334 Dr. T. H. Havelock on 
This series is convergent for k less than unity ; it converges 
rapidly, and in most cases we shall find the first three terms 
sufficient. 
Consider now the integral 
Jo 
(5) 
I£ p is large we can easily find a suitable series for I in 
ascending inverse powers of p ; we substitute for the Bessel 
function its equivalent series in ascending powers of //. and 
then integrate each term separately. Then, since we have 
Jl W - s =o (_1) .!(.+*)!{ (.+1)!}»W ' 
and 
o 
1 
-PtA 2S-. C2s) ! 
P 
we obtain the series 
i 
- ffl)M , -, y {(2«+2)'.}» i 
(6) 
?=0 
JlW 7",=o ( } 2 2s+2 5 !(, + 2)!l( 5 + l)!}V s+2 ' (0 
/ JlW ?".o ( ij 2 2s+2 ,!(, + 2)!{(, + l)!}y s+1 
" 4p 8/ "*" 32/ 128 p 7 + w 
The series are convergent forp>2. 
Further we have * 
Ji(/A)Ji(\/i-) = X 2 (-1) jfl^ + l)! V2J ' * ( ^ 
* Nielsen, Cylinderfunctionen, p. 20. 
