# nLab Laurent series

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

Laurent series generalize Taylor series and other power series by allowing negative indices. A Laurent series for the function $f\left(z\right)$ has the form

$f\left(z\right)=\sum _{n=k}^{\infty }{f}_{n}{z}^{n},$f(z) = \sum_{n=k}^{\infty}f_n z^n,

where $k$ is merely constrained to be finite and is often negative. Alternatively, we may write it as

$f\left(z\right)=\sum _{n=-\infty }^{\infty }{f}_{n}{z}^{n},$f(z) = \sum_{n=-\infty}^{\infty} f_n z^n ,

where all but finitely many of the negatively indexed terms are zero.

## Properties

### Algebraic closure

###### Theorem

If $K$ is algebraically closed and has characteristic 0, then the algebraic closure of the field of Laurent series over $K$ is the field of Puiseux series over $K$.

See at Puiseux series for more details.

Revised on February 4, 2013 09:56:02 by Urs Schreiber (82.113.99.102)