convergent, the researches of Lipschitz and subsequent writers arc 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 expansion was first given by W. B. Hamilton, Trans. K. Irish Acad. xix. A843), p. 313; his result was expressed thus: - f I cosB/3sin a) rf«= —^- 2 [0]"» ([--|]»J D/3)~)l cos ^-\i,n -±n), and he described the expansion as serai-convergent; the expressions [O]~"and [ —i}" arc to be interpreted as Ijn ! and { — i) (— §)... (- A result of some importance, which was obtained by Poisson in a subsequent, memoir*, is that the general solution of the equation is y = j4^ J e~hxaosu>da) + 5^ I e~hxcosu log (a; sin- «¦> Jo Jo where J. and 5 are constants. It follows at once that the general solution of the equation d*if 1 dy 7, _ ace- x ax ° is — A\ e~hxcos<° dco + B \ e-hxcoato\og {x sin-ca) g5gu. .'o Jo This result was quoted by Stokesf as a known theorem in 1850, and it is likely that he derived his knowledge of it from the integral given in Poisson's memoir; but the fact that the integral is substantially due to Poiswon has been sometimes overlooked^. * Journal de.. V&co\e R. Poly technique, xn. (eahier 19), A823), p. 476. The corresponding general integral of an associated partial differential equation was given in an earlier memoir, ibid, p. 227. t Cavib. Phil. Trans, ix. A856), p. [38], [Math, and Phys. Papers, in. A901), p. 42]. ? See Encyclopgdie des Sci. Math. n. 28 (§53), p. 213.
1'7] BESSEL FUNCTIONS BEFOKE 1826 13 1'7. The researches of Bessel. The memoir* in which Bessel examined in detail the functions which now bear his name was written in 1824, but in an earlier memoir^ he had shewn that the expansion of the radius vector in planetary motion is - = 1 + ? e2 + 2 Bn cos nM, where Bn = sin u sin (nu — ne sin u) clu \ nir J o v this expression for Bn should be compared with the series given in § 1'4. In the memoir of 1824 Bessel investigated systematically the function IJl defined by the integral J Ikh = Ty coa (hit — k sin u) du. lit He took h, to bo an integer and obtained many of the results which will be given in detail in Chapter ii. Bessel's integral is not adapted for defining the function which is most, worth study when h is not an integer (see §10*1); the • function which is of most interest for non-integral values of h is not I//1 but the function defined by Loinmcl which will be studied in Chapter ill. After the time of Bess<>l investigations on the functions became so numerous that it seems nmvenie.nt at this stage to abandon the. chronological account and to develop the theory in a systematic and logical order. An hiatoiMcal jKn-.ount. of lnsouivhen from the time of KouruM1 to 1858 lias boon compiled l>y Wagner, livni Mitthcil'uinfrn, 18!M, pp. 20-l--2(i(i; a briefer account of the early history wan given by Rlag»i, Atti <icUa li. Av.cud. <h-i Lined, (Transmit!), (:i) iv. AH80), pp. 2.r>{)—ii\\\. * lirrlhwr Abh. lH'2-1 [nublishod 1HUE|, pp. 1—52. The date o? this incmuir, ''UntcrsachuiiK ilc.H TlicilH dcf iiluiH'turisclK'ii Htorungcn, w<;lchei- aus der BewcKunj; <lcr Sount' ontsteht," is .Ian. li'.t, 1K24. •|- lin-hnvr Abh. 1KH1- 17 [published 181<)J, ])p. 4il—.">/>. ;[: This integral occurs in the cxnunnion of thu ocw.ntric anomaly; with the uolatiou oE § 1-4, "'lJt '* JU' ' a foiiuulii u\vvn by I'oi.smin, Cunnamance den Tern*, 1825 [publishod 1R2*2], p. 3H3.
CHAPTER 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 of defining 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- definition, and proved incidentally that the functions thus defined are equal to the definite integrals by which they had previously been defined by Bessel f. It should, however, be mentioned that the converse theorem that Bessel's inte- integrals are equal to the coefficients in the expansion, was discovered by Hansen| fourteen years before the publication of Schlb'milch's memoir. Some similar results had been published in 1836 by Jacobi (§ 2*22). The generating function of the Bessel coefficients is .KH). 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 A) eKH)-. I t»Jn(z). • «= —90 To establish this development, observe that eizt 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~*elt can be axpanded into an absolutely conver- convergent series of descending powers of t: When these series are multiplied together, 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 A), which is valid for all values of z and t, t = 0 excepted. * Ztittchrift fiir Math; wnd Phys. n. A857), pp. 137—165. For a somewhat similar expansion, namely that of ezC0B9t 8ee Frullani, Mem. Soc. Ital. {Modena), xvni. A820), p. 503. It muat be pointed out that Sbhldmiloh, following Haneen, denoted by Jlhn what we now write as JnBz); but the definition given in the text is now universally adopted. Traces of Hanson's notation are to be found elsewhere, e.g. Sohlafli, Math. Ann. m. A871), p. 148. + Berliner Abh. 1824 [published 1826], p. 22. X Ermittelung der Absoluten Stdrungen in Ellipses von beliebiger Excentricit&t und Neigung, i. theil, [Sohriften der Sternwarte Seeburg: Gotha, 1843], p. 106. See also the French transla- translation, Mimoire »ur la determination des perturbations absolues (Paris, 1845), p. 100, and Leipziger Abh. n. A855}, pp. 250—251.
2*1, 2*11] THE BESSBL COEFFICIENTS 15 If in A) we write — 1/t for t, we get rt" —oo on replacing n by — n. Since the Laurent expansion of a function is unique*, a comparison of this formula with A) shews that B) /-(*)« <-)»./»(*), whore n is any integer — a formula derived by Bessel from his definition of Jn (z) as an integral. From B) it is evident that A) may be written in the form C) ek*(t-iit) = Jo (Z) + v {p + (-.)«t-n] Jn (*). «-i A nummary of elementary results concerning «/„ B) has been given by Hall, The Analyst, 1. A874), pp. 81-—84, ami an account of elementary applications of these functions to problems of Mathematical Physics has been compiled by Harris, American Journal of Math, xxxiv. A912), pp. 3i)l—420. The function of order unity has been encountered, by Turriere, Nouv. Ann. de Math. D) IX. A1H5)), pp. 133—411, in connexion with the steepest curves on the surface z=-y E.i>2 -y*). 21. 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(J^) and exp( — ^zjt), thus Wlu'ii n is u po.sit.ivo integer or zero, tho only term of the first series on the right which, when associated with the general term of the second series gives rise t,o a term involving tn is the term for which r = n + m; and, since ?i ^ 0, then4 is always one (,enn for which r has this value. On associating these terms for all the values of m, we see that the coefficient of tn in the product is 3 (I*)'11- (-\z)m wif 0 (n + m) \ m \ We thei'efore have the result * For, if not, zero oouhl hv. oxpandod into a Laurent Heries in (, in which some of the coufliciuntH (nay, in inirticular, that of /"') woro uoL zero. If wo then multiplied the expansion by t-m-i anfl iulegrated it round a circle with oentru at the origin, we should obtain a contradiction. ThiH roBult was uotiood by Cauohy, Comptes Rendus, xm. A841), w. 911.
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 formula B) M»)—$r- In particular ?2 z4 z° To obtain the Bessel coefficients of negative order, we select the terms in- involving t~n in the product of the series representing exp (\zb) and exp(— Izjt), 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 m = n + r ; and so we have ~ ).==() f ! (n + v)! ' whence we evidently obtain anew the formula § 2*1 B), namely J-n{z) = (-)nJn(z). It is to be observed that, in the series A), the ratio of the (m + l)th term to the mth term is — \z"J\in (n + to)), and this tends to zero as on -*• 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 aa a finite combination of exponential, logarithmic and algebraic functions operated on by signs of indefinite integration. From A) we can obtain two useful inequalities, which are of some import- importance (cf. Chapter XVI) in the discussion of series whose general term in a multiple of a Bessel coefficient. Whether z be real or complex, we have and so, when n 5s 0, we have This result was given in substance by Cauchy, Comptes liendus, xm. A841), pp. 687, 854; a similar but weaker inequality, namely was given by Neumann, Theorie der BesseVschen Functionen (Leipzig, 1867), p. 27.
2*12] THE BESSBL COEFMCIENTS 17 By considering, all the terms of the series for Jn (z) except the first, it is found that E) jn(z)J&f where 6 <exp ±A—=• - 1 It should be observed that the aeries on the right in § 21 A) converges uniformly in tiny bounded domain of the variables z and t which does not contain the origin in the Z-plano. For if 8, A and It are positive constants and if the terms in the expansion of exp (^zt) exp (felt) do not exceed in absolute value the corre- corresponding terms of the product exp (?/2A) oxp (^/2/&), and the uniformity of the convergence follows from the test of Weierstrass. Similar considerations apply to the series obtained by term-by-term differentiations of the expansion 2tn Jn (z), whether the differentiations be performed with respect to z or t or both z and t. 2*12. The recurrence formulae. The equations* A) Jll_l(z) + Jn+1(z)^Jn(z)> B) Jn_1B)-/n+1(^) = 2/w'(^), which connect three contiguous functions are useful in constructing Tables of Bessel coefficients; they are known as recurrence formulae. To prove the former, differentiate the fundamental expansion of § 2-l, namely ,»*«-*">= i FJn{z), n- -so with respect to t; we get i«(l + l/P)e**('/0« 2 nt^Jn(z), ho that Jsa + l/P) 5 lnJn(z)= 5 ntn-\r»{z). ?l= — oo »= — oo If the expression on the left is arranged in powers of t and coefficients of tn~x are equated in the two Laurent series, which are identically equal, it is evident that | z {./¦„_, (z) + /?H, (*)} = nJn (z), which is the first of the formulaef. * Throughout the work primes are used to denote the derivate of a function with respect to its argument. f Differentiations are permissible because (§ 2*11) the resulting series ar? uniformly convergent. The equating of coefficients ie permissible because Laurent expansions are unique. w. u. p. 2
18 THEOBY OF BESSEL FUNCTIONS [CHAP. II Again, differentiate the fundamental expansion with respect to z; and then {t~xm= 2 t*Jn'(z), so that K*-V<) ^ tnJn{z)= t tnJnr(z). By equating coefficients of tn on either side of this identity we obtain formula B) immediately. The results of adding and subtracting A) and B) are C) sJn' (z) + nJn (z) = zJn_, 0), D) zJn 00 - nJn (z) = - z Jn+1 (z). These are equivalent to E) j?{*M*)} = *J»-i{*), F) §-z{z-nJn(z)} = -2~nJn+dz)- In the case n=0, A) is trivial while the other formulae reduce to G) Jo'(*) = -/,(*). The formulae A) and D) from wbich the others may be derived wore discovered by Bessel, Berliner Abh. 1824, [1826], pp. 31, 35. The method of proof given here is due to Schltimilch, Zeitschrift fur Math, und Phys. ir. A857), p. 138. Schlmnilch proved A) in this manner, but he obtained B) by direct diflerentiation of the scries for r/u (z), ,A. formula which Sehlomilch derived (ibid. p. 143) from B) is (8) 2r^#= 2 (-TrCm.Jn-m(z), where rQm is a binomial coefficient. By obvious inductions from E) and F), we have (9) () A0) [~) (r* Jn E)} = (-rz-»-™Jn+m (,), where n is any integer and in is any positive integer. The formula A0) is due to Bessel (ibid. p. 34). As an example of the results of this section observe that zJx (z) - 4/2 (z) - zJs (z) = 4/2 (z) - SJt (z) + zJ, (z) 2 n=l = 4 t (~-y-*nJm(z), n-i since zJiN+i (z)-*-0 as N-+ oo , by § 211 D).
2-13, 2-2] THE BESSEL COEFFICIENTS 19 The expansion thus obtained, A1) */,<*)« 4 2 {-f-'nJ^iz), is useful in the developments of Neumann's theory of Bessel functions (§37). 2*13. The differential equation satisfied by Jn(z). When the formulae § 2'12 E) and F) are written in the forms the result of eliminating* Jn-\ (z) is seen to be that is to say -n ^ <*J + „,-« Jn (Z)\ = - *.-«/„ (*), dz ) and so wo have Bessel's differential equation* A) ^'Q + ^W + v-^J.^o. az~ az The analysis i.s simplified by using ihi; oponitur ^ denned as z(d\d.z\ Thus the recurrence formulae are (S + a) Jn (Z) = zJw (Z), (Sr - W + 1) ,/"„_, («) = - Zjn (z), and ho (^ - n + 1) [2~l (^ + ;«) ./„ (z)] = - ^,/u (a), that is an<l the equation reduces at once to Bcssel's (Mjuation. Corollary. The .same differential equation i.s obtained if',/,,.,.,(.:) i.s eliminated from the formulae D + H+l) ./n + 1 {Z) = Z,ln B), (.9 - ») ./„ B)= -*J1I+ ,B). 2'2. Bessel'.s integral for the Bessel coefficients. Wo shall now prove that A) -/„ 0) = .-,1 I "' cos {n6-z sin 0) rf^. This cfjuation waa taken by Bessolf as the definition of Jn (z), and he derived the other properties of the functions from this definition. * Berliner Abh. 182J.[publi«lu'd 1820], p. H4; see also Frullani, Mem. Soc. ItaL.(Modeaa), xxm. AH20), p. 504. f Ibid. pp. 22 und 35. 2—2
2Q THEORY OF BBSSEL FUNCTIONS [CHAP. II It is frequently convenient to modify A) by bisecting the range of in- integration and writing 2ir—8- for 6 in the latter part. This procedure gives 1 f* (o\ Jn {z) = - cos {nd - z sin 6) dd. W 7T Jo Since the integrand has period Irr, the first equation may be transformed into C) Jn{z)-n~ \ cos (rtd — z sin 6) dd, v ' . 67? J a where a is any angle. To prove A), multiply the fundamental expansion of §21 A) by t~n~l and integrate* round a contour which encircles the origin once counterclockwise. We thus get The integrals on the right all vanish except the one for which w = n; and so we obtain the formula Take the contour to be a circle of unit radius and write t = e~i0, so that 0 may be taken to decrease from 2n + a to «. It is thus found that E) Jn{z)** ^- I ^V"*-"*') d6, a result given by Hansenf in the case a = O. In this equation take a = — tt, bisect the range of integration and, in the former part, replace 0'by —6. This procedure gives and equation B), from which A) may be deduced, is now obvious. Various modifications of Bessel's integral are obtainable by writing 1 f* 1 /'"¦ Jn (z) = — I cos nd cos (z sin d) dd + - sin nd sin (z sin 0) dd. TT JO ^ JO If 6 be replaced by tt- ^ in these two integrals, the former changes sign when n is odd, the latter when n is even, the other being unaffected in each case ; and therefore Jn {z) — — sin nd sin (z sin d) dd j F) Vrl («0dd)> 7T sin nd sin (^ sin 6) dd * Term-by-term integration ia permitted because the expansion is uniformly convergent on the contour. It is convenient to use the symbol J<a+) to denote integration round a contour encircling the point a ouce counterclockwise. t Ermittehuig der ahsoluten Stirrungen (Gotha, 1843), p. 105.
2-21] THE BUSSEL COEFFICIENTS 21 1 <<jr ~ I cos n$ cos (z sin 6) d6 I "I1" J 0 G) y° ^ (»even). = — / cos w# cos (z sin 0) eft? If 6 be replaced by \tr— jj in the latter parts of F) and G), it is found that 2 f*"" (8) Jn (z)= -(-)*(»-1) cos in] sin (z cos rj) drj (n odd), T Jo 2 r ^ir (9) «/M(s) =-(-)*« cos m?7 cos(z cos97) c??7 (?i even). The last two results are due substantially to Jacobi*. [Note. It was shewn by Parseval, Mem. des savans Strangers, I. A805), pp. 639—648, that 1 /•«• •... = I cos (a sin x) dx, IT J 0 ?2. 4a 22. 42 6Z and ho, in the special caBe in which « = 0, B) will be described as Parseval's integral. It will be seen in § 2*3 that two integral representations of Jn (z), namely Bessel's integral and Poisson's integral l>ecorno identical when n—O, so a special name for thi« case is justified.] The reader will find it interesting to obtain (after Bessel) the formulae § 2'12 (I) and S 2-12 D) from Bessol's integral. 2*21. Modifications of Parseval's integral. Two formulae involving definite integrals which aro clo.sely connected with Parseval's integral formula are worth notice. The first, namely A) t/o {\/{z2 — '/a)f — - I #''coa cos B sin 6) dB, IT J (I is due to Bcsael +. The simplest method of proving it is to write the expression on the right in the form 1 f* / /ill COB 0 \-lZ Bill 0 (JA lir J -n expand in powers of y cos d + iz sin 6 and use the formulae j.n1GH u>1 ' J-tt '/COfcJ u>i r(«+i) v the formula then follows without difficulty. The other definite integral, due to Catalan \, namely B) Jo Bi s/z) = - / " e«i-i-«) cob 0 COH {A - z) Hin 0} dd, ir J 0 ia a special caso of A) obtained by .s\ibstitutitfg 1 —2 and 1 +z for 2 and y respectively. * Journal fUr Math. xv. A336), pp. 12—18. [Gen. Math. Werke, vi. A891), pp. 100—102]; the integrals actually given by Jacobi had limits 0 and v with factors 1/w replacing the factors 2/w. See also An^er, Neueste Schriften der Naturf. Ges. in Danzig, v. A855), p. 1, and Oauohy, Comptes Ite.ndus, xxxvui. A854), pp. 910—913. + Berliner Abh., 1824 [published 1826], p. 37. See sIho Anger, Neueate Schriften der Naturf. Ges. in Danzig, v. A855), p. 10, and Lomrael, Zeitschrift filr Math, mid Phys. xv^A870), p. 161. % Bulletin de VAcad. R. de Belgique, B) xli. A870), p. 988.
22 THEORY OF BESSEL FUNCTIONS [CHAP. II Catalan's integral may be established independently by using the formula l 1 [to+) m! 2*u J m so that m »m r @ +) 2 so gin 1 m »m r by taking the contour to be a unit circle; the result then follows by bisecting the range of integration. 2'22. Jacobi's expansions in series of Bessel coefficients. Two 'series, which are closely connected with Bessol's integral, were dis- discovered by Jacobi*. The simplest method of obtaining them is to write t = ±ei$ in the fundamental expansion § 2*1 C). We thus got CO tt=l ~ = Jo (*) + 2 2 J2n(z)cos Zn6 ± 2i 2 J.ill+1 {z) sin Bn. + 1H. n=l n=0 On adding and subtracting the two results which are combined in this formula, we find 00 A) cos(zsin6) = Jo(z) + 2 2 Jm(z) cos B) sin 0 sin 0)= 2 2 Jm+l(z)xi Write \tt — 7/ for 6, and we get C) COS (Z COS 7]) = Jo (z) + 2 2 (-)n Jon (z) COS 2/G7, GO D) sin (z cos 97) = 2 2 (—)" «/2n+I B) cos B» + 1)?;. The results C) and D) were given by Jacobi, while the others weni obtained later by Angert. Jacobi's procedure was to expand com B cos 7) and sin (z t:os 17) into a series of coBines of multiples of rj, and use Fourier's rule to obtain the coedficionts in the form of integrals which are seen to be associated with Bessel's integrals. In view of the fact that the first terms in A) and C) arc nob formed according to the same law as the other terms, it is convenient to introduce Neumann's factor I en, which is defined to be equal to 2 when a is not zero, and to be equal to 1 when n is zero. The employment of this factor, which * Journal filr Math. xv. A836), p. 12. [Ges. Math. U'erke, vi. A891), p. 101.] t Ncueute Schriften der Naturf. Gvs. in Danzig, v. A855), p. 2. X Neumann, Theorie der BesseVschen Functianen (Leipzig, 18G7), p. 7.
2*22] THE BESSEL COEFFICIENTS 23 will be of frequent occurrence in the sequel, enables us to write A) and B) in the compact forms: GO E) cos B sin 6) = 2 em J2n {z) cos 2nd, 00 F) sin B sin 6) = X e2n+1 Jm+1 B) sin Bn +1) $. If we put 6 = 0 in E), we find (') I — Z emjm{2). If we differentiate E) and F) any number of times before putting 6 — 0, we obtain expressions for various polynomials as series of Bessel coefficients. We shall, however, use a slightly different method subsequently (§ 2'7) to prove that zm is expansible into a series of.Bessel coefficients when m is any positive integer. It is then obvious that any polynomial is thus expansible. This is a special case of an expansion theorem, due to Neumann, which will be investi- investigated in Chapter xvr. For the present, we will merely notice that, if @) be differentiated once before 6 is put equal to 0, there results (8) 2= § ea»+iB/i + l)./an+100, n-o while, if 0 be put equal to lir after two differentiations of E) and (C), then @) 2 sin 2 = 2 |22 J, (z) - 42 J, (z) + Gu J, (z) - ...}, A0) z com z = 2 11a J, B) - 3a ./„ \z) + 5- Jt B)-...}. These results are due to Lommel*. Notk. The expression exp \\z(t- l/t)\ introduced in $ 2-1 is not a generating function in the strict .Mcn.se. The generating functiont associated with enJn(z) is 2 fui"i/n(^)- If thi.s expression l>e called A', by using the recurrence formula $ 2*12B), wo have ?-JH)*+i(iH7 If we .solve thi.s differential ('([nation we get I / I \ f~ A1) • =es +2 [t+ ty- j j> '«Wu A result equivalent to this was given hy Brenke, Bull. American Math. Sac. xvi. A910), pp. 2ii.r;—230. * Studlen ilher di<< Be.tseVnc.hen Functional (Leipzig, 1868), p. 41. t It will be seen in Chapter xvi. that this is a form of " Lommcl's function of two variables."
24 , THEOEY OP BESSEL FUNCTIONS [OTIAP. II 23. Poisson's 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) sinm+1 Odd, \ cos (z cos 6) sinan 0*M< o Jo Avhere n is a positive integer or zero. He proved that these integrals are solutions of certain differential equations^ and gave the investigation, which has already been reproduced in § 1, to determine an approximation to the latter integral when z is large and positive, in the special case d — 0. We shall now prove that [' Jo and, in view of the importance of Poisson's researches, it seems appropriate to describe the expressions on the right§as Poisson's integrals lor Jn (c). In tho case n= 0, Poisson's integral reduces to Parseval's integral (§ -"-). Ifc is easy to prove that the expressions under consideration aiv (Mjual to Jn(z)] for, if we expand the integrand in powers of j and then integrate term-by-term||, we have - 5 ^—r [corf"" ^siir" 0<W 7Tm=0 Bm)! Jo ^2m 1^3. 5...B/; - l).l .:5..r) ... and the result is obvious. * Journal de I'ficole R. Poly technique, xn. (cahier 19), A823), pp. '249—10:5. t Ibid. p. 293, «t seq.; p. 340, et seg-. Integrals equivalent to thorn had proviouHly l«'»n examined by Euler, List. CaZc. I7(«. i/. (Petersburg, 1769), Ch. x. S 1036, but 1'oinmm'H foniiH ant more elegant, and bis Btudy of them i8 more eyBtematic. See also § ,'J\S. J E.g. on p. 300, he proved that, if R =rn+1 I cos (rp eos w) sin271 w </w, Jo then B satisfies the differential equation dra ,.2 Ji ~ ~ ^ «¦ § Nielsen, Handbuch der Theorie der Gylinderfunktionen (Leipzig, 190-i), p. 51, oallH them Basel's second integral, but the above nomenclature seems preferable. II The series to be integrated ia obviously uniformly convergent; the procedure adopted is duo to Poisson, ibid. pp. 814, 340.
2-3, 2-31] THE BESSEL COEFFICIENTS 25 Poisson also observed* that j eizam0 sin2n 0d6 = f * cos (z cos 6) sin2Jl Odd; JO JO this is evident when we consider the arithmetic mean of the integral on the left and the integral derived from it by replacing 6 by tt — 6. We thus get B) J (z\ = -*' pizcosO qin A slight modification of this formula, namely has suggested important developments (cf. § 61) in the theory of Bessel functions. It should also bo noticed that fir n-ir D) cos(seoH0)sin2n0d0 = 2 co Jo Jo = 2 cos(>sin0)cos2n0cZ0, jo and each of those expressions gives rise to a modified form of Poisson's integral. An interesting application of Besael's and Poisson's integrals was obtained by Lommelf who multiplied tho formula = 1 (-) ¦ - - ~«--x-,—-— -—smim6 Bnt)! by iH)H(zci)Hd) and integrated. Ii^thua follows that 231. Kes&el'x invedigutum of Poisson's integral. 'Vho. proof, i.hat ,/„ (z) is ocjual to Poisson's integral, which was given bj' BcHHcl.}, in HOinewhai/ elaborate; it is substantially as follows: it is scon on diltbrentiation that ' , cos 6 sin-" 6 com (z cor 0) — n " , »in8M+I 0 sin (z cos 6) A0 \_ J 2n + I J = Bm - 1) .sin8"-1-1 6 - In sin*1 ^ + 0~- —1 sinan+2 <9 cos (z cos 0), * PuiHKon ftfitimlly mado the statement (p. 293) concerning the integral whioU contains Hina"'¦' 0; but, as ho pointn out on p. MO, odd powers may be replaced by oven powers throughout hiH unalysiH. •j- Studien ilber die Iie»geV»rhen Functianen (Leipzig, 1868), p. 30. + lirrliner Abh. 1824 [published lH2fi], pp. 30—1O. Jacobi, Journal filr Math. xv. A886), p. 13, [Gen. Math. Werke, vi. AS(.J1), p. 102], when giving his proof (§ 2-32) of Poisson's integral formula, objected to Hie artificial character of Bg8ho1'b domonfltration.
26 THEOEY OF BESSEL FUNCTIONS [CHAP. II and hence, on integration, when ?i > 1, Bn -1) I"*" cos (z cos 0) sin291 Bdd - 2n \ cos (z cos 0) sin™ 6d0 Jo Jo + , z _ ("cos@cos0)sinBl+a0^0 = 0. ZW + I J o If now we write iifZ1 f ™* /• rtrt«» /)\ sinm Q^Q - ^ (?,); the last formula shews that z<f) (n - 1) - 2ncj6 (to) + s<f> (n + 1) = 0, so that <jb(?i) and «/n(^) satisfy the same recurrence formula. But, by using Bessel's integral, it is evident that 6 (I) = - (""" cos (z cos 0) sin2 6dd = -^ I' -%. {sin {z cos 0I sin ¦jrJo ttJo fto ( ] If". = - sin {z cos 0) cos Odd = — t/0' (z) = </, (^), ttJ o and so, by induction from the recurrence formula, wo have 4>{n) = Jn(z), when n = 0,1, 2, 3, .... 2'32. Jacohi's investigation of Poissons integral. The problem of the direct transformation of Poisson's integral into Bossel's integral was successfully attacked by Jacobi*; this method necessitates the use of Jacobi's transformation formula dn-i BJnan-! Q 1.3.5... B?i - 1) . Q ¦—-j-rri—=(-)n~1 --- 'sinw^, where /u. = cos 0. We shall assume this formula for the moment, and, .since no simple direct proof of it seems to have been previously published, we shall give an account of various proofs in §§ 2*321—2-323. If we observe that the-first n~ 1 derivates of A — /x8)'1"*, with inspect to fi, vanish when /u,= ± 1, it is evident that, by n partial integrations, we have zu I * cos (z cos 0) sinm Odd = zn ! cos (zft). A - /Aa)tt-l dp J0 J -1 J l COS EM - * Journal fiir Math. xv. A8S0), pp. 12—13. [Ges. Math. Werke, \i. A801), pp. 101—102.] See also Journal de Math. i. A836), pp. 195—196.
2-32, 2-321] THE BESSEL COEFFICIENTS 27 If we now use Jacobi's formula, this becomes cos (za — = 1.3.5... Bn — 1I cos (z cos 0 - %mr) cos ndcW Jo ()() by Jacobi's modification of § 22 (8) and (9), since cos(scos0 — ?n7r) is equal to (—)*'* cos {z cos 6) or (-i)**11) sin B cos 0) according as n is even or odd; and this establishes the transformation. 2*321. Proofs of JacohPs transformation. Jacobi's proof of tho transformation formula used in § 2'Zi consisted in deriving it as a special case of a formula due to Laoroix*; but the proof which Laeroix gave of his formula is open to objection in that it involves the use of infinite scries to obtain a result of an elementary character. A proof, based on the theory of linear differential equations, wan discovered by Liouville, Journal de Math. vi. AB41), pp. 69—73; this proof will be given in § 2-U22. Two yearn after Liouville, an interesting symbolic proof was published by Boole, i'anib. Math. Journal, in. AB43), pp. ^IC—22-1. xVn elementary proof by induction wan given by Gruniirt, Arc/do dor Math, und Plum. iv. A844), pp. 104— 109. This proof consists in shewing that, if tlion B,,,., — A — u2) -- ' — yjtuG,, — n Ui — \) I 9,. da. <? J i and that ( - )" "' 1 . li. f> ... ('¦In — I) (win nS)jn satisfies the same recurrence formula. Other proof's of this character have boon given by Todhunter, Differential Calculus (London 1H71), Oh. xxvm., and Orawford r, Proc. ICdinburgh Math. AW. xx. A902), pp. 11 IT), but all these proofs involve complicated algebra. A proof depending on the use of contour integration in due to Scbliifli, Ann. di Mat. B) v. (IK73), pp. 201—202. The contour integrals arc of the. type used in establishing Lagrange'.s expansion; and in vj ii*3lW we shall give the modification of Schliifli's proof, iii which the use of contour integrals is replaced by a use of Lagrauge's expansion. To prove Jambi'n formula, differentiate by Leibniz' theorem, thus: - 2 ( - )'" >iuC,m , i (sin \0)-m '1 (cos kdf'1'-- ~ ' m (i • sin ('In x kfl), and this is tin1 transformation required];. "¦ Tntitr du Ciilr. Dijl'. i. (Paris, IKK), 2nd edition), pp. 1H2—1HU. Sen also a note written by Catalan in 1H0H, MtUu. dc la Soc. It. den Sri. dr. Urge, B) xn. A8H.r)), j>p. IJ12—31tt. I Crawford iittribntoK tin; formula to Kodriguen, posHibly in couHotjuenoc of an incorrect state- statement by I'rdiK't, Herne11 d' Kxercivra (I'ai-iH, lK<iG), p. id, that it i« ^iven in EodriguoB' diBsertation, Currrxp. mi- PKcolc It. Poll/technique, in. AK14—1H1C), pp. :ilil—:iH.r). :[ I owe this proof to Mr C. T. I'roeco.
28 THEORY OF BBSSEL FUNCTIONS [CHAP. II 2*322. LiouvUle's proof of Jacobtfs transformation. The proof given by Liouville of Jacobi's formula is as follows: (l~fi2)w~* and let D be written for dfdfn then obviously Differentiate this equation n times; and then A - j*2) D>l+1y-fiD»y+n*Dn-1y=O: so that (? + iA Z>"-1?/=0. Hence * Dn~1y=A sn\n9 +Bcoan9> where A and D are constants ; since Dn~1y is obviously an odd function of 6, B is zero. To determine A compare the coefficients of 6 in the expansions of Dll~1y and A sinjitf in ascending powers of 0. The term involving 6 in Da~ly is easily seen to be ^"-M-)"-1 B»- 1) B»- 3) ... 3.1. A so that wyl = (-)»-1l .3.5... Bn-l), and thence we have the result, namely d*-isin«»-i0 .1.3.5...Bw- ()"~l 2*323. ScklqflCs proof of Jacobi's Iran* formation. We first recall Lagrango's expansion, which is that, if z~/x + kf (z), then *(•) = * 00 + | ? ^ so that </,' («) ^ = 1 ^ ^1 subject to the usual conditions of convergence*. Now take / («)a ~fc A -«•), 0'(«Ks/(l-a«), it being supposed that </>' B) reduces to ^A -/a2), i.e. to sin 6 when A-»-0. The singularities of z qua function of h are at A = e±ffl; and so, when 6 is real, tho ex- expansion of a/A — s2) in powers of h is convergent when both | h \ and | z \ are less than unity. NOW 2 = {l-v'(l I \n-l (jjn - 1 ojn 27i - 1 Q Hence it follows that ¦ \ ¦; ^r-r = = is the coefficient of A"-1 in tho ex- 2n.(n-l)! dfin~l pansion of *j{\ — z1). (Cz/d/i) in powers of h. But it ia evident that in A dz (l~heie)-i-(l-he-i6)^ ? 1.3.5... B^-1) enie - ni0 in A ? and a consideration of the coefficient of A11 in the last expression establishes the truth of Of. Modern Analysis, § 7*32.
2-322-2-33] THE BESSEL COEFFICIENTS 29 2*33. An application of Jacobi's transformation. The formal expansion r fir oo / (cos x) cos vxdx — 2 (—)mamf{n+m) (cos x) dx, .1A iiikH / . 0 .'0 j>i-0 in which <xm is the coefficient of Pl+im in the expansion of Jn (t)/J0 (t) in as- ascending powers of t, has been studied by Jacobi*. To establish it, integrate the expression on the left n times by parts; it transforms (§ 2'32) into W (cos m) sin and, when sin2na; is replaced by a series of cosines of multiples of x, this becomes 1 /¦"¦ ,. w x T- 2» ft 2n(rt-l) , "I 7 Q , ,. 7H—: /(n) (cos «) 1 —7 cos 2« + ¦'- -—y--r cos 4a; - ... dx. 2.4.f)... Bn).'o L " + 1 (n+l)(n+2) J We now integrate/(n)(cosa;) cos 2tf, /"" (cos a;) cos 4>x, ... by parts, and by continual repetitions of this process, we evidently arrive at a formal expansion of the type stated: When /(cos a;) is a polynomial in cos#, the process obviously terminates and the transformation is certainly valid. To determine the values of the coefficients a,,^ in the expansion I /(cos x) cos nxdx =f S (-)mam/"i+9in) (cos as) dx thiiH obtained, write /(cok a;) = (-)*" cos (t cos x), (—)*'«-J> sin (t cos x), according as n is even or odd, and we deduce from § 2-2 (8) and (9) that ./„(«)- 2 (-ra^*" {(-)»¦/,@}, so that am has the value stated. It has been stated that the expansion is valid when /(cos x) is a poly- polynomial in cok;/:; it can, however, be established when /(cos x) is merely re- restricted to be an integral function of cos x, say J bn co.s" x v provided that lirn a/| btl \ is less than the smallest positive root of the equation J0(t) = 0 ; the investigation of this will not be given since it seems to be of no practical importance. # Journal Jilr Math. xv. AHH6), pp. 25—2E [Ges. MaiA. IFerfce, vi. A891), pp. 117—118]. See uluo Jacobi, A«tr. Narh. xxvin. AB49), col. 94 [Ges. Math. Werke, vn. A891), p. 174].
30 THEORY OF BESSEL FUNCTIONS [CHAP. II 24. 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 Lommeli", is (l) Jn(y + *)« 2 J in- -oo The simplest way of proving this result is from the formula § 2*2 D), which gives 1 f@+) Jn (# + *) = 21 r"~1 e*l2/+Z1 tt'*' dt ¦|0+) oo t Jm.{y)\ V"-*-*eM m^-a> J oo = 2 Jm {y) Jn~m (z), TO- -co on changing the order of summation and integration in the third line of the analysis; and this is the result to be established. Numerous generalisations of this expansion will be given in Chapter xr. 2'5. Hansen's series of squares and products of Bessel coefficients. Special cases of Neumann's addition formula were given by Hanson J as early as 1843. The first system of formulae is obtainable by squaring the fundamental expansion § 21 A), so that By expressing the product on the right as a Laurent series in t, and equating the coefficient of tn in the result to the coefficient of tn in the Laurent ex- expansion of the expression on the left, we find that In particular, taking n = 0, we have§ A) JoBs) = Jf (z) + 2 ? (-Y Jr»(z) = ? (-)'• er J*(z). * Theorie der BcsseVgchen Functionen (Leipzig, 1867), p. 40. t Studienilber die BesseVschen Functionen (Leipzig, 1868), pp. 26—27; see also Sohlafli, Math. Ann. in. A871), pp. 135—137. t Ennitteluvg der dbsoluten St'drungen (Gotha, 1843), p. 107 et seq. Hansen did not give D), and he gave only the special case of B) in which n=l. The more general formulae are due to Loramel, Studien ilber die BesseVschen Functionen (Leipzig, 1868), p. 33. § For brevity, J,,2 (z) is-written in place of {Jn («)}2.
2-4-2-6] THE BESSEL COEFFICIENTS 31 From the general formula we find that B) Jn B*) - 2 Jr (z) Jn-r (*) + 2 2 (~)r Jr (z) Jn+r (z), when the Bessel coefficients of negative order are removed by using § 2*1 B). Similarly, since {rJ JrM*)}{j2 JL-y*v*J»(*)} = exp [\z (t - 1/0} exp [\z (- i + 1/0] it follows that 00 D) 2 (-)'1 Jr (Z) J*n-r (*) + 2 2 Jr (z) Jm+r (z) = 0. Equation D) is derived by considering the coefficient of tin in the Laurent expansion ; the result of considering the coefficient of tm+1 is nugatory. A very important consequence of C), namely that, when x is real, E) IJ.^I^l, |./r(*Oh where r = 1, 2, 3, ..., was noticed by Hansen. 2. Neumann's integral for </na (z). It is evident from § 2*2 E) that and so JH* (z) = - - -¦ I"" I eni^ ¦» e-wisi»0H-si" +j d0dcji. To reduce this double integral to a single integral take new variables defined by the equations 0 2 0f 2 ? % f so that It follows that " V ' 27T* where the field of integration is the square for which Since the integrand is unaffected if both y and ylr are increased by w, or if y is increased by it whilo yfr ia Himultaneously decreased by ir, the field of inte- integration may evidently be taken to be the rectangle for which 7T.
32 THEORY OF BESSEL FUNCTIONS [ <%" A I1. 11 Hence THEORY V J 27T _ 1 IT-. OF ¦!' 2io J Jo BESSEL 1 0ini\jj—a i-rr an, v,^^ COS FUNCTIONS izsin*cosx c^^ If we replace % by ?71- + 0, according as^ is acute or obtiiNr. we nbt.-tiit th» result J) ^n2(*) = - f** Jm Bz sin 6) dO. This formula may obviously be written in the form 2> Jn {z) = - f V^ B* si which is the result actually given by Neumann*. It was <lt-rivc-tl by him t»y some elaborate transformations from the addition-theorem winch will"},, ,av,-n v.Z- F°Of Which has Just been Siven is sugcttHlecl l>v l.h.« i»r,.«.f ..f that addition-theorem which was published by Graf and CJublci-f JVe obtain a different form of the integral if we-pcribrin Mir mt,Kr:i.i,.,, *lth r6SpeCt t0 * msfcead of wi^ respect to +. This pruccchuv Kiv,.s /n2 (*)»" in- ( «^o B« sin -Jr) e2"^ ^, so that J-'r C) Jn* (Z) = — I ^ jo ^z sb -1 f* - ~ Jo •/. B« sin f) a result which SchlafliJ attributed to Neumann. 2'61. Xeumunn's series for J 2 (*)
2'61, 2-7] THE BESSEL COEFFICIENTS This result was written by Neumann in the form 33 where B) 271 7",= w + 1) Bn + 3) Bn + 2) (In + 4)' ,p B7i + 1) Bw + 3) Bn + 5) This expansion is a special case of a more general expansion (due to Schlafli) for the product of any two Bessel functions as a series of powers with comparatively simple coefficients (§5-41). 27. JSchlomilch's expansion of zm in a series of Bessel coefficients. Wu shall now obtain the. result which was .foreshadowed in §2*22 con- concerning the expansibility of zm in a series of Bessel coefficients, where m is any positive integer. The result for in = 0 has already been given in §2*22 G). In the results §2-22A) and B) substitute for cos 2nd and sin Bn + 1N their expansions in powers of sina 8. These expansions are* ? l The results of substitution are (.,,q /> oiT, A)\ - / /"*\ 4. 9 V f sin (z sin ^) = ^ J.,tl+, h J JJ' B sin ,).,j , Tf we rearrange the series on the right as power series in sin 6 (assuming that it is permissible to do so), we have 2 (-)• B sin = i, --~ -¦¦- :, (n-s)\ * Cf. Hobson, Plane Trigonometry A918), §§80, 82. "W. IJ. I<\
34 THEORY OF BESSEL FUNCTIONS [CHAP. II If we expand the left-hand sides in powers of sin 0 and equate coefficients, we find that n=l _ ? The first of these is the result already obtained; the others may be com- combined into the single formula A) (?s)m = 2 * ^ - /TO+2rl D (m = 1, 2, 3, ...) , The particular cases of A) for which w = 1, 2,3, were given by Schlomilch *. He also shewed how to obtain the general formula which was given explicitly some years later by Neumann f and LommelJ. The rearrangement of the double series now needs justification; the rearrangement ia permissible if we can establish the absolute convergence of the double series. If we make use of the inequalities in connexion with the series for sin (z sin S) we see that oo I 9 oJn a [28 + 1 2 1 sin g ' . | ^ |28 +1 exp s=o Bs + l)! '2 ' t = siuh (| z sin 01) exp (^ 12 j2), and so the series of moduli is convergent. The series for cos (z sin 6) may be treated in a similar manner. The somewhat elaborate analysis which has just been given is avoided in Lommel's proof by induction, but this proof suflPers from the fact that it is supposed that the form of the expansion is known and merely needs verifica- verification. If, following Lornmel, we assume that \2) _ ^1 * Zeitschriftfilr Math, und Phys. n. A857), pp. 140—141. f Theorie der BesseVschen Functionen (Leipzig, 1867), p. 38. X Studien ilber die BesseVschen Functionen (Leipzig, 1868), pp. 35—36. Lommel's investigation is given later in this section.
2-71] THE BESSBL COEFFICIENTS 35 [which has been proved in §2*22 (8) in the special case m — 1], we have l (m+n—l)\) L+ } + 2n).(m + n)\ r ! »=0 Since (m + n)\ Jm+2n {z)fn\-*~ 0 as n -*- oo , the rearrangement in the third line of the analysis is permissible. It is obvious from this result that the in- induction holds for m = 2, 3, 4, An extremely elegant proof of the expansion, diu> to A. 0. Dixon*, is as follows:— Let t be a complex variable and lot u Iks defined by the equation u=.. —;,, no that when t describes a small circuit round tho origin (in.sido the circle 11 \~ 1), u does the wame. We then have 7I ¦ f *(l' * = 2^TT—z- I oxl when we calculate the sum of the residues at the origin for the last integral; the inter- interchange of the order of summation and integration is permitted because the series converges uniformly on the contour; and the required result is obtained. F-Ki tin • 1 duni . ; , , , , d lot»- u ~1 Notic. When m is zero, -¦• , has to be replaced by - - ,° . 1 titi (to (X f I 2 1. ScJdihnilch's ex^)a7isions of the type 2?i''«/M (-)• The formulae A) 2 Bnfr>J^{z)^ I /'I*'"*-, d') S (t&nJr lY^'"* Jn (z)— 2 /J ''^ z'im +' n=o m—(i in which p is any positive integer [zero included in B) but not in A)] and 1^ i.s a numeri- numerical coefficient, are evidently very closely connected with the results of § 2-7. The formulae * Messenger, xxxn. A903), p. 8; a proof on the Banie lines for the case ?«=1 had been pre- previously given by Kapteyn, Nieuw Archie/ voor Wiskunde, xx. A893), p. 120. 3—2
36 THEORY OF BESSEL FUNCTIONS [CHAP. II were obtained by Schlomilch, Zeitsohrift fiir Math,, und Phys. 11. A857), p. 141, and he gave, as the value of where wCfc is a binomial coefficient and the last term of the summation is that for which h is bn -1 or ?(m-1). To prove the first formula, take the equation § 2-22 A), differentiate 2p times with respect to 8, and then make 6 equal to zero. It is thus found that o Bm)! J^o" The terms of the series for which m > p, when expanded in ascending powers of 6, contain no term in tf2*, and so it is sufficient to evaluate o Bm)! J6=o m= m=0 since terms equidistant from the beginning and the end of the summation with respect to k are equal. The truth of equation A) is now evident, and equation B) is proved in a similar manner from § 2*22 B). The reader will easily establish the following special cases, which were stated by Schl5milch: 113 Jx<Ls) + 33 Jz v2.3.4J3B) + 4. 5. 6J6(«)-i-6. 7. 8^7(«)+. .. = ^3. 2*72. Neumann's expansion of z* as a series of squares of Bessel coefficients. From Schlbmilch's expansion (§ 27) of z*71 as a series of Bessel coefficients of even order, it is easy to derive an expansion of zim as a series of squares of Bessel coefficients, by using Neumann's integral given in § 2-6. Thus, if we take the expansion O) 2 and integrate with respect to $, we find that so that (when m > 0) Bm)! ?0
2*72] THE BESSEL COEFFICIENTS This result was given by Neumann*. An alternative form is 37 and this is true when m = 0, for it then reduces to Hansen's formula of § 2'5. As special cases, we have C) = ~ 2 ett. 4M2 D^ - 22) Jn2 (z), •5 . 4 ,i=2 ' f' fH • 4«2 D»ia - 22) n* B), 4. 5.6 n= If we differentiate A), use § 2T2 B) and thou rearrango, it is readily found that an expansion whoac exiatenco was indicated by Neumann. * Leipziger Berichte, xxi. A869), p. 226. [Math. Ann. in. A871), p. 585.]
CHAPTER III BESSEL FUNCTIONS 3*1. The generalisation of Bessel's differential equation. The Bessel coefficients, which were discussed in Chapter II, are functions of two variables', z and n, of which z is unrestricted but n has hitherto been required to be an integer. We shall now generalise these functions so as to have functions of two unrestricted (complex) variables. This generalisation was effected by Lommel*, whose definition of a Bessel function was effected by a generalisation of Poisson's integral; in the course of his analysis he shewed that the function, so defined, is a solution of the linear differential equation which is to be discussed in this section. Lommel's definition of the Bessel function Jv (z) of argument z and order v wasf B cos e) ^ m< and the integral on the right is convergent for general complex values .of v for which R(v) exceeds —\. Lommel apparently contemplated only real values of v, the extension to complex values being effected by Hanked; functions of order less than — \ were'defined by Lommel by means of an ex- extension of the recurrence formulae of § 2'12. The reader will observe, on comparing § 3 with § 1*6 that Plana and Poisson hadinvestigated Bessel functions whose order is half of an odd integer nearly half a century before the publication of Lommel's treatise. We shall now replace the integer n which occurs in Bessel's differential equation by an unrestricted (real or complex) number^ v, and then define <i Bessel function of order v to be a certain solution of this equation; it va of course desirable to select such a solution as reduces to Jtl(z) when v assumes the integral value n. We shall therefore discuss solutions of the differential equation which will be called Bessel's equation for functions of order v. * Studien iiber die BesseVschen Fwictionen (Leipzig, 1868), p. 1. + Integrals resembling this (with v not necessarily an integer) were studied byDuhamel, Coma cCAnalyse, ii. (Paris, 1840), pp. 118—121. % Math. Ann. i. A869), p. 469. § Following Lommel, we use the symbols v, y. to denote unrestricted numbers, the symbols n, m being reserved for integers. This distinction iB customary on the Continent, though it has not^yet come into general use in this country. It has the obvious advantage of shewing at a glance ¦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 A) which is valid near the origin; the form assumed for such a solution is a series of ascending powers of z, say y= 2 cmS where the index a and the coefficients cm are to be determined, with the pro- proviso that c0 is not zero. For brevity the differential operator which occurs in A) will be called V,, so that B) vrS5. *+*? + ,._,* dz- dz It is easy to see that* V, 2 cmza+m = 2 cm[(a + my-vl}g*+m + 2 cmz«+w+\ The expression on the right reduces to the first term of the first series, namely c{) (<xa — v~) za, if we choose the coefficients cm so that the coefficients of corresponding powers of 2 in the two series on the right cancel. This choice gives the system of equations /c, {(a + 1)9-!/8} =0 c,\(a + 2y-v~\+c0 =0 c!,j(a+3)a-vaj-r-e, =0 C) Cm {(a + niy -1/"} + Ck-2 = 0 If, then, these equations are satisfied, we have D) V, X cw3ft+m=Cb(a»-i/J)*\ wt - 0 From this result, it is evident that the postulated series can be a solution of A) only if a= ±p; for c0 is not zero, and za vanishes only for exceptional values of z. Now consider the with equation, in the system C) when wi> 1. It can be written in the form cm (a — v + m) (a+i/ + m) + cm_u = 0, and so it determines cm in terms- of cTO_2 for all values of m greater than 1 unless a— v or a+ v is a negative integer, that is, unless — 2v is a negative integer (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), and then (a + m)a — i/* does not vanish when m = l, 2, 3, .... It now * When the conutants o and cm have been determined by the following analysis, the series obtained by formal processes jh easily Been to be convergent and differentiate, so that the formal procedure actually produces a aolution of the differential equation.
40 THEORY OF BESSEL FUNCTIONS [CHAP. Ill follows from the equations C) that ol — ci = cs-= ... =0, and that cm is ex- expressible in terms of c0 by the equation <*» (a~ v + 2)(a- v + 4>) ... (oi-v The system of equations C) is now satisfied; and, if we take a = v, we see from D) that is a formal solution of equation A). If we take a = — v, we obtain a second formal solution In the latter, c0' has been written in place of c0, because the procedure of obtaining F) can evidently be carried out without reference to the existence of E), so that the constants c0 and c0' are independent. Any values independent of z may be assigned to the constants c0 and c</ '> but, in view of the desirability of obtaining solutions reducible to Jn {z) when v -*• n, we define them by the formulae* G) Co==2'T(i; + l)> Cd' = 2-T(-i/ + 1)' The series E) and F) may now be written In the circumstances considered, namely when 2v is not an integer, these series of powers converge for all values of z, {z = 0 excepted) and so term-by-term differentiations are permissible. The operations involved in the analysisf by which they were obtained are consequently legitimate, and so we have obtained two solutions of equation A). The first 'of the two series defines a function called a Bessel function of order v and argument z, of the first kind%; and the function is denoted by the symbol Jv (z). Since v is unrestricted (apart from the condition that, for the present, 2v is riot an integer), the second series is evidently J_v (z). Accordingly, the function Jv(z) is defined by the equation (8) J<~\- 1 ()m(i^+2m It is evident from § 2'11 that this definition continues to hold when v is a positive integer (zero included), a Bessel function of integral order being identical with a Bessel coefficient. * For properties of the Gamma-function, see Modern Analysis, eh. xn. + Which, up to the present, has been purely formal. X Fuuctions of the second and third kinds are denned in §§ 3-5, 354, 3-57, 3«6.
3*11] BESSBL FUNCTIONS 41 An interesting symbolic solution of Bessel's equation has been given by Cotter* in the form where D^.djdz while A and B are constants. This may be derived by writing successively [zD - 2v + D~xz] z"y = -! zD (z z- y 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 omitted from consideration, namely (i) when v is half of an odd integer, (ii) when v is an integer^. It will now be shewn that case (i) may be included in the general theory for unrestricted values of v. When v is half of an odd integer, let *» = (r + *)>, where r is a positive integer or zero. If we take a — r +•Iin the analysis of § 31, we find that .lBr + 2) =0, ami so ( \i /\ 0 = __ ___ S LJ!? K } am 2.4... Bm). Br + 3) Br + 5)... Br 4- 2m + 1)' which is the value of c»tn given by § 3'1. when a and v are replaced by r + ?. If we take 1 we obtain the solution which ih naturally denoted by the symbol Jr+i(z), so that the definition of § 3-1 (8) is still valid. If, however, we take a = — r — |, the equations which determine cm become (cl.l(-2r) =0, , > ( } \cm. m (m - 1 - 2r) + c^ => 0. {m }' As before, cx, cs, ...,csr-i are all zero, but tho equation to determine c^+i is ctr?ci ^/m* equation is satisfied by an arbitrary va/we of <ha-+u when to > r, cm+1 is defined by the equation 2m+1 Br + 3) Br + 6)... Bm + 1). 2 . 4 ... B?« - 2r)" * Proc. It. Irish, dead. xivn. (A), A909), pp. 157—161. f The oases combine to form the cane in which 2v is an integer.
42 THEORY OF BBSSEL FUNCTIONS [CHAP. Ill If /„ (z) be defined by § 3-1 (8) when v = -r - \% the solution now con- constructed is* c0 2-^ r A - r) J-r-i 0) + W2** r (r + f) Jr+h (z). It follows that no modification in the definition of Jv (z) is necessary when v = ±(r + ?); the real peculiarity of the solution in this case is that the negative root of the indicial equation gives rise to a series containing two arbitrary constants, c0 and c2r+1, i.e. to the general solution of the differential equation. 3*12. A fundamental system of solutions of BesseVs equation. It is well known that, if yx and y2 are two solutions of a linear differential equation of the second order, and if y-[ and yi denote their derivates with respect to the independent variable, .then the solutions are linearly inde- independent if the Wronskian determinant^ yl yi does not vanish identically; and if the Wronskian does vanish identically, then, either one of the two solutions vanishes identically, or else the ratio of the two solutions is a constant. If the Wronskian. does not vanish identically, then any solution of the differential equation is expressible in the form cx yx + c2 y2 where Cy and ca are constants depending on the particular solution under consideration; the solutions yx and y% are then said to form a fundamental system. For brevity the Wronskian of yx and, y^ will be written in the forms the former being used when it is necessary to specify the independent variable. We now proceed to evaluate OR {J, (*),/_<*)}. If we multiply the equations Vv /_„ {z) = 0, Vr Jv (z) = 0 ^ by Jv (z), J^v {z) respectively and subtract the results, we obtain an equation which may be written in the form * In connexion with series representing this solution, see Plana, Mem. della R. Accad. delle Sci. di Torino, xxvi. A821), pp. 519—538. | For references to theorems concerning Wronskians, Bee Eneyclopedie des Sci. Math. 11. 1& (§ 23), p. 109. Proofs of the theorems quoted in the text are given by Forayth, Treatise on Differential Equations A914), §§72—74.
32] BESSEL FUNCTIONS 43 and hence, on integration, (l) »{./„<*), J-Az)] = -, Z where G is a determinate constant. To evaluate G, we observe that, when v is not an integer, and \z\ is small, we have Jv (*} = r'(^l){l + 0 {z% J: {2)=Pp){l + 0 {z^ with similar expressions for J-V(z) and J'-v(z); and hence j. w .r , w - j-, w/; w=1 \ 2 sin vtr ,, . . = - + 0 (s). 7T2T ' If we compare this result with A), it is evident that the expression on the right which is 0(z) must vanish, and so* A) /¦()) 2sini'7r B) TTZ Since .sin vn is not zero (because v is not an integer), the functions Jv{z), J~v(z) form n fundamental system of solutions of equation § 3 A). i'ii i/ is an integer, «, we have seen that, with the definition of § 2*1 B), and when v is inude equal to —n in § 3'1 (8), we find thai r (z, _ y (-)"(l*y-*+- vSmce the first n terms of the last series vanish, the series is easily reduced to (—)n <l,i{z), so that the two definitions of ./_M(s) are equivalent, and the function.s >fn{z), J-n(z) do nut form a fundamental system of solutions of Vessel's equation for functions of order n. The determination of a fundamental system in this case; will be investigated in § 3E3. To sum up, the function Jv{z) is defined, for all values of v, by the expansion of § IV1 (8); and J,,(z), so defined, is always a solution of the equation V,,y— 0. When v is not an integer, a fundamental system of solutions of this equation is formed by the functions <lv{z) and J-V{z). A generalisation of the Beasel function has been effected by F. li. Jackson in his researches on " ba-sio numbers." Briefly, a bawic number [u] is defined iis--——, where piib the base, and the basic (Umtiiti function r,,(") is defined to .satisfy the recurrence formula The basio BoHHel function \n then defined by replacing the numbers which occur in the series for the BenHel function by basic numbera. It haH been shown that very many theoreniH * This refuilt is due to Loramnl, Math. Ann. iv. A871), p. 104. He derived the value of C by making z -*- oo and using the approximate formulae whioh will be inveatigated iu Chapter vii.
44 THEORY OF BESSEL FUNCTIONS [CHAP. Ill 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 main results are to be found in a aeries of papers, Proc. Edinburgh Math. Soc. xxi. A903), pp. 65-72; xxn. A904), pp. 80—85; Proc. Royal Soc. Edinburgh, xxv. A904), pp. 273—276; Trans. Royal Soc. Edinburgh, xli. A905), pp. 1—28, 105—118, 399-408; Proc. London Math. Soc. B) I. A904), pp. 361—366; B) II. A905), pp. 192—220; B) ill. A905), pp. 1—23. The more obvious generalisation of the Bessel function, obtained by increasing the number of sets of factors in the denominators of the terms of the series, will be dealt with in § 4. In connexion with this generalisation see Cailler, Mem. de la Soc. de Phi/s. de Genhve, xxxiv. A905), p. 354; another generalisation, in the shape of Bessel functions of two variables, has been dealt with by Whittaker, Math. Ann. lvil A903), p. 351, and Peres, Gomptes Rendus, clxi. A915), pp. 168—170. 313. General properties of Jv (z). The series which defines Jv{z) converges absolutely and uniformly* in any closed domain of values of z [the origin not being a point of the domain when R (v) < 0], and in any bounded domain of values of v. For, when lvl ^ JV and \z\ -$ A, the test ratio for this series is m (v + m) whenever m is taken to be greater than the positive root of the equation m? - mN - I As = 0. This choice of m being independent of v and z, the result stated follows from the test of Weierstrass. ifencef Jv (z) is an analytic function of z for all values of z (z = 0 possibly being 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- differentiations and integrations (with respect to z or v) of the series for Jv (z) are permissible. An inequality due to Nielsen \ should be noticed here, namely where j 6 \ < exp {|^j^j} -1, and | vq+1 I is the smallest of the numbers j v\-\ j, (v+2), | v+3|, .... This result may be proved in exactly the same way as § 2*11 E); it should be com- compared with the inequalities ¦which will be given in § 3. Finally, the function z", which is a factor of Jv (z), needs precise specifica- * Bromwich, Theory of Infinite Series, % 82. + Modern Analysis, § 5-3. % Math. Ann. iai. A899), p. 230; Nyt Tidsskrift, rx. B A898), p. 73; see also Math. Ann. ly. A902), p. 494.
3-13, 3*2] BESSEL FUNCTIONS 45 tion. We define it to be exp (v log z) where the phase (or argument) of z is given its principal value so that — 7r < arg z % 7r. When it is necessary to "continue" the function Jv{z) outside this range of values of arg z, explicit mention will be made of the process to be carried out. 3*2. The recurrence formulae for Jv (z). Lommel's generalisations* of the recurrence formulae for the Bessel co- coefficients (§ 2*12) are as follows: A) J^(z) + Jv+l(z) = ^Jv(z) B) Jv^(z)-Jv+1{z)^^JJ(z), C) zJv'(z) + vJ,.(z) = zJv^(z), D) zJ: (z) -vJv{z) = - zJv+l(z). These are of precisely the same form as the results of § 2'12, the only difference being the substitution of the unrestricted number v for the integer n. To prove them, we observe first that - 12* r (z\\ - — vKJl d ' K h dz ' "K h dz „,=„ 2"+2j" . m \V(v + m -t 1) oo ( \m ^v—\ +2»n. = nZo 2"-i+«" \ m\'V"(v~+'m) When we differentiate out the product on the left, we at once obtain C). In like manner, dz " dz ,M*l!() 2"+2m. m! F (y + m + 1) v . _L— .— » / \m-\ l ^aw+i = ,,,t«, 2^"^'"+r."mTT"r(V+ wi + 2) whence D) is obvious; and B) and A) may be obtained by adding and sub- subtracting C) and D). • StUdien ilber die tkascV»chen Fnnctionen (Leipzig, 1868), pp. 2, 6, 7. Formula C) waa given when v is half of an odd integer by Plana, Mem. delta R. Accad. ddle Sai. dl Torino, xxyi. A821), p. 538.
46 THEORY OF BESSEL FUNCTIONS [CHAP. Ill We can now obtain the generalised formulae F) by repeated differentiations, when m is any positive integer. Lommel obtained all these results from his generalisation of Poisson's integral which has been described in § 3*1. The formula A) has been extensively used* in the construction of Tables of Bessel functions. By expressing JV-i (s) ^nd J^v (z) in terms of J±v (z) and J'±v B) by C) and D), we can derive Lommel's formula f G) J, 0) /,_„ (,) + /_, {z) Jv_x (z) = 2 -^ from formula B) of § 3'12. An interesting consequence of A) and B) is that, if Qv {?) ¦= Jv2 B), then (8) Q»-i(*)-&+i(*)-y<2;(*);. this formula was discovei'ed by Lommel, who derived various consequences of it, Studien ilher die BesseVschen Functionen (Leipzig, 1868), pp. 48 et seq. See also Neumann, Math. Ann. hi. A871), p. 600. 3*21. Bessel functions of complex order. The real and imaginary parts of the function Jv+i^ (%), where 1/, /u, and x are real, have been discussed in some detail by Lommel+, and his results were subsequently extended by B6cher§. In particular, after defining the real functions KPiflL(x) and Sv>flL(x) by the equation || Lommel obtained the results ,,,, B) ^>(«) C) fifr+1,M («) = iSf * See, e.g. Lomnael, Milnchener Abh. xv. A884—1886^, pp. 644—647. t Math. Ann. tv. A871), p. 105. Some associated formulae are given in § 3-03. $ Math. Ann. m. A871), pp. 481—486. § Annals of Math. vi. A892), pp. 137—160. || The reason for inserting the factor on the right ia apparent from formulae which will be established in § 3-8.
3*21, 3'3] BESSEL FUNCTIONS 47 with numerous other formulae of like character. These results seem to be of no great importance, and consequently we merely refer the reader to the memoirs 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* many years ago. 3'3. LommeVs expression of Jv(z) by an integral of Poissons type. We shall now shew that, when 11 (v) > - ^, then A) ,/„ 0) = r - .j*^--. JJcOS (* COS 6) BUL»0d0. It was proved by Poissonf that, when 2v is a positive integer (zero in- included), the; expression on the right is a solution of Bessel's equation; and this expression was adopted by Loininel;J; as the definition of J,,(z) for positive values of v + ?. Lommel subscquontly proved that tins function, no defined, is a solution of Bessel's generalised equation and Unit it satisfies tlie recurrences formulae of § 32; and he then defined ./„ (z) for value* of v in the intervals (— J, — i|), ( —ij, - &), (-¦•?, -I), ¦¦• by suc- successive applications of ^ 3-2 A). To deduct' A) from UuMlefinition of Jv{z) adopted in this work, we trans- transform the general term of the series for ./",, (z) in the following manner: m\ 9.2m- r 1 Bm)!V provid<>d that R (v) > - |. Now when R(v)^{, the series conve.rgOH unifonuly with respect to t throughout the interval @, 1), and writ may be. integrated tenn-by-torm; on adding to the result the term for which * I'hil. Trans, of tin1 Roijul Km. 1814, p. SJIJS). See also 11 question set in the Mathematical Tripon, 1K<M. f Journal do. I'ikole U. l'olijtechnique, xn, (cahicr 19), A823), pp. 300 et seq., 340 et seq. Strictly Kponkmg, Poimion shewed that, when l2» is an odd integer, the expression on the right multiplied by Jz is a solution of the oquation derived from Besael'a equation liy the appropriate ohange of depondont variable. :|: Nlmlien iJber din Jieneel'nefien h'unctioncn (Leipzig, 18G8), pp. 1 et seq.
48 THEORY OF BESSEL FUNCTIONS [ <11! A T. IU m = 0, namely I tv-^(\-t)~^dt, which is convergent, we find (hat. wh.it im=o yzm). i whence the result stated follows by making the substitut ion / sin- 0 .it using the fact that the integrand is unaffected by writing ir — 0 in ]il;u-r «>i When -|<JS(i/)<|, the analysis necessary to establish the lunt. <«<|imti<.n i-« a \<.:\ more elaborate. The simplest procedure seems to be to take the .series with fh«- in .! • terms omitted and integrate by parts, thus m.2 Bm)! j0 • m=2V + %- Bm)T U=2 d 1 v-l I K ( — V'1 »2 I o U=2 Bm)! A~° J '"< on integrating by parts a second time. The interchain nt tv, i integration in the second line of analysis is permtibl o f' ;" 1:iIIIIIIiml l' convergence of the aeries. On adding the inteLT 0I^ ^«>unt .,( ,1,,. „,,„•,.,.,„„_, which are convergent), we obtain totZ^T"*^"* *° thl> tl<P" '" "''' * It follows that, when R („) > _ ^ then ,w. rt(:+«r («•'•sin -W.>t»?vahd only when
3*31] BESSEL FUNCTIONS 49 An expansion involving Beraoullian polynomials has been obtained from D) by Nielsen* with the help of the expansion in which <fin(f-) denotes the nth. Bernoullian polynomial and a=izt. [Note. Integrals of the type C) were studied before Poisaon by Plana, Mem. della It. Aecad. dellc Sci. di. Torino, xxvi. A821), pp. 519—538, and subsequently by Kumrner, Journal filr Math. xn. A834), pp. 144—147; Lobatto, Journal fiir Math. xvn. A837), pp. 3E3—371; and Duhamcl, Cours W Analyse, n. (Paris, 1840), pp. 118—121. A function, substantially equivalent to ./„ B), denned by the equation r\ e/(/x, x) = I A — ifiy* eo.s ox. di>, J 0 was investigated by Lommel, Archh> dcr Math, und Phys. xxxvii. A861), pp. 349—360. The converse problem of obtaining the differential equation satisfied by zK [P c'Uv-aT-Uv-BY-'dv ( was also discussed by Lommcl, Arc/uv der Math, und Phys. xl. A863), pp. 101—126. In connexion with llii.s integral see also Euler, Inst. Gale. Int. ir. (Petersburg, 1769), § 1036, ami Petzval, Integration der linearen Dijfcrentialgleichungen (Vienna, 1851), p. 48.] 3*31. Inequalities derived from PoLsson's inte<j7'al. From § 3'3 ((J) it follows that, if v bo real and greater than — •?, then (I) '.r,(z)\-; By using the recurrence formulae § ^A) and D), we deduce in a similar manner that B) By using the expressionf {2/(ttz)Y cos z for t/_j (z) it may be shewn that A) in valid when i> = —,}. These inequalities should be compared with the less stringent inequalities obtained in §«J"ltt. When v is complex, inequalities of a more complicated character can be obtained in the same manner, but they are of no great im- importance. * Math. Ann. lix. A<JO4), p. 108. The notation used in the text is that given in Modem Analysis, § 7'2; Nielsen uhob a different notation. t The reader nhould have no difficulty in verifying this result. A formal proof of a more general theorem will be given in § 'S-l. W. H. V. 4
50 THEORY OF BESSEL FUNCTIONS [CHAP. 3*32. Gegenbauer's generalisation of Poissoris integral. The integral formula in which 0/ (i) is the coefficient of an in the expansion of A - ?<xt + a'2) * in ascending powers of a, is due to Gegenbauer*; the formula is valid when R (v) > - $ and n is any of the integers 0,1, 2, .... When n = 0, it obviously reduces to Poisson's integral. In the special case in which v — §, the integral assumes tfye form B) Jn+i (z) = (-i)n (j-j* jVcos 9 Pn (cos 6) sin 6 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 and integrate n times by parts; the result is Now it is known thatj whence we have and Gegenbauer's result is evident. A symbolic form of Gegenbauer's equation is } " this was given by Rayleigh§ in the special case v = \. The reader will find it instructive to establish C) by induction with the ciid of the recurrence formula ¦ Wiener SiUuvgsberichte, lxvii. B), A873), p. 203; txx. B), A875), p. 15.. See also Bauer, ¦Milnchener Sitzungsberickte, v. A875), p. 262, and 0. A. Smith, Giornale di Mat. B) xn. A905), pp. 365—373. The function Cn" (t) has been extensively studied by Gegenbauer in a series of memoirs in the Wiener Sitzungsberichte; some of the more important results obtained by him are given in Modern Analysis, §15-8. + Proc. London Math. Soc. xxxv. A903), pp. 198-206. See §§ 6*17, 10'5. t Cf. Modern Analysis, § 15-8. § Proc. London Math. Soc. rr. A873), pp. 100, 263.
3-32,3*33] BESSEL FUNCTIONS 51 A formula which is a kind of couverse of D), namely* in which /*"** denotes a generalised Legendre function, is due to Filon, Phil. Mag. F) vr. A903), p. 198; the proof of this formula is left to the reader. 33. Gegenbauer's double integral of Poisson's type. It has been shewn by Gegenbauei-j- that, when R (v) > 0, C) Jv (V) = \T\ \ exp [iZ cos 9 - iz (cos <f> cos $ + sin <f> sin Q cos tlr)] 7rl \v) J oJ o sin2"-1 -\Jr sin12" Od-yjrdO, where ot2 = Z- + z* - 2 Zz cos <^> and Z, z, <j> are unrestricted (complex) variables. This result was originally obtained by Gegenbauer by applying elaborate in- integral transformations to certain addition formulae which will be discussed in Chapter XI. It is possible, however, to obtain tho formula in a quite natural manner by means of transformations of a type used in the geometry of the sphere;]:. After noticing that, when z = 0, the formula reduces to a result which is an obvious consequence of Pokson's integral, namely ,fv <Z) = VV ¦— eix™* sin2" 0. v I ()' sin2 I (i').'o we proceed to regard -v/r and 6 as longitude and colatitude of a point on a unit sphere; we denote the direction-cosines of the vector from the centre to this point by (I, m, n) and the element of surface at the point by dco. We then transform Poisaon'a integral by making a cyclical interchange of the coordinate axes in the following manner§ : .T\ If e Hina* 6 .sin8*-1 trl {JJ i-"-} dw TtI {V)J '«>() = &?)\ [ "f"" e'^"'"9*""*cos8' ^sin Od^dd. TTl (v).'o .'() ' It is supiioscd Unit f Wiener SitzitMtjsberichtr, lxx.iv. B), AH77), pp. 128—121). X Tliis method in offoolive in proving nmnerouH formulau of which analytical piools wore ivon by (Sugcnlumtsr ; uml it Hemns not unlikely that he discovered those funuulne by tho method i qUQHlion; of. SS Vl-12, 12-14. 'fho device* ih used by Boltrauii, Lombardn Jlcndlconli, (*i) xnt. 1«HO), p. !$2rt, for n fathor dilTorent iniriioso. § The symbol jj,,,^ moans that tho integration extends over the Riirfaco of the hemipphore on 'liich vi ifi positive. 4—i
52 THEORY OF BESSEL FUNCTIONS [CHAP. HI Now the integrand is an integral periodic function of ifr, and so the limits of integration with respect to ^ may be taken to be a and a + 2tt, where a is an arbitrary (complex) number. This follows from Cauehy's theorem. We thus get Jv(ib)- ("»'ar)> j | *'eivr sin 6cos* cos*y-i0sm6dylrd6 TlT (v) J o J a = (i™)" I *I ^i-orsinecos (*+a)cos2"8sin 8d<Ard8. ttT(v))o Jo We now define a by the pair of equations ta cos a = Z -¦ z cos <?, tsr sin a = z sin <?, so that jv ( o ) „ iW! I* f 2irexp [i (? __ ^ cos </>) sin 0 cos ^ - tar sin <? sin f sin #] wl (v) Jo Jo „ . ... ,„ cob2" 8 sin vdty dv. The only difference between this formula and the formula J» («) = ^S^ f * r exP [»w sin ^ cos ^1 c03'" e sin ^d^rf^ ttI (y)Jo Jo is in the form of the exponential factor; and we now retrace the steps of the analysis with the modified form of the exponential factor. When the steps are retraced the successive exponents are i{Z — zcos (f>)l — izsin <j). m, i(Z — z cos <?) n — iz sin <f>. I, i{Z—z cos </>) cos 6 — iz sin <f> cos ty sin 6 The last expression is iZ cos 8 — iz (cos </> cos 8 + sin <? sin 8 cos \Jr), so that the result of retracing the steps is AJ.jjjAv fir fir ? /-; ' exp [iZ cos 5 — i^ (cos <f> cos 9 + sin A sin 0 cos a!tI ^r(j/)JoJo sin2" aJt sin2 and consequently Gegenbauer's formula is established. [Note. The device of using transformations of polar coordinates, after the manner of this section, to evaluate definite integrals seems to be due to Legendre, M4m. de VAcad. des ScL, 1789, p. 372, and Poisson, Mem. de VAcad. des Sci. in. A818), y. 126.] 3*4. The expression of J±(»+j) (z) in finite terms. We shall now»deduce from Poisson's integral the important theorem that, when v is half of an odd integer, the function Jv(z) is expressible infinite terms by means of algebraic and trigonometrical functions of z. It will appear later (§ 4*74) that, when v has not such a value, then /„ {z) is not so expressible; but of course this converse theorem is of a much more recondite character than the theorem which is now about to be proved.
