Functions of a complex variable Definition

Laurent series

Builds on Power series in the complex plane

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.

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Complex numbers object
Power series in the complex plane definition
Laurent series definition

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Tag	`0X3XD2ATB0`
URI	`m2://0X3XD2ATB0`
Kind	definition
Writing	W1 stated
MSC	30B10
Defines	Laurent series
Uses	Power series in the complex plane

Laurent series

Definition

Remarks

Reading path

Where this leads

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