42 Mr Watson, Bessel functions 



Bessel functions of equal order and argument. By G. N. 

 Watson, M.A., Trinity College. 



[Received 1 November 1916: read 13 November 1916.] 



A proof of the approximate formula 



Jn{n)' 



TT 2» 3« w« 



(the order and argument of the Bessel function being equal and 

 large) was apparently first published by Graf and Gubler*, 

 although the formula had been stated by Cauchyf many years 

 before. The formula has been discussed more recently by 

 Nicholson J and by Lord Rayleigh§, while Debye|| has given a 

 complete asymptotic expansion of Jn{n) in descending powers 

 of 71 ; this expansion is obtained by the aid of the elaborate and 

 powerful machinery which is provided by the mode of contour 

 integration known as the "Methode der Sattelpunkteir"(Methode 

 du Col, method of steepest descents). 



The earlier writers, just mentioned, employed Bessel's formula 



1 /■'" 

 Jn (*') = — I COS (nO — X sin 6) dd, 

 ttJo 



valid when n is an integer, and it is by no mea.ns obvious to what 

 extent their methods of approximating are valid**. 



As the correctness of the approximation can be established 

 without the use of contour integration on the one hand and 

 without appealing to physical arguments ff on the other hand, 

 it seems to be worth while to write out a formal and rigorous 

 proof (based on comparatively elementary reasoning) that, when 

 n is large and real, then 



* Einleitung in die Theorie der Bessehcheii Funktionen, i. (1898), pp. 96 — 107. 



+ Comptes Rendus, xxxviii. (1854), p. 993; Oeuvres (1), xii. p. 163. 



J Phil. Mag., August 1908, pp. 273—279. 



§ Phil. Blag., December 1910, pp. 1001—1004. 



II Mathematische Annalen, lxvii. (1909), pp. 535 — 538. 



1[ This method of discussing Je"/W<^(s)(fi; consists in choosing a contour on 

 which If{s) is constant, and so Bf{s) falls away from its maximum as rapidly as 

 possible (/(s) being monogenic); it is to be traced to a posthumous paper by 

 Eiemann, Werke, 1876, p. 405. 



** See § 4 below. 



ft For example Kelvin's "Principle of stationary phase" {Phil. Mag., March 

 1887, pp. 252—255 ; Math. Paiyers, iv. pp. 303—306) is really based on the theory 

 of interference. See also Stokes, Camh. Phil. Trans, ix. (1850), p. 175, foot-note 

 (Math. Papers, ii. p. 341). 