3-4] BESSEL FUNCTIONS 53 [Note. Solutions in finite terms of differential equations associated with Jrn+i 00 were ob- obtained by various early writers; it was observed by Euler, Misc. Taurinensia, ill. A762— 1765), p. 76 that a solution of the equation for e^t/^.,., (z) is expressible in finite terms; while the equation satisfied by ^Jn+Az) was solved in finite terras by Laplace, Conn, des Tan*. 1823 [1820], pp. 245—257 and Mecavique Celeste, v. (Paris, 1825), pp. 82—84 ; by Plana, Mem. della It. Accad. delle Sci. di Torino, xxvi. A821); pp. 533—534; by Paoli, Mem. di Mat. e di Fis. (Modena), xx. A828), pp. 183—188; ami also l>y Stokes in 1850, Trans. Camb. Phil. Soc. ix. A856), p. 187 [Math, and Phis. Papers, n. A883), p. 350], The investigation which will now be given is based on the work of Lommel, Studien ilher din BesseVschen Functionen (Leipzig, 1868), pp. 51—56.] It is convenient to restrict » to bo a positive integer (zero included), and then, by § 3'3 D), ;, di* when we integrate by parts 2». + I times; since (I - f)n is a polynomial of degree 2n, the process then terminates. To simplify the last expression we observe that if dr(l — t2)n/dtr be cal- calculated from Leibniz' theorem by writing A — t-)n = A — t)n(l +t)n, the only term which does not vanish at the upper limit arises from differentiating n times the factor A —t)n, and therefore from differentiating the other factor /• — ii times; so that we need consider only the terms for which r^n. and similarly '¦ | = (—)r ".,.<?„.«!-¦ It follows that m\i\ hence A) ./„+ This result may bi>. written in the form*' B) ¦^( 2r+ 1) * A compact nicUiod of obtaining Huh formiiln \a (Ljivcn by do la Valk'e Pons.iin, Ann. de Iu Soc. Set. df llriiMllsH, xxix. (li)O.V, j>p. 1 -10—14:-t.
54 THEORY OF BESSEL FUNCTIONS [CHAP- In particular we have the former "of these results is also obvious from the power series for Jj. (z). Again, from the recurrence formula we have anfl hence, from AK e ,.r0 rT(n- r)! (Siy ,.r0 r\{n-r)l B*)" But, obviously, by induction we can express as a polynomial in 1/s multiplied by e±iz, and so tue must have ±iz % (±i)'-»( for, if-not, the preceding identity would lead to a result of the form where fa {?) and $%(z) are polynomials in \\z; and such an identity is obviously impossible*. Hence it follows thatf o r 1 (n - r)! B«)r ^ r=o r 1 (n - r)! B«)r ^o r t (n - r) 1 B^)r zdz) z Consequently D) J n ,( * Of. Hobson, Squaring the Circle (Carabtidge, 1913), p. 51. t From the series "* U r A) it is obvious that J, (e) = (—\
3*41] BESSEL FUNCTIONS 55 and hence E) /_,_, W = (_j [cos (. ^A^/ ,„ ( )( In particular, we have F) /_,(,)- (^) cos,, •/_,(,)_(_)(____„„,) We have now expressed in finite terms any Bessel function, whoso order is half of an odd integer, by means of algebraic and trigonometrical functions. The explicit expression of a number of these fimutioas can be written down from numerical results contained in a letter from Hcrmite to Giordan, Journal fur Math, lxvi. A873), pp. 303—311. 31. Notations for functions whose order is half of an odd integer. Functions of the types ./.i-oHj) C2) occur with such frequency in various branches of Mathematical Physics that various writers have found it desirable to denote them by a special functional symbol. Unfortunately no common notation has been agreed upon and none of the many existing notations can be said to predominate over the others. Consequently, apart from the summary which will now be given, the notations in question will not be used in this work. In hi.s researches on vibrating spheres .suitouihWI by a gas, Stokes, Phil. Trans, of the llo<f<d Soc. ciA'in. (IH()8), p. 4.01 [Math, and Phys. Papers, iv. A901), p. 300], made use of the series n (Wjf 1) (it-!)«(» + !)(»+ 2) + SS.iW a. 4. (/«»•)* which in annihilated by the operator frJ .„. d ii (n+l) dr* dr /•" This series Stokes denoted by the symbol /„(?") and he wrote where »S'H and *S',,' are zonal .surface harmonics; so that \^,, i.s' annihilated by tho total operator d~ 'A d ,, /' (/i -f-1) - -)- -)- in- — , dr- r dr r- and by the partial operator ¦ .,+ - + - • ., -,* {Hill fK-: In this notation Stoker wan followed by Haylei;^h, I'rov. London, Math. >So<\ IV. ^1^73), pp (jn—io;}, 2."i3 '283, and ugniu /Vof. //»?/<// Sor. i.xxir. (HKK), pp. -10—11 \Sricntijlc y'f//ic/'.S v. (l!)l'2), pp. 112 -114|, aj)art iVotu the csomparatively trivial change that Haylcigh would have written/„ (imr) where Stokes wrote/,,(r).
56 THEORY OF BESSEL FUNCTIONS [CHAP. Ill In order to obtain a solution finite at the origin, Bayleigh found it necessary to take #n'= (- )"+1 Sn in the course of his analysis, and then It follows from § 3-4 that -—&& > = (i -4- ) — , ° rn+l \^ rcrJ r ' and that In order to have a simple notation for the combinations of the types e*u' ftl (± w*) which are required for solutions finite at the origin, Lamb found it convenient to write in his earlier papers, Proc. London Math. Soc. xni. A882), pp. 51—66; 189—212; XV. A884), pp. 139—149; xvi. A885), pp. 27—43; Phil Trans, of the Royal Soc. clxxiv. A883), pp. 519—549; and he was followed by Bayleigh, Proc. Royal Soc. lxxvii. A, A906), pp. 486—499 [Scientific Papers, v. A912), pp. 300—312], and by Love*, Proc. London Math. Soc. xxx. A899), pp. 308—321. With this Dotation it is evident that -(-)• »• 3¦ 5 ... Subsequently, however, Lamb found it convenient to modify this notation, and accord- accordingly in his treatise on Hydrodynamics and also Proc. London Math. Soc. xxxn. A901), pp. 11—20, 120—150 he used the notation t * fcw_i_ [~1 *2 , ywW 1.3.3... Bn + l)L 2B»+3)^2.4Bn (d \n e~iz - —j- I = ^n B) - ityn (z), zcLzJ z sothat *„(«), g)t+^ * , «M*)- -S+-J-"- • while Bayleigh, Phil. Trans, of the Royal Soc. ccni. A, A904), pp. 87-110 [Scientific Papers, v. A912) pp. 149—161] found it convenient to replace the symbol fn{z) by Xniz)- Love' Phil. Trans, of the Royal Soc. ccxv. A, A915), p. 112 omitted the factor (-)" and wrote while yet another notation has been used by Soramerfeld, Ann. der Phytik und Chemie, D) xxviii. A909), pp. 665—736, and two of his pupils, namely March, Ann. der Phy&ik und Chemie, D) xxxvu. A912), p. 29 and Rybczynaki, Ann. der Physik und Chemie, D) xi.r. A913), p. 191 ; this notation is d \n sin2 ) Ca B) = faz? [Jn + j B) + ( -)» iJ_n _ j B)], and it is certainly the best adapted for the investigation on electric waves which was the subject of their researches. * In this paper Love defined the function En{z) as (-)n.l .3 ... Bn-l) ( — j — , but, as stated, he modified the definition in his later work. t This is nearer the notation used by Heine, Handbuch der Kugelfunctionen, 1. (Berlin, 1878), p. 82 ; except that Heine defined $n(z) to be twice the expression, on the right in his treatise, but not in his memoir, Journal fur Math. lxix. A869), pp. 128—141.
3*5] BESSEL FUNCTIONS 57 Somiuerfeld's notation is a slightly modified form of the notation used by L. Lorenz, who used vH and vn + ( - )n iwn in place of \0>tt and fn; see his memoir on reflexion and refraction of light, K. Danake Videnskobernes Sehkabs Skrifter, F) vi. A890), [Oeuvres scientifiques, I. A898), pp. 405—502.] 3*5. A second solution of BesseVs equation for functions of integral order. It has been seen (§ 3*12) that, whenever v is not an integer, a fundamental system of solutions of Bessel's equation for functions of order v is formed by the pair of functions Jv (z) and ./_„ (z). When v is an integer (= n), this is no longer the case, on account of the relation »/_„ (z) = (-r-)n Jn (z). It is therefore necessary to obtain a solution of Bessel's equation which is linearly independent of Jn (z); and the combination of this solution with Jn (z) will give a fundamental system of solutions. The solution which will now be constructed was obtained by Hankel*; the full details of the analysis involved in the construction were first published by Bocherf. An alternative method of constructing Hankel'n solution was discovered hy Forsyth; his procedure, is based on the general method of Frobonius, Journal fiir Math, lxxvi. A874), pp. 214—235, for dealing with any linear differential equation. Forsyth's solution was contained in hi.s lectures on differential equations delivered in Cambridge in 189-1, and it has since been published in his Theory of Differential Equations, iv. (Cambridge, 1902), pp. 101—10*2, -mid in his Treatise on Differential Equations (London, 1903 and 1914), Chapter vi. note 1. It is evident that, if v be unrestricted, and if n be any integer (positive, negative or zero), the function /,(*)-(_)«./¦_(,) is a solution of Vessel's equation for functions of order v\ and this function vanishes when /> = /(. Consequently, so long as v ? v, the function J,,(z) -(-)»./-„ {z\ v — n is also a solution of Bessel's equation for functions of order v; and this function assumes an undetermined form* when v — v. We shall now evaluate ,im/.w.-<::>¦_•(.-? W, ,. -*. rt v - n and we shall shew that it is a solution of Bessel's equation for functions of * Math. Ann. i. A8(i<)), pp 4fi!) -472. t A ninth of Math. vi. A892), pp. 8/5—DO. See aim Niemoller, Zeituchriftfilr Math, und Phys. xxv. <1HHO), pp. (i.r) - 71 :\: The crhcucc of Hiuikcl's invuBtigatiou is the construction of an expression which satisfies the equation when v in not an integer, which assumes an undetermined form when v is equal to the integer n and which han a limit when v-*-u.
58 THEORY OF BESSEIr FUNCTIONS [CHAP. Ill order n and that it is linearly independent of Jn{z)) so that it may be taken to be the second solution required*. It is evident that Jv(z) - H" J- (z) _ Jv jz) - Jn(z) XnJ__v(z)-J_n(z) : z ~~ "" z :—\ j : v — n as v-^n} since both of the differential coefficients existf.. Hence v n exists; it is called a Bessel function of the second land of order n. To distinguish it from other functions which are also called functions of the second kind it may be described as Hankel's function. Following Hankel, we shall denote it by the sym"bolj Y?l<V) so that n (z) = lim n and also B) ^ } v~n dv x ' dv It has now to be shewn that YOT (z) is a solution of Bessel's equation. Since the two functions J±v (z) are analytic functions of both z and v, the order of performing partial differentiations on J±v(z) with respect to z and v is a matter of indifference §. Hence the result of differentiating the pair of equations with respect to v rnay be written w. o. When we combine the results contained in this formula, we find that v. * The reader will realise that, given & solution of a differential equation, it is not obvious that a limiting form of this solution is a solution of the corresponding limiting form of the equation. t See § 8*1. It is conventional to write differentiations with respect to z as total differential coefficients while differentiations with respect to v are written as partial differential coefficient-. Of course, in many parts of the theory, variations in v are not contemplated. X The symbol Y%(z), which was actually used by Hankel, is used in this work to denote a function equal to 1/w times Hankel's function (§ 3*54). § See, e.g. Hotson-, Functions of a Real Variable A921), §§ 312, 318.
3-51] so that ~dJv(z) , ,n dv { } BESSEL FUNCTIONS dJ-v (z)' dv dJv (z) dv )J-V(z)' dv 59 Now make z^ ~^n. All the expressions in the last equation are continuous functions of v, and so we have dJv{z) — ( -v (z) = 0, where v is to be made equal to n immediately after the diiferentiations with respect to v have been performed. We have therefore proved that \ *-* / It, If, \ / v ) so that YnB") is a solution of BesseVs equation for functions of order n. It is to be noticed that J?^ B,\ 7TT 1 1 1 11 ^ ' / V -*¦ — n V -j- 11 _ —— ii I ii n whence follows a result substantially due to Lonnnel*, Again, dv while, because Jv(z) is a monogenic function of v at v = 0, we have dv V =30 dJv{ a (— z^)_ )ws that — ^ a^ ~dJv (z)~ dv E) A result equivalent to this Vas given by Duhamelf as early as 1840. 3*51. The expansion oJ"Y0(z) in an ascending series. Before considering the expansion of the general function Y,^), it is con- convenient to examine the function of order zero because the analysis is simpler and the resulting expansion is 'more compact. We use the formula just obtained, * Studien iiber die HeaseVachen Functional (Leipzig, 1868), p. 87. Lommel actually proved thiH reault for what is sometimes called Neumann's function of the second ldnd. See § 3'58 (8). t Court d'Analyse, n. (Paris, 1840), pp. 122—124.
60 THEORY OF BESSEL FUNCTIONS [CHAP. ITI and the result of term-by-term differentiation is Y(,(V) = 2 ? — -J, /¦ Jloei" (¦k-z) — s- log F (v + m ¦+ 1) [ \_m*omW(v + m+1) { ° dv ° j_ v-0 where ty denotes, as is customary, the logarithmic derivate of the Gamma- function*. Since 0 <\^ (m +1) < m when ?n = l, 2, 3, ... the convergence of the series for Yo(z) may, be established by using D'Alembert's ratio-test for the series in which ^(w+1) is replaced by m. The convergence is also an immediate consequence of a general theorem concerning analytic functions. See Modern Analysts, § 5*3. The following forms of the expansion are to be noticed : A) Y0(z »i=0 K'''' ¦) r oo (—)™(XzYm 1 B) Yo(*)= 2 log(**).«/,(*)- S V , ,* t(m + 1)h L »«=o \m ¦) J C) Yo (z) = 2 [7 + log (\z)} Jo (z) - 2 JS (~r (^^r A ' l ¦ ¦ 1] The reader will observe that is a solution of Bessel's equation for functions of order- zero. The expansion of this function is (log*) 2 (-V^} 2 ( I V 7 + 5 + -+-[- This function was adopted as the canonical function of the second kind of order zero by Neumann, Theorie der Bessel'sohen Functionen (Leipzig, 1867), pp. 42—44; see § 3-57. But the series was obtained as a .solution of Bessel's equation, long before, by Euler f. Eulor's result in his own notation is that the general solution of the equation xx cdy + x dx cy + gxny ex2 — 0 _2yliy n GAg- 2 '22Ag3 3n 100%4 W V~ HFX ~ 1. 8a*X" + 1.8.W *"" " r8.27.'64M» + «/i" 1 . 4w' ' 1.4.9%°' 1.4.9.l6ns" + a «naf+1.4»i*-1! 1.4.»»«lC +1.4.9. * Modern Analysis, Gh. xn. It is to be remembered that, when m is a positive integer, then "/'A)=-7, ^(m + l)=\ + h + ••• +--7, where 7 denotes Eider's constant, 0-5772157 t Imt. Gale. Int. 11. (Petersburg, 1709), §977, pp. 233—235. See also Ada Acad. Petrop. v. A781) [published 1784], pars 1. Mathematics, pp. 186—190.
3*52] BESSEL FUNCTIONS 61 where A and a are arbitrary constants. He gave the following law to determine successive numerators in the first line: 6 = 3.2-1.0, 22=5.6-4.2, 100 = 7.22-9.6, 548 = 9 .100-16 . 22, 3528 = 11. 548-25 . 100 etc. this law is evidently expressed by the formula 3'52. The expansion of Yn (z) in an ascending series and the definition of We shall now obtain Hankel's* expansion of the more general function Yn(z), where n is any positive integer. [Cf. equation D) of § 3'5.] It is clear that dv when v-^-»?, where ?? is a positive integer. That is to say The evaluation of [dJ_v {z)ldv\-n is a little more tedious because of the pole of yjr (- v 4- m 4-1) at i> = n. in the terms for which m = 0,1, 2, ..., n — 1. We break the series for «/_„ (z) into two parts, thus , M( ) »! T(- v +m 4- and in the former part we replace 1 , F (v — m) sin (v — m) it Now, when 0 ^ m < n, 3 \(\ z)~v+*m F (v — w) sin (j» - m) ir ) v — m) sin (v—mOr 4- cos (v—in) ir —ir~l log (^) sin (v—inOr}]v»n F (n - m) cos (n - nt) 7r. * itfat/j. Ann. i. A869), p. 471.
62 THEORY OF BESSEL FUNCTIONS [CHAP. Ill Hence '?. — m) dv La,, m~o ml that is to say Wl -( ^(n-m-l)! - when we replace wi by ?? + m in the second series. On combining A) and B) we have Hankel's formula, namely (8) Y(i)— s ( )'a ( ]}**> x ! Ill 1 ) ^ 2 ^ <>-^ w "*" 1 ^ 2 ?(+mf In the first terra (m = 0) of the last summation, the expression in { } is 1 1 1 =-+5 + ... +-. 12 n It is frequently convenient (following Lommel*) to write D) %(z)J^-Jv so that when i/ is a negative integer, ^v (z) is defined by the limit of the expression on the right. We thus have F) Yn (*) = 2 Jn (z) \ogz + <$n @ + (-)« 3_7( (*). The complete solution of x -rK + ay=O was given hi the form of a series (part of which contained a logarithmic factor) by Euler, Inst. Cede, Int. II. (Petersburg, 1769), 4^§ 935, 936; solutions of this equation are #* Jx BaM), ** Yt BaU'i). Euler also gave (ibid. §§ 937, 938) the complete solution of %$ — + ay=O; solutions of this equation are Studicn ilber die Be&seVschen Functional (Leipzig, 1868), p. 77.
3-53,3-54] BESSEL FUNCTIONS 63 3-53. The definition o/Yv (z). Hitherto the function of the second kind has been defined only when its order is an integer. The definition which was adopted by Hankel* for un- unrestricted values of v (integral values of 1v excepted) is A) Y^) This definition fails both when v is an integer and when v is half of an odd integer, because of the vanishing of sin 2v7r. The failure is complete in the latter case; but, in the former case, the function is defined by the limit of the expression on the right and it is easy to reconcile this definition with the definition of § 3*5. To prove this statement, observe that ,. __ ,, v f ire'* v-n Jv(z)cos vir-«/"_„0I hm Y,, (z) = hm . . —-- z±-* ,,-».n ,,-*.)! [cos vk sin vrr v — n J v-n ~ , . , • |"(-)n COS I»7T - 1 T . /] = Yn (z) + Inn > -' Jv{z) i'-*-n L V — n J and so we have proved that B) It is now evident that Y^^), defined either by A) or by 1' limiting form of that equation, is a solution of Bessel's equation for functions of order v both when (i) v has any value for which 2v is not an integer, and when (ii) v is an integer: the latter result follows from equation B) combined with § 3*5 C). The function Y,, (z), defined in this way, is called a Bessel function of the second kind (of Hankel's type) of order v; and the definition fails only when v +1 is an integer. Notb. The reader should bo careful to observe that, in wpito of the change of form, the function Yv (z), qua function of v, is continuous at v = n, except when z ia zero; and, in fact, Jv {?) and ~Yv(z) approach thoir limits ¦/„ B) and Yn (z), aw u-*-n, uniformly with respect to z, except in tho neighbourhood of 2 = 0, where n in any intogor, positive or negative. 3'54. The Weber-Schldfii function of the second kind. The definition of the function of tho second kind which was given by Hankel (§ 33) was modified slightly by Webcrf and SchlafliJ in order to avoid the inconveniences produced by tho failure of tho definition when the order of the function is half of an odd integer. * Math. Ann. 1. A869), p. 472. ¦\ Journal fUr Math, lxxvi. A873), p. 9; Math. Anv. vi. A873), p. 148. These papors are dated Sept. 1872 and Oct. 1872 respectively. In a paper written a few months before these, Journal filr Math. lxxv. A873), pp. 75—105, dated May 1872, Weber had used Neumann's function of the second kind (see §§ 8-57, 3-58). % Ann. di Mat. B) vn. A875), p. 17 ; thiH paper is dated Oot. 4, 1872.
64 THEOBY OF BESSEL FUNCTIONS [CHAP. Ill The function which was adopted by Weber as the canonical function of the second }rind is expressible in terms of functions of the first kind by the formula* Jv (z) cos viT — J.,, (z) sin vtt (or the limit of this, when v is an integer). Schlafli, however, inserted a factor \tr\ and he denoted his function by the symbol K, so that, with his definition, Jv(z)cosvrr-J-V(z) Jti v (Z) — ?7T : . x ' - sinv7r Subsequent writers, however, have usually omitted this factor %-ir, e.g. Graf and Gublerin their treatisef, and also Nielsen, so that these writers work with Weber's function. The symbol K is, however, used largely in this country, especially by Physicists, to denote a completely different type of Bessel function (§ 3-7), and so it is advisable to use a different notation. The procedure which seems to produce least confusion is to use the symbol Yv(z) to denote Weber's function, after the manner of Nielsen}, and to adopt this as the canonical function ol the second kind, save in rare instances when the use of Hankel's function of integral order saves the insertion of the number tt in certain formulae. We thus have A) Yv {z) = jvWcoBiHr-J.,,^ = cosj^r v ' s ' sin vw ire'1 B) Fn<*)« Km ^(^tt-J.^I [Note. Schlafli's function has been used by B6cher, Annals of Math. vi. A892), pp. 85—90, and by McMahon, Annals of Math. vm. A894), pp. 57—61; ix. A895), pp. 23—30. Schaf heitlin and Heaviside use Weber's function ¦with the sign changed, ho that the function which we (with Nielsen) denote by Yv (z) is written as - Yv B) by Schaf heitlin § and (wheu v — n) as -Gn{z) by Heaviside||. Gray and Mathews sometimesif use Weber's function, and- they denote it by the symbol Yn. * Weber's definition was by an integral (see §6-1) which is equal to this expression; the expression (with the factor \v inserted) was actually given by Sohlatli. t Einleitung in die Theorie der Bestel'schen Funhtionen, 1. (Bern, 1898), p. 34 et xeq. % Nielsen, as in the case of other functions, writes the number indicating the ordoi- as an index, thus Y"(z), Handbuch der Thearic der Cylinderfunktionen (Leipzig, 190-i), p. 11. There are obvious objections to suoli a notation, and we reserve it for the obsolete function used by Neumann (§3-58). § See, e.g. Journal filr Math. cxiv. A895), pp.31—44, and other papers; also Die Theorie der BesscV'schen Funktionen (Leipzig, 1908). || Proc. Royal Soc. liv. A893), p. 138, and Electromagnetic Theory, 11. (Loudon, 1899), p. 255; a change in sign has been made from hifi Electrical Papers, n. (London, 1892), p. 445. If A Treatise on Bessel Functions (London, 1895), pp. 65—66.
3-55] BESSEL FUNCTIONS 65 Lommel, in his later work, used Neumann's function of the second kind (see § 3f57), but in his Studien iiber die Bes&eVschen Functionen (Leipzig, 1868), pp. 85—86, he used the function \* rn (z)+{i (n+4)+log 2) jn (z), where YH(z) is the function of Weber. One disadvantage of this function ia that the presence of the term \js (n + $) makes the recurrence formulae for the function much more complicated; sec Julius, Archives JV&erlandaises, xxvni. A895), pp. 221—225, in this connexion.] 3*55. Heine's definition of the function of the second hind. The definition given by Heine* of the function of the second kind possesses some advantages from the aspect of the theory of Legendre functions; it enables certain generalisations of Mehler's formula (§ 6*71), namely lim Pn (cos 6In) = Jo F), n-*- ¦¦*> to be expressed in a compact form. The function, which Heine denoted by the symbol Ktl B), is expressible in terms of the canonical functions, and it is equal to — ^irYn{z) and to — ?Yrt(.s); the function consequently differs only in sign from the function originally used by Schlafli. The uso of Heine's function aeeniH to have died out on the Continent many years ago ; the function was occasionally used by Gray and Malhews in thoir treatise+, and they term it (/„ B). In this form the function has boon extensively tabulated first by Aldis:|: and Airey§, and mibscquontly in British Association Reports, 1913, 1914 and 1916. This revival of the uko of Hoino'a function acorns distinctly unfortunate, both on account of the existing multiplicity of functions of the second kind and also on account of the fact (which will become moro apparent in Chapters vi and vn) that the relations between the functions ./„ B) and Yn{z) present many points of resemblance to the relations between the cosine and sine; mo that the adoption|| of Jn{z) and (.?n(z) a.s canonical functions is com- comparable to the use of cosz and -\tth\i\z as canonical functions. It must also be pointed out that the symbol On(z) ban boon uwed in .senaes other than that just explained by at least two writera, namely Ileaviaide, I'roo. Royal Hoc. uv. A893), p. 138 (as was stated in § 34), and Dougall, Proc. Edinburgh Math. Sac. xvni. A900), p. 3E. Notb. An error in sign on p. 245 of Ileme'w treat mo has been pointed out by Morton, Nature, r,xni. A901), p. 29; the error ia equivalent to a change in the sign of y in formula § 351 C) supra. It was also stated by Morton that thi.s error had apparently boon copied by various other writers, iuoluding (as had been previously noticed by Gray IT) J. J. Thomson, Recent Researches in Electricity and Magnetism (Oxford, 1893), p. 2G3. A further error * Handbuch der Kuyelfunctionen, 1. (Berlin, 1878), pp. IRS—218. t A Treatise on Basel Functions (London, 1B'.M), pp. 5I, 147, '242. I 1'rvc. Royal Soc. lxvi. A900), pp. 32—43. § Phil. Mag. (C) xxn. A011),.pp. 658—608. II Erotn the historical point of view there in something to bo said for using Hankel's function, and alno for using Neumann's function; but Heine's function, boing more modern than either, has not oven thia in its favour. II Nature, xlix. A894), p. 359. w. b. F. 5
66 THEOBY OF BESSBL FUNCTIONS [CHAP. Ill noticed by Morton in Thomson's work seems to be due to a most confusing notation employed by Heine; for on p. 245 of his treatise Heine uses the symbol Ko to denote the function called -?*rF0 in this work, while on p. 248 the same symbol Ko denotes -?tt (Fo- iJ0). 3* 56. Recurrence formulae for Yv (z) and Yv {z). The recurrence formulae which are satisfied by Yv (z) are of the same form as those which are satisfied by Jv (z); they are consequently as follows: A) 7^l(z)+7^(z)^7p(z), B) YP^(z)-Yv+l(z) = 27/0), C) zY9'{z) + v7,{z)-z7,-l(z), D) ZY;(z)-vYv(Z) = -zYv+l(z), and in these formulae the function Y may be replaced throughout by the function Y. To prove them we take § 3'2 C) and D) in the forms jz [zv Jv (*)} = ** J~ (*) > jz \*v J- (*)} = -*v J-v+i 00; if we multiply these by cot vir and cosec vrr, and then subtract, we have whence C) follows at once. Equation D) is derived in a similar manner from the formulae ^ [z- Jv («)} = - s- Jv+1 (z), ^ [z-* «/_„ (^)] = z-" /_,_, (*>. By addition and subtraction of C) and D) we obtain B) and A). The formulae are, so far, proved on the hypothesis that v is not an integer ; but since Yv (z) and its derivatives are continuous functions of v, the result of proceeding to the limit when v tends to an integral value n, is simply to replace v by n. Again, the effect of multiplying the four equations by ire™ sec vtt, which is equal to ire{v±'vOri sec [y ± 1) v, is to replace the functions Y by the functions Y throughout. In the case of functions of integral order, these formulae were given by Lommol, Stud-ten iiber die Bessel'schen Functionen (Leipzig, 1868), p. 87. The reader will find it instructive to establish them for such functions directly from the series of § 3-52. Neumann's investigation oonnected with the formula D) will be discussed hi § 3'58.
3-56,3*57] BBSSEL FUNCTIONS 67 3*57. Neumanns function of the second kind. The function which Neumann* adopted as the canonical function of the second kind possesses the advantage that it is represented more simply by integrals of Poisson's type than the functions of the second kind which have been hitherto discussed; but this is its only merit. We first define the function of order zerof, which will be called F(o) (z). The second solution of Bessel's equation for functions of order zero being known to contain logarithms, Neumann assumed as a solution the expression Jo (z) log z + w, where w is a function of z to be determined. If this expression is to be annihilated by V0) we must have VoW^-Vo {./,(*) log*} = -2*Jr0'(s). Bat, by §212 A1), - 2z /„' 0) = 2z J, (z) = 8 2 (-)»-' n Jm (z); and so, since Vo Jin {z) = 4n2 J.M (z), we have the change of the order of the operations ? and Vo is easily justified. Hence a possible value for w is 2S(-^4(«)/«, and therefore Neumann's function F@) (z), defined by the equation A) F« (*) = Jo (*) log z + 2 I (-)»->J^ , n-1 n is a solution of Bossel's equation for functions of order zero. Since w -*- 0 as z ¦*¦ 0, (the series for w being an analytic function of z near the origin), it is evident that /„ (z) and Ym (z) form a fundamental system of solutions, and hence Yo (z) is expressible as a linear combination of Jo (z) and Ym(z); a comparison of the behaviours of the three functions near the origin shews that the relation connecting them is B) Y* {z) = ?Yo (*) + (log 2 - 7) Jo {t). * Theovie der Bessel'schenFunctionen (Leipzig, 18E7), pp. 42—14. "Neumann calla this function Iiessel's associated function, and he doscriboa another function, On{z), fts the function of the second kind (g 9-1). But, because O)l(z) h not a solution of BqshcI'h equation, this dcnci-iplion is un- undesirable and it has not survived. + Neumann's function is distinguislied from the Webei'-Sohlafli function by the position of the suilix which indicates the order. 5—2
68 THEORY Oi1 BESSEL FUNCTIONS [CHAP. Ill 3*571. The integral of Poissoris type for Y& (z). It was shewn by Poisson* that is a solution of Bessel's equation for functions of order zero and argument x; and subsequently Stokes obtained an expression of the integral in the form of an ascending series (see § 3*572). The associated integral 2 ft* - cos (z sin 0). log Dz cos2 0) d0 Tf j 0 was identified by Neumannt with the function 7<°> (z); and the analysis by which he obtained this result is of sufficient interest to be given here, with some slight modifications in matters of detail. From § 2-2 (9) we have (-)">/"»»(*) = A f 4rrcos (z cos 6) cos 2n6dO, and so, if we assume that the order of summation and integration can be changed, we deduce that o v (-)" Jm (*) 4.. f ** . m 2 cos 2n0 ,a 2 X ^_i—?i-v_/ — _ cos u cos q\ 2 do 2 2 [l* — I cos (z cos 6). log D sin2 8) dd; from this result combined with Paxseval's integral (§ 2'2) and the definition of Y(o) (^t), we at once obtain the formula A) F«°» 0) = - f ^cos (* cos 0). log Da sin2 (9) dd, from which Neumann's result is obvious. The change of the order of summation and integration haa now to be examined, because 2n~1 cos 2nd is non-uniformly convergent near 6=0. To overcome this difficulty we observe that, since 2 ( -)» J2n (z)jn is convergent, it follows from Abel's theorem J that 2 (-}n^2nB)/n= Km 2 (-)*a*/*.(*)/»« lim - 2 I * Journal de. v?cole B. Polytechnique, xn. (cahier 19), A823), p. 476. The solution of an associated partial differential equation had been given earlier [ibid. p. 227). See also Dnhamel, Cours d?Analyse, n. (Paris, 1840), pp. 122—124, and Spitzer, Zeitschrift filr Math, und Phys. n. A857), pp. 165—170. •j- Theorie der Bessel'tchen Functionen (Leipzig, 1867), pp. 45—49. See also Niemoller, Zeit- Zeitschrift Jilr Math, und Phys. xxv. A880), pp. 65—71. X Cf. Bromwich, TJieory of Infinite Series, § 51.
3-571,3-572] BBSSEL FUNCTIONS 69 Now, since a is less than 1, 2 (a'v coa 2nd)/n does converge uniformly throughout the range of integration (by comparison with 2an), and so the interchange is permissible; that is to say 2 S - 2 " / As a* COS 2nd JA 2 /"I* . » a1lcos2%(9 ,. cos (a cos 8) dd«=- I cos (z cos 0) 2 dd ' n it J 0 n.t ?i 1 /i* = - - I cos B cos 0) log A - 2a cos 2(9 + a2) dd. "^ / 0 Hence we have ~YJ (z) 1 /"i lim _ We now proceed to shew that* (YJ (z) 1 /i 2 S_i—aw = _ lim _ 008 ^ cos Qj log (x _2a oos 26 + a2) =«l n w y lim I *l-0 J 0 lim 0 J 0 It isevidont that 1 -2acos 25 + a'2- and so log A - 2a cos 20 -fa2) ^ log Da sin2 d). Hence, if A he the upper bound + of | cos (z cos8) \ when 0^0 ^7r, we have I (*" co.s (z cos (9) (log (I-2a cow 2(9 + a2) - log Da sin2 (?)} dd J * {log A - 2a coh 2(9 + a2) - log Da sin2 d)} dd 0 0 torin-by-torin integration being permissible .since a<l. Hence, when a<l, I cos (z cos 6) {log (l-2a cos 28 -I- a'J) - log Da sin'2 d)) dd < Jinvl log (l/a)-»-O, as «-^l -0; and this is the result to bo proved. Consequently I (~ )nJ'*1 ^ = - lim - I ^ cos (z cos (9). log Da sin2 (9) dd JI---1 W a-*-l-07r y () - __ / cos (zoos(9).log Dsin2(9)^, "¦ J o and tlie interchange is finally justified. The reader will find it interesting to deduce this result from Poisson's intogral for Jv{s) combined with § 3*5 (B). 3'572. Stokes' scries for the Poisson-Neumann integral. Tho differential oquation considered by Stokes J in 1850 was -A + - j-'ni2y=O, where m is a constant. This is Bosael's equation for functions of ordor zero and argument imz. Stokes statod (presumably with reference to Poisson) that it was known that the general •solution waH y =, f n {C+ Dibg (z sin2 d)} cosh (tu cos (9) dd. J o * The value of this limit was assumed by Neumann. t If z is real, 4 = 1; if not, A =$ oxp {| 2 (z) |}. t Tram. Camb. Phil. Soc. ix, A856), p. [38]. [Mathematical rtn& Physical Papers, in. A901), p. 42.]
70 THEORY OF BESSEL FUNCTIONS [CHAP. Ill It is easy to see that, with Neumann's notation, the value of the expression on the right iir {0-2) log Dm,)} Jo (imz)+faDri°) (xmz). The expression was expanded into a series by Stokes; it is equal to z) Jo {imz) + 22) 2 V^r cos2n 6 log 8in ddd> 71=0 (*«,} ' J 0 and, by integrating by parts, Stokes obtained a recurrence formula from which it may be deduced that 3#58. Neumann's definition of F(n> (z). The Bessel function of the second kind, of integral order n, was defined by Neumann* in terms of F(o) (z) by induction from the formula A) z dYfa^- which is a recurrence formula of the same type as §2*12D). It is evident from this equation that B) F<»> <*) = (-*)» (^ f<°> Now F@) (z) satisfies the equation j and, if we apply the operatorf —j- to this equation n times, and use Leibniz' zdz theorem, we get C) * (l)"+! rm¦<*>+Bn+2) and so z*(J^j [z-"YW (z)) + Bn + 2) (~~\ \z-nY™ {»)) + z-*Y*> (z) = 0. This equation is at once reducible to D) *srw(«)-0, and so F(n) (z) is a solution of Bessel's equation for functions of order n. Again, C) may be written in the form {-*1 F<n+1> ()} - B» + 2)#-n-i F^1)(*) + z~nF<w> (*) = 0, * Theorie der Bessel'schen Funclionen (Leipzig, 1867), p. 51. The function is undefined when its order is not an integer. t The analysis is simplified by taking i«2=f, bo that d _d zdz *~df
3-58,3-581] BESSEL FUNCTIONS 71 so that z whence we obtain another recurrence formula E) z ~jf^ + nYW (z) = *F^' («> When avc combine A) with E) we at once deduce the other recurrence formulae F) F' (*) + F««+1> (z) = — F<«> (*), z G) F<w-]> (*) - F"l+1J (*) = 2 Consecjuently F(Jl) B^) satisfies the same recurrence formulae as ./„ {?), Yn and Yn (*). It follows from § 3'57 B) that (8) YM (z) = ^7rYn(z) +(log 2 - A solution of the equation V,t(y) = 0 in the form of a definite integral, which reduces'to the integral of § 3'571 when « = U, has boon constructed by Spitzor, Zeitschrift filr Math, und Phys. ill. A858), pp. 244-246; of. {} 3-583. 3*581. Neumann's expansion of Y{n) (z). The generalisation of the formula § 3-57 A) has been.given by Neumann*; it is u -1 cjrn—m—t ^ J T I v\ + s — 111 1 where sn — T + ^ + -tz -t- ... 4- - , so — (). 12 3 n To establish this result, we first define the functions Ln (z) and Un (z) by the equations C) f/,,w=s,,/),w+ I (r)^^ so that F'°)(^) = X0^)-f70D We shall prove that Ln(z) and CTn^) satisfy the recurrence formulae D) ?M+1 (*) = - /-»' (*) + (n/z) Ln (*), ff,*, (*) = ~ l/»' <*) + (n/a) 0^ («), and then A) will be evident by induction from § 3*58 B). * Theorie der Bessel'schen Functional (Leipzig, 1867), p. 52. See also Lommel, Studien ilber die BesseVschen Functioncn (Leipzig, 1868), pp. 82—84; Otti, BernMittheilungen, 1898, pp. 31—35; and Haentzschel, Zeitschrift/Ur Math. undPhrja. xxxi. A886), pp. 25—38.
72 THEORY OF BESSEL FUNCTIONS [CHAP. Ill It is evident that 1[ T and the first part of D) is proved. To prove the second part, we have dz\ zn ] ndz\ zn J (z) 1 °° ^ + J and the second part of D) is proved. It follows from §358B) that . Y»*> (z) - Ln+1 (z) + Un+1 (z)_ d[ If* (z) - Ln (*) + U« (z)\ zn dz\ and since the expression on the right vanishes when n = 0, it is evident by induction that it vanishes for all integral values of n. Hence and the truth of equation A) is therefore established. 3*582. The power series for Un (z), The function Un (z), which was defined in § 3-581 C) as a series of Bessel coefficients, has been expressed by Schlafli* as a power series with simple coefficients, namely ( Y A To establish this result, observe that it is true when n — 0 by § 3*51 C) and §3*57 A); and that, by straightforward differentiation, the expression on the right satisfies the same recurrence formula as that of § 3*581 D) for Un (z); equation A) is then evident by induction. Note. It will be found interesting to establish this result by evaluating the coefficient of (?s)n+2m in the expansion on the right of § 3-581 C). * Math. Ann. in, A871), pp. 146—147.
3-582-3-6] BESSEL FUNCTIONS 73 The reader will now easily prove the following formulae: <*) 3» (*) = G - log 2} Jn (z) - Un (z), C) • F<») (z) = Ln (z) + 3, (z) + (log 2 - 7) Jn (z), D) 3-583. The integral of Pomon's type for F('») (z). The Poisaon-Neumann formula of § 3-571 for F(°) (z) was generalised by Lommel, Shidien ilber die BesseVsehen Functionen (Leipzig, 1868), p. 86, with a notation rather different from Neumann's; to obtain Lommel's result in Neumann's notation, we first observe that, by differentiation of Poisson's integral for Jv (z), we have G^F^)f 6 (kg (* cos2 Q- and so, from § 3-582 C), *"cos (*ain and hence, since ^ (^) = ^ A) — 2 log 2 = - y - 2 log 2, we have the formula A) n»)(z) = -T1&I lhirco 1 [n + -%) 1 {it) J 0 {*<+i)K4)}n()+M), in which it is to bo remembered that Ln B) i« expressible as a finite combination of Bessel eoeflieiontu and powers of z. 3. Functions of the third kind. In numerous developments of the theory of Bessel functions, especially those which are baaed on Hankel's researches (Chapters VI and vii) on integral representations and asymptotic expansions of Jv {z) and Yv (z), two combina- combinations of Beasel functions, namely ./„ (z) ± i Yv (z), are of frequent occurrence. The combinations also present themselves in the theory of "Bessel functions of purely imaginary argument" (§ 3'7). It has consequently seemed desirable to Nielsen* to regard the pair of functions Jv{z) ±iYv(z) as standard solutions of Bessel'a equation, and he describes them as functions of the third, hind; and, in honour of Hankel, Nielsen denotes them by the symbol //. The two functions of the third kind are defined by the equations^ A) Bll\z) = Jv(z) + iYv{z), Hf\z) = Jv{z)-iYv{z). From these definitions, combined with § 3'54 A), we have B) a...w_-r_-A>Y=>rniz®, /,<»w.l-M-r1.*®.. v ' v v ' % sin vk " —iB}nv7r When v is an integer, the right-hand sides are to be replaced by their limits. Since Jv{z) and Y,,(z) satisfy the same recurrence formulae (§§3, 3*56), in which the functions enter linearly, and since the functions of the third kind * Ofversigt over det K. Damke Videmkabernen Setekabs Forfiandlinger, 1902,. p. 125. Hand- buck der Theorie dcr Cylinderfunktioiien (Leipzig, 1904), p. 16. f Nielsen uses the symbols ^(z), H^{z).
74 THEORY OF BESSEL FUNCTIONS [CHAP. Ill are linear functions (with constant coefficients) of Jv (z) and Yv (z), it follows that these same recurrence formulae are satisfied by functions of the third kind. Hence we can at once write down the following formulae: C) I <*) * <„ .f^ F) ,f^ G) << (8) V,tt Note. Kayleigli on several occasions, e.g. Phil. Mag. E) xliii. A897), p. 266 ; F) xiv. A907), pp. 350—359 [Scientific Papers, rv. A904), p. 290; v. A912), pp. 410—418], has used the symbol Dn (z) to denote the function which Nielsen calls $niH®' (z). * 3*61. Relations connecting the three kinds of Bessel functions. It is easy to obtain the following set of formulae, which express each function in terms of functions of the other two kinds. The reader will observe that some of the formulae are simply the definitions of the functions on the left. A) J,{M) B) J^ (*) (to Y (.w^( E) g^M^^ F) h B) (*). *" From E) and F) it is obvious that G) H™ (z) - e- If ?» (*), if (_8i (z). e r. sin sin 2» v{z)-e sin VTT VK )-e-" -viti y VTT ml Y ( > w
3-61-3-63] BBSSEL FUNCTIONS 75 3*62. Bessel functions with argument — z and zemni. Since Bessel's equation is unaltered if z is replaced by — z, we must expect the functions J±v(— z) to be solutions of the equation satisfied by J±v(z). To avoid the slight difficulty produced by supposing that the phases of both of the complex variables z and — z have their principal values*, we shall construct Bessel functions of argument zemiT\ where m is any integer, arg z has its principal value, and it is supposed that arg (zemni) — nnr + arg z. . Since Jv {z)\zv is definable as a one-valued function, it is obviously con- convenient to assume that, when the phase of z is unrestricted, Jv(z) is to be denned by the same convention as that by which zv is defined; and accordingly we have the equations A) Jv(z(Fwi)*=emv*iJv(z)> B) J_v(zemvi)^e-mvjriJ^v(z). The functions of the second and third kinds will now be defined for all values of the argument by means of the equations § 3*54 A), § 3*6 A); and then the construction of the following set of formulae is an easy matter: C) Yv (zemH) = e- mvni Yv (z) + 2i sin mv-ir cot vtt Jv (z), D) Y_,, (ze"lwi) = e~mynt F_„ (z) + 2i sin mm cosec vn Jv (z), E) II {t\ze»™) = e--- JffW (*) - 2e--* ^^ J? (z) = !Ea rjate? h<1) (z) - e—*6in™^ H«(z\ BinvTT " sini;7r " ainQ tw^5 B) sin "^ w (z) sini>7r Of theao results, C) was given by Hauled, Math. Ana. vm. A875), p. 454, in tho special case when m— 1 and i» is an integer. Forniulao equivalent t» E) and F) were obtained by WolKjr, Math. Ann. xxxvn. A890), pp. 411, 412, when ?»=1; sco § 6*11. And a memoir by («raf, Zeitichrift fiir Math, und Phya. xxxvm. AH93), pp. 116—120, contains tho general formulae. 3*63. Fundamental systems of solutions of Bessel's equation. It has been seen (§ 3*12) that Jv (z) and «/_„ (z) form a fundamental system of solutions of Bessel's equation when, and only when, v is nob an integer. We shall now examine the Wronskians of other pairs of solutions with a view to deter- determining fundamental systems in the critical case when v is an integer. • For Arg(-z) = Argitt, according as I(z) $ 0.
76 THEOBY OF BESSEL FUNCTIONS [CHAP. Ill It is clear from § 3'54 A) that m [Jv (*), Tv (z)} - - cosec m. Mi [Jv (*), J-v (*)} vz' This result is established on the hypothesis that v is not an integer; but con- considerations of continuity shew that A) «{/,(*).*,(*)}-2/(t*), whether v be an integer or not. Hence Jv(z) and Yv(z) always form a funda- fundamental system of solutions. It is easy to deduce that B) ?l and, in particular*, C) When we express the functions of the third kind in terms of Jv (z) and Yv {z\ it is found that D) Oft {H ^ (*), H? (z)} = - 2t m {Jv (z), Yv (*)} = - 4i/M, so that the functions of the third kind also form a fundamental system of solutions for all values of v. Various formulae connected with A) and C) have been given by Basset, Proc. London Math. Soc. xxi. A889), p. 55; they are readily obtainable by expressing successive differ- differential coefficients of Jv(z) and Yv{z) in terms of Jv{z), Jv' (z\ and Yv(z\ Yv' (z) by re- repeated differentiations of Bessel's equation. Basset's results (of which the earlier ones are frequently required in physical problems) are expressed in the notation used in this work by the following formulae: E) Jv (z) Yv" {z)-Yv {z)Jv" (»)« - Ij, F) JJ (z) JV' (*)- F/ (*) J» (*) = G) Jv (z) Y,'» (e) - Yv (z) JJ" (•)-! (^~-1) (8) j; (») Yy"' (z)- f; (z)Jv'" (?)= A0) Jv (t) A1) j;W Throughout these formulae Yv may be replaced by J_v if the expressions on the right are multiplied by -sim/ir; and Jv> Yv may be replaced by E^\ H^ throughout if the expressions on the right are multiplied by - 2i*. ¦ Of. Lommel, Math. Ann. tv. A871), p. 106, and Hankel, Math. Ann. vm. A875). p. 457.
3-7] BESSEL FUNCTIONS 77 An associated formula, due to Lommol*, Math. Ann. iv. A871), p. 106, and Hankel, Math. Ann. vin. A875), p. 458, is A2) •/¦„(*) rr+1@-J,+tw f,(Z)=-^. This is proved iu the same way as § 3*2 G). 3'7. Bessel functions of purely imaginary argument. The differential equation which differs from Bessel's equation only in the coefficient of y, is of frequent occurrence in problems of Mathematical Physics; in such problems, it is usually desirable to present the solution in a real form, and the fundamental systems Jv (iz) and /_„ (iz) or Jv (iz) and Yv (iz) are unsuited for this purpose. However the function e-»"Tri Jv (iz) is a real function of z which is a solution of the equation. It is customary to denote it by the symbol Iv(z) so that When 2 is regarded as a complex variable, it is usually convenient to define its phase, not with reference to the principal value of argi.gr, as the consideration of the function Jv {iz) would suggest, but with reference to the principal value of arg z, ho that /„ (z) = e~*"ni Jv (ze*vi), (— 7r < arg z ^ \it), /„ (z) = e*"ni Jv (ze~^% (^7t < arg z ^ it). The introduction of the symbol Iv (z) to denote "the function of imaginary argument" is due to Bassetf and it is now in common use. It should be men- mentioned that four yeans before the publication of Basset's work, Nicolas+ had suggested the use of the symbol l<\ (z), but this notation has not been used by other writers. Tho rohitive positions of I'ure and Applied MathoinaticH on tlio Continent as compared with thia country are remarkably ilhmlralud by tho fact that, in Nielsen's standard troatiNO^, neither tho function Iv B), nor the aocond solution Ky(z), which will bo defined immediately, i« even mentioned, in npito of their importance in physical applications. The function I~v{z) is also a solution- of A), and it is easy to prove (cf. §8-12) that ' C) 2^ * Lommel gavi! the correHponding formula for Neumann'H function of the second kind. I Proc. Cam!). Phil. Sac. vi. A889), p. 11. LTm's paper wa« first published in 1880.] Basset, in this puper, defined the funotion of integral order to be ?+n Jn (iz), but ho subsequently changed it, in his Hydrodynamics, 11. (Uambridgo, 1H88), p. 17, to that given iu the text. The more recent definition in now universally used. X Ann. tici. de VJ&cole norm. sup. B) xi. A882), eupplement, p. 17. § Handbuch der Theorie. der Cylinderfunktionen (Leipzig, 1904).
78 THEOEY OF BESSEL FUNCTIONS [CHAP. Ill It follows that, when v is not an integer, the functions /„ (z) and /_„ (z) form a fundamental system of solutions of equation A). In the case of functions of integral order, a second solution has to be con- constructed by the methods of §§ 3*5—3'54 The function Kn (z), which will be adopted throughout this work as the second solution, is denned by the equation An equivalent definition (cf. § 3'5) is It may be verified, by the methods of § 3^5, that Kn (z) is a solution of A) when the order v is equal to n. The function Kv(z) has been defined, for unrestricted values of v, by Macdonald*, by the equation F) JT.W-^W-^W and, with this definition, it may be verified that G) Kn(z) = Mm Kv(z). It is easy to deduce from F) that (8) ' Kv (z) = iTrt'e^ Hl? (iz) = ^ie"^ H^l (iz). The physical importance of the function Kv(z) lies in the fact that it is a solution of equation A) which tends exponentially to zero as z-+.<x> through positive values. This fundamental property of the function will be established in § 7-23. The definition of Kn(z) is due to Baaaot, Proc. Gamb. Phil. Soc. vi. A889), p. 11, and his definition is equivalent to that given by equations D) and E); the infinite integrals by which he actually defined the function will be discussed in §§ 6-14, 6*15. Basset subse- subsequently modified his definition of the function in his Hydrodynamics, n. (Cambridge, 1888), pp. 18—19, and his final definition is equivalent to ^-—. —^—^ ~^- 2n+1|_ ov dv Jv~n In order to obtain a function which satisfies the same recurrence formula© as Iv (z), Gray and Mathews in their work, A Treatise on Beisel Functions (London, 1895), p. 67, omit the factor l/2n, so that their definition is equivalent to 2J_ dv Ov The only simple extension of this definition to functions of unrestricted order is by the formula K, iz) s \v cot V* {/_„ (z) - Iv («)}, * Proc. London Math. Socxxx. A899), p. 167.
3*71] BESSEL FUNCTIONS 79 (cf. Modern Analysisy § 17*71) but this function .suffers from the aerioua disadvantage that it vanishes whenever 2i» is an odd integer. Consequently in this work, Macdonald's function Will be used although it has the disadvantage of not satisfying the same recur- recurrence formulae as Iv (z). An inspection of formula (8) shews that it would have been advantageous if a factor \rr had been omitted from the definition of Kv(z); but in view of the existence of extensive tables of Macdonald's function it is now inadvisable to make the change, and the presence of the factor is not so undesirable- as the presence of the corresponding factor in Schliifli's function ($ 3*54) because linear combinations of Iv (z) and K„ (z) are not of common occurrence. 3*71. Formulae connected with Iv{z) and Kv{z). We shall now give various formulae for Iv(z) and Kv(z) analogous to those constructed in §§ 3*2——3*6 for the ordinary Bessel functions. The proofs of the formulae are left to the reader. o,. 9j> A) /,_, (*) - /„+, (z) = L} L (*), Kv-r (z) - Kv+i (*) = ~^Kv(z), z z B) /,_! (z) + /,.H 0) = 2/; (z), Kv-X (z) + Kv+l (z) = - 2KV' (g), C) zlj (z) + vlv (z) = zlv^ (z), zK; (z) + vKv (z) = - zK^ (z), D) zi; (z) - vlv (z) = zl,+l (z), zK; (z) - vliv (z) = - zKy+1 (z), d V" (/,.<*)) _ frim(z) ( d \»<KP(*)\ _, ^K^(z) G) V(*) - 11 (a), ^.' (*) = ~ Ar, («), (8) /_„ {z) = /„ (^), Ar_, (^) = 7C (^). The following integral formulae are valid only when R(v +• \) > 0 : :o.sh (z cos 0) wii •1 V m C0Hh (^ cos "*cosh
80 THEORY OF BBSSEL FUNCTIONS [CHAP. Ill These results are due to Basset. We also have A0) 1 n+i (*) - -—j \t ^ -1 (n _ r), B0)r (n) r ,_x - ,^v (~Y(n + r)\ -* ^ A4) iT0 (*) = - log (\z) . /. (z) + 2 ^ f (tn + 1), A5) ^.W-gS, m, (i^-» m A6) Z"o(*) = -^ f Iv (zemvi) = e"^ I A8) Zv (^e-) = e-™** Kv (z) - iri ^^ Iv (z), sm vtt A9) »(!,(«), ^r(*)}—1/*, B0) i9 oo zv+1 (*> + /„_,! (*) jr, (o = \\z. The integral involved in A6) has been discussed by Stokes (cf. § 3-572). The integrals involved in (9) and the series in A4) were discussed by Riemann in his memoir "Zur Theorie dev Nobili'schen Farbenringe," Ann. der Pkysik und Chemie, B) xcv. A855), pp. 130—139, in the special case in which v=0\ he also discussed the ascending power series for 70 B). The recurrence formulae have been given by Basset, Proc. Camb. Phil. Soc. vi. A889), pp. 2—19; by Macdonald, Proc. London Math. Soo. xxix. A899), pp. 110—115; and by Aichi, Proc. Phys. Math. Soc. of Japan, C) n. A920), pp. 8—19. Functions of this type whose order is half an odd integer, as in equations A0) and A2), were used by Hertz in his Berlin Dissertation, 1880 [Qes. Werke, 1. A895), pp. 77—91]; and he added yet another notation to those described in § 3*41.
3-8] BESSEL FUNCTIONS 81 3'8. Thomson's functions ber (z) and bei (z) and their generalisations. A class of functions which occurs in certain electrical problems consists of Bessel functions whose arguments have their phases equal to \ir or ? it. The functions of order zero wore first examined by W. Thomson*; they may be defined by the equationf A) ber (a:) + i bei (x) = Jn (ari \Ji) = /„ {x ^i), where x is real, and ber and bei denote real functions. For complex argu- arguments we adopt the definitions expressed by the formulae B) ber (z) ± i boi (z) = /„ (zi V+ i) = Jo (* V ± »)• Hence we have (8) ber (,)-!-«?<$ Extensions of these definitions to functions of any order of the first, second and third kinds have been effected by llusscllj: and Whitehead§. The functions of the second kind of order zero were defined by Russell by a pair of equations resembling B), the function /„ being replaced by the function Ko, thus E) kcr (z) ± i kei (z) - Ko (z«/± i). Functions of unrestricted order v were defined by Whitehead with reference to Bessel functions of the first and third kinds, thus F) ber,, (z) ± i bei, (z) •¦= ./„ (ze'iiri), G) her,, (z) ± i hoi, (z) = //„"> {ze^ni). It will be observed that|| (8) ker {z) = - hrr hei {z), kei (z) = ^ir her (z), in consequence of § ^(8). The following Heries, due to Russell, are obtainable without difficulty: (<)) ker 0) = - log D*). ber (z) + \tt bei (z) * Preflidentiiil AddrcH« to the Instituto of Kloofcrioul Enp;inoei'u, IH80. [Math, and Plnjs. PapiTH, in. A8»O), p. 4<J. ] f In the caao of Eunctiona of zoro ordor, it iH ciiHtonmry to omit tho anflix which indicates the order. X Phil. Mag. @) xvn. A909), pp, 524—r>82. § Quarterly Journal, xui. A911), pp. 31C—342. || Integrals equal to ker B) and koi (z) oocnr in a memoir by Hertz, Ann. der Physik und Chemie, C) xxii. A884), p. 450 [Gem. Werke, 1. A89G), p. 2891- w. b. if. 6
82 THEORY OF BBSSEL FUNCTIONS [CHAP. HE A0) kei (z) = - log (\z). bei (z) - \ir ber (z) It has also been observed by Russell that the first few terms of the expansion of ber2 B)+bei2 {z) have simple coefficients, thus but this result had previously been obtained, with a different notation, by Nielsen (cf. § 5-41); the coefficient of (%s)irn in the expansion on the right is l/[(m !J. Bm)!]. Numerous expansions involving squares and products of the general functions have been obtained by Russell; for such formulae the reader is referred to Russell's memoir and also to a paper by Savidge*. Formulae analogous to the results of §§3*61, 362 have been discussed by Whitehead; it is sufficient to quote the following here: A2) ber_v (z) = cos wr. ber,, (z) — sin wrr. [heiv (z) — bei,, (z)], A3) bei_v (z) = cos vrr. beiv (z) + sin vrr. [herv (z) — ber,, (z)], A4) her_v (z) =* cos vir. herv (z) — sin vrr. heik (z), A5) hei_v (z) = sin vjt . herv (z) + cos vir. heiv (z). The reader will be able to construct the recurrence formulae which have been worked out at length by Whitehead. The functions of order unity have recently been examined in some detail by B. A. Smithf. 3*9. The definition of cylinder functions. Various writers, especially SonineX and Nielsen§, have studied the general theory of analytic functions of two variables ^v (z) which satisfy the pair of recurrence formulae 913 A) ».-»(*)+ ».**(*)« 7 *,(*), in which z and v are unrestricted complex variables. These recurrence formulae are satisfied by each of the three kinds of Bessel functions. Functions which satisfy only one of the two formulae are also discussed by Sonine in his elaborate memoir; a brief account of his researches will be given in Chapter x. * Phil. Mag. F) xix. A910), pp. 49—58. f Proc. American Soc. of Civil Engineer^ xlvi. A920), pp. 375—425. t Math. Ann. xvi. A880), pp. 1—80. § Handbuch der Theorie der Cy UnderfunktiantH. (Leipzig, 1904), pp. 1, 42 et seq.
3'9] BESSEL FUNCTIONS 83 Following Sonine we shall call any function 9Bv{z), which satisfies both of the formulae, a cylinder function. It will now be shewn that cylinder functions are expressible in terms of Bessel functions. When we combine the formulae A) and B), we find that C) z<®J (z) + v<@v (z) = *$Lx (z), D) zc@; (z) - v<$v (z) = - zK+l {z), and so, if & be written for z (d/dz), we deduce that E) (* + *) #,(*)-*«•_,(*), F) (^-^K(z)^-z%,+1(z). It follows that #,(), that is to say G) V^rE)-0. Hence <#„ (z) = <*„/„ (z) + &„ K, (*), where a,, and bv are independent of z, though they may depend on v. When we substitute in C) we find that «„«/„_, 0) + hvYv^ (z) = a,,-!J,._i (z) + by-iYr-i (z), and so, since ./„_! (z)/ Yv_t (z) i.s not independent of z, we must have Hence a,, and bv must be periodic functiiona of v with period unity; and, conversely, if they arc. such functions of v, it is easy to see that both A) and B) arc satisfied. Hence the general solution of A) and B) is (8) #'„ (z) = or, („) J,. (z) + «r2 (v) Yv (z), whore, ¦n-i(v) and ¦nry(r) arc arbitrary periodic functions nf v with period unity. It may be observed that an equivalent solution is (9) K (z) = W| (,) //„<" B) + flr4 (») ^<4' («). A difference oquutioii, which is more general thiin A), him been cxatuiiied by Messenger, xxxiv. A905), pp. 52—71 ; in cortnin circuin.stancoa the solution i.s expressible by BeaMol functious, though it UHiially involves hyporgeoinotric functions. Notk. The name cylinder function i.s used by Nielsen to denote Jv (s), \\, (i), //,,<') (:"> and i/v('i|) (z) sus well as the more general function.s diweuHaed in thin section. This procedure ia in accordance with the principle laid down by Mittag-Lefflor that it irt, in general, undesirable to associate functioim with the names of particular mathematicians. The name cylinder function in derived from the fact that normal solution* of Laplaoc'« equation in cylindrical coordinates are (cf. § 4"8 and Modern Analysis, § 18'5).
THEORY OE BESSEL FUNCTIONS [CHAP. Ill writers*, following Heinet who called Jn{z) a Fourier-Bessel function, call Jn(z) • function. >ugh Bessel coefficients of any order were used long before the time of Bessel 3, 1*4), it seems desirable to associate Beasel's name with them, not only because :come generally customary to do so, hut also because of the great advance made by a the work of his predecessors in the invention of a simple and compact notation functions. Bessel's name was associated with the functions by Jacobi, Journal fiir Math. xv. A836), p. 13 [Qes. Math. Werke, vi. A891), p. 101]. "Transcendentium /** naturam varios- que usus in determinandis integralibus definitis exposuit ill. Bessel in commentatione celeberrima." A more recent controversy on the name to be applied to the functions is to be found in a series of letters in Nature, lx. A899), pp. 101, 149, 174; lxxxi. A909), p. 68. * E.g. Nicolas, Ann. Sci. de VEcole norm. sup. B) xi. A882), supplement, f Journal fur Math. lxix. A868), p. 128. Hoine also soems to be responsible for the term cylinder function.
CHAPTER TV DIFFERENTIAL EQUATIONS 4*1. Daniel Bernoulli's solution of Riccati's equation. The solution given by Bernoulli* of the equation A) %-a.-+by consisted in shewing that when the index )i has any of the values 0; -f-*; -8,-5; -V. -V-; -?,-?; •••> while a and 6 have any constant vainest, then the equation is soluble by means of algebraic, exponential and logarithmic functions. The. values of vi just given are comprised in the formula 4w <2> " " - 2,,,. ± I • where m is zero or a positive integer. Bernoulli's method of solution is as follows : Tf u bo called the index of the equation, it is first proved that the general equation'! of index n is transformable; into the general equation of index N, where C) N=- " ; and it is also proved that the general equation of index n is transformable into the general equation of index v, where; D) v = - h - 4. The liiccati equation of index zero is obviously inUigrable, because the variables are separable*. Hence,by D), the e'.epialiem of ineicx — [< is inU'-grabh1. Hence by (tf), the eejuation of itidesx — 1j is integrabl«\ If Liu's pn»ec;ss be con- continued by using the transformat.iems C) and D) alternately, we1 arrive1 at I.be set of soluble cases given above1, and it is eiasy te> see1 thai, these ease's ;in> comprised in the general lonnula B). * lixercitationcs quaeiiam nuithematicar (Yemen, 1721), pp. 77- -Hi); Act<i fr'.nidilonnn, 17'Ji5, pp. 473—47E. The notation us«d by Bernoulli Han Ixiun uliKlitly modified; and in this unalyaia n ta uot reatricteel to be an integer. + It ia assumed that neither a nor b in zero. If oithor were zoro tho vaviahlcm would obviuuHly be separable. $ That is, the eejuation in which a ami h havu arbitrary viUuoh.
86 THEORY OF BESSEL FUNCTIONS [CHAP. IV 4*11. Daniel Bernoulli's transformations of Riccati's equation. Mow that the outlines of Bernoulli's procedure have been indicated, we proceed to give the analysis by which the requisite transformations are effected. Take § 4'1 A) as the standard equation of index n and make the substitutions n+l ' * 7 [Note. The substitutions are possible because -1 is not included among the values of n. The factor n+1 in the denominatoi was not inserted by Bernoulli; the effect of its presence is that the transformed equation is more simple than if it were omitted.] The equation becomes 1 dY h that i is where N — — nj(n + 1); and this is the general equation of index JV. Again in § 4'1 A) make the substitutions The equation becomes d% V' where v = — n — 4; and this is the general equation of index v. The transformations described in § 4*1 are therefore effected, and so the equation is soluble in the cases stated. But this procedure does not give the solution in a compact form. 4*12. The limiting form of Riccati's equation, with index —2. When the processes described in §§41, 4'11 are continually applied to Riccati's equation, the value to which the index tends, when m -*- qo in § 4*1 B), is - 2. The equation with index ~ 2 is consequently not soluble by a finite number of transformations of the types hitherto under consideration. To solve the equation with index — 2, namely write y — v/z, and the equation becomes dv . , „ z -j- — a + v + bv2; dz ' and this is an equation with the variables separable. • Hence, in this limiting case, Riccati's equation is still soluble by the use of elementary functions.
4-11-4-13] DIFFERENTIAL EQUATIONS 87 This solution was implicitly given by Euler, Inst. Calc. Int. n. (Petersburg, 1769), § 933, p. 185. If we write (cf. § 4-14) y = —j-rri the equation which determines y is which is homogeneous, and consequently it is immediately soluble. Enler does not seem to mention the limiting case of Riccati's equation explicitly, although he gave both the solution of the homogeneous linear equation and the transforma- transformation which connects any equation of Riccati's type with a linear equation. It will appear subsequently (§§ 4*7—4*75) that the only cases in which Riccati's equation is soluble in finite terms are the cases which have now been examined; that is to say, those in which the index has one of the values 0; -f -*; -§, -|; .... -2, and also the trivial cases in which a or b (or both) is zero. This converse theorem, due to Liouville, is, of course, much more recondite than Bernoulli's theorem that the equation is soluble in the specified cases. 4*13. Eider's solution of Riccati's equation. A practical method of constructing a .solution of Riccati's equation in the soluble cases was devised by Euler*, and this method (with some slight changes in notation), will now be explained. First transform Riccati'a equation, § 4*1 A), by taking new variables and constants as follows: A) yss — Tf/b, ab — ~dl, ?i = 2<7-2; the transformed equation is B) ^ + r? - c'z2"-* = 0 ; and the soluble cases are those in which l/q is an odd integer. Define a new variable w by the equation C) ,-^+1*?, so that the equation in w is D) ^ + 2c**-1 dy + (q - 1) czi-'w = 0. A solution in series of the last equation is r-0 provided that A (M + G ^lKV ±<1~ 1) * Nov. Comm. Acad. Petrop. vm. A700—1701) [1703], pp. 'A—03 ; and ix. A762—1768) [1764], pp. 154—169.
88 THEORY OF BESSEL FUNCTIONS [CHAP. IV and so the series terminates with the term Amz~qm if q has either of the values ± l/Bm + 1); and this procedure gives the solution* examined by Bernoulli. The general solution of Riccati's equation, which is not obvious by this method, was given explicitly by Hargreave, Quarterly Journal, vn. A866), pp. 256— 258, but Hargveave's form of the solution was unnecessarily complicated; two years later Cayley, Phil. Mag. D) xxxvi. A868), pp. 348—351 [Collected Papers, vii. A894), pp. 9—12], gave the general solu- solution in a form which closely resembles Euler's particular solution, the chief difference between the two solutions being the reversal of the order of the terms of the series involved. Cayley used, a slightly simpier form of the equation than B), because he took constant multiples of both variables in Riccati's equation in such a way as to reduce it to E) ^+^_^-2=0. 4" 14. Gayley's general solution of Riccati's equation. We have just-seen that Riccati's equation is reducible to the form dz given in § 4)*13 B); and we shall now explain Cayley'sf method of solving this equation, which is to be regarded as a canonical form of Riccati's equation. When we make the substitution| v) = d(\ogv)/dz, the equation becomes A) g_c2^-aw==0; and, if Uj, and U2 are a fundamental system of solutions of this equation, the general solution of the canonical form of Riccati's equation is where Cj and <72 are arbitrary constants and primes denote differentiations with respect to z. To express Ux and U3 in a finite form, we write v = w exp (czQjq), so that the equation satisfied by w is § 4-13 D). A solution of this equation in w proceeding in ascending powers of & is (g-PCg-P and we take Ux to be exp (czQ/q) multiplied by this series. • When the index n of the Biccati equation is - 2, equation D) is homogeneous. t Phil. Mag. D) xxxvi. A868), pp. 848—351 [Collected Papers, vn. (-1894), pp. 9—12]. Of. also the memoirs by Euler which were cited in § 4*13. t This is, of course, the substitution used in 1702 by James Bernoulli; cf. § l-l;
4-14] DIFFERENTIAL EQUATIONS 89 Now equation A) is unaffected by changing the sign of c, and so we take and both of these series terminate when q is the reciprocal of an odd positive integer. Since the ratio f/j : 272 is the exponential function exp Bc2q/q) multiplied by an algebraic function of 2$, it cannot be a constant; and so Uu Us form a fundamental system of solutions of A). If q were the reciprocal of an odd negative integer, we should write equation A) in the form whence it follows that where yx and y» are constants, and The series which have now been obtained will be examined in much greater detail in §§ 4*4—4'42. The reader .should have uo difficulty in constructing the following solutions of Ricoati's equation, whon it is soluble in finite terms. (i) (iii) (i) (ii) (iii) Equation (drjjdz) +7,2=1 (drfjdz)+7,2 => z ~ yb Equation (drjIdzW-Z-* (drj^+rj^Z-W Values of f/1( C/2 exp (±z) A + r>21('6 + Xtf-apfi) exp (+ bzm) Values of Vif V% z exp (+ \jz) z(l + 32-'/3)exp(±32-1/3) 2.(!!5.::.!.!?.a:!'.):i^.±6.::r.) It is to be noticed that the series Dlf U^ (or Vlt V2, as the case may be) are BUpposed to terminate with the term before the first term which has a zero factor in the numerator; see § 4*42 and Glaishor, Phil. Trans, of the Royal Soc. olxxii. A881), p. 773.
90 THEORY OF BESSEL FUNCTIONS [CHAP. IV Among the writers who have studied equation A) are Kummer, Journal fUr Math. xn. A834), pp. 144—147, Lobatto, Journal par Math. xvn. A837), pp. 363—371, Glaisher (in the memoir to which reference has juist been made), and Suchar, Bull, de la Soc. Math, de France, xxxn. A904), pp. 103—116; for other references see § 4-3. The reader will observe that when ?=0, the equation A) is homogeneous and imme- immediately soluble; and that the second order equation solved by James Bernoulli (§ 1*1) is obtainable by taking 2=2 in A), and so it is not included among the soluble cases. • 4*15. Schlafli's canonical form of Riccati's equation. The form of Riccati's equation which was examined by Schlafli* was This is easily reduced to the form of § 413B) by taking -t~ala as a new independent variable. To solve the equation, Schlafli wrote dloi and arrived at the equation the general solution of the equation in y is y = c1F(a> t) + G»t-*F (- a, t). The solution of A) is then = <hV**F (a + l>t)+c2F(-a-1, t) U ~ ClF(a, t) + c2t~aF (- a, t) The connexion between Riccati's equation and Bessel's equation is thus rendered evident; but a somewhat tedious investigation is necessary (§ 4*43) to exhibit the connexion between Cayley's solution and Schlafli's solution. Note. The function <? : s, denned as the series 1+2 + 1 °2 . 1 ^l + z+2" 2(z+l)T2. + which is evidently expressible in terms of Schlafli's function, was used by Legendre, Mdments de Odometrie (Paris, 1802), note 4, in the course of his proof that n ia irrational. Later the function was studied (with a different notation) by Clifford; see a posthumous fragment in his Math. Papers (London, 1882), pp. 346—349. • Ann. di Mat. B) i. A8B8), p. 232. The reader will see that James Bernoulli's solution in series (§ 1*1) ia to be associated with Schlafli's solution rather than with Cayley's solution, f Tais notation ahould be compared with the notation of § 4*4.
4*15,4-16] DIFFERENTIAL EQUATIONS 91 It is obvious that Jv (z)=(\z)» F (v, - ?z2), and it has recently been suggested* that, because the Schlafli-Clifford notation simplifies the analysis in the discussion of certain problems on the stability of vertical wires under gravity, the standard notation for Bessel functions should be abandoned in favour of a notation resembling the notation used by Schlafli-Cliftbrd :—a procedure which seems com- comparable to a proposal to replace the ordinary tables of trigonometrical functions by tables of the functions 4*16. Miscellaneous researches on Riccatts equation. A solution of Riocati's equation, which involves definite integrals, was given by Murphy, Trans. Camb. Phil. Soc. in. A830), pp; 440—443. The equation which he considered is ~ + Au* = Blm, at ' and, if a be written for l/(m+2) and A~l d (log y)/dt for u, his solution (when ABa?**!) is A-» [(f> (A) exp (^//z) +e/> (I/A) exp (A*1/")] dh, where d> (h) = ehh~a P e-^ha~l dh= 2 *" rv ; Jo «»0a If \jh be written for h in the second part of tho integral, then the last expression given for y reduces to irit multiplied by tho residue at the origin of h~l (p (h) oxp (tl'ajh), and the connexion between Murphy's solution and Schliifli's solution (§ 4'15) is evident. An investigation w.-is published by Clmllis, Quarterly Journal, vil. A806), pp. 51—58, which shewed how to connect two equations of tho type of § 4-13 B), namely in one of which lfq is an odd positivo integer, and in tho other it is an odd negative integer. This investigation is to be associated with the discovery of the two types of solution given in § 4-14. mi • du an , ., „ ,. The equation -,- + —h 6z" uz - czm~0, 1 dz z which is easily transformed into <ui equation of Riccati's type by taking 2»-« + 1 and z?u na new variables, was investigated by Ruwsou, Messenger, vn. A878), pp. 69—72. Ho trans- transformed it into the equation ctz by taking bu*=czaly; two such equations are called cognate Riccati equations. A somewhat similar equation wan reduced to Riccati's typo by Bimsinne, Journal de Muth. XVI. A851), pp. 255—256. The connexions between the various types of equations which different writers have adopted as canonical forms of Riccati's equation havo been set out in a paper by Greenhill, Quarterly Journal, xvi. A879), pp. 294—298. * Greenhill, Engineering, cvn. A1I9), p. 334; Phil. Mag. F) xxxvin. A919), pp. 601—528; see also Engineering, oix. A920), p. 851.
92 THEOBY OF BESSEL FUNCTIONS [CHAP. IV The reader should also consult a short paper by Siaoci, Napoli Rendiconti, C) vn. A901), pp. 139—143. And a monograph on Riccati's equation, which apparently contains the majority of the results of this chapter, has been produced by Feldblum, Warschau Univ. Nach. 1898, nos. 5, 7, and 1899, no. 4. 4*2. The generalised Riccati equation. An obvious generalisation of the equation discussed in § 4*1 is A) ^ where P, Q, R are any given functions of z. This equation was investigated by Euler*. It is supposed that neither P nor R is identically zero; for, if either P or R is zero, the equation is easily integrable by quadratures. It was pointed out by Enestrom, Encyclopddie des Soi. Math. n. 16, § 10, p. 75, that a special equation of this type namely nxx dx - nyy dx+xx dy=xy dx was studied by Manfrediua, De constructione aequationwn differentialum primi gradus (Bologna, 1707), p. 167. "Sed tamen haec eadem aequatio non apparet quomodo construi- bilis sit, neque enlni videmus quomodd illam integi-ernus, nee quomodo indeterminatas ab invicem separemus." The equation A) is easily reduced to the linear equation of the second order, by taking a new dependent variable u denned by the equation f The equation then becomes 1 dR\ du Conversely, if in the general linear equation of the second order, /A\ dhb du n (where pQ, p1} p2 are given functions of z), we write E) u=ebldz, the equation to determine y is dz p0 p0J y' which is of the same type as A). The complete equivalence of the generalised Riccati equation with the linear equation of the second order is consequently established. The equations of this section have been examined by Anisimov, Warschau Univ. Nach. 1896, pp. 1—33. [Jakrbuch fiber die Fortschritte der Math. 1896, p. 256.] * Nov. Gomvi. Acad. Petrop. vni, A760—1761) [1763], p. 32; see also a short paper by W. W. Johnson, Ann. of Math. m. A887), pp. 112—115. f This is the generalisation of James Bernoulli's substitution (§1*1). See also Euler, Inst. Calc. Int. n. (Petersburg, 1769), §§ 831, 852, pp. 88,104.
4-2,4-21] DIFFERENTIAL EQUATIONS 93 4*21. Eiders theorems concerning the generalised Riccati equation. It has been shewn by Euler* that, if a particular solution of the generalised Riccati equation is known, the general solution can be obtained by two quadratures; if two particular solutions are known the general solution is obtainable by a single quadrature f. And it follows from theorems discovered by Weyr and Picard that, if three particular solutions are known, the general solution can be effected without a quadrature. To prove the first result, let y0 be a particular solution of and write y = yo+ \/v. The equation in v is dv dz of which the solution is v exp [f(Q + 2Ry0) dz} + JR exp {J(Q + 2%,) dz}. dz = 0, and, since v = 1/B/ —yo), the truth of the first theorem is manifest. To prove the second, let yQ and yx be two particular solutions, and write ~ y - yi' The result of substituting (yiW — yo)/(w — 1) for y in the equation is (w — 1 )a (^ w — 1 rf^ w - 1 ^2 w - 1 \ w -1 and, when we substitute for (dyjdz) and (dyo/dz) the values P + Qy^ + Ry^ and P+ Qyo + -Z2y,,2, the last equation is reduced to 1 dw tv n - -y- = lv?/o — iit/i, so that w = c exp j/Giy0 - i^i) <i*j, where c is the constant of integration. Hence, from the equation defining w, we set; that y in expressed as a function involving a single quadrature. To prove the third result, let ;</„ and y, bo the solutions already specified, let 2/a be a third solution, and let c' be the value to be assigned to c to make y reduce to y... Then and this is the integral in a form free from quadratures. * Nov. Comrn. Ac.ad. Petrop. vni. A760—17C1) [1763], p. 32. t Ibid. p. 59, and rx. A76-2—1763) [1764], pp. 163—164. See also Minditig, Journal fiir Math. xr,. A850), p. 361.
THEORY OF BESSEL FUNCTIONS [CHAP. IV It follows that the general solution is expressible in the form Hence it is evident that, if jfc, ytl y,, jfc be any four solutions, obtained by giving G the values 0» Ct, Q%, d respectively, tAen *te crow-rofro is independent of z; for it is equal to In spite of the obvious character of this theorem, it does not seom to havo been noticed until some forty years ago*. Other properties of the generalised Riccati equation may be derived from properties of the corresponding liDear equation (§ 4-2). Thus Raffy \ has gi von two methods of reducing the Riccati equation to the canonical form these correspond to the methods of reducing a linear equation to its normal form by changes of the dependent and independent variables respectively. Various properties of the solution of Riccati's equation in which P, Q, ft arc rational functions have been obtained by C. J. D. Hill, Journal filr Math. xxv. A843), pp. 23 - 37 ; Autoune, Comptes Rendus, xcvi. A883), pp. 1354—1356; cxxvin. A899), pp. 410—412 ; ami Jamet, Comptes Rendut de VAssoc. Frangaise (Ajaccio), A901), pp. 207—228 ; Ann. dr la Fac. des Sci. de Marseille, xn. A902), pp. 1—21. The behaviour of the solution near singularities of P, Q, R has been studied by Kulkon- hagen, Jfieuw Archief voor Wiskunde, B) vi. A905), pp. 209—248. The equation of the second order whose primitive is of the type where eu c2, c% are constants of integration (which is an obvious generalisation of the primitive of the Riccati equation), has been studied by Vessiot, Ann. de la Fac. dc* Sci. dr. Toulouse, ix. A895), no. 6 and by Wallenburg, Journal fiir Math. exxt. A900), pp. 210 217; and Comptes Rendus, exxxvn. A903), pp. 1033—1035. • Weyr, Abh. bnhm. Ges. Wm. F) vin. A875-1876), Math. Mem. x. p. 30 ; Pioanl ^nn Sri de VEcole uorrn. tup. B) vi. A877), pp. 342-343. Picard's thesis, in which the result -j» con- contained, « devoted to the theory of sui-faces and twisted curves-a theory in which Hicoali'H equation has various applications. t Nouv. Ann. de ilpth. D) n. A902), pp. 529—545.
4'3] DIFFERENTIAL EQUATIONS 95 4*3. Various transformations of Bessel's equation. The equations which we are now about to investigate are derived from Bessel's equation by elementary transformations of the dependent and inde- independent variables. The first type which we shall consider is* ,n d'u p(p+l) A) _?_cHt = /!-iL__u, • where c is an unrestricted constant. The equation is of frequent occurrence in physical investigations, and, in such problems, p is usually an integer. The equation has been encountered in the Theory of Conduction of Hoat and the Theory of Sound by Poisson, Journal de V&cole Poly technique, xn. (cahier 19), A823), pp. 249—403; Stokes, Phil Trans, of the Royal Soc. 1868, pp. 447—464 [Phil. Mag. D) xxxvi. A868), pp. 401—421, Math, and Phys. Papers, IV. A904), pp. 299—324]; Rayleigh, Proc. London Math. Soc. iv. A873), pp. 93—103, 25S--283 [Scientific Papers, I. A899). pp. 138, 139]. The special equation in which p = 2 occurs in the Theory of the Figure of the Earth; see Ellis, Camb. Math. Journal, u. A841), pp. 169—177, 193—201. Since equation A) may be written in the form Q d? (fur*) d (usr*) , ., „ , ..., . rt aW + Z" dz " ^ + (P + &) 1 • uz = °- its general solution is B) u = z*%H{ciz\ Consequently the equation is equivalent to Bessel's equation when p is unrestricted, and no advantage is to be gained by studying equations of the form A) rather than Bessel's equation. But, when p is an integer, the solu- solutions of A) nre expressible "in finite termsf" (cf. § 3 4), and it is bhen frequently desirable to regard A) as a canonical form. The relations between various types of solutions of A) will be examined in detail in §§ 4*41—4>'4>']. The second type of equation is derived from A) by a transformation of the dependent variable which makes the indicial equation have a zero root. The roots of the indicial equation of A) are p + 1 and -p, and so we write u = vz~p; we arc thus led to the equation C) __^__c2w:=0, of which the general solution is D) fl = ^+K^,+i(cu). * See Plana, Mem. della II. Acctid. delle Sci. di Torino, xxvi. A821), pp. 519—538, andPaoli, Mem. di Mat. e di Fis. della Soc. ltaliana delle Sci. xx. A828), pp. 183—188. t This was known to Plana, who studied equations A) and E) in the paper to whioh referenoe has just been made. ,
96 THEORY OF BESSEL FUNCTIONS [CHAP. IV Equation C), which has been studied in detail by Bach, Ann. Sci. de Vtlcole norm. sup. B) in. A874), pp. 47—68, occurs in certain physical investigations; see L. Lorenz, Ann. der Physik und Chemie, B) xx. A883), pp. 1—21 [Oeuvres Scientifiques, I. A898), pp. 371— 396]; and Lamb, Hydrodynamics (Cambridge, 1906), §§ 287—291. Solutions of equation C) in the form of continued fractions (cf. §§ 5-6, 9 5) have been examined by Catalan, Bulletin de I'Acad. R. de Belgique, B) xxxi. A871), pp. 68—73. See also Le Paige, ibid. B) xli. A876), pp. 1011—1016, 935—939. Next, we derive from C), by a change of independent variable, an equation in its normal form. We write z — fijq, where q—l/Bp +1), the equation then becomes E) ~ and its solution is F) v- When a constant factor is absorbed into the symbol 9B, the solution may be taken to be Equation E), which has already been encountered in § 4-14, has been studied by Plana, Mem.della R. Aocad. delle Sci. di Torino^ xxvi. A821), pp. 519—538; Cayley, Phil. Mag. D) xxxvi. A868), pp. 348—351 [Collected Papers, vh. A894), pp. 9—12]; and Lommel, Studien itber die BesseVschen Functioned (Leipzig, 1868), pp. 112—118. The system of equations which has now been constructed has been dis- discussed systematically by Glaisher*, whose important "memoir contains an interesting account of the researches of earlier writers. The equations have been studied from a different aspect by Haentzschel f who regarded them as degenerate forms of Lame's equations in which both of the invariants g2 and g3 are zero. The following papers by Glaiaher should also be consulted: Phil. Mag. D) xmr. A872), pp. 433—438; Messenger, via. A879), pp. 20—23; Proc. London Math. Soc. ix. A878), pp. 197—202. It may be noted that the forms of equation A) used by various writers are as followa: (Glaisher). Equation E) has been encountered by Greenhill J 5n his researches on the stability of a vertical pole of variable cross-aection, under the action of gravity. When the oroas-section is constant, the special equation in which y=| is obtained, and the solution of it leads to Bessel functions of order +•?. * Pl\il. Trans, of the Royal Soc. clxxh. A881), pp. 759—828; see also a paper by Curtis, Cam- Cambridge and Dublin Math. Journal, xx. A854), pp. 272—290. t Zeitschriftfilr Math, und Phy*. tlxxl. A886), pp. 25—33. t Proc. Catnb. Phil. Soc. iv. A883), pp. 65—73.
4'3l] DIFFERENTIAL EQUATIONS 97 4#31. Lommel's transformations of Bessel's equation. Various types of transformations of Bessel's equation were examined by Lommel on two occasions; his earlier researches* were of a somewhat special type, the laterf were much more general. In the earlier investigation, after observing that the general solution of is B) y = z"%\,{z\ Lommel proceeded by direct transformations to construct the equation whose general solution is z^v~°-%'v (yz1*), where a, /3, 7 are constants. His result, which it will be sufficient to quote, is that the general solution of C) z* ~ + Ba - 2/3v + l)z j- + (#V^a + « (« - 2?i/)} u = 0 is D) tt = ^"-«K(Y23)- When # = 0, the general solution of C) degenerate's into anil when 7 = 0, it dogonerutcs into unlo.s.s fiu i.s zero. The .solution of C) wan given explicitly by Lomnuil in numerous special eases. It will be .sufficient to quote the following for reference: (f>) d?l + - -d- + 4 ^ - ?,J u = 0 ; a = %'v (f). , ^ dru du («) + + (8) @) A1) (d^±ZU~()' u = An account of Stokes' researches on the solutions of equation A1) will be given in §§ 6-4, 10-2. * Studien ilher die BcaseVachen Functionen (Leipzig, 1868), pp. 98—120; Math. Ann. in. A871), pp. 475—487. t Math. Ann. xiv. A879), pp. 510—586. W. B. F. 7
98 THEORY OF BESSEL FUNCTIONS [CHAP. IV Lommel's later researches appeared at about the same time as a memoir by Pearson*, and several results are common to the two papers. Lommel's procedure was to simplify the equation f of which the solution is (§ 4*3) A2) *-* On reduction the equation becomes . Now define the function <jf> (z) by the equation It will be adequate to take A4) *(*)« If we eliminate %(^), it is apparent that the general solution of 4 ^j IS As a special case, if we take <? (z) = 1, it is seen that the general solution of is A8) y Next, returning to A3), we take ^ («) = {-^r (z)}*-", and we find that the general solution of IS B0) Messenger, ix. A880), pp. 127—131. j- The inactions x («) and ^(z) are arbitrary.
4-32] DIFFERENTIAL EQUATIONS 99 The following are special cases of A7): B1) g + (^_^),/ = 0; y = <&,(#), B2) d+ /y = 0] y-'^W The independent researches of Pearson proceeded on very similar lines except that he started from Bessel's equation instead of t from the modified form of it. The reader will find many special cases of equation A7) worked out in his paper. A partial differentialjequation closely connected with G) and (8), namely has been investigated by Kcpinski, Math. Ann. r,xi. A906), pp. 397—405, and Myller- Lebedcfif, Math. Aim. lxvi. A909), pp. 325—330. • The reader may verify that Kepinski's formula —r j oexp x r } &\ t is ii .solution, whon/(w) denotes an arbitrary function of w. The special case of the equation when v= — 1 wa« also investigated by Kepinski, Bull, int. de I'A cad. den Svi. de Uracovie, lUOfl, pp. L9H—2iM. 4'32. Malnmti'n's differential equation. Twenty year.s before Lnnmicl published Iuh rosenrehes on traiisformation.s of Bessel's equation, Malin.steu* investigated conditions for the integrability in finite terms of the equation d~ it r d>j which is obvioimly a gonerali.sation of Be-ssol's equation ; and it is a special case of § 4'31 A5). To reduce the equation, Malmston chone new variables defined by the formulae *-CJ, .'/=--«?"¦', where p and q are constants to be suitably chosen. Tlie transformed equation is We ehoo.se p and q so that this may reduce to the equation of § 4%3 A) considered by Plan a, and therefore we tako 2pq - q +1+ q>' = 0, (m + 2) q - 2, so thatp= -hr-\m. The equation then reduces to de u^+aI1 *c J' * Camb. and Dublin Math. Journal, v. A8.10), pp. 180 — 182. The oase in which « = 0 had been previously considered by MalmBt6n, Journal fUr Math, xxxix. A850), pp. 108—115. 7—2
100 THEORY OF BESSEL FUNCTIONS [CHAP. IV By § 4*3 this is integrablc in finite terms if ¦where n is an integer; so that The equation is also obviously integrable in the trivial cases A=0 and m— - 2. notation of Pochhammer for series of hypergeometric type. A compact notation, invented by Pochhammer* and modified by Barnes f, is convenient for expressing the series which are to be investigated. We shall write now and subsequently («)„ = a (a + 1) (a + 2) ... (a + n - 1), (aH = 1. The notation which will be used is, in general, («i)» (g«)n • • • Mn n In particular, „ n . ~\ _ ? (i)» («)n Mn n pt, ..., pq, Z)= X . 7~y Z . n=0 "! \Pi)n\PQ)n ••• \Pq)n i (p; ^) = The functions defined by the first three series are called generalised hyper- hypergeometric functions. It may be noted here that the function & (a; p\ z) is a solution of the differential equation and, when p is not an integer, an independent solution of this equation is *1-<\1JP1(a-p + l; 2 -p; z). It is evident that Various integral representations of functions of the types xJ^, ^F^ 0i^3 have been studied by Pochhammer, Math. Ann. xli. A893), pp. 174—178, 197—218. * Math. Ann. xxxvi. A890), p. 84 ; xxxvm. A891), pp. 227, 586, 587. Cf. § 4-15. t Proe. Londmi Math. Soc. B) v. A907), p. 60. The modification due to BarneB is the insertion of the suffixes p and q before and after the F to render evident the number of sets of factors.
4*4,4*41] DIFFERENTIAL EQUATIONS 101 4*41. Various solutions in series. We shall now examine various solutions of the equation d-u „ p(p + 1) dz1 z- and obtain relations between them, which will for the most part be expressed in Pochhammcr's notation. It is supposed for the present that p is not a positive integer or zero, and, equally, since the equation is unaltered by replacing p by — p— 1, it is supposed that /) is not a negative integer. It is already known (§ 4*3) that the general solution* is ?*r^+i (caV), and this gives rise to the special solutions ^ • o*\ G> + 8; i e1*9); r-P. 0Ft (* - p; } <W). The equation may be written in the forms dz* ~ * dz z'2 which are suggested by the tact that the functions ei:CZ are solutions of the original equation with the right-hand side suppressed. When ^ is written for z (d/dz), the last pair of equations become (S - p - 1) ($ + p). (ue*n) ± 2cz% (m*cz) = 0. When we solve these in series we an: led to the following four expressions for u: jt^ (p + i; 2p + 2\ - -2cs); z~i'6RZ. ,Ft(~p; -2p; -2cz); z ^ (p + i . 2p + 2; -2c2); • z~ve~u .Jt\{-p; - 2p; Icz). Now, by direct tnultiplication of series, the two expressions on the left are expansible in ascending series involving z>'+\ zp+l\zp+i, .... And the expressions on the right similarly involve z~>', z^~l't z""l\ Since none of the two sets of powers are the same when 2p is not an integer, we must have A) ew.,/'', 0> + 1 i 2p + 2 ; - 2cz) = er" .J<\(}> + 1; '2p + 2 ; 2cz) B) <?¦'. tF, (- p ; - 2p ; - 2cz) = <?-«. ,F, {-p\ - 2;>; = n^,(i-;»; i^2;- Those formulae are due to Kuminerf. When (I) has been proved for general /alues of p, the truth of B) is obvious otv replacing p by — p — 1 in (I). We now have to consider the cases when 2p is an integer. * It follows Jrom § 3*1 that a special investigation is also neoosnary when ]> is hall of an odd nteger. t Journal filr Math. xv. A830)), pp. 18b—111.
102 THEORY OF BESSEL FUNCTIONS [CHAP. IV When p has any of the values $,$,$,..., the solutions which contain z~p as a factor have to he replaced by series involving logarithms (§§ 3'51, 3'52), and there is only one solution which involves only powers of z. By the previous reasoning, equation A) still holds. When p has any of the values 0,1, 2, ... a comparison of the lowest powers of z involved in the solutions shews that A) still holds; but it is not obvious that there are no relations of the form -p; -2p; - x{~p\ ~2p; where ku kz are constants which are not zero. We shall consequently have to give an independent investigation of A) and B) which depends on direct multiplication of series. Note. In addition to Rummer's researches, the reader should consult the investiga- investigations of the series by Cayley, Phil. Mag. D) xxxvi. A868), pp. 348—351 [Collected Papers, vn. A894), pp. 9—12] and Glaisher, Phil. Mag. D) xliii. A872), pp. 433—438; Phil. Trans, of the Royal Soc. clxxii. (] 881), pp. 759—828. 4*42. Relations between the solutions in series. Th-3 equation ecziF1 (p + 1; 2p + 2; -2cz) = e-^F,(p + l; 2p + 2; 2cz), which forms part of equation A) of § 4*41, is a particular case of the more general formula due to Kummer* (i) i^(«; p; O = «^(p-«; p; -?),, which holds for all values of a and p subject to certain conventions (which will be stated presently) which have to be made when a and p are negative integers. We first suppose that p is not a negative integer and then the coefficient of ?n in the expansion of the product of the series for e$ and lJP1 (p — a; p ; — ?) is if we first use Vandermonde's theorem f and then reverse the order of the factors in the numerator; and the last expression is the coefficient of ?w in & (a; p; ?)• The result required is therefore established when a and p have general complex values j. * Journal filr Math. xv. A836), pp. 138—141; see also^Baoh, Ann. Sci. de'l'ficole norm. sup. B) in. A874), p. 55. t See, e.g. Chrystal, Algebra, n. A900), p. 9. X Another proof depending on the theory of contour integration has been given by BarneB, Tram. Camb. Phil. Soc. xx. A908), pp. 254—257.
4*42] DIFFERENTIAL EQUATIONS 103 When p is a negative integer, equation A) is obviously meaningless unless also a is a negative integer and | a \ < \ p j. The interpretation of A) in these circumstances will be derived by an appropriate limiting process. First let a be a negative integer (= — N) and let p not be an integer, so that the preceding analysis ia valid. The series ^ (— iV; p; f) is now a terminating series, while XF{ (p + N ; p; — ?) is an infinite series which con- consists of JV+ 1 terms followed by terms in which the earlier factors p + N, p 4- N + 1, p + N + 2, ... in the sequences in the numerators can be cancelled with the later factors of the sequences p, p -f 1, p -f- 2, ... in the denominators. When these factors have been cancelled, the aeries for ^(-iV; p ; ?) and xFi (p ¦+ N; p ; — ?) are both-continuous functions of pne&v p = — M, where M ia any of the integers iV, N +1, N + 2, .... Hence we may proceed to the limit when p -*• — M, and the limiting form of A) may then be written* B) ^(-if; -1/; C)-i=flflJf'l(tf-.3f;-4f;-?I, in which the symbol ~] means that the series ia to stop at the term in ?A', i.e. the last term in which the numerator doos not contain a zero factor, while the symbol 1 moans that the sories is to proceed normally as far as the term in ^'w~iV, and then it is to continue with henna in ^i1/+l, %'v+ii ...,the vanishing- factors in numerator and denominator being cancelled as though their ratio were one of equality. With this convention, it is easy to see that C) lFl(-N\ -M- ?I=i*U-*V; -#;?)! NUM- N)\ When we replace N by M - N and ? by — ?", we have D) ^(N-M- -M; -fI -^(N-M; -i)/; As an ordinary case of A) we have ,F,{M-N^ 1; J/+2; f) = e^Fx (N + 1; M + 2; - ?)> and from this result combined with B), C) and D) we deduce that E) xF^-N; -M- ^^e^Fx{N-M- -M;-ZI. This could have been derived directly from A) by giving p — a (instead of a) an integral value, and then making p tend to its limit. *' Cf. Cayloy,Messenger (old series), v. A871), pp.77—8'2 [Collected Papent,viu. A895),pp.458— 462], and Glaishor, Messenger, vni. A879), pp. 20—23.
104 THEORY OF BESSEL FUNCTIONS [CHAP. IV We next examine the equation F) fl-^Cp + l; 2p + 2; - Zcz) = «/x(p +1; ?<?*«), which forrn9 the remainder of equation A) in § 4*41, and which is also due to Kummer*. If we suppose that 2p is not a negative integer, the coefficient of (cz)n in the product of the series on the left in F) is c,m m)\ Now — t 2m. nGm (p + l)m (- n - 2p -1)^ is the coefficient of tn in the expansion of A - 2t)-*-1 A - t)n+w+\ and so it is equal to 1 f(o+) 1 /¦(»+) ?p\ A - 2?)-p-1 A - t)n+2P+1 ir^1 dt = ~\ A - v?)-?-1 u-n~l du, where u = t/(l — t) and the contours enclose the origin but no other singularities of the integrands. By expanding <the integrand in ascending powers of u, we see}that the integral is zero if n is odd, but it is equal to ,, > |?t when n is even. Hence it follows that and this is the result to be proved. When we make p tend to the value of a negative integer, — N, we find by the same limiting process as before that lim 1F1{p + l; 2p + 2] -2cz)^1F1(l-N\ 2-2F; -2c*) 1 N It follows that 0*\(f-i\r; &**) = &*.Mil-N; 2-2iV; ~2czI f—WrAr—1M AT' If we change the signs of c and 2 throughout and add the results so obtained, we find that G) 2.^x(t-iV; ic^2)=e«.1F1(l-iV; 2-2ZV; ~2czI * Journal filr Math. xv. A836), pp. 138—141. In connexion with the proof given here, see Barnes, Trans. Camb. Phil. Sac. zx. A908), p. 272.
4*43] DIFFERENTIAL EQUATIONS 105 the other terms on the right cancelling by a use of equation A) This is, of course, the expression for J_#+j {icz) in finite terms with a different notation. For Barnes' proof of Rummer's formulae, by the methods of contour inte- integration, see § 6. 4 3. Sharpe's differential equation. The equation A> 'g + g + C' + ^-o. which is a generalisation of Bessel's equation for functions of order zero, occurs in the theory of the reflexion of sound by a paraboloid. It has been investigated by Sharpe*, who has shewn that the integral which reduces to unity at the origin is rftn B) y = C cos (z cos d + A log cot ?0) dd, J 0 where C) 1 = C f "cos (A log cot hd) dd. Jo This is the appropriate modification of Parseval's integral (§ 2'3). To in- investigate its convergence write cos d = tanh <j>, and ib becomes D) tf~wJ0 cosh</> It is easy to sec from this form of the integral that it converges for (complex) values of A for which 11 (A)' < 1, andf 2 C— - cosh hirA. 7T The integral has been investigated in great detail by Sharpe and he has given elaborate rules for calculating successive coefficients in the expansion of y in powers of z. A simple form of the solution (which was not given by Sharpe) is The reader should have no difficulty in verifying this result. * Messenger, x. A881), pp. 174—185 ; xn. A884), pp. 66—79; Proc. Camb. Phil. Soe. x. A900), pp. 101—136. t See, e.g. Wataon, Coviplex Integration and Cauchy}s Theorem A914), pp. 64—65.
106 THEORY OF BESSBL FUNCTIONS [CHAP. IV 4*5. Equations of order higher than the second. The construction of a differential equation of any order, which is soluble by means of Bessel functions, has been effected by Lommel* ; its possibility depends on the fact that cylinder functions exist for which the quotient ^v iz)\^_v (z) is independent of z.. Each of the functions Jn (z) and Yn (z), of integral order, possesses this property [§§ 231, 3*5]; and the functions of the third kind Hv® (*), Hv® (z) possess it (§ 3'61), whether v is an integer or not. Now when § 3*9 E) is written in the form Am. A) ^^ G Vf> = {h)m^(v-m) #-» G V*), the cylinder function on the right is of order — v if m = 2v. This is the case either (i) if v is an integer, n, and m = 2n, or (ii) if v = n +¦ ^ and m = In +1. Hence if <$n denotes either Jn or Yn, we have From this equation we obtain Lommel's result that the functions zin Jn G \/z), n(y \/z) are solutions off \*) fern zn 1 where 7 has any value such that <ym— (—)nom, so that 7 = ic exp (rvi/n). (r = 0, 1, 2, ..., n - 1) By giving 7 all possible values we obtain 2n solutions of B), and these form a fundamental system. Next, if ^n+i. denotes H{1]n+h we have (^-{n+h) = e(n+^ni%lH> so that and hence ^in+lJffA>n+j G *Jz) is a solution of where 7 has any value such that 72n+1 = c27l+1 e-'"*^**, so that 7 = - io exp {r7ri/(n + \)), (r = 0, 1, % ..., 2w) and the solutions so obtained form a fundamental system. * Studien ilher die Bessel'schen Functionen (Leipzig, 1868), p. 120; Math. Ann. 11. A870), pp. 624—635. f The more general equation has been diacussed by Molina, Mem. de I'Acad. de* Sd. de Touloute, G) thi, A876), pp. 167—189.
4-5] DIFFERENTIAL EQUATIONS 107 For some applications of these results, see Forsyth, Quarterly Journal^ xix. A883), pp. 317—320. In view of A), which holds when m is an integer, Lotnmel, Math. Ann. n. A870), p. 635, has suggested an interpretation of a "fractional differential coefficient." Thus he would interpret (^-) oxp (±y Jz) to mean ($o(yijz). Tho idea has been developed at some length by Heaviside in various papers. Lommel's formulae may be generalised by considering equation C) of § 4'31, after writing it in the form C» + a) (»• + a - 2l3v) u = - ^fs^u, the solution of the equation being tt = z^v-a(&v{ryzfi). For it is easy to verify- by induction that, with this value of u, II (^ + a - 2r/3) (^ + a - 2/3v - 2r/3) a = (-)'l/92)l( and so solutions of w-l 4) II (? + a - 2r/3) (^ + a - 2/3v - 2r/3) u = (-)n/3m< are of the form u = z$v~a<$v {^}, where 7 = cexp (vTri/n). (r = 0, 1, ..., n — I) By giving 7 these values, we-obtain "In solutions which form a fundamental system. In the special case in which n = 2, equation D) reduces to ($¦ + a) (S- + a - 2?) (^ + a - 2^v) (^ + a - 2/9v - 2/8) it = /9JcJ^ m. This aiiuation reseinblns an equntion which has been encountered by Nicholson* in tho investigation of tho shapes of Spongo Hpiculon, namely * U*"| @) that is to aay 3C If we identify this with tho special form of D) we obtain the following four distinct sets of values for a, #, /x, v : a 0 s » -I p 1 * -A 0 1 * 3 V \ 1 10 * Proc. Royal Soc. xam. A A917), pp. 50E—619. See also Dendy and NicholHon, Proc. Royal Soc. lxxxix. B A917), pp. 573—587; the special oases of E) in which ,u, = 0 or 1 had been solved previously by Kirchhofl, Berliner Monatsherichte, 1879, pp. 815—828. [Ann. der Physik und Chemie, C) x. A880), pp. 501—512.]
1Q6 THEORY OF BESSEL FUNCTIONS [CHAP. IV These four cases give the following equations and their solutions: F) ^=«; « (9) 37 These seem to be the only equations of Nicholson's type which are soluble with the aid of Bessel functions; in the case y*=2, the equation E) is homogeneous. Nicholson's general equation is associated with the function /3,-2M 2 + 2;* 1+2^ z*"^ \ 0 3 \4-3j*' 4-2j*' l-2/i' D-2/*O" 4*6. Symbolic solutions of differential equations. Numerous mathematicians have given solutions of the equation § 4'3 A) namely in symbolic forms, when p is a positive integer (zero included). These forms are intimately connected with the recurrence formulae for Bessel functions. It has been seen (§ 4*3) that the general solution of the equation is and from the recurrence formula § 39 F) we have Since any cylinder function of the form ^j (oiz) is expressible as where a and /3 are constants, it follows that the general solution of A) may be written A modification of this, due to Glaisher*, is C) u{) where a' == a/c, ft'——/3/c. This may be seen by differentiating oc'ef:!:+ft'e~cz once. * Phil. Trans, of the Royal Soc. clxxh.. A881), p. 813. It was remarked by Glaisher that equation C) ib substantially given by Earnahaw, Partial Differential Equations (London, 1871), p. 92. See also Glaisher, Quarterly Journal, xi. A871), p. 269, formula (9), and p, 270..
4-6] DIFFERENTIAL EQUATIONS 109 Note. A result equivalent to B) was set by Gaskin as a problem* in the Senate House Examination, 1839; and a proof was published by Leslie Ellis, Camb. Math. Journal, II. A841), pp. 193—195, and also by Donkin, Phil. Trans, of the Royal Soc. oxlvii. A857), pp. 43—57. In the question as set by Gaskin, the sign of c2 was changed, so that the solu- solution involved circular functions instead of exponential functions. Next we shall prove the symbolic theorem, due to Glaisherf, that \zdz) -*p+4v dz) In operating on a function with the operator on the right, it is supposed that the function is multiplied by 1/z2p~'- before the application of the operators z* (d/dz). It is convenient to write and then to use the symbolic formula E) /(*). (e"°Z) = ea0 ./(* + a) Z, in which a is a constant and Z is any function of 2. Tho proof of this formula presents no special difficulties when /'(.!)) is a polynomial in 3, as is the caso in tho present investigation. See, o.g. Foi\syth, Treatise on Differential Equations A5I4), $ 33. It is easy to see from E) that zdz 2p + 2) C> - '2p + 4) (^ - 2p + 6) ... S, when we bring the successive functions e""-0 (beginning with those on the left) past the operators one at a time, by repeated applications of E). We now reverse, the orderJ of the operators in tb<; last result, and by a reversal of the previous procedure we. got ^ + 2^ - 2) (^ + 2p - 4)... (^ + 2) ^ dz " Tho problom wa« tlie Hecond pjirt of qucHtion ft, Tuonday aftornoon, Jan 8, 1839; Hec tho Cambridge. University Calendar, 1SB9, p. 315). f Nuuvelle Coir. Muth. n. A870), pp. 240—243, iM'J—!iC0 ; mid Phil Trans, of the Royal Soc olxxii. A881), pp. HOB—805. f. It waa remarked by Ctvyley, Quarterly Journal, xn. A872), p. 132, in a footnote to a paper by Glaishor, that differential operators of the form a"*1 yz~a, i.o. ^-a, obey the commutative law.
110 THEORY OF BESSEL FUNCTIONS [CHAP. IV and this is the result to he proved. If we replace p by p •+• 1, we find that When we transform B) and C) with the aid of D) and F), we see that the general solution of A) is expressible in the following forms: The solutions of the equation d* z dz cv~°' [C) of §4-3], which correspond to B), C), G) and (8) are A different and more direct method of obtaining G) is due to Boole, Phil. Trans, of the Royal Soc. 1844, pp. 251, 252 ; Treatise on Differential Equations (London, 1872), ch. xvn. pp. 423—425; see also Curtis, Cambridge and Dublin Math. Journal, ix. A854), p. 281. The solution (9) was first given by Leslie Ellis, Camb. Math. Journal, n. A841), pp. 169, 193, and Lebesgue, Journal de Math. xr. A846), p. 338; developments in aeries were obtained from it by Bach, Ann. Sci. de I'jtcole norm. sup. B) in. A874), p. 61. Similar symbolic solutions for the equation -7-5 — cizi<1~iv=0 were discussed by Fields, CtZ" John Hopkins University Circulars, vi. A886—7), p. 29. A transformation of the solution (9), due to Williamson, Phil. Mag. D) xi. A856), pp. 364—371, is A3) i>=c2p (I -Y (o^+pfl—). This is derived from the equivalence of the operators -*-,--, when they operate on functions of cz. We thus obtain the equivalence of the following operators it being supposed that the operators operate on a fuuction of cz; and Williamson's formula is then manifest.
4*7] DIFFERENTIAL EQUATIONS 111 4*7. Liouville's classification of elementary transcendental functions. Before we give a proof of Liouville's general theorem (which was mentioned in § 4-12) concerning the impossibility of solving Riccati's equation "in finite terms " except in the classical cases discovered by Daniel Bernoulli (and the limiting form of index — 2), we shall give an account of Liouville's* theory of a class of functions known as elementary transcendental functions] and we shall introduce a convenient notation for handling such functions. For brevity we write j" , Jx (*) = *(*) slog*, l,(z)=>l(l(z)), h(*)= I &(*)), ..., ex (z) = e (z) = ez, e., (z) = e (e (z)), e3 (z) = e (e2 (*)), }, «,/(*)-5 {«,/(*)} A function of z is then said to be an elementary transcendental function^ if it in expressible as an algebraic function of z and of functions of the types lr<f>(z), e,.ty(z), <s,xB)> whore- the auxiliary functions 4>(z), ^fr(z), x(~) ar<-1 expressible in terms of z and of a second set of auxiliary functions, and so on; provided that there exists a finite number n, such that the »ith set of auxiliary functions are all algebraic functions of z. The. order of an elementary transcendental function of z is then defined inductively as follows: (I) Any algebraic function of z is of order zero§. (II) Tf fr{z) denotes any function of order r, then any algebraic function of functions of the types lf.(z)y ef.{z), 9/,(s), /,(,), /r_, <*)..../.(*) (into which at least one of the first three enters) is said to be of order r + 1. (III) Any function is supposed to be expressed as a function of the lowest possible order. Thus elfr(z) is to be replaced by f,-(z), and it is a function of order ¦/•, not of order r + 2. In connexion with Uiin and tho following Hoct.ions, tin*, render should study Hardy, Ordiws of fv/ini(i/ (Cunl). Mulli. Trncts, no. 12, 1910). The function.s ili.scuascd by Hardy wcro of ¦: uli^lit.ly moru restricted cliamct.or hhan those now under i-onsidonitimi, Hinca, 1'ur ]»i.H pui'pones, l,h(! symliol v is not required, and also, for liin purposes, it is oonvuniont to po.s1ul.ite the reality of thu functions which ho. investigates. It may be noted that Liouville did not study properties of the. Hynibol s in detail, but merely remarked that it had many properties akin to tho.se of the .symbol I. * Journal, tie Math. n. A837), pp. 5E—105 ; in. A838), pp. 523—547 ; iv. A839), pp.423—456. t It is supposed that the integrals aro all indefinite. X "Une fonction iiuio exylioito." § For the purposoH of this invoHtigatiou, irrational powers of z, such us z*, of course inu«t not be regarded as algebraic functions.
112 THEORY OF BESSEL FUNCTIONS [CHAP. IV 4*71. Liouville's first theorem* concerning linear differential equations. The investigation of the character of the solution of the equation A) 3?-«*(*)> in which % (z) is a transcendant of orderf n, has been made by Liouville, who has established the following theorem : If equation A) has a solution which is a transcendant of order m + 1, where m^n, then either there exists a solution of the equation which is of order], n, or else there exists a solution, u1} of the equation expressible in the form B) u1 = <jili(z).efli(z)) where fy.{z) is of order fx, and the order of ^^(z) does not exceed fxy and /u. is such that n ^ /* ^ m. If the equation A) has a solution of order m + 1, let it be /m+1 (z) ; then fm+i (z) is an algebraic function of one or more functions of the types lfm (z)y qfm (z), efm (z) as well as (possibly) of functions whose order does not exceed m. Let us concentrate our attention on a particular function of one of the three types, and let it be called 6, ^ or © according to its type. (I) We shall first shew how to prove that, if A) has a solution of order m + 1, then a solution can be constructed which does not involve functions of the types 6 and ^. For, if possible, let /,,l+1 (z) = F(z, 6), where F is an algebraic function of 8; and any function of z (other than 6 itself) of order m+1 which occurs in F is algebraically independent of 6. Then it is easy to shew that m *?.-F W,) = ^+ 2 d^(z) d*F W dz, *-X W - dz2 +fm(z) dz \fm{z) dz \ d6> \_dz\fm(z) dz it being supposed that z and 8 are the independent variables in performing the partial differentiations. The expression on the right in C) is an algebraic function of 6 which vanishes identically when 6 is replaced by lfm(z). Hence it must vanish identically for all values of 6 \ for if it did not, the result of equating it to zero would express lfm{z) as an algebraic function of transcendants whose orders do not exceed m together with trans^endants of order m+1 which are, ex hypothesi, algebraically independent of 8. * Journal de Math. iv. A839), pp. 435—442. t This phrase is used a8 an abbreviation oi " elementary transcendental function of order m." ¦f Null solutions are disregarded; if u were of order less than n, then , would be of order U uZ less than n, which is contrary to hypothesis.
4*71] DIFFERENTIAL EQUATIONS 113 In particular, the expression on the right of C) vanishes when 8 is replaced by 0 + c, where c is an arbitrary constant; and when this change is made the expression on the left of C) changes into d"F(z, 0 + c) „, A x , . ^i -FB,e + c).x (*), which is therefore zero. That is to say When we differentiate D) partially with regard to c, we find that dF(z, 6 + c) &F (z, 6 + c) dc ' dc2 > ••• are solutions of A) for all values of c independent of z. If we put c = 0 after performing the differentiations, these expressions become dF(z,6) d*F(z,6) d6 ' d&> ' '"' which are consequently solutions of (I). For brevity they will be called Fo> Fee, •••• Now either F and Fe form a fundamental system of ¦solutions of A) or they do not. If they do not, we must have* where A is independent both of z and 9. On integration we find fchat where <1> involves transcendants (of order not exceeding m + 1) which are algebraically independent of 0. But this is impossible because eA0 is not an algebraic function of 6; and therefore F and Fe form a fundamental systein of solutions of A). Hence F90 is expressible in terma of F and Fe by an equation of the form Fee = AFe + BF, where A and B are constants. Now this may be regarded as a linear equation in 6 (with constant coefficients) and its solution is F = <l\ea0 + ^eW or F= e*° {<!>, + <J>a0}, where <PX -and ^ are functions of the same nature as <X>, while a and ft are the'roots of the equation a?-Ax-? = 0. The only value of F which is an algebraic function of 9 is obtained when a = ft — 0 ; and then F is a Linear function of 0. Similarly, if /,»+, (z) involves a function of the type 'S-, we can prove that it must be a linear function of ^. * Since F mint involve 0, Fe cannot be identically zero. w. B. v. 8
114 THEORY OF BESSEL FUNCTIONS [CHAP. IV It follows that, in so far as fm+x (z) involves functions of the types 0 and S-, it involves them linearly, so that we may write f^) = $ei(z)O2(z)...ep(z).%(z)%(z)...\(z).fp,q(z), where the functions ^p,q(z) are of order m +1 at most, and the only functions of order m +1 involved in them are of the type 0. Take any one of the terms in /m+iC?) which is of the highest degree, qua function of 8U 9.,,...%,%, ..., and let it be Then, by arguments resembling those previously used, it follows that 9 9 9 9 i d$i dO2 ddp d^i 9^2 9^q is a solution of A); i.e. typ,Q B) is a solution of A). But typtQ (z) is either a function of order not exceeding m, or else it is a function of order m+1 which involves functions of the type © and not of the types 6 and ^r. In the former case, we repeat the process of reduction to functions of lower order, and in the latter case we see that some solution of the equation is an algebraic function of functions of the type ©. WTe have therefore proved that, if A) has a solution which is a transcendant of order greater than n, then either it has a solution of order n or else it has a solution which is an algebraic function of functions of the type ef,j.{z) and </>m (¦&)» where /M (z) is of order /u, and 4v (z) is of an order which does not exceed /a. (II) We shall next prove that, whenever A) has a solution which is a transcendant of order greater than n, then it has a solution which involves the transcendant ef^z) only in having a power of it as a factor. We concentrate our attention on a particular transcendant © of the form e/M(^), and then the postulated solution may be writter in the form G(z} ©), where G is an algebraic function of ©; and any function (other than © itself) of order ft 4-1 which occurs in G is algebraically independent of ©. Then it is easy to shew that /KX dG n /x dG , ««,, E) -& - G. X W - W + 28/, The expression on the right is an algebraic function of © which vanishes when © is replaced by ef^(z), and so it vanishes identically, by the arguments used in (I). In particular it vanishes when © is replaced by c ©, where c is independent of z. But its value is then
4*71] DIFFERENTIAL EQUATIONS 115 so that <Pg(*,o6) When we differentiate this with regard to c, we find that dG (z, c©) &G (z, c©) OC 00 are solutions of A) for all values of c independent of z. If we put c = 1, these expressions become Hence, by the reasoning used in (I), we have BO© — AG or else where A and B are constants. In the former case G = <t>©/1, and in the latter (r has one of the values where cl:>, (l>i, 4>2 are functions of ^ of order fi + 1 at most, any functions of order yu +1 which are involved being algebraically independent of B; while 7 and 8 are the roots of the equation *(*- I)- Ax- B = Q. In any case, G either contains B only by a factor which is a power of (F1) or else G is the sum of two expressions which contain H only in that manner. In tin; latter case*, G (z, (;©) - c*V (*, H) is a solution of (I) which contains © only by a factor which is a power of (H). By repetitions of this procedure, we see that, if ©,, ^X, ... ©r are all the transcendants of order fi + 1 which occur in the postulated solution, we can derive from that solution a sequence of solutions of which the sth contains ©i) ^a> ••• ^« only by factors which are powers of H,, (»\, ... <*)„; and the rth member of the sequence consequently consists of a product of powers of <F)U CwJ) ... Hr multiplied by a transcendant which is of order fx, at most; this solution is of the form which is of the form <^ {z). ef^ (z). * If <I>1 is not identically zero; if it ia, then %QS ia a solution of the specified type. 8—2
n6 THEORY OF BESSEL FUNCTIONS |_CHAP. IV 4-72. LiouvilWs second theorem concerning linear differential equations. We have just seen that, if the equation d?u_ . . A) 'dz2~U^ [in which %{z) is of order n] has a solution which is an elementary tran- transcendant of order greater than n, then it must have a solution of the form where fi>n. If the equation has more than one solution of this type, let, a solution for which p has the smallest value be chosen, and let it be called «|. Liouville's theorem, which we shall now prove, is that, for this solution, the order of d (log u^/dz is equal to n. Let d log «i _ t and then t is of order /* at most; let the order of t be N, where iV <: p. If N=n, the theorem required is proved. If N>nt then the equation satisfied by t, namely B) -+*2= (z) has a solution whose order iV is greater than n. Now t is an 'algebraic function of at least one transcendant of tin; types [ts-i(z), sfx-\(z\ efs-\{z) and (possibly) of transcendants whose order docs not exceed JV- 1. We call the first three transcendants 8, Sv, H respectively. If ? contains more than one transcendant'of the type 0, we concentrate our attention on a particular function of this type, and wo write ¦* TP / *. Z)\ t =" X* ( Z) U j. By arguments resembling those used in § 4-71, we find that, if N > tt, then is also a solution of B). The corresponding solution of A) is and this is a solution for all values of c independent of z. Hence by differentiation with respect to c, we find that the function u2 defined as is also a solution of A); and we have .so that diu diu lhdz Un-dI-u*F'-
4*72,4-73] DIFFERENTIAL EQUATIONS 117 But the Wronskian of any two solutions of A) is a constant*; and so vrh\ = C, where C is a constant. If C = 0, F\s independent of 6, which is contrary to hypothesis ; so G ^ (), and Hence ux is an algebraic function of 6 ; and similarly it is an algebraic function of all the functions of the typos 6 and S- which occur in L Next consider any function of the type H which occurs in t; we write t = G (z, 0), and, by arguments resembling those used in §4-71 and those used earlier in this section, we find that the function u. donned as i ~] - cxp [(} (z, c:B) dz fa Jr-l is a solution of A)-; and we have so that dil-x dllx »r\/t This Wronskian is a constant, C\, and .so Consequently vt, is an algebraic function, not only of all the transcendants of the types 0 and ^, but also of those of type M which occur ini; and therefore ?/, is of order N. This is contrary to the hypothesis that ux is of order (i+ 1, where /* ^ iVr, if Ar > n. The contradiction shews that JV cannot be gn^ater than n\ hence the order of rf(log u^/dz is n. And this is the theorem to b<» established. 4*73. Liouville's theorem ~f that Besnel's equation has no algebraic integral. We shall now shew that the equation dz" dz has no integral (other than a null-function) which is an algebraic function of z. We first reduce the equation to its normal form by writing y-uz~l, p = ±v — \. * See e.g. Forsyth, Treatise on Differential Equations A914), § 65. t Journal dc Math. iv. A839), pp. 429—435 ; vi. A841), pp. 4—7. Liouvillo's first investigation was concerned with the general caao in •which x B) iR any polynomial; the application (with various modifications) to B<?88eFs equation was given in his later paper, Journal de Math. vi. A841), pp. 1—13, 3C.
118 THEORY OF BESSEL FUNCTIONS [CHAP. IV This is of the form dht , . d?=lL*>{2)> where If possible, let Bessel's equation have an algebraic integral; then A) also has an algebraic integral. Let the equation which expresses this integral, u, as an algebraic function of z be C) cS0(u,s)*O, where ?& is a polynomial both, in u and in z; and it is supposed that ?#¦ is irreducible*. Since u is a solution of A) we have D) MuvM? - ZS€uz<s4uMz + Mzz -S4U* + MuX (*) = 0. The equations C) and D) have a common root, and hence all the roots of C) satisfy D). For, if not, the left-hand sides of C) and D) (qua functions of u) would have a highest common factor other than S€ itself, and this would be a polynomial in u and in z. Hence &4> would be reducible, which is contrary to hypothesis. Let all the roots of C) be Wj, u^, ... u3I. Then, if s is any positive integer, is a rational function of z; and there is at least one value of s not exceeding M for which this sum is not zerof. Let any such value of s be taken, and let m=l M /(\u \r Also let Fr=s(s-l)..,(s-r + lJC7 , m = \ \ (iZ ) where r = 1, 2, ... s. Since u1} uit... un are all solutions^ of A), it is easy to prove that E) g dW F) -jf-'Wr+i + ris-r + VxWWn, (r = 1, 2,... s- 1) G) ^ * That is to say, ?4> has no factors whicli are polj'nomials in u or in z or in both w and z. f If not, all the roots of C) would be zero. + Because D) is satisfied by all the roots of C), qua equation in u.
4*73] DIFFERENTIAL EQUATIONS 119 Since Wo is a rational function of z, it is expressible in partial fractions, sothafc W-2AP+2 —"•» n= - k n,q \" ^q) where An and Bn<q are constants, « and A are integers, n assumes positive integral values only in the last summation and aq=f= 0. Let the highest power of l/(z - aq) which occurs in Wo be \\{z - aq)p. It follows by an easy induction from E) and F) that the highest power of l/(z - aq) in Wr is I/O - a(J)p+r, where r = 1, 2,... s. Hence there is a higher power on the left of G) than on the right. This contradiction shews that there are no terms of the type Bn>q {z — aq)~n in Wo and so ,„ A . We may now assume that A\ j= 0, because this expression for Wo must have a last term if it does not vanish identically. From E) and F) it is easy to see that the terms of highest degree in z which occur in Wu, Wlt Wo, W3, ... are* AKz\ \AK2*-\ AKsz\ \/lACs-2)^-\.... By a simple induction it ia possible, fco show that the term of highest degree in W,r is ylA^ . 1 . 8 ... Br-]).s (s- 2) ... (s- 2r + 2). An induction of a more complicated nature is then necessary to shew that the term of highest degree in W.ir+X is \Ai*-l2.4,...{2r).(8-l)(8-S)...(8-2r + l).aF1{h,-l*'> 4 - i* ; l)r+i, where the suffix r + 1 indicates that the first r + 1. terms only of the hyper- geometric series are to be taken. If s is odd, the terms of highest degree on the left and right of G) are of degrees X — 2 and A, respectively, which is impossible. Hence TF0 vanishes whenever s is odd. When s is even, the result of equating coefficients of z1"'1 in G) is XAi.sl^-XA^.s}^^, -?s; i-k; l)i8. That is to say \AK. s\ ,F, (i ,-^s; ? - ±s; 1) = 0, and so, by Vandermonde's theorem, The expression on the left vanishes only when A, is zerof. * It is to be remembered that the term of highest degree in x(^) ia - !• •|- Tho analyais given by Liouville, Journal de Math. vi. A841), p. 7, seems to fail at this point, because he apparently overlooked the possibility of \ vanishing. The failure seoms in- inevitable in view of the fact that J^+. (z) + </'in_i (^) is an algebraic function of z, by § 3-4. The subsequent part of the proof given liore is based on a suggestion made by Liouville, Journal de Math. iv. A889), p. 435; see also Genocchi, Mem. Accad. delle Sci. di Torino, xxni. A866), pp. 299—362; Comptes Rendus, lxxxv. A877), pp. 391—394.
120 THEOEY OF BESSEL FUNCTIONS [CHAP. IV We have therefore proved that, when 5 is odd, WQ vanishes, and that, when s is even, Wn is expressible in the form where AQ)8 does not vanish. V A z~n n=0 From Newton's theorem which expresses the coefficients in an equation in terms of the sums of powers of the roots, it appears that M must be even, and that the equation ?4 (u, z) = 0 is expressible in the form (8) uM+ 2u*- where the functions "$r are polynomials in \\z. When we solve (8) in a series of ascending powers of ljz} we find that each of the branches of u is expressible in the form m=0 where n is a positive integer and, in the case of one branch at least, c0 does not vanish because the constant terms in the functions ^r are not all zero. And the series which are of the form CmZ ttl=0 are convergent* for all sufficiently large values of z. When we substitute the series into the left-hand side of A), we find that the coefficient of the constant term in the result is c0, and so, for every branch, c0 must be zero, contrary to what has just been proved. The contradiction thus obtained shews that Bessel's equation has no algebraic integral. 44. On the impossibility of integrating Bessel's equation in finite terms. We are now in a position to prove Liouville's theoremf that Bessel's equation for functions of order v has no solution (except a null-function) which is expressible in finite terms by means of elementary transcendental functions; if 2v is not an odd integer. As in § 4*73, we reduce Bessel's equation to its normal form A) 3?-"*<*>» where x(JS)=z~^JrP.(P + *)/*2 an(*P = ± v~\- Now write d (log u)jdz = t, and we have B) . *+,+¦!_?<?+!> _o. * Gcmrsat, Goun d'Analyse, n. (Paris, 1911), pp. 273—281. Many treatises tacitly assume the convergence of a series derived in this manner from an algebraic equation, t Journal de Math. vi. A841), pp. 1—13, 36.
4*74] DIFFERENTIAL EQUATIONS 121 Since ^(z) is of order zero, it follows from § 472 that, if Bessel's equation has an integral expressible in finite terms, then B) must have a solution which is of order zero, i.e. it must have an algebraic integral. If B) has an algebraic integral, let the equation wnich expresses this integral, t, as an algebraic function of z, be C) ?€(t,z) = O, where S€ is an irreducible polynomial in t and z. Since t is a solution of B), we have D) 642+{x(z)-t*}at = 0. As in the corresponding analysis of § 4*73, all the branches of t satisfy D). First suppose that there are more than two branches of t, and let three of them be called tlt t>, ts, the corresponding values of m (denned as exp jtdz) being ult t^, us. These functions are all solutions of A) and so the Wronskians dus du% dux dus dv2 dux dz dz dz dz dz dz are constants, which will be called 0,, C», Ca. Now it is easy to verify that n _ du:i d'iu _ . and t3 — ti is not zero, because, if it were zero, the equation C) would have a pair of equal roots, and would therefore be reducible. Hence 6\ ^ 0, and so Therefore u.zu3 (and similarly u3ux and uxu2) is an algebraic function of z. But Wi and therefore uy is an algebraic function of z. This, as we have seen in§4'73, cannot be the cane, and so t has not more than two branches. Next suppose that t has two branches, so that t9^ (t, z) is quadratic in t. Let the branches be U ± \/ V, where U and V are rational function^ of z. By substituting in B) wo find that Let V be factorised so that where A is constant, tcq and \ are integers, and Kq and«a9 are not zero.
122 THEOBY OF BESSEL FUNCTIONS [CHAP. IV From the second member of E) it follows that tj •< kq and then by substituting into the first member of E) we have Now consider the principal part of the expression on the left near aq. It is evident that none of the numbers rcq can be less than — 2, and, if any one of them is greater than — 2 it must satisfy the equation ^+4^ = 0, so that Kq is 0 or — 4, which are both excluded from consideration. Hence all the numbers Kq are .equal to — 2. Again, if we consider the principal part near oo, we see that the highest power in V must cancel with the — i in ^ (z), so that \ = — 2 kq. q It follows that *JV is rational, and consequently ?? (t, z) is reducible, which is contrary to hypothesis. Hence t cannot have as many as two branches and so it must be rational. Accordingly, let the expression for t in partial fractions be A Bn<, *- 2 4n*»+2 where An and Bn> q are constants, k and. \ are integers, n assumes positive values only in the last summation and aq ^ 0. If we substitute this value of t in B) we find that »=»-« 11,9(^ — 09) U--IC (z-aq)n) z2 If we consider the principal part of the left-hand side near aq we see that lj(z — aq) cannot occur in t to a higher power than the first and that ¦B1)9~-B2li9=0, so that jBlt9= 1. Similarly, if we consider the principal parts near 0 and 00, we find that Since p = + v — \, we may take A^ = -p without loss of generality. It then follows that Accordingly, if we replace u by z~pe±izw in A), we see that the equation must have a solution which is a polynomial in 2, and the constant term in this polynomial does not vanish.
4*75] DIFFERENTIAL EQUATIONS 123 When we substitute 2 cmzm for w in G) we find that the relation connecting successive coefficients is m (m — 2p — 1) cm ± 2icm_! (m — p — 1) = 0, and so the series for w cannot terminate unless m—p — \ can vanish, i.e. unless p is zero or a positive integer. Hence the hypothesis that Bessel's equation is soluble in finite terms leads of necessity to the consequence that one of the numbers + v — \ is zero or a positive integer; and this is the case if, and only if, 2v is an odd integer. Conversely we have seen (§ 3*4) that, when 2v is an odd integer, Bessel's equation actually possesses a fundamental system of solutions expressible in finite terms. The investigation of the solubility of the equation is therefore complete. Somo. applications of this theorem to equations of the types discussed in § 4 have been recorded by Lebesguc, Journal de Math. xr. A846), pp. 338—340. 4*75. On the impossibility of integrating Riccati's equation infinite terms. By means of the result just obtained, wo can discuss Riccati's equation dz azn + bii- with a view to proving that it is, in general, not integrable in finite terms. It has been seen (§ 4'21) that the equation in reducible to where ?i=2<y —2; and, by § 4-3, the last equation is reducible to Bessel's equation for functions of order l/Bg) unless q = 0. Hence the only possible cases in which liiccuti's equation is integrable in finite terms are those in which q is zero or \/q is an odd integer; and these are precisely the cases in which n is equal to — 2 or to (m = 0, 1, 2, ...) 2??;, ± I" Consequently the only cases in which Riccati's equation is integrable in finite terms are the classical cases discovered by Daniel Bernoulli (cf. § 4*1 i) and the limiting case discussed after the manner of Euler in §4*12. This theorem was proved by Liouville, Journal de Math. vt. A841), pp. 1—13. It seoms impossible to establish it by any method which is appreciably more brief than the analysis used in the preceding sections.
124 THEORY OF BESSEL FUNCTIONS [CHAP. IV 4*8. Solutions of Laplace's equation. The first appearance in analysis of the general Bessel coefficient has been seen (§ 1*3) to be in connexion with an equation, equivalent to Laplace's equation, which occurs in the problem of the vibrations of a circular membrane. We shall now shew how Bessel coefficients arise in a natural manner from Whittaker's* solution of Laplace's equation v ; 3a; dy* dz2 The solution in question is B) V=\ f(z + ix cos u + iy sin n, u) die, J — ir in which / denotes an arbitrary function of the two variables involved. In particular, a solution is (n ¦ • • ek (i + izGOBU + iysinu) cos mu du> — IT in which k is any constant and in is any integer. If we take cylindrical-polar coordinates, defined by the equations x — p cos <?, y = p sin </>, this solution becomes = 2ekz eikpC0*v cos mv cos vi6 dv, J o = 2trim e*z cos m<p. Jm (kp), by § 2'2. In like manner a solution is f "" gk B+wcoBtt + iysinM) sin mU(lU) J —W and this is equal to i-iri™ ekz sin m.(j>. Jm (kp). Both of these solutions are analytic near the origin. Again, if Laplace's equation be transformed^ to cylindrical-polar coordi- coordinates, it is found to become * Monthly Notices of the R. A. S. lxh. A902), pp. 617—620; Math. Ann. lvii. A902), pp. 883—341. t The simplest method of effecting the transformation is by using Green's theorem. See W. Thomson, Camb. Math. Journal, iv. A845), pp. 33—42.
4*8,4*81] DIFFERENTIAL EQUATIONS 125 and a normal solution of this equation of which ekz is a factor must be such that is independent of <f>, and, if the solution is to be one-valued, it must be equal to — m% where m is an integer. Consequently the function of p which is a factor of V must be annihilated by da Id and .therefore it must be a multiple oiJm{kp) if it is to be analytic along the line p - 0. We thus obtain anew the solutions ^l^mf.Jnikp). olll These solutions have been derived by liobsou* from the solution e^'J^^p) by Clerk Maxwell's method of differentiating harmonics with respect to axes. Another solution of Laplace's equation involving Besael functions has been obtained by Hobson (ibid. p. 447) from the equation in cylindrical-polar coordinates by regarding 8/82 as a symbolic operator. The solution ho obtained is sin where/(s) is an arbitrary function; but the interpretation of this solution when <$m involves a function of the second kind is open to queution. Other solutions involving a Bessel function of an oporator acting on an arbitrary function havo been given by Hobson, Proc. London Math. Hoc. xxiv. A893), pp. 55 -A7 ; xxvi. A895), pp. 4<J—494. 4*81. Solutions of the equations of wave motions. We shall now examine the equation of wave motions A) "?+--%-v=1.~^, in which t represents the time and c the velocity of propagation of the waves, from the same aspect. Whittaker'sf solution of this equation is B) V = f{n; sin u coh v + ymnu sin v + z cos u + ct, u, v) dudv, J -IT J 0 where /denotes an arbitrary function of the three variables involved. In particular, a solution is V=\ I eikix»i"ut''mi'+y«UluaU>v+zrwu+el)F(u, u)dudv, J -vJ 0 where F denotes an arbitrary function of u and v. * Proc. London Math. Hoc. xxn. A892), pp. 431—449. t Math. Ann. uvu. A902), pp. 342—345. See also Havelock, Proc. London Math. Soc. B) n. A904), pp. 122—137, und Watson, Messenyer, xxxvi. A907), pp. 98—106.
126 THEORY OP BESSEL FUNCTIONS [CHAP. IV The physical importance of this particular solution lies in the fact that it is the general solution in which the waves all have the same frequency kc. Now let the polar coordinates of («, y, z) be (r, 9, <?), and let (a>, -f) be the angular coordinates of the direction (u, v) referred to new axes for which the polar axis is the direction @, tp) and the plane >Jr = 0 passes through the 2-axis. The well-known formulae of spherical trigonometry then shew that cos co = cos 0 cos it + sin 9 sin u cos (v — <?), sin u sin {v — </>) = sin a> sin -x/r. Now take the arbitrary function F (u, v) to be Sn {u, v) sin u, where Sn de- denotes a surface harmonic in {u, v) of degree n; we may then write Sn (u, v) = 8n,@, ?; », "f), where Sn is a surface harmonic* in (o>, \Jr) of degree n. We thus get the solution Vn = eilcet r T eikra0Hw ^n (9, (/>; w, -Jr) sin o> rf«d-^r. J -irJ 0 Since Sn is a surface harmonic of degree n in (&>, i|r), we may write S« {0, +; a), t) = A» F>, <f>). PM (cos a)) + S {^n l where jln @, ^>), Jln(m) @, <f>) and J5n<m> @, </>) are independent of w and •x/r. Performing the integration with respect to ty, we get Vn = 27reifceJ^lw @, ^>) (^-«»B-Pn (cos «) sin by § 3-32, Now the equation of wave motions is unaffected if we multiply x, y, z and t by the same constant factor, i.e. if we multiply r and t hy the same constant factor, leaving 6 and </> unaltered; so that An @, </>) may be taken to be in- independent^ of the constant "k which multiplies r and t. Hence lim (k~n Vn) is a solution of the equation of wave motions, that is to say, rnAn(9, <?) is a solution (independent of t) of the equation of wave motions, and is consequently a solution of Laplace's equation. Hence An @, ?) This follows from the fact that Laplace's operator is an invariant for changes of rectangular + This is otherwise obvious, because Sn may be taken independent of k. axes.
4-82] DIFFERENTIAL EQUATIONS 127 is a surface harmonic of degree n. If we assume it to be permissible to take An @, cp) to be any such harmonic, we obtain the result that eihctr~ijn+i (kr) P,™ (cos 6)C™ mcf> is a solution of the equation of wave'motions*; and the motion represented by this solution has frequency Ice. To justify the assumption that An(d, $) may be any surface harmonic of degree n, we construct the normal solution of the equation of wave motions 3 / „ i V\ 1 5 / . . c V\ 1 . ?fl V r* 32 V [ n*ZI J/ (l fj 1 3 / „ i V\ 1 5 / . . c V _ [ n*Z I «J_, / (Jill fj r V dr ) + ain & d6 \ d6 _ [ «J_, fj 1-1- . — -_ dr V dr ) + ain & d6 \ d6 ) + am* 6 dp ~ cl dt which has factors of the form eikel .' md>. The factor which involves 6 must then be of 8111 the form Pnm(con 6); and the factor which involves r is annihilated by the operator dr \ dr) ao that if this factor in to bo analytic at the origin it inu.st be a multiple of </rt.(.j {kr)l*Jr. 4182. Theorems derived from solutions of the equations of Mathematical Physics. It in possible to prove (or, at any rate, to render probable) theorems con- concerning Bossel functions by a comparison of various solutions of Laplace's equation or of the equation of wave motions. Thus, if we take the function ekzJ0 [k*/{f? + a9 - 2ap cos $)}, by making a change of origin to the point (a, 0, 0), we see that it is a solution of Laplace's equation in cylindrical-polar coordinates. This solution has ekz as a factor and it is analytic at all points of space. It is therefore natural to expect it to be expansible in the form r oo "I ekz \ Ao Jo (kp) + 2 2 (Am cos m<f> + Bm sin m<f>) Jm(kp) . Assuming the possibility of this expansion, we observe that the function under consideration is an even function of (f>, and so Bm — 0; and, from the symmetry in p and a, Am is of the form cmJm(/ca), where cm is independent of p and a. We thus get oo Jo )/cV(pa + a" ~ 2ap cos </>)} = 2 emcmJm{kp) Jm(ka) cos m<f>. If we expand both sides in powers of p, a and cos <j>, and compare the coefficients of {k-pa cos <?)m, we get Cf. Bryan, Nature, lxxx. A909), p. 309.
128 THEORY OF BESSEL FUNCTIONS [CHAP. IV and so we are led to the expansion* 00 Jol&vV + a2 - 2ap cos </>)) == 2 emJm(kp) Jm(ka) cos m<j>, of which a more formal proof will be given in § il*2. Again, if we take eifc(ct+z\ which is a solution of the equation of wave motions, and which represents a wave moving in the direction of the axis of z from + oo to - oo with frequency he and wave-length Zirjk, we expect this expression to be expansible! in the form where cn is a constant; so that gifcrcosfl __ I \ ^ q i" If we compare the coefficients of (At cos 0)w on each side, we find that (n + |)" 2». (n\y' and so cn = n +1; we are thus led to the expansion^: \fer/ ,l=o of which a more formal proof will be given in § 11'5. 4*83. Solutions of the wave equation in space of p dimensions. The analysis just explained has been extended by Hobson§ to the case of the equation dxf + Zxf + " * + datf " ca dt- ' A normal solution of this equation of frequency kc which is expressible as a function of r and t only, where must be annihilated by the operator 9 .JP13 , 9?2 r 8r and so such a solution, containing a time-factor eiict, must be of the form * This is due to Neumann, Theorie der BeaeVichen Functionen (Leipzig, 1867), pp. 59—65. t The tesBeral barmonicB do not occur because the function is symmetrical about the azie of z. J ThiB expansion is due to Bauer, Journal filr Math. vn. A859), pp. 104, 106. § Proc. London Math. Soc. xxv. A894), pp. 49—75.
4-83] DIFFEEENTIAL EQUATIONS 129 Hobson describes the quotient W\(p-.>i)(k?')/(Jcr)klp~s) as a cylinder function of rank p ; such a function may be written in the form By using this notation combined with the concept of ^-dimensional space, Hobaon succeeded in proving a number of theorems for cylinder functions of integral order and of order equal to half an odd integer simultaneously. As an example of such theorems we shall consider an expansion for / j/l *j(r- + a" — 2a/1 cos <?) j p), where it is convenient to regard c/> as being connected with xp by the equation Xp — r cos cj>. This function multiplied by eikct is a solution of the wave equation, and when we write p — r sin 0, it is expressible as a function of p, (ft, t and of no other coordinates. Hence eikotJ [/.' VO'" + «"" - 2a?1 cos <f>) \ p] is annihilated by the operator dp- p op aXy that is to say, by the operator d* p-l d (/)-2)cos0 d I 3» ,2 „ -t- ¦— ^ -t ~'~ ; " -, ¦" "T ., x~.- -T it> ¦ or- r or r- sm <p d<p r- <)<p- Now normal functionH which are annihilated by this operator are of the form where JJn (cos 0 ¦ p) is the coefficient* of an in the expansion of A - 2a cos </> + cpy-b'. By the reasoning used in § 4*82, we infer that J \/c V(/"'J +- a2 — 2(W cos <?) | p\ Now uxy)and all the JBes.se 1 functions and equate the coefficients of (k'-ar cos <j>)"- on each side; we find that so that /ln=2^-'(;? + ^-l)r(iy>-- 1). * So that, in Ge^enbauer's notation, W. B. l'\
130 THEORY OP BESSEL FUNCTIONS [CHAP. IV We thus obtain the expansion '2 + a2 - 2ar cos <f>)} + a2 - 2ar cos ?i=0 An analytical proof of this expansion, which holds for Bessel functions of all orders (though the proof given here is valid only when p is an integer), will be given in § 11*4. 4*84. Batemav's solutions of the generalised equation of wave motions. Two systems of normal solutions of the equation 3F 3F VV &V IPV *¦ ' daf dxj dxi dccf ~ c2 dt2 have been investigated by Batenlan*, who also established a connexion between the two systems. If we take new variables p, or, %, -^ denned by the equations xx = p cos ^, sc3 = <t cos i/r, xz — p sin x, #4 = cr sin i/r, the equation transforms into A normal solution of this equation with frequency kc is /M (kp cos O) Jv (ho- sin ^>) ei b»x+f*+*<?Oj where <l> is any constant. Further, if we write p = r cos (j>, <r=r sin ^>, so that (r, %, ^3 <^>) form a system of polar coordinates, equation B) transforms into 3r2 r dr 2 r dr r2 8</J + 'r2 cos2 ^ dx2 + r2 sin2 Now normal solutions of this equation which have e*0*x+">H-*<rf> as a factor are annihilated by the operator Meuenger, xxxni. A904), pp. 182—188; Proc. London Afatfc. ^oc. B) m. A905), pp. 111—123.
