This equation simply states that the corresponding source at the origin is obtained 

 by integrating the distributed source along the contour with the weighting function 

 of 2e cos Kri. 



DERIVATION OF THE KERNEL FUNCTION 



14 

 The inverse Fourier transform of Equation (42), we can express f (x) as 



f(x) = ln{2K) 6(x) + — —+ TT G^Cx.O.G) - ifi 6 (x) (106) 



2 Ix I 



where 6 (x) is Dirac's delta function and is defined by 1 where x = 0, and when 

 x ?^ 0. From Equations (C3 through C4) and (35), G (x,0,0) is 



00 ot 



G3(x,0,0) = -^ e"^^ dk I 



K di 



2 2^/2 2 2 ^/2 

 -^ (k +£ ) [(k +£ ) -k] 



1 I J I (K +m u cos 9) exp(-ixu cos 9) j„ /-,^-,n 

 = — ^ du '=- — d9 (107) 



2tt J J ,,,1/2. 1/2 



u - (K-'-^^+m-'-^'u cos 9) 



2 



where m = U /g. 



The poles of Equation (107) are 



/''2\ 1/2 1/2 ^/^ 



/ \ 1 - 2(mK)^ cos 9 ± [1-4 (mK)^ cos 9] 



\ Ut / 2m cos 9 



(108) 



The integral path of Equation (107) is determined by the value of cos 9 as follows; 



28 



