APPENDIX D 

 DERIVATION OF G^Cx.O.G) FOR NUMERICAL EVALUATION 



We change the contour of the integral path with respect to u in Equation (115) 

 as shown in the following cases. 

 1. X > 



a. First Integral of Equation (115) 



Evaluate the following integral in the complex plane of w 



K exp(-iwx cos 9) 



w - [(K)-'-^V(m)''"^^w cos 0] 



dw 



j + j + f = - 27Ti (residue) 



(D.l) 



^1 "2 ^3 



Because of 



w-PLANE 



1/2 1/2 2 — — 



- [ (K) + (m) u cos 6] = - m cos 6 (w-u^ ) (w-u ) 



the residue becomes 



K exp(-iu X cos 9) -iK exp(-iu x cos 9) 



- m cos 9 (u -u ) ,, , „xl/2 „ .. 

 2 1 L'^(InK) cos 9-lJ 



57 



