1550251515-Classical_Complex_Analysis__Gonzalez_

522 Chapters

Since the f n are analytic in R and C is homotopic to a point in R ( R being

simply connected), we have

J fn(z) dz= 0

c

for eacl:i n by the Cauchy-Goursat theorem. Hence

J F(z)dz = 0 (8.1-1)
G
The function F( z) is continuous at each point zo E R, and so continuous
on R, since any such point can be enclosed in a compact subset of R, and
on that subset Fn = Ji + · · · + fn ~ F. In addition, (8.1-1) holds for
every closed contour C contained in R. Hence, by Morera's theorem, F
is analytic in R.

Note Fbllowing S. Saks and A. Zygmund [32], a sequence (or a series) of
functions defined in an open set A is called almost uniformly convergent
in A if the sequence (or series) converges uniformly to a function on every
compact subset of A. Also, the phrase locally uniformly convergent is
sometimes used.
Corollary 8.3 Suppose that:

1. The functions Fn(z) are analytic on a simply connected region R.


2. Fn(z) ~ F(z) on compact subsets of R.
   Then F( z) is analytic in R.

Proof If suffices to let

fn(z) = Fn(z) - Fn-1(z),

for all z in R. Then

Fo(z) = 0

Fn(z) = fi(z) + · · · + fn(z)

so that

00

L fn(z) = F(z)

n=l

the convergence being uniform on compact subsets of R. Now the
conclusion follows from Corollary 8.2.

Corollary 8.4 Let F( z) = 2::;:'= 1 anzn, and suppose that the power series

converges for lzl < r. Then F(z) is analytic in the disk lzl < r.