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THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



phase delay. This method is of general application in changing the phase 

 delay, and the corresponding networks are easy to obtain. 



11. The Inversion Theorem for a Circular Contour 



An alternative derivation of the exterior stream function for a circular 

 contour of radius coo is based on the method of inversion, in which p is re- 

 placed by oil/pj Fig. 14. This transformation maps the region inside C on 

 the region outside C and vice versa. Points on the circle remain on the circle 

 but are transformed to the conjugate complex points. 



Now suppose that the transmission function Yi{p) is defined inside the 

 circle as an analytic function of p, and that it satisfies the conditions for 

 physical realizability. Then if we have a unit charge at some complex point, 



Fe(P) = Fi,(a;g/p) 



Fig. 14 — The inversion theorem for a circular contour. 



p on the circle there must be a hke charge at the conjugate complex point, 

 />*, while the total charge must be non-negative. For simpHcity we may 

 assume that the total charge is zero and then the exterior function Fe{p) 

 must be analytic outside C. We wish to show that Fi(col/p) may be inter- 

 preted as the exterior function. Obviously since Fi{p) is analytic inside C 

 we must have Fi{o}l/p) analytic outside C. Hence it will represent the ex- 

 terior function outside C if it represents a function whose potential is the 

 analytic continuation of the potential inside C. 

 On the circle we have 



\p\2^ pp* = ^l ^ or /»* = cooV/'. (58) 



But we have akeady seen that when the complex zeros and poles occur in 

 conjugate pairs we must have \Fi{p)]* = Fi{p*). Hence on the circle 



FiifA/p) = Fiip^) = \Viip)]* = Vi - i<^i . (59) 



