K (s) - <T* m (s + k ) 2 , K (a) - e* 1/4 (s - k ) 2 

 + o — o 



In order that (10.8) will be satisfied. 



B. Decompositions Related to the Square Root Function y 



~~ — — — — ^— — — — — — — ^— — — — — ^ 



As Juat noted, the multiplicative decomposition of y - (s 2 - k 2 ) J 



into factors analytic, non-zero and algebraic at infinity is 



l 

 K + (s) - a (s + k Q ) T 



K (s) - a -1 (s - k > T (10.9) 



— o 



for any constant a. If it is useful t"> require that K + (-a) - K_(s) 

 then a should be chosen aa e 



For the convected wave equation (with subsonic convection veloci- 

 ties) Y is replaced by 

 s 



£ - [s 2 - (k + Ms) 2 ] 2 



for which the multiplicative split is 



l k l 

 K + (s) - a(l - M*) T (a + -gtfp 



k 1 

 K.(s) - « _, (8 -^) 2 (10.10) 



for any constant a. Her* of course ve cannot sake K <-e) ■ K_(s). 



Because the factors K, (s) K (s) for K(s) are analytic and non-zero, 

 the split 'or 1/K(s) ia given by (L/K + (o)), (1/K_(s)). 



The additive decomposition of y and functions related to it arises 

 very frequently. Noble [5] gives several ways of calculating the split 

 functions and several representations of those functions. Here we will 

 Just verify that if 



75 



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