15. LAURENT SERIES
15.1
Analytic Functions in a Punctured Disk
Suppose that
is an analytic function on a closed disk
.
Then
where
and
is the circle
.
That is, an analytic function on
can be expressed there as a power series. But what about
those functions that are only analytic in the punctured
disk
?
It turns out, as we shall prove, that those functions
can be expressed in what is called Laurent series that
has some resemblance to power series:
Suppose then that
is analytic in the punctured disk
and has a pole of order
at
We then recall that there is an analytic function
on
with
and such that
LetThen
15.2
Principal and
Analytic Parts and Connection with Power Series
The sum of the terms involving
negative powers of
is a polynomial in powers of
of degree
, denoted by
is called the principal part
of
at
The other part is called the analytic part of
at
It
is standard to write
then as:
which is the Laurent series
of
at
In short, we can write
where
is analytic on
Suppose now that
is an analytic function on a domain
except at poles
in
of orders
respectively. By repeated application
of the above procedure we immediately see the following:
Theorem 1.
. If
is an analytic function on a domain
except at poles
in
of orders
respectively, then
where
,
are polynomials in
of degrees
respectively and
is analytic on
