Let
be a domain,
, and
a holomorphic function. A Laurent expansion of
around
is a representation of
as a Laurent Series:

where
, and the series converges on an annular region around
(i.e., excluding the point
).
A slightly more general form of the expansion above is the following: Let
be two radii (the expansion around a point corresponds to
), and let
be an annular region around
, and let
be a holomorphic function. Then the Laurent Series

with
is a Laurent expansion of
on
, provided the series converges for all
.
Every holomorphic function on
has a Laurent expansion around
, and the coefficients
in the expansion are given by:

for a radius
with
.
The coefficients are uniquely determined by:
Proof of Existence and Uniqueness of the Laurent Representation
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Uniqueness follows from the Identity Theorem for Laurent Series. To prove existence, choose a radius
such that
and choose
so that
. Let
be arbitrary. "Cut" the annular region
at two points using radii
and
such that the cycle
is represented as the sum of two closed curves
and
in
that are null-homotopic. Choose
and
so that
is encircled by
. By the Cauchy Integral Theorem, we have:

and

since
does not encircle
. Thus, because
, we have:

For
, we have:

The series converges absolutely because
, and we obtain:

Now, consider the integral over the inner circle, which is analogous to the above for
:

Thus, due to
, the series converges, and we obtain:

Thus, it follows that for
:

which proves the existence of the claimed Laurent expansion.
This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:
https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Laurententwicklung