Expansions of Bessel Functions. 249 



provided that 



p=±,r+§(*-»)t(J)* (84) 



when a is less than z. 



The corresponding values from expansion A are, retaining 

 the leading terms 



R= 



( ^-nt)l 



pss^-w^+nBin-^-JiMr+^r, . . ( 8 5) 



But when « and z are nearly equal, if sin -1 - = — — e. 



Then 



n . 1 . N , , v 3 



cos e=- ? sin e= (: — ra)3 (; + np ; 



and 



p=2(sin e — e cose) + J.7T 



2\* 



+«(*-*)*g)* (86) 



on reduction, which is the value furnished by expansion C. 



Accordingly there exists a region in which either A or 

 C may be used. Similarly, such a region exists for C 

 and B. 



Finally, the scheme of expansions A, B, C is complete 

 for a real argument. For the case of purely imaginary 

 argument, only one set of expansions is necessary. These 

 may be expressed in terms of the same coefficients (X, fi) as 

 AandB*. 



Trinity College, Cambridge. 



* Cf. British Association Report, Dublin, 1908. 



