1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences and Series 203

Remark In general, a test for convergence becomes a test for uniform
convergence in some set Di if it is satisfied independently of z E Di.

Theorem 4.18 Suppose that:

1. The functions Un ( z) ( n = 0, 1, 2, ... ) are all defined on D.

2. F(z) = I:;~=O un(z), the convergence being uniform on BCD.

3. The functions un(z) are continuous at z 0 E B (in a restricted sense,
   unless B is open).

Then F(z) is continuous at zo (in the same sense).

Proof Let Sn(z) = uo(z) + · · · + un(z). Then Sn(z) ~ F(z) on B, and

each Sn(z) is continuous at z 0 EB. By Corollary 4.4 (see also the Remark

following Corollary 4.5), F(z) is continuous at z 0 • Clearly, it follows that if
the terms of the series are continuous on B, then F(z) is continuous on B.

Note Since limz-+zo,zEB F(z) = F(zo), we have

00 00 00

z-+zo lim ~ ~ un(z) = ~ ~ un(zo) = ~ ~ z-+zo lim un(z)

zEB n=O n=O n=O zEB

4.11 Power Series

Series of functions of a complex variable with terms of the form un(z) =

anzn or, more generally, of the form un(z) = an(z -zor, where {an} is

a sequence of complex constants, are called power series (more precisely,

series of powers of z - z 0 with nonnegative integral exponents).

Since the second form can be reduced to the first by the substitution

z - zo = z', in this section we consider only power series of the form

00

L anZn = ao + aiz + · · · + anzn + · · · (4.11-1)

n=o

We note that in a power series all terms are defined and continuous on

C. In what follows our main object is to determine sets of convergence

(pointwise, absolute, uniform), and of divergence, of the series (4.11-1).

Example The simplest example of a power series is obtained by choosing

an = 1 for all n. This gives the so-called geometric series

00

L Zn = 1 + z + z^2 + ... + Zn + ... (4.11-2)

n=o