336 THE BELL SYSTEM TECHNICAL JOXIRNAL, APRIL 1951 



Then the charge distribution is determined by the discontinuity in^ across 

 C If we measure the charge from the real axis, t? = 0, we find from equa- 

 tion (24), 



lirqid) = 2 OmOiS sin wt? — [— &o^ — S ^mco^"* sin md] 

 Hence we have two alternative formulations for q: 



qW = -^ \- - zl bm coT^ sin m^ 



got? 1 A 



= — r> 1 I / / 1 + - 2l^ gmcoo sin wt? . 



ZtT log I OJo/Wo I TT m=l 



(45) 



We have substituted bo = bo log | coo | for the constant bo , where coo is an 

 undetermined frequency. The total charge on the contour is 



q(2T) = bo = — flo/log I ojo/coo | • (46) 



Since the total charge must be non-negative this implies that bo ^ 0, but 

 ao may be either positive or negative, according as coo is greater or less 

 than coo . 



The gain and phase are determined by the values of F on the real fre- 

 quency axis, p = io)j hence, inside C, 



«.• = oo + Z (-)" a2nCo^ /3, = E (-)" a2n+ico^"+' , (47) 



and outside C, 



2/1+1 W 

 n=l n=0 



Oie = —^0 log I £0/C0o | + S (" )" ^2nC0 



(48) 



^. = ±*o ^ - Z (-)" *. 



-2n-l 

 r, .^^ N X -'2n+lW 



-^ n=0 



where the minus sign in /3e refers to points on the positive half of the co-axis, 

 and the plus sign to points on the negative half. 



We note that a is an even function, of a> while /8 is an odd function. This 

 agrees with equation (7) and it means that if only the gain is prescribed 

 we know directly only the even coefficients, a2n , in the power series ex- 

 pansion, oi Fi . Hence we know only the even part of Fi{p). But we have 

 seen that the singularities in the logarithmic expression for a occur in 

 pairs, one of each pair being the negative of the other. To determine the 

 complete transmission function Fi(p) we must assign one of each pair of 

 singularities to F(p) and the other F{—p) in such a way that equations 

 (7) and (8) are satisfied. 



