+ k )\K_<8) ~K_(-k )/ + (s + k o )\(-lc o ) (3 ' 4) 



(8 + k ) K (s) (s 



x O - .v.— - X)V o - o 



The first term no longer has a pole at s ■ -k , for near there 

 - (s + k )x (function analytic at -k ) , 



K_(s) K_(-k Q ) 



and it is therefore a Qf unction G (s) aay, while the correction term 



i/(s + k ) K (-k ) is evidently a© function, G,(s) say. Ncte that 

 o - o ^ + 



this cdditive split is not restricted to any particular form of K_(s) , 



but turns on the presence of a pole term (s + k )~ l only. 



o 



We now rewrite the equation (3.3) as 



Y (s) 

 K + (s) Y + ( 8 ) - G + (s) - G_(s) - £^-y , (3.5) 



and consider the function E(s) defined by 



S(s) - K + (s) Y + (s) - G + (s) . (3.6) 



This is a function defined and analytic throughout R and of algebraic 

 beh&vior (algebraic growth or decay) at °° in R . E(s) is not defined 

 by (3.6) except in R . However, within D, E(s) can equally well be 

 defined by 



y (s) 



E(s) -0_(.) -fjfi (3.7) 



and this definition then CONTINUES ANALYTICALLY the function E(s), 



defined originally only in R by (3.6), through Cue strip D of overlap 



into the lower half-plane R , and there E(s) is also analytic end of 

 algebraic growth or decay at infinity. 



15 



*■""--'-■■ r "V ■-' 



