344 



THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



We are interested particularly in applying (67) to a point just outside 

 the unit circle in the 2£;-plane. To do this we place the point w on the circle 

 and then keep it just outside the actual contour by introducing an infinitesi- 

 mal semicircular indentation as shown in Fig. 16. Over this semicircle the 

 integral may be evaluated by writing \ — w = 8e^", where 5 is the infinitesi- 

 mal radius, and assuming that FeiX) is practically constant; then its value is 





iFXwl 



(68) 



' ^ r/- 16*' 16 W 16 w3 

 3r(w) = ^W-AJ.4.J5J5 



Fig. 15 — Illustrative contours for the transformation. 



p = T(w) = aw 1 ; 



w vf 



Then over the contour C which is the unit circle excluding the indentation, 

 (67) becomes 



in Jc'\ — w 



(69) 



If we now interpret Fe(X) as the exterior complex potential of our charge 

 distribution problem, on the circle, and introduce angular coordinates 



X = e^\ w = e»>, (70) 



