618 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



Substitutions in the exponential ecjiiivalents of the other sine and cosine 

 in (3) give: 



T, = 



K-->i) 



A; even 



n = \ [z' - 3 ) , k odd 



(6) 



Network appUcations depend upon the nature of the relationship be- 

 tween the variable p, and the variable z. The relationship is illustrated 

 in Fig. 3, which indicates corresponding contours in the p and z planes. 



Since angle is real in the useful interval, z, as defined by (4), has 

 unit magnitude. In equivalent conformal mapping terms, the unit circle 

 in the z plane maps onto a segment of the axis of real frequencies in 

 the y plane — namely the segment extending from p = —iwc to -\-iwc . 

 Hereafter, we shall say merely that the useful interval in the z plane is 

 the unit circle, or | 2 | = 1. The real frec[uency intervals outside the 

 useful interval map onto the imaginary axis in the ^-plane. 



2-plane circles with radii other than unity map onto p-plane ellipses, all 

 with foci at p = zb iuic ■ This is reminiscent of filter theory using Tcheby- 

 cheff polynomials, and in fact such a filter may be obtained by spacing 

 2-plane mappings of natural modes uniformly around such a circle. f 



p-PLANE 



2-PLANE 



+ La;c 



.p = 



La;cSiN9!> I 



REAL p 



Fig. 3 — The complex planes for p and z. 



t The filter theorj^ is developed in detail in a monograph by Wheeler^, which 

 also includes an extensive bibliography. 



