10. CONSTRUCTION OF W-H SPLIT FUNCTIONS 



In thia section we first outline a general method for effecting 

 either the additive or the multiplicative decomposition of a function 

 analytic in a strip, and then we set down some properties of the functions 

 which arise frequently in acoustics problems. Finally, we record the 

 corresponding properties for strictly incompressible flow problems. 



A. Cauchy Integrals 



Let F(s) be analytic in some strip D; e ( < Im s < C 2 and let R + 

 denote the domain Im s > e , R_ the domain Im s < e . Suppose also that 

 |F(s)| ■* uniformly as fast as |s|~ for some X > as |s| ■*■ » within 

 any closed region within D, i.e., as |s| ■*■ «■ with 



Then 



F(s) - F + (s) + F_(s) (10.1) 



for s in D, where 



is analytic and bounded in R , 



I 



».«■» -Hf^* a0 - J> 



is analytic and bounded in R_. 



The path \j runs from - • to + «• in D below t ■ e, while 



—/~\—tvrb from - <*> to + «• in D above t ■ s. 



Suppose further now that |F(s)|-*0 uniformly as fast as |s| 

 for soma u > as |s| -*-ooin the strip D. Then j F(t)dt converges 

 absolutely, end 



71 



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