Laurent series

A Laurent seriesMathworldPlanetmath centered about a is a series of the form


where ck,a,z. The principal part of a Laurent series is the subseries k=--1ck(z-a)k.

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



R1:=lim supk|c-k|1/k


R2:=1/(lim supk|ck|1/k).

Every Laurent series has an associated function, given by


whose domain is the set of points in on which the series converges. This function is analytic inside the annulus D, 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 formulaPlanetmathPlanetmath.

Title Laurent series
Canonical name LaurentSeries
Date of creation 2013-03-22 12:04:52
Last modified on 2013-03-22 12:04:52
Owner djao (24)
Last modified by djao (24)
Numerical id 12
Author djao (24)
Entry type Definition
Classification msc 30B10
Synonym Laurent expansion
Related topic EssentialSingularity
Related topic CoefficientsOfLaurentSeries
Defines principal part