Laurent series

Definition

Let $c \in \mathbb{C}$ and let $(a_n)_{n \in \mathbb{Z}}$ be complex numbers. The Laurent series with center $c$ and coefficients $a_n$ is the series $$\sum_{n = -\infty}^{\infty} a_n (z - c)^n$$ in the complex variable $z$. It converges at a point $z \neq c$ if both series $\sum_{n \ge 0} a_n (z - c)^n$ and $\sum_{n \ge 1} a_{-n} (z - c)^{-n}$ converge. Its sum at $z$ is then the sum of the two limits.

Remarks

A power series is a Laurent series with $a_n = 0$ for each $n < 0$. If the series converges on a nonempty open set, then it converges absolutely on an annulus $r < |z - c| < R$, and uniformly on each smaller closed annulus. For example, $e^{1/z} = \sum_{n \ge 0} \frac{1}{n!} z^{-n}$ for each $z \neq 0$. The part with negative powers is the principal part.

Explanations

L1 human

A Laurent series is like a power series, but it can also have negative powers of $z - c$. The negative powers allow it to describe functions that blow up at $c$, such as $1/z$ or $e^{1/z}$.