354 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



at zero and infinity in the >-plane. There will be no branch points if the 

 original transmission function Fi{p) is an even function of p, for then the 

 exterior function Fe{p) will also be an even function. In this case the simple 

 closed curve analysis does not have to be modified. The usual method can 

 be used to determine ^e(w) in the le-plane, and the charge distribution on 

 C\ determined. 



When Fi{p) is an odd function, however, we have to proceed more care- 

 fully, since the transformation now introduces branch points in the ^e'-plane 

 corresponding to a /ac/oy \/Ti{w). In this case we assume that Vi{w) is given 

 in analytic form, and determine the root Wi of Ti{w) = which lies outside 

 Ci . Then it will be possible to express Fe(w) in the form 



V 1 — w/wi 



where G(w) is analytic outside Ci , and has the proper behavior at infinity. 

 From the conditions imposed on Ti it can be shown that Wi is real; hence we 

 introduce a rationalizinoj factor 



MM = a/(i-^-)(i-±), (101) 



y \ Wi/\ WWiJ 



and multiply both sides of equation (100) by M{w). This leads to 



M{w)F'e{u^ = A/l - — G{w) = H{w), (102) 



y wwi 



where H{w) is analytic outside Ci . On Ci , | w | = 1, so that M{w) is real 

 and on Ci the potential and stream functions are defined by 



Miw) V[{w) = Re H(w), 



(103) 

 M(w) "^eM = Im H{w). 



Thus the real part of H{w) is determined by the known potential Veiw); 

 this determines in turn the imaginary part of H(w) and hence ^^(2^) is de- 

 termined. 



When Fi{p) is neither even nor odd we divide it into even and odd parts 

 and treat each part separately. If only the gain is important we need retain 

 only the even part, or if only the phase is important we consider only the 

 odd part. 



18. Examples 



So far we have been describing the potential analogue method in general 

 terms, and developing a systematic design procedure applicable to a wide 

 range of problems. The method involves a certain arbitrariness, in the initial 

 choice of contour, and there may also be some doubt in the reader's mind as 



