106 BELL SYSTEM TECHNICAL JOURNAL 



From Hobson's contour integral definition ^ of PiJ^iO it can be 

 shown that if R(m) < 1/2 



i^i;2W - Vx r(l/2 - m)~" J, V.^^^^^ ' 



where arg (w — 1) = ip, arg (w — /) = — tt + (^, where <p is the angle 

 measured counter-clockwise from the positive direction of the real axis 

 to the line directed from w = I to w = t. Setting m = -\- Xjl — n 

 where n is a positive integer we obtain 



Ji Vw^ — 1 \ 2 



Thus Equation (12) becomes 



= pil^ir 





(13) 



where in passing from the first to the second line we have set w = 

 in ^ 



1 / / _ 1 \ «/2-l/4 / 1 — / \ 



^-'- « = mr+l^A—x) K'/2, 1/2; „+ 1/2;^), 



and have summed the resulting series to show that P-ii2iki/s) 

 = ^IJttt. The function K(k2) may be obtained from (13) by inter- 

 changing k\ and ^2. 



Combining (13) and (11), and using r = ki/k^ gives the convergent 

 series for ni(r, 0) given in the statement of results as equation (14). 



Another Power Series for / 



Here we obtain an expression for / somewhat similar to the one 

 obtained in the previous section. The first step is to deform the 

 contour joining the points A and B (w — kjs and w — kijs). The 

 deformation is carried out in two steps shown in Figs. 2a and 2h, 

 respectively. 



In Fig. 2(a) the contour joining A to B has been pulled around the 

 point + 1 and looped over itself. The point // is destined to move 



3 E. W. Hobson, loc. cit., p. 188. 



