Cox 



Simplification of U3(r*)p^ and ^(r*, 0*)^^ 

 By applying Eq. (28) to Eq. (18) 



"a('-*)ps 



\ 2 Q r M r ■" r r*" 



q=l m=l 



dr 



(B5) 



M N r_^ ^9^ r ^m 



,CTq,T) dT 



d^drV , 



where 



f (r,r*,cr ,t) = 



[\.2t2 - 2rr* cos ^ + v^ + r*^]' 



and 



di - T + a 

 ^ q 



The integral associated with b^ can be integrated with respect to r to give 



(r^- r* cos i/;) 



r . j- 



J (\,2T2+r*2 sin2 0) |j-^_: 



^r^ - 2r r* cos li + r ^ + r*^1 

 m r- m J 



(r^.j- r* cos 0) 



(B6) 



dr 



[K,^r^- 2r^_,r* cos ^ + r^.^+ r*^]' 



For the integral associated with c^^, 



d^ = A0 J f (r,r*,a-q,^) d^ 



J J f(r,r*,aq,T) dr 



(B7) 



f 



+ J (^-^n-i) f(r,r*,a^,^) dd 



after integration by parts. Although the integrations with respect to r can be 

 solved analytically as elliptic integrals, it may be more straightforward to 

 simplify f (r, r*, cr^,d) such that integrations with respect to can be per- 

 formed analytically [18]. For the integral with finite limits, this can be achieved 

 by replacing with 0^.i + a and substituting cos a =^ 1 - (a^/2), sin a - a. 

 Hence, 



956 



