# 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