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THE BELL SYSTEM TECHXICAL JOURXAL, JULY 1952 



Actually, the definitions must \'ary ^vith the nature of the useful fre- 

 quency iuter\'al. For the present, however, it will be assumed that the 

 useful interval extends from co = to coc ; or more precisely', from 

 CO = — Wc to +Wc (in accordance with the symmetries of gain and phase 

 functions). Useful intervals which do not include w = require changes 

 in the definitions, which will be taken up in Section 28. 



For our present purposes, Tchebycheff polynomials Tk may be defined 

 as follows: 



p = ico = ioic sin <^ 

 Tk = cos /:(/), /: even 

 Tk = i sin k<t), k odd 



(3) 



The first equation defines an auxiliary angle variable, ^, in terms of 

 which Tk is especially simple. The imaginary scale factor i, associated 

 vnth. poljttiomials of odd order, simplifies later analysis. In addition, it 



Fig. 2 — Tchebycheff poh-nomials. 



