Expansion of Bessel Functions of High Order. 699 

 Substituting this expression in the equation 

 d\f 2df , / n.n + l\,_ f) 



it appears that 



EE ;/ -iR /2 + (l-^^ L \.2R 2 -2 = 0, 



where the accent indicates differentiation with respect to z. 

 If this be again differentiated it becomes linear, and yields 



n ; 



./ 1 n. 7i+l\ -pv, , , n . n-\- 1 -r, „ , . 



+ 4\1 ? — JR'+4 — ^— R = 0, . (8) 



which may be integrated in series. 



By reference again to Lomrael's formula*, it appears that 

 when z is very great in comparison with ?i, R takes the 

 value unity. 



The series solution of (8) satisfying this condition is 



-p ., n . n+1 1 n — 1 . n .n + 1 .n + 2 1.3 , /n . 



R = l+ — 2 — .^+ --4 -'2A+-" ( 9 > 



When rc is of order ~, and c — n is not small, this leads at 

 once to 



R= 



V^-n-i-i 2 ' 



as proved by Lorenz. The value of $ in (4) follows by (6)- 

 We proceed to obtain a definite integral for the function R. 

 Writing m = 2w + l, so that m is an odd integer, 



R=l 4- ™ 2 ~ 12 1 m*- l 2 .m 2 -3 2 L3 



4; 2 '2 + (4* 2 ) 2 '2.4 + •" 



where 



a(0) = l + — r -..- sr + .(-&r) +.-(10) 



But by a well-known result |, since m is odd, 



sinhmi m 2 -l 2 . m 2 -l 2 .m 2 ~3 2 . , < 



= sinh*+ — — p- smh 3 /+ ^ smh 5 £-f-... 



* Zoc. c«V. f Ftrfe Chrystal's Algebra, Part II. p. 180. 



