*:? > ; >'«p/«-.^^v-> ■ . t : > - -sf. -:■ > : *■?■*: t «w#jr» 



and the original radiation condition of outgoing waves is seen to be 

 equivalent, for k 2 > and s in D, to the condition that $(s,y) ■* as 



y ■*■ + m. 



The boundary condition (8.6) involves *' (8,0), where the ' indicates 

 3/dy, and therefore we differentiate (8.19) to get the pair of equations 

 (equivalent to the differential equation plus the radiation condition) 



#(s,0) - * + (s,0) + *_(s,0) - A(s) 



*'(s,0) - 4>!(s,0) + * , (s,0) - - Y A(s) 



+ - 6 



where, for example, 



*_(s,0) - J <>(x,0) e isx dx , 



(8.20) 



*;(s,0) - J f* (x.0) e iBi dx, etc. 



In (8.20) two of the functions aie known. From (8.6) we have 



*7s,0) - (for o in R_) (8.21) 



while from (8.7) we have 



V 8 '°>-ys^-qT (8 ' 22) 



for In s > - q, , i.e., for sc.V . Eliminating A(s) between the two 



equations in (8.20) and using (8.21) and (8.22) gives 



- iv 

 K(s) 0;(s,0) + 4_(s,0) - - (B ° + q) (8.23) 



56 



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