j SSfgSK »;.«s»*ws»>.w» 



2ik 



v f a \ » _ 2 



V s ' (6 + k)(k + k) 



10 l 



4 21k 



V (s> 



s+k (s+k)(k +k 



l> V' k o) 



(3.12) 



Before proceeding to the inversion of the Fourier transform we 



return to the question of the uniqr.t .-.ess of the K (s) . Let one specific 



factorization be K (s) K (s). Then in any other factorization, the 

 factors can be written 



[A + (s) K + (s)][A_(s) K_(s)] 



where, since K (s) are analytic and free of zeros in R + , A + (s) must 

 also be analytic aud free of zeros in R + , respectively. Further, 



A + (s) A_(s) - 1 



for seD, and because A (s) has uo zeros in R_ 



V B> " a4sT 



again for seD. Define 



F(a) - A + (s) 



F(S) " AlsT sea - 



Then F(s) is analytic throughout the whole s-plana (on ENTIRE function) . 

 Suppose further that the factors are required to have some .. >*cified 

 algebraic behavior at infinity in that respective half-plane. Thea A + (s) 

 each tend to constant values at infinity in R + , and the entire function 

 F(s) ha3 constant values everyvhera at infinity. By Liouville'a theorem 



18 



