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The Cauchy Integral formulas (10.2, 10.3) enable the product 

 decomposition K(s) - K + (s) K_(s) to be effected by taking logarithms. 

 Suppose K{s) is analytic in the atvip, and that |K(s)| ■*■ 1 uniformly 

 as |s| •*■ » in the strip. Suppose further that K(s) j in the strip. 

 Then F(s) * Jin K(s) is analytic in the strip for any branch of the 

 logarithm, and can be decomposed as in (10.1). Define 



K + (s) - exp F (s), K_(s) - exp F_(s) . (10.6) 



Then 



K(s) - K + (s) K_(s) (10.7) 



for s in D, and K + (s) are analytic and non-zero throughout R + , respectively. 



This decomposition is unique up to multiplication of say K (s) 

 by a ncn-zero entire function and division cf K (s) by the same function. 

 It may be necessary to use this freedom to remove non-algebraic behavior 

 at infinity of K + (s) in certain cases. Noble [5] gives several examples 

 of this minor difficulty. 



When K(o) is even, the factors K + (s) as defined by (10.6), (10.2) , 

 and (10.3) have the property 



K + (-s) - K_(s) (10.8) 



If the split is achieved by some way other than use of Cauchy integrals 



it may be necessary to adjust the functions before (10.8) holds. For 



l 



exaov'e* *■£ K(«) ■ (a 2 - k 2 ) T then the "obvious" split is 



i l 



K. (s) - (s +k ) T , K (s) - (s - k ) T , 

 + o — o 



wi/2 



but K.(-s) ■ e (s - k ) , so that we need to redefine K. (s) as 

 + o • x 



74 



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