# Definition:Laurent Series

Jump to navigation
Jump to search

*Not to be confused with Definition:Formal Laurent Series.*

This article needs to be linked to other articles.convergesYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{MissingLinks}}` from the code. |

## Definition

Let $f: \C \to \C$ be a complex function.

Let $z_0 \in U \subset \C$ such that $f$ is analytic in $U \setminus \set {z_0}$.

A **Laurent series** is a summation:

- $\ds \sum_{j \mathop = -\infty}^\infty a_j \paren {z - z_0}^j$

such that the summation converges to $f$ in $U \setminus \set {z_0}$.

## Source of Name

This entry was named for Pierre Alphonse Laurent.

## Historical Note

The Laurent series expansion of an analytic function, was established by Carl Friedrich Gauss in $1843$, but he never got round to publishing this work.

Karl Weierstrass independently discovered it during his work to rebuild the theory of complex analysis.