642 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



ditions, provided none are pure imaginaries. Since an imaginary z^ is a 

 negative real 2<, : 



There must he no negative real Zg . 



There will be no negative real zl if the polynomial ^ KkZ^ in (55) is 

 non-zero at all negative real z". If the requirement is violated, initially, 

 one or more Kk , of the highest orders, must be modified. Graphical 

 methods are likely to be useful for this, combining plots of the original 

 polynomial, and proposed changes. An approximation of the form 

 C2k = Cik , k ^ m, will still be realized, but with m < n. 



The error function corresponding to m < n is as follows, in place of 

 (67): 



H(z) = ^^^, (08) 



18. ACCURACY 



The accuracy of match again may be estimated by the means of (41), 

 using H{z) of (67) or (68). H{z), however, may not be so easily cal- 

 culated as for the previous example. 



A simpler but less reliable estimate of accuracy is furnished by the 

 error in the first unmatched coefficient in the Tchebycheff polynomial 

 series. If Kk = Kk through k = m, the leading terms in various series 

 are as follows. First, from (64) and (68), 



Kl n (l - I) Ri^)Ri-z) = 1 + ^-^'-^^-^' .-^^ . . . (69) 



using this in (52) gives : 



{Cu - C2k)z = -log 1 -f I ^ ) z 



Then, from the properties of logarithms, 



Zfn n ^,2fc Am+l — -Km+l 2m+2 /^iN 



(C2jfc — ^ik)^ = — i> z •" wi; 



-ft-o 



Consequently (also from (52)) : 



K^+l — Km+l , . 



a — a = ^ 1 27H-2 • • • K'^J 



Ao 



This is the same as the leading term of H{z), except that z'"''^^ is re- 

 placed by — T2m+2 . If m = n, the same equation holds with /v,„+i = 0. 



(70) 



