APPENDIX A 

 DERIVATION OF G (y,z) WITH EXPONENTIAL FUNCTION 



Equation (30) can be rewritten as 



G2i,<y,z) . 2 



L. r 



2TT J 



kz / iky, -ikyv 

 i (e -^+8 -^ ) 



k-K 



dk 



(A.l) 



For the first term of Equation (A.l), we change the variable as 



w = -i(k-K) (y-iz) 



(A. 2) 



and 



dk = 



-dT, 



i(y-iz) 



(A. 3) 



The contour of the integral path will be different depending upon y < 0. When y > 0, 

 we take the integral path in the following figure 



iK(v-iz) 



K 



k-PLANE 



w-PLANE 



By substituting Equations (A. 2) and (A. 3) into Equation (A.l), the first term 

 becomes 



kz+iky 



k-K 



dk 



= 1 



iK(y-iz) 



iK(y-iz) -w 



(A. 4) 



49 



