.-:;s , .'-:*a.-.tBas«rj»»vsn>»> i 



with V + (s) - 0(a -1 ) at infinity in R + . On the other hand, Che function 

 occurring en the right of (7.24) is a© function — but it is exponentially 

 large, like exp (s 2 £) in the uppei half-plane, and so it is necessary 

 to make a split 



ie 



-is* 



(s + k ) K x (s) " V 8) + H - (8) 

 o + 



(7.26) 



with H + (s) ■ 0(s~ ) in R + in order to remove this exponential increase. 



Now we can split (7.24) into©andQparts, each of which is 0(s -1 ) 

 at infinity in R + and each of which therefore vanishes identically. 

 Thus we get 



H (s) - V (s) - e is£ K (s) Y, (s) - , 



(7.27a) 



Z,(s) 

 ^OO + V + (b)-H + (.)-0 



(7.27b) 



The equations (7.23a,b; 7.27a,b), obtained by the W-ll argument, are not, 



in general, solutions to fhe problem, but they constitute a pair of 



integral equations which are in a form suitable for approximate solution 



in the "high frequency limit" (i.e., for k I » 1, k.fc » 1 in general). 



o * 



To see this we have to use the general formula (see Section 10) for 

 expressing a function F(s), analytic in the strip D and with suitable 

 behavior at infinity in D, as the sum of functions analytic in R + and 

 0(s _1 ) at infinity there. If 



F(s) - F + (s) -t- F_(s) 



where the functions are to have the e'eated properties, then 



42 



^^'^■'■""V'ii-'-./i'^ffBiaffi.-m^'-i 



