15 



Introducing dimensionless coordinates x = a£ and the expressions 



o(x) = <r Q o*U) 



various forms of the integral for R can be derived for purposes of numerical 

 evaluation. 



We confine ourselves to the source-sink integral. 10 Splitting up 

 <r*(§) into a main antisymmetrical and a symmetrical part 



**U) -**('*)'+»*(«') 



and remembering that an integral taken over an odd integrand between limits 

 of equal absolute value but opposite sign vanishes, we obtain with the desig- 

 nation 



R = 4C 2 7rpg |~y o J exp[-4 j- y Q sec 2 0] sec 3 0. [11 ] 



J^fU) sin(y Q |sec )d|l 2 + j J^^f^) cos (y Q f sec )d*l J 



const J ' 2 exp[-4-£- y Q sec 2 ©] sec 3 0[p* 2 + q* 2 ]d0 [12] 

 We introduce further polynomials for 



/!*(*) = 1 -Z^^ [13] 



n 



hence 



<r*U) = ^n a^ 1 [14] 



or 



**U) = **U) + **U) = -X2ka * 2fe_1 - T (2m+1)a ? 2m [14a] 



a S Ji-d 2k *-* 2m+l 



k rn 



with k, m as integers. 



For the main antisymmetrical part the intermediate integral p* becomes 



P* = J" 1 «7*(0 sin (y $ sec )d$ = -^2k a fc J o £ 2/c_1 sin (y Q | sec0)d£ 

 = - X2k a M t i (y sec 0) [15] 



■^-' ft 2ft - l O 



d0 = 



with 

 M 



_ i (y sec 0) = M 2fc _ i (y) = J"V _1 sin(y Q | sec 9 )d| = J 1 ^ 2 * _1 sin(y^)d| 



[16] 



