182 Miss Dorothy Wrinch on a G eneralized 



sense, and therefore by considerations which are sufficiently 

 well known in applications of this method, the important 

 "groups of errors" in the divergent development of the 

 functions all arise from 1 1% We shall, however, show 

 independently that I 2 is negligible compared with Ij. 



In considering I 2 we again choose a circle with centre £ = 

 and radius 8 as the contour, but are free to choose 8 as we 

 please provided only that 8 and \z\/8 are large when \z\ is 



g e - 



Then 





) e ntr n t- s n X ~dt 



5 









K> C ~ 







< 











Z r Jc 







< 



9«- ^i n • I S 



— e vo 8~ n e~ n n 8\ 



-. i r 



<* 1 



But let 8 be so chosen that 

 Then 



-r, — ** 



C Jr S t r , I -p. „ „[2(»+U „ 2(»fl) 2(^T) 



Jc Z 1 i 



Whatever the value of r, this is negligible compared with 

 any of the terms 



e {n+lZr) 



in which R(r r )>0. Hence I 2 is negligible compared with Ii 

 and the asymptotic expansion for F n results in the form 



n-r(i+^) 

 oF.(x +«i, . . . i +«» ; z) - -J^vn+i 



|mk(»41*,.)| < g 



This result is of a similar form to the one assumed for 

 F n _i and already proved for F 3 , and is therefore now 

 established in the general case. The complete form is 



