Laurent Series

Named after Pierre Alphonse Laurent, a French mathematician and Military Officer, published in the series 1843.

The Laurent series is a representation of a complex function f(z) as a series. 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.

=–

see https://math.stackexchange.com/questions/1126321/proof-of-laurent-series-co-efficients-in-complex-residue

=—
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.

Embedded software developer
Passionately curious and stubbornly persistent. Enjoys to inspire and consult with others to exchange the poetry of logical ideas.

This site uses Akismet to reduce spam. Learn how your comment data is processed.