348 



THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



k < 1. The eccentricity of the ellipse varies with k; in the hmit ^ ^ 1 the 

 elhpse degenerates to the segment of the real frequency axis | co | < coo . 



Now for a given transmission function inside C, Fi(p), the complex po- 

 tential inside Ci is Fi{w) = Fi[T(w)]. In general this function will have 

 singularities inside Ci , but when Fi(p) may be expanded in a power series 

 in p we may use the separation theorem of the last section to obtain a simple 

 formula for the charge distribution on Ci . For instance, let Fi{p) be a poly- 

 nomial in p, 



Fi(p) = T.anp\ (85) 



n 



then 



.:.w = i:4»yt-y". (86) 



When the binomial is expanded in a power series the terms involving positive 



p PLANE 



Fig. 17 — Elliptic contour in the />-plane. 



powers of w will belong to Fa{w)f while the terms involving negative powers 

 will belong to Fb(w). Hence the parts of Fi{w) analytic respectively inside 

 and outside Ci are 



..« = r..(|)-[©--.©-% 



n(n — 1) 



2! 



(r-] 



n-2 





+ 



2! 



