A similar expression can be given to the potential (J). 



By substitution of Equations (52) and (53) into Equation (46), the outer approxi- 

 mation of V . is 

 J 



■"j =^j Sd + ^j Sd-^^j« (^jSd-^jV 



= ^j ^2D + ^j Sd + ^j(^) (^j+^3^ Sd + ^j(^> (Sd-Sd^ ^j (5^> 



From Equation (34) , the last terra of Equation (54) is 



- „ ., Kz-iK yl Kz+iK y L „ . Kz ^ _. 

 2D ~ 2D " ^^^ ' '+e i-'i) = 2ie cos Ky = 2i 



By substitution of the above expression into Equation (54) , the outer approximation 

 is finally given by 



"P. = [a.-KJ.+C.(x) {.0 .M5 .)\ G„^ + 21 C.(x) a. (55) 



J 3JJ Jj2D J J 



The Fourier transform of Equation (55) becomes 



^ .* = [a.+a.+C.(x) (a.-hj.)]* G2J) + 2i[C. (x)a.]* (56) 



MATCHING 



The inner and outer solutions are matched in a suitable overlap domain to 

 determine the unknown source strength, q., of the outer solution and the coefficients 

 C. in the inner solution. From Equations (45) and (56) we have the following 

 relation 



13 



