Page 735 - Advanced_Engineering_Mathematics o'neil

POWER SERIES THE LAURENT
EXPANSION
CHAPTER 21

Series

Representations

of Functions

There are two types of series expansions that are important for working with complex functions.
The first is the power series.

21.1 Power Series

We will precede a discussion of power series with some facts about series of complex numbers.
Sequences and Series of Complex Numbers
We will assume some familiarity with real sequences and series.
Suppose z n is a complex number for each positive integer n. If we write z n = x n + iy n , then
the complex sequence z n converges to L = c + id exactly when

lim x n = c and lim y n = d.
n→∞ n→∞
In this case, we write
lim z n = L.
n→∞
This reduces every complex sequence to a consideration of two real sequences.
Complex series are treated similarly in terms of their real counterparts. Suppose ∞ c n is
n=1
a series of complex numbers. Write c n = a n + ib n . Then
∞

c n converges to L = A + iB
n=1
if and only if
∞ ∞

a n = A and b n = B.
n=1 n=1
This reduces questions about series of complex numbers to questions about real series to which
standard tests (comparison, integral test, ratio test, and others) may apply. In particular, if ∞ c n
n=1
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