232 Dr. J. W. Nicholson on the Asymptotic 



when it ceases 1o be small. As defined above, 3 n (z) admits 

 the integral formula* 



■J»W=f cos(*sin0-w0)rf0--sin»wrf d0 .e-"*-***', (17) 



z being positive, whilst ?rJ_ M (z) is represented by the same 

 expression with the sign of: n changed. 



If n be an integer we select, as standard solutions, 



n= (^fj,U), (18) 



»— (s#*.w. (19) 



where Y w (z) is HankePs second solution of BessePs equation, 

 defined by 



r»(*)=(^^-(-)»5%^) . ■ • • (20) 



v ' \ On v ' on ]u- integer 



By proceeding to the limit when n is integral in the 

 former case, it is at once obvious that this substitution is 

 the natural continuation of the first. Thus it is again true 

 that 



Wl'-*/l*//=l; (21) 



and, therefore, when n is integral, the expansions deduced 

 for (J- n (z)— cos mr J n (z)) cosecn7r will remain valid for 



— -Y„ (Y). This is an obvious property of HankePs expan- 

 sions when n is small. When z is small, Y n (z) may be 

 written 



Y„(,)=2 JK (,){ 7+ lo g i}-(J)--(^ + ^(|) S + ...) 



AYS, , /a\'+» 8,+.S»n ™ 



-\2)nl + \2) « + l!l!~-' ' {Zl > 

 where 



7 =— 577..., S.-l+J+*+-+S 



and negative factorials are to be taken as zero. 

 The asymptotic expansion when z is large isf 



(0Y< Z )= -U.W cos (*- f + J) + V*) sin (,- f + j),(»j| 



where U«, V» are as before. 



* e. g. vide Whittaker, Modem Analysis, p. 281. 

 f Hankel, Math. Ann. i. p. 494. 



