Expansions of Bessel Functions. 243 



Expansions when n is greater than z. 

 When n is non-integral, in the notation oil (36), 



T x = -z — - — I sin 2nx . J Q (2z sin x)dx 



I — COS ZllTT J n 



= ^(r-cos'2;r 7 r)j C*+W> • 



(62) 



whe 



re 



('it 

 sin 2/i(£ : | : /a sin #)<?#> . . . (63) 



v 



r sin 6 

 //.= > 1 



ft 



Now ~-(x±fi sin .?;) is never zero, or a very small quantity 



lor any possible values of x and ^ in the double range or 

 integration. Therefore the integration of I\ and L may be 

 effected by the method of the last section, and in so far as 

 the leading terms are concerned, 



C n 1 

 I 1} L= 1 ----- — (1 +yL6 cos x) sin 2n(x ±fi sin x)dx 



Jo ■ L i^' 



i C n , 1— cos2>*7r 



= x .1 sm2ntdt— j——, — r— r« 



is 



Tl= ir#( ] . ^+--1^ 



4ttL r V^ + ^ m< /> >i-~sin<f>/ 



'o 



_1 s_ 



- 2 '(,rW)i' 



on reduction. 



The leading term of — is therefore (>t 2 — ~ 2 )~*. Let its 

 expansion as a series be of the form 



5 X X* X° 



where x — (n 2 —~ 2 )K -j^ 



It is a solution of the same differential equation as — 

 which had the value when n<z, 



R 1 , \-2 , 



~ (* 2 _w 8 )i (s 2 -ft 2 )i 



K2 



