and each side is the representation, in R or R as the case may be, 

 of a single entire function which behaves like s -1 everywhere at 

 infinity, and is therefore identically zero. Hence 



K + (s) Y^s) - G + (s) + U + (s) - (7.23a) 



Y_(s) 

 G.(s) - ^j - U_(s) - . (7.23b) 



Now return to the generalized W-H equation, and this time, instead 

 of dividing by K (s), we divide by exp(is£) K (s) to get 



-let 



Z (s) e "* Y_(s) , . -1st 



MsT + -nT(sl + e M')M')- (.; k )r (> ) < 7 ' 2 '>> 



T T O + 



The first term is analytic in R^ end 0(s -1 ) at infinity there whil s 

 the third is analytic in R and 0(s -1 ) there only because of the factor 

 exp(-is£). If we had not divided by exp is£ we would have been left 

 with a third term which was analytic in R_, but exponentially large 

 at infinity in R_ (eee Eq. 7.11) and the function theoretic argument 

 would not go through (in particular, the entire function would not be 

 zero, but would be exponentially large in R_, and there is no general 

 way of constructing functions of this kind). , 



The function appearing second on the left in (7.24) has mixed 

 properties, so we again try to split it as 



e Y (s) 



YJfi - V + (s) + V_(s) (7.25) 



41 



Saaa &i&aiiaKMaa^^ 



