1550251515-Classical_Complex_Analysis__Gonzalez_

574 Chapter^8

lies outside all Jensen circles,

P'( ) n 1

Im P(z) =Im L --= -2yK

Z k=l Z - CXk

K being positive. Hence if z is nonreal and outside all the Jensen circles,
then P'(z) -:/= 0.
Next, we proceed to prove a proposition that was stated without proof
in Section 5.14.

Theorem 8.28 If P(z) is a polynomial of degree n > 1, there are at most

n points in the extended w-plane with less than n distinct inverse images
under the mapping w = P(z).

Proof We have already noted in 5.14 that the point w = oo has just one

inverse image, and so fewer than n distinct inverse images. If the number of

distinct inverse images of w 1 -:/= oo is less than n, the equation P( z) = w1
must have a multiple root. By Theorem 8.20 any multiple root of this
equation must satisfy also the equation P'(z) = 0. But this equation is of

degree n - 1, and so has no more than n - 1 distinct roots /31, /32, ... , f3k

(1 ::::; k ~ n - 1). Hence the numbers

(8.10-4)

are the only values of w for which P(z) = w has a multiple root. Since at
most n of the numbers in (8.10-4) are distinct, the theorem follows.

Exercises 8.3

1. Find the set of zeros of f(z) = z + iz. More generally, find the set of
   zeros of f(z) = az + bz, where !al = lbl -:/= 0.


2. Find the order of the zero at z = 0 for each of the following functions.

(a) zsinz (b) sin^3 z

( c) tan z^3 ( d) z^3 ( ez

2

• 
  
  
  
  1. (e) 1-cosz (f) e•inz - etanz
     (g) 6 sinz^3 - 6z^3 + z^9 (h) z-^4 / 15 sinz+^1 / 15 tanz-8/s tan^1 / 2 z
  
  
  
  

3. If f is analytic in a region Rand J(n+l)(z) = 0 in R, prove that f is

a polynomial of degree ::::; in R.

4. Prove that any circle that encloses all zeros of a polynomial also encloses
   all zeros of its derivative.


5. Let J(z) =I:: anzn for all lzl < R, where R is the radius of convergence
   of the series, and suppose that a 0 -:/= 0.

(a) Prove that f has no zeros in the region {z: lf(z) - aol < laol}.