630 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



without regard to phase. It reflects the more specific functional forms of 

 our present a and a. 



The formulas show that the coefficients in the Tchebycheff poly- 

 nomial expansion of our present a — a are fixed by the logarithm of a 

 polynomial in z\ of degree /;- + 2. Since the Tchebycheff polynomial 

 series is simply one representation of the function a — a, this means 

 that a — a itself is determined by the polynomial in z". Out of the ri + 2 

 zeros, in terms of z, n are subject to arbitrary choice, but the other two 

 are required to be at e = 2o • 



To arrive at a useful choice of the zeros, one may start with the ex- 

 panded form of the polynomial, which replaces the second equation of 

 (33) by: 



Z {C-,, - C-2,)z'' = -log \K, + K,i + • • • A',„+22""''l (34) 



All but two of the coefficients Kk may be assigned arbitrary values, pro- 

 vided the remaining two are then adjusted to give the required two 

 zeros at z~ = zl . The corresponding zeros zl may then be found by 

 ordinary root extraction methods. 



The coefficients may be chosen in such a way that the complex poly- 

 nomial approximates unity, when \z\ = 1. Then the logarithm approx- 

 imates zero, the coefficients in the power series (34) are small, and since 

 these are also the coefficients in the Tchebycheff polynomial series in 

 (33), a — a is small. 



11. TCHEBYCHEFF POLYNOMIAL SERIES MATCHED THROUGH U TERMS 



A special choice of coefficients, which meets these requirements fairly 

 well, is the choice determined by (28), with m = n. The function on the 

 left side of (28) is here the polynomial in (34). For our present purposes, 

 therefore, (28) becomes: 



{A'o + Kiz' + • • • Kn+'zz'"'-'} = 1 (35) 



This requires /vo = 1, and Kk = for k = 1 to n. Then Kn+i and Kn+2 

 must be adjusted to give the two reciuired zeros at z' = z'(, . This gives: 



^0 + • • • /?„+2^'"+' = 1 - ( n + 2) (^Q""""" + (n + 1) (^^y (36) 



In accordance with Section 9, this special choice of coefficients corre- 

 sponds to a match of Tchebycheff polynomial series, a to a, through 

 terms of order 2n: 



C-2k = Cu- , A- ^ n (37) 



