nLab
Laurent series

Idea

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

$$f(z)=\sum _{n=k}^{\mathrm{\infty }}{f}_n{z}^n,$$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

$$f(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_n{z}^n,$$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.