Laurent series

Explains the Laurent series, a representation of a complex function f(z). Named after Pierre Alphonse Laurent, a French mathematician and Military Officer, published in the series 1843.

Unlike the Taylor series which expresses $$f(z)$$ as a series of terms with non-negative powers of $$z$$, a Laurent series includes terms with negative powers. Therefore, a Laurent series may be used in cases where a Taylor expansion is not possible.

$$f(z)=\sum _{n=-\infty }^{\infty }a_{n}(z-c)^{n}$$

where the $$a_n$$ and $$c$$ are constants defined by

$$a_{n}={\frac {1}{2\pi i}}\oint _C{\frac {f(z)\,\mathrm {d} z}{(z-c)^{n+1}}}$$

The contour $$C$$ is counterclockwise around a closed, enclosing $$c$$ and lying in an annulus $$A$$ in which $$f(z)$$ analytic.

To calculate, use the standard and modified geometric series

\frac{1}{1-z}= \left\{ \begin{align} \sum_{n=0}^{\infty}&\ z^n,&&|z|\lt1\nonumber\\ -\sum_{n=1}^{\infty}&\ z^{-n},&&|z|\gt1\nonumber \end{align}\nonumber \right.\nonumber

Here $$f(z)=\frac{1}{1-z}$$ is analytic everywhere apart from the singularity at $$z=1$$. Above are the expansions for $$f(z)$$ in the regions inside and outside the unit circle, centered on $$z=0$$, where $$|z|\lt1$$ is the region inside the circle and $$|z|\gt1$$ is the region outside the circle.