4-84] DIFFERENTIAL EQUATIONS 131 and since such solutions are expressible as the product of a function of r and a function of <f> they must be annihilated by each of the operators dr2 r dr 32 . . , , , ,, 9 . ,. /. . , s v? v2 ^.; 4-(cot(D —tan 5y + 4\ (X + 1) - -—-r-r-r , dty d<f> K cos'</> sin3 <^ where X, is a constant whose value depends on the particular solution under consideration. The normal solutions so obtained are now easily verified to be of the form (kr)'1 JA+l (kr) cos^ sin"<f> 2 2 It is therefore suggested that J^ (A;?1 cos <f> cos •$) /v (A:r sin <? sin is expressible in the form 2 aK (kr)~x J.d\+i (/c?') cos'4^) sin" 6 . ojP, (^—^r X, -— where the summation extends over various values of X, and the coefficients depend on X and <t>, but not on r or 0. By symmetry it is clear that +\ + 1; v + l; sin-<J>V where b\ is independent of <J>. It is not difficult to see that and Bate man has proved that We shall not give Bateman's proof, which is based on the theory of linear differential equations,*but later (§ 11/6) we shall establish the expansion of J,, (kr cos <j> cos <1>) Jv (kr sin <j> sin <t>) by a direct transformation. 9—2
CHAPTER V MISCELLANEOUS PROPERTIES OF BESSEL FUNCTIONS 5'1. Indefinite integrals containing a single Bessel function. In this chapter we shall discuss some properties of Bessel functions which have not found a place in the two preceding chapters, and which have but one feature in common, namely that they are all obtainable by processes of a definitely elementary character.. We shall first evaluate some indefinite integrals. The recurrence formulae § 3"9 E) and F) at once lead to the results . A) B) j z~"+} <@v (z) dz = - To generalise these formulae, consider jV»/(*) #,( let this integral be equal to where A (z) and B (z) are to be determined. The result of differentiation is that z"^f(z)<$v(z) = z»» U*(z)<@v(z) + A(zJ^±1^V(Z)-A(z)^v+1 (z) I z + «•+» \B' (z) <@v+1 (z) 4 B (z) <@v (*)}. In order that A (z) and B(z) may not depend on the cylinder function, we take A (z) = B' (z), and then f{z)~ A' (z) + ^±-2 A(z) + B{z). z Hence it follows that C) JV+1 \b"{z) + 2-?±l B' (z) + B («)| K (z) dz = z^ [B1 {z) <@v (z) + B (e)%sv+, (z)}. This result was obtained by Sonine, Math. Ann. xvi. A880), p. 30, though an equivalent formula (with a different notation) had been obtained previously by Lominel, Studien ihber di-e Bessel'schen Funclionen (Leipzig, 1868), p. 70. Some developments of formula C) are due to Nielsen, Nyt Tidsskrift, ix. A898), pp. 73—83 and Ann. di Mat. C) vi. A901), pp. 43—46. For some associated integrals which involve the functions ber and bei, see Whitehead, Quarterly Journal^ xlii. A911), pp. 338—340.
5-1, 5-11] MISCELLANEOUS THEOREMS 133 The following reduction formula, which is an obvious consequence of C), should be noted: D) |**+*c@v(z) dz = -<ji*- v-)\z»~lK(z) dz 5*11. Lo-mmel's integrals containing two cylinder functions. The simplest integrals which contain two Bessel functions are those derived fr6m the Wronskian formula of § 312 B), namely r , •. T,, •. 2 sin vv irz which gives dz v «/_„ (z) A) ] z,T*{z) 2sinj77r Jv(z) ' '¦ dz it J^v(z) z'Jjz) J~-*(z)= ~ 2 sin w g Tv{z) ' and similarly, from § 3'6:i A), : (h _7T YAS) zJ;- (z) ~ 2 </„ \z)' -, ~ dz v Jv(z) (o) I sY/(z) = ~2 r:V)' The reader should liave no difficulty in evaluating the .similar integrals which contain any two cylinder functions of the mune order in the denoiuinator. The formulae acLunlly given are duo to Lommel, Math. Ann. lv. AH71), pp. J03—11(>. The reader should compare C) with the result duo to Euler which wan quoted iu ij 1 -52. Some more interesting results, also due to Lommel*, are obtained from generalisations of Bessel's ucjnation. It is at once verified by differentiation that, if y and 77 satisfy the equations then * Math. Aim. xtv. A879), pp. 520—5315.
134 THEORY OF BESSEL FUNCTIONS [CHAP. V Now apply this result to any two equations of the type of § 4*31 A7). If ^»> ^ denote any two cylinder functions of orders fi and v respectively, we have where <f) (z) and >/r (z) are arbitrary functions of z. This formula is too general to be of practical use. As a special case, take $ (z) and sfr (z) to be multiples of z, say kz and Iz. It is then found that G) JS j(# - P)z - ~^\ K (kz) tv {lz)dz (Jfc.8) tv («*) - ^ (^) ^v+1 (^)) -(ji-v) %\ (kz) *@v (Iz). The expression on the left simplifies still further in two special cases (\)(jl=v, )ife=.«. If we take fi = v, it is found that (8) f . % (fa) *, (U) iz = & — t This formula maj'.be verified by differentiating the expression on the right. It becomes nugatory when k — l, for the denominator is then zero, while the numerator is a constant. If this constant is omitted, an application of l'Hospital's rule shews that, when l-^k, (9) I*z^.(kz)<$»(kz)dz = -~ {kz<^+:(kz)t/ (A?*) .- luV^{he) W^Xkz) - ^ (kz) %\+, (kz)}. The result of using recurrence formulae to remove the derivates on the right of (9) is A0) je^ikz)'^ (kz) dz = \e* {2^M (kz) ^M (kz) - ^ (kz) r#M+1 (kz)
5-11] MISCELLANEOUS THEOREMS 135 Special cases of these formulae are: A1) J'z ^ (kz) dz = \z* [<@* (kz) - %Vi (&*) K+i (kz)} l ~ ^) Vj (kz) + V <& v fS — — — A2) ?<$„. {kz) %% {kz) dz - \z> [2<&n (kz) r$_M (kz) + <$„_, {kz) c$-»-x (kz) J the latter equation being obtained by regarding e~tL1ticS ^(kz) as a cylinder function of order — /j,. To obtain a different class of elementary integrals take k = I in G) and it is found that A3) W (kzW (kz) d- = - kZ W'+l {kz) J *^ ^ z z f fi + v The result of making v -*-/x in this formula is A4) j ^M (&») ^v(/^) = 2 The last equation is also readily obtainable by multiplying the equations 1 *yCAi / \ *j by "" a ' "^mC^) respectively, subtracting and integrating, and then re- placing z by kz. As a special case we have A5) J ^Vfa)" = ^{^I(^)^(fa)-^(fa)^I(fo))+-~^»(A»). An alternative method of obtaining this result will be given immediately. Results equivalent to A1) are as old as Fourier's treatise, La Th&orie Analytique do la Chaleur (Paris, 1822), §§ 318—319, in the case of functions of order zero ; but none of the other formulae of this section seem to have been discovered before the publication of Lommel's memoir. Various special cases of the formulae have been worked out in detail by Marcolongo, Napoli Rendiconti, B) in. A889), pp. 91—99 and by Chessin, Tram. Acad. Sci. of St Louis, xii. A902), pp. 99—108.
136 THEORY OF BESSEL FUNCTIONS [CHAP. V 52. Indefinite integrals containing two cylinder functions; Lommel's second method. An alternative method has been given by Lomnael* for evaluating some of the integrals just discussed. By this method their values are obtained in a form more suitable for numerical computation. The method consists in adding the two results ~ {z» ^(z) %\ (*)} = - z> [% (z) Wv+1 (z) + <g?M+1 (*) %\ (z)} (z) %,+1 (*)} = z» (#„ (*) K+l (z) + fy+l (z) %\ (z)} ^.(p-fM-v-2) zo-i ^ {z) W,,+l (z), so that = z' [^ (z) Wv (g) + Y^+l (z) (SJv+l (z)}, and then giving special values to p. Thus we have A) B) As special cases of these C) /**-*-» ff Vh 00 dz = - 5— [Vf (z) + ^ fl 00}, D) j z^ W (z) dz = 4— _ {^ Again, if p be made zero, it is found that (/* + v)J K(*)®,@ --Qi + v+2)j r<fM+1 (z) %°v+l (z) -Z- so that, by summing formulae of this type, we get = r^ (z) @y (z) 4- 2 "S VM+m (Z) r&v^n 0) + %+n (Z) &v+n (Z). m=l * Math. Ann. xiv. A879), pp. 530—536. z
5-12-5-14] MISCELLANEOUS THEOREMS 137 In particular, if /x = v = 0, _ A? F) J ( = - i [V. (*) $. 0) + 2W?1 ^ (*> $„ <*) + % (,) where n — 1, 2, 3, .... But there seems to be no simple formula for For a special case of A) .see Rayleigh, Phil. May. E) xi. A881), p. 217. [Scientific Papers, I. A899), p. 516.] 53. Soni.ne'a integrals containing two cylinder functions. The analysis of $ Tr\ has been extended by Souine, Math. Ann. xvi. A880), pp. 30—33, to the disc:iiHnion of conditions that J f(z) r$ W* B)} *$'*• W' iz)\ dz may be expressible in tho form A iz) Vr'a {</> B)} 'fi> v ty (z)} + B (z) Y/>'M+\ {</* (z)) ((" v tyi2)) + o B) '{/'•'in {'P (z)\ 'ft 1/+1 fy (i)' ¦+¦ d (z) '^v+i '*/* B)i' r^v4i 'iV' C)}' but the reaultH aru too complicated and not. sufficiently important to justify their insertion here. 5'14. Schaflieitlvn's reduction formula. A reduction formula for which is a natural extension of the formula § 51 D), has been discovered by Schafheitlin* and applied by him to discuss the rate of change of the zoros of c@v{z) as v varies (§ 15). To obtain the formula we observe that 5} *r. %\ (z) Wv'(z)] + * [zwtf,1* (z) + (fi + l) z»+l %', (z) '<C {z)} dz. Now, by a partial integration, jjl + 3) f V+9(&V* {z) dz = [^ +:t '$;* (z)] + 2 * Berliner Sitmngsberichte, v. A906), p. 88.
138 THEORY OF BESSEL FUNCTIONS [CHAP. V and so 0* + 1) | V+2 ^/2 (*) dz = |>+8 ^/2 (*)] + 2 J V+1 (*2 - *2) #„ (*) #/ 0) dz. Hence, on substitution, (fi + 1) f V O2 - v2) ^/ (*) d? («) dz + {(jjl + If - 2v2j [V+1^, («) ^'(^ dz 0) By rearranging we find that + 2) f and this is the reduction formula in question. 5*2. Expansions in series of Bessel functions. We shall now discuss some of the simplest expansions of the type ob- obtained for Qz)m in § 2-l7. The general theory of such expansions is reserved for Chapter XVI. The result of § 2*7 at once suggests the possibility of the expansion A) (W..i(?±*^»> which is due to Gegenbauer* and is valid when fi is not a negative integer. To establish the expansion, observe that is a series of analytic functions which converges uniformly throughout any bounded domain of the s-plane (cf. § 3*13); and since d*lva ' it is evident that the derivate of the series now under consideration is w] -o. ¦ TFicner Sitzungsberichte, lxxiv. B), A877), pp. 124—130.
5-2,5*21] MISCELLANEOUS THEOREMS 139 and so the sum is a constant. When we make z -*-0, we see that the constant is unity; that is to say wr0 n! (W-^m«(*)- I- and the required result is established. The reader will find that it is not difficult to verify that when the expansion on the right.in A) is rearranged in powers of gt all the coefficients except that of #¦ vanish; but this is a crude method of proving the result. 5*21. The expansion of a Bessel function as a series of Bessel functions. The expansion X ,~0 n\ T(v + l-fi- n)f(v'+ n + 1) is a generalisation of a formula proved by Sonine* when the difference v — ju, is a positive integer; it is valid when /a, v and v — n are not negative integers. It is most easily obtained by expanding each power of z in the expansion of {^zy~vJv(z) with the aid of § 5, and rearranging the resulting double series, which is easily seen to be absolutely convergent. It is thus found that r{? + m+ n) r (u+w) r (y + i = .* »ff (, +! - m-• i) r\ by Vanderrnonde's theorem; and the result is established. If we put v = fx -f m, we find that which ia Sonine's form of the result, and is readily proved by induction. * Math. Ann. xvi. A880), p. 22.
140 THEORY OF BESSEL FUNCTIONS [CHAP. V By a slight modification of the analysis, we may prove that, if k is any constant, x gi^j Qi + n, -n\ v + 1; k-) (fi + 2n) JM+an(z). This formula will be required in establishing some more general expansions in § 11-6. 5*22. Lommel's expansions of (z + li)±)svJv [>J(z + ^)}- It is evident that (z + h)~^vJv {\J(z + h)}, qua function of z 4- h, is analytic for all values of the variable, and consequently, by Taylor's theorem combined with | 3-21 F), we have oo Uiri Jm A) (z + h)~*J,y(z +h)} « 2 "~ %- \z-*Jv (V*)} Again, (z 4- h) VJV {\J(z + h)} is analytic except when z+ h = 0; and so, provided that | h \ < I 2}, we have B) " f „ (v) These formulae are due to Lommel*. If we take v = ~\ in A) and »' = ^- in B) we deduce from § 3'4, after making some slight changes in notation, C) D) equation D) being true only when |?|<?J3|. These formulae are due to Glaisherf, who regarded the left-hand sides as the generating functions associated with the functions whose order is half of an odd integer, just as exp [\z (t — Ijt)} is the generating function associated with the Bessel co- coefficients. Proofs of C) and D) by direct expansion of the right-hand sides have been given by Glaisher; the algebra involved in investigations of this nature is somewhat formidable. * Studien iiher die liesseVschen Functionen (Leipzig, 1868), pp. 11—16. Formula A) was given by Bessel, Berliner Abh. 1824 [1826], p. 35, for the Bessel coefficients. + Quarterly Journal, xti. A873), p. 136 ; British Association Report, 1878, pp. 469—470. Phil. Trans, of the Royal Soc. clxxii. A881), pp. 774—781, 813.
5-22] MISCELLANEOUS THEOREMS 141 We shall now enumerate various modifications of A) and B). In A) replace z and h by z- and Icz'2, and then E) Jv {W(l + k)) = A + A)*' 2 U 7/1 = 0 and, in particular, If we divide E) by A -f &)*" and then make & -*. — 1, we find that ^ r(v + l)"m:(> m! l/v+JttW In like manner, from B), (8) Jv [z*J{\ + /,;)} = A + k)-*> % ^'CZf- J_n (z), ¦m.~o i*1- provided that J Ic \ < 1. If we make A- -*- — 1 + 0, we find, by Abel's theorem, Jim [A +k)*J, {WO +^I] = S ^-^.^_WI(A provided that, the series on the right i.s convergent. The convergence is obvious when v is an integer. If v is not an integer, then, for large values of ?», t^""./_ (,) - t-^JK^'"-'_>„!„ (,„ _ ,), „ Hence the condition for convergence! is H(v)>0, and if the condition i.s satisfied, the convergence i.s absolute. Consequently, when A'(V)>0, and also when v is any integer, (9) 2 S---?-.lf*> J,_.w,.(^) = 0. ,„ _ i, //A ! In like manner, if R(i>) > — 1, and also when v is any integer, we have A0) J,(*V2) = 42-J" % V*'T-*-**{*)• It should be observed that functions of the second kind may be substituted for functions of the first kind in A), B), E) and (8) provided that i h. \ < j z , and | h \ < 1; sfo that A1) (« +A)"*' Yv W(* + h)} - S ^"^')Wi8-^+»> F^Cvfc), 00 ( 1 / w=0 7
142 THEORY OF BESSEL FUNCTIONS [CHAP. V These may be proved by expressing the functions of the second kind as a linear combination of functions of the first kind; by proceeding to the limit when v tends to an integral value, we see that they hold for functions of integral order. By combining A1)—A4) with the corresponding results for functions of . the first kind, we see that we may substitute the symbol "gf for the symbol Y throughout. . These last formulae were noted by Lommel, Studien, p. 87. Numerous generalisations of them will be given in Chapter xi. It has been observed by Airey, Phil. Mag. F) xxxvr. A918), pp. 234—242, that they are of some use in. calculations connected with zeros of Besael functions. When we combine E) and A3), and then replace \/(l + k) by X, we find that, when |X2-1|<1, A5) ff. (\z) = X" 2 U {* U*«i <#v+m (s), and, in particular, when X is unrestricted, These two results are frequently described* as multiplication theorems for Bessel functions. It may be observed that the result of treating A4) in the same way as (8) is that (when v is taken equal to an integer n) (H) -(»-l)! B/*)»«,r 2 m^ *»-«(*). m=0 m • An alternative proof of the multiplication formula has bee.n given by Bbhiner, Berliner Sitzungsberichte, xin. A913), p. 35, with the aid of the methods of complex integration; see also Nielsen, Math. Ann. lix. A904), p. 108, and (for numerous extensions of the formulae) Wagner, Bern Mittheilungen, 1895, pp. 115—119; 1896, pp. 53—60. [Note. A special case of formula A), namely that in which v = l, was discovered by Lommel seven years before the publication, of his treatise; see Archiv der Math, xxxvn. A861), p. 356. His method consisted in taking the integral — M cos (?r cos 6 + yr sin 6) d? dr) over the area of the circle ?2+»j2=l, and evaluating it by two different methods. The result of integrating with respect to rj is -^ j cos frcoa 6)via U(l-?). rain ? See, e:g. Sohafheitlin, Die Th?orie der Besselschen Funktimen (Leipzig, 1908), p. 83.
5*23, 5*3] MISCELLANEOUS THEOREMS 143 and the result of changing to polar coordinates (p, (p) is =- I I cos {rp cos (cp — 9)} pdpdcp = — I j cos (rp cos <p) pdpd<p Alt J -ir J o &1T J —V J 0 = i- J J cos (?r) (*?<*, = ^ ? a - |*)icos (?r) rff-J^ (r)/r. If we compare these equations we obtain A) in the case v = \ with z and h replaced by ?-2 cos2 0 and r2sin20.] 5'23. The expansion of a Bessel function as a series of Bessel functions. From formula § 5 22 G), Lommel has deduced an interesting series of Bessel functions which represents any given Bessel function. If fx, and v are unequal, and //, is not a negative integer, we have The repeated series is absolutely convergent; consequently we may re- rearrange it by replacing p by m — n, and then we have and hence, by Vandermonde's theorem, A) </,(-)= ^---^^r^^ m! Thia formula was given by Loinmcl, Studien ubar die Bes&efschen Functionen (Leipzig, 1868), pp. 22—23, in tho wpecial caso fx = O; by differentiating with respect to v and then putting 1^ = 0, it is found that B) M'o(*Wo(*)l"g(^)-rr(f+V ^ ^fenCW-^^+.W • and, when f*=0, we liave Lommol'a formula C) ^Y0(z) = .r0(z).{y+\og(hz)}+ 2 2 . =l "* • '*»• This should be compared with Neuinann'B expansion given in § 3'57. 5*3. An addition formula for Bessel functions. An extension of the forraula of § 2 to Bessel functions of any order is A) Jv(z + t)= 2 Jv- where | z\ < 111, v being unrestricted. This formula is due to Schlafli* ; and the similar but more general formula B) %\(z + t)= X <$v- is due to Sonine-f. * Math. Ann. in. A871), pp. 135—137. f Ihid- XVI- A880), pp. 7—8.
144 THEORY OF BESSEL FUNCTIONS [CHAP. V It will first be shewn that the series on the right of A) is a uniformly convergent series of analytic functions of both z and t when z | < r, R «; j t \ ^ A, where r, R, A are unequal positive numbers in ascending order of magnitude. When m is large and positive, Jv-m(t)Jm(z) is comparable with sin vir. {\R)V. (r/R)m ~^ and the convergence of the series is comparable with that of the binomial series for A — rjKy. When m is large and negative (= — n), the general term is comparable with T(v'+n-hl).nl and the uniformity of the convergence follows for both sets of values of m by the test of Weierstrass. Term-by-term differentiation is consequently permissible*, so that •" 7ft= -00 1 " -5 X J",_m (t) (/w_i (ar) - «/¦,„+! {z)\, " in = - oo and it is seen, on rearrangement, that all the terms on the right cancel, so that 00 Hence, when | z \ < \t\, the series 2 Jv_m (t) Jm (z) is an analytic function 7tt= -00 of 2- and ^ which is expressible as a function of z + t only, since its derivates with respect to z and t are identically equal. If this function be called F {z +1), then TO=—00 If we put 2 = 0, we see that F(t)=J?(t), and the truth of A) becomes evident. Again, if the signs of v and m in A) be changed, we have m= —oo and when this result is combined with A), we see that C) 5 1»= -CO * Cf. Modern Analysis, § 5*3.
5'4] MISCELLANEOUS THEOREMS 145 When this is combined with A), equation B) becomes evident. The reader will readily prove by the same method that, when | z \ < i t |, D) Jv{t-z)= I Jv+m{t)Jm{z), 7I= — 00 E) #„(*-*)= S %\+m{t)Jm{z), »)! = —00 F) Yv{t-z)= S F,+ Of these results, C) was given by Loinmcl, Studien uber die BesseVschen Functionen (Leipzig, 1868), when v is an integer; while D), E) and (G) were given* explicitly by Graf, Math. Ann. xliii. A893), pp. 141—142. Various generalisations of these formulae will be given in Chapter xi. 5*4. Products of Bessel functions. The ascending series for the product J^ (z) Jv (z) has been given by various writers; the expansion is sometimes stated to be clue to Schonholzerf, who published it in 1877, but it had, in fact, been previously published (in 1870) by SchlarliJ. More recently the product has been examined by Orr§, while Nicholson]j has given expansions (cf. § 5-42) for products of the forms M*)Y*(z) and Ym(z)Yn(z). In the present section we shall construct the differential equation satisfied by the product of two Bessel functions, and solve it in series. We shall then (§ 541) obtain the expansion anew by direct multiplication of scries. Given two differential equations in their normal forms if y denotes the product vw, we have y" = v"w + 2v'iv' + vw" where primes indicate differentiations with respect to z. * See also Epstein, Die vicr Rechnungtoperationen mil Beasel"when Fimctionen (Bern, 1894), [Jahrbuch iiber die l'ort»chritte tier Math. 1H93-181I, pp. 845—846]. t Ueber die Auxwerthung he.atimmterlntegrale.init Iliilfc von Veriindettmgendea Integrutionsioeyes (Bern, 1877), p. 13. The authorities who attribute the expansion to Schonholzer include Graf and Guhler, Einlcihtng in die Thcorie tier HeaaaVacheii Funktioncn, u. (Bern, 15H0), pp. 85—87, and Nielsen, Ann. Sci. dc vAcole norm. sup. C) xvm. A901), p. 50; Handbuch der Theoric der Gyliu- dcrfunktionen (Leipzig, 1904), p. (JO. According to Nielsen, Nouv. Ann. de Math. D) n. A902), p. 3'J6, MeiBael obtained some series for products in the Iserlohn Progmmm, 18G2. X Math. Ann. in. A871), pp. 141—142. A trivial defect in Schlafli's proof is that lie uscb a contour integral which (as he points out) converges only when § Proc. Camb. Phil. Soc. x. A900), pp. 93—100. || Quarterly Journal, xuir. A912), pp. 78—100. w. B. F.
146 THEORY OF BBSSEL ITJNCTIONS [CHAP. V It follows that j!~ [y" + (I + J)y} = 2v"w' + 2v w" and hence y'" + 2 (/ + J) ?/ + (I' + /') y = (/ - /) (vfw - vw'). Hence> in the special case when I = J,y satisfies the equation but, if J *fc J, it is easy to shew by differentiation that B) *L W— This is the form of the differential equation used by Orr; in connexion with A), see Appell, Comptes Rendw, xoi. A880), pp. 211—214. To apply these results to Bessel's equation, the equation has to be reduced to a normal form; both Orr and Nicholson effect the reduction by taking z ^v(z) as a new dependent variable, but, for purposes of solution in series, it is simpler to take a new independent variable by writing so that (~ni$ + (e29 - v*) Jv (z) = 0. atf Hence the equation satisfied by J^ (z) Jv (z), when ft? ^ v2, is that is to say C) O4 - 2 O2 + v2) Sa + (m2 - v*f] y + 4e2S (^ +1) (% + 2) y = 0, and the equation satisfied by Jv (z) J±v (z) is D) ^(^-4^ Solutions in series of C) are m=0 where a — + p + v and 4 (a 4- 2m -1) (a + 2m) cm-\ m~ (a+ ijl + v + 2m) (a + /Lt — v + 2m) (a — /t + 7/ + 2m) (a - /a — v + 2w/) ' If we take a = /u + v and we obtain the series 00 2 - v + m+ 1)T (/t + m + 1)T (v + m+ 1)' and the other series which are solutions of C) are obtained by changing the signs of either fi or v or both /x and v.
5-41J MISCELLANEOUS THEOREMS 147 By considering the powers of z which occur in the product J^ (z) Jv (z) it is easy to infer that, if 2/z, 2v and 2 (fx + v) are not negative integers and if /a2 j= n8, then In like manner, by solving D) in series, we find that, when 2v is not a negative integer, then and, when 7/ is not a negative integer, then By reasoning which resembles that given in § 4*42, it may be shewn that F) holds when v is half of an odd negative integer, provided that the quotient F Bv + 2m + 1)/F Bv + in + 1) is replaced by the product Bv + m + l)m. 5*41. Products of series representing Bessel functions. It is easy to obtain the results of § 5-4 by direct multiplication of series. This method has the advantage that special investigations, for the cases in which /xa= v2 and those in which j^ + v is a negative integer, are superfluous. The coefficient of (-)"l(^y+"+2»l in the product of the two absolutely convergent series 2 (-) H) ) r\T( 2 v mfo m! r (/* + »i + 1) nr0n\T(v+n + is i n=0«! r (" + « + 1) • (»« - n)! I" (/* + W - M + 1) m! r C/jl + m + 1) T (p + m + 1) (fjL + v+m + 1 )m ~ m! T (/u, + m + 1) P {y + to 4-1)' when Vandermonde's theorem is used to sum the finite series. Hence, for all values of /x and v, a,nd this formula comprises the formulae E), F) and G) of § 5*4. 10—2
148 THEORY OF BESSEL FUNCTIONS [CHAP. V This obvious mode of procedure does not seem to have been noticed by any of the earlier writers; it was given by Nielsen, Math. Ann. lii. A899), p. 228. The series for J0(z) cos z and J0(z)a'mz were obtained by Bessel, Berliner Abh. 1824, [1826], pp. 38—39, and the corresponding results for Jv{z) cos z and Jv(z) aim were deduced from Poisson's integral by Lonimel, Studien iiber die Bessel'sohen Functionen (Leipzig, 1868), pp. 16—18. Some deductions concerning the functions ber and bei have been made by Whitehead, Quarterly Journal, xlii. A911), p. 342. More generally, if we multiply the series for /M (az) and Jv (bz), we obtain an expansion in which the coefficient of (-)m a* 6" (\zy+v+m is 3on\ r(i/ + ?i + l).(m -n)! F(> + m-n + l) _ a2m BF, (- m, - ji - m; y + 1; 67a2) "" m! r(fi + m + l)T(v + l) and so B) ^M^F,) = (i^M" 1 (-) (|oa)8w afi (- m, - /Ji - m; v + 1; 6'/ and this result can be simplified whenever the hypergeometric series is expressible in a compact form. One case of reduction is the case b — a, which has already been discussed ; another is the case b — ia, provided that fj? == v-. In this case we use the formula* and then we see that / N r / \ V (~)W (\CLZ)-V^m COS C) iW=o w! r (y + m + 1) I> + 2m + 1)' DA / ( )I ( \- 5 (~)m (j^Jm cos (\v - ^m) 00 (~\ T / \7" / \ *C* If we take a = e^ in C) we find that F) ber,2 (z) + bei,2 («) = S — (^)a>+4m m=o ml 1 (v + w + 1) F (» + 2m + 1)' an expansion of which the leading terms were given in § 3-8. * Cf. Kutnmer, Journal fUr Math. xv. A836), p. 78, formula E3).
•5-42] MISCELLANEOUS THEOREMS 149 The formulae C), D), E) were discovered by Nielsen, Atti delta R, Accad. dei Lincei, E) XV. A906), pp. 490—497 and Monatshefte fiir Math, und Phys. xix. A908), pp. 164—170, from a consideration of the differential equation satisfied by Jv{az) J±v{bz). Some series have been given, Quarterly Journal, xli. A910), p. 5ft, for products of the types Jj,3 (z) and J^ (z) «/_„ («), but they are too cumbrous to be of any importance. By giving fi the special values + \ in B), it is easy to prove that G) e"°->/,_ i « {g ( i t* ? The special case of this formula in which 2v is an integer has been given by Hobson*. 5'42. Products involving Bessel functions of the second hind. The series for the products «/)»(z) Yn (z), Jm (z) Yn (z), and Ym (z) Yn (z) have been the subject of detailed study by Nicholsonf; the following is an outline of his analysis with some modifications. We have irJ,(z) Yn (z) = 1 [J,(z) Jv («)} - (-)»I [J, (z) J_v (,)}, where v is to be made equal to n after the differentiations have been performed. Now and ^ {J, (Z) J_ {Z)\ = - log QZ). J. (Z) J-v (Z) - " +"»"' + l)'r'O* + r +1)'!1 (- v + r + 1) n-l We divide the last series into two parts, 2 and % . In the former part we have * Proc. London Hath. Soc. xxv. A894), p. G6; see also Cailler, M6m. de la Soc. de Phys. de Genlve, xxxiv. A902—1905), p. 310. f Quarterly Journal, xvau A912), pp. 78—100. The expansion of J0(z) Yq(z) bad been given previously by Nielsen, Handbuch der Theorie der Cyliiiderj'unktionen (Leipzig, 1904), p. 21.
150 THEORY OF BESSEL FUNCTIONS [CHAP. V while in the latter part there is no undetermined form to be evaluated. When r is replaced in this part by n + r, it is seen that ttJ.(z) Yn(z) = - 2o rlTOi + | (-)rg !( x {g(?) ^(/ --^(/ju + n + r + l)- a//(> + r +1) - ^ (ft + r + 1) - i|r (r + 1)}. The expression on the right is a continuous function of //, at //, = in where m— 0,1, 2, ..., and so the series for irjm{z) Yn(z) is obtained by replacing /j, by m on the right in A). The series for Tm (z) Yn (z) can be calculated by constructing series for in a similar manner. The details of the analysis, which is extremely laborious, have been given by Nicholson, and will not be repeated here. 5 3. The integral for Jp{z)Jv(z). A generalisation of Neumann's integral (§ 2*6) for Jn2 (z) is obtainable by applying the formula* to the result of § 5*41; the integral has this value when m —0,1,2,..., provided that R(jm + v)>-1. It is then evident that J()J() 2 so tha.t.1 when JR (/a + v) > — 1, 2 f* A) JMB)JV(^) = - 7T J 0 the change of the order of summation and integration presents no serious difficulty. * This formula is due to Cauchy; for a proof by contour integration, see Modern Analysis, p. 263.
5-43-5-51] MISCELLANEOUS THEOREMS 151 If n be a positive integer and R (ji — n) > — 1, then B) j; (z) Jn (z) = -fc? ( */„_» B* cos 0) cos (i* + n) 0d0, T Jo and this formula is also true if yu and w are &otf/i integers, but are otherwise unrestricted. Formula A) was given by SchltiBi, Math. Ann. in. A871), p. 142, when p±v are both integers; the general formula ia due to Gegenbauer, Wiener Sitzungsberichte, oxi. Ba), A902), p. 567. 5*5. The expansion oftyzY*" os a series of products. A natural generalisation of the formulae of Neumann (§2*7) and Gegenbauer (§ 5-2) is that Ml-0 "i! The formula is true if /a and v arc not negative integers, but the following proof applies only if R(/x +u + l)> — 1. From § 5-2 we have 1 ^±"+2«)^+"±^/,+^,, B,003^). If we multiply by cos (/j, — v) 6 and integrate, it is clear from § 5'43 that ^ + „ + 2m) T (/* + v + m) J ^ cos (/x — vNdd= S o m-0 Wl X and the result follows by evaluating the integral on the left; for other values of fx and v the result may be established by analytic continuation. The formula in at once deducible from formulae givon by Gegeubauer, Wiener Sitzungs- berichte, lxxv. B), A877), p. 220. 5*51. Lommel's series of squares of Bessel functions. An expansion derived by Lommel* from the formula z dz so that 1 f -, (z) dz = [ ^ (v + 2n) /» * The results of this seotion will be found in Math. Ann. h. A870), pp. 632—633; xiv. A878), p. 532 ; MUnchener Abh. xv. A886), pp. 548—549.
152 THEORY OF BESSEL FUNCTIONS [CHAP. V Hence, by § 511 A1), we have A) \* \.1\_, (if) -</„_, E) Jv (z)} = I (v + 2n) J\+m (*), n-0 on taking zero as the lower limit when R(v)>0; by adding on terms at the beginning of the series, it may be seen that the restriction R,(v) > 0 is super- superfluous. If we take in turn v — \, v = f, and add and subtract the results so obtained, we have (§ 3*4) B) -= X T 71=0 sin 2# °° {6) —x = -^ \—) l Jinr n~o while, by taking v = l, we see that D) \z* {«/02 (z) + Jf (z)} = 2 Bh 4- Another formula of the same tvpe is derived by differentiating the series for it is evident that 5 c 7"s /»\_9 V c T (AT' (A ~t~ & cji o fi^.n x*1) — •" *-i en <J v+n \") " v+n \*/ 00 — S e T fr\ \ T (v\ T /iy\\ and so,, when R(v) >0, we obtain a modification of Hansen's formula (§2-5), namely E) 2 e»A^(*)~2J J,2@t- «=o Jo t An important consequence of this formula, namely the value of an upper bound for j Jv(%) j, will be given in § 1342. By taking v-\, it is found that ~[ ^t Jo 0 F) 2 " Z dt sin2i . o t and so »=0 where, as usual, the symbol Si denotes the "sine integral." This result is given by Lommel in the third of the memoirs to which reference has been made.
5-6] MISCELLANEOUS THEOREMS 153 5*6. Continued fraction formulae. Expressions for quotients of Bessel functions as continued fractions are deducible immediately from the recurrence formula given by §3*2A); thus, if the formula be written Jv(z) \z\v it is at once apparent that \ ) 7 7~T\ = t This formula is easily transformed into B) Jv^ 1 1 - -1 «/m+«l±i(?) . W ' J?_x (z) 1v\z - 2\v +1)/z - ... - 2 {v + mjlz - j;,+m (z) " These results are true for general values of v; A) was discovered by Bessel* for integral values of v. An equivalent result, duo to Schlomilchf, is that, if Q,, {z) = Jv+l(z)/\}jZ Jv (z)}, then KP) V, W - „ +1 _ J7+ 2 -7+8 - ... - v + m - 4 ' Other formulae, given by Lommel^, are Jv+l(z) z z2 z" z" *J,+m+i(z) D) Jv(z) W; Jv (z) + 2 (i; + 1) - 2 (v + 2) - ... - 2 (iJ + m) - ^,,+m B) " The Bessel functions in all these formulae may obviously be replaced by any cylinder functions. It was assumed by Bessel that, when in -*• co, the last quotient may be neglected, so that J.(z) _\z\v \s*l\v(v+l)) \z*l[(v JZJ—r F) r- 1 * Berliner Abh. AS'24), [18'2()]t p.})]. Formula B) neeron not to have been given by the earlior writers; soo Encyclopedic den Sci. Math. n. 28, § .38, p. 217. A slightly different form is used by- Graf, Ann. di Mat. B) xxin. A895), p. -17. •(• Zeitschrift fttr Math, und Phys. 11. A857), p. 142 ; Schlouiilch considerud integral values of v only. % Studien tibev die Be&seVxchcn Functionen (Leipzig, 18C8), p. 5 ; see also Spitzer, Archiv der Math, und Phys. xxx. A858), p. 332, and Giinther, Archiv der Math, und P/iys. Lvr. A874), pp. 292—297.
154 THEORY OF BESSEL FUNCTIONS [CHAP. V It is not obvious that this assumption is justifiable, though it happens'to be so, and a rigorous proof of the expansion of a quotient of Bessel functions into an infinite continued fraction will be given in § 9*65 with the help of the theory of " Lommel's polynomials." [Note. The reason why the assumption is not obviously correct is that, even though the fraction pm\iw. tends to a limit as m-*-oo, it is not necessarily the case that n^m w+* tends to that limit; this may be seen by taking The reader will find an elaborate discussion on the representation of Jv (z)/Jv-i (z) as a continued fraction in a memoir* by Perron, Miinchener Sitzungsberichie, xxxvu. A907), pp, 483—504; solutions of Riccati's equation, depending on such a representation, have been considered by Wilton, Quarterly Journal, xlvi. A915), pp. 320—323. The connexion between continued fractions of the types considered in this section and the relations con- connecting contiguous hypergeometric functions has been noticed by Heine, Journal filr Math. lvii. A860), pp. 231—247 and Christoffel, Journal filr Math, lviii. A861), pp. 90—92. 5*7. Hansen's expression for Jv{z) as a limit of a hypergeometric function. It was stated by Hansenf that A) J,(,)- lim - /l We shall prove this result for general (complex) values of v and z when X and H tend to infinity through complex values. If X —1/5, n = 1/rj, the (m + l)th term of the expansion on the right is This is a continuous function of 8 and 7); and, if S0,t]0 are arbitrary positive numbers (less than 2 | z I), the series of which it is the (m + l)th term con- conrH. | I ( ) verges uniformly with respect to 8 and 7} whenever both | 8 | <S0 and 1771 ) For the term in question is numerically less than the modulus of the (w + 1 )th term of the (absolutely convergent) expansion of and the uniformity of the convergence follows from the test of Weierstrass. Since the convergence is uniform, the sum of the terms is a continuous * This memoir is the subject of a paper by Nielsen, Miinchener Sitzungsberichte, xxxvm. A908), pp. 85—88. + Leipziger Abh. 11. A855), p. 252 5 see also a Halberstadt dissertation by F. Neumann, 1909. [Jakrbuch Uber die Fortsehritte der Math. 1909, p. 575.]
5-7,5*71] MISCELLANEOUS THEOREMS 155 function of both the variables (B, rj) at @,0), and so the limit of the series is the sum of the limits of the individual terms; that is to say lim (**} - F (l 1 • v + 1 • - Z* \n — o/ and this is the result stated. 5'71. Bessel functions as limits of Legendre functions. It is well known that solutions of Laplace's equation, which are analytic near the origin and which are appropriate for the discussion of physical problems connected with a sphere, may be conveniently expressed as linear combinations of functions of the type rnPn (cos 0), rnPnm (cos 0) °°S m <j>; Sill these are normal solutions of Laplace's equation when referred to polar coordinates (r, 0, <f>). Now consider the nature of the structure of-spheres, cones and planes associated with polar coordinates in a region of space at a great distance from the origin near the axis of harmonics. The spheres approximate to planes and the cones approximate to cylinders, and the structure resembles the structure associated with cylindrical-polar coordinates; and normal solutions of Laplace's equation referred to such coordinates are of the form (§ 4'8) e-ikzJm(kp) . vi(b. r sin ^ It is therefore to be expected that, when r and n are large* while 9 is small in such a way that r sin 0 (i.e. p) remains bounded, the Legendre function should approximate to a Bessel function; in other words, we must expect Bessel functions to be expressible as limits of Legendre functions. The actual formulae by which Bessel functions are so expressed are, in effect, special eases of Hanson's limit. The most important formula of this type is A) lim Pn (cos-)=/„(*). This result, which hccuis to have been known to Neumannt in 1862, has been investi- investigated by Mohlor, Journal fur Math. Lxvm. A868), p. 140; Math. Ann. v. A872), pp. 136, 141—144; Heine, Journal fur Math. lxix. A869), p. 130; Raylcigh, Proc. London Math. Soo. ix, C1878)> 1>P- 61—64 ; Proc. Royal Hoc. xcn. A, A916), pp. 433—437 [Scientific Papers, i. A899), pp. 338—341 ; vi. A920), pp. 393—397]; and Giuliani, Giorn. di Mat. xxn. A884), pp. 236—239. The result ban been extended to generalised Lcgundre functions by Heine and Raylcigh. It has usually been assumed that n tends to infinity through integral values in proving A); but it is easier to prove it when n, tends to infinity as a continuous real variable. * If ?i were not large, the approximate formula for Pnm (cos 6) would be (8him0)/»il. + Cf. Journalfilr Math. lxii. A863), pp. 36—49.
156 THEORY OF BESSEL .FUNCTIONS [CHAP. V We take Murphy's formula Pn (cos zjn) = aFj (- n, «• +1; 1; sin'2 \zjn); and the reasoning of the preceding section is applicable with the slight modification that we use the inequality when j z | ^ 2 | n \, and then we can compare the two series 2F, (- n, n + 1; 1; sin2 \z\n\ .F, A/fi,, l/*0 +1; 1; ^8021 z |«), where So is an arbitrary positive number less than § | z \~x dnd the comparison is made when \n\> l/80. The details of the proof may now be left to the reader. When n is restricted to be a positive integer, the series for Pn (cos z/n) terminates, and it is convenient to appeal to Tannery's theorem * to complete the proof. This fact was first noticed by Giuliani; the earlier writers took for granted the permissibility of the passage to the limit. In the case of generalised Legendre functions (of unrestricted order m), the definition depends on whether the argument of the functions is between ¦f1 and — 1 or not; for real values of cg (between 0 and tt) we have n ™ cos - ) = -. . :,¦ 2^i (— ft) w +1; m + 1; sin3 ? cc n), \ 11/ F(m + 1) v so that B) lim »»Pn-« (cos ^ = /m (a;), but otherwise, we have Pn~m (cosh ~) = ta^2^^ oFi (- n, n +1; m + ¦ 1 ; - sinh2 \ z/n), so that C) lim nmPn-m (cosh -) = Jm («). The corresponding formula for functions of the second kind may be deduced from the equation which expresses! Qnm in terms of Pnm and Pn~m; it is D) lim 2n(m + ri)nr $"* V°sh n) = Km(^)- This formula has been given (with a different notation) by Heine:};; it.is most easily proved by substituting the integral of Laplace's type for the Legendre function, proceeding to the limit and using formula E) of § 6'22. * Cf. Brornwich, Theory of Infinite Series, § 49. + Cf. Barnes," Quarterly Journal, xxxix. A908), p. 109 ; the equation is in Barnes' notation, which is adopted in this work. t Journal fUr Math. lxiz. A868), p. 131.
5-72] MISCELLANEOUS THEOBEMS 157 Another formula, slightly different from those just discussed, is <«> Km P. this is due to Laurent*, and it may be proved by using the second of Murphy's formulae, namely Pn (cos 0) — cos'1 \0. ^ (- n, - n; 1; - tan2 \6). [Note. The existence of the formulae of this section must be emphasized because it used to be generally believed that there was no connexion between Legendre functions and Bessol functions. Thus it was stated by Todliuntor in his Elementary Treatise on Laplace's Functions, Lame's Functions and BcsseVs Functions (London, 1875), p. vi, that " these [i.e. Bessel functions] are not connected with the main subject of this book."] 5*72. Integrals associated luith Melder's formula. A completely different method of establishing the formulae of the last section was given by Mehler and also, later, by Bayleigh ; this method depends on a use of Laplace's integral, thus 1 fn 1\ (cos 6)-- (cos 8 + i sin 6 cos 6)n d6 1 f" — _j 7T .' 0 Since n log (cos (z/n) +¦ % sin (z/n) cos (f>) -*- iz coa § uniformly as n ¦->- oo when 0 ? 0 < 7r, we have at once lim Pn (coa zjn) - f eizcos¦ dej) = Jo {2). Heinej" and de Ball;]: have made similar passages to the limit with integrals of Laplace's type for Legend re functions. In this way Heine has defined Bessel functions of the second and third kinds; reference will be made to his results iu § (r22 when we deal with integral representations of Yv(z). Moliler ha.s also given a proof of his formula by using the Mchlor-Dirichlct integral 1 n (OOH 0) _ -¦ j ^ - If ?«/>=>!/¦, it may be wliewn that but the pa-saago to the limit })resci)tH some little difficulty bccau.se the integral is an im- improper integral. Various formulae have been given recently which exhibit the way in which * Joumal </e Math. C) 1. A875), pp. 384—385; the formula actually given by Laurent is erioneous on account of an arithmetical error. t Journal fiir Math. lxix. }18G8), p. 181. See also Sharpe, Quarterly Journal, xxtv, A890), pp. 383—386. X Asir. Nacli. exxvni. A891), col. 1—4.
158 THEORY OF BESSEL FUNCTIONS [CHAP. V the Legendre function approaches its limit as its degree tends to infinity. Thus, a formal expansion due to Macdonald* is A) Pn-™(cos0) «(n+?rm (cos \ey™ [Jm («) + sin4 \6 {^x J^+a(x) - Jm+2 (a) +$xr1 Jw+1 (a?)} + ...], where a? = Bn +1) sin \0. Other formulae, which exhibit an upper limit for the error due to replacing a Legendre function of large degree by a Bessel function, aref B) Pn (cos 7}) ± iir~l Qn (cos rj) = V(sec tf). «±(»+««i-tM,) ryo {(n + ^ tan*/} + %Y0 {(n +1) tan ^J] v) D) Qn (cosh ?) = e-(^+i)<f-tanh « ^(sech ^ . ^o ^(n + $ tann |j where, in B), 0 ^ rj < ^ir, and, in C) and D), f ^ 0; the numbers 9Xl 62, 6% are less than unity in absolute magnitude, and n may be complex provided that its real part is positive. But the proof of these results is too lengthy to be given here. 5' 73. The formulae of Olbricht The fact that a Bessel function is expressible by Hansen's formula as a limit of a hypergeometric function has led OlbrichtJ to investigate methods by which Bessel's'equation is expressible as a confluent form of equations associated with Eiemann's P-functions. If we take the equation of which a fundamental system of solutions is the pair of functions and compare the equation with the equation defined by the scheme !a, b, c, \ «, A 7, *[. «'. 0, 7. > * Proc. London Math. Soe. B) xm. A914), pp. 220—221; Borne associated results had been obtained previously by the same writer, Proc. London Math. Soc. xxxi. A899), p. 269. + Watson, Trans. Gamb. Phil'. Soc. xxn. A918), pp. 277—308; Measenrjer, xlvu. A918), pp. 151—160. X Nova Acta Caes.-Leop.-Acad. (Halle), 1888, pp. 1—48.
5-73] MISCELLANEOUS THEOREMS 159 namely fry | fl-tt-tt^ 1 -fi-fi' ^ l-Y-y|dy | — a 2 — 6 z — c ) dz x \act'(a-b)(a-c) | ffff F - c) F ~ a) t 77' (c - a) (c - { -2 — a z — b z — c (z - a)(z— b) (z — c) ' we see that the latter reduces to the former if a = 0, a— v — fi, a' = — v — p, while b, c, fi, fi', 7, 7' tend to infinity in such a way that fi + fi' and 7 H- 7' remain finite (their sum being 2/j. + 1) while fifi' =77' = \b* and b + c — O. We thus obtain the scheme f 0, 2ifi, -2ifi, j liin Pi v-fi, fi, 7, z\, l-z/-/i, -fi, 7', J where 7, 7' = /i + ^ ± ^{0"- + ?)a + /3al- Another similar scheme is {0, ifi, 00, ) v-fi, fi, 7, z\ -v-fi, -fi, <y', J with the same values of 7 and 7' as before. A scheme for Jv (z) derived directly from Hansen's formula is ( 0, oo, -4a/9, >, (aAZZ) [-\V) fi-\v, v+l-a-fi. J Olbricht has given other schemes but they are of no great importance and those which have now been constructed will be sufficient examples. Note. It has been observed by Haentzsohcl, Zeitschrift fiir Math, und Pkys. xxxi. A886), p. 31, that the equation du* whoee solution (§ 4-3) is k^ (Aw), may be derived by confluence from Lamp's equation when the invariants g2 and ^3 of the Weierstrassian elliptic function arc made to tend to zero.
CHAPTER VI INTEGRAL REPRESENTATIONS OF BESSEL FUNCTIONS 6*1. Generalisations of Poisson's integral. In this chapter we shall study various contour integrals associated with Poisson's integral (^ 23, 3'3) and Bessel's integral (§ 2*2). By suitable choices of the contour of integration, large numbers of elegant formulae can be obtained which express Bessel functions as definite integrals. The contour integrals will also be applied in Chapters vii and viii to obtain approximate formulae and asymptotic expansions for Jv (z) when z or v is large. It happens that the applications of Poisson's integral are of a more elementary character than the applications of Bessel's integral, and accordingly we shall now study integrals of Poisson's type, deferring the study of integrals of Bessel's type to § 6*2. The investigation of generalisations of Poisson's integral which we shall now give is due in substance to Hankel *. The simplest of the formulae of § 3'3 is § 3*3 D), since this formula contains a single exponential under the integral sign, while the other formulae contain circular functions, which are expressible in terms of two exponentials. We shall therefore examine the circumstances in which contour integrals of the type eiztTdt are solutions of Bessel's equation; it is supposed that T is a function of t but not of z, and that the end-points, a and b, are complex numbers independent of z. The result of operating on the integral with Bessel's differential operator Vti denned in § 3\L, is as follows: V, \zv \ eizt Tdt\ = zv+* \ elzt T{\-12) dt + Bv + l)izv+> [ J a ) J a J eizi Ttdt * Math. Ann. i. A869), pp. 473—485. The discussion of the corresponding integrals for Iv(z) a&d Kv{z) is due to Sohlafli, Ann. di Mat. B) i. A868), pp. 232—242, though Schlafli's results are expressed in the notation explained in § 4-15. The integrals have also been examined in great detail by Gubler, Zurich VierteljahrsschrLft, xxxm. A888), pp. 147 —172, and, from the aspect of the theory of the linear differential equations which they satisfy, by Graf, Math. Ann. xlv. A894), pp. 235—262; lvi. A903), pp. 432—444. See also de la Vallee PouBBin, Ann. de la Soc. Sci. de BruxelUs, xxix. A905), pp. 140—143.
6-1] INTEGRAL REPRESENTATIONS 161 by a partial integration. Accordingly we obtain a solution of Bessel's equation if T, a, b are so chosen that %j\T(?- 1)} =- Bz/ +1) Tt, IV* T(V - 1I* = 0. The former of these equations shews that T is a constant multiple of (t' — iy-i, and the latter shews that we may choose the path of integration, either so that it is a closed circuit such that eizt(t" —1)" + * returns to its initial value after t has described the circuit, or so that efat(i2- 1)" + * vanishes at each limit. A contour of the first type is a figure-of-eight passing round the point t = 1 counter-clockwise and round t = — 1 clockwise. And, if we suppose temporarily that the real part of z is positive, a contour of the second type is one which starts from + ooi and returns there after encircling both the points — 1, + 1 counter-clockwise (Fig. 1 ami Fig. 2). If we take a, b = ± 1, it is Pig. 2. necessary to suppose; that. R(v -\- !,)>(), and we merely obtain Poisson's integral. To make the many-valued function (?'- — 1 )"""* definite*, we take, the phases of i — 1 and t+1 to vanish at the point A when? the contours cross the real axis on the right of t-1. We therefore proceed to examine the contour integrals •(l-h ( ) zv etzt (i9 -])"-* dt, zv J A J +• c eizt f - I)"~J dt. * It is supposed that y has not one of the vuIuoh \, %, J, ... ; for then the intugrands &vcanalytic at ±1, and both integrals vanish, by Canohy's theorem. w. u. if. 11
162 THEORY OF BESSEL FUNCTIONS [CHAP. VI It is to be observed that, when R (z) > 0, both integrals are convergent, and differentiations under the integral sign are permissible. Also, both integrals are analytic functions of v for all values of v. In order to express the first integral in terms of Bessel functions, we expand the integrand in powers of z, the resulting series being uniformly convergent with respect to t on the contour. It follows that ro=O '"¦: Now tm (t2 — II""* is an even or an odd function of t according as m is even or odd; and so, taking the contour to be symmetrical with respect to the origin, we see that the alternate terms of the series on the right vanish, and we are then left with the equation zv I ' 4gtCP — l)w-*dt*=2 ? , Z 1 &**(&— ly^dt Ja • «-o Bm)! Jo = 2 m-o Bm)! Jo on writing t = \/u; in the last integral the phases of u and u — 1 vanish when u is on the real axis on the right of u = 1. To evaluate the integrals on the right, we assume temporarily that R(v + ^)>0; the contour may then be deformed into the straight line from 0 to 1 taken twice; on the first part, going from 0 to 1, we have u—l-=(l~u)e~tri, and on the second part, returning from 1 to 0, we have u — 1 = A — u) e+ni, where, in each case, the phase of 1 — u is zero. We thus get Now both sides of the equation -l(» - I)-* du = Kcos„ o are analytic functions of v for all values of v\ and so, by the general theory of analytic continuation*, this result, which has been proved when i2 {v + •?) > 0, persists for all values of v. * Modern Analysis, § 5-5. Tue reader will also find it possible to obtain the result, when R (" + ?) <0, by repeatedly using the recurrence formula Jo U ' v+n+b Jo U which is obtained by integrating the formula the integral is then expressed in terms of an integral of the same type in which the exponent of m- 1 has a positive real part.
6-1] INTEGRAL REPRESENTATIONS 163 Hence, for all * values of v, = 2" +l I T (i) r (V + |) COS V7T . Jv (z). Therefore, if v + $ is not a positive integer, A) and this is Hankel's generalisation of Poisaon's integral. Next let us consider the second type of contour. Take the contour to lie wholly outside the circle \t\ = 1, and then (t-— I)"-* is expansible in a series of descending powers of t, uniformly convergent on the contour; thus we have v ; m=o rn ! 1 (| - i') and in the series the phase of t lies between — %tt and +• 4-tt. Assuming! the permissibility of integrating term-by-tcrm. we have But r<o-B <*> (ixj) la where a is the phase of z (between ± A77-); and, by a well-known formula*, the last integral equals - 2tti/V Bhi — 2v -f 1). Hence ,'(-! + , 1+) <* when we use the duplication fonnula§ to express V Bm — 2y -H i) in terms of r(h ~ v + m) and V (—/'+- in + 1). * If v - \ is a negative integer, the BnnpleKl wuy of evaluating the intcynil in to ouloulute tho residue of the integrand at m = 1. t To justify the tcrm-by-term integration, obuerve that 1 ' | eizt dt \ ibcouvur^cul; lot J 00 i ita value be A". Siuce the expansion of (t*-l)"~l converges uniformly, it follows that, when wo are given a positive number e, we cau find an integer Mo independent of t, Huoh that tho remainder after M terms of the expansion does not exceed efK in absolute valuo when M 5: Jl/0. Wo then have at once ooi aud the required result follows from the definition of the Bum of an infinite serios. % Cf. Modern Analysis, § 12-22. § Cf. Modern Analysis, § 12-1». 11—2
164 THEOBY OF BESSEL FUNCTIONS Thus, when B (z) > 0 and v + \ is not a positive integer, [CHAP. VI This equation was also obtained by Hankel. Next consider J ooiexp(-t«>) where &>is an acute angle, positive or negative. This integral defines a function of z which is analytic when — \tr + ax arg z < \tt + <o) and, if z is subject to the further condition that | arg#| < ^ir, the contour can be deformed into the second of the two contours just considered. Hence the analytic continuation of j-v{z) can be defined by the new integral over an extended range of values of arg z\ so that we have P/l_ ,A pvvi A ~ C) e*{r-iy-*dt, V i 1 {%) J ooiexp(-ia.) , where arg z has any value between — ^ir + a> and \ir + oo. By giving « a suitable value*, we can obtain a representation of «/_„ (z) for any assigned value of arg z between — it and tt. When R (z)>0 and R (v + ?) >0 we may take the contour to be that ahewn in Fig. 3, \/ /\ in which it is supposed that the radii of the circles are ultimately made indefinitely small. By taking each straight line in the contour separately, we get i) A -py-l dt - t2y-l dt + 1 -fiy-idt . * If | w | be increased in a series of stages to an appropriate value (greater than sentation of J-v (z) valid for any preassigned value of arg z may be obtained. a repre-
repre6-11] INTEGRAL REPRESENTATIONS 165 On bisecting the third path of integration and replacing t in the various integrals by it, —/, ±t, t, it respectively, we obtain a formula for J_v(z), due to Gubler*, which corre- corresponds to Poisson's integral for Jv(~); the formula is D) J-^s^Tjv2+lfr(ii) [>Sinr („ e~J' and, if this be combined with I'oisaon's integral, it is found that E) Yv (z) = r (^? ^) [ J' sin (zt). A -««)*" 4 dt - a formula which was also discovered by Gubler, though it had been previously stated by Weber t in the case of integral values of v. After what has gone before the reader should have no difficulty in obtaining a formula closely connected with A), namely F) Jv{z) = l in which it is supposed that the phase of t'2 — 1 vanishes when t is on the real axis on the right of t — \. 6*11. Modifications of Ilankel's contour integrals. Taking R(z) > 0, let us modify the two contours of §6 into the contours shewn in Figs. 4 and 5 respectively. Fig. 4. Fit?. <r). By making those portions of the contours which are parallel to the real * Zurich Vicrtcljnhraschrift, xxxur. A88H), p. 155). Hue also Graf, ZciUchriJl fiir Hath, und Phys. xxxviii. A803), p. 115. f Journal filr Math, lxxvi. A873), p. 9. Cf. Hayashi, Nyt TUlsskrift for Malh. xxnr. n, (li)J^), pp. 8G—(J0. The formula was examined in tho cuso ^ = 0 by Esohorioh, Moiiutshe/tc. filr Math, und Phys. m. A892), pp. 142, 231.
166 THEOBY OF BESSEL FUNCTIONS [CHAP. VI axis move off to infinity (so that the integrals along them tend to zero), we obtain the two following formulae: eizt (p _ ] y-j dt + eizt (t2 - 1)"-J dt 1+ooi ' J -1+ooi J U(l+) f(-l+) _ efe« ^_ iy-h dt l+ooi .'-1 + ooi In the first result the many-valued functions are to be interpreted by taking the phase of t2— 1 to be 0 at J. and to be + v at B, while in the second the phase of t- — 1 is 0 at A and is —tt at B. To avoid confusion it is desirable to have the phase of t" - 1 interpreted in the same way in both formulae; and when it is supposed that the phase of ir — 1 is + 7T at B, the formula A) is of course unaltered, while B) is replaced by /qx T (,\ r <*-">•(*«)' r r L J J-1+ooi In the last of these integrals, the direction of the contour has been reversed and the alteration in the convention determining the phase of i2 —1 has necessitated the insertion of the factor e-2(K-sOrt On comparing equations A) and C) with § 3'61 equations A) and B), we see that unless v is an integer, in which case equations A) and C) are not independent. We can, however, obtain D) and E) in the case when v has an integral value (n), from a consideration of the fact that all the functions involved are continuous functions of v near v = n. Thus ¦#«A) (*) - Km HJu (z) and similarly for Hn{2) (z).
6'12] INTEGRAL REPRESENTATIONS 167 As in the corresponding analysis of § 61, the ranges of validity of D) and E) may he extended by swinging round the contours and using the theory of analytic continuation. Thus, if — -|7r < co < f 7r, we have F) #„<» (,) while, if — f 7r < « < \tt, we have G) HM.T3JZIU f provided that, in both F) and G), the phase of z lies between — \ir + <o and \rr + o). Representations are thus obtained of //„"> (z) when arg z has any value between — it and 2tt, and of Hv® (z) when arg s has any value between — 2-7T and rr. If co be increased beyond the limits .stated, it i.s necessary to make the contours coil round the singular points of the integrand, and numerical errors are liable to occur in the interpretation oi' the integrals unless great eare ia taken. Weber, however, him adopted this procedure, Math. Ann. xxxvn. A890), pp. 'Ill—412, to determine the for- formulae of § 3-62 connecting IIv(l)(-z), IIvV)(-z) with //„<>>(s), HvW(z). Notk. The formula l2iYv(z) = HvM{z) — IIvV)(z) makes it possible to oxpress Yv(z)'m terms of loop integrals, and in this manner Ilankel obtained the scrioH of § 3\r>2 for Yn (z); this investigation will not be reproduced in view of the greater simplicity of Hankel'a other method which has been described in fcj 3-52. 6#12. Integral representations of functions of the third kind. In the formula §6*1"J ((i) suppose that the phase of z has any given value between — it and 2ir, and define /3 by the equation arg z = <y + ft, so that — \ir < ft < \ir. Then we shall write t- L = e-J^s-i (- u), so that the phase of — n increases from — it + ?? to vr + ft as t describes the contour; and it follows immediately that A) //,<- 00 = 1LLLJ^2 • ( r« (_ tt)-» (i +1) dn, where the phase of 1 + \iujz has its principal value. Again, if ft be a given acute angle (positive or negative), this formula affords a representation of Hv{1} (z) valid over the sector of the .z-plane in which - \ir + ft < arg z<\tca- ft.
168 THEORY OF BBSSBL FUNCTIONS [CHAP. VI Similarly*, from §611 G), B) ? ^ ^ where ^9 is any acute angle (positive or negative) and - 17r + ft < arg z < \tt + ?. Sincef, by § 3-61 G), #_„« (s) = e"« jffrW (*), it follows that we lose nothing by restricting v so that R (v + \) > 0; and it is then permissible to deform the contours into the line joining the origin to oo expi/3, taken twice; for the integrals taken round a small circle (with centre at the origin) tend to zero with the radius of the circle |. On deforming the contour of A) in the specified manner, we find that C) #<»(,) where /? may be any acute angle (positive or negative) and R(v + \)>0, - \tt + 13 < arg z < |tt + J3. In like manner, from B), TTZj V(v + $) Jo V 1Z) where ft may be any acute angle (positive or negative) and R (v + |) > 0, - f 7T + j3 < arg z < \tr + /3. The results C) and D) have not yet been proved when 2v is an odd positive integer. But in view of the continuity near v=n + % of the functions involved (where » = 0, 1,2, ...) it follows, as in the somewhat similar work of § 6*11, that C) and D) are true when v = \, f, |, .... The results may also be obtained for such values of v by expanding the integrands in terminating series of descending powers of z, atid integrating terni'by-term; the formulae so obtained are easily reconciled with the equations of § 3'4. The general formulae C) and D) are of fundamental importance in the discussion of asymptotic expansions of J±v{z) for large values of \z\. These applications of the formulae will be dealt with in Chapter vii. A useful modification of the formulae is due to Schafheitlin§. If we take arg z = /3 (so that arg zt is restricted to be an acute angle), and then write u — iz cot 6, it follows that rin^tf * To obtain this formula, write f There seems to be no simple direct proof that is an even function of v. X Cf. Modern Analysis, § 12*22. § Journal fur Math. cxrv. A894), pp. 31—44.
6'13] INTEGRAL REPBESENTATIONS 169 and hence that •sin (*r ** These formulae, which are of course valid only when iJ (v + |) > 0, were applied by Schafheitlin to obtain properties of the zeros of Bessel functions (§§ lS't^i—15*35). They were obtained by him from the consideration that the expressions on the right are solutions of Bessel's equation which behave in the appropriate manner near the origin. The integral I e~uz «""& A + ?«)M~* dti, which is reducible to integrals of the types J o occurring in C) and D) when /j. — v, htx» been studied in soiho detail by Nielsen, Math. Ann. lix. A904), pp. 89—102. The integrals of thia section are alao discussed from the aspect of the theory of asymp- asymptotic solutions of differential equations by limjtzew, Warschau Polyt. Inst. Nach. 1902, nos. 1, 2 [Jahrbuch ilber die Fortschritte der Math. 1903, pp. 575—077]. 6" 13. The geneudised Mehler-flonine integrals. Some elegant definite integrals maybe obtained to represent Bessel functions of a positive variable of a suitably restricted order. To construct them, observe that, when z is positive (= w) and the real part of v is less than -|, it is per- permissible to take co = \tr in § (M] ((>) and to take w = — \tt in § 611G), so that the contours aro those shown in Fig. 0. When, in addition, the real part of v is greater than — \, it is permissible to deform the contours (after the manner of § 612) so that the first contour consists of the real axis from + ] to + oo taken twice, while the second contour consists of the real axis from — 1 to - oo taken twice. 1'ig. 6. We thus obtain the formulae ^ 9 A - e*) fV' (t* - 1 )»-* dt, P (a) = - rA~.-~l;-^- (I - e2'"-^) ffl-** (t* -1)""* dt, 7T-41 (^) ,'l the second being derived from § 6*11 G) by replacing t by -1.
170 THEORY OF BESSEL FUNCTIONS [CHAP. VI In these formulae replace v by — v and use the transformation formulae given by § 3*61 G). It follows that, when x > 0 and - \ < R (v) < |, then A) H" (*) -^j—j^jj-^j j i — ^, so that - v) r Of these results, C) was given by Mehler, J/aitA.. Ann. v. A872), p. 142, in the special case v-=0, while Sonine, Math. Ann. xvi. A880), p. 39, gave both C) and D) in the same special case. Other generalisations of the Mehler-Sonine integrals will be given in §• 6*21. 6'14. Symbolic formulae due to Hargreave and Macdoncdd. When R (z) > 0 and R (v + §) > 0, it ia evident from formula § 6'11 F) that tr(i)/^— \iz)" j fii*tC\—ti'V-idt r(v+t)T(t) J l+cci where the phase of 1 — ft lies between 0 and — \n. If D denotes (dfdz) and/is any polynomial,-then and so, when v+^ is a positive integer, we have JM When i/ + ^ is not a positive integer, the last expression may be regarded as a symbolic representation of HJM B), on the understanding that / (D) (e±«/s) is to be interpreted as Consequently M and similarly so that C) D\ Y (z) {W (I l D^-h ~
6-14,6-15] INTEGRAL REPRESENTATIONS 171 The series obtained from D) by expanding in ascending powers of D does not converge unless it terminates; the series obtained in a similar manner from C) converges only when R(v)>\. The expressions on the right of C) and D), with constant factors omitted, were given by Hargrcave, Phil. Trans, of the Royal Soc. 1848, p. 36 as solutions of Bessel's equation. The exact formulae are duo to Macdonald, Proc. London Math. Soc. xxix. A898), p. 114. An associated formula, valid for all values of v, is derivable from § 6*11 D). If n is any positive integer, we see from tho equation in question that J i+*« 71111 ('*) so that A similar equation hold.s for the other function of the third kind, and so This result, proved when R(z)>i), is easily extended to all values of z by the theory of analytic continuation; it was discovered by Sonino, Math. Ann. xvi. A880), p. 66, when v = ?i, and used by Stein thai, Quarterly Journal, xvm. (L882), p. 338 when v = n+h; in the case when v = n + i the result was givon slightly earlier (without the use of the notation of Beasel functions) by Glaisher, Proc. Cavih. Phil. Soc. m. A880), pp. 269—271. A proof basod on arguments of a physical character has boon given by llavclock, Proc. London Math. Soe. B) n. A904), pp. 124—125. 6*15. Sclddjii's* integrals of Poissoti's type for Iv{z) and Kv{z). If we take co = \ir in § 6*1 C) and then replace z by iz, wo find that, when ! arg * | < \ir, r (% ) i (l and the phase of t2 — 1 at the point where t crosses the negative real axis is — 2tt. Fig. 7. If we take R(v +^)>0 to secure convergence, the path of integration may be taken to be the contour of Fig. 7, in which the radii of the circles may be made to tend to zero. We thus find the forraulaf J_p (z) = liizi^l^-. [A _ <r—0 ( V< (f - 1)-* dt + t (e-"^ + e~avni) [ e-zt A - i3)""* dt , * Ann. di Mat. B) i. A8E8), pp. 239—241. Schliifli obtained the results A) and B) direofcly by the method of § 6-1. t Cf. Serret, Journal de Math. ix. \1844), p. 204.
172 THEORY OF BESSEL FUNCTIONS [CHAP. VI in which the phases of t2 — 1 and of 1 — t2 are both zero. Now, from § 3*71 @), we have B) /»= T^Twii) r.r(l - ^*¦ and so C) /_ (.) -1. (,) - ?&=&??" W />- (' - 1^ * that is to say* D) Kv 0) = ~$^j^ jV« (*> - 1)-* <ft, whence we obtain the formula E) 7C (*) = rr a result set by Hobson as a problem in the Mathematical Tripos, 1898. The formulae are all valid when R(v + J)>0 and ;arg.2J< \tr. The reader will find it instructive to obtain D) directly from § 611 F). 6*16. Basset's integral for Kv (xz). When x is positive and z is a complex number subject to the condition | arg z j < \tt, the integral for H{^v{xze^1) derived from § 611 F) may be written in the form Now, when R(v)~^ — \, the integral, taken round arcs of a circle from p to p giitri-iargz^ ^en(js ^0 Zero as p -*» oo , by Jordan's lemma. Hence, by Cauehy'a theorem, the path of integration may be opened out until it becomes the line on which R (zt) — 0. If then we write zt = iu> the phase of — (uPfz2) — 1 is — tt at the origin in the u-plane. It then follows from § 3'7 (8) that TV / \ 1 * — JL utr» TT (I.1 / A-rvV \ e~xzt dt ar(t) _r(-v+i-).B^ and so we have Basset's formula valid when R (v + |) > 0, « > 0, | arg z\<\ir The formula was obtained by Bassetf, for integral values of v only, by regarding K0(x) as the limit of * The integral on the right was examined in the case f=0 by Riemann, Ann. der Phy-uk und Chemie, C) xcv. A855), pp. 130—139. t Proc. Camb. Phil. Soc. vi. A889), p. 11; Hydrodynamics, n. (Cambridge, 1888), p. 19.
6'16, 6*17] INTEGRAL REPRESENTATIONS 173 a Legendre function of the second kind and expressing it by the corresponding limit of the integral of Laplace's type (Modern Analysis, § 15'33). The formula for Kn (xz) is obtainable by repeated applications of the operator —j-. zctz Basset also investigated a similar formula for /„ (xz), but there is an error in his result. The integral on the right in A) was studied by numerous mathematicians before Basset. Among these investigators were Poisson (ace § 6'32), Journal de I'ltcole Folgtechniqne, IX. A813), pp. 239—241; Catalan; Journal de Math. v. A840), pp. 110—114 (reprinted with some corrections, Mem. de la Soc. R. des Sci. de Liege, B) xn. A885), pp. 26—31); and Serret, Journal 'de Math. vni. A843), pp. 20, 21; ix. A844), pp. 193—210; Schlomilch, Analytischen Stvdien, n. (Leipzig, 1848), pp. 96—97. These winters evaluated the integral in finite terms when v + A is a positive integer. Other writers who must be mentioned are Malra.stdn, A'. Svcnstcn, V. Akud. Handl. lxii. A841), pp. 6f> —74 (see $ 7'23); Svanbcrg, Nooa Ada Reg. Soc. Sui. Upsala, x. A832), p. 232 ; Leslie Ellis, Trans. Ca/u.b. Phil. Sue, vni. A849), pp. 213-210; Euuepor, Math. Ann. VI. A873), pp. 360—30;") ; Glaisher, Phil. Trans, of the Royal Soc. clxxii. A881), pp. 792— 815; J. J. Thomson, Quarterly Journal, xvur. A882), pp. 377—381; Coate.s, Quarterly Journal, xx. A885), pp. 2f>0—2(iO; and Oltmmare, Comptcs Jtendus de VAxsoo. Fmnpaise, xxiv. A895), part u. pp. 167—171. The laat named writer proved by contour integration that j cos .nt .dii _ (-)"-' 7T Jk* + z*? " 2^' .'{n- I)! •"-'(l+zO". The. former of these results may be obtained by differentiating the equation .da Tre~'rs^v and the latter is tlion obtainable by using Lagrang«'s expansion. 6*17. Wlnttdker'fi* yenerulisations of llavkel's integrals. Formulae of the type contained in § 32 suggest that solutions of Bessel's equation should be constructed in the form zi ( V' Tdt. It may be shewn by the methods of § (y I that and so the integral is a solution if Tin a solution of Lcgondre's equation for functions of order v— \ and the values of the integrated part are the same at each end of the contour. * Proc. London Math. Soc. xxxv. A903), pp. 198—20G.
