Lee 



for X ^ 0. This integral is the last term in Eq. (C-7), and we observe 

 that lim I,= - ire'^ysin K|x|. This implies the existence of a sinu- 



Ixl—oo 

 soidal wave with wave number K in the far field. It obviously vio- 

 lates the radiation condition that the outgoing waves have wave number 

 K'. However a careful examination of the integral I shows that it 

 is just one of the homogeneous solutions of the problem which can be 

 discarded, if desired, because of the radiation condition. Thus sub- 

 stituting the expressions obtained above for Ij and Ig into Eq. (C-7) 

 and discarding the last integral in that equation, we find that 



= ■ A Re, re"'''E|(iK'z) , e'^^'^E, (-JK'z) I , AeJ!! 3,„ k' |x | . 

 IT ' L K+K' K' -K J K -K 



(C-9) ^ 



We combine Eqs. (C-4) and (C-9) to finally obtain 



^l^^-y) = K' - K 



■ jA r e''^''E,(iK'z) ^e-'^'^,(-iK'z)1 ^ 

 •' TT ' L K+K' K - K' J 



±iz 

 Since lim e E,(±iz) = 0, we see that 



x|— oo 





as is required. 



950 



