or in the way associated with the forcing field exp iqx, 



$ ~ exp {- q 2 x} (9.13) 



It follows from these that 



R is Im s > - min 



R is Im s < + 



\1 + M ' S -/ 



1 - M 

 and that the strip is 



/ k 2 ) 



\1 + M ' q 2/ 



K, 



5 ' ^ttm • V K *■ 8 < + r-r-s < 9 - 14 > 



provided M < 1. 



Take Fourier transforms of (9.7) for y > to get 



*(s,y) = A(s) exp (-£ y) + B(s) exp(+£ y) (9.15) 



c s 



£ is defined in (9.5). We take a cut from s « + k 7(1 - K) to infinity 

 s o 



above the strip D and one from s - -k /(l + M) to infinity below D, and 

 1 ° X » 



define [s - (k +Ms)] 2 to be the branch which behaveo like + (1 - M) 2 s T 



° l 



when s is large, real, and positive, [s + (k +Ms)] T to be the branch 



l 1 ° 



behaving like + (1 + M) T s T for large real positive s. With these branch 



cuts and choices of branch, ?» £„ > for all s in D, just as for y 



in $8. Then since (9.15) only holds in D we have to havs B(s) - 0, 



and now we car. differentiate (9.15) with respect to y, put y - 0+ 



and eliminate the functirn A(e). Thus 



65 