174 THEORY OF BESSEL FUNCTIONS [CHAP. VI If T be taken to be the Legendre function Qp_j (t), the contour may start and end at +ooiexp(— ia>), where co is an acute angle (positive or negative) provided that z satisfies the inequalities — %ir + co <argz<\tc + co. If T be taken to be Pv_j(<), the same contour is possible; but the logarithmic singularity of Pv~i (t) at t~ — 1 (when v~\ is not an integer) makes it impossible to take the line joining — 1 to 1 as a contour except in the special case considered in §8*32; for a detailed discussion of the integral in the general case, see § 10'5. We now proceed to take various contours in detail First consider /¦( J o where the phase of t is zero at the point on the right of t = 1 at which the contour crosses the real axis. Take the contour to lie wholly outside the circle 1*1 = 1 and expand Q „_$(?) in descending powers of t. It is thus found, as in the similar analysis of § 6*1, that (hz)* g40W (+,+) 17 L (?) ' octexp(-tui) and therefore B) /_, (z) = ^^tk- ** Q-,-i @ dt If we combine these formulae and use the relation* connecting the two kinds of Legendre functions, we find that C) Hy'2> («) = ^4. Again, consider , ra+) z*\ e^Qy^(t)dt; J ooiexp(—fio) this is a solution of Bessel's equation, and, if the contour be taken to lie on the right of the line R(t) — a, it is clear that the integral is 0 [z* exp(— a\z\)} as z—~ + <x>i. Hence the integral is a multiple of #„<•) (z). Similarly by making z -*~ — oo i, we find that J o ooiexpt— iu) * Tibe relation, discovered by Schlafli, iB cf. Hobson, Phil. Trans, of the Royal Soc. cixxxyu. A896), p. 461.
6-2] INTEGRAL REPRESENTATIONS 175 is a multiple of Hv('l) (z). From a consideration of A) it is then clear that Bs^<?--iO'+iW* "A+) D) i cxp(~iu>) lri f(-1+> <t* Q._ * it) dt, ¦ *i cx)() Bz E) 2T,« (*) - ^ and hence, by§ 3'61 combined with Schliifli's relation, w w wl w wl ($) cos y i«!»(/«) this is also obvious from C). The integral which differs from (G) only by encircling the point 4-1 instead of — 1 is zero since the integrand is analytic iiuside such a contour. In E) and (G), arg (t + 1) vanishes where the contour crosses the real axis on the right of" — 1, and, in E), arg {t — L) is — tt at that point. 6*2. Generalisations of Bessel's integral. We shall next examine various representations of Bessol functions by a system of definite integrals ami contour integrals due to Sonine* and Schlaflif. The fundamental formula which will be obtained is easily reduced to Bessel's integral in the case of functions whose order i« an integer. We take Hankel's well-known generalisation} of the second Eulerian integral 1 i r (o-i-) — _.__ I f-v-m-i A ,]f (v+m + l) l2,7nJ-M in which the phase of t increases from — nv to tt as i describes the contour, and then Consider the function obtained by interchanging the signs of summation and integration on the right; it is •(on , g,\ .-4* This is an analytic function of z for all values of z, and, when expanded in ascending powers oi' z by Maclaurin's theorem, the coellicients may be obtained by differentiating with regard to z under the integral sign and making z zero after the differentiations^. Hence r'-'expu- .¦[ dt— y, —¦ •- * Mathematical Collection, v. (Moscow, 1870) ; Math. Ann. xvi. A880), pp. 0—29. t Ann. di Mat. B) v. A873), p. 204. His memoir, Math. Ann. in. AH71), pp. 134—149, should also be consulted. In addition, see Graf, Math. Ann. lvi. A003), pp. -123—432, and Chexain, Johnx Hopkins University Circulars, xiv. A895), pp. 20—21. t Cf. Modern Analysis, § 12-22. § Of. Modern Analysis, §§ 5-32, 4-44.
176 THEOEY OP BESSEL FUNCTIONS [CHAP. VI .and so we have at once This result, which was discovered by Schlafli, was rediscovered by Sonine; and the latter writer was the first to point out its importance. When | arg z\<\ir, we may swing round the contour about the origin until it passes to infinity in a direction making an angle arg z with the negative real axis. On writing t — \zu,y we then find that, when | arg z j < \nr, This form was given in Sonine's earlier paper (p. 335). Again, writing u = ew, we have i roo+7rt C) . /„(*) = «—. eZBiabw~vw dw, Z7TI J oo—wi valid when | arg 2 | <\ic. This is the first of the results obtained by Schlarli. In this formula take the contour to consist of three sides of a rectangle, as in Fig. 8, with vertices at oo — iri, — iri, tri and x + iri. ni \ -171 Pig. 8. If we write t + iri for w on the sides parallel to the real axis and ± id for w on the lines joining 0 to + in, we get Schlafli's generalisation of BesseVs integral 1 f IT ' /* OO D) /„($) = — cos{yQ - zsin &)dd — e~vt~zsinht dt, 7T J 0 IT Jo valid when j arg z \ < \ir. If we make arg z -*- + \tt, the first integral on the right is continuous and, if R (v) > 0, so also is the second, and /„ (z) is known to be continuous. So D) is still true when z is a pure imaginary if R{y) is positive. The integrals just discussed were examined methodically by Sonine in his second memoir; in that memoir he obtained numerous definite integrals by appropriate modifications of the contour. For example, if \|r be an acute angle (positive or negative) and if
6'21] INTEGRAL REPRESENTATIONS 177 the contour in C) may be replaced by one which goes from oo — (ir — ty) % to oo + (tt + y!jf)i. By baking the contour to be three sides of a rectangle with corners at oo — (rr — -v/>) i, — (tt — yfr) i, (ir + yfr) i, and oo + (nv + yfr) i, we obtain, as a modification of D), E) /, (z) = e—^ r e"81"*008"cos (v$-zcosyjr sin d) dd 7T J o ^ Jo Again, if we take ^ to be an angle between 0 and ir, the contour in C) may be replaced by one which passes from oo — (|-7r -f yfr) i to oo + (|tt 4- ty) i, and so we find that 1 fJf F) /, (z) = - cos (j/0 - ^ sin ^ C0S ^ - 1 w - Vty) dt, _ j "VJu provided that | arg z\ is less than both -^ and nr — yfr. When R(i') > 0 and z is positive (=#)> we niay take\^ = O in the last formula, and get* G) Jv(x)= - cos{yS — a;sin ^)dd + tf'^Hin (xuoaht — \vrr)dt. Another important formula, derived from A), is obtained by spreading out the contour until it is parallel to the imaginary axis on the right of the origin; by Jordan's lemma this is permissible if lt(v) > — 1, and we then obtain the formula in which c may have any positive value; this integral is the basis of many of Sonine's investigations. Integrals which msnmblc those given in this section nro of importance in tho investiga- investigation of the diffraction of light by a prism; hoo (Jarnlnw, Proc. London Math. Soc. xxx. A899), pp. 121--] ill ¦ W. 11. Jackson, Proc. London Math. Soc B) I. A904), pp. 393—414; Whipple, Proc. London Math. Soc. B) xvr. A917), pp. 94—111. 6*21. Integrals which represent functions of the second and third kinds. If we substitute Schliirli's integral § 62 D) for both of the Bessel functions on the right of the equation Yv (z) = Jv (z) cot vtr — »/_„ (z) cusec vn, we find that rrr Trr 7r Yv (z) — cot vnv \ cos(vd — z&\T\6)d6-coseci>'Tr\ cos(r0 +sain 6)dd h) Jo - cos vir f*e~vl-zs[nh'¦dt-l evt-zahlhldt Jo. i» * Of. Gubler, Math. Ann. xlix. A897), pp. 583—584. W. B. V. 12
178 THEORY OF BESSEL FUNCTIONS [CHAP. VI Eeplace 6 by tt - 6 in the second integral on the right, and it is found on re- reduction that A) Yv (z) = - ("sin (z sin 0 - v6) d9-~ [*\evt + e~vt cos v-rr) e~zainht dt, a formula, practically discovered by Schlafli (who actually gave the correspond- corresponding formula for Neumann's function), which is valid when | argz\< ^tt. By means of this result we can evaluate irij _oo when | arg z \ < -J-tt ; for we take the contour to be rectilinear, as in Fig. 9, and ni X o Fig. 9. write — t, id, t + iri for w on the three parts of the contour; we then see that the expression is equal to 1 f« ] fit ±_l 6vt-zsmht(lt + _ ei TTtJo TTJO O and this is equal to Jv (z) + iF,(z) from formula A) combined with §6D). Hence, when | argz\<^tt, we have B) H,® (z) = —. •TTl _«, 1 rao-m" C) H® (z) = - —.A gisinh«-,« ^w> T^J -oo Formulae equivalent to these were discovered by Sommerfeld, Math. Ann. xlvii. A896), pp. 327—357. The only difference between these formulae and Sommerfeld'a is a rotation of the contours through a right angle, with a corresponding change.in the parametric variable; see also Hopf and Sommerfeld, Archiv der Math, und Phys. C) xvm. A911), pp. 1-16. By an obvious change of variable we may write B) and C) in the forms D) H,W (z) = —. tt-»-i exphe (u--)\ du, 1 f«> exp(—iri) ( / 1 \ ^ E) Hv& (z) - - -U «--i exp hi (u -~)\ du;
6*21] INTEGRAL REPRESENTATIONS ' 179 the contours are those shewn in Fig. 10, emerging from the origin and then bending round to the left and right respectively; results equivalent to these were discovered by Schlafli. Fig. 10. [Note. There is no difficulty in proving these results for integral values of v, in view of the continuity of the functions involved ; cf. § Gil.] We proceed to modify the contours involved in D) and E) to obtain the analytic continuations of the functions on the left. If co is an angle between — ir and tt such that | w — arg21 < \tt, we have F) and 1 roooxp V J TTiJoexp/o ' exp hz[u--)\ du, oxp the contours being those shewn in Fig. 11 and Fig. 12; and these formulae give the analytic continuations of the functions on the left over the range of Fig. 11. Fig. 12. values of z for which to — ^nr < arg#< «o 4- \tt\ and co may have any value between * - nv and ir. '' If | w I were iucreased beyond these limita, diflSculties would arise in the interpretation of the phase of u. 12—2
180 ' THEOBY OF BESSEL FUNCTIONS [CHAP. VI Modifications of B) and C) are obtained by replacing w by w ± \tri; it is thus found that* g—J VTtl I" CC • (8) Hv^ (Z) = r- eiz cosh w (josh vW,dw, 0 (9) W W—S-j. provided that |argz\<\ir. Formulae of special interest arise by taking z positive (=%) in F) and G) and -1 < E {v) < 1. A double application of Jordan's lemma (to circles of large and small radius respectively) shews that, in such circumstances, we may take co= \ir in F) and w = — \ir in G). It is thus clear, if u be replaced by + iel, that ^ 7T» J_oo Tti JO A1) iLW (a;) = - ^-^ ( e~i^oaht-vt fa _ f_ I g-tecosht cosh ^ . dt, and hence, when x > 0 and — 1 < E (i/) < 1, we have 2 /*°° A2) ,/„ (x) = - sin (a; cosh t — •jU7r) •cosn v^> • dt, A3) Fv (x) = I cos (# cosh i — -J ptt) . cosh vt.dt\ TT J 0 and, in particular (cf. §63), when we replace cosh t by i. The last two formulae are due to Mehler, Math. Ann. v. A872), p. 142, and Sonine, Math. Ann. xvi. A880), p. 39, respectively; and they have also been discussed by Basset, Proc. Camb. Phil. Soc. vni. A895), pp. 122—128. A slightly different form of A4) has been given by Hardy, Quarterly Journal, xxxn. A901), pp. 369—384; if in A4) we write a? = 2 J(ab), xt = au+blu, we find that A6) f" sin («« + -) ^ = Wo {2 J{ab)}. Note. The reader will find it instructive to obtain A4) from the formula n . .. 2 fw sin combined with the formula § 5*71 A). This was Mehler's original method. * Cf. Coates, Quarterly Journal, xxi. A886), pp. 183—192.
6-22] INTEGRAL REPBESENTATIONS 181 6*22. Integrals representing Iv{z) and Kv(z). The modifications of the previous analysis which are involved in the dis- discussion of Iv (z) and Kv (z) arc of sufficient interest to be given fully; they are due to Schlafli*, though he expressed his results mainly in terras of the function F (a, t) of §415. The analysis of § 6 is easily modified ho as fco prove that and hence, when | argz | < \nr, C) IA z) = --}-. f" "e*11"' The formulae B) and C) are valid when arg z = ± |7r if R (v) > 0. If in C) the contour is taken to be three; sides of a rectangle with corners at oo — iri, — 7ti, 7ri, oo + 7ri, it is found that D) so that and hence, E) j when \i\rgz\ ezlimOcoHi'0d6 IT |<.\7T, e~ZC0Hlw cosh vt. dt, Jo ^conhvt.dt, ( Jo a formula obtained by Schlaflif by means of somewhat elaborate transforma- transformations. From the. results just obtained, we can evaluate when i arg^ \< lir. For it is easily seen that ZTTl J - oo * /(nn. di Mat. B) v. A873), pp. 190—205. f Ann. di Mat. B) v. A873), pp. 109—201; tins formula was lined by Heine, Journal fur Math, i.xix. A868), p. 131, aa the definition to whioh reference waa made in § 5'72.
182 THEORY OF BESSEL FUNCTIONS [CHAP. VI and hence 2tsini/ir Again, we may write E) in the form G) Kv{z) = and hence, by the processes used in § 6'21, (8) Kv (z)=t vr«-> exp - \z (u + -) du, when — 7r < «< 7r and — \tt + a>< arg ^ < \ir + <o. Similarly (9) e™ /_, (*) - e-"« Jv (^) = =^' m-"-1 exp U^ (i* + -) \du; 7T JOexp(-w+o,)i (V W/J this is valid when 0 < w< 27r and — \ir + &>< arg ^ < |7r + w. The contours for the formulae'(8) and (9) are shewn in Figs. 13 and 14 respectively. Pig. 13. Fig. 14. Further, when z is positive (= x) and — 1 < R (v) < 1, the path of integra- integration in (8) may be swung round until it becomes the positive half of the imaginary axis; it is thus found that ri | v-"-1 exp {- \ix [v - -) \ dv, Jo (. \ v/) Kv (a) = |e-} so that A0) Kv(x) = %e-*v»ir e-ixa™ht~vt dt, J and, on changing the sign of v, A1) Kv{x) =
6*23] INTEGRAL REPRESENTATIONS 183 From these results we see that A2) 2coa$vir.Kv(a)) = l e-ixainht cosh vt.dt, so that 1 f00 A3) Kv (x) = r— / cos (x sinh t) cosh vt. dt, 7 cosfvirJo and these formulae are all valid when x > 0 and — 1 < R(v) < 1. In particular A4) Ko (x) = cos (.-«sinh 0 d« = -775-jriV> .'o Jo v(" + J) a result obtained by Mehler* in 1870. It may be observed that if, in G), we make tho substitution hzel = T, we find that A5) *„(*) = $ (i«)^exp {-r— } ~ provided that /i(s'-)>0. The integral on the right has been studied by numerous mathe- mathematicians, among whom may be mentioned Poianon, Journal de VEcole Polytechniqne, ix. (cahier 16), 1813, \\ 237; Ulaisher, British Association Report, 1872, pn. 15—17; Proc. Camb. Phil. Sue. in. AH80), pp. 5—12; and Kaptoyn, Bull, des Sci. Math. B) xvr. A892), pp. 41—44. Tho integrals in which v has the special values ? and \ wore discussed by Euler, Inst. Gala. Int. iv. (Petersburg, 1794), p. 415; and, when v is half of an odd integer, the integral has been evaluated by Legcndro, Exercicxs de Catcul Integral, I. (Paris, 1811), p. 366; Cauchy, Rvercires des Math. (Paris, 1826), pp. 54—56; and Schlomilch, Journal fllr Math. xxxm. A84fi), pp. 2C8—280. Tho integral in which tho limits of inte- integration art) arbitrary has boon examined by Binct, Co tuples Ilendun, xn. A841), pp. 958— 962. 6*23. Hardy s formulae for integrals of Du Bois Reyinond's type. The integrals TOO ?.2 F JO rfO. sin t. sin — . t"~x dt, cos t. cos — . tv~l dt, Jo t Jo t in which x > 0, — 1 < R{v) < 1, have been examined by Hardyf as examples of Du Bois Reyinond's integrals I™ f(tfmt.t^dt, Jo cos in which f(t) oscillates rapidly as t -*- 0. By constructing a differential equation of the fourth order, Hardy succeeded in expressing them in terms of Besael functions; but a simpler way of evaluating them is to make use of the results of §§6'21, 6-22. * Math. Ann. xvnt. A881), p. 182. + Messenger, xl. A911), pp. 4=4—51.
184 THEORY Or BESSEL FUNCTIONS [CHAP. VI If we replace t by o?e{, it is clear that Too g,1 reo sin t sin —. tv~l dt = x" sin (xel) sin (aer*). evt at JO t J _oo fe2ia;cosh« _|_ e~Uxcos\it _ giixuinht _ g-2issinhtj Qvt fit — 00 ^[ Bx) - Trie*"* i/(_2)r Ba-) „ () and hence we have A) I™ sintsin^ .tv-'dt = -r-~~i- [ Jo < 4sm|i/7rL and similarly B) [" v.os«cos j. t"-> dt - ^[V7r W-u B«) - ^ Ba?) + /_ Ba;) - /, Ba;)]. When v has the special value zero, these formulae become C) rBin«sin?.^ = i Jo t t D) f °° cos t cos ^. ^ = - \nr Yo Ba?) + J5f0 Jo it 6'24. Theisinger's .extension of BesseVs integral. A curious extension of Jacobi's formulae of § 2-2 has been obtained in the case of «/0 (x) and Jx(x) by Theisinger, Monatshefte fur Math, und Phys. xxiv. A913), pp. 337—341; wo Bhall now give a generalisation of Theisinger's formula which is valid for functions of order v where - \ < v < |. If a is any positive number*, it ia obvious from PoiBson's integral that 2 fi-'c"I' fi" J (x) = ——v»*/ , , <?-aEsin 8 cos (x cos 6) sin2" 6dd r(v+^)r(^) Jo v + r(p2,(!fwiv I ' (l-e-**««<>)cos(d Now 2 I " A— e~c Jo ¦ IT 1 _ g—OIBillfl o sinh (a? Bin 6) z-ljzy dz -2 •. ,"Vi .-/--,ti sinh (-ta;?). where the contour passes above the origin. Take the contour to be the real axis with an indentation at the origin, and write z— ±tan faf> on the two parts of the contour; we thuB find that the last expression is equal to /i^l-expC-a.wcotrfI) . . . ... ,_. ,„„ , d .-. - — . Y sin (x tan id>). eVWl cot2" d> -r o sin(a;cot^>) ^ ~^ ^si . f iv 1 - exp (axicot d>) . + i J r-~s J a si cos (i^ cot $-v1r)s sm = 4 f ** sin (i Jo * In Theisinger's analysis, a is au even integer.
6-24-6-31] INTEGRAL REPRESENTATIONS 185 and therefore A) lii^iilii) Jv (A.) „/"*"¦ c-«x«in0 COH (,v coa 0) Hjn* flrffl + 2 fi\sin(tocot0)co3(^o(,t</,-^)Sin(A>-tan^c«)t^0 ^- . Jo " ' sin (a?cot<]!>) ' Bin <j> The transformation fails when v^?, because the integral round the indentation does not tend to zero with the radius of the indentation. The form given by Theiainger in the case v = l differs from A) beoau.se he works with ij 3*3 G) which gives r (v—&) r CA'i Hir B) —XTTTr-i 'h 0») = I " e-ax»in• .sin (.r cos 8) sin2" 0 cos 8d6 . . P17 . ,, , .> ,, . , . sin2(^tan-i0) ,,„ ., , dd> + i I sin (mx cot (i) cos (ifw.1 cot d)-vir) - ---= r-fr^ cot2" c/> -r—J-v, Jo sin (#uot (/>) 'Hin2))!) provided that §<d<$. 6'3. TAe equivalence of the integral representations of Kv{z). Three different types of integrals which represent Kv{z) have now been obtained in §§6M5D), 0-22E) and 010A), namely _. I e-z™»htcosh vt.dt, Jo /r r (w 4-!}). {2z)v r* coh a,™. dtt "(K)" x'V(\) Jo («» + *')•'+*• The equality of the first and necond was directly demonstrated in 1871 by Schliirli*; but Poishou proved the equivalence of the second and third as early as 1815-1, while M alms ten gave a less direct proof of the equivalence of the second and third in IK41. We proceed to describe the three transformations in question. 61. tSchldflVs tranafonnufioH. Wo first give an abstract of the analysis used by Schliifli, Ann. di Mat. B) v. A873), pp. 19?)--201, to provo the relation 1\w)'(-,2?" t' «"s( (<2 - 1)"- * <it = [ " e~"™uo cosh v6d6 1 (" + 2) J ) Jn which arises from a comparison of two of the- integral rnpresontationw of Kv (z), and which may be established by analysis resembling that of $ 2'323. We have, of course, to suppose that li (z) >0 to secure convergence, ami it is convenient at first to tal<e+ - i < li (v) < 1. * An earlier proof is duo to Kmntnor, Journal fiir Math. xvn. A837), pp. 228—242, but ifc h much more elaborate tlian Schliiili's invusUgation. I The result is established for larger values of R (i>) either by the theory of analytio continua- continuation or by the use of rocurreuoo formulae.
186 THEORY OF BESSEL FUNCTIONS [CHAP. VI Now define S by the equation where x^l; and then> if t—x — (# -1) it, we have "I f w [r,— \S\ v~i «->(l-n)"-Hl 1 o /" Jo on expandingthe last factor of the integrand in powers of u and integrating term-by-term. Replacing x by cosh 6, we see that so that, by a partial integration, r(i) Jo vr(i) Jo z r(i-v)JO dudt -*' /""«-f«(*a- the inversion of the order of the integrations presents no great theoretical difficulty, and the transformation is established. 6'32. Poisson's transformation. The direct proof that is due to Poisson*, Journal de VEcole Polytechnique, ix. A813), pp. 239—241. The equation is true when \axgz\<\ir, x>0 and R(p)> — ?, but it is convenient to assume in the course of the proof that R (v) > \ and j arg z \ < {ir, and to derive the result for other values of z and v by an appeal to recurrence formulae and the theory of analytic continuation. If we replace t by a new variable defined by the equation v=<s"e~vi, we see that it is sufficient to prove that * See also Paoli, Mem. di Mat. e di Fit. della Soc. Italiana delle Sci. xx. A828), p. 172.
6-32,6-33] INTEGRAL REPRESENTATIONS 187 Now the expression on the left is equal to /o jo (v* + zy+\"dtdu==]0 j0 s"~i Q^V {-H^2 + z2)}cos.vu.dsdu = I I {exp (- sm2) cos #?t. du} .nv~\ exp (— sz2) ds, Jo Jo when we write t=s <>2+-22) and change the order of the integrations*. Now / exp (- m'2) cos xu. du=\T (i) s- i exp (- %x*/s), and so we have which establishes the result. [Note. It is evident that s^^ve-'/z^lvV/z. Tho only reason for modifying 1 fM A J —00 by taking »asa parametric variable in to obtain an integral which is ostensibly of the «anie form as the integral actually investigated by Poisson ; with his notation the integral is I exp (— ./.•" - a*.v~ ") dx.] Jo 6'33. Mulmslen's transformation. Tho method employed by Malmstcnt in proving that, when fi{z)>Q and R(v)> -i, then P"r(i")J, {u*+z*y + h- r(,+i) J, is not ko direct as the analysis of $$ 6*31, G-32, ina.sruuch an it involves an appeal to the theory of linear differential equations. It is first shewn by Malnistdn that the three expressions cos (xu) du f" _ tl 2 _ 2, „ _. . /¦x ?(, . _ 2 a functions of .r, arc annihilated { by the opemtor and that as %-*- +oo, the third is ^(e11) while the first and second are bounded, provided that R (v) > 0. It follows that the .second and third expressions form a fundamental system of solutions of the equation Cf. Bromwioh, Theory of Infinite Series, § 177. K. Srenska V. Akad. Handl. lxii. A841), pp. Co—74. The reader should have no difficulty in supplying a proof of this.
188 THEORY OP BESSEL FUNCTIONS [CHAP. VI and the first i8 consequently a linear combination of the secoud and third. In view of the unboundednesB of the third as #-*-+<»> it is obvious that the first must be a constant multiple of the second bo that f ° jo c _ where 0 is independent of x. To determine C, make x-*~0 and then so that aud the required transformation follows, when R(v)>0, if we use the duplication formula for the Gamma function. f '*" cos xu . du An immediate consequence of Malmsten's transformation is that / -.- a'—r.r.n J 0 \U -\-Z~) expressible in finite termB; for it is equal to IS method of evaluating the integral is simpler than a method given by Catalan, Journal de Math. v. A840), pp. 110—114; and his investigation is not rigorous in all its stages. The transformation is discussed by Serret, Journal de Math. vni. A843), pp. 20, 21; IX. A844), pp. 193—216; Bee also Cayley, Journal de Math. xn. A847), p. 23E {Collected Papers, I. A889), p. 313.] 6*4. Airy's integral. The integral r°° cos (t3 ± at) dt Jo which appeared in the researches of Airy* "On the Intensity of Light in the neighbourhood of a Caustic" is a member of a class of integrals which are expressible in terms of Bessel functions. The integral was tabulated by Airy by quadratures, but the process was excessively laborious. Later, De Morgan f obtained a series in ascending powers of a? by a process which needs justification either by Stokes' transformation (which will be explained immediately) or by the use of Hardy's theory of generalised integrals^ * Trans. Gamb. Phil. Soc. vi. A838), pp. 379—402. Airy used the form Cos \tr (m3 - mw) dw, f Jo 0 but this is easily reduced to the integral given above. t The result was communicated to Airy on March 11, 1848; Eee Trans. Comb. Phil. Soc. vm. A849), pp. 595—599. J Quarterly Journal, xxxv. A904), pp. 22—66; Trans. Camb. Phil. Soc. xxi. A912), pp. 1—48.
6'4j INTEGRAL REPRESENTATIONS 189 Stokes observed* that the integral satisfies the differential equation d"v . . and he also obtained the asymptotic expansions of the integral for large values of x, both positive and negative. As was observed by Stokes (loc. cit. p. 187), thin differential equation can be reduced to Bessel's equation ; cf. § 4*3 E) with 2q = 3. The expression of Airy's integral in terms of Besael functions of ordorst + ^ was published first in a little-known paper by Wirtinger, Berichtc des natur.-med. Vereins in .Innsbruck, xxm. A897), pp. 7—15, and later by Nicholson, Phil. Mag. F) xvn. A909), pp. 6—17. Subsequently Hardy, Quarterly Journal, xu. A910), pp. 226—240, pointed out the con- connexion between Airy's integral and the integrals discussed in ^ 0*21, 6*22, and ho then examined various generalisations of Airy'a integral (§§.10*2—10*22). To evaluate Airy's integral+, we observe that it may be written in the form - I exp (i7,:l + iirt) dt. ^ J -co Now consider this integrand taken along two arcs of a circle of radius p with centre at the origin, the arcs terminating at p, pe^1 and pe1"', pe*1 respectively. The integrals along these arcs tend to zero asp -*¦ cc , by Jordan's lemma, and hence, by Cauchy's theorem, we obtain Stokes' transformation [ f co exp Jiri foo [ fooexpjiri cos (f ± xt) dt = ~\ exp (if ± irt) dt J() ? J 00 HXp JtT1 I f [e^[ exp (- t-1 ± e^xr) + e *w exp (- r1 ± e" ^xr)} dr] the contour of the second integral consists of two rays emerging from the origin and the third integral is obtained by writing re^vl, re^vl for t on these rays. Now, since the resulting scries are convergent, it may be shewn that§ w - 0 m ¦ * Trim*. Ctnufi. J'/iil. Svc. ix. A8011), pp. IGli—187. [Math, mid PIiijh. Pn^-rx, n A883), pj). 32U—349.] Hce also Stokes1 letter of May 1'2, 1848, to Airy, Sir (!. G. Stokcx, Memoir and Scientific Correa-pondcnce, i\. (Cambrid^o, 1007), pp. 15'J—100. t l-'or other occurrences of theso t'unctiouB, see Bayleigh, Phil. May. (G) xxvni. A914), pp. 609—619; xxx. A915), pp. 829—388 [Scientific Papers, vi. A920), pp. 266—275; 341—349] on stability of motion of a viscous fluid; also Weyl, Math. Ann. i-xviu. A910), p. 267, and, for approximate formulae, §8*4*3 infra. X The integral is convergent. Cf. Hardy, he. cit. p. 228, or de la Valloo Poubbiu, Ann. de la Soc. Set. <te Jinixelles, xvt. A892), pp. 150—180. § Bromwich, Theory of Infinite iSeries, § 176.
190 THEOBY OF BESSEL FUNCTIONS [CHAP. VI and so f m-o m\ Jo = 5 5 ( _ r This is the result obtained by De Morgan. When the series on the right are expressed in terms of Bessel functions, we obtain the formulae (in which x is to be taken to be positive) due to Wirtinger and Nicholson: 6*5. Batoned integral representations of Bessel functions. By using integrals of a type introduced by Pincherle* and Mellinf, Barnea+ .has obtained representations of Bessel functions which render possible an easy proof of Kuramer's formula of § 4-42. Let us consider the residue of - T Bm - s). (izy at s = 2m 4- r, where r - 0,1, 2,.... This residue is (-)r (iz)im+rlr\, so the sum of the residues is (—)mzlme~iz. Hence, by Cauchy's theorem, ^T^T^ + mTTi) *' if the contour encloses the points 0, 1, 2, .... It may be verified, by using Stirling's formula that the integrals are convergent. Now suppose that JR (*»)> — \, and choose the contour so that, on it, R (v + s) > — ?. When this last condition is satisfied the series ? r Bm - s) is convergent and equal to * Eend. del E. Ittituto Lombardo, B) xix. A886), pp. 559—562 ; Atti della II. Ace ad. dei Lineei, ser. 4, Efindtconfi, iv. A888), pp. 694—700, 792—799. f Mellin has given a summary of hia researcheB, Math. Ann. lxviii. A910); pp. 305—337. { Camb. Phil. Tram. xx. A908), pp. 270—279. For a bibliography of researches on integrals of this type, see Barnes, Proc. London Math. Soc. B) v. A907), pp. 59—65.
6*5] INTEGRAL REPRESENTATIONS 191 by the well-known formula due to Gauss. If therefore we change the order of summation and integration * we have r (\ -iz-SML f@+)r(-s)rp, + s + ?).(^)g^ Jv{z)e ~ a** L T(v + i« + i)r(v + i* +1)• The only poles of the integrand inside the contour are at 0, 1, 2, When we calculate the sum of the residues at these poles, we find that so that A) J, (t) e-» = which is Kummer's relation. In like manner, we find that B) J, (z) e« = Y^T)liPl (V + *; 2i; + X These formulae, proved when jK (v) > — \, are relations connecting functions of v which are analytic for all values of v, and so, by the theory of analytic continuation, they are universally true. It is also possible to represent Bessel functions by integrals in which no exponential factor is involved. To do this, we consider the function qua function of 5. It has poles at the points 5 = 0, 1, 2, ...; - v, - v + 1, - v 4- 2, .... The residue at s = m is sin vir ' ?n\ V (v + m + 1)' while the residue at s = — v + m is sin vir' vi! V (v + in + 1)' so that C) vre-i^+n-t #„«) (Z) = - — I r (- v - s) V (- s) {\iz)v+w ds, j&TT'l J and, in like manner, D) -/re*<"+'»« Hv<" (z) = - ^. f r (- v - s) r (- «) (- Iizy+°-» ds, where the contours start from and return to + oo after encircling the poles of the integrand counter-clockwise. When | arg iz \<\ir in C) or | arg (— iz) j < \ir in D) the contours may be opened out, so as to start; from ocu and end at — oo i. If we reverse the directions of the contours we find that E) 7re-*("+1>« ff,121 0) = =i-. rc+c°V (- v - s) V (- s) (|»>)-+M rf», Z7TI J -c-aoi * Cf. Bromwich, Theory of Infinite Series, § 170.
192 THEOEY OF BESSEL FUNCTIONS [CHAP. VI provided that J arg iz I < \ir; and F) 7rei<"+1)« H® (z) = ~ f""""""V (- v - s) T (- a) (- ^V)v+28 (fa, provided that j arg (-iz) !<?•""> and, in each integral, c is any positive number exceeding R (v) and the path of integration is parallel to the imaginary axis. There is an integral resembling these which represents the function of the first kind of order v, but it converges only when R(v) > 0 and the argument of the function is positive. The integral in question is G). ,(M and it is obtained in the same way as the preceding integrals; the reader will notice that, when j s | is large on the contour, the integrand is 0 (| s I"""). 6*51. Barnes' representations of functions of the third kind. By using the duplication formula for the Gamma function we may write the results just obtained in the form a) j (z)«.*» Consider now the integral in which the integrand differs from the integrand in A) by a factor which is periodic in 5. It is to be supposed temporarily that 2v is not an integer and that the path of integration is so drawn that the sequences of poles 0,1, 2, ...; — 2v, 1. — 2v, 2 — 2v, ... lie on the right of the contour while the sequence of poles — v — \, —v — \% -v — §, ... lies on the left of the contour. In the first place, we shall shew that, if | arg iz j < f 7r, the integral taken round a semi- semicircle of radius p on the right of the imaginary axis tends to zero as p -*¦ oo ; for, if s — p ei9t we have and, by Stirling's formula, g ~ peie log Biz) - (v + p&*) (log p + id) + pei6 - % log Bir) ; and the real part of this tends to — co when — %ir<6<lTr, because the dominant term is - p cos 6 log p. When 6 is nearly equal to ± \ir, \ sin sir | is comparable with \ exp [prr \ sin 61] and the dominant term in the real part of the logarithm of s times the integrand is p cos 8 log j Iz \ - p sin 8. arg 2iz - p cos 6 log p + p8 sin 8 + p cos 8- Ipir | sin 01, and this tends to — <x> as p -*• co if j arg iz \ < §7r.
6*51] INTEGRAL REPRESENTATIONS 193 Hence s times the integrand tends to zero all along the semicircle, and so the integral round the semicircle tends to zero if the semicircle is drawn so as to pass between (and not through) the poles of the integrand. It follows from Cauchy's theorem that, when j arg iz \ < §tt and 2v is not an integer, then may be calculated by evaluating the residues at the poles on the right of the contour. The residues of r (- s) r (- 2u - s) r (v+s + d . {lizy at s — m and s = — 2v 4- m are respectively 7T r (v + m +1). Biz)m _ 7T r (~v +m ¦+-\). Biz)~*v+m sin tvrv m! V Bv + m + 1) ' sin Ivir m\ V (— tv + m + 1) and hence r' r (- *) r (- iv - «> r (v +«+ j>. B«)« <is J -w1- "- snT^-'¦r(i-2v)-^(i'1 l~2"; 2"> It follows that, when | arg is | < ilir, B) i/,w(^) = e"ltt"VirI""u/l"n'li ^>ftV 7T-! cri and similarly, when ! arg (— w) j < I ir, ("^N H w(A— e~ ((>s f;7r ' x ["' r (~«) r(-2v-s) V(v+s + l).(~lizyds. - OC ( The restriction that vis not to bo an integer maybe removed in the usual manner by a limiting process, but the restriction that 2i> must not be an odd integer cannot be removed, since then poles which must be on the right of the contour would have to coincide with poles which must be on the left. w. b. p. 13
in cases CHAPTER VII ASYMPTOTIC EXPANSIONS OF BESSEL FUNCTIONS 7-1. Approximate formulae for Jv (z). In Chapter in various representations of Bessel functions were obtained the form of series of ascending powers of the argument z, multiplied in sonic by log z. These series are well adapted for numerical computation when ^ is not large compared with 4(i/ + l), 4(i/ + 2), 4(i/ + 3), .... since the series converge fairly rapidly for such values of z. But, when | z \ is large, the series converge slowly, and an inspection of their initial terms affords no cluo to the approximate values of Jv{z) and Yv(z). There is one exception to this .state- .statement; when v + ? is an integer which is not large, the expressions for J±v (~) in finite terms (§3-4) enable the functions to be calculated without difficulty. The object of this chapter is the determination of formulae which render possible the calculation of the values of a fundamental system of solutions of Bessel's equation when z is large. There are really two aspects of the problem to be considered ; (.lie invest.i- gation when v is large is very different from the investigation when v i.s neH. large. The former investigation is, in every respect, of a moro recondite character than the latter, and it is postponed until Chapter vm. It must, however.be mentioned that the first step towards the. solution ol' the more recondite problem was made uy Carlini* some years before Poi.sson'.s-f- investigation of the behaviour of JQ(x), for large positive values of .r, was published. The formal expansion obtained by Poisson was /„(*)« — cos(tf-J-7r).-M—i-i?_ + i • 6 •& • ' _ I \-7rx) I K i J \ 2!(8a;)« 4!(8«L "") ("I2 12 '¦}- <V~ l " + sm (a; - M. -V - —~-~ + I V ? ; |l!8a: 3! (8a;)8 +'"*J ' when x is large and positive. But, since the series on the right are not con- convergent, and since Poisson gave no investigation of the remainders in t,h<> series, his analysis (apart from his method of obtaining the dominant term) is to be regarded as suggestive and ingenious rather than convincing. ! C°niuer9enza ddla serie ch* «"« «fl« ooluzione delproblema di Keplero (Milan, 1617). An account of these investigations has already been given in § 1-4 t Journal de VEcole Poly technique, xu. (cahier 19), A823), pp. 350- see Functions (London, 1895), pp. 34—38.
7*1] ASYMPTOTIC EXPANSIONS 195 It will be seen in the course of this chapter that Poisson's aeries arc asymptotic; thin has been proved by Lipschitz, Hankel, Sehlafli, Weber, Stioltjos and Barnes. It must be mentioned that Poisson merely indicated tho law of formation of Hiioeessivo terms of the series without giving an explicit expression for the gunoml tonn ; such an expression was actually obtained by W. R. Hamilton* (cf. § 1'6). The analogous formal expansion for Jy (x) is due to HaiiHo.nf; and a few years later, Jacobij obtained the more general formula which is now usually written in the form Jn (#) ~ ( —) cos (x — -|n 7r — I rr) yrrxJ [_ J Dna - 1«) D?i2 - 3") Dn9 - I") Dn9 - 3a) Dw« - 5*) D/ia - 7a) ( 2!"('8i)a + 4!(8^ " - sin (^ - These expansions i'orJ0(x) and ./, (.v) wm; uwjd by llan/stiu for purpowm of nunuirioal computation, and a comparison of the results so obtained for isolated values of .v with Mje results obtained from the ascending norioH 1<><1 Hanson to infer that tlid expansions, nltliough not convergeut, could safely be used for purposes of computation §. Two years before the publication of .Jacobi's c.x])ansion, PlanaH luid din- covered a-method of transforming Paiwval'.s inLcgral which placed thcicxpan.sioii of Ja(x) on a much more, satisfactory busi.sH. Hi.s work was followed by l.lw1 researches of Lipschitz**, who gave the first rigorous invcstigalion of I,lie asymptotic expansion of J^[z) with the aid of \\\v. theory of contour integra- integration; Lipschitz also briefly indicated how his results could Unapplied l.o ./„ (;), The general formulae for J,.(z) and Yv{z), where /' haw any assigned (com- (complex) value and z is large and complex, were obtained in the great memoir by Hankel ft. written in 1868. * Some information concerning W. It. HiitiiiUoii'H rorfoarohoH will \w found in Sir (Senriit' Gabriel Stokes, Memoir and Scientific Correspondcnrt', i. (CamliridKO, M)(O), pp. 1110- ¦ I ilfi. t Krmittelung tier absolutcn Stiirunijen [Schriftni tier Stornwurte Svrlninj], ((iol.liu, IHl.'t), pp. ny—123. %A8tr.Na.ch. xxvin. A849), col. «L. [O«. Math. lVeriw, vir. (lH')l), p. 17-1.1 Jmiol.i'n rtmiilt is obtained by making the Bubatitutions J2 . nin (* - 4«7r - |tt) = ( - lL»(»l-l)Hin x _ ( _ i jin(n- i; C()H ^ in the form quoted. § See n, note by Niemoller, Zeitschrift fUr Math, wid Phye. xxv. A8H0), pp. .]1- -4H. II Mem. delta R. Accad. delle Sci. di Torino, B) x. (lB-ii)), pp. 275—292. If Analysis of Plana's type wan used to obtain the aaymptotio oxpanmonH of J,, (z) and IV (s) l)y ilcMahon, Annals of Math. via. A894), pp. 57—E1. ** Journal fttr Math. lvi. A859), pp. 189—19G. tt Math. Ann. i. A869), pp. 467—501. 13—52
196 THEORY OF BESSEL FUNCTIONS [CHAP. VII The general character of the formula for Yn(z) had been indicated by Lommel, Studien iiber die Bessel'schen Functionen (Leipzig, 1868), just before the publication of Haukol'a memoir; and the researches of Weber, Math. Ann. vi. A873), pp. 146—149 must also be mentioned. • The asymptotic expansion of Kv{z) was investigated (and proved to be asymptotic) at an early date by Kummer*; this result was reproduced, with the addition of the corresponding formula for /,, (z), by Kirchhoff f; and a little- known paper by Malmst6n| also contains an investigation of the asymptotic expansion of Kv (z). A close study of the remainders in the asymptotic expansions of Jo (.?), Yo G^X ^ (x) and Kd(%) has been made by Stieltjes, Ann. Sci. de VEcole norm. sup. C) in. A886), pp. 233—252, and parts of his analysis have been extended by Callandreau, Bull, des &ci. Math. B) xiv. A890), pp. 110—114, to include functions of any integral order; while results concerning the remainders when the variables are complex have been obtained by Weber, Math. Ann. xxxvn. A890), pp. 404—416. The-expansions have also been investigated by Adamotf§, Petersbtirg Ann. Inst. polyt. 1906, pp. 239—265, and by Valewink|| in a Haarlem dissertation, 1905. Investigations concerning asymptotic expansions of Jv (z) and Yv (z), when \z\ is large while v is fixed, seem to be most simply carried out with the aid of integrals of Poisson's type. But Schlaflill has shewn that a large number of results are obtainable by a peculiar treatment of integrals of Bessel's typo, while, more recently, Barnes** has discussed the asymptotic expansions by means of the Pincherle-Mellin integrals, involving gamma-functions, which were examined in §§ 6*5, 6*51. 7'2. Asymptotic expansions of H^ {z) and Hv® (z) after Hankel. We shall now obtain the asymptotic expansions of the functions of the third kind, valid for large values of \z\\ the analysis, apart from some slight modifications, will follow that given by Hankelff. Take the formula § 612 C), namely valid when - \ir < j3 < \-rr and - \tr + /3 < arg z < |tt + /3, provided that The expansion of the factor A + ^iujz)v-^ in descending powers of z is * Journal filr Math. xvii. A837), pp. 228—242. + Ibid, xlviii. A854), pp. 348—370. $ K. Svenska V. Akad. Handl. ixn. A841), pp. 65—74. § See the Jahrbuck iiber die Fortschritte der Math. 1907, p. 492. II I6W.190S, p. 328. IT Ann. di Mat. B) vi. A875), pp. 1—20. ** Tram. Camh. Phil. Soc. xx. A908), pp. 270—279. ft Math. Ann. i. A869), pp. 491—495.
7-2] ASYMPTOTIC EXPANSIONS 197 but since this expansion is not convergent all along the path of integration, we shall replace it by a finite number of terms plus a remainder. For all positive integral values of p, we have* It is convenient to take p so large that R (v — p — |) ^ 0; and we then choose any positive angle 8 which satisfies the inequalities The effect of this choice is that, when 8 is given, z is restricted so that - 7T + 28 < arg z < 2-n- - 28. When the choice has been made, then ut 2iz sin 8, <7T, for the values of t and u under consideration, and so say, where Ap is independent of z. On substituting its expansion for A + ^iujz)v'^ and integrating terra-by- term, we find that 2\> .. , , Hto(z\-BY " ^ ; \-irz) where { fo<i - ^ * r where jBp is a function of v, p and $ which is independent of z. Hence, when R (v — p — |) < 0 and # (v + |) > 0, we have (i) ff«•> W = (AI ^.-w- when z is such that — tv + 2S ^ arg z <2ir — 2B, 8 being any positive acute angle;,and the symbol 0 is the Bachmann-Landau symbol which denotes a function of the order of magnitude f of z~v as | z \ -*¦ oo. The formula A) is also valid when R (v — p — |) > 0; this may be seen by * Of. Modern Analysis, § 5-41. The use of this form of the binomial expansion seems to bo due to Graf and Gubler, Einhitung in die Theorie der BesseVschcn Funktionen, I. (Bern, 1898)» pp. 86—87. Of. Whittaker, Modern Analysis (Cambridge, 1902), §161; Gibson, Proc. Edinburgh Math. Soc. xxxvni. A920), pp. 6—9; and MaoRobert, ibid. pp. 10—19. t Of. Modern Analysis, § 2-1.
198 THEORY OF BESSEL FUNCTIONS [CHAP. VII supposing that R(v- $>-%)> 0 and then taking an integer q so large that R(v-q-%)<0; If the expression which is contained in [ ] in A) is then rewritten with q in place of p throughout, it may be expressed as p terms followed by q -p +1 terms each of which is 0 (*-*) or o (*"*); and the sum of these q — p + 1 terms is therefore 0 (z~v). In a similar manner (by changing the sign of i throughout the previous work) we can deduce from § 6*12 D) that provided that R (v + %) > 0 and that the domain of values of z is now given by the inequalities -2?r + 28^args^7r-2&. If, following Hankel, we write _ {4>v* - I2} {<k»* - 32)... {4i/a - Bm - I)'2} 2«».m! these expansions become For brevity we write these equations thus: E) Bfi W F) jr.-w Since (»/, m) is an even function of v, it follows from the formulae of § 3*61 G)j> which connect functions of the third kind of order v with the corre- corresponding functions of order — v, that the restriction that the real part of v exceeds - \ is unnecessary. So the formulae A)—F) are valid for all values of v, when z is confined to one or other of two sectors of angle just less than 37r. In the notation of generalised hypergeometric functions, the expansions are G) i« w~(i)'«<-¦•*-»¦>. jr (8) of which G) is" valid when — w < arg * < 27r, and (8) when - 27r < arg ^ < tt.
7*21] ASYMPTOTIC EXPANSIONS 199 7'21. Asymptotic expansions o/Jv(z), J-V(z) and Yv(z). If we combine the formulae of § 7'2, we deduce from the formulae of § 3*01 (which express Bessel functions of the first and second kinds in terms of functions of the third kind) that / 2 (i) JW( <„ ,.w()[ +¦ COS (z ~ } 2<7T - } 7r) . i - »¦ ¦ -, ¦ , m-o V-~y J D) F_rW~f2y V + cos (z + }, vir - }tt) . i, - --' •-- ¦--,- and (in the casi; of funct-ions of integral order u onlij), These formalin* arc all valid for largv values of j z \ provided thai j arg z\<ir\ and the error dm1, to stopping at any term is obviously of the order of magni- magnitude of that term multiplied by \\z. Actually, however, this factor \\z may be replaced by i/z-; this may be. seen by taking the expansions of IIJ^ (z) and HJd (z) to two terms further than the. last term required in the particular combination with which we have to deal. As has been seen in § 7-2, the integrals which are dealt with when H(v)>-1 represent-, H® (z) and H®(z\ but, whe.n lt(v)<-l, the integrals from which the asymptotic expansions are derived are those ¦which represent HM-,(z) and H®-r(z). This difference in the mode of treatment of Jv{z) and Yv(z) for such values of v seems to have led some writers to think* that formula A) is not valid unless lt[v) > — \> * Cf. Sheppard, Quarterly Journal, xxm. A889), p. 223 ; Searle, Quarterly Journal, xxxix. A908), p. 60. The error appears to have originated from Todhunter, An Elementary Treatise on Laplace's Functions, Lam^n Functions and BesneVs Function* (London, 1875), pp. 312—Blii.
200 THEORY OF BESSEL FUNCTIONS , [CHAP. VII The asymptotic expansion of J0(z) was obtained by Lipschitz* by inte- integrating e*** A - i2)~J round a rectangle (indented at ± 1) with corners at ± 1 and ± 1 + oo i. Cauchy's theorem gives at once f eizt A - #)-* dt + e$™ re^-w tr* B + tu)-* du J-i .'o TOO - eiwi e~{i+u)z u~* B - iu)~* du ==¦ 0, .0 and the analysis then proceeds on the lines already given; but in order to obtain asymptotic expansions of a pair of solutions of Bessel's equation it seems necessary to use a method which involves at some stage the loop integrals discussed in Chapter vi. It may be convenient to note explicitly the initial terms in the expansions involved in equations A)—D); they are as follows: « (-)m . (v, 2m) _ Dya ~ ~ B)» " wo () 118* The reader should notice that - I2 (^2jri!)J4il^) (^ _?) ~3T(87 a formula given by Lomuiel, Studien, p. 67. Note. The method by which Lommel endeavoured to obtain the asymptotic expansion of Yn{z) in his Sticdien, pp. 93—97, was by differentiating the expansions of J±v(z) with respect t?> v; but of courae it is now known that the term-by-term differentiation of an asymptotic expansion with respect to a parameter raises various theoretical difficulties. It should be noticed that Lomrael's later work, Math. Ann. iv. A871), p. 103, is free from the algebraical errors which occur in his earlier work. These errors have been enumerated by Julius, Archives Neerlandaises, xxviii. A895), pp. 221—225. The asymptotic expansions of Jn(z) and Tn{z) have also been studied by McMahon, Ann. of Math. vrn. A894), pp. 57—61, and Kapteyn, Monatshefte fiir Math, und Phys. xiv. A903), pp. 275—282. A novel application of these asymptotic expansions has been discovered in recent years: they are of some importance in the analytic theory of the divisors of numbers. In such investigations the dominant terms of the ex- expansions are adequate for the purpose in view. This fact combined with the consideration that the theory of Bessel functions forms only a trivial part of the investigations in question has made it seem desirable merely to mention the work of Voronoif and WigertJ and the more recent papers by Hardy §. * Journal fiir Math. lvi. A859), pp. 189—196. t Ann. Sci. de VEcole norm. sup. C) xxi. A904), pp. 207—268, 459—534; Verh. des Int. Math. Kongresses in Heidelberg, 1904, pp. 241—245. % Ada Mathematica, xxXvu. A914), pp. 113—140. § Quarterly Journal, xlvi. A915), pp. 263—283; Proc. London Math. Soc. B) xv. A916), pp. 1—25.
7-22] ASYMPTOTIC EXPANSIONS .201 7*22. Stokes' phenomenon. The formula § 721 A) for Jv{z) was established for values of z such that | arg z | < 7T. If we took arg z to lie between 0 and 2tt (so that arg ze~vi lies, between — it and ir) we should consequently have so that, when 0 < arg z < 2tt, -2) - sin (« + 1-.V7T + ,|tt) ^ V^^' and this expansion is superficially quite different from -the expansion of § 7*21 A). We shall now make a close examination of this change. The expansions of § 7 21 are derived from the formula and throughout the sector in which — tt < arg z < 2tt, the function Hv^ (z) has the asymptotic expansion The corresponding expansion for 11 v^ (z), namely (!) ^(^ is, however, valid for the sector - 2-7T < arg z < tt. To obtain an expansion valid for the sector 0 < nrgz < 2ir we use the formula of § 3'62F), namely IIW (z) = 2 cos vir. H,® (ze-**) + evrti Hv^ (ze^1), and this gives Wz The expansions A) and B) are both valid when 0<arg^<7r; now the difference between them has the asymptotic expansion and, on account of the factor eiz which multiplies the series, this expression is of lower order of magnitude (when \z\\s large) than the error due to stopping
THEOBY OF BESSEL FUNCTIONS [OHAP. at any definite term of the expansion A); for this error is 0 (e'iz *r*~*) we stop at the pth term. Hence the discrepancy between A) and (-), wluc ^ occurs when 0<arg*<ir, is only apparent, since the series in A) has to I used in conjunction with its remainder. Generally we have where the constants c,, c2 have values which depend on the domain of valuos assigned to arg z. And, if arg .2 is continually increased (or decreased) whil<< \z\ is unaltered, the values of c, and c2 have to be changed abruptly at various stages, the change in either constant being made when the function wliie.U multiplies it is negligible compared with the function multiplying the other constant. That is to say, changes in d occur when I (z) is positive, whil^ changes in c2 occur when I(z) is negative. It is not difficult to prove that the values to be assigned to the constant* cx and c-2 are as follows: |2p(+i)>ri) [Bp _ l) 7t < arg z < B/> + 1) tt], i)" [2p7T < arg « < ( 2p + 2) 7r'|, where p is any integer, positive or negative. This phenomenon of the discontinuity of the constants was discovered by Stokes and was discussed by him in a series of papers. It is a phenomenon which is not confined to Bessel functions, and it is characteristic of integral functions which possess asymptotic expansions of a simple type*. The fact that the constants involved in the asymptotic expansion of the arudytic function Jv (z) are discontinuous was discovered by Stokes in (March?) 1857, and the discovery wuh apparently one of those which are made at three o'clock in the morning. Seo Sir Uvonje Gabriel Stokes, Memoir and Scientific Correspondence, i. (Cambridge, 1907), p. (W. Th<>. papers in which Stokes published his discovery are the followingt: Trans. Camb. Phil. Soc. x. A864), pp. 106—128; XI. A871), pp. 412—425; Ada Math. xxvi. A902), pp. »i>3 ¦ 397. [Math, and Phys. Papers, IV. A904), pp. 77—109; 283—298; v. A905), pp. 283- iW7. J The third of these seems to have been the last paper written by Stokes. 7*23. Asymptotic expansions of Iv (z) and Kv{z).- The formula §7*2E) combined with equation §3*7(8), which coime<U,s Kv (z) and H^ (iz), shews at once that * Cf. Bromwich, Theory of Infinite Series, § 133. t Stokes illustrated the change with the aid of Bessel fnnctions whose, orders are 0 and the latter being those associated with Airy's integral (§ 6-4).
7-23, 7*24] ASYMPTOTIC EXPANSIONS 203 when | argz j < § ir. And the formula /„ (z) = e*vrri Jv {e~^vi z) shews that provided that — \ir < arg 2 < |7T. On the other hand, the formula Iv(z) = e~^vniJv (<?**' z) shews that (d) "( } ~ B^7)* ^« ~2^r + 12^)T Mto BFr ' provided that — §7r < arg 2; < \tt. The apparent discrepancy between B) and C) when z has a value for which arg z lies between —\tt and |tt is, of course, an example of Stokes' phenomenon which has just been investigated. The formulae of thi.s section wore ntated explicitly by Kummer, Journal filr Math. XVII. A837), pp. 228—242, and Kirchhnff, Journal fur Math, xlviii. A854), pp. 348—376, except that, in B) and C), the negligible second series is omitted. The object of the retention of the negligible scries is 1,0 make A) and C) formally consistent with $3'7(G). The formulae are also given by Rictnann, Ann. der Pkysik und C/temie, B) xcv. A855), p. 135, when v = 0. Proofs are to bo found on pp. 49E—498 of Hankel's memoir. A number of extremely interesting symbolic investigations of the formulae are to be found in Heaviaide's* papers, but it is difficult to decide how valuable such researches are to be considered when modern standards of rigour are ado])ted. A remarkable memoir is due to Malmste'rif, in which the formula cos ax. dx -we~a xfBa)n + nG\(n+l).Ba)*-1 +nGa(n+ l)(n+2).Ba)M~2+ ...] is obtained (cf. § 6'3). This formula is written symbolically in the form (m cos ax. dx _ Trcra ( I )n the [ ] denoting that [n]" is to bo replaced by (/<)_,„, and this, in Malmsten's notation, means It will be observed that this notation is different from the notation of § 4*4. 7'24. The asymptotic expansions of her (z) and bei B). From the formulae obtained in §§ 721, 7*23, the asymptotic expansions of Thomson's functions ber (z) and bei (z), and of their generalisations, may be written down without difficulty. The formulae for functions of any order have been given by WhiteheadJ, but, on account of their complexity, they will not * Proc. Eoyal Soc. mi. A893), pp. 504—529; Electromagnetic Theory, n. (London, 1899). My thanks are due to Dr Bromwich for bringing to my notice the results contained in the latter work. + K. Svemka. V. Akad. Handl. lxii. A841), pp. E5-74. $ Quarterly Journal, xlii. A911), pp. 829—338.
204 THEORY OF BESSEL FUNCTIONS [CHAP. VII be quoted here. The functions of zero order had heen examined previously by Russell*; he found it convenient to deal with the logarithms^ of the functions of the third kind which are involved, and his formulae may be written as follows: lber & - exPaO)cos a (A {~ z) sin "w (~g) cos o / j() V(/tt) sin' K where " z 1 25 13 0) + 384a8 V2 128s4 7T 1 1 25 The ranges of validity of the formulae are | axgz | < \ir in the case of A) and | arg z | < \<k in the case of B). These results have been expressed in a modified form by Savidge, Phil. Mag. F) xix. A910), p. 51. 7*25. Hadamard's modification of the asymptotic expansions. A result which is of considerable theoretical importance is due to Hadamard^; he has shewn that it is possible to modify the various asymptotic expansions, so that they become convergent series together with a negligible re- remainder term. The formulae will be stated for real values of the variables, but the reader should have no difficulty in making the modifications appropriate to complex variables. We take first the case of Iv{x) when v > — \. When we replace sin \6 by u, we have the last result being valid because the series of integrals is convergent. We may write this equation in the form where 7 denotes the "incomplete Gamma-function" of Legendre§. ¦ Phil. Mag. F)'xvn. A909), pp. 531, 537. t Cf. the similar procedure due to Meissel, which will be explained in § 8-11. X Bull, de la Soc. Math, de France, xxxvi. A908), pp. 77—85. § Cf. Modern Analysis, § 16-2.
7-25, 7*3] ASYMPTOTIC EXPANSIONS 205 For large values of x, the difference between is O(xv+n+h e~2x) which is o(l) for each integral value of n. In the case of the ordinary Bessel functions, we take the expression for the function of the third kind / 9 \i oi&— iwir—Jir) Too / o'tj\"~i V \7T^.' l(l/ + $) Jo V 2xJ \) J „ so that B> //,<¦)(«) - (A)! ^^.-1., 5 (^1"'Vr(»+-';iti^+o( 7 \7ra7 „,-(, I (y+J). w!B?a')'n x and similarly S" 'r From these results it is easy lo dcrivo convergent scries for the functions of the first and .second kinds. Hadaimiril guvc. tho formulae, for fuiuitioii.s of order /.iv.r»> only ; but t.ho extension to ftmotioriH of any order oxcoeding - I in obvimi.s. 7*3. Formulae for the remainders in the (tsi/)ii}ttotic expansions. In §7'2 we gave an investigation which shewed that the remainders in the asymptotic expansions of Hvw {z) and H^ {z) are of the same order of magnitude as the first terms neglected. Jn the case of functions of the first and second kinds, it is easy to obtain a more exact and rather remarkable theorem to the effect that when v is real* and x is positive the remainders after a certain stage in the asymptotic expansions of J±v (x) and Y±v (x) are numerically less than the first, terms neglected, and, by a slightly more re- recondite investigation (§ 72), it can be proved that the remainders are of the same sign as the first terms neglected. Let us write * We may take v^O without losing generality.
206 THEORY OF BESSEL FUNCTIONS [CHAP. VII so that — ) [cos (x + \vtt -\-n) P (x, v) - sin (x + \v-rr - \tt) Q (x, v)], IT 00/ B) Y±v (x) = (J~) [sin{x + %vk-\ir) P {x, v) + cos (x + \vtt - \ir) Q(x, v)]. Now 7(i') = 0 and, in the analysis of §7*2, we may take 8 to be %tt since the variables are real, and so Aip — 1. It follows that, if p be taken so large that 2p > v — \, there exists a number 6, not exceeding unity in absolute value, such that ± 2x) mto ml \2i»/ Bp)! V2^/ and, on adding the results combined in this formula, we hav.e* +2/ ^V 2a?; mt0 Bm)! UW + Bp)\ where | ^01 ^ 1; and, since ^Q is obviously real, -1 ^ ^0< 1. It follows on integration that 1 {X, V) — i and since I <90 e~u u"+2^-i du Jn = V (v + 2p we see that the remainder after p terms in the expansion of P (%, v) does not exceed the (p + l)th term in absolute value, provided that 2p > v — \. From the formula ml 2; B we find in a similar manner that the remainder after p terms in the expansion of Q (x, v) does not exceed the (p + l)th term in absolute value, provided that 2p$* v — |. These results were given by Hankel, Math. Ann. i. A869), pp. 491—494, and were reproduced by Gray and Mathews in their Treatise on Bessel Functions (London, 1895), p. 70, but small inaccuracies have been pointed out in both investigations by Orr, Trans. Camb. Phil. Soc. xvn. A899), pp. 172—180. In the case of Kv (x) we have the formula ( + rJ **• * Thi9 result was obtained in a rather different manner by Lipschitz, Journal fUr Math. lvi. A859), pp. 189—196.
7*31] ASYMPTOTIC EXPANSIONS 207 and and, when j)^v— ?, the last term may be written where 0 < ^ •§ 1, and so, on integration, v where 0 < #2 -S ] when p ^ v — \. This is a more exact result than those obtained for P (.r, v) and Q(x,v) by the same methods; the reason why the greater exactness is secured is, of course, the fact that A + \ at/r)v~i'~* is positive and does not oacilhite in sign after the manner of A + \iut!xY~v~* ± A - \iiitjx)v~P-^. 71. The researches of titieltjes on </„(.<"), Vu(x) and A"o (.•*:). The results of § 7"<S were put into a more precise form by Sticltjes*, who proved not only that the remainders in the asymptotic expansions associated with Jo (.'/;), Yn (:v) arid Ar0(a;) are numerically less than the first terms neglected, but also that the remainders hare the tianw- si(/n as those terms. Sticltjcs also examined /(i(.r), but, his result is complicated and wo shall not reproduce it+. It i.s only to be expected that /„ (.r) is intractable because in the dominant expansion the toruiH all have the same, sign whereas in the other throe asymptotic expansions the terms alternate in si^n. It i.s evident from the definitions of §7%'J that T (a; 0) = ^:'' I i' -"* ir k- \{ I + Jr«)-* + A - \in) *} dti, Q(-'\ 0)- i~ I & "¦*¦»/"* ((I +J/m)-!-(! -\iu)¦'}(/». In these formulae rejilace A + lji<)~- by 2 '*"¦ d(j> W. o !• ± i?" sina</> * ylnn. 6Yct. (Z« VEcole norm. sit}). C) in. (lHfi()), pp. 233—2f>'2. t The function Iv(x) has alfio been oxamined by Sclial'hcitlin, Jahresbericht tier Deulschen Math.-Vereinigwig, xix. A930), pp. 120—129, but he appears to use Lagrange's form for the remainder in Tajlor'a thoorom when it is inapplicable.
208 THEORY OF BBSSEL FUNCTIONS [CHAP. VH It is then evident that (**{1 \? i* / + + (y-1 (im2 sin4 - \v? sin* </> + ...+ (-y-1 (im2 sin4 tf)*'-1 in4 )/( i 2 sin where p is any positive integer (zero included). Now, obviously, sin. o l+i^sm8^ ^ Jo 0 h'es between 0 and 1; and hence If we multiply by the positive function e~wa: u~h- and integrate, it is evident that Bp-2)!(&»)ap~a +( - + ^ > 2) where 0 < 6i < 1, and p is any positive integer (zero included); and this is result which had to be proved for P(x, 0). Similarly, from the formula ^ |i we find that where 0 < ^2 < 1> and p is any positive integer (zero included); and this is the result which had to be proved for Q (%, 0). In the case of K»{x)y Stieltjes took the formula Ko(x) = ~ I" e~™it-*A+ \u)~Hu, v^ Jo I and replaced A + \u)~i by - | -— ^. - ; the procedure then follows the J 7rJo 1+fusina<f> * method just explained, and gives again the result of § 7'3. By an ingenious device, Callandreau* succeeded in applying the result of Stieltjea to obtain the corresponding results for functions of any integral order; but we shall now explain a method which is eftective in obtaining the precise results for functions of any real order. * Bull, des Sci. Math. B) xiv. A890), pp. 110—114.
7-32] ASYMPTOTIC EXPANSIONS 209 7*32. The signs of the remainders in tlie asymptotic expansions associated with Jv (x) and Yv (x). It has already been seen that /„ (%) and Yv (x) are expressible in terms of two functions P(x,v) and Q(x,v) which have asymptotic expansions of a simpler type. We shall now extend the result of Stieltjes (§7-31) so as to shew that for any real value* of the order v, the remainder after p terms of the expansion of P (x, v) is of the same sign as (in addition to being numerically less thanf) the (p + l)th term provided that 2p > v — \\ a corresponding result holds for Q (a, v) when 2p > v — $. The restrictions which these conditions lay on p enable the theorem to be stated in the following manner: In the oscillatory parts of the series for P {x, v) and Q (x, v), the remainders are of the same sign an, and numerically less than, the first terms neglected. By a slight modification of the formulae of § 7*3, we have P ) Q {x'v)=2~i and, exactly as in § 7 3, we may shew that l _ typ- It{{[+1 iuty-:P-H _ A _ ?lltT-^ i-ij dt The reader will see that we can establish the theorem if we can prove that, when 2p > v — |, the last term on the right is of fixed .sign and its sign is that of It is clearly sufficient to shew that . -— -. f \\ -tfi'~* \i 1A + JwtO"-*+i-(L -iittO'-^"-*} dt Zp — v—^'o is positive. Now this expression is equal to} * As in § 7*3 we may take v^:0 without loss of generality, t This has already been proved in § 7'3. i Since |sinD\u«) |^\u/, the condition 2p>v-^ secures the absolute convergence of the infinite integral. w. B. F. 14
210 THEORY OF BESSEL FUNCTIONS [CHAP. VH Now A — ?)^-2 is a monotonic decreasing function of t\ and hence, by the second mean-value theorem, a number ?, between 0 and 1, exists such that I (I- tyv* sin (\\ut) eft = [ sin {\\ut) dt > 0. Jo Jo Since T Bp - v +1) is positive, we have succeeded in transforming i _ typ-^i {A + |iu*)"-*H-(l -\iut)v^^) dt into an infinite integral in which the integrand is positive, and so the expression under consideration ic positive. That is to say, ,~o Bm)l +* Bp)! where 0>O when 2p> v — \. And it has already been seen (§7'3) that in these circumstances | 01 ^ 1. Consequently 0 ^ 6 $ 1; and then, on multiplying the last equation by e~uxuv~^ and integrating, we at once obtain the property stated for P {x, v). The corresponding property 'for Q (x, v) follows from the equation the details of the analysis will easily be supplied by the reader. Note. The analysis fails when ~^<v<-| if we take p=0, but then the phase of (l±im)"-4 lies between 0 and ±%(v-%)tt, and so J {(l-j-^'u)"-4 + (l - }iu)v-l} has the same sign as unity, and, in like manner, i{(l-t-?m)"~4-(l -^i«)"~4}/i has the same sign as i(v-|-)M, and hence P(#, v) and <2 (^> ") have the same sign as the first terms in their expansions, so the conclusions are still true; and the conclusion is true for Q(x> v) when 7*33. Weber's formulae for the remainders in the expansions of functions of the third kind. Some inequalities which are satisfied by the remainders in the asymptotic expansions of Hvll) (z) and Hv® (z) have been given by Weber*. These inequali- inequalities owe their importance to the fact that they are true whether z and v are real or complex. In the investigations which we shall give it will be supposed for simplicity that v is real, though it will be obvious that modifications of detail only are adequate to make the mode of analysis applicable to complex values * Math. Ann. xxxvn. A890), pp. 404—416.
7-33] ASYMPTOTIC EXPANSIONS 211 of v. There is no further loss of generality in assuming that v^0, R(z)^0. We shall write | z \ = r, and, since large values of | z | are primarily under con- consideration, we shall suppose that 2r^ v — \. If v -1 > 0, we have*, by § 6'12 C), (z) | - — "tw—7 n e uv * l + o- w*/ r(l/ + f) .'o \ 2* —) p,- --f ' 2 \iel'(*~*'"r~i'r)l Jo V ZrJ du v — i du i / o \t \\irz) i ! If 0 ^ v < •?, we use the recurrence formula and apply the inequality just obtained to each of the functions on the right. It is thus found that and similarly wheret C) G = 1 - 2r The results may be called Weber's crude inequalities satisfied by HV{1) (z) and Hv{"] (z). By an elegant piece of analysis, Weber succeeded in deducing more refined inequalities from ihiun in the following manner: Take the first /) terms of the series involved in Hankel's two expansions and denote them by the. symbols ^,,A) (z; -p), S,,'2' (z; p), so that 7 (—\m (v It is easy to verify that We regard this as an equation to be-solved by the method of variation of para- parameters; we thus find that 2,ll) {*; p) = Q**? e-<^i'-4-) {il («) jyrw (z) + j5 E) * In the third lino of analysis the inequality ex^ 1 + x (x^O) haB been used, t When k i we take 2r ¦> v + S. 14—2
212 THEORY OF BESSBL FUNCTIONS [CHAP. VII where A (z) and B (z) are functions of z so chosen that (A' (z) Hvw (z) + B' (g) #„<»(z) = 0, \a'(z) ^H* (,)+*'(*)?B.» (z) - - (*»*)-*rf«M--w_?ifc^. It follows that and so where ^1 is a constant. We obtain a similar expression for B (z), and hence it follows that tj]) (z; p) = {AHvw (z) + BHV® (z)} D 7r^)i e-«-(*-^*-i*) By considering the behaviour of both sides of this equation'as z -*- + oo , it is not difficult to see that A = 1 and i? = 0. Hence we may write Hankel's formulae in the forms rw (z) = (—) 6««-i-»-l») {S/» (z;p) + RpV], \7rz/ where the remainder itlj,111 may be defined by the equation Sinee R (z) > 0, we have \z + t\^ a/(^2 + ^2)» and so, by using the crude inequalities, we see that the modulus of the last integrand does not exceed ( Hence », p) | f (r2 + f )-i<p+D dt, J o f J o and so, when p > 1, we have and similarly These are the results obtained by Weber; and it will be observed that in the analysis no hypothesis has been made concerning the relative values of v and p; in this respect Weber's results differ from the results obtained by other writers.
7*34] ASYMPTOTIC EXPANSIONS 213 7-34. Approximations to remainders in tite asymptotic expansions. When the argument of a Bessel function is not very large*, the asymptotic expansion is not well adapted for numerical computation because the smallest term in it (with the remainder after the smallest term) is not particularly small; at the same time the argument may be .sufficiently large for the ascending series to converge very slowly. An ingenious method for meeting these numerical difficulties was devised by Stieltjesf; we shall explain the method in detail an applied to the function K0(x) and state the results which were obtained by Stieltjes by applying the method to Jo {%) and Yo (x). We apply the transformation indicated in § 7"81 to the formula §0*15D), so that p—X TOO p—XU //,, o .' o «*(? -t- jvt sin- 0) - v o Jo u* (I + luHin20) J That is to say, where Now the value of m for which @, •/n)/B.r)m is Uiast in nearly e<]iial Ui 2./- when .* is large; accordingly, in order U> c.(Hisi(itT l.ho ronmindor al'tior the smallest term of the series for Ar,( (.-/.•). wo ^Iioohi' ^ ho that x = lp + a, where a is numerically less than unity; and then ri Too /V «--/U 7TSJo Jo ?/>(! -I- JftHin*^) Now, as u increases from 0 to oo , \ ue~ *" iiiereaHcs from 0 up U) a maxinium ?-1 (when m = 2) and then decreases t<j z(M-u ; ho we write where f increases from — oo to oo, and, for similar reaaoiiH, we? write sin2 6 — e~i\ * The range of valueB of x uuder contempliilion for tho function.1! Jn(.r), 1 '0 (x) and A'«(.rJ in rom about 4 to about 10. t Ann. Sci. dc I'Ecole norm. avp. (8) m. A886), pp. 241—252.
