1550251515-Classical_Complex_Analysis__Gonzalez_

572 Chapter^8

(See Fig. 8.5.) Since imaginary parts of reciprocal complex numbers are
opposite in sign, we have

b

Im--<0

z - O:k

Then, multiplying (8.10-1) by band taking imaginary parts, we get

Im bP'(z) =~Im _b_ < 0

P(z) L.J z - o:k

k=l

which shows that P' ( z) f=. 0 if z E H'. Therefore, all zeros of P' ( z) lie in H.

Corollary 8.12 The smallest convex polygon containing the zeros of a
polynomial also contains the zeros of its derivative.

Corollary 8.13 If all zeros of a polynomial P(z) lie on a line L, then all

zeros of P'(z) lie on L. In particular, if all zeros of P(z) are real, then all
zeros of P'(z) are real also.

Proof ~uppose that there is a zero /3 of P'(z) which does not lie on L.
Consider a parallel L' to L halfway between /3 and L. Then all zeros of
P(z) will lie in that half-plane determined by L' which contains L, while
there would be a zero of P'(z) lying in the opposite half-plane.

In the next theorem we consider polynomials P(z) with real coefficients.

From Exercises 1.6 (29) it follows that the nonreal zeros of P(z) occur in

conjugate pairs so that if P( o:) = 0 we have P( ii) = 0 also. The circles

with centers on the real axis and diameters lo: - &I are called the Jensen

circles of P(z) (Fig. 8.6). The following theorem was stated without proof

by J. L. W. Jensen (18] in 1913 and proved by J. L. Walsh (38] in 1920.

For further generalizations of this theorem see Walsh ((39] and (40]).

y

L

&1

H l'-2

(^0) • x
<Xn
Fig. 8.5