Mathematics of Physics and Engineering

216 Singularities of Complex Functions

of ZQ. The proof of this series representation essentially relies on the fact
that a neighborhood of a point is a simply connected set. It turns out
that a somewhat similar expansion exists in domains that are not simply
connected. This expansion is known as the Laurent series, the study of
which is the main goal of this section. The French mathematician PIERRE
ALPHONSE LAURENT (1813-1854) published the result in 1843.
Before we state the result, let us make a simple yet important observa-
tion that will be an essential part of many computations to follow.
Recall that the geometric series formula
.. oo

fc=0

is true for \z\ < 1. On the other hand, writing

1 1

1-z *(l-i)

and replacing z with 1/z in (4.4.1), we can write

1 - Z ~ £"fc=0 Zk+l '

which is now true for \z\ > 1. Therefore, we have two representations for
1/(1 — z), one, for small \z\, and the other, for large.
We will now state and prove the main result of this section. Recall that
the Taylor series is written in an open disk {z : \z — ZQ\ < R}, which is the
basic example of a simply connected domain; we allow R = oo to include
the whole plane. Similarly, the basic domain that is not simply connected
is an annulus, a set of the type {z : R\ < \z — ZQ\ < R}, where ZQ is a fixed
complex number and, for consistency, we allow i?i = 0 and/or R = oo; the
Latin words anulus and anus mean "a ring". The Laurent series expansion
is written in an annulus.

Theorem 4.4.1 Laurent series expansion. If a function f = f (z) is
analytic in the annulus G = {z : R\ < \z — ZQ\ < R}, then, for all z £ G,
oo
f(z)= J2 ^(z-z 0 )k
k= — oo
7 r^ + 7 r + c 0 + ci(z - z 0 ) + c 2 (z - zQ)^2 + ...,
(Z - Z 0 y (2 - Zo)
(4.4.2)