214 THEORY OF BESSBL FUNCTIONS [CHAP. VIl The domain of integration becomes the whole of the (?, n) plane; and it is found that where o«.| o-ix c°° r °° ( °° ^ 7T* J-ooJ-oo |r,*=0 J by some rather tedious arithmetic. It follows* that the dominant terms of the asymptotic expansion of Rp for large values of p are so that B) 0, 2 "f ,0 It is easy to verify by Stirling's theorem that Ba;)" v ; * so that the error due to stopping at one of the smallest terras is roughly half of the first term omitted. In like manner Stieltjes proved that, if P (x, 0) and Q (x, 0) are defined as in § 7'3, then where provided that ^) is chosen so as to be nearly equal to x, and t is defined to be x — jp. Results of this character are useful for tabulating Bessel functions in the critical range; some similar formulae have been actually used for that purpose by Airoy, Archiv der Math, und Phys. C) XX. A913), pp. 240—244-; C) xxn. A914), pp. 30—43 ; and British Association Reports, 1913, 1914. It would be of some interest to extend the results, which Stieltjes has established for Bessel functions of zero order (as well as for the logarithmic integral and some other functions), to Bessel functions of arbitrary order. * Of. Bromwich, Theory of Infinite Series, §§133, 137, and 174, or the lemma which will be proved in § 8-3,
7*35, 7'4] ASYMPTOTIC EXPANSIONS 215 7*35. Deductions from SchafheiUin's integrals. If we replace u by 2 tan 6 in the formulae of § 7'32, we deduce that ** sin"-* 6 cos (v-\)Q which resemble Schafheitlin's integrals of §6'12, It is obvious from these results that P(x,v)>o, (-i<"<4) Q (a, v)>0, (? < v < f) Q (#, v) < 0. (— \ < v < \) An interesting consequence of these results is that we can prove that Q{x,v)IP{xtv) is an increasing function of x when — ?< v <\ and that it is a decreasing function of x when | < v < f. For we have Q' (x, v) P (x, v) - F (x, v) Q (x, v) _ . o .'o where jp (ut (j))= - - -'- - (tan v — tan (?) coa {v — \) a sin {v — so that ----- (tan 6 — tan ^>) sin (| —r v) | If we interchange the parametric variables 6, <j> in the double integral and add the results ho obtained wo, set! that, when —\<v< •§, the double integral has the same sign as \ — v ; and this proves the result. 7*4. Schlo-Jii's investigation of the asymptotic expansions of JBessel functions. In a memoir which seems hardly to have received the recognition which its importance deserves, Schliifli* has given a very elegant but soincwKat elaborate investigation of the asymptotic expansions of the functions of the third kind. The integral formulae from which lie derived these expansions are generalisations of Bessol's integral; although Bessel's integral is not so well adapted as Poisson's integral for constructing the asymptotic expansion of * Ann. di Mat. B) vi. A875), pp. 1—20. The only standard work oa Beasol functions in which the importanco of this memoir is reoognised is the trisaUne by Graf and G-ubler.
216 THEORY OF BBSSBL FUNCTIONS [OHAP. VII /„ (z) when z is large and v is fixed, yet Schlafli's method not only succeeds in obtaining the expansion, but also it expresses the remainders in a neat and compact form. Schlafli's procedure consisted in taking integrals of the type =—.-. vr"'1 exp \ + h'eia (u + - ] \ du, 27TIJ f f2 V WJ and selecting the contour of integration in such a way that, on it, the phase* of lreia (m - 2 + lju) is constant. He took two contours, the constants for the respective contours being 0 and rr; and it is supposed that r is positive and a is real. (I) Let us first take the phase to be tt; write u = l +pei\ where p is positive and 6 is real, and then reiap2e2ie/(l + peie) is negative, and is consequently equal to its conjugate complex. Hence we have /i\ sin (a+ 20) sin 6 .,„,,, Next choose a new parametric variable <? such that <f> = 20 + a - ir, and then /o\ _ cos i (a ~ <?) id, . to, (u — IJ __ —r sin2 0 » cos \(a + 0) w cos ? (a — <?) cos |(a + <j>)' Now, as <j> varies from — Gr — a) to Gr — a), u traces out a contour emerging from the origin at an angle — (ir - a) with the positive real axis and passing to infinity at an angle (tt — a) with the positive real axis, provided that 0 < a < 2tt. If this restriction is not laid on a the contour passes to infinity more than once. We shall now lay this restriction on a; and then the contour is of the type specified for formula § 6'22 (9), provided that we give a> and arg z the same value a, as is permissible. It follows that eV7liI-v(reia)-e~vH Iv(reia-\- 1 r"~a ZT-—= — — « : I U~"~1 exp 1% sin vtt 2*rn J _(„._,») x where u is defined in terms of (f> by equation B). * The reader will find it interesting to compare the general methods of this section with the "method of steepest descents" which in applied to obtain various asj'mptotic expansions in Chapter vm.
7*4] ASYMPTOTIC EXPANSIONS 217 Changing the sign of $ is equivalent to replacing u by 1/u, and so, replacing the expression on the left by its value as a function of the third kind, we have "I fir—a C / 1 \ > f] \f\cf 11 C) ei"H Hvu (*•«'«—»*>) = —. (irv + w) exp \ W» U + - H. —?- d<b. rrt J o (. \ wj a<p From B) it follows that — reia(u — iy/u increases steadily* from 0 to + co as cf> varies monotonically from 0 to tt — a.; and, if we write - reia (u - Vfju ~ t, so that t is positive when u is on the contour, we have du dt _ dt u ^ - 7'ei<r(u -I/it) = F*(^-«)» (m* + t the range of values of arg u being less than ir. Next, by Cauchy's theorem, it is convenient to take the point ?=1 inside the contour, but ? = 0 must be outside the contour because the origin is a branch-point. It follows that ) Hence Now it is evident that where p ia any positive integer (zero included). It will be convenient subse- subsequently to suppose that -p exceeds both R(v— \) and R(— v — ^). On making this substitution in the laat integrand and observing that 27rtJ C (G U tK~r(i/-w+i).Bm!) Bw)! ' (with the notation of §7), we deduce that where t«+,if) e-\t tP J „. d sin2 <f> _ sin </> A + 2 ooa a cos </> + cob2 <p) d<f> cos a + cos <f> ~ (cos a+ oos </>J
218 THEOBY OF BESSEL FUNCTIONS [CHAP. VH First consider 1 /•(«+, 1/M+, 1 Whenp is so large that it exceeds both R(v — {) and ?(— v — %), we take the contour to be as shewn in Fig. 15; and when the radii of the large and small circles tend to oo and 0 respectively the integrals along them tend to zero. If now we write on the two rays (which are all that survives of the contour), we find that _ (~)p cos v-w f1 .vP-"-i A - x)P+" T~ 1 T^~T^ Fig. 16. Now the numerator of the integrand is positive (when v is roal), and the modulus of the denominator is never less than 1 when {it < a < -jtt ; for other values of a it is never less than | sin a |. Therefore where | 0O | is 1 or | cosec a | according as cos a is negative or positive. When v is complex, it is easy to see that G) -V COS VTT cos R (vrr)
7*4] ASYMPTOTIC EXPANSIONS 219 Hence, finally, when — $tt < arg z < $rr, )e where | 0! | does not exceed 1 or | sec (arg z) \ according as I(z) is positive or negative, provided that v is real and p + ? > | v \. When v is complex, the modi- modified form of the remainder given by G) has to be used. Since R {1 -te(l -a;)j$reia)} 2s 0 when R(e-ill)$Q, we see that, in (8),' Bx has its real part positive when v is real and I(z) ^0. If z be replaced by iz in (8) we find that, when | arg 2 | < 7r, and, when v is real, (i) 72 @a) > 0 and | 03 \ < 1, if R {z) > 0, (ii) j #a I < I coscc (arg z) \, if R (z) < 0. The modifications necessary for complex values of v are left to the reader. (II) We next discuss the consequences of talcing the phase of to be zero. As before, we write u = 1+ pei0, and then reu p~ e"!<>/( I -\-pei0) is positive, and therefore equal to its conjugate complex, so that we obtain anew equation (I). We then diverge from the preceding analysis by writing so that I (a + <p) ., . (a—I)" rsin2^ „=i@)e..i,,. reia (y=_.^ sin .J (a - ^>) ' « sin Now, as <f> varies from — a to a, u traces out a contour emerging from the origin at an angle a with the positive real axis and passing to infinity at an angle —a with the positive real axis, provided that a lien between —tt and tt. The contour is then of the type specified for formula § (r22 (8) if, aa is per- permissible, we give o> and argxr the same value a. It follows that, when — 7r < a. < it, Kv (reia) = }2 cos vir \ w"" exp 1 - }2 reia lu+ ¦) I ? d(j>, where ii is defin<id as a function of <j> by equation A0); and therefore A1) //„« (re'<«-»">) = w^.l-\\u-v + u")exp|^~^reia fu + ^1 -
220 THEORY OF BESSEL FUNCTIONS [CHAP. VII and hence, if now t = reia(u-'iy/u, we find that (lto H <* (reiW) - ^ 6XP t^3 IK rA"t+) *—*- ^'"*ifH.1)^ A2) i/,>(t«i ;)_——-^^ JJ (S-~Ty-&Kreu) ' We have consequently expressed a second solution of Besael's equation in a form from which its asymptotic expansion can be deduced; and the analysis proceeds as in the case of Hv[l) (z), the final result .being that, when where | #2 j does not exceed 1 or | sec arg^ | according as I {z) ^0 or I(z) ^ 0, provided that v is real and p + \ > | v \; and R ($») ^ 0 when / (z) ^ 0. If v is complex the form of the remainder has to be modified, just as in the case of (8). It should be observed that, since the integrands in C) and A1) are even functions of v, it is unnecessary in this investigation to suppose that R (v) must exceed — \, as was necessary in investigations based on integrals of Poisson's type. 7*5. Barnes' investigation* of asymptotic expansions of Besnel functions. The asymptotic expansions of functions of the third kind follow immediately from Barnes' formulae which were obtained in §§6'5, 651. Lot us consider V (- s)V (- 2v - s)T (v + s + l)(- 2iz)° ds —001— V-p If | arg (— iz) | ^ f it - B, we have -s + v+p)V(-s~v+p) r(s~ coi aai ^ \T (- s + v + p) V (- s - v + p) r(s~p + %)e^-Vslih , J -oai and the last integral is convergent and so the first integral of all is 0 [(- 2u)-"-*'}. But, by the arguments of §6-51, the first integral is -2iri times the sum of the residues at the poles on the right of the contour, and so it is equal to - 7T- Hvu B)/!>l"(z-J"r) cos v-nB*)"] plus - 2iri times the sum of the residues at s = _j/_.i.)-.I,_|j .,, t _ „ _p _|_ ?. The residue at - v - m - ^ is * Trans. Camb. Phil. Soc. xx. A908), pp. 273—279.
7*5, 7-51] ASYMPTOTIC EXPANSIONS 221 and so, when j arg (— iz) and this is equivalent to the result obtained in § 7*2. The investigation of HJ® (z) may be constructed by replacing i by - i throughout. The reader should notice that, although the determination of the order of magmtude of the remainders by this method is transparently simple, it is not possible to obtain concrete formulae, concerning the magnitude and the sign of the remainders, which are ultimately supplied by the methods which have been previously considered. 7*51. Asymptotic expansions of products of Bessel functions. It does not seem possible to obtain asymptotic expansions of the four products J±n(z) J±v{z) in which the coefficients have simple forms, even when yu,= v. The reason for this is that, the products /I/' (z) H^ (z) and H^w (z) Hvw (z) have asymptotic expansions for which no simple expression exists for the general term; the leading terms in the two expansions are 7TZ 1 q: + 4 The products H^ (z)!!^ (z) and ///> (z)Hvil)(z),however, do possess simple asymptotic expansions; and from them we can deduce asymptotic expansions for M*)M*) + YMYM and for JM (z) Yv (z) - Yh (z) Jv (z). The simplest way of constructing the expansions is by Barnes' method, just explained in § 7. A consideration of series of the type obtained in § 5*41 suggests that we should examine the integral the contour is to be chosen so that the poles of F Bs + 1) lie on the left of the contour and the poles of the other four Gamma functions lie on the right of the contour; and it is temporarily supposed that /*, v and fi±v are not integers, so that the integrand has no double poles. The integral is convergent provided that | arg(w) | < §7r.
222 THEOBY OF BESSEL FUNCTIONS [CHAP. VII First evaluate the integral by swinging round the contour to enclose the sequences of poles which lie to the right of the original contour; the expression is equal to minus the sum of the residues at these poles, and the residue at m + |(/i + v) is 9r»e*iH-')** r (fi + v + 2m + 1) ¦ (-)m (%zy+v+zm sin fiir. sin vrr. sin (/* + v)ir'm! V(fi + m + 1) F (v + m +1) V (/jl + v + m +1)' It follows that _v(z) sin fnr sin vrr \ sin (fx + v) it sin (jm- _v (z) _ sin (v — fi) ir sin (fi + v) ir sinGt +v)tt ^^'^ **w J "w< >™ {2 cos fj,7T cos vrr +i sin (/* + v) - cot 10* - v) tt {J. {z) Yv (z) - Y. (z) Jv {z)\ ] By writing —i for i throughout the analysis we deduce that, if both |argiz| and |arg(— iz) \ are less than f 7r, i.e. if |argz\<tt, then
7*51] ASYMPTOTIC EXPANSIONS 223 and <2> t tan i (^ - ») 7T. (/„ («) T. (z) - Y r (--"'* - s) r (- ^p - *) sin 57r. (*,)¦ a. These results hold for all values of /a and v (whether integers or not) provided that, in the case of the former fi+v and fi — v are nofc even integers, and. in the case of the latter /x + v and fj, — v are not odd integers. We now obtain the asymptotic expansions of the functions on the left of A) and B) after the manner of § 7'5. We first take p to be an integer so large that the only poles of the in- integrands on the left of the line R (s) = —p — \ are poles of FB6* + 1); and then J -<X>< ./ - X/-J)- J (when either integrand is inserted) is equal to 2?n' times the sum of the residues at the poles between the contours. Since we deduce that the asymptotic expansions, when |arg2l< it, are C) [Jh (z) Jv (z) + V. (z) \\ (,) j - cot \(p-v)ir. [J. (z) Yv <*) - Jv (z) . U + V Sill -t- - 7T —^—x 7tz2 sin $((A — v) 7r V 2 2 2 2 2 z-j and D) [/M(z) Jv (z) + TM(z) Yv (z)} + tan \(n-v)ir. [Jh(z) Yv (z) - ./, (z) Y^z)] ttzcos i (/a — v)rrr"i \ 2 ' 2 ' 2 ' 2 ' 2' 07
THEORY OF BESSEL FUNCTIONS [OHAP. VII In the special case when /a = p, the last formula reduces to E) JS(*)+YS(*)~§-g ? A.3.5...Bm-l))^, and, in particular, Formula E) seems to have been discovered by Lorenz, K. Danske VidensL Sehkabs Skrifter, F) vi. A890). [Oeuvres seientifiques, l. A898), p. 435], while the moro general formulae C) and D) were stated by Orr, Proe. Camb. Phil. Soc. x. A900), p. 99. A proof of E) which depends on transformations of repeated integrals was given by Nielsen, Hand- biich der Theorie der Cylinderfunktionen (Leipzig, 1904), pp. 245—247; the expansion E) is, however, attributed to Walter Gregory by A. Lodge, British Association Report, 1906, pp. 494—498. It is not easy to estimate exactly the magnitude or the sign of the re- remainder after any number of terms in these asymptotic expansions when this method is used. An alternative method of obtaining the asymptotic expansion of JJ* (z) + IV {z) will be given in § 135, and it will then be possible to form such an estimate.
CHAPTER VIII BESSEL FUNCTIONS OF LA RUE ORDER 8*1. Bessel functions of large order. The subject of this chapter is the investigation of descriptive properties, including approximate formulae, complete asymptotic expansions, and in- inequalities of various types connected with Bessel functions; and the pro- properties which will Le examined are of primary importance when the orders of the functions concerned are large, though many of the results happen to be true for functions of all positive orders. We shall first obtain results which arc. of a purely formal character, associated with Carlini's formula (§ 1*4). Next, we shall obtain certain approximate formulae with the aid of Kelvin's* "principle of stationary phase." And finally, we shall examine the contour integrals discovered by Debyef; these will be employed firstly to obtain asymptotic expansions when the. variables concerned are real, secondly, to obtain numerous inequalities of varying degrees of importance, and thirdly, to obtain asymptotic expansions of Bessel functions in which both the order and the argument are complex. In dealing with the function ./,, (./;), in which v and x are positive, it is found that the problems under consideration have; to he divided into three classes, according us u-jv is less than, nearly equal to, or greater than unity. Similar sub-divisions also have to be made in the corresponding theorems concerned with complex variables. The trivial problem <>f determining the asymptotic. expansion of Jv [z\ when v i.s largo and z in fixed, may he noticed here. It. is evident, l>y applying Stirling's theorem to the expansion of i^ 3*1, that Jv (z)~exp [>' + .< log (kz) -(.'-hi) log •'} • <\> + '' +'; + -.. , L " »'* J when* rns=\i\lBTr); this result lion heen pointed out hy Horn, Math. Ann. I.II. AH99), p. 3.W. [No'i'K. For physical applications of approximate formulae for functions of largo order, the following writers niiiy l>e consulted: Maedonald, Proo. ttvynl Soc. i,xxi. A903), pp. 251— 2fi8 ; LXXti. A004), pp. r>!> -UH ; x.<\ A A9M), pp. fH—(il; Phil. Trans, of the Royal Sue. CXX. A (li)lu), pp. 113—111; Dehye, Ann. dor 1'kysik it/td (Jhnnie, Dj xxx. A1H9), pp. &7—130"'; March, Aim. der l'/rt/xik and dhemic, D) xxxvn. A912), pp. 20—f>0; Rylxayriski, A-itv, der * Phil. May. (R) xxm. AHH7), pp. 252—255. [Math, and Phys. Pupero, iv. A'JlO), pp. 803—806.] In connexion witli tlio principle, hoo Htoken, Trtiwt. Vamb. I'hil. Sov. ix. A85A), p. 176 footnote. [Math. undPhys. Poper*, u. AHH3), p. IMl.J i1 Math. Ann, i.xvii. (l'.IOi)), pp. t~>'.l5—558; Milnchencr Hitzuvgsderichte, xh. [5], A910).. W. H. P. 15
228 THEORY OF BESSEL FUNCTIONS [CHAP. VIII where* A) P,=?^Dsec2/3 + sec</3) Sf^Li. C2 sec2 ft + 288isec* /3 + 232 sec" ft 4 13 sec8 ft) 2^ + 4128° sec4^+14884 seC/3 +17493 sec8/3 + 4242 sec10 ft + 103 sec13 /3) B) ^==p A6 -1512 sec2 ft - 3654 sec4 ft - 375 sec8 /3) 8 B56 + 78720 sec2/3 + 1891200 sec4/3 + 4744640 secB/3 322560j/b 4-1914210 sec8/3 + 67599 sec10 ft) To determine Jv {v sec ft) in terms of these expansions, we take ft to tend to ^tt, and compare the results so obtained with the expansions of Hankel's type given in § 7*21; we see that, as ft-^-^Tr, and we infer that cot C) ^ D) Hvv (vsec/8) = It follows that E) Jv (v sec ft) = . / f J . e'1'" cos (Qv /a\ v / o\ //2cot/3\ _„ . n (o) xv{vsecp) = A/ I ).e ^"sin(y^ where P^ and Q,, are defined by A) and B). It will appear subsequently (§ 8'41) that these formulae are actually asymptotic expansions of Jv (v sec /3) and Yv (y sec ft) when v is large and ft is any assigned acute, angle. Formulae which are valid for small values of ft, i.e. asymptotic expansions of Jv (z) and Yv (z) which are valid when z and y are both large and are nearly equal, cannot easily be obtained by this method; but it will be seen in § 8*2 that, for such values of the variables, approximations can be obtained by rigorous methods from SchlafU's extension of Bessei's integral. * The reader will observe that the approximation has been carried two stages further than in the corresponding analysis of § 8*11.
8-2] FUNCTIONS OF LARGE ORDER 229 Note. The dominant terms in the expansions E) and (t>), which may be written in the form G) Jv {x)=Mv cos (Q, - iff), Yv (x)*=Mv sin (Qv - ?*¦), where ^ Qv "" iJifP ~ V<1) ~ hV7T + '' arC Sm ("/¦*')> had been obtained two years before the publication of Meisaol's paper by L. Lorenz in a memoir on Physical Optics, K. Danske Videnskab&rnes Selskahs Shifter, F) vi. A890). [Oeuvres Seimtifiques, i. A898), pp. 421—436.] The procedure of Lorenz was to take for granted that, as a consequence of the result which has been proved in § 7*51, 2 .j;2 2.4 and then to use the exact equation (8) dQ^- 2 which is easily deduced from the Wron.skian formula of $ 3*63 A), to provo that J [ [rr.vMv~ whence the approximation stated for Qt, follows without difficulty. Subsequent roHoarchoH on the linen laid down by Loronz are duo to Macdonald, Phi/. Trans, of the. Jtoi/af Soc. ccx. A A!IO), pp. 131- HI, and Nicholson, Phil. Mag. (G) xiv. A907), pp. 697—707 ; («) xix. (MHO), pp. ±}H- d-IS); r»I(J -r>37; Proc. London Math. Soo. {i) IX. A911), pp. G7—HO; B) xi. (IMS), pp. KM--I2C. A romilt conooming Qv.,x- Qu, wliich in closely connected with (H), haa been publi.shod l>y A. Lodge, liritixh Association Report, 1906, pp. 494—498. 8. The principle of stationary phase. Hessel functions of equal order and argument. The principle of stationary j)ha.s(i was formally enunciated by Kelvin* in connexion with a problem of Hydrodynamics, though the. essence of the principle is to be found in .some much earlier work by Stokesf on Airy'.s integral (§ ()*4) and Parseval's integral (§ 2), and also in a posthumous paper by Riemann.|. The problem which Kelvin 'propounded wan to find an approximate) expression for the integral •«==¦ I cos \m {.>•- tf(m)\] dm, which exj>ressos the effect at place and time (x, t) of an impulnivo disturbance at place and time @, 0), when /(?«) is the velocity of propagation of two-dimensional waves in water corresponding to a wave-length 2ir/m. The princ'plo of intorforonce set forth by Stokes * Phil. Mag. F) xxm. A887), pp. '252—255. [Math, and Phys. Papers, iv. A910), pp. 803—306.] t Camh. Phil. Trans, ix. A85E), pp. 175,183. [Math, and Phys, Payerr, II. A888), pp. 341, 851.] % Ges. Math. Werhc (Leipzig, 1870), pp. 400—400.
230 THEORY OF BESSEL FUNCTIONS [CHAP. VIH and Rayleigh in their treatment of group-velocity and wave-velocity suggested to Kelvin that, for large values of x—tf(m), the parts of the integral outside the range (fi -a, p + a) of values of m are negligible on account of interference if p is a value (or the value) of m which makes In the range (/x-a, /i+a), the expression m {x-tf(m)} is then replaced by the first three terms of its expansion by Taylor's theorem, namely ^{x-tf (ji)} + 0. (m - fx) - ?* {fif" (/i) + 2/' (/*)} (m - /02, and it is found that, if* \/2 then In the last integral the limits for <r, which are large even though a be small, have been replaced by - oo and + oo . It will be seen from the foregoing analysis that Kelvin's principle is, effectively, that in the case of the integral of a rapidly oscillating function, the, important part of the integral is due to that part of the range of integration near which the phase of the. trigonometrical function involved is stationary^. It has subsequently been noticed| that it is possible to give a formal mathematical proof of Kelvin's principle, for a large class of oscillating functions, by using Bromwich'a generalisation § of an integral formula due to Pirichlet. The form of Bromwich/s theorem which will be adequate for the applica- applications of the principle to Bessel functions is as follows : Let F(x) be a function of x which has limited total fluctuation when cc^Q; let yhea function of v which is such that py-^oo as v-*¦ co . Then,if—\<\x,<\> v" Paf~xF(as)sinvx. dx — F(+ 0) f* t"~lsint.dt=F(+ 0) V (fi) sin \/nr; Jo Jo and, ifO<fi< 1, the sines may be replaced by cosines throughout. The method which has just been explained will now be used to obtain an * This is the appropriate substitution when m {x-tf (m)\ has a minimum at m=/u.; for a maximum the sign of the expression under the radical is changed. t A persistent search reveals traces of the use of the principle in the writings of Cauahy. See e.g. equation A19) in note 16 of his Theorie de la propagation ties Ondex, crowned Sept. 1815, M6m.pr4se.nt6s par divers savants, I. A827). [Oeuvres, A) i. A882), p. 280.] $ Proc. Camh. Phil Soc. xix. A918), pp. 49—55. § Bromwioh, Theory of Infinite Series, § 174.
8-2] FUNCTIONS OF LARGE ORDER 231 approximate formula for Jv(v) when v is large and positive. This formula, which was discovered by Cauchy*, is A) Jv{ 2* . 3^ 7TP^ This formula has been investigated by means of the principle of stationary phase, com- comparatively recently, by Nicholson, Phil. Mag. (G) xvi. A909), pp. 276—277, and Rayleigh, Phil. Mag. F) xx. A910), pp. 1001—1004 [Scientific Papers, v. A912), pp. 617—620]; see also Watson, Proe. Camb. Phil. Soo. xix. A918), pp. 42—48. From § 62 D) it is evident that Jv (v)- I cos {v F - sin 6)\ dd - —— e-^+sinh*) dt, TT J 0 TT Jv> and obviously sin vir I00 • , •> If — -¦it lit, ^ — i o no = u \ i / v). 7T • 0 TTJi) Hence ,/„ („) = l- ("cos [v {d - ain d)\ d6 + 0 (l/v). TT J I) Now let 0 = 0 — ain #, and then / cos [v F — urn d)\ d0 = I y '^-^d<}>. But lim = -9, 0 -*.()! — COH ^ 6 * and hence, i/ </>7A — cos 6) has limited total fluctuation in the interval @, v), it follows frmn Bromwich's theorem that "" coa vd> 2 f00 _ i . aw <^j „ j <p 3 coh v<bd(b and then A) follows at once. It still has to bo proved that (/^/(l -coh 0) ha.s limitud total Iliiotuation ; to establish this result wo observe that d % ^ where </ (tf) - 2 ^ T ^-^ - 3 F - sin 0), til 11 U so that // @) = 0, g (tt - 0) = + oo , (f @) = A - coa ^)»/(l -hcoH 6) > 0, and therefore, by integration, g(8)^Q when O^^^rr. Consequently <j)*/(l-cos 8) i» monotonic and it is obviously bounded. The result required is therefore proved. * Comptes Jtendus, xxxvm. A854), p. 998. [Oeuvres, A) xn. A900), p. 608.] A proof by Cauchy's methods will be given in § 8'21.
232 THEORY OF BBSSBL FUNCTIONS [CHAP. VIII By means of some tedious integrations by parts*, it is possible to obtain a second approximation, namely B) /rW and it may also be proved that C) ^(,) an associated formula is The asymptotic expansions, of which the.se results give the dominant terms, will be investigated with the aid of more powerful analytical machinery in § 8-42. 8*21. M&isseV-s third expansion. The integral just discussed has been used by Cauchy f and Meissel + to obtain the formal asymptotic expansion of /„ (n) when n is a large integer. It will now be explained how this expansion was obtained by Cauchy and (in a more complete form) by Meissel; the theoretical justification of the pro- processes employed will be investigated in § 8*42. Taking the formula Jn (n) = - fcos [n F - sin 0I dd, T J 0 let us write 0 — sin 0 — ^ts; it then follows that, for sufficiently small values of t, and Xo = 1j It follows that 1 P7 ( Jn(n) = -\ e, $nt3 is larg Jn(n) ~ - i Bm + 1) \m. # I °° «2m cos (%nts) dt, IT «i=0 .' 0 \ ( ) .'0 ( wi=0 When n is large, $nt3 is large at the upper limit, and Meissel inferred that * SeeProc. Cam6. Phil. Soc. xix. A918), pp. 42—48. t Comptes Rendus, xxxvm. A854), pp. 990—993, 1104—1107. [Oeuvres, A) xii. A900), pp. 161—164, 167—170.] t Astr. Nach. cxxvn. A891), col. 359—362; cxxvm. A891), col. 145—154. Concerning formula A), Meisael stated "Sehon vor dreissig Jahren war ich zu folgenden Formel gelangt."
8*21,.8-22] FUNCTIONS OP LABGE ORDER 233 where Q is the sign indicating a "generalised integral" (§ 6'4); and hence, by integrating term-by-term and using Euler's formula, Meissel deduced that 1 OC A) ./»(n)~- 2 A™ F(§m+ Meissel also gave an approximation for Xnl, valid when m is large; and this approxima- approximation exhibits the divergent character of the expansion A). The approximation is obtainable) by the theory developed in tho memoir of Darboux, "Sur ^approximation dea functions dc tres granda nombrea," Journal de Math, C) iv. A878), pp. 5—56, 377—416. We consider the singularities of 8 qua function of t; the singularities (where 6 fails to be monogenic) are the points at which 6 = 2rjr and C = A2nr)", where ?• = +1, +2, +3, ...; and near* t— ±.{\%tys the dominant terms in the expansion of 6 arts By the theory of Darboux, an approximation to Am in the sum of the coefficients of fim + i }n fc]J0 expansions of the two function!-, oompri.sed in the last formula ; that is to way that Am~2. (.ion-)!* v»rri Bjw + 1)! (I^tt)"'"'1 ¦» and so, by Stirling's formula, (IH)i r(Ji)(H»+il)*(ia7r)'m' Thi.s is Moiasel'.s a]))»r<>ximnt,ioii; an approximation of the name character was obtained by Cauchy, lor. cit., p. 1 IO(>. 8*22. The ajqriimtion of Kelni-n's principle tu / The principle of Ht.atioiifiry jjlianu ha.s 1k:c>ii applied by Ray]eight to obtain an approximate formula for ,/,. (/> sec /3) where ^ is a fixed positive acute angle, and v is large. \. As in § S'2 we have Jv(v s»;c /8) = -1- Tcos {vF-se.c j3 .sin tf)} <W + 0 (IIv), W . 0 and # — seC/S sin ^ ia stationary (a minimum) when 6 = /?. Write 0 — ace /3 sin 9 — ft — tan ft -{¦ (f>, ho that </> decreases to zero as # in- increases from 0 to ft and then increases as # increases from /3 to 7r. • These are the HinKularitios whioh are uoaroat to tho origin, t Phil. Mag. («) xx. A910), p. 1004. [Hcievtijic Papers, v. A012), p. 620.] X See also MacdonaliJ, Phil. Trans, of the Royal Soc. cox. A A910), pp. 181—144; and Proc. EoyalSoc. lxxi. A908), pp. 261—258; i.xxn. A904), pp. 59—68.
234 THEORY OF BESSBL FUNCTIONS [CHAP. VIII Now cos {vF — sec /3 sin 6)} dO In cos \y (<f> + ^8 — tan f3)\ -5-7 a<^>, Jo J a9 -K 0 rir-p + tan/5"] tan/3-/3 Jo and <20 1 as 6 VB tan ?) Hence, if <ft (ddjd<j)) has limited total fluctuation in the range 0 ^ 0 ^ ir, it follows from Bromioictis theorem that cos [v (8 — sec y3 sin 8)} d& ^ 2 cos [v (9 + p — tan /3)} Jo Jo and so T cos {1/ (tan fl-&)- \tt] The formula B) Yv (v sec /3) <^ sl- is derived in a similar manner from § 6*21 A). The reader will observe that these are the dominant terms in Meissel's expansions § 8*12 E), F). To complete the rigorous proof of these formulae we have to shew that </>- (d8/d<f>) has limited total fluctuation. Now the square of this function, namely <fi (dBjdcf))'*, is equal to 0 - sec j3 sin 6 -18 + tan fl _ , .. (l-secpcostfJ"" W) say. But ,,. . _ cos fl coseo 6 A - see ft cos 6f - 2 {6 - sec>.Jj sin 6 - j3 + tan |Q) cos j3 coaec ^ A - The numerator, h{$\ of this fraction has the differential coefficient — cos /3 cos 6 cosec2 6 A - sec /3 cos 6)*, and so kF) decreases steadily as 6 increases from 0 to -im-, and then increases steadily as 8 increases from ^n to n; since k(9) = 0 when d = p<$n, it follows that A'(8)^0 when 0^0</3 and /i'@) changes sign once (from negative to positive) in the range Hence | Jh F) j is monotpnie (and decreasing) when 0^ 6</3, and it has one stationary point (a minimum) in the range /3 < 8 < jt ; since | ^A (^) | is bounded and continuous when 0 ^ 0 ^v it consequently has limited total fluctuation when 0 ^ 6 ^ n, as had to be proved.
8*3] FUNCTIONS OF LARGE ORDER 235 8*3. The method of steepest descents. A development of the theory of contour integration, called the method of steepest descents*, has been applied by Debyef to obtain integral representa- representations of Bessel functions of large order from which asymptotic expansions are readily deduced. If, in general,, we consider the integral in which j v | is supposed to be large, the contour is chosen so that it passes through a point w0 at which f'(w) vanishes ; and the whole of the contour is then determined by the assumption that the imaginary part of f(w) is to be constant on it, so that the equation of the contour may be written in the form //(W) =//(<>. To obtain a geometrical conception of the contour, let w — u + iv, where u, v are real; and draw the surface such that the three coordinates of any point on it are u, v, Rf(w). If Rf\w) — z, and if the s-axis be supposed to be vertical, the surface has no absolute maxima or minima except where f{w) fails to be monogenic; for, at all other points, du? dv- The points [ui)y v0, Kf(wn)] are saddle points, or passuH, on the surface, so that the contour of integration is tin; plan of a curve on the, surface which goes through one of the passes on the surface. Thi.s curve possesses a further property derived from the equation of the contour; for the rate of change of f(w), at any given value of w, haw a definite modulus, since f(w) is supposed to be monogenie ; and since I/{w) does not change as w traverses the contour, it follows that Rf(w) must change, as rapidly us possible; that is to Kay, that the curve is charactermed by the property that its direction, at any point of it, is so chosen that it is the steepest curve through that point and on the surface. It may happen that we have, a freedom of choice in selecting a pass and then in selecting a contour through that pass; our choice is to be determined from the consideration that the curve must descend on both sides of the pass; for if the curve ascended, Rf{u>) would tend to + oo (except in very special cases) as w left the pass, and then the integral would diverge if R(v) > 0, * French "Mcthode du Col," Gorman "Methotlo der Suttelpunkte." t Math. Ann. lxvii. A009), pp. 535—558; MUnchener SitzuntjKbcrichte, XL. [6], A910). The method is to be traced to a posthumous paper by Riemaim, Werke, p. 405; and it has recently been applied to obtain asymptotic expansions of a variety of funotions.
236 THEOEY OF BESSEL FUNCTIONS [CHAP. VIII The contour has now been selected* so that the integrand does not oscillate rapidly on it; and so we may expect that an approximate value of the integral will be determined from a consideration of the integrand in the neighbourhood of the pass: from the physical point of view, we have evaded the interference effects (cf. § 8*2) which occur with any other type of contour. The mode of derivation of asymptotic expansions from the integral will be seen clearly from the special functions which will be studied in §§ 8*4—8*43, 8*6, 81; but it is convenient to enunciate at this stage a lemmaf which will be useful subsequently in proving that the expansions which will be obtained are asymptotic in the sense of Poincare. Lemma. Let F{r) be analytic when \ r J ^ a + 8, where a > 0, 8 > 0; and let m-l when | t | ^ a, r being positive; also, let \F (r)\< Kebr, where K and b are positive numbers independent of r, when r is positive and r^a. Then the asymptotic expansion f00 00 e~vrF{r) dr~ S amT (m/r) v~mlr Jo m = l is valid in the sense of Poincare when \v\is sufficiently large and arg v [ ^ \tt — A, where A is an arbitrary positive number It is evident that, if M be any fixed integer, a constant 7ij can be found such that M-l whenever t ^ 0 whether t ^ a or t > a; and therefore e-"r^(T)d;T= 2 e-VTamT{m/r)~ldr + RM, JO m-\ J 0 /"* where " | RM j < I I e~VT I. K, r{Mlr]~x e&T dr Jo provided that R (y) > 6, which is the case when j v j > b cosoc A. The analysis remains valid even when b is a function of v such that R (v) — b is not small compared with v. We have therefore proved that f er^F (t) dr = Mt \m Y (m/r) v~™* + 0 (v~ JO m-l f O and so the lemma is established. * For an acoount of researches in which the contour is the real axis Bee pp. 1343—1350 of Burkhardt's artiole in the Encyclopddie der Math. Wiss. u. 1 A916). + Cf. Proc. London Math. 8qc. B) xvn! A918), p. 133.
8-31] FUNCTIONS OF LAItGE ORDER 237 8*31. The construction of Debyes contours* when the variables are real. It has been seen in §§ 6'2,6'21 that the various types of functions associated with Jv (%) can be represented by integrals of the form taken along suitable contours. On the hypothesis that v and so are positive, we shall now examine whether any of the contours appropriate for the method of steepest descents are of the types investigated in §§ 6*2, 6'21. In accordance with the principles of the method of steepest descents, as explained in § 8*3, we have first to find the stationary points of x sinh w — viu, qua function of w, i.e. we have to solve the equation A) x cosh w — v =¦ 0; and it is at once seen that wo shall have three distinct cases to consider, in which xjv is less than, greater than, or equal to 1, respectively. We con- consider these three cases in turn. (I) When x\v < 1, we can find a positive number a such that B) x — l/.sech a, and then the complete solution of A) is iv — ± a + 2 It will be sufficient to confine our attention to the stationary pointsf ± a; at these points the imaginary part of a; sinh w — my i.s zero, and so the equation of the contour to be discussed is / (.v sinh w — vw) — 0. Write w =. a +iv, where it, v are n«il, and this equation becomes cosh u .sin v — v cosh a = 0, so that v = 0, or , v cosh a C) cosh u = - . v ' sm v The contour v = 0 gives a divergent integral. We therefore consider the contour given by equation C). To values of v between 0 and rrr, corre- correspond pairs of values of u which are equal but opposite in sign; and as v increases from 0 to 7r, the positive value of u steadily increases from a to 4- oo . * The contours investigated in thiH section are thofio which were diaousacd in Debye's earlier paper, Math. Ann. i.xvn. A1H9), pp. 585—558, except that their orientation is different; cf. § 01. t The effect of taking stationary points other than ±a would be to translate the oontour parallel to the imaginary axiB.
238 THEORY OF ,BESSEL FUNCTIONS [CHAP. VIII The equation is unaltered by changing the sign of v and so the contour is symmetrical with regard to the axes; the shape of the part of the contour between v = — ir and v = tt is shewn in Fig. 16. TTi -7TI \ Fig. 16. If t = sinh a —a cosh a — (sinh w — w cosh a), it is easy to verify that r (which is real on the curves shewn in the figure) increases in the directions indicated by the arrows. As w travels along the contour from oo — nri to oo + wi, r decreases from + oo to 0 and then increases to + oo ; and since, by § 6-2 C), 1 f«>+7rt J/,v.\ I pxs'mhw—vw,!,,,, we have obtained a curve from which we can derive information concerning Jv(x) when x and v are large and x\v< 1. The detailed discussion of the integral will be given subsequently in §§ 8, 8'5. The contours from — oo to oo ±nri give information concerning a second solution of Bessel's equation; but this problem is complicated by Stokes' phenomenon, on account of the two stationary points on the contour. (II) When xjv > 1, we can find a positive acute angle ft such that and the relevant stationary points, which are now roots of the equation cosh w — cos /3 = 0, are w = + ift. When we take the stationary point ift, the contour which we obtain is I (sinh w — w cos ft) = sin ft — ft cos ft, so that, replacing w by u + iv, the equation of the contour is , sin. ft+(v- ft) cos B cosh ii=—-——.—¦—'-- • . E) sin?;
8-31] FUNCTIONS OE LARGE ORDER 239 Now, for values of v between 0 and rr, the function sin ft + (v — ft) cos ft — sin v has one minimum (y = ft) at which the value of the function is zero ; for other values of v between 0 and it, sin ft + (v — ft) cos ft > sin v. Hence, for values of v between 0 and tt, equation E) gives two real values of u (equal but opposite in sign), and these coincide only when v — ft. They are infinite when v is 0 or it. The shape of the curves given by equation E) is as shewn in the upper half of Fig. 17 ; and if r — i (sin ft — ft cos ft) — (sinh w — w coh /3), it is easy to verify that t (which is ival on the curves) increases in the directions indicated by the arrows. As w travels along the contour from — oo • to oo + 7rt, t decreases from + oo to 0 and then increases to + oo and so we Fig. 17. have obtained a curve from which |§ EL D)j we can derive information con- corning IIVU) (a:) wluin ./¦ and v are large and xjv > 1. The detailed discussion of the integral will be given in $ S'41, 1;V8. If we had taken the stationary point ~ift, we should have obtained the curves shewn in the lower half of Fig. 17, and the curve going from — oo to oo — Tri gives an integral associated with Hv['iy{x); this also will be discussed in § 8'41. The two integrals now obtained form a fundamental system of solutions of Bessel's equation, so that there is a marked distinction between the case xjv < 1 and the case oojv > 1.
240 THEORY OF BBSSEL FUNCTIONS [CHAP. VHI (III) The case in which v=*x may be derived as a limiting case either from (I) or from (II) by taking a or ft equal to 0. The curves now to be con- considered are v — 0 and F) cosh u = w/sin v, and they are shewn in Fig. 18. Fig. 18. We obtain information concerning H}^ (v) and HJ'2] (v) by considering the curves from — oo to oo + iri, while information concerning Jv (v) is obtained from the curve which passes from oo — tt% to oo + iri. The detailed investiga- investigation will be given in §§ 8'42, 8*53, 8#54. 8*32. Geometrical properties of Debye's contours. An interesting result which will be found to be important in dealing with zeros of Besael functions (§ 15 "8), and which is also used in proving certain approximate formulae which will- be stated in § 83, is assooittted with the second of the three contours just dis- discussed (Fig. 17 of § 8'31). The theorem in question is that the dope* of (he branch from — oo to oo + Tti is positive and does not exceed */2. It is evident that, for the curve in question, (v — /3) cos o cos ,3 . , du sin (v — sinh u -7- = ^ 0 av sm'v But sin (y — /3) sec v - (v - /3) cos /3 has the positive derivative coa j3 tan2 v, and hence it follows that sin (v — 3) — (v — /3) cos v cos ?f has the same signt as v—/3. Therefore since v — /3 and v are both powtive or both negative for the curve under consideration, dvjdu is positive. * Proc. Camb. Phil. Soc. xix. A918), p. 105. Since, in the limiting case (Fig. 18) in whioh C=0, the slope is 0 on the left of the origin and is ^3 immediately on the right of the origin, no better results of this type exist. t This is obvious from a figure.
8'32, 8*4] FUNCTIONS OF LARGE ORDER 241 Again, to prove that dv\du does not exceed JS, wo write and then it is sufficient to prove that .•ty'2(v)-%p(v) + 1^0. Now the expression on the left (which vanishes when v—fi) has the derivate 0-*(«)] ¦ ; u--- \(v — B) {sin2 w+ 3 cos2 v) cos)9-|-siuiJj;sm/9 — 3 cos v sin (v-8)\ sin3 v L ^ K/J " 3 coa*sin (t> ~ ^ ,i -i.- j • A 4 sin4 »cos/S ....... has the positive denvato .— -.,— . a j v,, and so, since it is positive when v~0, it is ^sin v -p o cos vj positive when 0 < v < tt. Therefore, sinco ^' (v) has the same sign as v - /9, it follows that has the same sign as ?' - /3, and consequently has y = )9 for ita only minimum between v=0and v~ir; and therefore it is not negative. This proves the result stated. 8. The asymptotic expansion* of Jv (v scch a). From the results obtained in §8*31 wo shall now obtain the asymptotic expansion of the function of the first kind in which the argument is lesa than the order, both being largo and positive;. We retain the notation of § 8'31 (J); and it is clear that, corresponding to any positive value of t, there are two values ofw," which will be called w-^ and m,; the values of wx and w.2 differ only in the sign of their imaginary part, and it will be supposed that I(wl)>0, I(w,)<0. We then have f,v (timll a-«) fee (,/,,,, fill)) Jv {v sech a) = -¦ . e~XT < -, , (It, 2.7TI J o ( (IT (IT ) where x = v sech a. Next we discuss the expansions of ?/;, and w% in ascending powers of t. Since r and dr/dw vanish when w — a, it follows that the expansion of t in powers of w — a begins with a term in (w — a)a; by reverting this expansion, we obtain expansions of the form wl-a= S --^ Ti(mfl)> w»-«= 5 (-)'»+' -^-Ti^1), ,»=(,m+l »0o wi + 1 * The aaymptotio expansinnH contained in thiB Beotion and in §§ 8'41, 8*42 were established by Debye, Math. Ann. lxvii. (l<H<)), pp. 535—55H. w. B. V. 16
242 THEORY OF BESSEL FUNCTIONS [CHAP. VIII -xnd, by Lagrange's theorem, these expansions are valid for sufficiently small 'allies of j r j. Moreover 1 2iri *+> dw l(m+i) * The double circuit in the T-plane is necessary in order to dispose of the ractional powers of r; and a single circuit round a in the w-plane corresponds o a double circuit round the origin in the T-plane. From the last contour ntegral it follows that am is the coefficient of l/(w — a) in the expansion of ,-j-{m+i) in ascending powers of w — a; we are thus enabled to calculate the ?oefficients am. Write w — a = W and we have t = — sinh a (cosh W - 1) — cosh o (sinh W — ^Y) vhere c0 = — | sinh a, Ci = — $• cosh a, c2 = - fa sinh a,.... Therefore am is the coefficient of Wm in the expansion of (c0 + c, H^ + c2 F2 + ...j-1 ("l+>). The coefficients in this expansion will bo called ao(?n), a^m), a,o(?)i), .... ind so we have 1) m + 1 c2 (^ + l)B» + 3) cf ' 2.1! "ci,4" 2s!"" " "cV2 ra + 1 c;t (m + l)(m + 3) 2c, c2 (wi +1) (m + 3) (m +J3) ^) (m + 3) (w + 5) 3(j,aca v.. , 1)(//t+ 3)(w4-5)G) c,*j 2) jn substitution we find that ao = ao @) = 4- (-1 sinh a)-*, aj = a, A) = - (- | sinh a) {? coth a], a2 = a2 B) = - (- | sinh a)~» {? - ^ coth2 a], a3 = a3 C) = — (— \ sinh a) {-^ coth a — ^47 coth3 a}, a4 = a4 D) = 4- (- i sinh «)-«{T|H - ^ coth2a 4 $& coth*a}
8*4] FUNCTIONS OF LAEGE ORDER 243 JNow j -= 2, a«mrm* dr w -0 " when I t 1 is sufficiently small; and since -7— = cosh a — cosfh w, aw it follows that d {wx — w^/dr tends to zero as t tends to + 00 . Hence the conditions stated in the lemma of § 8'3 are satisfied, and so Jo \dr dr) asymptotic expansion y u2w F (m + |) is large. Since ar# {(a/, - a)/r* 1 -*¦ J ir as t -*- 0, it follows that, in B), the phase of a0 has to be interpreted by the convention arg «„ = + l-rr, and hence. C) J",(i/secha)~ - e v L (»* + *>. Al» .. 7 ^irv tanh a) Ul _0 F(|) (Ji/tanha)™' where (A „ = 1, ^4, = ? - 5^ coth" a, ^* = 1 !i r - rtVfl coth'J a + ^ c»th* a, The formula C) givc^H th(^ afl>'niptoti(^ oxpatiHion of •/„(!>secha) valid when a is any fixed positive number and i/ is largo and positive. The corresponding expansion for Uiu function of tlio hccoikI kind, obtained by taking a contour from -oo to oo ±niy is (V r (V**ha)~- ""(n"tmiha) ^ r(m+4) (-)»'.'i,(, The position of the singularities of d(wx — w^jdr^ qua function of the complex variable r, should be no tied. Tlu\so singularities correspond to tho points where w fails to bo a tnonogenic function of t, i.e. the points where dr/div vanishes. Hence the singularities correspond to the values ±a.-h%nri of w, ho they are the points when; r = Imri cosh a, r = 2 (sinh a- a cosh a) + 2'H.tti cosh a, and ?h assumes all integral values. It in convenient to obtain a formula for dwjdr in tho form of a contour integral; if (w0, t0) bo a pair of corresponding valuon of (w, r), then, by Oauchy's theorem, /dw\ 1 [(*»+) dw dr 1 F(«>o+) dw_ \drjo"^2iri J drr-m" 2jrt J r-tq' wliero the contour includes no point (except wQ) at which r haa the value r(). 10—2
244 THEORY OF BESSEL FUNCTIONS [CHAP. VIII 8*41. The asymptotic expansions of Jv (v sec /3) and Yv (v sec /3). In § 8 we obtained the asymptotic expansion of a Bessel function in which the argument was less than the order, both being large; we shall now obtain the asymptotic expansions of a fundamental system of solutions of Bessel's equation when the argument is greater than the order, both being large. We retain the notation of § 8-31 (II); it is clear that, corresponding to any positive value of t, there are two values of w lying on the contour which passes from — oo to co + tri; these values will be called Wj and w2, and it will be supposed that B (wO > 0, R (w2) < 0. We then have v sec 8) « : e~xr \ -j1 - -j- \ dr, ^ in Jo [dr dr) where x—vsec/3. The analysis now proceeds exactly on the lines of §8*4 except that a is replaced throughout by i/S, and the Bessel function is of the third kind. It is thus found that dwj [* _w(^ dwj e {—? ?—r Jo [dr dr\ To determine the phase of a0, that is of (— \i sin /3)~*, we observe that arg {(iVx — ift)fr} -*- + ? 7r as t -*- 0, and so Consequently A) H a> (v sec 8) In like manner, by taking as contour the reflexion of the preceding contour in the real axis of the w-plane, we find that B) JT In these formulae, which are valid when fi is a fixed positive acute angle and v is large and positive, we have to make the substitutions: If we combine A) and B), we find that D) Jv (v sec ft) ~
8-41,8-42] FUNCTIONS OF LARGE ORDER 245 E) F,(»secj8)«» Tain (v tan 8 vB Utan# The dominant terms in these expansions are those obtained by the principle of stationary phase in § 8'21. 8*42. Asymptotic expansions of Bessel functions whose order and argument are nearly equal. The formulae which have been established in §§ 8'4, 8#41 obviously fail to give adequate approximations when a (or /?) is small, that is when the argument and order of the Bessel function concerned are nearly equal. It is, however, possible to use the same method for determining asymptotic ex- expansions in these circumstances, and it happens that no complications arise by supposing the variables to be complex. Accordingly we shall discuss the functions where z and v are complex numbers of large modulus, such that | z — v \ is not large. It will appear that it is necessary to assume that z — v = o(z^), in order that the terms of low rank in the expansions may be small. We shall write i'«*(l-e), and it is convenient to suppose temporarily that iarg2'< |tt. We then have ^ roo-l-iTi A) Hv{l) (z) — ¦ . exp \z (.sinh w - w) -f- sew) dw, where the contour is that shown in Fig. 1<S; on this contour sinh^w — w is real and negative. Wo write r = w — si nh w, and the values of w corresponding to any positive value of r will be called wx and ?-y2. of which ?«, is a complex number with a positive real part, and w., i* a real negative number. We then have B) HJn 0) = —-. | e~M \exp (««/,) -4^ - exp (z€W2) -~\ dr. v/ 7rc J o [ wr ar)
246 THEORY OF BESSEL FUNCTIONS [CHAP. VIII The expansion of t in powers of w begins with a term in vfi, and hence we obtain expansions of the form exp\zewi) -j— — t z* omrs , ex CubU 2 = t~3 2 6* {m+i) H bm t*"», wi=O and these are valid when | r \ is sufficiently small. To determine the coefficients bm we observe that __Lf " tori] @+) As in the analogous investigation of § 8*4, a single circuit in the r-plane is inadequate, and the triple circuit is necessary to dispose of the fractional powers of t; a triple circuit round the origin in the r-plane corresponds to a single circuit in the w-plane. It follows that bm is equal to leiim+l)rri multiplied by the coefficient of wm in the expansion of exp (zew). {(sinh w — w)/w3}~^m+1). The coefficients in this expansion will be called bo(vv), 6j(w), b.2(m),... so that It is easy-to shew that C) For brevity we write
8*42] so that* FUNCTIONS OF LARGE ORDER 247 B, (ez) = B, {ez) = ? eV We then have exp (zeWjj exp //.111 OO and [exp (zeiv). (dw/dr)] satisfies the conditions of the lemma of § 8'3. It follows from the lemma of §8*3 that E) #.«<*) — .?. 5 e^^iBm(ez)sm j (wi + 1O and similarly F) //„« (*) ~ - .^ J^-I«»-h)« /iw(€5)Hin i (wi -f-1) 7T. We deduce at onc<^ l.hab r From the C'auchy-Mci.sHc.l formula §S'2l B), it is to be inferred that, when in is large, (-)» «,„ («,) ™,. j (»+1)». but there seems to lie no very simple approximate formula for Iitll(ez). Tin; dominant terms in G) were, obtained l>y Me.iH.s«l, in »i Kin? Pnn/rammi, 1892; and .some similar results, which neein to roHOtnblo thowc stated in ^ H*43, woro obtained by Ivoppe in a Merlin Prayramm :|, 1H!)O. 'M\u\ dominant teruiH in (H) as well aw in G) woro -also investigated by Niehol.son, Phil. May. ((>) xvi. A«K)H;, j>p. 271—279, shortly bofore tho appearance of Debyo'H nioinoir. * Tho vnlnoH of /?„(()), /'u(O), .../*io@) word ^ivon by MoibhuI, AMr. Nach. oxxvn. A891), col. 351>—362; apart from tho uhc of tho dontourH MriiHHel'H analywiH (of. § H-21) iH HtiliBtnntially the (same ah the analymH Rivmi in thin Hoction. The object of iiBhi« tho mothoda of contour integration in to evado the diffiuultiuH produced by UHing gencraliHcd integrala. The vnlueH of 11^ ('~)> lh (ez) an<' "o iez) w'" b*1 found in a paper by Airoy, Phil. Mag. ((i) xxxi. A91E), p. fi'i-t. t See tlie Jahrbuch illter die Forttchntte. der Math. 1R92, pp. 47E—478. % Sec the Jahrbuch ilber die FartHchritte der Math 1899, pp. 420, 421.
248 THEORY OF BESSEL FUNCTIONS [CHAP. VIII We next consider the extent to which the condition | arg#| < |7r, which has so far been imposed on formulae E)—(8), is removable. The singularities of the integrand in B), qua function of t, are the values of t for which wx (or w2) fails to be a monogenic function of t, so that the singularities are the values of t corresponding to those values of w for which dr/dw = 0. They are therefore the points T = 2W7TI, where n assumes all integral values. It is consequently permissible to swing the contour through any angle r/ less than a right angle (either positively or negatively), and we then obtain the analytic continuation of Hvin (z) or Hv{i) (z) over the range — |7r - t) < arg?<?7r — 7}. By giving 77 suitable values, we thus find that the expansions E)—(8) are valid over the extended region — ir < arg z < it. If we confine our attention to real variables, we see that the solution of the problem is not quite complete; we have determined asymptotic expansions of Jv (x) valid when ,r and v are large and (i) xjv <1, (ii) xfv> 1, (iii) | x — v \ not large compared with x\ But there are transitional regions between (i) and (iii) and also between (ii) .and (iii), and in theho transitional regions xjv is nearly equal to 1 while | x — v \ is large. In these transitional regions simple expansions (involving elementary functions only in each term) do not exist. But important approximate formulae have been discovered by Nicholson, which involve Bessel functions of orders ± \. Formulae of this type will now be investigated. 8*43. Approximate formulae valid in the transitional regions. The failure of the formulae of §§ 8*4—842 in the transitional regions led Nicholson* to investigate second approximations to Bessel's integral in the following manner: In the case of functions of integral order n, J,i(#) = - cos (nd — x sin 6) dd. it Jo and, when x and n are nearly equal (both being large), it follows from Kelvin's principle of stationary phase (§ 8*2) that the important part of the path of integration is the part on which 6 is small; now, on this part of the path, sin 6 is approximately equal to 6 — \6%. It is inferred that, for the values of- x and n under consideration, Jn (*) ~ - f cos (nd -%0 + \xdz) dd if ¦ Phil. Mag. F) xix. A910), pp. 247—249; see also Emde, Archiv der Math, und Phys. C) xxiv. A916), pp. 239—250.
8*43] FUNCTIONS OF LARGE ORDER 249 and the last expression is one of Airys integrals (§6*4). It follows that, when as < n, () ' it and, when x > n, /9\ t / \ 1 f 2 (a; — n B) JOOj where the arguments of the Bessel functions on the right are ? {2 (cc- The corresponding formula for Yn (x) when x > n was also found by Nicholson ; with the notation employed in this work it is C) r.(.)~ The chief disadvantage of these formulae is that it seems impossible to determine, by rigorous methods, their domains of validity and the order of magnitude of the errors introduced in using them. With a view to remedying this defect, Watson* examined Debye's integrals, and discovered a method which is theoretically simple (though actually it is very laborious), by means of which formulae analogous to Nicholson's are obtained together with an upper limit for the errors involved. The method employed is the following: Debye's integral for a Bessel function whose order v exceeds its argument *'(= v sech a) may be written in the formf Jv (v sech a) = —s—r— / e~XTdw, where t = — sinh a (cosh w — 1) — cosh a (sinh w — w), the contour being chosen so that t is positive on it. If t is expanded in ascending powers of w, Carlini's formula is obtained when we approximate by neglecting all powers of w save the lowest, -|w2sinha; and .when a = 0, Cauchy's formula of §8 2A) is similarly ob- obtained by neglecting all powers of w save the lowest, - $w3. These considerations suggest that it is desirable to examine whether the first two terms, namely — \w"sinh a- iws cosh «, may not give an approximation valid throughout the first transitional region. The integral which we shall investigate is therefore e~xrdW, where r = — -| W2 sinh a. - \ W3 cosh a, * Proc. Camb. Phil. Soc. xix. A918), pp. 96—110. t This is deducible from §8-31 by making a change of origin in the ir-plane.
250 THEORY OF BESSEL FUNCTIONS [CHAP. VIII and the contour in the plane of the complex variable W is so chosen that r is positive on it. If W = U -f- iV, this contour is the right-hand branch of the hyperbola and this curve has contact of the third order with Debye's contour at the origin. It therefore has to be shewn that an approximation to i e~XTdiv is is These integrals differ by j/\J- [J ao JO dW) , -j- \dr, dr and so the problem is reduced to the determination of an upper bound for {d(w—W)/dr}\. And it has been proved, by exceedingly heavy analysis which will not be reproduced here, that \d(w-W) d < sech a, and so />/," Hence dw dW\ ' dr j "ooexp(j7rO ( exp (-!«•; where | Qx \ < 1. To evaluate the integral on the right (which is of the type discussed in § 6*4), modify the contour into two lines starting from the point at which W = - tanh a and making angles ± $tt with the real axis. If we write W= - tanh a + f-e^™ on the respective rays, the integral becomes elm exp (^v loxfcz aj / exp j_ ^vf& _ ^v^e^i tanh2 a} dg Jo — e"!^ exp (|y .tanh1'' a) exp j— I v%3 — | v%e~^ tanh2 a} d%. Expand the integrands in powers of tanh2 a. and integrate term-by-term—a procedure which is easily justified—and we get on reduction § 7ri tanh a exp (^v tanh3 a). [/"_ j (-|y tanh3 a) — /j (^ tanh3 a)], and hence we obtain the formula D) Jv (v sech a) = - -— exp [v (tanh a -f ^ tanh3 a - a)} ifj (?i> tanh8 a) + 3^r-lexp {i/(tanh a-a)}, where | 021 < 1. This is the more precise form of Nicholson's approximation A).
8'43] FUNCTIONS OF LARGE ORDER 251 It can be shewn that, whether \.v tanh3 a be small, of a moderate size, or large, the error is of a smaller order of magnitude (when v is large) than the approximation given by the first term on the right. Next we take the case in which the order v is less than the argument x (= v sec ft). We then have gvi(tanj3-/3) ,-oo+zGr-/3) Hvw (v sec ft) = :— I e~aT dzv, TTl J _ a, -ip where T = — i sin 13 (cosh w — 1) — cos ft (sinh xu — w), the contour being so chosen that r is positive on it. The process of reasoning already employed leads us to consider the integral je-^dW, where T = - $»' Tf2 sin ? - \W3 cos /3, and the contour in the plane of the complex variable W is such that t is positive on it. If W= U+iV, this contour is the branch of the cubic (?/-F2)tan/3 + ?FCtf3- V*) = 0 which passes from — go — itan/3 through the origin to oo exp^7n". It therefore has to be shewn that an approximation to e~xrdw is j e~XTdW. The difference of these integrals is and it has been proved that, when* ft ? \ir, then - W) < 127r sec 6 dr Hence it follows that 4>d e~XTdW+ , To evaluate the integral on the right, modify the contour into two lines meeting at W= — i tan ft and inclined at angles .$ tt and 7r respectively to the real axis. On these lines, write W = - i tan ft - f, - i tan ? + ?eK * The important values of /S are, of course, small values. If /S if" not small, Debye's formulae of § 8-41 yieli effective approximations. The geometrical property of Debye's contour which was proved in § 8*32 is used in the proof of the theorem quoted.
252 THEORY OF BESSEL FUNCTIONS [CHAP. VIII expand the integrands in powers of tan* & integrate term-by-term, and it is found that e~XT dW = $iri tan 0 exp (- ? vi tan3 0) J — co— itan/3 x [e~ J-* J_ j (? v tan8 ?) + «** Jj ($ i> tan8 0)] exp (_ jOT- tan3 0) FjW (|» tan8 V3 On equating real and imaginary parts, it is at once found that E) Jv (v sec 0) = ?tan 0 cos {» (tan 0 - ?tan3 0-0)}. [J- j + Jj] + 3 -* tan /S sin (i; (tan )9 - i tan3 ? - 0)}. [J. j - J{] + 2402/i/, F) Yv(vsec/?) = ?tan0sin {»(tan0-$tan80-0)}. [J-j + Jj] - 3"* tan /8 cos [v (tan ? - ? tan3 ? - ?)}. [J_ j - /j] + 24^,/», where the argument of each of the Bessel functions J±j on the right is ?i>tans/3; and | 6^\ and | 6t\ are both less than 1. These are the more precise forms of Nicholson's formulae B) and C); and they give effective approxima- approximations except near the zeros of the dominant terms on the right. It is highly probable that the upper limits obtained for the errors are largely in excess of the actual values of the errors. 8*5. Descriptive properties* of Jv (vx) when 0 < x ^ 1. The contour integral, which was obtained in § 8*31 (I) to represent Jv (v sech a) was shewn in § 8-4 to yield an asymptotic expansion of the function. But the contour integral is really of much greater importance than has hitherto appeared; for an integral is an exact representation of a function, whereas an asymptotic expansion can only give, at best, an approximate representation. And the contour integral (together with the limiting form of it when x — 1) is peculiarly well adapted for giving interesting information concerning Jv(vx) when v is positive. In the contour integral take v to be positive and write so that u = log r, v — 9. With the contour selected, x sinh w — w is equal to its conjugate complex, and the path of integration is its own re- reflexion in the real axis. Hence 1 Too +ni Jv (vx) = jr—A ev iXBinh w~w) dw 27nJoc-,rJ r -If, * The results of this section are investigated in rather greater detail in Proc. London Math. Soc. B) xvi. A917), pp. 150-174.
8*5] FUNCTIONS OF LARGE ORDER s 253 Changing the notation, we find that the equation of the contour is r 1 _ 26 r x sin 6' so that and, when this substitution is made for r, the value of (w — x sinh w) is log =—f. cot 6, VC^2 — *'2 sm2 6). ° x sin u ' This last expression will invariably be denoted by the symbol* F{6, x)} so that (.1) Jv{vx) = and by differentiating under the integral sign (a procedure which is easily justified) it is found that This is also easily deduced from the equation JJ (vx) ^^r—.r Vle»(*^v>-w sinh wdw. Before proceeding to obtain further results concerning Bessel functions, it is convenient to set on record various properties f of F{6ix). The reader will easily verify that D) so that E) h F (*¦ •) - Srrjzs and also Next we shall establish the more abstruse property G) FF, x) 2s F@, x) + \ (& - x2 sin2 0)f»J(l + a?). To prove it, we shall first shew that \ 0 - a? sin 0 cos 6 * This function will not be confused with Schlafli's function defined in § 4'15. ¦\ It is supposed throughout the following analysis that 0<;e<1,
254 THEORY OF BESSEL FUNCTIONS [CHAP. VIII It is clear that so thatj if gF,x), qua function of d, attained its greatest value at 0 or tr, that value would be less than V(l + *2)* I? however, g F, sc) attained its greatest value when 0 had a value 60 between 0 and tt, then 0o)__ @o"-a!sBinB0o)* (df-tfsirfO,)* ~ and therefore g (9, as) $ g @O, as) = V(l - & cos 20O) so that, no matter where g{9,x) attains its greatest value, that value does not exceed V(l + °^)- Hence and so ffl^(^) Z"9<9-x2sin whence G) follows at once. Another, but simpler, inequality of the same type is (8) F@, a) > F(Q, x) + ^02 J(\ - a?). To prove this, observe that - * sin2 d) and integrate; then the inequality is obvious. From these results we are now in a position to obtain theorems concerning Jv {yx) and JJ (vx) qua functions of v. Thus, since the integrand being positive by E), it follows that Jv(vx) is a -positive de- decreasing function of v\ in like manner, «/„' (vx) is a positive decreasing function ofv. Also, since 3 \evF(H,Z) T AJT\\ 1 [t 97^ ^r I the integrand being positive by E), it follows that e"*1'0-*) Jv (vx) is a decreasing function ofv; and so also, similarly, is evFifi>x) Jv' (vx).
8-51] FUNCTIONS OF LARGE ORDER 255 Again, from (8) we have p—vF@, X) /•«• "ic T JO 0,x) r< I n 7T Jo so that e-vF @,*) (9) Jv (vx) < {1 - The last expression is easily reduced to Carlini's approximate expression (§§ T4, 8*11) for Jv(vx); and so Carlini's expression is always in error by excess, for all* positive values of v. The corresponding result for JJ (vx) is derived from G). Write and replace 0 F, x) by G for brevity. Then e-vF@,x) roo IT 7(| and so A0) xJJ (vx) < e~" W- *) A + .x-2)V<\/B7ri/). The absence of the factor \/(l —a;2) from the denominator is remarkable. It is possible to prove the formulaf [j(vtut~ ?L - Joi/lrWdtBwv»)*(l-a?)* {1 + ^/A-«¦))- in a very similar manner. This concludes the results which we shall establish concerning a single Bessel function whose argument is less than its order. 8'61. Lemma concerning F@,x). We shall now prove the lemma that, when 0 ^a;^: 1 and 0 ^0 <;ir, then A) *Z#_-> _ lm «, - F @, ,)| t-*™?™* > 0. The lemma will be used immediately to prove an important theorem con- concerning the rate of increase of ,/„ (vx). * It i8 evident from Debye'B expansion that the expression is in error by exooas for sufficiently large values of v. f Cf. Proc. London Math. Soc. B) xvi. A917), p. 157.
256 THEORY OF BBSSEL FUNCTIONS [CHAP. VIII If V@2 - «* sin2 0) s 27@, a), we shall first prove that dF@,x) idH@,x) dd I dd is a non-decreasing function of 0 ; that is to say that A-0 cot 0J + 02-x- sin2 0 0~-a? sin 0 cos 0 is a non-decreasing function of 0. The differential coefficient of this last function of 6 is @ - x* sin 0 cos 0)~2 [@2 cosec2 0 - 1 - ? sin2 0) A - a;2) -f 2 @2 cosec2 0 - 0s cot 0 cosec2 0 - | sin2 0) A - x*) + 2za A-0 cot 6) @ cosec 0 - cos 0J + sin2 0 A - a-2K], and every group of terms in this expression is positive (or zero) in consequence of elementary trigonometrical inequalities. To establish the trigonometrical inequalities, we first observe that, when 0 < 0 $ n, (i) 8+'sin 0 cos 6 - 20-x sin2 0^0, (ii) 8+s'm 8cos 6-282cot 6^0, (iii) sin 0 - 8 cos 8 - $ sin3 8^0, because the expressions on the left vanish when 0=0 and have the positive differential coefficients (i) 2 (cos8-6~x sin8f, (ii) 2 (cos 8-8 cosec 8)'\ (iii) sin 8{8-sin 8 coa0), and then 62 cosec2 (9 - (9s cot 8 cosec2 0 - \ sin2 0 = (<92 cosec2 8 -1) A - 8 cot (9) + cosec (9 (sin 0 - 8 cos (9 - ? sin3 8) ^ 0, <92cosec20-l-Jsin20 =8 cosec2 0 @ + sin 8 cos (9 - 20~l sin2 0) + cosec 0 (sin 0 - 0 cos 0 - ? sin3 0) > 0> so that the inequalities are proved. It has consequently been shewn that where the. variables are understood to be 6 and x, and primes denote differ- differentiations with regard to 0. It is now obvious that d \F'H F)ffd (F*) deXHr-F) = HT0\n] and, if we integrate this inequality from 0 to 0, we get Since F' and H/H' vanish when 0 = 0, this inequality is equivalent to and the truth of the lemma becomes obvious when we substitute the value of H @, x) in the last inequality.
8-52] FUNCTIONS OF LARGE OEDEE 257 8*52. The monotonic property of Jv (vx)jJv (v). We shall now prove a theorem of some importance, to the effect that, if x is fixed, and 0 < x < 1, then. Jv {vx)jJv (v) is a non-increasing function of v, when v is positive. [The actual proof of the theorem will be valid only when 8^x%l, (where S is an arbitrarily small positive number), since some expressions introduced in the proof contain an x in their denominators; but the theorem is obvious when 0 ^ x ^ 5 since e^fo.*) Jv {vx) and e~vF{o, x)/jv (v) are non-increasing functions of v when x is sufficiently small; moreover, as will be seen in Chapter xvir, the theorem owes its real importance to the fact that it is true for values of x in the neighbourhood of unity.] It will first be shewn that m T (&lJ(VX) dJ(vx)dJ{vx) Ovdx ox dv To establish this result, we observe that, with the usual notation, Jv(vx)= - dJv{vx)_ v dx and, when we differentiate under the integral sign, {0 w .)]-* !i?ff..')rf», de l v ' /J dd if we integrate by parts the former of the two integrals. Hence it follows that Jv{vx)*{>(»*)_ VA™) dJ>W=_i^ ' dvdx dx dv 27r where by using the inequality F(yfr, x) ^F(Q, x) combined with the theorem of § 8*51. w. b. f. 17
258 THEORY OF BESSEL FUNCTIONS [CHAP. VIII Since ?1 @, -ty) is not negative, the repeated integral cannot be negative; that is to say, we have proved that 7-/ ,d'2Jv(vx) dJv{vx)dJv{vx) Jv (yx) ——-= ^ — ^ U dvox ox ov so that dv I "v Integrating this inequality between the limits x and 1, we get so that dJv(vx) I _ . dJv(v) I T Since Jv(yx) and Jv(y) are both positive, this inequality may be written in the form and this exhibits the result which was to be proved, namely that Jv(vx)IJv{v) is a non-increasing function of v. 83. Propeviies of Jv(v) and Jv'(v). If, for brevity, we write F@) in place of FF, 1), so that A) F ($) = log - —-i^7lS1-- - cot 9J(&Z- sin* 0), ° 8in0 the formulae* for Jv{v) and Jv'(v) are B) jv(u)=i The first term in the expansion of F (d) in ascending powers of 6 is 403/(9 ^3); and we shall prove a series of inequalities leading up to the result that F (d^O3 is a non-decreasing function of 6. We shall first shew that To prove this we observe that ^ . #$ + ^ ^ _ rin. <k It is to be understood that J"v' (v) means the value of dJv (x)/dx when x haa the particular value v.
