o-n(e,;;€) de d; 



Ogilvie 



expansion is just the sun? of these: 



♦(x,y,z;c)~Ux-^ Z 3 3 TTlfV^mfF ' ^^' ^^ 



^ ^-(o^-cD L(x-§) +y + (z-0 J 



This is the most general possible outer expansion for this problem. 



It will be necessary presently to know the inner expansion of 

 the above outer expansion. To find it, define an inner variable: 



Y = y/€, (2-9) 



substitute for y in the outer expansion, and re -order the resulting 

 expression with respect to e. A direct approach to this process is 

 difficult, but the following method, in four steps, allows us to obtain 

 the desired results to any number of terms in a fairly simple way: 



1) Take the Fourier transform of ^^ with respect to x: 



■ikx 



* 1 r"' * r 



4>n(k;y,z;c) = - ^ \ d^ a^ (k;^;c) \ 



dx e 



-00 [x'^ +y^ + (z-Cn' 

 = - ^ £^d^ cr*(k;;;c)Ko(|k|V[y2 + (z-02]) 



where Kq is the modified Bessel function usually denoted this 'VJa.Y t 

 and (r*(k;z;€) is the Fourier transform of the function o-^{x,z;c). 

 The convolution theorem was used in the first step above. 



2) Take the Fourier Transform next with respect to z: 

 (|>n (k;y;m;c) = - -^ij^ (Tn (k,m;c)j dz e ^KQ(|k|Ly +zj ) 



** /. 2 2,1/2 , , 



0-n (k,m;€) -(k + m) lyl 



- ■ ... 2 . 2.1/2 ® 

 2(k Tm ) 



where <x^ (k,m;€) is the double transform of (rn(x,z;6). 



3) Substitute: y = cY and expand the exponential function 

 into a power series: 



682 



