up to °°. As s ■> °° with Im s < we need to write x ■ £ - z, to get 



Yj(s) - exp(is£) J y(£. - z) exp(-isz) dz 



and because of the exponential exp(-isz) we can expand about z ■ again 

 to get 



-isz 



Yj(s) - exp is£ J [y(Jl-) - zy' (*-) + .--W 

 ^o 



dz 



exp(isJl) - f y(£-) + 0(- 



(7.11) 



Thus the entire function Y^s) is algebraically small , 0(s _1 ) or smaller, 

 in 1m s > 0, but exponentially large like exp(isH), in Im s < 0. 

 The differential equation 



4-Z + k 2 y = 



dx 2 l 



(where y is the total displacement in < x < I) gives 



(s 2 - k 2 ) Y,(s) - [y'(H-) e is£ - y'(0+)] 



- isly(Z-) e ls£ - y(0+)] 



(7.12) 



and confirms (7.10) and 7.11). Further, Yj(s) can have no singularities 

 for any finite value of s, so that the right side of (7.12) must vanish 



for both s ■ k and s - -k , giving 



[y'(Jl-) e ik i* - y'(0+)] - ik^yU-) e 11 ^ - y(0+)]» , 



(7.13) 



[y'(Jl-) e" lk i £ - y'(0+)]+ ikjy^-) e~ ik > £ - y(0+)] - . (7.14) 



37 



^^^±££~.^^^^^~-^ ri . ^ ,^ — -mmsmmm 



