# Definition:Laurent Series

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

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## 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 $U \setminus \set {z_0}$.

## Also known as

A **Laurent series** is also commonly known as a **Laurent expansion**.

## Also see

- Results about
**Laurent series**can be found**here**.

## 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.