Laurent decomposition

Description

We notice that the function $f(z)=e^{\frac{1}{z}}$ doesn’t allow for a Power series around the value $0$. But it does allow for an inverse power series $f(z)=\sum \frac{1}{k!}(\frac{1}{z}{)}^k$ on the outside of $0$. This “inverse power series” describes the function “at infinity”. Or in other words, in a disc around infinity. Assuming (again) the disc doesn’t contain any singularities.

This gives us the idea to split up the singularities up into singularities inside a circle and singularities outside a circle. We can then use the power series and inverse power series to describe the function around that circle.

Definition

Let $f:K_{r,R}$ be a map holomorphic on a annulus $K_{r,R}$ (possibly with $r=0$ and $R+\infty$). Then there is a decomposition $f=f_n+f_h$ into a Holomorphic function $f_n$ defined on an inner disc $K_{0,R}$ and a holomorphic function $f_h$ defined on an outer disc $K_{r,\infty }$ such that the two functions do not have any singularities on their respective domains. $f_h$ and $f_n$ called the Hauptteil and Nebenteil of $f$ respectively.

The two domains interlock in an annulus. Note: $f_n$ stands for “Nebenteil” of the Laurent series and behaves just like a Power series.

Properties

Tip

Examples

Example