where u^ and u are given in Equation (110). The first integral in Equation (D.l) 

 is the same as the first in Equation (115). The second integral along the contour 

 c becomes zero 



-it/ 2 ^g 



I , , , T . r K exp(-iRe x cos 6) ^ 10 . ,„ . „ 

 I ( ) dw = lim ^ .Q „ Re i d6 ^ 



J R-Ko j Re^^ - ( )2 



^2 



The last integral of Equation (D.l) becomes 



f f K exp(-iivx cos 6) i dv __ j iK exp(-vx cos 6) dv 



^ ^ '^^ " 1/9 1/9 ^ J 1/? 1/? ^ 



J J iv - [(K)-'^^+(m)^^iv cos 0] o iv+[(K)^^^-i(m)^ v cos 0] 



C3 



Therefore, the first integral of Equation (115) is given by 



exp(-iu x cos 0) 



J 1/? 1/? ^ 1/2 "*" 



o u - [(K) '^+(m) ' u cos 0] [4(mK) '^ cos -1] 



K exp(-iux COS0) , „,, 2 

 *- du =- 2ttK 



j IK exp(-vx COS 0) J 



J 1/2 1/2 ^ 



o iv +[(K)^ -i(m)^ v cos 6] 



(D.2) 



b. Second Integral of Equation (115) 



Evaluate the following integral in the complex plane of w 



K exp(-iwx cos 0) , 1,1,1 -^ ■ r ■ a \ 

 *^ — — ^ dw = I +1 + j = - 2tti (residue) 



1/2 1/2 J J J 



7 - [(K)-''^+(m)^ w cos 0] c^ C2 c^ 



58 



