*(s,y) - A(s) e" Y s y + B(s) e Y s y (8.16) 



that will only be possible if the branch cut3 from ± k do not enter 



the strip D. Thus the cut from +k must go to infinity above the strip, 



that from -k to infinity below the strip. No further specification of 



the cuts need be made at this stage, because we shall find a solution 



for s in D, and the values of y a *e already fixed for s in D by the 



i. -I s 

 requirement (s ± k ) 2 - + s 2 as s -•• + °> and by the general location 



of the cuts. 



We can now see that 



< arg(s + k ) < Ti 



(8.17) 



-ti < arg(s - k ) < 

 o 



for all s in D, and therefore 



A 



IT . ,2 ,2.2 ,7T 



- 2 < arg(s 2 - k 2 ) 2 <- 

 or equivalently 



Re Y > for all s in D (8.18) 



which is the essential property of Y . It is possible to choose the 

 branch cuts so that (8.18) holds throughout the entire complex (cut) 

 plane, but there is no need for this since at the moment we are concerned 

 only with values of s in D. Thrn the only possible form for B(s) in 

 (8.16) is B(s) • 0, otherwise * would be infinite as y -*• + «. Thus 



*(s,y) - A(s) exp(-Y 8 y) (8.19) 



55 



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