            Sessions # 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 Let Then 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  