Expansions of Bessel Functions. 235 



Some of the special cases in which 2n is an odd integer 

 are very interesting, as the integral can then be expressed in 

 terms o£ trigonometric functions. 



Since by (4, 16) 



J P _1 J 



d: ~ R 



and when z is very great in comparison with n, the usual 

 expansions yield 



it follows that in general 



Second Solutions. Values ofT, t when n is not integral. 



The second series solution, when n U not integral, is o£ 

 the absolutely convergent form 



-?> 1 ° 

 M -. 4.J - . X '° * 4. 



*nr—V 2.4r-h?r-3' 

 Now if 



c m = I sin -//.'■ ^'m- , \cdx, 



where n is not, and m is an integer, then it is readily shown 

 that 



— 2m.2m— 1 



= - - — — -1 sm2wfl^, 



^ 2 "';r — nr. n' — m — l'-. ... vr — 1\" 



whence 



2nz C*(i (2zsinv) 2 (2.:sin.r) 4 \ . ' , 

 1— cos27i7Tj \ 2^ 2-4- / 



= = — ^ — I sin 2twb . J f ,(2r sin a?) dx. (36) 



1 — cos2imtJ u 



It remains to identify n 2 . Now making a substitution 

 (a modification of the third asymptotic substitution), 



y\ 



\t) Jn ^' i ' /s= Cy) cosec > i7r - J -»( : )> 2/i2/s =Ti. (37) 



