266 Chapter5- For w = (z^2 - 3z + 2)/(z^2 + 1), p = 2, so it has two zeros: 1, 2, and
two poles: i, -i. The point oo is neither a zero nor a pole. In fact,
w(oo) = 1. - For a polynomial w = P(z)/1 = ao + alz + · · · + anzn (an ¥= 0), we
have p = n. It has n zeros and n poles, namely, oo with multiplicity
n. Note that in this case
Rl(z) = P ( ~) = ao + alz-
1
+ · · · + anz-naozn + alzn-l +···+an
=which has z = 0 as a pole of order n.
Next, we wish to consider briefly a rational function of order 2 of the formw=------ao + alz + a^2 z^2
b0 +biz+ b 2 z^2
(5.16-2)where a2 ¥= O, b 2 ¥= O, and numerator and denominator have no common
factor of the form z - c. The condition for this to happen is that
By using bilinear transformations of both the dependent and indepen-
dent variables in (5.16-2), the function defined thereby can be reduced to
either of two simpler forms, depending on whether the function has a double
finite pole f3 or two distinct simple poles /3i, /32.1. ·Suppose that b 0 +biz+ b 2 z^2 = b 2 (z - (3)^2 • Then (5.16-2) can be
written asw=
Ao + Aiz + A2z^2
(z -(3)2where Aj = aj/b 2 (j = 0, 1, 2). If we now let z = f3 + 1/(, we have
Ao+ Ai(/3 + 1/() + A 2 ((3 + 1/()^2
w = 1/(2= Ao(^2 + Ai((/3( + 1) + A2(/3( + 1)^2
= B2(^2 +Bi(+ Bo