8'53] FUNCTIONS OF LARGE ORDER 259 and that A. f* ~0c°t0) _ fl2cosec2fl-f-flcotfl-2 _ (^cosec» (9 + 6 cot fl - 2) sin»- 9 Hence it follows that d (F'@)) 0-sin0cos<9 . „ n \ = e>W-^rW( sec2 e + ecot 6~2) by inequalities proved in § 851. Consequently C) 6F"F)-2 that is to say A (^' (^) _ ZF @)} > 0. If we integrate this inequality from 0 to 0 we get D) dF'(ff)-3F(d)>0, and this is the condition that F(8)/03 should be a non-decreasing function of 6. It follows that F{e, i) > lim F(d) 4 and therefore 2*3*^*' so that Cauchy's approximation for /^ (i/) is always in error by excess. An iueqnality which will be required subsequently is E) 2 F* - sin2 6) F' F) -3F- sin 6 cos 6) F{6) > 0. The truth of this may be seen by writing the expression on the left iu the form (ff*-2sin2d+tf sin6cos 6) F' F) + F -sin 6cos 0) {0/"@)-3FF)}, in which each group of terms is positive (cf. § 81). [Note. A formula resembling those which have just been established is- r\ i ^ F) / Jv(vt)dt~±- Jo *»> 3 .see Phil. Mag. f&j&xv. A918), pp. 364—370.]
260 THEOBY OF BESSEL FUNCTIONS [CHAP. VIII 8'54. Monotonic properties of Jv (v) and Jv' (v). It has already been seen (§ 8#5) that the functions Jv (v) and JJ (v) are decreasing functions of v. It will now be sKewn that both v*Jv (v) and v%Jv'(v) are steadily increasing* functions of v. To prove the first result we observe that _ t rF(d) e~VF (., de ?- ^ [6F' F) - SF@)} e~"F^ dO o7T J o since the integrated part vanishes at each limit and (§ 8*53) the integrand is positive. Hence v$ Jv (v) is an increasing function of v; and therefore A) v*Jv{v)< lim \v\Jv{v)} =r(i)/B'3M = 0-44731. In connexion with this result it may be noted that ^A) = 0-44005, 2.78 (8) = 0-44691. To prove the second result, by following the same method we find that by § 8*53 E), and so v* Jv' (v) is an increasing function of v. Hence B) v* JJ (v) < lim {v* JJ (v)} = 3^ T (^)/B*tt) = 0-41085. It is to be noted that J{ A) = 0-32515, 4JV (8) = 0-38854. 8'55. T/ie monotonic -property of v*JJ (v)/Jv(v). A theorem which is slightly more recondite than the theorems just proved is that the quotient WW + WMv)} is a steadily increasing function of v. * It is not possible to deduce these monotonic properties from the aBymptotic expansions. If, aB»»-*-oo, J'(p)~(f>(»), and if <p(v) is monotonic, nothing can be inferred concerning monotonic properties of/(v) in the absence of further information concerning/^).
8-54, 8-55] FUNCTIONS OF LARGE ORDER 261 To prove this result we use the integrals already mentioned in §§ 8'53, 8 4 for the four functions (v), Taking the parametric variable in the first and third integrals to be -f in place of 0, we find that [, (v)) 7P J o J 0 where - sin> 0) - -FW\ - sin2 + ^sin5cos ^ - 2sin2 (9) by §8-51. The function Cl^O,^) does not seem to be essentially positive (cf. § 8-52); to overcome this difficulty, interchange the parametric variables S and i|r, when it will be found that ? {??! ^ ff(n, (tf. IV [V*jv{v)) 7TZJ o J 0 Now, from the inequality just proved, ^ J-_1JC_ yjr2 + ty sin yfr cos ^ — 2 sm2 -^ ,, ^ „, ,^ _ — sin2 Since. 0-i V(^ - sin2 0) and J 0JT@) - ,P@) are both (§ 8-68) increasing functions of 0, the factors of the first term in the sum on the right are both positive or both negative; and, by §§8-51, 8-53, the second and third terms are both positive. Hence Ox @, ^) +- H, M e) is positive, and therefore which establishes the result stated,
262 THEOEY OF BESSEL FUNCTIONS [CHAP. VIII 8*6. Asymptotic expansions of Bessel functions of large complex order. The results obtained (§§8'31—82) by Debye in connexion with /„(&•) and Yv (x) where v and x are large and positive were subsequently extended* to the case of complex variables. In the following investigation, which is, in some respects, more detailed than Debye's memoir, we shall obtain asymptotic, expansions associated with «/„ (z) when v and z are large and complex. It will first be supposed that | arg z\<\rtr, and we shall write v — z cosh 7 = z cosh (a + ift), where a and ft are real and 7 is complex. There is a one-one correspondence between a + ift and v/z if we suppose that ft is restricted to lie between f 0 and 7r, while a may have any real value. This restriction prevents z\v from lying between — 1 and 1, but this case has already (§ 8*4) been investigated. The integrals to be investigated are TTlJ- ~ nl e-'fi*» dw = - --. [" e**^ dw, 7TI.I _«, TTlJ -oo+rti where f(w) = w cosh 7 — sinh w, A stationary point of the integrand is at 7, and we shall therefore in- investigate the curve whose equation is If we replace w by u + iv, this equation may be written in the form (v - ft) cosh a cos /? + (u — a) sinh a sin /3 — cosh u sin v + cosh a sin /8 = 0. The shape of the curve near (a, /3) is {(u - «J ~(v- ftJ} cosh a sin ft + 2 (u - a) (v - ft) sinh a cos ft = 0, so the slopes of the two branches through that point are \ tr + ^arc tan (tanh a cot ft), — ? 7r + ^arc tan (tanh a cot ft), where the arc tan denotes an acute angle, positive or negative; Rf(tu) in- increases as w moves away from 7 on the first branch, while it decreases as w moves away fro.m 7 on the second branch. The increase (or decrease) is steady, and Rf(w) tends to + 00 (or - 00 ) as w moves off to infinity unless the curve has & second double-point J. * Miinchener Sitzungsherichte, xl. [5], A910); the asymptotic expansions of Iv(x) and Kv(x) were Btated explicitly by Nicholson, Phil. Mag. F) xx. A910), pp. 938—943. f That is to say 0</3<tt. X As will be seen later, this is the exceptional case.
8*6,8-61] FUNCTIONS OF LARGE ORDER 263 If (i) and (ii) denote the whole of the contours of which portions are marked with those numbers in Fig. 19, we shall write Svw (*) = --. ! e-*-»> dw, Sv® (z) = - —. I e»/<«> dw, and by analysis identical with that of § 8'41 (except that i/8 is to be replaced by y)> it is found that the asymptotic expansions of SV{1} (z) and 8V® (z) are given by the formulae S B) where arg (— ? j/7rt tanh 7) = arg z + arg (— 1 sinh 7), and the value of arg (— i sinh 7) which lies between — f 7r and |tt is to be taken. Fig. 19. The values of Ao, A1} Ai} ... are 'Ao = 1, Ai - % — s\ coth2 7, 7 + i^fffV cothJ 7, C) It remains to express ?TVA' (^) and Hv® (z) in terms of 8VW (z) and /S,,1-' B); and to do this an intensive study of the curve on which //(«;) «//(y) is necessary. 8*61. The form of Debyes contours when the variables are complex. The equation of the curve introduced in the last section is A) (v — /3) cosh a cos ft + (u — a) sinh a sin y8 — cosh u sin v + cosh a sin /S = 0, where («, u) are current Cartesian coordinates and 0< y8< 7r. Since the equation is unaltered by a change of sign in both u and a, we shall first study the case in which a ^ 0; and since the equation is unaltered when 7r — v and ir - y8 are written for a and 0, we shall also at first suppose that 0 < /3 <^7r, though many of the results which will be proved when # is an acute angle are still true when [3 is an obtuse angle.
THEORY OF BESSEL FUNCTIONS [CHAP. VIII For brevity, the expression on the left in A) will be called <?(w, v). Since ??0*j1) = sinh a sin ? - sinh u sin v, ou it follows that, when v is given, dcji/du vanishes for only one value of u, and so the equation in u, <j> (% v) - 0, has at most two real roots; and one of these is infinite whenever v is a multiple Of 7T. When 0 < v< tt, we have* <f>(~ 00,1)) =-OO, <f>(+CC ,V) = ~ CO , '(f> (a, v) = cosh a {(» - j3) cos /3 - sin v + sin 0} > 0, and so one root of the equation in u, (j> (u, v) — 0, is less than a and the other is greater than o, both becoming equal when v = /?. By considering the finite root of the equations <? (u, 0) = 0, (f> (u, it) = 0, it is seen that, in each case, this root is less than a, so the larger root tends to + oo as v tends to + 0 or to it - 0, and for values of v just less than 0 or just greater than tt the equation <? (it, v) = 0 has a large negative root. The shape of the curve is therefore roughly as shewn by the continuous lines in Fig. 20. Next consider the configuration when v lies between 0 and — tt. v- ) -r TTi f \ — TTi 0 \ y-2iri Fig. 20. Wh6n v is - ft d(f> (u, v)/du vanishes at u - '- a, and hence 0 (u, - /3) has a minimum value 2 cosh a sin /3 A — /3 cot /3 — a tanh a) at u = — a. There are now two cases to consider according as 1 — /3 cot /3 — a tanh a is (I) positive or (II) negative. * Since d<f> [a, t;)/9u=cosha (cos j3-cosv), and this has the same sign as v-/3, <f> (a, t>) has a minimum value zero at v=j9.
8-61] FUNCTIONS OF LARGE ORDER 265 The domains of values of the complex 7 = a -f i@ for which 1 — ft cot ft — a tanh a is positive (in the strip O^ft^-rr) are numbered 1, 4, 5 in Fig. 21; in the domains numbered 2, 3, 6a, 66, 7 a, 76 the expression is negative;, the cor- corresponding domains for the complex vjz = cosh (a 4- iff) have the same numbers in Fig. 22. 6b TTi —¦—^> 7b la ^ 5 v 3 A V 0 Fig. 21. 4 2 ^_p_-_—-—• 6b 6a Fig. 22. (I) When 1 — ft cot ft - a tanh a is positive, (f> (u, — ft) is essentially positive, so that the curve never crosses the line v = -ft. The only possibility therefore is that the curve after crossing the real axis goes off to — 00 as shewn by the upper dotted curve in Fig. 20. (II) When 1 -/Scot /3- a.tanh a is negative, the equation <?(-a, w) = 0 has no real root between 0 and ft - 2ir, for 50 (— a, v)/dv = cosh a (cos ft — cos v). Therefore <?>(--a, v) has a single maximum at -ft, and its value there is negative, so that $ (— a, v) is negative when v lies between 0 and /3 — 2ir. Also 4>(u,ft— 2-77-) has a maximum at u = a, and its value there is negative, so that the curve $ (u, v) = 0 does not cross v = ft—2ir; hence, after crossing the real axis, the curve must pass off to 00 - iri, as shewn by the dotted curve on the right of Fig. 20. This completes the discussion of the part of the curve associated with &V]) (z) when a > 0, 0 < ft ^ \ir. Next we have to consider what happens to the curve after crossing the line v t= + 7r. Since $ (a, v) — cosh a {(v - /3) cos ft — sin v + sin ft], and the expression on the right is positive when v > ft, the curve never crosses the line u = a; also <j> (u, mr) = (u — a) sinh a sin ft + (mr — ft) cosh a cos ft + cosh a sin ft,
266 THEORY OF BESSEL FUNCTIONS [CHAP. VIII and this is positive when u > a, so that the parts of the curve which go off to infinity on the right must lie as shewn in the north-east corner of Fig. 23. When l-«tanha + Gr-/3)cot/3>0, i.e. when (a, /3) lies in any of the domains numbered 1, 2 and 3 in Fig. 21, it is found that the curve does not cross v = 2nr — /3, and so the curve after crossing v = 7r passes off to — oo + iri as shewn in Fig. 23 by a broken curve. We now have to consider what happens when (a, /3) lies in the domain numbered 6a in Fig. 21. In such circumstances 1 - a tanh a + (tt - ?) cot y8 < 0; and </> (— a, v) has a maximum at v ~ 2ir — fi, the value of <j> (- a, 2tt - /S) being negative. The curve, after crossing v = tt, consequently remains on the right of u = — a until it has got above v «= 2-tt — #. Now $ (— a, v) is increasing in the intervals (/S,27r-y8), Btt + /S, 4tt-/9), D7T + /9, 6-rr-?), let the first of these intervals in which it becomes positive be Then <? (w, 2Mir + 2ir — /3) has a minimum at u — — a, at which its value is positive, and so the curve cannot cross the line v=> 2Mw + 27r-/3; it must therefore go off to infinity on the left, and consequently goes to -oo +BJf+lOri; it cannot go to infinity lower than this, for then the complete carve would meet a horizontal line in more than two points.
i*6l] FUNCTIONS OF LARGE ORDER When (a, ft) is in 6a, the curve consequently goes to infinity at 26' ~oo ¦/here M is the smallest integer for which ] — a tanh a + {(M + 1) tt — ft] cot /? s positive. We can now construct a table of values of the end-points of the contours or SyM (z) and Sv® (z), and thence we can express these integrals in terms o' /vA> (z) and Hv® (z) when (a, ft) lies in the domains numbered 1, 2 and 6a it. Jig. 21; and by suitable reflexions we obtain their values for the rest of th<- •omplete strip in which 0 <ft< nr. The reader should observe that, so far ar he domain 1 is concerned, it does not matter whether ft is acute or obtuse. If M is the smallest integer for which 1 - a tanh a + {(M + l)ir-/3] cot /3 s positive when cot /3 is positive, and if JV is the smallest integer for which 1 - a tanh a - (Ntt + ft) cot ft ^s positive when cot/3 is negative, the tables of values of Svm(z) and Sv® (z xre as follows: Regions 1,3,4 2,6a 5,76 66 7a End-points — oo , oo + ni oo — ni, co -f ni — oo, — oo.+ 2tti -oo - 2Nni,co + ni -oo,oo + BJ/+l)m 2./, B) 2e-»«iJ_v(z) fiAW#,,(i)(8<.-AM) Regions 1,2,5 3, la 4, 66 6a 76 End-points - oo + ni, oo — oo + iri, — oo — ni oo + 2ni, oc -oo + BJ/ + l)»t, oo — oo-f Trt, oo- 2Nni 2 J, («) 2e"Ti /_ „ (z) eMvni ffjp) (zeMri) e-Nnri ffvm (zeNH) From these tables asymptotic expansions of any fundamental system o~ ;ojiutions of Bessel's equation can be constructed when v and z are both arb- rarily large complex numbers, the real part of z being positive. The range o 'alidity of the expansions can be extended to a somewhat wider range of values »f arg z by means of the device used in § 8*42.
268 THEORY OF BESSEL FUNCTIONS [CHAP. VIII The reader will find it interesting to prove that, in the critical case j8=^7r, the contours pass frojn — oo to co 4- nri and from - oc + iri to co , so that the expansions appropriate to the region 1 are valid. Note. The differences between the formulae for the regions 6a and 6b and also for the regions la and lb appear to have been overlooked by Debye, and by Watson, Proc. Royal Soc. xcv. A, A918), p. 91. 8*7. Kapteyn's inequality for Jn (nz). An extension of Carlini's formula (§§ 8'11, 8) to Bessel coefficients in which the argument is complex has been effected by Kapteyn* who has shewn that, when z has any value, real or complex, for which z- — 1 is not a real positive numberf, then znex-p{n\/(l — z* A) \Jn(nz] This formula ia less precise than Carlini's formula because the factor Btt?i) - A —z2)* does not appear in the denominator on the right, but nevertheless the inequality is sufficiently powerful for the purposes for which it is required J. To obtain the inequality, consider the integral formula Jn (nz) = ^—. I t1'1 exp [\nz (t - 1/t)} dt, in which the contour is a circle of radius eu, where u is a positive number to be chosen subsequently. If we write t = eu+{a, we get Jn (nz) = 5- [" exp [n [%z (euei9 - e-"le~i6) -u- id}] dO. AIT J —n Now, if M be the maximum value of | exp [\z (eueie - e-ue~i9) - u - id] | on the contour, it is clear fchat But if z = peia-, where p is positive and a is real, then the real part of \z(euei9 - e~ue-ie) -u~i6 is %p{eu cos(a+ 6) — e~u cos (a - 6)} — u, and this atfain$ its maximum value when tan 6 — — coth u tan a, and its value is then p V(sinh8 u + sin2 a) —' u. * Ann. Set. de VEcole norm sup. C) x. A893), pp. 91—120. f Since both sides of A) are continuous when z approaches the real axis it follows that the inequality is still true when z- -1 ia positive: for such values of z, either sign may be given to the radicals according to the way in which z approaches the cuts. % See Chapter xvn.
8-7] FUNCTIONS OF LARGE ORDER 269 Hence, for all positive values of u, | Jn (npeu) j ^ exp [np «/(sinh2 u + sin2 a) — mo]. We now choose u so that the expression on the right may be as small as possible in order to get the strongest inequality attainable by this method. The expression p V( sinh2 it + sm2 ot) — u has a minimum, qua function of u, when u is chosen to be the positive root of the equation* sinh n cosh u 1 V(sinh2 u + sin2 a) p' With this choice of u it may be proved that 2 V(l — z*) • sinh u cosh u = ± (cosh 2t* — e-ia), and, by taking ^ to be real, it is clear that the positive sign must *be taken in the ambiguity. Hence 2 A + a/A — z2)} sinn u cosh u = em — eM"a, and so log _ i and it is now clear that n (m) | sinh2 u 4- sin2 a sinh u cosh w p V(sinh2 u + sin2 a) — w, J^ exp V(l — ¦s;2)] An interesting consequence of this inequality is that j Jn (nz) | ^ 1 so long as both j z 11$ 1 and !*expVO-_5»): -, To construct the domain in which the last inequality is satisfied, write as before z — peia, and define u by the equation sinh u cosh u 1 V(sinh2 u + sin2 a) p' The previous analysis shews at once that, when z exp then /? \/(sinh2 w 4- sin2 a) — u = 0. * This equation is a quadratic in sinha u with oue positive root.
270 THEORY OF BESSEL FUNCTIONS [CHAP. VIII It follows that 2u sinh 2w' sin2 a = sinh u (u cosh u — sinh u). As u increases from 0 to 1-1997 ..., sin2a increases from 0 to 1 and p de- decreases from 1 to* 0-6627434.... It is then clear that i V*** ffi "" f? 1 + y(l ~ z) inside and on the boundary of an oval curve containing the origin. This curve Fig. 24. The domain in which j fn (nz) | certainly does not exceed unity. is shewn in Fig. 24; it will prove to be of considerable importance in the theory of Kapteyn series (Chapter xvn). When the order of the Bessel function is positive but not restricted to be an integer wo take the contour of integration to be a circle of radius eu terminated by two rays inclined ±ir — arc tan (coth u tan a) to the real axis. If we take [ t | = ev on these rays, we get iSinvjr /"* P i cosh (u+v) — cos 2a cosh (v-u) ~] , x — ' - vv \ dv and so 3inv7r| f- . . ...\ ~V l^exp{-«/('y-M)}rfv| * This value is given by Plummer, Dynamical Astronomy (Cambridge, 1918), p. 47.
CHAPTER IX POLYNOMIALS ASSOCIATED WITH BESSEL FUNCTIONS 9*1. The definition of Neumann's polynomial 0n(t). The object of this chapter is the discussion of certain polynomials which occur in various types of investigations connected with Bessel functions. The first of these polynomials to appear in analysis occurs in Neumann's * investigation of the problem of expanding an arbitrary analytic function f(z) into a series of the form XanJn (z). The function 0n (t), which is now usually called Neumann's polynomial, is defined as the coefficient of enJn (z) in the expansion of l/(t — z) as a series of Bessel coefficients!, so that A) -i- = Jo (*) 00 (t) + 2,7, (*) 0, @ + 2/2 (z) 0, (t) + ... b "~ Z From this definition we shall derive an explicit expression for the function, and it will then appear that the expansion A) is valid whenever \z\< \t\. In order to obtain this expression, assume that | z \ < \ 11 and, after expanding 1/D - z) in ascending powers of z, substitute Schlomilch's series of Bessel coefficients (§27) for each power of z. This procedure gives 1 1 °° z" - _ — _ -I- V _ t-z t sfi t*+1 2" f (a + 2m).(a + ml)! r I 1 ^ ^ () f • ) Assuming for the moment that the repeated series is absolutely convergent J, * Theorie dor Bessel'schen Functional (Leipzig, 18E7), pp. 8—15, 'Hi; sec also Journal fiir Math, lxvii. A867), pp. 310—314. Neumann's procedure, after assuming the expansion A), iH to derive the differential equation which will be given subsequently (§ 9#12) and to solvo it in series. t In anticipation of § 1611, we observe that the expansion of an arbitrary function is obtained by substituting for l/(t - z) in the formula % Of. Pineherle's rather more general investigation, Rcndiconti R. 1st. Lomhardo, B) xv. A882), pp. 224—225.
272 THEORY OF BESSEL FUNCTIONS [CHAP. IX we effect a rearrangement by replacing s by n — 2m, and the rearranged series is a series of Bessel coefficients; we thus get n=l () *> 11=1 {m-O Accordingly the functions On (t) are denned by the equations (8) It is easy to see that D) ««0»@BS-^r and the series terminates before there is any possibility of a denominator factor being zero or negative. We have now to consider the permissibility of rearranging the repeated series for \ftt — z). A sufficient condition is that the series T should be convergent. To prove that this is actually the case, we observe that, by § 2-11 D), we have < 2 D | * |)«+** {exp (i 1 0 Hence _ 1 z | exp (fr ! z P) The absolute convergence of the repeated series is therefore established under the hypothesis that \z\ < \ t j. And so the expansion A) is valid when '^rj<|i|, and the coefficients of the Bessel functions in the expansion are defined by B) and C). It is also easy to establish the uniformity of the convergence of the ex- expansion A) throughout the regions 111 >R, \z\^r, where R >r > 0.
9-1] ASSOCIATED POLYNOMIALS 273 When these inequalities are satisfied, the sum of the moduli of the terms does not exceed J - (s + 2m).(s + m-l)\ (?rI+aw exp (|r2)\ exp (frr2) U2o" w! * "¦ (* + 2m)!' J * S-r '' Since the expression on the right is independent of z and t, the uniformity of the convergence follows from the test of Wcierstrasa. The function 0n(i) was called by Neumann a Bessel function of the second kind*; but this term is now used (cf. §§ 3*53, 3'54) to describe a certain solution of Bessel's equation, and so it has become obsolete as a description of Neumann's function. The function On (t) is a polynomial of degree n + 1 in 1/t, and it is usually called Neumanns polynomial of order n. If the order of the terms in Neumann's polynomial is reversed by writing |n-mor \{n — \) — m for m in B), according as n is even or odd, it is at once found that /k\ n ^ 1 *4 n. (In + m -1)\ , , ~~t + !?+ I6 + t7 ' 0n (t) = 4 J? ^^jyp^ (» odd) These results may be combined in the formula The equations E), F) and G) were given by Neumann. By the methods of § 2'11, it is easily proved that (») I c« 0n {t) U J • (n!) • (| 111)"" exp (J | where | 0 j ^ [exp (? | * |2) - l]/B« - 2). From these formulae it follows that the series %a.nJn(z) 0n(t) is convergent whenever the series 1an(z]t)n is absolutely convergent; and, when z is outside the circle of convergence of the latter series. <tn Jn(z) 0n(t) does not tend to zero as n -+¦ oo , and so the former series does not converge. Again, it is easy to prove that, as n -+¦ oo , en Jn {*) On (t) -~l{l~ ^ + 0 («-) j , * By analogy with the Legendre function of the second kind, Qn (t), which is such that -i-= 2 (in + l) Pn(z)Qn(t). l~Z 71=0 Cf. Modern Analysis, % 15-4. W. B. F. 18
274 THEORY OF BESSEL FUNCTIONS [CHAP. IX and hence it may be shewn* that the points on the circle of convergence at which either series converges f are identical with the points on the circle at which the other series is convergent. It may also be proved that, if either series is uniformly convergent in any domains of values of z and ?, so also is the other series. Since the series on the right of A) is a uniformly convergent series of analytic functions when |z \ < \ t \, it follows by differentiation| that ^ g r where p, q are any positive integers (zero included). It may be convenient to place on record the following expressions: 02 (t) = l/t + 4/i3, 08 (t) = 3/P + 04 (t) = l/t + 16/?» + -192/i8, OB (t) = 5/P + The coefficients in the polynomial 0n (t), for w=0, 1, 2, ... 15, have been calculated by Otti, Bern Mittheilungen, 1898, pp. 4, 5. 9*11. The recurrence formulae satisfied by 0n{t). We shall now obtain the formulae A) (n -1Hn+1@ + (n +1) Qn-^0-gfo^±> On(t) = B) 0^(t)- 0n+1 @ = 20n'(t), C) _01(*)=Oo/@- The first of these was stated by Schlafli, Math. Ann. in. A871), p. 137, and proved by Gegenbauer, Wiener Sitzungsberichte, lxv. B), A872), pp. 33—35, but the other two were proved some years earlier by Neumann, Theorieder BessePschen Functionen (Leipzig, 1867), p. 21. Since early proofs consisted merely of a verification, we shall not repeat them, but give in their place an investigation by which the recurrence for- formulae are derived in a natural manner from the corresponding formulae for Bessel coefficients. Taking \z \ < 11 \, observe that, by § 91 A) and § 2*22 G), 00 CO (t-z)l enJn (z) 0n (t) = 1 = 2 en cos2 \rnr. Jn (z), n=0 »=0 * It is sufficient to use the theorems that, if Zbn is convergent, so also is 2bjn, and that then S6K/n2 is absolutely convergent. t This was pointed out by Pincherle, Bologna Memorie, D) ni. A881—2), p. 160. + Cf. Modern Analysis, § 5*33.
9'11] ASSOCIATED POLYNOMIALS 275 and hence z I enJn (z) On (t) = I enjn E) {tOn (f) - cos* \n-rr) n=0 n=0 = 2 enJn (z) [tOn (t) - COS2 |W7TJ, n=l since tOa(t)= 1. If now we use the recurrence formula for J7i(z) to modify the expression on the right, we get I enJn (z) On (t) = 2 (J^ (*) + /„+, 0I [tOn (t) - cos2 %mr)ln. If we notice that Jn+l(z)[tOn(t) — co82^inr)/n tends to zero as n-*-oo, it is clear on rearrangement that Jo E) {00 («) - *0, @1 + ^ (*) {20, (t) - \tO, (t) +1} a) tOn+'{t) tOn-'{t) + 2n sin2 * Now regard z as a variable, while t remains constant; if the coefficients of all the Beasel functions on the left do not vanish, the first term which does not vanish can be made to exceed the sum of all the others in absolute value, by taking \z\ sufficiently small. Hence all the coefficients vanish identically* and, from this result, formula A) is obvious. To prove B) and C) observe that \dt dzj t — z ' and so, \z\ being less than \t\, we have 2 €»/„ (*) On' (t) + 1 enJn (*) 0n (t) = 0. n - 0 n - 0 By rearranging the series on tbo left we find that 1 enJn E) On' («) = /, («) 00 («) - 2 |^., <5> - /M+1 E>| On (t) n=0 n=l = - Jo (Z) 0, @ - 2 -/„ (Z) @n+1 @ - On-, (*)}. that is to say, ¦/o («) {Oo' (t) + 0, (t)} + 2 Jn(z) [Wn (t) + Ott+l (t) - On_, («)} S 0. On equating to zero the coefficient of J,x(z) on the left, just as in the proof of A), we obtain B) and C). * This is the argument used to prove that, if a convergent power series vanishes identically, then all its coefficients vanish (cf. Modern Analysis, §3-73). The argument ie valid here because the various series of Bessel coefficients converge uniformly throughout a domain containing z = 0. 18—2
276 THBOBY OF BESSEL FUNCTIONS [CHAP. IX By combining A) and B) we at once obtain the equivalent formulae D) ntOn-j (t) - (to2 - 1) 0n (t) = (n - 1) tOn' (t) + n sin2 \n-rr, E) ntOn+1 (t) - (n2 - 1) On (t) = - (n + 1) tOn' (t) + n sin2 \n-K. If & be written for t (d/dt), these formulae become F) (n - 1) (^ + n + 1) 0n (t) = n \tOn^ (t) - sin2 \n-rr), G) (n +1) (^ - n + 1) 0w («) = -» {tOn+1 (t) - sin2 |wtt} . The Neumann polynomial of negative integral order was defined by Schlafli* by the equation (8) 0-(*)-<-)" ?«@- With this definition the formulae A)—G) are valid for all integral values of n. 9*12. The differential equation^ satisfied by On(t). From the recurrence formulae §9'11 F) and G), it is clear that (S + n + 1) O - n +1) On (t) = r (fc + » + 1) {- ntOn+i (t) + naina Jtittj fit = —.- (^ + n + 2) On+1 (t) + n sin2 %nir = — t {tOn (t) — cos2 \n7r) + n sin2 \nir, and consequently On(t) satisfies the differential equation (^ + IJ On (t) + («2 - TO2) On (t) = t COS2 ^UTT + W Sin2 \fiir. It follows that the general solution of the differential equation d'*y 3 dy f n2 —1\ _cos2-|-n7r nsin2|//7r is ^OnW + r^ntf), and so the only solution of A) which is expressible as a terminating series is On{t). It is sometimes convenient to write A) in the form where (neven) i2. (nodd) * Jtf«tfe. ^«n. m. A871), p. 138. t Neumann, Theorie der BesseVschm Funetionen (Leipzig, 1867), p. 13; Journal fur Math. ixvh. A867), p. 314.
9*12, 9'13] ASSOCIATED POLYNOMIALS 277 Another method of constructing the differential equation is bo observe that andso 2 <nJn(z)On(t)= 2 <nri>Jn n=0 „=(, {t-zf ' it-zf^ t-z Now 1 = 2 BnJ2n (z), 2= 2 e2n +! (in +1) Jin +1 (z), k=o n=o and hence t+z=t2 S en,<7n («) ./„ C). 71=0 Therefore ^2Fn,/n (*) 1^1^ ^ +3< - + 1 + *2 - »»| 0n (<) - t*ffn (OJ 350. On equating to zero the coefficient of Jn (z) on the loft-hand aide of this identity, just as in § 9*11, we obtain at once tho differential equation satisfied by 0n(t). 9' 13. Neumanns contour integrals associated with On{z). It has been shewn by Neumann* that, if G be any closed contour, A) I 0m CO 0n (z) dz ~ 0, {in = n and m j= n) J c B) C) f Jn J a where k is the excess of the number of positive circuits of the contour round the origin over the number of negative circuits. The first result is obvious from Cauchy's theorem, because the only singu- singularity of Om{z) On{z) is at the origin, and the residue there is zero. The third result follows in a similar manner; the only pole of the inte- integrand is a simple pole at the origin, and the residue at this point is l/en. To prove the second result, multiply the equations V«y« (*) = 0, Vn [z0n (z)\ - *gn (*) by z0n{z) and Jm {z) respectively,and subtract. If U{z) be written in place of Jm{z) dz Z°n{Z) "dz ' the result of subtracting assumes the form *• TJ'{z)+zU{z) + {m* - n2)zJm{z) On{z) = z3gn{z) Jm{z)> * Theorie der BetseVtchcn Functionen (Leipzig, 1867), p. 19.
278 THEOBY OF BBSSEL FUNCTIONS [CHAP. IX and hence [zU(z)]c+(m*-n*)[ Jm{z)On{z)dz = \ z*gn(z)Jm(z)dz. Jo Jo The integrated part vanishes because U(z) is one-valued, and the integral on the right vanishes because the integrand is analytic for all values of z; and hence we deduce B) when m* ^ n8. Two corollaries, due to Schlafli, Math. Ann. in. A871), p. 138, are that D) The first is obtained by applying B) and C) to the formula § 2-4 A), namely and the second follows by making an obvious change of variable. 9*14. Neumann's integral for On(z). It was stated by Neumann * that a) cm - r|u+V(" l J 0 We shall now prove by induction the equivalent formula B) On (z) = \\ [[t + V(l + t2)}n + {W(l + *2)}n] e~« dtt Jo where a is any angle such that | a + arg z \ < \ir; on writing ? = u/z, the truth of A) will then be manifest. A modification of equation B) is C) On (z) = l ^ *Ve + {-ye-1*6) fi-*«w»» cosh 0d0. Jo To prove B) we observe that ("Mexpta faa expta Jo Jo and so, by using the recurrence formula § 9*11 B), it follows that we may write (• oo exp ia 0B(*)- <f>n(t)e-*dty Jo where D) <?n+1 (t) - 2«^n 00 - ^ (t) = 0, and E) *.(«)« * Theorie der BesseVsehen Functionen (Leipzig, 1867), p. 16; Journal filr Math. lxvu. A867), p. 312.
9*UJ ASSOCIATED POLYNOMIALS 279 The solution of the difference equation D) ia <j>n (t) = A{t + V(«2 + l)}n + B [t - V(l + *2)}n, where A and B are independent of n, though they might be functions of t. The conditions E) shew, however, that A = B = \; and the formula B) is established. This proof was given in a symbolic form by Sonine*, who wrote <j>n (Z>). A/z) where we /°o expia <\>n @ e""zt dt, D standing for (djdz). 0 A completely different investigation of this result is due to Kapteynf, whose analysis is based on the expansion of § 91 A), which we now write in the form When j ^I < I z |, we have if p be so chosen that It follows that (fa 1 , = M S /*•/« (S) \ z .'0 (»=-«, J .' 0 L»= -» We shall wow shew that tho interchange of summation and integration ia justifiable; it will be sufficient to shew that, for any given values of ? and z (such that | f | < | z |), m r n=N-\-\ Jo z can be made arbitrarily small by taking N sufficiently large J ; now amlso (O\ f Jo * Math. Ann. xvi. A880), p. 7. For a similar symbolic investigation aee § 6-1-1 supra. t Ann. Sci. de I'ticole norm. slip. C) x. A893), p. 108. X Cf. Bromwich, Theory of Infinite Seriea, § 176.
280 THEORY OF BBSSEL FUNCTIONS [OHAJP. IX Therefore, since | i j < | z |, we have M r n=N+\ J 0 and the expression on the left can be made arbitrarily small by taking iV sufficiently large when 2 and ? are fixed. Hence, when | ? j < j z |, we have JL I p^l L I where On (z) is defined by the equation e du, and it is easy to see that 0n(z), so denned, is a polynomial in 1/z of degree ?i+ 1. When the integrand is expanded* in powers of z and integrated term by term, it is easy to reconcile this definition of On(z) with the formula § 9'1 D). 9*15. Sonine's investigation of Neumann's integral. An extremely interesting and suggestive investigation of a general type of expansion of l/(a — z) is due to Soninef; from this'general expansion, Neumann's formula (§9'1) with the integral of § 94 can be derived without difficulty. Sonine's general theorem is as follows: Let yjr (w) be an arbitrary function of w; and, if -ty (w) = %, let w = ^ {°°)> so that 41 is the function inverse to ty. Let Zn and An be defined by the equations% Then — = S ZnAn, a — z „=_«, it being assumed that the series on the right is convergent. Suppose that for any given positive value of x, \ w \ > \ -fa («) | on a closed curve C surrounding the origin and the point z, and | w \ < \ ^ (a) \ on a closed * Cf. Hobson, Plane Trigonometry A918), § 264. t Mathematical Collection (Moscow), v. A870), pp. 323—382. Sonine's notation has been modified slightly, but the symbols \p and ^ are his. X This is connected with Laplace's transformation. See Burkhardt, EneyclopSdie der Math. Wiss. ii. (Analysis) A916), pp. 781—784.
9-15,9*16] ASSOCIATED POLYNOMIALS 281 curve c surrounding the origin but not enclosing the point z. Then oo -I ao rm /• 2 ZtlAn = ~ S e^w w dwdx W — = ( Jo () o provided that i? (z) < R (a); and the result is established if it is assumed that the various transformations are permissible. In order to obtain Neumann's expansion, take yjr (¦«/) = -| O - 1/w), 4- (*) = * ± V(*'J + 1), and then >* « n0 Since An + (-)» ^l_rt = ("e--* [{« ± .v/(al2 + l)}n + (-)B {a ± *J(a? + l)}~n] dx, J o we at once obtain Neumann's integral. Sonine notes qj. 328) that Jn (z) ~ {\zYJn!, <rt 0B («) ~ n ! (i«)-», so that the expansion of l/(a —2) converges when |a|<|a|; and in tile later part of hiw memoir he gives further applications of his general expansion. 9'16. The generating function of 0n (s). The series 2 (-)nentn0n(z), which in a generating function associated with 0n(z), does not converge for any value of t except zero. Kaptoyn*, however, haa "summed" the series after the method of Borel, In the following manner: nJ~Y (ntn°n B)=«- + nl, »i (i"-ttin)T(i«)^~t _ v " (w +i) • (n+m) ' P" '"' 2 (»+i).(»+»«)i^-^1 1 » Bm)! ^"/ _ * Nieuw Archiefvoor Wiskunde B), vi. A905), pp. 49—55.
282 THEOBY OF BESSEL FUNCTIONS [CHAP. IX /"> e~udu is convergent so long as A - f) zjt is not negative. There is no great difficulty in verifying that the series 2 ( — )nfntn0n(z) is an asym- ptotic expansion of. the integral for small positive values of t when | arg z | < n, and so the integral may be regarded as the generating function of 0n (z). Kapteyn has built up much of the theory of Neumann's function from this result. 9'17. The inequality of Kapteyn's type for 0n(m). It is possible to deduce from Neumann's integral an inequality satisfied by On{nz) which closely resembles the inequality satisfied by Jn (m) obtained in § 8-7. We have the path of integration being a contour in the w-plane, and so where that value of the radical is taken which gives the integrand with the greater modulus. Now the stationary point of \w + V(w2 + «*)} e~w is V(l — 22), and so A) I On(nz) I < ^ |L±^^pj| (W + W + *•)} 6-1.1 cZti; |, where the path of integration is one for which the integrand is greatest at the stationary point. If a surface of the type indicated in § 8-3 is constructed over the w-plane, the stationary point is the only pass on the surface; and both w = 0 and w — + oo are at a lower level than the pass if Hence, since a contour joining the origin to infinity can be drawn when B) is satisfied, and since the integral involved in A) is convergent with this contour, it follows that, throughout the domain in which B) is satisfied, the inequality is satisfied for some constant value of A ; and this is an inequality of the same character as the inequality of § 8'7.
9*17, 9-2] ASSOCIATED POLYNOMIALS 283 9*2. Gegenbauer's generalisation* of Neumann's polynomial. If we expand zvj{t — z) in ascending powers of z and replace each power of z by the expansion as a series of Bessel functions given in § 5, we find on rearrangement that zv _ « z"+' _ Z 2"+* { 5 (y + < + 2m). T (y + s + m) } the rearrangement has been effected by replacing s by n — 2m, and it presents no greater theoretical difficulties than the corresponding rearrangement in § 9']. We are thus led to consider Gegenbauer's polynomial A1hv(t), defined by the equation (i) AnAi)^ this definition is valid whenever v is not zero or a negative integer; and when | z\< \t\, we have B) .— = 2 An>v(t)Jv+n(z). t — z n^o The reader should have no difficulty in proving the following recurrence formulae: C) ^v + 7l)i1] n + E) (i; F) (i/ + «) ^n+l, „ @ - {v + n + G) AM(*)- * Wiener Sitzungsberichte, lxxiv. B), A877), pp. 124—130.
284 THEOBY OF BESSEL FUNCTIONS [CHAP. IX The differential equation of which AniV(t) is a solution is <8> 5? + where (9) 9nAt)= The general solution of (8) is J.n, „ @ + tv~l c$v+n (t). Of these results, C), D), (8) and (9) are due to Gegenbauer; and he also proved that A0) ~. Jl0+) An, „ @ «« <*i = 2" »* r (j/). (v + n) Gn" (z), where Gnv (z) is the coefficient of an in the expansion of A — 2az + a*)~"; this formula is easily proved by calculating the residue of (izt)m An> v (t) at the origin. The corresponding formula for Neumann's polynomial is 1 rio+) A1) — -. 0H (t) eizt dt = i11 cos {n arc cos z\. The following formulae may also be mentioned : A2) I Am y (z) An v (z) dz = 0, (m = n and m j= n) J c ' A3) f z~" Jv+m(z) An<v{z) dz = 0, (m2^n2) ¦I G f A4) *-" Jv+n (z) An „ (z) dz = 2irik, .' c where G is any closed contour, n — 0,1, 2, ..., and h is the excess of the number of positive circuits over the number of negative circuits of (J round the origin. The first and third of these last results are proved by the method of § 9*13 ; the second is derived from the equations V,+w Jv+m (z) = 0, Vu+n [zl~v An> „ (*)} = s?-* gn> „ (z), whence we find that (m - n) Bv + m + n) I z~v Jv+m (z) An, v (z) dz = I z-~v gn „ (z) Jv+m (z) dz = 0. Jo J c 9'3. Schldfli's polynomial Sn (t). A polynomial closely connected with Neumann's polynomial On (t) was investigated by Schlafli. In view of the greater simplicity of some of its properties, it is frequently convenient to use it rather than Neumann's poly- polynomial.
9-3] ASSOCIATED POLYNOMIALS 285 SchlaflTs definition* of the polynomial is <bt (lit _ im — 1 \\ A) flf»@ = 2 (ll-^L_l)j(^)-n+,W) (n ^ 1} B) S,@-0. On comparing A) with § 9'1 B), we see at once that C) \n Sn (t) = tOn (t) - cos2 \mc. If we substitute for the functions 0n (t) in the recurrence formulae. § 91 A) and B), we find from the former that D) Sn+i (t) + ?n-i (t) - 2nt~l SH (t) = 4* cos'J 1- nir, and from the latter, | (n - l)8n^ (t) - | (n + 1) Sn+1 (t) = nSn' (t) - ntr' 8n (t) - 2t~l cos2 frnr. If we multiply this by 2 and add the result bo D), we got E) Sn_1{t)-Sn+l(t) = 2Sn/(t). The formulae D) and E) may, of course, be proved by elementary algebra by using the definition of Sn(t), without appealing to the properties of Neumann's polynomial. The definition of Schlafli's polynomial of negative order is F) <8Ln(*)-(-)n+1&@, and, with this definition, D) and E) are true for all integral values of n. The interesting formula, pointed out by Schliifli, G) fifM_, CO+ iSll+1 @-40B(«), is easily derived from C) and D). Other forms of the recurrence formulae which may be derived from D) and E) are (8) t8n-i @ - nSn @ - t8n' (t) = 2 cos2 \ wir, (9) tSn+l (t) - nSn (t) + tS.n' (t) = 2 cos91 mr. If we write Sy for t (d/dt), these formulae become A0) (S + n) Sn @ = t$n-i @-2 cos" I nir, A1) (^ - n) Sn @ = - tXn+i @ + 2 Gon*$nir. It follows that (^2 - >r) Sn (t) = t (Sr + 1 - m) Sn_, @ + 2/i cos2 ^nvr = - Pfin (t) + 2t sin2 \ nir + 2//. cos" \ nir, and so *Sn@ is a solution of the differential equation A2) t* -f- + t -j + {t* - ?<a) y = 2« sin21 nir + 2n cos'^ ? nvr. Cot" C^6 • * Math. Ann. in. A871), p. 138.
286 THEORY OF BESSEL FUNCTIONS [CHAP. IX It may be convenient to place on record the following expressions : 5,@-2/*, ?2(?) = 4/?a, St (t) = 2jt + 16/?, S4 (t) = 8/t* + 96/t\ S6 (t) = 2jt + 48/?3 + 768/P, SC) (t) = 12/t2 + 384/f* + 7680/*8: The general descending series, given explicitly by Otti, are _2n 2n(w!»-2') 2n (»s - 22) (it8 - 42) ~t2+ P ~+ t6 - ? 2 (?t2 - I2) 2 (h2 - I2) (w2 - 32) ~1+ ? + i5 +'"" The coefficients in the polynomial Sn (t), for n — 1, 2, ... 12, have been calculated by Otti, Bern Mittheilungen, 1898, pp. 13—14; Otti's formulae are reproduced (with some obvious errors) by Graf and Gubler, Einleitung in die Theorie der BesseVschen Funktionen, n. (Bern, 1900), p. 24. 9*31. Formulae connecting the polynomials of Neumann and Sc/ddfli. We have already encountered two formulae connecting the polynomials of Neumann and Schlafli, namely \n Sn (t) = t0n (t) - cos2 -|?i7T, of which the former is an immediate consequence of the definitions of the functions, and the latter follows from the recurrence formulae. A number of other formulae connecting the two functions are due to Crelier*; they are easily derivable from the formulae already obtained, and we.shall now discuss the more important of them. When we eliminate cos2-|->wr from § 9*3 C) and either § 9*3 (8) or (9), we find that A) Sn_l(t)~Sn'(t) = 2On(t), B) S»+,@+5«'@-2OB@. Next, on summing equations of the type § 9*3E), we find that <D»-D C) #„(<) = -2 2 S'n-2m-i(t) + sin2 imr.SAt), and hence D) Sn (t) + Sn^ (t) = - 2 "?2 S'n.m-i (t) + Sx (t). ¦m=0 * Covyptes Rendw, cxxv. A897), pp. 421—423, 860—863; Bern Mittheilungen, 1807, pp. 61—96.
9*31,9-32] ASSOCIATED POLYNOMIALS 287 Again from § 9'3 G) and E) we have 4 {()„_: (t) + 0B+1 (*)} = ?n~2 @ + %Sn (t) + Sn+i (t) = <A-2 (*) - Sn (t)} - {Sn (t) - Sn+2 («)} 4,Sn (t) so that E) Sn" (t) + Sn (t) = 0n_x @ + 0M+1 (i). This is the most interesting of the formulae obtained by Crelier. Again, on summing formulae of the type of §911 B), we find that < 4G1-1) F) 0n(t) = -2 2 O'^-i(*) + sin?\mr . O1 (t) + cos2\mr.Oo(t), JM=*O and hence G) 0n (t) + 0n-, (t) = - 2 ^ O'n^-x («) + 0, («) + 0Q (t). 9*32. Grafs expression, of Sn (z) as a sum. The peculiar summatory formula A) Sn(z) = 7T i [Jn(s)Ym(s)-Jm(z)Yn(z)} m— —n was stated by Graf* in 189&, fchc proof being supplied later in Graf and Gubler's treatisef. This formula is most readily proved by induction; i% is obviously true when ?i = 0, and also, by §3*63A2), when n—\. If now the sum on the right be denoted temporarily by <j>n{z), it is clear that <?n+1 (z) + <?„_, (z) - Bn/s) <fin {z) m+1 n+l = irjn+1(z) s rm(«)-«-yw(«) 2 ./„(*) m=-7i-l m=-?i-l + ^M(*) ^ Y^W-irY^z) ^ Jm(z) m=— n+l m= — n+l -{2mriz)Jn{z) I Fm(^) + Bn7r/^) Yn(z) 2 JwD m=—w »t=-n Now modify the summations on the right by suppressing or inserting terms at the beginning and end so that all the summations run from -nton; and we then see that the complete coefficients of the sums XJm(z) and 2 Ym(z) vanish. It follows that <f>n+i (*) + (f>n-i (*) - Bn/s) <?„ 0) - wJW (*) (y.+» (*) + F_n_, (^} - 7T Fw (Z) [Jn+1 (Z) + /.n-x (Z)\ - »/»., (*) {Yn(») + Y_n (z)} + irYn^ («) {./w (*) + JLn (^)j = - 7T {1 + (- 1)W1 {Jn-i (Z) Yn (Z) - Jn {Z) IV* (*)} = 4^~1COS2^W7T, by §363A2); and so 4>nB) satisfies the recurrence formula which is satisfied by Sn(z), and the induction that <$>n{z) = Sn(z) is evident. * Math. Ann. XLin. A893), p. 138. t Einleitung in die Theorie der BesseVschen Funklionen, n. (Bern, 1900), pp. 34—41.
288 THEORY OF BESSEL FUNCTIONS [CHAP. IX 9*33. Crelier's integral for 8n(z). If we take the formula §914B), namely r oo exp ta On («) = * [{t + V(l + *2)jw +{t~ V(l + *2)}Jl] «' dt, and integrate by parts, we find that On(z) = i JITT ft + Hence it follows that This equation, which was given by Schlafli, Math. Ann. in. A871), p. 146, in the form B) Sn(z) = Jo {«•*-(-)»« «fl}e 'i-nhfldfl, is fundamental in Crelier'a researches*, of which we shall now give an outline. We write temporarily and then so that T rn — """ /yi ; rp > ¦in J- 71/ J n—l and therefore the continued fraction having n elements. It follows that T?H.i/7'n is the quotient of-two simple continuants-\ so that Tn+i __ iTB^ 2^, ...,2^^ Tn the suffixes w, n — 1 denoting the number of elements in the continuants. It follows that^: Tn(KBt)n-\ is independent of n; and since yi = 2V(l + <»), we have * Comptet Rendus, cxxv. A897), pp. 421—423, 860—863; Bern Mittheilunyen, 1897, pp. 61—96. t Chrystal, Algebra, n. A900), pp. 494—502. $ Since all the elements of the continuant are the same, the continuant may be expressed by this abbreviated notation.
9-33,9*34] ASSOCIATED POLYNOMIALS 289 and hence r« oxpio C) ?*(*)= 2 K{2t)n^e~^dt. J n From this result it is possible to obtain all the recurrence formulae for 8n(z) by using properties of continuants. 9*34. Schl&fli's expansion of Sn(t + z) as a series of Bessel coefficients. We shall now obtain the result due to Schlafli* that, when \z\<\t , Sn (t + z) can be expanded in the form A) 8n(t + z)= 2 Sn_m{t)Jm(z). VI ~ — 00 The simplest method of establishing this formula for positive values of n is by induction f. It is evidently true when n — 0, for then both sides vanish : when n — 1, the expression on the right is equal to Sx(t)J0(z)+ 2 [S^ 2 111 = 1 = 2 2 *mOm(t)Jm(-z) = 2/0 + z) = 8, (t + z), by §9-1 (I) and §9-3 G). Now, if we assume the truth of A) for Schlafli's polynomials of ordens 0, 1, 2, ... n, we have Sn+l (t + z) = Sn^ (t + z)~ 2Sn' (t + z) = t 6'n_.,rt_x (t)Jm(z)~2 t S'n^n(t)Jm(z) = ? /SfB+1_m(O«/mCs), Mi = — r> and the induction is established; to obtain the second line in the analysis, we have used the obvious result that * Math. Ann. in. A871), pp. 139—141; the examination of the convergence of the aeries is left to the reader (cf. § 9-1). + The extension to negative values of n follows on the proof for positive value?, by § 9-3 (G). W. B. F. 19
290 THEOBY OF BESSEL FUNCTIONS [CHAP. IX The expansion was obtained by Schlafli by expanding every term on the right of (I) in ascending powers of z and descending powers of t. The investigation given hero is due to Sonine, Math. Ann. xvi. A880), p. 7; Sonine's investigation was concerned with a more general class of functions than Schlafli's polynomial, known as hemi-cylindrical functions (§ 10-8). When we make use of equation § 9*3 G), it is clear that, when | z \ < 11 \, B) On(t + z)= ? On.m{t)Jm{z). TO = —oo This was proved directly by Gegenbauer, Wiener Sitzungsberichte, lxvi. B), A872), pp. 220—223, who expanded 0n (t+z) in ascending powers of z by Taylor's theorem, used the obvious formula [cf. § 9*11 B)] and rearranged the resulting double series. It is easy to deduce Grafs* results (valid when [ z I < i t ]), D) Sn{t-z)= 1 8n+m(t)Jm(z), E) 0H(«-*)- 5 OW,(«)J.(«). 9-4. TAe definition of Neumanns polynomial ?ln (t). . The problem of expanding an arbitrary even analytic function into a, series of squares of Bessel coefficients was suggested to Neumannf by the formulae of § 2*72, which express any even power of z as a series of this type. The preliminary expansion, corresponding to the expansion of l/(t — z) given in § 91, is the expansion of l/(?2—.z2); and the function D.n (t) will be defined as the coefficient of en Jn{z) in the expansion of \j{tl — 22), so that A) n~Z~* = ^°2 (*) ^° (*)+ '^i ^) ^! (*)+ 2^2 (*) ^MO + ¦ • • = i enjn2B)nw(i). To obtain an explicit expression for Cln(t), take xz\ < \t |, and, after ex- expanding l/(?2 — z2) in ascending powers of z, substitute for each power of z the * Math. Ann. xliii. A893), pp. 141—142; see also Epstein, Die vier Rechnung<operationen mit BesseVschen Functionen (Bern, 1894). [Juhrbuch liber die Fortschritte der Math. 1893—1891, pp. 845—846.] t Leipziger Berichte, xxi. A869), pp. 221—256. [Math. Ann. in. A871), pp. 581—610.]
9-4] ASSOCIATED POLYNOMIALS 291 series of squares of Bessel coefficients given by Neumann (§ 2*72). As in § 91, we have 1 ? z™ 5=0 1 2 T " t/i! fl 4- when we rearrange the series by writing n — s for in; this rearrangement presents no greater theoretical difficulties than the corresponding rearrange- rearrangement in § 9. Accordingly the function Cln @ is defined by the equations C) O0 @ = l/?2. On reversing the order of the terms in B) we rind that D) n.(o- i I n^ - V-fe"^!1' ¦ (» > i) while, if B) be written out in full, it assumes the form .2.3 4NaD?r-22)Dn--4") . _2- _1_ 4 t* 4.5.0 Also 2-n(n!K I 2n where 2n ' * 2hBh-2) Bn-l)Bw-3)Bn-5) nB«-2)B»-4) ' ^ m 2»Bw-2j...2 ' Since 0 < ©2n< 1, it is easy to shew by the methods of § 211 that (8) j en Hn (t) | < 2-'1111-2»-" (n!)'J exp (| and, when n > 0, (9) ennn (t) = 2"" «-^-'J («!)• A + tf), where | 6\ $ (exp|i|2 - l}/B»i- 1). 19—2
292 THEOBY O!F BESSEL FUNCTIONS [CHAP. IX By reasoning similar to that given at the end of § 9*1, it is easy to shew that the domains of convergence of the series 1.an Jn* (z) Xln (t) and %an (zftf" are the same. The reader should have no difficulty in verifying the curious formula, due to Kapteyn*, 91. The recurrence formulae for Hn (t). The formulae corresponding to § 9*11 B) and C) are B) C) B/0 n; (t) = - 2n, («) + 2n0 (*). There seems to be no simple analogue of § 9*11 A). The method by which Neumannf obtained these formulae is that described in § 9'11. Take the fundamental expansion § 9*4 A), and observe that and that, by Hansen's expansion of § 2'5, 2Ji(m)J,'(*)--z 2 {J*n-i{z)-<K+i(z)\ln. n = l We find by differentiations with regard to t, and with regard to z, that n=0 - *aJ = 2 Jo (*) Jo' (z) Ho («) + « 2 (JV: (*) - JVi (*)} ^n (*) On comparing these results, it is clear that *-> t enjn>(z)nn'(t)+ 2 (jrv,(*)-^a»+i(«)}.{«n(*)-/ On selecting the coefficient of Jna(z) on the left and equating it to zero (cf. § 91), we at once obtain the three stated formulae. * Ann. Sei. de I'Acole norm, xup. (8) x. A893), p. 111. t Leipziger Beriehte, xxi. A869), p. 251. [Math. Aim. in. A871), p. 606.]
9*41, 9*5] ASSOCIATED POLYNOMIALS 293 9*5. Gegenbauer's generalisation of Neumanns polynomial On (?). If we expand z** j{t — z) in ascending powers of z and replace each power of z by its expansion as a series of products of Bessel functions given in § 5, we find on rearrangement (by replacing s by n — 2m) that efl -{- V oo ~u. ¦+• v 4- & _ 5? ^ ) ? ¦*¦ \A" i 2' «r<) i8+1 (»i=o r(/ti + j/+s+l) w! x 00 ^> ?h— am + i (At+1/ + n) r(fj. + bi-m+l)V (v + In -m + l)T(fi + v + n - wt) wi! r((tt + i/ 4- n — 2//i + 1 j it is supposed that | 21 < 111, and then the rearrangement presents no greater theoretical difficulties than the corresponding rearrangement of § 9'1. We consequently are led to consider the polynomial Bn.ylK)V(t)t denned by the equation ~^r.,-ln — m + l)V(v + %H—m + \)r(fjL + v + n — m)n .wn X ,Mt0 w !~T (/* + w + w - 2//i + 1) ^' This polynomial waa invostigated by Gegenbauer*; it satisfies various recurrence formulae, none ol which arc of a simple character. It may be noted that The following generalisations of Gogenbauer's formulae are worth placing on record. They are obtained by expanding the Bessel functions in ascending series and calculating the residues. ] <o-f-) <3> .A '"' D) 5^-.|"'+)r»./.B«sin x J<\(- w,^ + 1, fM + v + n; %f* + ±v + {, \fi + \v + 1; sina In the special case in which (jl^v, this reduces to E) This formula may be still further specialised by taking </> equal to \ir or ^tt. ¦ Wiener Sitzungsberiehle, lxxv. B), A877), pp. 218—222.
294 THEORY OF BBSSEL FUNCTIONS [CHAP. I 9*6. The genesis of Lommel's* polynomial Rm<„ (z). The recurrence formula may obviously be used to express Jv+m(z) linearly in terms of Jv{z) anc Jv~i(z); and the coefficients in this Hnear relation are polynomials in \\: which are known as Lommel's polynomials. We proceed to shew how t< obtain explicit expressions for them. The result of eliminating Jv+l (z), Jv+i (z),... Jv+m-i (z) from the system o'. equations Jv+p+i (z) - B (v + p)l*\ Jv+P (z) + Jy+p-i (z) = 0, (p = 0,1, ... to - 1 is easily seen to be l), 1, 0 0, 1, -22-1(i/ + m-2), 0 0, 0, 1, 0 0, 0, 0, 1 J?{z), 0, 0, -fc-^+ Jv-l(z)-Bpjz)Jv(z), 0, 0, 1 By expanding in cofactors of the first column, we see that the cofactor o" Jv+m (z) is unity; and the cofactor of (—)w*~1 /„ (z) is 1, o, 1, -22 (v+m-3K o, o, o, 0 0 0 0, 0, 0, -2«-1(i' + l), 1 0, 0, 0, " 1, -<2z-1v The cofactor of (—)m~l Jv^ B) is this determinant modified by the suppressioi of the last row and column. The cofactor of (—)m~lJv(z) is denoted by the symbol (—)m Rm>„ (z); anc Rm,v(z)> thus denned, is called Lommel's polynomial. It is of degree m in 1/r and it is also of degree m in v. The effect of suppressing the last row and column of the determinant by which Rm>v(z) is defined is equivalent to increasing v and diminishiag m b} unity ; and so the cofactor of (-)9M~1 Jv-.x (z) is (-)m~1 Rm_it,+1 (z). Hence it follows that Jv+m (z) ~ J* (*) Rm, v (z) + J*-i (z) Rm-h v+, (z) = 0, * Math. Ann. iv. A871), pp. 108—116.
9-6,9*61] ASSOCIATED POLYNOMIALS 295 that is to say A) Jv+m 0)« Jv (z) Rm, „ (z) - /„_! (z) Rm-it v+1 (z). It is easy to see that* Rm>v(z) is the numerator of the last convergent of the continued fraction (v ± m — \\ The function Rm>v{z) was defined by Lotnmel by means of equation A). He then derived an explicit expression for the coefficients in the polynomial by a somewhat elaborate induction; it is, however, simpler to determine the coefficients by using the series for the product of two Bessel functions in the way which will be explained in § 9'61. It had been observed by Bessel, Berliner Abh., 1824, p. 32, that, in consequence- of the recurrence formulae, polynomials An.l (z), #n_i (z) exist such that where [cf. § 92 (8)] It should be noticed that Graft nnd CrelierJ use a notation which differs from Lommel's notation ; they writo equation A) in the form 9" 61. The series for Lommel's polynomial. It is easy to see that {—)mJ-v-m {z), qua function of the integer m, satisfies the same recurrence formulae as Jv+m(z)\ and hencu the analysis of § 9"E also shews that A) (~)m/_,_M(*) =./_,(*)Rm>v(z) + J_VH(*) /iw_lir+I (z). Multiply this equation by ./„_, (z) and § 9*6 A) by J-v+l (z), and add the results. It follows that B) Jv+m (z) J_,+l (z) + (-)- /_v_m («) /r_, (*) = Rm<v{z) \J,(b) J-v+l (z) + J_v(z)./,_, (z)} * Of. Chrystal, Algebra, n. A900), p. 502. t Ann. di Mat. B) xxm. A896), pp. 45—05; Einleitung in die Theoric tier BeaaeVtclien Funk- tionen, n. (Bern, 1900), pp. 98—109. t Ann. di Mat. B) xxiv. A896), pp. 131—163.
296 THEORY OF BESSEL FUNCTIONS [CHAP. IX by § 3 G). But, by § 51, we have (A T t,\-l (-F( +>n{Z)J-v^{Z)~ Z\r( + fi-o n! r(- p - w + n + 1) r (v + n) when we replace n in the last summation by m + p + l. Now it is clear that (m+p+l)\ p\ (m + p + iji! p\ ' and so, when we combine the series for the products of the Bessel functions, we find that 2 sin VTT n . , '» (~)m+n(_ m . n\ (X2\-m+m-i sin vk <*• V nt() 71! (ill - inj\ V(v + 71) ' the terms for which n>\m vanish on account of the presence of the factor (—m + n)n in the numerator. When v is not an integer, we infer that n=o >i! (?/i - 2n)! F (V + w) n=0 r(i/ + ?i) x^ 7 But the original definition of Rm<v(z), by means of a determinant, shews that R,^(z) is a continuous function of i, for all values of v, integral or not • and so, by an obvious limiting process, we infer that C) is a valid expression' tor Rm<v{z) even when „ is an integer. When „ is a negative integer it may be necessary to replace the quotient *> t™ -JL> bv r-V" ^ W-T w + !> r(i;+W) '^ rT-i;-«iT#7Ti) in part of the series. The herie,, C; wa.s giveu by Lommel, Math. Ann. iv. A871), pp. 108-111- a An interesting result, depending on the equivalence of the quotients just mentioned, was first noticed by Graf*, namely that D) R Ann. di Mat. B) xxm. A895), p. 56.
9*62] ASSOCIATED POLYNOMIALS 297 In the notation of Pochhammer (cf. §§ 4*4, 4*42), we have E) B,w>v{z) = {v)m{\z)-m.J\{\-lm, -|m; v,-m, \-v-m; -*»). Since RM,v{z)/z is a linear combination of products of cylinder functions of orders v + hi and v — 1, it follows from § 5'4 that it is annihilated by the operator O4 - 2 {(w + in)- + (v - l)-j ^J + {0 + m)B ~(v-l )8]-] + 4s" (V + 3^ + 2) ; where <& = z(d/dz); and so Rwv(z) is a solution of the differential equation F) [O + w) O + 2v + vi - 2) O - 2i/ - »i) (S - wt - 2)] y An oiiuatiou equivalent to this was stated by llurwitz, Math. Ann. xxxiil. A889), p. 251; ami a lengthy proof of it waa given by Nielsen, ^1?^?^. di Mat. C) VI. A901), pp. 332—334; a .simple proof, differing from the proof just given, may be obtained from formula E). 9'62. Various properties of Loinmel's polynomial. We proceed to enumerate .some theorems concerning Rmt,(z), which were published by- Lomiwel in his memoir of 1871. In the first place, § (J'u' A) holds if the Bessel functions are replaced by any other functions satisfying the same recurrence formulae; and, in particular, A) Yv+m{z) = K(z) RMtv (*)- F,..., (*) Rm.hv+1 (z), whence it follows that B) Y,,,.M B) /„_., (s) - JVTin B) }•;_, (z) = R,,,,v{z) 1 Yv{z)./,_, E) - Jv(z) F,,_1(.2))= - mm>v(z)l(irz). Next, in §9"C1B), takevn to be an even integer; replace in by 2m, and v by v - vi. The equation then becomes C) Jv+m{z)Jm+,_v(z) + «/_„_„, (*),/_,„_,+„(*) = 2 (-) sin vir. Jttm,?^,i(e)/(irz), and, in the .special case v—\, we get D) J',^.i (z) + J*- w_i {z) = 2 (-)« iiam, i,,H («)/M, that is to say This is the special case of the asymptotic, expansion of § 751 when the order is half of an odd integer. In particular, we have F) 225
298 THEORY OF BESSEL FUNCTIONS [CHAP. IX Formula E) was published in 1870 by Lommel*, who derived it at that time by a direct multiplication of the expansions (§ 3'4) followed by a somewhat lengthy induction to determine the coefficients in the product. As special cases of § 9'6 A) and § 9-61 A), we have 2\* . ^ . , / 2 B \^ / 2 \^ —) sinz.Rmii(z)~ I —) cos*.2^,9(e), TTZ/ XnZ) (-r J-m-i (*) = [^f cos *. Rm> i (*) + (j^)* sin z. Rm By squaring and adding we deduce from D) thatf (8) R\n,i(z) +J?Ml|W»(-r^H.W Finally, if, in § 9*61 B), we replace m by the odd integer 2m + 1 and then replace v by v — m, we get (9) Jv+m+1 (z) J_v+m+1 [z) — J_|/_m_1 {z) Jv.-m-\ \z) = 2 (-)m sin virRzn^v_m (*)/(ir*). An interesting result, pointed out by Nielsen, .4?mi. di Mat. C) v. A901), p. 23, is that n if we have any identity of the type 2 >fm (z) Jv + TO (z) = 0, where the functions fm (z) are m=0 algebraic in z, we can at once infer the two identities S /mCL,,(#0, 2 AWClhiW^O, m=o m=0 by writing the postulated identity in the form 2 /„ (*) {Jv (z) Hm, v iz) - Jv-1 (?) An-i, ,+1 (*)}s0, and observing that, by § 474 combined with § 3*2 C), the quotient Jv-i(z)lJv («) is not an algebraic function. Nielsen points out in this memoir, and its sequel, ibid. C) vi. A901), pp. 331—340, that this result leads to many interesting expansions in series of Lommel's polynomials; some of these formulae will be found in his Handbuch der Theorie der Cylinderfunktionen (Leipzig, 1904), but they do not seem to be of .sufficient practical importance to justify their insertion here. 9*63. Recurrence formulae for LommeVs 'polynomial. In the fundamental formula replace m and v by m +1 and v — 1; on comparing the two expressions for Jp+m (z), we see that * Math. Ann. n. A870), pp. 627—632. t This result was obtained by Lommel, Math. Ann. iv. A871), pp. 115—116.
9*63] ASSOCIATED POLYNOMIALS 299 Divide by /„_! (z), which is not identically zero, and it is apparent that A) Rmr*.*i 2(l71} To obtain another recurrence formula, we replace m in § 92 B) by m + 1 and m — 1, and use the recurrence formula connecting Bessel functions of orders v + m — 1, v + m and v + m + l; it is then seen that \&) J^ni-l v \z) + -"-m+i i/ B) = - -",rt v \z)> and hence, by combining A) and B), Again, write § 92 B) in the form — Z~m~* Rm< „ B) = [Z~"~m Jv+m \z)\ \Zv~l J „_! (z) j — [z~"~m Yy+m \Z)] \ZV 1 */t^-i \Z)\, and differentiate it. We deduce that D) R\n> „ (s) = /2m, „ (z) + Rm+lt v_j (*) - 2i!w+li „ («), z and so, by C), A) and B), /k\ 7?' (A — — 7—Ti (z\4-'R (tA—R (A F) ii!',n „ (z) = —-+-OT ^ „ (*) - /i,n_1,+, (z) - Rm+l v (z), G) Rm,p(z)=- V m~ Rm,v{z)+Rm+\,*-i{*) + -Rm-\,v{*)' The majority of these formulae were given by Lomrnel, Math. Ann. IV. A871), pp. 113— 116, but F) is duo to Nielsen, Ann. di Mat. C) vi. A901), p. 332; formula B) haa boon UHod by Porter, Annals of Math. B) in. A901), p. EE, in discuHsing the zoroa of llm< v («)• It is evident that B) may be uaod to define Rmv(z), when bho parameter m is zero or a negative integer; thus, if B) is to hold for all integral valuew of wi, we find in succession from the formulae jl> / 4i/(j» + 1) „ . . 2v that and hence generally, by induction, (9) R v (z) = (—)W~J Rrn-2 v(z)- This formula was given by Graf, Ann. di Mat. B) xxni. A895), p. 50. If we compare (9) with Graf's other formula, § 9*61 D), we find that When the functions of negative parameter are defined by equation (9), all the formulae A)—G) are true for negative as well as positive values of m.
300 THEORY OF BESSEL FUNCTIONS [CHAP. IX 9'64. Three-term relations connecting Lommel polynomials. It is possible to deduce from the recurrence formulae a class of relations Avhich has been discussed by Crelier*. The relations were obtained by Crelier from the theory of continued fractions. First observe that § 9*63 B) shews that Jv+m (z) and Rm„ (z), qua functions of m, satisfy precisely the same recurrence'formula connecting three contiguous functions; and so a repetition of the arguments of § 9*6 (modified by replacing the Bessel functions by the appropriate Lommel polynomials) shews that A) J^m+n,v \z) — -t^m, v\z) Rn,v+m \Z) ~ R-m—\v \z) -"-«—i.v+m+i \Z)- Next in §98B) replace m by m — 1 and v by v+l, and eliminate 2 (in + v)\z from the two equations; it is then seen that Rm, v iz) Rm, v+i (z) — i?m+i, v (?) Rm-i, v+i (z) = Rm-\, v (z) ¦fim-l.t+i (z) — Rm, v (z) Rm-1, v+\ (z), and so the value of the function on the left is unaffected by changing m into m — 1. It is consequently independent of m; and, since its value when m — 0 is unity, we have Crelier's formula B) Rm>v (z) Rm< u+1 (z) —'Rm+i,v (z) Rm-i, *+i (z) = 1) a result essentially due to Bessel (cf. §96) in the special case v = 0. More generally, if in § 9*63 B) we had replaced m by m — n and v by v -f n, we should have similarly found that ¦"wi, v \Z) J^"m— n+i,v+n \Z) J-hn+i,» \Z) -t^m—n^+h \Z) — -tlm—i, v \Z) Hrn—n,v-\-n \Z) J-hn,v \Z) tlm—n—i, v+n \Z), and so the value of the function on the left is unaffected by changing m into m — 1. It is consequently independent of m ; and since its value when m = n is Rr-i,A*), we find from § 93 A0) that C) Kmtv \Z) Hm—n+i,v+n \Z) Rtn+i,v \z).Mm—n,v+n \Z) — -K"tu-i,v \2)> a result given in a different form by Lommel+. Replace m and n by m — 1 and n + 1 in this equation, and it is found that D) Hm-.itV\Z) tiyn—n—\, v+n+\ \Z) — ¦tlm,v \z) ¦K"m-n—2,v+n+i \z) == -t^n,v \Z)- If we rewrite this equation with p in place of n and eliminate Rm^1>v(z) be- between the two equations, we see that Rn, v (z) Rm-p-i,v+p+\ (z) — Rp< „ (z) Rm-n_h v+n+1 (z) — Rm,v\Z)[Rm—p-2,v+p+i \z) Rm-n-i,v+n+\ \z) — Rra—n-i,v+n+i \Z) R7n—p—ilV+p+i \z)] — Rm,v \Z) i^n-p—\,v+p+i \z)y by C). If we transform the second factor of each term by means of § 9'63 A0), we obtain Crelier's result (loc. cit. p. 143), E) Rn>v (z) JBp_m_1) v+m+i (z) — RPt v (z) Rll—m-h v+m+i (z) ~ -^m,c \z) -"jj—n—\,v+n+\ \z)- * Ann. di Mat. B) xxiv. A896), p. 136 et seq. f M"th> Ann- IV-
9-64] ASSOCIATED POLYNOMIALS 301 This is the most general linear relation of the types considered by Crelier; it connects any three polynomials Rm<v(z), Rni/(z), Rp>v{z) which have the same parameter v and the same argument z. The formula may be written more symmetrically (o) J^n,v \?) Rp-m~i,v+m+i \z) + Rp,v \z) Rm~n-\,v+n+i \z) "T -tim, v \z) -»''«—p—\, v+p+i \z) = "> that is to say 0) . 2 Rn,v(z)Rp-m-i,i>+m+i(z) = 0- in, n, p A similar result may be obtained which connects any three Bessel functions whose orders differ by integers. If we eliminate Jv+m-\ (z) between the equations* )" v-\-n \z) ~ " v+m \z) -"ji—»h,h-»i \z) " v+m—l \z) ¦^/n—m—x,v-i-m+i \z)i "v+p \z) ~ *Jv+m \z) ¦lip—m,i>+iii \z) ~~ J v+m—i \z) -^p—m-i.v+m+i \z)> we find that " v+n \Z) ?*>p—m—\,»+m+\ \z) ~~ •'v+p \z) J?,i—>n—\,i>+m+i \z) ~ " v+m \z) l^n—7n,v+in \z) l*p—in—i, v J-?/t-t-i \z) -"^i—m, *+m \z) ^*?i—w—i,?H-m+i \z)\ — Jv+m \z) J^p-n-^.v+n+i \z) \ the last expression is obtained from a special case of E) derived by replacing m, n, p, v by 0, n — m, p — m, v -f- m respectively. It follows that (8) ? Jv+n (z) Rp-m-i, v+m+i (z) = 0, m, n, p and obviously we can prove the more general equation (9) S f^+»U)^p-m-i.v+fn4i(^) = 0. where *2f denotea any cylinder function. The last two forrnulao aecin nevor to have, licon previously statod explicitly, though Graf and Gnhlcr hint at the. oxi.stonoc of sucli oqiiation.s, h'inleitunf/ in die. Theorie dor Besset'scheri Funktionen, n. (JJcni, 1!)(K)), jtp. 108, 105). [Note. If we eliminate «^_i B) from the equations and use B) to nininlify the resulting equation, wo find that •A-B)= -'A. + m(z) /'»-!,nl (?)+./^ + M_iB) /("„,-!, •-+ l(^), and ho, replacing «< hy v-w, wo have. «/v_„, B) = - ./„ (z) Rm _ jj, v _ m M C) + ./v _ ! (c) Rm - lt v_m + 1 B)- By using § 9-G3 A0), we deduce that ./„-,„ {*)~>h (z) li.w,„ (z) -•/,.-, (s) A'.-,,,-,, n . B), that is to say that the equation $ 9fj A), which has hitherto beon ewnsidored only for positive values of the parameter m, in atill true for negative values.] * It ifl supposed temporarily that m is tho Hinallest of the intern in, m, j); but since the final result is symmetrical, this restriction may be removed. See also the note at the end of tho Hootion.
