V*"^^9SWW»W)»*» 



By the usual arguments we then have the solutions 



K + (s) Y + (s) - G + (s) - P(s) 



G - (8 > - Hi} " p(s) 



where P(s) is a polynomial. And because G + (s) » 0(8 -i ), K + (s) - 0(1) 

 and Y ± (s) - 0(s~ l ) (because y is finite at x - 0) at infinity in their 

 respective half-planes of analyticity, the polynomial P(s) must be 

 identically zero, so that 



(6.18) 



Y ( S ) m J Z-J 



V s ' (8 + k,)(8 + ik,) 



a s + b 



Y (s) 



(s - k )(s - ik ) 



(6.1?) 



The inverse transforms can thea be performed explicitly, the poles 



s - k and s n ik in R gives rise to the reflected wave and a 

 o o + 



decaying mode in x < 0, the poles s - -k, and -Jkj gives the trans- 

 mitted wave and a decaying mode in x > 0. Specifically we find 



(a,k, - b.) 

 y(x > 0) - i - ^> + 1 exp(ik,x) 



(a,ik, - b,) 

 + l k, (1 - i) «*P<- k ,x) 



(a k + b ) 

 >(x < 0) - i — — - — exp(-ik x) 



k o (l - i) 

 (a ik - b ) 



- i 



exp(+k x) 



k o (.l - i) 



which is of the anticipated form (6.2). For any specified boundary 

 conditions the values of y and its first three derivatives are known 

 at x ■ + , so that the coefficients a , b , a , b in (6.20) can be 



30 



(6.20) 



ft i .«wiih il i | -Yhtt « iV«< i , l > . ti-,r' 1 MiVai n .v^ 



