[§U
SOME BASIC THEOREMS 3
For this purpose, we shall write
(1.4) F(z) = wQ(z), w = 1 + [P(z)lQ(z)l
tfq denotes the number of zeros of Q(z) in C, then according to Th. (1,2)
(1.5) A
c
arge(z) = 27rfl.
Since \P(z)lQ(z)\ 1 on C, the point w defined in eqs. (1,4) describes (see Fig.
(1,2)) a closed curve Y which lies interior to the circle with center at w = 1 and
radius 1. Thus, point w remains always in the right half-plane. The net change
u-plane
FIG.
(1,2)
in arg w as w varies on Y is therefore zero. This means according to eqs. (1,4)
and (1,5) that
A
„, .
A
,
A
-,
N
~
' A(; arg F(z) = A
c
arg w + Ac arg Q(z) = 2nq
and according to Th. (1,2) that F(z) has also q zeros in C.
We shall now apply Rouche's Theorem to a proposition which we shall often
use either explicitly or implicitly. It is the proposition that the zeros of a poly-
nomial are continuous functions of the coefficients of the polynomial. In more
precise language, it may be stated as
THEOREM (1,4). Let
f(z) = a0 + axz + +
anzn
= an IT (z -
z,)w
, an * 0,
F(z) = (a0 + €0) + (a, + Cl)z + + (an_x +
€9^zn'1
+
anzn
and let
(1,6) 0 rk min \zk - z,|, j = 1, 2, , / ; - 1, k + 1, •••,/?.
There exists a positive number e such that, if |ej ^ for i = 0, 1, •, n 1,
then F(z) has precisely mk zeros in the circle Ck with center at zk and radius rk.
Previous Page Next Page