NKTWOKK SYXTIIKSIS ()41 



2 ^'^'' = iT^iw = s (^ + « (ly 



(02) 



thus Kk and Kk become 



(63) 



If these Kk are used to evahiate the polynomial on the left hand side of 

 (53), in accordance with (5o), and if the polynomial is then multiplied 

 by the above special R(z)R( — z), the result is exactly (36). 



The error function H(z), of (30) and (42), may now be defined as 

 follows : 



Kl n (l - I) Riz)R(-z) = 1 + Hiz) (64) 



The error a — a is again: 



a - a = - log I 1 + H(z) 1 (G5) 



If (64) is used to express (53) in terms of H{z), a "1" mny be sub- 

 tracted from each side of the relation, to get 



me 



H(z) = (66) 



^^'hen tn = n, this requires an H(z) of the following form, in terms of 

 the coefficients Kk derived from R{z)R{ — z): 



fr 2n+2 , ^ 2n+4 , 



H{z) = -^l^ "t^in, ^ '" (67) 



17. CHARACTERISTICS OF Z„ 



As in the previous example, \z„\ > I when the approximation, a to 

 a, is at all reasonable. The z^ are again zeros of 1 + H(z); but now 

 H{z) is defined by (64). If | H(z) | < 1, when \z\ = 1, there will be 

 the same number of zeros of 1 + H(z) as poles, in the region \ z \ < 1. 

 Any poles would have to be poles of R{z)R{ — z). In Section 8, we noted 

 that this function is regular in the region \z\ ^ 1. Hence, there will be 

 no poles, and there wdll be no | 2, | < 1, luider ordinary accuracy con- 

 ditions. 



As before, the 2<, can be chosen in accordance with the physical con- 



