1550251515-Classical_Complex_Analysis__Gonzalez_

526 Chapter 8

Proof Let B be any compact subset of A. As in the proof of Theorem 8.2,

let p__ = d(B,BA), and for any z E B consider the open disk D(z) =

{ (: I( - zl < r }, where r = p/3 if p is finite, and r = an arbitrary fixed

positive number if p is infinite. Let G = { D(z)} zEB be .an open covering

of B, and let G 1 = {D(z;)};: 1 be a finite subcovering of B.

Consider the circle e;: ( - z; = 2reit, 0 ::::; t ::::; 27!", and for any given

E > 0 choose 0 < E^1 < rk e/2k!. Since er is compact and er c A, there is

a positive number N;( e') such that

n

I: lfv(()I < e' (8.1-5)

v=m+l

for m,n > N;(e') for all ( E e;, by assumption (2) and the Cauchy

condition for uniform convergence for series.

For any z E D(z;) we have

J<k)( ) = ~ j fv(O d(

v z 27ri ((-z)k+l

G;

and

IJ<k)( v z )I< - ~ 27r j J( lfv(C)I - zJk+l I d(I

G;

so that, for m, n > N;( e').

t lf~k)(z)I::::; :~ j L:~(i:_+;

1

~; 1 COl I d(J

v=m+l G;

k! E^1 2k!e'

::::; 27r rk+l(47rr)=----;:;;- <e (8.1-6)

Hence the series L:;: 1 IJ~k)(z)I converges uniformly on each D(z;). By

ta.king m,n > max(N1,N 2 , ••• ,Nm) we see that (8.1-6) holds for all the

disks in G 1 , so the series converges uniformly on B.

Remark This theorem is applied in Selected Topics, Section 3.4.

8.2 ANALYTIC FUNCTIONS DEFINED BY REAL

IMPROPER INTEGRALS

As an application of Theorem 8.2, we prove the following property.

Theorem 8.4 Suppose that: