Expansions of Bessel functions. 237 



Now near r = 0, e 2t = — becomes zero like 



~2n e 



y 2 ' 2 2 T(»r(n + iy 



or near ,s = 0, 



1 7T 



t=zn lo g -J - + - log- 



82 2 6 7lT 2 (7l) f 



and therefore in general 



'- NnT^T" Nf»+*l«8*+} 9 (a! " J)*S 1*1) 



this being the only possible function satisfying all the con- 

 ditions. In a similar way, if t x corresponds to T t as t to T, 



^l l °S^--^+nlo g:+ ^ r iy, (42) 



These two relations cease to be distinct when n is very 

 great in comparison with z, for it will appear that T 2 and T 

 only differ by an amount which is exponentially evanescent 

 when n tends to infinity. 



The third series satisfying the associated equation is 



u _-,. +I fl_±L±l *' 2n + 1.2n + 3 ** x 



8 V » + 12.:'»+r n+l.n + 2 2 .4 . 2n + l .2 .n + 2 '"J 



- r( ",a r y i) -r j -< 2 " k ^ <«> 



But 



,2n+l 



y 



TTZ 2 TT.Z 2n+l ( Z 2 \ 



and therefore by comparison, and by the differential equation, 



2/i 2 = 2S+IT% + 1) = *■£" J *» (^ sin *) d0 ' 



by the use of a well-known property of gamma functions. 



Here 2n must be greater than - 1, in order that the 

 integral may be finite. Thus for all real values of z, and 

 values of n greater than — i, 



7rjJ»= f J 2n (2z sin d)d0,i . . . (44) 



which is a known result for integral values of n. 



