PlanetMath: Laurent series

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Laurent series

(Definition)

A Laurent series centered about is a series of the form

$\displaystyle \sum_{k=-\infty}^\infty c_k (z-a)^k$

where $c_k, a, z \in \mathbb{C}$ . The principal part of a Laurent series is the subseries $\sum_{k=-\infty}^{-1} c_k (z-a)^k$ .

One can prove that the above series converges everywhere inside the (possibly empty) set

$\displaystyle D := \{z \in \mathbb{C} \mid R_1 < \vert z-a\vert < R_2 \}$

where

$\displaystyle R_1 := \limsup_{k \rightarrow\infty} \vert c_{-k}\vert^{1/k}$

and

$\displaystyle R_2 := 1/\left(\limsup_{k \rightarrow\infty} \vert c_{k}\vert^{1/k}\right).$

Every Laurent series has an associated function, given by

$\displaystyle f(z) := \sum_{k=-\infty}^\infty c_k (z-a)^k,$

whose domain is the set of points in $\mathbb{C}$ on which the series converges. This function is analytic inside the annulus

, and conversely, every analytic function on an annulus is equal to some unique Laurent series. The coefficients of Laurent series for an analytic function can be determined using the Cauchy integral formula.

"Laurent series" is owned by djao.

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Other names:

Laurent expansion

Also defines:

principal part

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Cross-references: Cauchy integral formula, coefficients of Laurent series, annulus, analytic, points, domain, function, converges, series
There are 14 references to this entry.

This is version 8 of Laurent series, born on 2001-12-28, modified 2007-03-31.
Object id is 1152, canonical name is LaurentSeries.
Accessed 16233 times total.

Classification:

30B10 (Functions of a complex variable :: Series expansions :: Power series )

Pending Errata and Addenda

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