302 THEORY OF BESSEL FUNCTIONS [CHAP. IX 9*65. Hurwitz' limit of a Lommel polynomial. We shall now prove that A) J1* -J(*> This result was applied by Hurwitz, J/a^A. Ann. xxxin. A889), pp. 250—252, to discuss the reality of the zeros of Jv{z) when v has an assigned real value (§ 15-27). It has also been examined by Graf, Ann. di Mat. B) xxin. A895), pp. 49—52, and by Crelier, Bern Mittheilungen, 1897, pp. 92—96. From 19-61 C) we have y (m-2n)! T(» + m+1)" Now write (m-n)\T(u + m-n + 1) . (m-2n)\V(v + m + l) ~tf^m>n)> so that dim n)= v ' ; (v + m)(v + m-l) ...(v + m-n + l)' If now N be the greatest integer contained in \v\, then each factor in the numerator of 0(m, ri) is numerically less than the corresponding factor in the denominator, provided that n > N. Hence, when n > N, and m > 2N, \6(m,n}\<l, while, when n has any fixed value, lim 6 (m, n) = 1. Since 2 ¦ fJ(* } is absolutely convergent, it follows from Tannery's theorem* that 7T and the theorem of Hurwitz is established. 00 ( — )n (iz)v + 2n Again, since the convergence of 2 - ,¦ ¦¦ r is uniform in any bounded domain t o«! T{v + n+\) J of values of z (by the test due to Weierstrass), it follows that the convergence of to its limit is also uniform in any bounded domain of values of z. * Cf. Bromwich, Theory of Infinite Series, § 49. t An arbitrarily small region of which the origin is an internal point must obviously be excluded from this domain when E (v) ^ 0.
9*65, 9'7] ASSOCIATED POLYNOMIALS 303 From the theorem of Hurwitz it is easy to derive an infinite continued fraction for /„_] (z)/Jv (z). For, when </„ [z) ^ 0, we have ~f-^- = lira = lim L^-l/^-^-, by § 9'63 A). On carrying out the process of reduction and noticing that we find that iW (>).,_.__ 1 I 1_ iC,,+, (^) 2 (i/ + 1) z-1 - ~i(v + 2) s-1- ... - 2 (./+ »i) i-- ' and hence ,2\ •/.^.^)=9^-1 1 . l . K) Jv{z) 2 (i/+I)*-2A'+ 2) *-'-... Thi.s procedure avoids the necessity of proving directly that, when «i-*-oo, the lust element of the continued fraction ,/, B) 2 (v+1 )i::l -... may bo neglected; the method is due to (huf, Ann. di Mat. (~1) xxni. A Ht)T>), p. .">2. 9*7. The modified notation for IjOiumel polynomials. In order to discuss properties of the zeros of Lonnnel polynomials, it i.s convenient to follow Hurwitz by making a change in the notation, for the reason that Lomrnel polynomials contain only alternate powers of the variable. Accordingly we define the modified Lonnnel polynomial </,„,,{z) by the equation* <*'* p (-)» I1 (r + wi - u + I) ^ A) </lll<v{z)-^m-n^n r(|/ + n+l) .so that B) ^l,,H^)=(br"ii/M,1-({4 By making the requisite changes in notation in §§ (Witt, 9"(>4, the i'(>ader will easily obtain the following formulae: C) gm+l%?(z) = (p + vl+\)gm,,(z) -zgm.,,„(z), [§ i)'O3 B)] (*) //m+i.,-1 («) = vgm.Az) - Zf/m-i,^! B), [§ 9'()S A)] E) ^--I^{*|l^,(*)H^m-...(*)+^,,.H1 >-!(*). [§» («) *w« ^ l^" flr«,, (*)} = ^h-,,-! (*) -«/*+,,, (z), [§ 9-68 D)] [A special case of § 9'64 E).J * This notation differs in unimportant detailn from the notation used by llurwit/,.
304 THEORY OF BESSEL FUNCTIONS [CHAP. IX These results will be required in the sequel; it will not be necessary to write down the analogues of all the other formulae of §§ 9'6—964. The result of eliminating alternate functions from the system C) is of some importance. The eliminant is (v + m) gm+2i v (z) = cm (z) gm> „ (z) - (v + m + 2) z*gm_^ „ (z), where cin (z) = (v + m + 1) {(v + m)(v + m + 2) — 2z\, We thus obtain the set of equations: ' 0 + 2)gitV (z) = c%(z)g%v(z) -(v + 4>) z*g0>v {z), (v + 4>)gs>v(z) = c4 (z) gi>v {z) - (v 4- 6) z2gi>v {z), (v + 2s) ^+2i, (z) = Ca, {z) gUiv (z) - (v + 2s + 2) z^g^^„ (z), (v + 2m-2) g^,, v {z) = cm^ («) gm_2> „ («) - (i; + 2m) ^m (8) 9*71. TAe reality of the zeros of gimv{z) when v exceeds — 2. We shall now give Hurwitz' proof of his theorem* that when v > — 2, the zeros of gm,,v(z) are aM f^al; and also that they are all positive, except when — l>v>-2,in which case one of them is negative. After observing that gwitV{z) is a polynomial in z of degree m, we shall shew that the set of functions gimtV{z), g-M^viz), ... g-iiV(z), gn,v(z) form a set of Sturm's functions. Sufficient conditions for this to be the case are (i) the existence of the set of relations § 9*7 (8), combined with (ii) the theorem that the real zeros of g-im-^vi2) alternate with those of gzmv(z). To prove that the zeros alternate, it is sufficient to prove that the quotient gim,t>(z)l9vm-2,i>(z) is & monotonic function of the real variable z, except at the zeros of the denominator, where the quotient is discontinuous. We have f^ (z) ± dz where Wir,» = 9r,v(*) g\,„ (z) - g,t„(z)g'r%„(*); and. from § 9'7 C) it follows that R (V + 2m) Kft (v + 2m - 2)g so that m-l 2I2a2m,2w-2 = 5'22m-2,,(^+(^ + 2m) 2 (v + 2r)fzrhv(), r=l and therefore, if m > 1, 5EKami!!nir-2 is expressible as a sum of positive terms when v > — 2. * Afaift. Ann. xxxin. A889), pp. 254—256.
V71, 9-72] ASSOCIATED POLYNOMIALS 305 The monotonic property is therefore established, and it is obvious from a jraph that the real zeros of gmi-v, v (^) separate those of g.m< v (z). It follows from Sturm's theorem that the number of zeros of gmv(z) on my interval of the real axis is the excess of the number of alternations of sign n the set of expressions g^,lt„ (z), g^-^„(z)> ...,go>v(z) at the right-hand end of he interval over the number of alternations at the left-hand end. The reason why the number of zeros ia the excess and not the deficiency is that the inotient g2mi v{z)/ffim-2, v B) 'a & decreasing function, and not an increasing function of z, -.s in tho usual version of Sturm's theorem. Sec Burnsido and Panton, Theory of Equations, . A918), § 9Q. The arrangements of signs for the set of functions when z has the values - ^ 0, oo are as follows: -oo 0 oo 2?/j + ± 2w - 2 + ± 2m - 4 + ± 2 + ± - 0 + + + .; urmer or lower signs are to be taken according as y + 1 is positive or yaiive; and the truth of Hurwitz' theorem is obvious from an inspection i this Table. 9'72. Negative zeros- of g.an<v(z) when v < — 2. Let v be less than — 2, and let the positive integer s be defined by the nequalities - 2,v > v > - 2,s- - 2. It will now be shewn that*, when v lies between — In and— 2s - 1, g^^z) iasno negative zero; but that, ivhen v lies between. — 2s—I and — 2s — 2, gwn,iV(z) i,as one negative zero. Provided that, in each case, m is taken to be so large hat v + 2m is positive. It will first be shewn that the negative zeros (if any) of g&n<v(z) alternate vith those of g<t,n-%v(z)- * This proof differs from the proof given by Hurwitz; Beo Proc. London Math. Soc. B) xix. 1921), pp. 266—272. W. B. F. 20
306 THEORY OF BESSEL FUNCTIONS [CHAP. IX By means of the formulae quoted in § 9*7, it is clear that () ± i = (v + = (y + 2m) g provided that v + 2m is positive and z is negative. Therefore, in ,the circum- circumstances postulated, the quotient is a decreasing function, and the alternation of the zeros is evident. The existence of the system of equations § 9*7 (8) now shews that the set of functions form a set of Sturm's functions. The signs of these functions when z is — oo are +,+, .... +, +,->+> .-., (-)', and there are s alternations of sign. When z is zero, the signs of the functions are the upper signs being taken when —2s>v> — 2s— 1, and the lower signs being taken when — 2s- I > v> — 2s- 2; there are s and s + 1 alternations of sign in the respective cases. Hence, when —2s>v>-2s — 1, ffm.,vB) hfls no negative zero; but when - 2s - 1 > v'> - 2s - 2, g2m,iV(z) has one negative zero. The theorem stated is therefore proved. 9*73. Positive and complex zeros ofg2m>v(z) when v< — 2. As in | 9*72, define the positive integer s by the inequalities -2s>p>~2s-2. It will now be shewn* that when v lies between - 2s and - 2s — 1, gWttlV{z) has m — 2s positive, zeros; but that, when v lies between — 2s — 1 and — 2s — 2, gm,v(z) has m — 2s—\ positive zeros. Provided that, in each case, m is so large that m+ v is positive. * This proof is of a more elementary character than the proof given by Hurwitz; see the paper cited in § 9*72.
9'73j ASSOCIATED POLYNOMIALS 307 In the first place, it follows from Descartes' rule of signs that, in each case, 9zm,Az) cannot have more than the specified number of positive zeros. For, when v lies between - 2s and —25 — 1, the signs of the coefficients of 1, z, z\ ..., z\ z**\ z*+\ z^\ ...,zm in gWt?(z) are +, +, +, ..., +, -, +, -, ..., (-)»; and since there are m — 2s alternations of sign, there cannot be more than m — 2.s% positive zeros. When v lies between — 2s — 1 and — 2s — 2 the corre- corresponding set of signs is _ i / \m • > } > • ••> > » ~i > "•> \ ) j and since there are m — 2s — 1 alternations of sign there cannot be more than m — 2s — 1 positive zeros. Next, we shall prove by induction from the system of equations § 9*7 (8) that there are as many as the specified number of positive zeros. When v lies between - 2s and — 2s — 1, the coefficients in g^tV{z) have no alternations of sign (being all +) and so this function has no positive zeros. On the other hand 9*»+*,u@) > 0, gu+,,v(+ M ) = - oo , and so gi8+iiV(z) has one positive zero, o^, say; and, by reasoning already given, it has no other positive zeros. Next, take gM+ilV(z); from § 9'7 (8) it follows that its signs at 0, ahl, + oo are +,—, + ; hence it has two positive zeros, and by the reasoning already given i t has no others. The process of induction (whereby we prove that the zeros of each function separate those of the succeeding function) is now evident, and we infer that <j.,m>v{z) has m— 2.9 positive zeros, and no more. Again, when v lies between — 2s — 1 and — 2.s' — 2, the coefficients in #48+2,v{z) have no alternations in sign (being all —), and so this function ha.s no positive zeros. On the other hand <7«+4,.(O) < 0, gu+4,A+ oo ) » + oo , and so gmi>v{z) has one positive zero, and by the reasoning already given it has no other positive zero. By appropriate modifications of the preceding reasoning we prove in suc- succession that .<7«+fll„(.?), g4n+n,v(z), ••• have 2, 3, ... positive zoroa, and in general that g-2m)V{z) has 7n - 2.v — 1 positive zeros. By combining these results with the result of § 972, we obtain Hurwitz' theorem, that, when v < — 2, and in is so large that in + v is positive, //?,>,,,„(?) has 2s complex zeros, where s is the integer such that -2s>v>- 2s - 2. 20—2
CHAPTEE X FUNCTIONS ASSOCIATED WITH BESSEL FUNCTIONS 10*1. The functions 3v{z) and E,(^) investigated by Anger and H. F. Weber. In this chapter we shall examine the properties of various functions whose definitions are suggested by certain representations of Bessel functions. Wo shall first investigate functions defined by integrals resembling Bessel's inte- integral and Poisson's integral, and, after discussing the properties of several functions connected with Yn (z) we shall study a class of functions, first defined by Lommel, of which Bessel functions are a particular case. The first function to be examined, Jv(z), is suggested by Bessel's integral. It is defined by the equation A) J, (z) = - I "cos (v6 - z sin 6) dO. 7T Jo This function obviously reduces to Jn (z) when v has the integral value n. It follows from § 6'2 D) that, when v is not an integer, the two functions are distinct. A function of the same type as Jv(z) was studied by Anger*, but he took the upper limit of the integral to be 2-7r; and the function J^(^) is conveniently described as Anger's function of argument z and order v. A similar function was discussed later by H. F. Weberf, and he also investigated the function E,, (z) defined by the equation B) E, (z) = - ["sin {v6 - z sin 0) d0. 7T J 0 In connexion with this function reference should also be made to researches by Lommel, Math. Ann. xvi. A880), pp. 183-208. It may be noted that the function — / cos (i/0 —asin 8) dd which was actually dia- ln J 0 cussed by Anger is easily expressible in terms of JV (z) and 13, (z); for, if we replace 8 by 2rr — 8 in the right-hand half of the range of integration, we get \ fin 1 fn 1 [k — I cos(v0-z3in0)o?0 = r- I cos(vQ ~ZH\nd)dd + -cr cosBvtt-v& + z sin 0) dfi in J o ^tt J o «jo = cos2j/»r . Jp B) +sin vtt cos vn. E,, B). * Neueste Schriften der Naturf. Ges. in Danzig, v. A855), pp. 1—29. It was shewn by PoiRson that Vv I cos (vd - z sin 6) dO = (z - u) sin vir, Additions a la Conn, des 1 emps, 1836, p. 15 (cf. § 10*12), but as he did no more it seems reasonable to give Anger's name to tbe function. t ZUrich Vierteljahrmhrift, xxiv. A879), pp. 33—76. Weber omits the factor l/rrjn his defi- definition of Ev (a).
10'1] ASSOCIATED FUNCTIONS 309 To expand Jv {z) and E,, (z) in ascending powers of z, write \nr + <f> for 6 in the integrals and proceed thus: • [ln a\nm 6 sin vd dO = I cosm <? sin (\vrr + vcf)) dj> J - Jt = 2 ain \vir cosm <?> cos v<bd<b 'o .'o . mlsin by a formula due to Cauchy*. In like manner, sin1'1 ^ cos v6d6 = r r~TT o 2wr(Awj-4»'+1) But, evidently, 1 oo /_\m »!ttn Ttt 1 oo /_\m 2-m+l JAz)=- ^ --,;--Tr sin"»l^cosy(9^+ - ^ ,-/"-i\7 'w 7rm=0 (-2m)! Jo 7rma.0B?»+l)I so that oo (—)m(if2)- IS) Ju (Z) = COS >rl'7r Jw rr;; 7~ ¦" , .", > / \ ~,~tX oo / \7« + ainii'7r ^ -tS-; - , - and similarly oo i )m (irz\''n D) ¦.W-HinW^p^^^^/f^^^^ ,„.,() I1 {m — |v + ij) 1' (>il + kv + |) These results may be written iu the alternative forms * r~ *i i h 1 , „. M , . mil vir I _ z z & sin j'7t f ^ ^3 2D F) 1^,,- -w- . 1 + cos vir Results equivalent to the.se were given by Anger and Weber. Tho formula corresponding to E) was given by Anger (boforo tho publication of his memoir) in a lotter to Cauchy which was communicated to the French Academy on July 17, 1854; soe Comptes Rendun, xxxix. A854), pp. 128—135. * Mem. sur les intigrales difinies (Paris, 1825), p. 40. Cf. Modern Analysis, p. 263.
310 THEORY 6F BESSBL FUNCTIONS [CHAP. X For a reason which will be apparent subsequently (§ 10*7), it is convenient to write /•7\ , /Vi= — ?l 4. fL V) VW-ii^ A2A)C22) (P3)C22)E2^) "•' (8) «->,*(*) = - -2 + y2 ^2. _ „«) ~ v* B2 _ „«) D,9 - „«)+ •';» and, with this notation, we have (9) J, (a) 0>„ () IT 7T /ia\ « / \ 14-COSV7T 1/A — COS I/7r) A0) E, (a) = : So, „ (z) - -+ } *_,, „ (z). It is easy to deduce the following formulae from these results: A1) cos v6 . cos {z sin 6) dO = — v sin vrr. s_, v (^), Jo tic A2) sin v^ . cos (z sin 0) d# = — v A — cos vir). 5_j v (^), Jo ' A3) I sin v6. sin (a sin 6) dO — sin vir . s0 v (z), Jo r-rr A4) I cos v6 . sin B; sin 0)d^ =A + cob vrr). s0 v (z), Jo • A5) cos v(j>. cos B; cos </>) d^> = — 1/ sin \vk . 5_j ^ (z), Jo A6) cos vcp. sin (z cos </>) d<^> = cos ^7r. s0 v (z). Jo Integrals somewhat resembling the integrals discussed in this section, namely fe™»°?*(nO-coad)ddt J sin have been examined by Unferdinger, Wiener Sitzungsberichte, lvii. B), A868), pp. 611—620. Also, Hardy, Messenger, xxxv. A906), pp. 158—166, has investigated the integral f °° dd \ a\n(v6-zwine) ~-. Jo o and has proved that, when v is real, it is equal to in- 2 rjnJn (z), where »;„ is 1, 0 or - 1 n= -00 according as v — n is positive, zero, or negative. 10*11. Weber's formulae connecting his functions with Anger's functions. It is evident from,the formulae § 101 (9), A0), A5) and A6) that /i\ - / % „ / \ 4 COS il/7T /"i7r , , A) Jv (z) 4- J_,, (z) = — I cos v<fi cos ( B) Jv (^) - J_v (z) = ^— cos v<f} si IT JO (z cos
10-11,10-12] ASSOCIATED FUNCTIONS 311 C) e,(,) + e_(,)»-!~!±!![ [*"cosp<}>Bin(zcos<f>)d<b, IT .' 0 D) Ey (z) - E_y (z) = S1 ^t/7r cos vrf» cos (s cos <?) dd>. 7T .' 0 It follows on addition that J, (a) = \ cot Jvtt (E, (z) - E_, (*)} -1 tan Jw {E, (*) + E_v (a)}, so that E) sin i»7r. Jv (z) = cos V7r . Ev (z) - E_v (z), and similarly F) sin vtr . Ev (^) = J_v (z) — coa vtt . J,, (z). The formulae E) and F) are due to Weber. 10*12. Recurrence formulae for Jv (z) and Ep {z). The recurrence formulae which are satisfied by the functions of Anger and Weber have been determined by Weber. It is evident from the definite integrals that J,-! (z) + J,+1 (z) - 2v Ju (z) = -? f * (coa d--) cos (vd - z sin 6) dd Z IT JO \ Z/ in 0)}de J TTZ I o dd 2 Hill I/7T and 2i/_ , . 2 f'r 2v 2 f'7 v\ E,,_, B) + E,+, E) - — E, (z) - - I f cos 0 - - J sin (v6? -2 sin 0) d0 2 r-^ , 2A — cos vrr) It i.s also very easy to prove that From these results we deduce- the eight formulae /1 \ T / \ . -r 1 \ 2v _ , x 2 sin vtt A) J_, (^ -t- JH.t (z) = -JM(^) , Z TTZ B) J^1(*)-Jr+1E)«2Jr/(*)l C) (^ + v) J, (z) = ^J,_, (*) + (sin vrr)/w, D) (^ - v) Jv (z) = - zJv+1 (z) - (sin wr)/
312 THEORY OF BESSEL FUNCTIONS [CHAP. X -2vT (A = — *» \z> F) Bv-1(«)-Bv+,W = 2Er (*), G) (^ + v) Ev (Z) =^EM (z) + A - COS V7r)/7T, (8) O - v) E, (z) = - *E,+1 (z) - A - cos V7r)/7r, where S, as usual, stands for z(d/dz). Next we construct the differential equations; it is evident that (e) = (*-v) {zJv^ (z) + (sin vjr)/ir} — (v sin i>7r)/7r (z) + (z sin w7r)/7r - (v sin vjr)f7r, so that We also have (W - v2) B, (z) = (%-v) {zE^ (z) + A - cos i«r)/ir} =>z(& + l-v) Ev_j (?r) ~ i/A - cos V7r)fir = — Z2 E,, (^) — ? A + COS l/7r)/7T — 1/ A — COS V7r)/TT, so that A0) 7T Formulae equivalent to (9) and A0) were obtained by Anger, Neueste Schriften tier Naturf. Ges. in Danzig, v. A855), p. 17 and by Weber, Zurich Vierteljahrssckrift, xxiv. A879), p. 47, respectively; formula (9) had been discovered earlier by Poisson (of. § 101). KV13. Integrals expressible in terms of the functions of Anger and H. F. Weber. It is evident from the definitions that A) Jv(z)±iEv(z) = - J"exp {± i(V0 -zsin Q)\ dd. By means of this result, combined with formulae obtained in §§ 6'2—622, it is possible to express numerous definite integrals in terms of the functions of Bessel, Anger and Weber. Thus, from § 6'2 D) we have B) I fl-^-nh* dt « JL^ {J, (,) _ Jv {z)]> when |arg0|<|7r; the result is valid when \oxgz\=-\ir, provided that R (v) > 0. Again, we have
10-13, 10* 14] ASSOCIATED FUNCTIONS 313 so that, when we combine B) and C), D) |" *-*»»"»>«cosh vt dt= h-rr tan Ji/tt {J, (s) - Jv{z)} - \tr {E,(z) + F,(«)}, E) |M <r"iHl1' sinh ^ d« = |tt cot Jwr {/„ (z) - Jv (z)) - ?tt {E, (z) + Yv {z)\. The integral g-zcoshf ^g^ vifa ^as already been evaluated (§6-3); but j o ' I™ e-zcoalxtsinhvtdt h does not appear to be expressible in a simple form; its expansion in ascending powers of z can be obtained from bhe formula of § 6*22 D), 0) = - f^^ocos v$dO + 2 S1--y7r re-z™»usinhi/i bub, since fn «> z^i /i //i (~)w win V7r CT / i/ + m _ v + m J. C0S " C0S •*«" " W(, + ») ¦<Fl (- "¦ -»- ; a - • g ; - the integral under consideration cannot be evaluated in any simple form*. The formulae B)—E) are nugatory when v is an integer, but from §§ 6*21, 9-33 we have F) I" 0»'-»l»" ' d< = \ [Sn (Z) - 7TEn (Z) - 7T YH (Z)}, J (I G) (" e-'"-"i"i" dt = i (-)» '•' |»SY(, (a) + ttE71 (z) + nr \\ (z)\. Jo Tho asH(K;iat,(^i intcgraLs ./ ii niii Jo mn have been nnticftri by (loiitiw, Quarterly Journal, xx. AHH5), p. 2(!0. VarioiiH integrals of tho.so types occur in rCHoarchcs on dittraction by a prism; noe, e.g. Whipplo, Prw.. London Math. ,%<:. B) xvi. A1I7), p. 101$. 10*14. Asymptotic expansions of Anger- Weber functions of large argument. It follows from § KM3 B) that, in order to obtain the asymptotic expansion of J±v(z) when \z\ is large and |arg^|<^7r, it is sufficient to obtain the asymptotic expansion of the integrals fjT-vt—znh\h t (]f ll To carry out this investigation we shall first expand cosh vi/cosh t and sinh j>?/cosh t in a series of ascending powers of sinh t. * See Anding, Sechsxtelliye Tafeln der lictmcliicheii b'unktionen imaifiniiren Arguments (Leipzig, 1911) [Jahrbuch ilber die Fortschrilte der Math. 1911, pp. •11K—494], and Tukeuchi, TShokuMath. Journal, xvtn. A020), pp. 295—296.
314 THEORY OF BESSBL FUNCTIONS [CHAP. X If eil = u, we have, after the manner of § 7*4, so that cosh vt J flu+> 1/M+<1+) J ~ 27riJ (C- l)a - 4fsi cosh« (C (?-lJp-i {(?_ I)* _4? sinh2 Now 7T * Bm)! and, if we take p so large that R(p +^±%v)>Q, and then take the contour to be that shewn in Fig.'15 of § 7*4, we find that 7T Jo If v and t are real, the last expression may be written in the form (-)pcosfri where 0^.6x^1, since 1 + 4a; A — a>) sinh21 > 1. It follows that, when R (p + fr + fr v) > 0, we have cosh y< = cos i wr T^1 (-ynT{m + \ + \v)Y(m ±}Z\v) coshtf 7r |_m=0 Bm)! )r(fH-W B sinh .2W For complex values of v and t this equation has to be modified by replacing the condition 0 < 6X < 1 by a less stringent condition, in a way with which the reader will be familiar in view Qf the similar analysis occurring in various sections of Chapter vii. Similarly we have /"(-!«+, 1/M + ) ( 1 1
10*14] ASSOCIATED FUNCTIONS 315 so that sinh^_ 1 /•(«+. i/«+,i+) g»df whence it follows that, if we take p so large that R (p + 1 + \v) > 0, then sinh vt sin i v cosh* 7T |_TO==o Bm+ 1)! /(" j"JlH1^ B sinh t)*+l I . On integrating these results, it follows that r J0 I m=0 s (- If v is real and 2 is positive, these asymptotic expansions possess the property that the remainder after p terms is of the same sign as, and is numeri- numerically less than, the (p + l)fch term when p is so large that R (p + I + \ v) > 0. It follows from §§ 1013 B) and C) combined with § 1011 @) that A) sin vtt TTZ V V B- - li") M B* - I'") Da - l/J) e ¦. 1 - cos vtt \v _ j/ Bs - i/a) 7/ B- - z/'J) Da - v'J) _ -wz [z z* zn TIiohc rosults wore .stated without proof by Wc.bcr, Zurich Yie.rtolju/irm'.hi'ift, xxiv. A870), p. 48 and by Lommel, Math. Ann. xvi. A880), pp. 180— 1H8. Tlioy wore proved tin .special casos of much more goncrul formuku by Nieln(!n, llandlmch de.r Theorio. der Cylindarfunktionen (Leipzig, 1904), p. 228. Tho proof of thin action doiw not hcohi to have been given previously. Since the only singularities of coah vt/couht and ainh vtjcon\\t, qua functions of sinh^, arc at sinh t = ± i, it is possible to change the contours of integration into curves in the i-plane on which arg (ainh t) ia a positive or negative acute angle; and then we deduce in the usual manner (cf. §6'1) that the formulae A) and B) are valid over the sector | arg z \ < it.
316 THEORY OF BESSEL FUNCTIONS [CHAP. X 10*15. Asymptotic expansions of Anger- Weber functions of large order and argument. We shall now obtain asymptotic expansions, of a type similar to the expansions investigated in Chapter viii, which represent J,, (z) and Ev (z) when | v | and \ z j are both large. In view of the results obtained in^ § 1013, it will be adequate to obtain asymptotic expansions of the two integrals 1 /"" _ / gTW-zsinht^ TTJO As in Chapter viii, we write v — z cosh (a + i/9) = z cosh 7, where 0 ^ ft ^ 7r and 7 is not nearly equal* to ttx. (I) We first consider the integral 1 f° 1 r*> — I e~vt~zainhtdt=: — I g-ztfcosh y-fsinh t) fy TTJO T J 0 in which it is supposed temporarily that v\z is positive. When cosh 7 is positive, t cosh 7 + sinh t steadily increases from 0 to 00 as t increases from 0 to 00 ; wo shall take this function of t as a new variable t. It is easy to shew that t is a monogenic function of t, except possibly when t = Bw +¦ lOrt cosh y± sinh y + y cosh y, where n is an integer; and, when coshy is positive, none of these values of r is a real positive number; for, when y is real, B?i + 1) tricosh y does not vanish, and, when y is a pure imaginary ( = i/3), the singularities are on the imaginary axis and the origin is not one of them since y is not equal to iri. The expansion of dt/dr in ascending powers of t is where and so am is the coefficient of 1/t in the expansion of r in ascending- powers of t. In particular we have _ 1 1 _^~ cosn 7 1+cosh 7' 1 2 A -f cosh 7)*1 2 24 A + cosh <yf ' _ 225 — 54 cosh <y + cosh2 7 as ~ 720A +cosh7I0 " From the general theorem of § 8'3, we are now in a position to write down the expansion A) ^ Expansions valid near y=xi are obtained at the end of this section.
10*15] ASSOCIATED FUNCTIONS 317 This expansion is valid when v/z is positive; it has, so far, been established on the hypothesis that | arg z j < \ir, but, by a process of swinging round the contour in the T-plane, the range of validity may be extended to cover the domain in which | arg z\< it. Next, we consider the modifications caused by abandoning the hypothesis that cosh 7 is real. If we write t — u + iv, the curve on which t is real has for its equation u sinh a sin /3 + v cosh a cos /9 ¦+ cosh u sin v = 0. The shape of this curve has to be examined by methods resembling those of § 81. For brevity we write u sinh a sin j3 + v cosh « cos /3 4- cosh a sin v = <t> (u, v). Since 4> (w, v) is unaffected by a change of sign of both u and a, we first study the curve in which a ^ 0. It is evident that the curve has the origin as its centre. Since d<$> (u, v)ldu = sinh a sin /3 + sinh u sin v, it follows that, when v has any assigned value, d<i>/du vanishes for only one value of u, and so the equation in u cfc (u, v) = 0 has, at most, two real roots; and one of these is infinite whenever v is a multiple of it. When 0 > v > — it, we have <I> (— oo ,¦?;) = — oo , <t> (+ oo , v) = — oo ; and, when v = ft — 7r, the maximum value of ^((t, w), qua function of a, is at u = a, the value of <1? (u, ?;) then being - cosh a sin /9 {1 — a tanh a + (ir - /3) cot ^j. If this is negative, the equation <1> (u, /9 — rr) = 0 has no real loot, and ho the contour does not meet the line v = $ — it or (by symmetry) the line V — ir — jS. Hence provided that the point (a, /?) lies in one of the domains num- numbered I, 2, :i in Fig. 21 of § 8#(I, the contour <t> (u, v) = 0 lies as in Fig. 25, the continuous curve indicating the shape of the contour when a is positive Fig. 25. and the broken curve the shape when a is negative; the direction in which t increases is marked by an arrow.
318 THEORY OF BESSEL FUNCTIONS [CHAP. X It follows that the expansion A) is valid when (a, ft) lies in any of the domains 1, 2, 3. Next, we have to consider the asymptotic expansion when («, ft) does not lie in any of these domains. To effect our purpose we have to determine the destinations of the branch of the curve <J> (u, v) = 0 which passes through the origin. Consider first the case in which a is positive and ft is acute. The function <t> (a, v) has maxima at v = Bn •+• 1) ir — ft and minima at v = B?? + 1) tr + ft, each minimum being greater than the preceding; and since <f> (a, ft — tt) is now positive, it follows that <j> (a, v) is positive when v is greater than — it. Hence the curve cannot cross the line u = a above the point at which v — - w, and similarly it cannot cross the line u - — a below the point at which v=ir. The branch which goes downwards at the origin is therefore confined to the strip — a < u < a until it gets below the line v — — 2Ktt + rrr — #, where K is the smallest integer for which 1 - a tanh a + {Bif + 1) 7T -/9} cot ? >0. The curve cannot cross the line v = — BK + 1) 7T+./3, and so it crosses the line u = a and goes off to infinity in the direction of the line v = — IKtt. Hence, if o is positive and $ is acute, we get while, if a is negative and /9 is acute, we get By combining these results with those obtained in § 861, we obtain the asymptotic expansions for the domains 6 a and 7 a. If, however, C is obtuse and a is positive, the branch which goes below the axis of u at the origin cannot cross the line u = a below (a, ir — @) and it does not cross the M-axis again, so it must go to — oo along the line v = — BL + 1) 7r, where L is the smallest integer for which 1 - a tanh a - {BX + 1) it + /3\ cot /3 > 0. Hence, if a is positive and ft is obtuse, we get while, if a is negative and ft is obtuse, we get co+BL+l)»ri 1 r- E) - TTJu
10'15] ASSOCIATED FUNCTIONS 319 By combining these results with those obtained in § 8*61, we obtain the asymptotic expansions for the domains 4, 5, 66 and 76. Since formula A) is the only one which is of practical importance, we shall not give the other expansions in greater detail. An approximate formula for a^, when m is large and y is zero, namely was obtained by Cauchy, Comptes Rendus, xxxvin. A854), p. 1106. (II) Next consider the integral 1 /ta0 1 f _ I evt-zain\it fa _ _ I e-z(-tcoshy+B\nht) fa ttJo ttJo The only difference between this and the previous, integral is the change in the sign of cosh 7; and so, when 7 lies in any of the regions numbered 1, 4, 5 in Fig. 21 of §8-61, we have where aOT' is derived from aw by changing the sign of cosh 7; so that ,_ 1-cosh 7' -(I-cosh 7)*' ^ 24A -ooah^O' This expansion fails to be significant when 7 is small, just as the previous ex- expansion A) failed when 7 was nearly equal to iri. To deal with this case we write v = z A — e), 'r — t — sinh t, after the method of § 8*42. It is thus found that i r°° 1 f~°° dt t gct-zsinht dt — - e?r e~'zt j- dr 7rJ0 7rJo <ZT 1 r-00 00 and hence A result equivalent to this has been given by Airoy, Proc. Royal Soc. xciv. A, A918), p. 313.
320 THEORY OF BESSEL FUNCTIONS [CHAP. X 10*2. Hardy's generalisations of Airy's integral. The integral considered by Airy and Stokes (§ 6*3) has been generalised by Hardy* in the following manner: If s = sinh (j>, then f 2 cosh 2<jb = 4s2 + 2 I 2 sinh 3</> = 8s3 + 6s I [ 2 sinh o<f> = 32sB + 40s3 + 10s, and generally 2 l™l n<j> = Bs)\F, (- *«, i - in; 1 - n; -1 the cosh or sinh being taken according as n is even or odd. Now write Tn(t,ot) = tn.,Fl(-^n,^~^n;l-n-) so that ( T2(i,a) = i2 + 2a T3<?, a) = t* + %at T, (t, a) = t4+ 4ai2 + 2a2 Then the following three integrals are generalisations! of Airy's integral: A) Ci»(o)= I™cos Tn(t, a)dt, J o B) Sin(a) = j sin Tnit, a) dt, C) ^"n («) = f exp {- Tn (t, a)} dt. Jo It may be shewnj that the first two integrals are convergent when a is real (whether positive or negative) if n — 2, 3, 4,.... But the third integral converges when a is .complex ; and it is indeed fairly obvious that Ein{oi) is an integral function of a. When n is an even integer, the three functions are expressible in terms of Ressel functions; but when n is odd, the first only is so expressible, the other two involving the function of H. F. Weber. Before evaluating the integrals, we observe that integral functions exist which reduce to Cin (a) and Sin (a) when a is real; for take the combination Gin («) + iSin (a) = f" exp [iTn(t, a)} dfc JO * Quarterly Journal, xli. A910), pp. 226—240. t The sine-integral in the case n=3 was examined by Stokes, Camb. Phil. Trans, ix. A856), pp. 168—182. [Math, and Phys. Papers, n. A883), pp. 332—349.] X Hardy, loc.cit., p. 228.
10*2, 10-21] ASSOCIATED FUNCTIONS 321 By Jordan's lemma, the integral, when taken round an arc of a circle of radius jR with centre at the origin (the arc being terminated by the points with complex coordinates jK, Re*™1*1), tends to zero as R-*-oo. And therefore roooxp(J«7«) Cin (a) + iSin (a) = exp {* Tn (t, a)} dt Jo = eW» | °°exp {- Tn (t, ae~Hln)} dr, Jo where t = te~^iriln; and the last integral is an integral function of a. The combination Gin(a) — iSin(a) may be treated in a similar manner, and the result is then evident. 10*21. The evaluation of Airy-Hardy integrals of even order. To evaluate the three integrals Gin(a), Sin(a), Ein(a) when n is even, we suppose temporarily that a is positive, and then, making the substitution t = 2a* sinh (u/ri) in the integrals, we find that, by § 6'21 (.10), n J o 2«1 f Gin («) + iSin (a) = — 0XP Bain i cosh u) cosh (u/n) du n J o u Bai?t), that is to say Gin(a) + iSin(a) = .™h ~{flW/« J_xlnBa»»)- er™* JVn H Hill yTTl'lt) If we equate real and imaginary parts, we have In a similar manner, 2a^ f Eh, (a) = exp (— 2a*71 cosh w) cosh (ajn) du, so that, by §(J-22E), C) Kin(a) = B<*/n)KlhgBa**). These results havo. been obtained on the hypothesis that a is positive; and the expressions on the right are the integral functions of a which reduce to Cin (a), Sin (a) and Ein (a) when a is real, whether positive or negative. Hence, when a is negative the equations A), B), C) are still valid, so that, for example, we have (H <a\= - * J V (-)»O"« nV } 2nain(i/n)Ut m!f(m+li/») whether a be positive or negative. w. b. f. 21
322 THEORY OF BESSEL FUNCTIONS [CHAP. X Hence, replacing a by — ft, we see that, when ft is positive and n is even, then D) E) F) O»(-/8)- &in(-ft) = Ein(-ft) = (/-„„ W - /„. B/3-)], It follows from § 4'31 (9) that, when n is even, the functions Cin (a) and 8in (a) are annihilated by the operator and that Ein (a) is annihilated by the operator In the case of the first two functions it is difficult to obtain this result* directly from the definitions, because the integrals obtained by differentiating twice under the integral sign are not convergent. 10*22. The evaluation of Airy-Hardy integrals of odd order. To evaluate Cin(a) when n is odd, we suppose temporarily that a is positive, and then, by § 6'22 A3), 2a* r Gin (a) = — cos Ba*71 sinh u) cosh (uln) du n Jo 7b That is to say, A) Using the device explained in § 1021, we see that, when /9 is positive, It follows that the equation § 10'21 D) is true whether n be even or odd; and, whether n be even or odd, Oin(a) is annihilated by the operator dl (_)nnt «-2 for all real values of a. * It has been proved by Hardy, loc. cit., p. 229, with the aid of the theory of "generalised integrals."
10'22] ASSOCIATED FUNCTION 323 Next we evaluate Ein (a) when a is positive; making the usual substitution, we find that, by § 1013 D), 2a* f" Ein (a) = — exp (— 2a*n sinh u) cosh (u/n) du w J o = ~ {tan (WO Ji/n B«»») - B1/n Ba*-)] + -^7r-~-r- {JL^ Ba*») - J1/n Ba*-)}. n sin (tr/n)l ' v 7 ' v /J Hence the series which represents Ein (a) when n is odd and a may have any value is J n ~ n cos (\Tr/r\)m=0 F (m + f —1/%) F (?ft + -| + ^-/w) ~- f oo / \m «tnti co / \m nmn \ + J 51 .l—iL.W „ V _A~J__I L w sinX7r/«) (mt0 m! F (m + 1 - l/?i) m^0 w! F (?» + 1 + l/n)J ' and hence it follows that D) — + n>or-*\ EiH («) = ^a*1 (da j Next consider Cin(a) + i Sin(a), where a is temporarily assumed to be positive. From § 103 D) we deduce that 2gj /•*> Gin (a) + i >S'in (a) = — exp Ba*.n i sinh ?^) cosh (u/n) du {tan (^Tr/n) J1/n (- 2a*'lt) - E1/u (- 2a»«t)} « n cos (^tt/«) m r (w + | - i/n) F (w + f + ™* F*«/» /_1/n Ba*-) - e-*-V« / r/J!I and therefore whence it follows that, when /9 > 0, y ±ZL-.ii n-o F (to + & — ^/u) F Gfi • 21—2
324 THEORY OF BESSEL FUNCTIONS [CHAP. X and hence, for all real values of a, G) 1-^L - nB an-*\ Sin (a) = - nob <»-»>. (da2 ) This equation was given by Stokes in the case n =¦ 3. It should be noticed that (8) 8in(a) + (-)*«n+D Ein(a) = nf?*v/n) {sin <**/ Wit {sin (iw/n)+ (-!)»*+"} where /9 = — a, and a and /9 are real. The formulae of the preceding three sections are due to Hardy, though his methods of obtaining them were different and he gave some of them only in the special case n = 3. 10*3. Conchy's numbers. In connexion with a generalisation of Bessel's integral which was defined by Bourget, and subsequently studied by Giuliani (see § 10'31), it is convenient to investigate a class of functions known as Cauchy's nawhers. The typical number, JV_nijt|W, is defined by Cauchy* as the coefficient of the term independent of t in the expansion of H)K)* in ascending powers of t. It is supposed that n, k, and in arc integers of which the last two are not negative. It follows from Cauchy's theorem that A) 2 2'n+A: jm r-n 2m+ktfn r 1*4 {e~ni0 + (~)m enie] cos* 8 sin OdB Jo IT Jo It is evident from the definition that N-.n>ktm is zero if — n + k + m is odd or if it is a negative integer. * Comptes RenduK, xi. A840), pp. 473—475, 510—511; xn. A841), pp. 92—93; xnr. A841), pp. 682—687, 850—854.
10'3] ASSOCIATED FUNCTIONS 325 From A) it is seen that C2~\ AT , — (—\m AT , — (-\n-k i\r , V-6/ -iv — n,k,m— V / xv«,J:,m— \ ) -ivM,Jk,m- These results, together with recurrence formulae from which successive numbers may be calculated, were given by Bourget*. The recurrence formulae are C) -ZV-m,k, in, = -"->i+-i, Jfc-i,m + -N-n-\,k-i,m > and they are immediate consequences of the identities rn (t +1 /tf (t -1 ft)m = v~n(t + i /t)k~i (t -1 /t)m + rw-x (t + i/tf-1 (t - i/t)m, t~n (t + i/tf (t - i/om = t1-* (t + i/t)k (t - i/t)m~i - t-n~i (t + i/t)k (t - i/t)m-\ By means of these formulae any Cauchy's number is ultimately expressible in terms of numbers of the types i\T_njti0, iV_wom. A different class of recurrence formulae, also due to Bourget, owes its existence to the equation 4 "dtV-t It follows that @+) by a partial integration. On performing hhe differiitifciatiou we see that E) (in + I) N-n,k, m = *'^-«,*-i, rn li ~ (^ ^ and similarly F) (^ -I- 1) JV-«,it,hi = nN_Kk+liin_! - (m - Developments duo to (JhcHsiu, Annals of Math. x. AH05---<>), j)p. I—2, aro I- 0 («) iV _ n, ,. 1(i = s (-)'' A-. N . ui. _.,,, ,, ,„ ..,. r-l) Theso may bo deduced by induction from C) and (-1). Another formula due to (Jh<\snin is (9) ff-n,*, .»=» 2 ( -)rtCp..r.mCr, whore p — \ (k+m — n). Thiw i« proved by .selecting the coufficiuut of tn in the product (<+l/0*x(«-l/0Bl. * Journal de Math. B) vi. A861), pp. 88—54.
326 THEORY OF BESSEL FUNCTIONS [CHAP. X 10*31. The functions of Bourget and Giuliani. The function Jn<k(z) is defined by the generalisation of Bessel's integral where n is an integer, and k is a positive integer. It follows that o T eXP {- * (n6 - Z Sin 6)\ • B C0S 6f d6> Z7T J -.„ and therefore B) «7n * CO = - T B cos e)k <=os (n& ~ z sin 0) de> 7T .1 o The function Jn,k(z) has been studied by Bourget, Journal de Math. B) vi. A861), pp. 42—65, for the sake of various astronomical applications; while Giuliani, Giornale di Mat. xxvi. A888), pp. 151—171, has constructed a linear differential equation of the fourth order satisfied by the function. [Note. An earlier paper by Giuliani, Giornale di Mat. xxv. A887), pp. 198—202, contains properties of another generalisation of Bessel's integral, namely 1 /¦"¦ - / coa (n6 - 2" sin" 6) de, it J a but parts of the analysis in this paper seem to be incorrect.] If we expand the integrand of A) in powers of z, we deduce from § 10'3 that C) /«,*(*)= 2^f # ; and it is evident from A) that D) Jn<a(z) = Jn(z). Again from § 10B) and C) it is evident that E) J-n,k(*) = (-Y^kJn,*(*). F) Jn,k (z) = /„-,,*_, (*) + «/¦„+,,*_, B); and, if we take Ic— 1 in this formula, G) ^,..W-v^('). z These results were obtained by Bourget; and the reader should have no difficulty in proving that (8) 2J'M <*) - J*-i,k (*) - /.+,,*(*)• Other recurrence formulae (due to Bourget and Giuliani respectively) are (9) A0) 4J"n,*_a E) . Jn>, («) - 4J^ (z).
10-31] ASSOCIATED FUNCTIONS 327 The differential equation is most simply constructed by the method used by Giuliani; thus 1 f"¦ d Vn Jn k (z) = - -ja {— (n + z cos 0) sin (nB - z sin 0)] B cos Of dd ' "".'(> **C7 2& P1 = (w + 2 cos 0) sin (n0 - z sin 0) B cos 0)*-1 sin 0d0 T Jo 2k f* (d ) w t B) + — i ja cos (w^ - z sin 0) f B cos d)k~l sin 0d0 TT .!(, [atf j 'n fc B) cos (n0 - z sin 6) ~ {B cos 0)*-1 sin 01 ^0, 7T J 0 wC 7T J 0 and so d* Operating on this equation by -^- + 1, and using A0), it follows that n Jn'k{Z) + 2kzJ'n'k& + ^J^(z)\ = k(k~ l) JnA*)> and hence we have Giuliani's equation A1) *J\k(g) + Bk + 5)zJ'\k(g) + [2z* + (k + 2Y -n2} J\k(z) + B/c + 5) gJ\ fc (g) + (z* + k + 2 - na) Jnt k (z) = 0. It was also observed by Giuliani that A2) e'*Bi"oBcos0)*= 25 €.mJ.zn>k(z)cou2nd 71 = 0 00 + i % em.HJm+l<k(z)ainBn+l)d; this is verified by applying Fourier's rule (of. § 2*2) to the function on the right. A somewhat similar function J(z\ v, k) ban boon utudiod by Bruna, A»tr. Nach. CIV. A883), col. 1—8. Thitt function is duiined by tho aoricn 00 i' _ <ni /jL-»\»' + afc+aT»i. 9., The most important property of this function is that A4) /(,,., *)-/(.; ¦whence it follows that
328 THEORY OF BESSEL FUNCTIONS [CHAP. X 10'4. The definition of Struve's function I3.v{z). Now that we have completely examined the functions defined by integrals resembling Bessel's integral,, it is natural to investigate a function defined by an integral resembling Poisson's integral. This function is called Struve's function, although Struve investigated* only the special functions of this type of orders zero and unity. The properties of the general function have been examined at some length by Siemonf and by J. Walker ? Struve's function lrLv{z), of order v, is denned by the equations sin2" Odd, provided that R(v) > - \. By analysis similar to that of § 3'3, we have ? f tf^1 A - <2)"~i ^^ M J o H.W-,7^ r (v +1) r (J) w.o Bm + 1)! so that The function Hv(,z) is denned by bhis equation for all values of v, whether R(u) exceeds — f or not. It is evident that Hv (z) is an integral function of v and, if the factor (\z)v be suppressed, the resulting expression is also an in- integral function of z. It is easy to see [cf. §§ 211 E), 3121 A)] that where and | vo + f | is the smallest of the numbers 11> +f|, * Mim. de VAcad. Imp. des Sci. de St Pitersbourg, G) xxx. A882), no. 8; Ann. der Physik, C) xvn. 11882), pp. 1008—101G. See also Lommel, Archiv der Math, und Phys. xxxvi. A861), p. 399. t Programm, Luisenschule, Berlin, 1890. [Jahrbuch iiber die Fortschritte der Math. 1890, pp. 340—342.] X The Analytical Theory of Light (Cambridge, 1904), pp. 392—395. The results contained in this section, with the exception of C), D), A0) and A1), are there given.
10-4] ASSOCIATED FUNCTIONS 329 We can obtain recurrence formulae thus: d_ , d \Z and similarly = ~ ^+1 T (w + f) r A/ + m + $ ^^a"r (m +1) r (i; + m + On comparing these results, we find that E) "SS-v-i {z) + Hv+1 (z) = — Hv B) + p V F) EU (*) - Hr+I (*) = 2H/ (jr) { G) (^ + ,)H,B) = Ah(^ (8) (* - „) Hv {z) = r(-J^ In particular we have (9) A{*HI(*)} = *H0(*)) 1{H Again, from G) and (8), we have (*» - v>) H, (*) - (Sr -1,) {^ so that H,,B) satisfies the diflferential equation A0) V, Hv (*) = p^-^y i^jy • Tho function Lv (s) which boars the Hamo rolation to Struve's function aa Iv (z) bears to Jv{z) haw been studied (in tho caso v=0) by* Nicholnon, Quarterly Journal, XLIL A911), p. 218. This function is denned by the equation (U) U («)= ^fSTrF the integral formula being valid only when R(v)> — \. The reader Bhould have no difficulty in obtaining the fundamental properties of this function. * See also Gubler, Zilrich Vierteljahrmchri/t, xi.vii. A902), p. 424.
330 THEORY OP BESSEL FUNCTIONS [CHAP. X 10*41. The loop-integral for M.v (z). It was noticed in § 1O4 that the integral definition of Hv(z) fails when U(v)<-$, because the integral does not converge at the upper limit. We can avoid this disability by considering a loop-integral in place of the definite integral. Let us take (P - 1)"-* sin zt. dt, Jo where the phase of i2 -1 vanishes at the point on the right of t = 1 at which the contour crosses the real axis, and the contour does not enclose the point If we suppose that R(v) > — ?, we may deform the contour into the seg- segment @, 1) of the real axis, taken twice, and we find that /¦(i+) f1 (t2 - 1)"-* sin zt.dt** % cos vrr\ A - i2)""* sin zt. dt, Jo Jo where the phase of 1 -1 is zero. Hence, when R (v) > — ?, we have Both sides of this equation are analytic functions of v for all * values of v; and so, by the general theory of analytic continuation, equation A) holds for all values of v. From this result, combined with § 61 F), we deduce that B) j. (z) + iav (z) = r (* ~l\'r^ f1+) «* («2 -1)-* dt To transform this result, let a> be any acute angle (positive or negative), and let the phase of z lie between — \tt + to and \ir + a>. We then deform the contour into that shewn in Fig. 26, in which the four parallel lines make an angle — « with the imaginary axis. It is evident that, as the lines parallel to the real axis move off to infinity, the integrals along them tend to zero. The integral along the path which starts from and returns to 1 + oo ie~iu> is equal to Hvw (z); and on the lines through the origin we write t = iu, so that on them (f -1)--4 = eT (»-i)« (i + u*y-l. It follows that jv (z)+iuv (z) - H* (z) + r^ffr (i) i e~m A + uiy~k du> * The isolated values }i $,$,... are excepted, because the expression on the right is then an undetermined form.
10*41] ASSOCIATED FUNCTIONS where the phase of 1 + u2 has its principal value; and hence 331 This result, which is true for unrestricted values of u, and for any value of 2 for which - it < arg z <ir, will be applied immediately to obtain the asym-, ptotic expansion of Hv (z) when | z | is large. Fig. 2A. A result equivalent to B) was obtained by J. Walker*, who assumed that 11 (v)> — I, Ji(z)>0, so that to might be taken to be zero. In the case v—0, the result had previously been obtained by Rayloighf with the aid of the method ol' Lipschita (§ 7*21). If, as in §(M2, we replace w by arg 2-/3, it is evident that C) may be written in the form > ex}) « where — \ir < y8 < iir and - \rr + /3 < arg z < In + /3. This equation*gives a representation of Hv (^) when j arg z \ < tv. To obtain a representation valid near the negative half of the real axis, we define H,,(.z) for unrestricted values of arg z by the equation E) H,, (zemni) = e» ("+*>** Hv (,?), and use D) with z replaced by ze^vi. * The Analytical Theory «/ Light (Cambridge, 1904), pp. 894—395. t Vroc. London Math. Soc. xix. A889), pp. 504—507. [Scientific Papers, in. A902), pp. 44—46.]
332 THEORY OP BESSEL FUNCTIONS [CHAP. X If vie write z**ix in C), where x is positive, we see bhat, when R(v)<i, and, by considering imaginary parts, we deduce that a result given by Nicholson, Quarterly Journal, xlii. A911), p. 219, in the special case in which i»=O. 10*42. The asymptotic expansion of Hv {z) when \z\is large. We shall now obtain an asymptotic expansion which may be used for tabu- tabulating Struve's function when the argument z is large, the order v being fixed. Since the corresponding asymptotic expansion of Yv (z) has been completely investigated in Chapter vn, it follows from § 10*41 D) bhat it is sufficient to determine the asymptotic expansion of L As in | 7'2, we have We take p so large that i? (v -p — |) <0, and bake 8 bo be any positive angle for which so that if is confined to the sector of the plane for which - it + 28 < arg z^tt — 28. We then have so that t'tfc. ^ sin h, I(Sin say, where Ap is independent of z. It follows on integration that > OXp tK where 0 (-BTV). a exp i|3
10-42, 10-43] ASSOCIATED FUNCTIONS 333 We deduce that, when j arg z | < ir and | z j is large, provided that R (p — v + .]) ^ 0; but, as in § 7*2, this last restriction may be removed. This asymptotic expansion may also be written in the form B) H. (,)- Y. W + I?F(, + [^;(}>y^ + 0 (*>—). It may be proved without difficulty that, if v is real and z is positive, the remainder after p terms in the asymptotic expansion is of the same sign as, and numerically less than the first term neglected, provided that R (p + I- —v)> 0. This may be established by the method used in § 7-32. The asymptotic expansion* was given by Rayloigh, Proc. London Math. Soc. XIX. A888), p. 004 in the cawe v = 0, by Struve, Mem. de I'A cad. Imp. das ScL de St Petersbourg, G) xxx. A882), no. 8, p. 101, and Ann. der Pkys. und Okemie, C) xvn. A882), p. 1012 in the case i> = 1; the result for general valuer of v was given liy J. Walker, The Analytical Theory of Light (Cambridge, 1904), pp. 31L—395. If v has any of the values ?, '•},, ..., then A + iP/z1)"'* i.s expressible as a terminating series and Yv(z) i.s also expressible in u finite form. It follows that, when v is half of an odd positive integer, Ht.(z) is expressible in terms of elementary functions. In particular 10'43. The asymptotic expansion of Ntrave's functions of large order. We shall now obtain asymptotic expansions, of a type, similar to the expansions investigated in Chapter Viti, which represent Struvc'.s function Hv (z) when | v \ and | z \ are both largo. Am usual, we, shall write v = z cosh (a + i/9) = z cosh 7 and, for simplicity, we shall confine (he investigation to the special case in which cosh 7 is real and positive. The. more general case, in which cosh 7 is complex may be investigated by the methods used in § <S-(i and § 1015, but it is of no great practical importance and it involves some rather intricate analysis. * For an asymptotic; expansion of the. HHHOi'.iutiul inti^'al „ ( 1 -I- j ) du, see Raylcigh, Phil. Mag. ((i) vm. (l!H4), pp. 481—487. [Scientific Papas, v. A912), pp. 200—211.]
334 THEORY OP BESSEL FUNCTIONS [CHAP. X The method of steepest descents has to be applied to an integral of Poisson's type, and not, as in the previous investigations, to one of Bessel's type. In view of the formula of § 10*41 C), we consider the integral dw which we write in the form f „ dw where r = w- cosh 7. log A + w3). It is evident that r, qua function of tv, has stationary points where w = e**, so that, since 7 is equal either to a or to ift, two cases have to be considered, which give rise to the stationary points (I) e±a, (II) e±l*. Accordingly we consider separately the cases (I) in which z/v is less than 1, and (II) in which z/v is greater than 1. (I) When 7 is a real positive number a, r is real when w is real, and, an w increases from 0 to 00 , t first increases from 0 to e~a — cosh a. log (J + e~2a), then decreases to ea — cosh a. log A + e211) and finally increases to + 00 . In order to obtain a contour along which r continually increases, we. suppose that w first moves along the real axis from the origin to the point e~a, and then starts moving along a certain curve, which leaves the real axis at right angles, on which t is positive and increasing. ' To find the ultimate destination of this curve, ..it is convenient to make a change of variables by writing ?= % + iy, e~a = sinh ?0, where ?, 77 and ?0 are real. The curve in the ?-plane, on which r is real, has for its equation cosh f sin <r\ = 2 cosh a arc tan (tanh f tan ?/), and it has a double point* at ?„. We now write v 1 y \ _ 2 arc tan (tanh ? tan rj) ~ cosh ? sin y and examine the values of F(%, 17) as ? traces out the rectangle whose corners 0, A, B, C have complex coordinates 0, arc sinh 1, arc sinh "I + J tti, I iri. As ?goes from 0 to A, F(t;, rj) is equal to 2 sinh f/cosh8 ^, and this .steadily increases from 0 to 1. * Except when a = 0, in which case it has u triple point.
10-43] ASSOCIATED FUNCTIONS. When ? is on AB, F(t;, rj) is equal to V2.arc tan [—jcr)• cosec % 335 and this steadily increases from 1 to tt/^/2 as r) increases from 0 to \tr. ) j2 and observe that arc tan *\ " Notb. To establish this result, write tan t) d It T because - arc tan «, which vanishes with i, has the positive derivate When ? is on J?C, i^1 (f, i;) is equal to 7r sech |, and this increases steadily from 7r/V2 to it as ? goes from B to 0; and finally when ?" is on GO, F(j;, rj) is zero. Hence the curve, on which F (?, if) is equal to sech a, cannot emerge from the rectangle OABG, except at the double point on the side CM; and so the part of the curve inside the rectangle must pass from this double point to the singular point G. The contours in the w-plane for which a ha,s the values 0, \ are shewn in Fig. 27 by broken and continuous curves respectively. Fig. '27. Consequently a contour in the w-phuu', on which t is real, consists of the part of the real axis joining the origin to e~a and a curve from this point to the singular point /'; and, aa w traces out this contour, r increases from 0 to + oo . It follows that, if the expansion of d^jdr in powers of r is dt °° ar w=o
336 THEORY OF BESSEL FUNCTIONS [CHAP. X then (V™ A + w2)"-* duo = and hence, by 10*4 A), we have (i) H.w-.j.w It is easy to prove that b0 = 1, &x = 2 cosh 7, 52 = 6 cosh2 7 — \, b3 = 20 cosh3 7 — 4 cosh 7, .... (II) When 7 is a pure imaginary (= i/9), t is real and increases steadily from 0 to 00 as w travels along the real axis from 0 to 00; and so /•oo foo f I JO JO (\/(l+ Hence, from § 10-41 C) it follows that provided that j arg z \ < \ir. This result can be extended to a somewhat wider domain of values of arg z, after the manner of § 8*42. From the corresponding results in the theory of Bessel functions, it is to be expected that these results are valid for suitable domains of complex values of the arguments. In particular, we can prove that, in the caae of functions of purely imaginary argument, C) lip (vx) ~ /„ (vx) when I v I is large, | arg v \ <^n-, % is fixed, and the error is of the order of magnitude of times the expression on the right. [Note. If in (I) we had taken the contaur from w—0 to w=e~a and thence to w— -i, we should have obtained the formula containing %JV (z) in place of - i,fv (z). This indicatcn that we get a case of Stokes' phenomenon as y crosses the lino # = ().] 10*44. The relation between Hn (z) and En (z). When the order n is a1 positive integer (or zero), we can deduce from § 10*1 D) that En (z) differs from - Hn (z) by a polynomial in z; and when n is a negative integer, the two functions differ by a polynomial in 1/z.
10-44, 10*45] ASSOCIATED FUNCTIONS 337 For, when n is a positive integer or zero, we have and _ and therefore, since Jw B) = ./„ (z), we have n q\ {m-1) iri A ^yt-Trt. E'1 ^*>=tx r (f- w r (« +1 •} ~ H & that is to say A) B,w^ rlM! »,\ In like manner, when — n is a negative integer, B) B_,(*)- ^ 10*45. !/V<e sign, of Struve's function. We shall now prove the interesting result that !!„(&¦) is positioe when x is positive and v has any positive value greadisr than or (ujual to \, This result, which was pointed out by St.ruve* in the caso u=l, is derivable from a definite integral (which will be established in § 115*47) which is of con- considerable importance in the Theory of Diffraction. To obtain the result by an elementary method, we integrate § H)-4(l) by parts and then we see that, for values of v exceeding1 \, r (J¦ +1) v (i ck ri r™"" - Bi/ - I) I cos (j; cos 6) m\"v-20 cos 0d0\ r4ir — (w —I) <'.oh (."¦ ctis (^) sui""""*^ c< Jo is^jl -cos(.eco«^)J (^ since the integrand is positive. * A"«»i. <'c VAcad. Im\i. den Sri. de St Pctcrdwurif, G) xxx. AB82), no. 8, pp. 100—101. The proof given here is the natural (ixtension of NtnivcV proof. w. ». v.
338 THEORY OF BESSEL FUNCTIONS [CHAP. X When v is less than h, the partial integration cannot be performed ; and, when i/ = A, we have and the theorem is completely established. A comparison of the asymptotic expansion which was proved in § 1O42 with that of Yv (x) given in § 7 1 shews that, token x is sufficiently large and positive, H,, (.«) is positive if i>>? and that Hv (x) is not one-signed when v <\; for the dominant torm of tho asymptotic expansion of H,, (x) is according as v > \ or v < ^. The theorem of this section proves the more extended result that Struve's function is positive for all positive values of x when v >-J and not meroly for sufficiently large values. The theorem indicates an essential difference between Struve's function and Bessel functions; for the asymptotic expansions of Chapter vn shew that, for sufficiently large values of x, Jv (x) and Yv (x) are not of constant sign. 10*46. Theisinger's integral. If we take the equation \ j = / jhn e-*™o log J -h* log 1 + hdz and choose the contour to be the imaginary axis, indented at the origin*, and then write z— ±ttan^</>, we find that and so 7" Vo (X)-I*o (.*)} = / cos (x cot <f>) log tan (|tt+J0) -j-^, A) /0 (a?) - Lo (a1) = -5 I ^ cos («tan 0) log-cot (^) -^ , it j o coscp a formula given by Theisinger, Monatshefte fiir Math, und Phys. xxiv. A913), p. 341. If we replace x by asintf, multiply by win (9, and integrate, we find, on changing the order of the integrations in the absolutely convergent integral on the right, / "Ei (> tan <p) log cot (?cp) -^- = J ( *" {/„ (a; sin (9) - Lo (.« aiu 6)\ sin fldd J o cos <p 2 J o cos <p so that B) f **?!(*tan 0)lqgoot A0) ^ = | .1~'"*, y 0 COS ^5 ¦" '<- on expanding the integrand on the right in powers of x. This curious roault is also due to Theisinger. * The presence of the logarithmic factor ensures the convergence of the integral round the indentation.
10-46,10*5] ASSOCIATED FUNCTIONS 339 10. Whittakers integral. The integral l which is a solution of Bessel's equation only when 2v is an odd integer, has been studied by Whittaker*. It follows from § 617 that, for all values of v, A) V, L* [X e^iVj (t) dt\ = - lim |>*e&« A - &) P'.-M)] I .'-l J f-*--i+o = — cos vir. IT " If we expand the integrand (multiplied by eiz) in ascending powers of z and integrate term-by-term + it is found that (A\ 1 I ~'-i It /.\ 7. r» 1 _."«»» \m1iti) • I/O i The formula of § 3*32 suggests that we write and then it is easy to verify the following recurrence formulae, either by using the series B), or by using recurrence formulae for Legendre functions: 2v f C) Wv_, {z) + Wv+1 (z) = - \WV (z) - D) Wv_, (s) - -v z~i e~iz 2»r) r (|- F) (» _ „) W. (,) — ,ww (,) An asymptotic expansion of W, {z) i'or large values of | z | may be obtained by deforming the path of integration after the manner of Lipschitz (§ 71). * Proc. London Math. Soc. xxxv. A903), pp. 198-200. t By a uso of Legondre'H equation tho recurrence formula j_{(i + 0* ^-i @'"=(-M- +?-t J_t (i + 0«- /Vi @ ^ /I 2 P^, («)d(=———\-p75i v by expand hit,' -1 I ($ +V) L [ft V) F (i - vi i + v > 1» i - 4*) iQ ascending powers of 1 - t, and integrating torm-by-term. 22—2
340 THEORY OF BESSEL FUNCTIONS [CHAP. X The function is thus seen to be equal to Now it is known that*, near t= 1, Py-iit^tFiQ-v^ + v, 1; J-iO. cos x {log f ---) - { \ A J {g f ) ty ( + l) + f (m- v { \ A J and since we obtain the asymptotic expansion G) Wr(«)~lJ5T,«(*) fir+i*) cosi/tt f « (v, rn) 1)— log 25 — ?7™} . Some functions which satisfy equations of the same general type as A) have been noticed by Nagaoka, Journal of the Coll. of Sci. Imp. Univ. Japan, iv. A891), p. 310. 10*6. The functions composing Yn{z). The reader will remember that the Bessel function of the second kind, of integral order, may be written in the form (§ 3*52) w=0 «l! The series on the right may be expressed as the sum of four functions, each of which has fairly simple recurrence properties, thus A) irYn(z)=2{\og(lz)-f(l)}Jn(z)-Sn(z) + rn(z)-2Un(z), * Cf. Barnes, Quarterly Journal, xxxix. A908), p. 111.
10*6] ASSOCIATED FUNCTIONS • 341 where B) Tn (z) = - 'S1 (n -^rJ* (.b)-n+2w and (cf. § 3-582) C) Un (z) = S {JU?J^ {+ (n + m +1) - f A)}. m=0 The functions 7'nB) and Un(z) have been studied by Schlafli, Math. Ann. in. A871), pp. 142—147, though he used the slightly dift'erent notation indicated by the equations Sn(z)=-2Gn(z), 7;B)=2//MC), Un(z)=-En(z); more recent investigations are clue to Otti* and to Oraf and Gublert. The function Tn (z) is most simply represented by the definite integral D) Tn (z) = - r$nr - 6) sin {z sin 6 - nd) dd. To establish this result, observe that m> in-i r (m + L+«) r (n + m -I-1 - e) J t)n+4i»i \og_t . _ 2 r* ^ e"l0(-?:^in^y^^._(^_:-|7r) ¦""'i J o m> -l,i -1 "(n + 2m)! where i has boon replaced by ei"r)i. It follows that S (- ^ .sin 6IH im _ (cosh (- iz sin ^) (?i even) V Now m>-in-i \n + 2w)! Vsinh (— iz sin t9) (?i odd) and so Tn (a) = --. F - W) U»w-f*Hhi« + eMi(«-^)-»-i*Bia«j ^t 7TtJo * i/er« MittheilunriP.n, 1898, pp. 1—56. t Einleitung in die Theork der Bessel'schen t'uaktionen, n. (Born, 1900), pp. 42—69. Loramel'B treatise, pp. 77—87, should also be consulted.
342 THEORY OF BESSEL FUNCTIONS [CHAP. X If 6 is replaced by it — 6 in the integral obtained by considering only the second of the two exponentials, the formula D), which is due to Schlafli, is obtained at once. The corresponding integral for Un (z) is obtained by observing that n (a-- [I 5 (-)w(^)n+2 n W " |> m-o -m! r (n + and so, from § 6 D), we deduce that E) ff»(*) = {16g(i«)-*(l)}/n(*) + - f "(9 sin (n0 - z sin 0) <20 + (-)» f ° vr Jo Jo 10*61. Recurrence formulae for Tn{z) and Un{z). From § 10-6 D) we see that - 0) sin 0 sin 0 - w0). B cos 6 - 2n/z] d6 ?7r - ^) ^ (cos (*sin Q-v^de = -cos2|??.7r-- Jn(z), z z on integrating by parts and using Bessel's integral. Thus A) T,» (z) + Tn+1 (t) = Bn/*) Tn (z) + 4 [cos^nTr - Jn (z)}/z. Again Tn' (z) = - ['(W - d) sin 0 cos (s sin 6 - w0) d0, 7T Jo and so B) Tn_1(z)-Tn+l(z) = 2Tn'(z). From these formulae it follows that C) O + n) Tn (z) - ^f^ (z) - 2 cos21-Mr + 2J« (^), D) (* - n) Tn {z) = -z Tn+1 (z) + 2 cos2 \ nrr - 2Jn (z), and hence (cf. § 1012) we find that E) Vn Tn (z) = 2\z sin2 \rnr + n cos2 \mr\ - 4m- Jn (z).
10*61, 10*62] ASSOCIATED FUNCTIONS 343 With the aid of these formulae combined with the corresponding formulae for Jn{z)t Yn(z) and Sn (z), we deduce from § 10*6A) that F) U-x (z) + Un+1 (z) - {2n/z) Un (z) -B/*) Jn {z), (?) Un.x (z) - Un+1 (z) = 2Un' (z) - B/*) Jn(*), (8) (* + n) Un (z) = zUn^ (z) 4- %TH (z), (9) (^-n)Un(z) = -zUn+1 (z), [cf. §§358A), 3'58B)] i I IM \/ / / ( c \ —*• ..i, / v I I o'x ixvi M **^ "ft V*1/ ~~ **& *J 9t-|-i I *• /• The reader may verify these directly from the definition, § 10*6 C). It is convenient to define the function T_n (z), of negative order, by the equivalent of § 10 D). If we replace 0 by ir — 6 in the integral we find that T_n (z) = • - ['(<? vr - d) sin (z sin <9 + n^) dd 7T Jo 7T J {) and so (ii) r_nB) = (-y^Tn(z). We now define U_n(z) by supposing § 10'fi A) to hold for all values of n ; it is then (bund that A2) tf_B(*) = (-)" \Un(z) - Tn(z) + Stt(z)\. 10-62. Series for Tn(z) and Uu(z). We shall now shew how to derive the expansion A) T (z) = 1& \J ., (z) — J .. (z)\ from § 10'G D). The method which we shall use is to substitute .