Mr Watson, Bessel functions of large order 97 



The approximations which I shall obtain are derived by 

 shewing that certain integrals of Airy's type* are effective 

 approximations to the integrals which occur in Debye's analysis. 

 It will be assumed that the reader is familiar with Debye's 

 memoirs, although it seems desirable to modify the notation to 

 a considerable extent. The formula for Jn (nx), when x^l, is 

 of importance in connexion with the maxima of the Bessel 

 function f. 



The two formulae which will be obtained in this paper are as 

 follows : 



(I) When a ^ 0, 



Jn ( n sech a) ~ 27r~^ 3 " ^ tanh a 



X exp [n (tanh ol + ^ tanlr' a — a)} . if ^ {^n tanh* a), 



where the error is less than 37i"^ exp [?i(tanh a — a)|, and K,,,{2) 

 denotes the Bessel function of Basset's type (see § 6). 



(II) When ^ /3 ^ \7r, 



Jn {>i sec /3) ~ ^ tan /3 cos [n (tan ^ — ^ tan^ yS — /3)} 

 X [J_ I (-3-*^ tan=^ ^) + /, (i/i tan* /3)] 



+ 3" ^ tan yS sin [n (tan ^- ^ tan* j3 - /3)} 

 X [/_ , (iw tan* /S) - J^ O tan* y8)], 



where the error is less than 24/?i. 



Part I. The value of Jn(n.v) ivhen O^^-^l. 

 2. We take Sommerfeld's integral 



The stationary points of x sinh w — w, qua function of w, are 

 given by cosh w — l/x ; accordingly we replace x by sech a, where 

 a ^ ; and then, putting w^ot + t, we have 



jiTTl J 00 -ni 



+ n (sinh t — t)] dt. 



The exponent has a stationary point at ^ = 0, and the method 

 of steepest descents provides us with the contour whose equation is 



/ {tanh a (cosh t-\)-\- (sinh t-t)] = 0. 



* These integrals have been expressed in terms of Bessel functions by Nicbolsou, 

 Phil. Mag., July 1909, pp. 6—17, and by Hardy, Quarterly Journal, xli. (1910), 

 pp. 226—240. 



+ Proc. London Math. Soc. (2), xvi. (1917), p. 169. 



