124 BELL SYSTEM TECHNICAL JOURNAL 



and we see that E is purely real. Hence 



/ = 2bir-Kl - h-Y [ (If- - T-)~"(l - t2)-i dr 

 J-h 



is an equation from which /; may be determined as a function of (. Setting 

 r' = ft^x, expanding (1 — h-x)~^ in powers of //- and integrating termwise 

 leads to 



I TT 1 (1 — Oi) f^ ,1\ai \—'la T^f. i 'j ;2\ 



lb 1 ( 2 ~ <^) 



= ""j/^ - "^ A--F(^ - a, 1 - a; I - a; /r) (Al-2) 



1_ _^ 'T^'rC-a) /,i-2a(i _ /,2)a^(j^ i; 1 + «; 1 - h') 



sin xa r(| — a) 



where we have used relations from the theory of hypergeometric functions. 

 The term 1/sin ira is the reduced form of an original term containing a 

 hypergeometric function which has been evaluated by the binomial theorem. 

 The second and third expressions are suited to calculation when A- < 1/2 

 and Iv^ > 1/2 respectively. 



Now that the guide of Fig. 1 has been transformed into the upper half of 

 the f plane, the next step is to transform this upper half into the straight 

 guide of Fig. 2. We want f = — 1, i.e. Ss , to go into v = —x and f = 1, 

 i.e. Zs , to go into i' = -\- ^ . Again using the Schwarz-Christofifel formula 

 with w = r -\- id (the exterior angles at r == ±=o are equal to ir) 



w = A + £i f (r + 1) \t - 1)"-' dr (Al-3) 



We take the point So in Fig. 1 to correspond to r = 0, ^ = in Fig. 2. Since 

 this corresponds to i' = 0, Di must be zero. Also dw/d^ is real because w 

 traverses the walls of the guide of Fig. 2 as f moves along the real axis in the 

 .t plane. Hence £i is real. As f goes from 1 — e to 1 + e around a small 

 circular indentation above .t = I, w changes from x to x + /tt. Thus 



iir = Ex2-'{-iir) or £1 = -2 (Al-4) 



When (Al-3) is integrated, (Al-4) inserted, and the result solved for f 

 we obtain 



i" = tanh w/2 (Al-5) 



