XETWORK SYNTHESIS ()23 



When U I = 1, 



n(i-.T)] 0") 



rSamo Ha.ioual \ ^^J-T^^ 



[ r unction HI —\'z\ \ z 



The relation (18) dtu's not, l)y itself, require (19) to be true. (19) fol- 

 lows from (18) if and only if the function of z has a power series expan- 

 sion in^'ol^■ing only i)ositive powers of z, and the function in —1/z has 

 a power series expansion in — l/z, with the same coefficients. This added 

 condition, howe^'er, is readily established for the useful interval. f 



Combining (19) and (16) yields a most useful relationship connecting 

 the z-plane mappings z^ , of the network singularities />„ , and the 

 coefficients Ct , of the TchebychefT polynomial expansion of a + z/3: 



Eife' = log A'.- ^ "'^ 



(20) 



n 1 



In more qualitative terms: 



The transfornmiion from variable p to variable z 

 converts an expansion in Tchebycheff polynomials 

 in p into an expansion in a power series in z. 



Thus, b}' working with the Za , in place of the Pa , one may use a 

 power series sort of analysis in calculating, or in choosing, the coeffi- 

 cients C'k in the Tchebycheff polynomial series. 



The relations (20) refer to the combined gain and phase function. 

 Exactly similar relations can readily be obtained, however, for gain and 



t As defined in (8), | s„ | > 1. In the useful interval, \ z \ =1. Hence | z/z, | < 1. 

 It follows that log (1 — z/z^) has a power (MacClauren) series expansion in posi- 

 tive powers of z, convergent on and within the circle | 2 | = 1. Finally the first 

 logarithm in (19) may be expressed as a sum of logarithms of this simple type, 

 each of which may be expanded separately. Sul)stituting —l/z for z maps the 

 unit circle onto itself. It follows that the second logarithm in (19) has an expan- 

 sion in positive powers of —l/z, in the useful interval, provided the first loga- 

 rithm has an expansion in positive powers of z; and the coefficients in the two 

 series will be the same. 



