uniqueness of Laurent expansion

The Laurent seriesMathworldPlanetmath expansion of a function f(z) in an annulusr<|z-z0|<R  is unique.

Proof.  Suppose that f(z) has in the annulus two Laurent expansions:


It follows that


where ν is an integer.  Let now γ be an arbitrary closed contour in the annulus, going once around z0.  Since γ is a compact set of points, those two Laurent series converge uniformly (http://planetmath.org/UniformConvergence) on it and therefore they can be integrated termwise (http://planetmath.org/SumFunctionOfSeries) along γ, i.e.

n=-anγ(z-z0)n-ν-1𝑑z=n=-bnγ(z-z0)n-ν-1𝑑z. (1)


γ(z-z0)n-ν-1𝑑z={2iπifn=ν,0  ifnν,

when integrated anticlockwise (see calculation of contour integral).  Thus (1) reads

2iπaν= 2iπbν,

i.e.  aν=bν,  for any integer ν, whence both expansions are identical.

Title uniqueness of Laurent expansion
Canonical name UniquenessOfLaurentExpansion
Date of creation 2013-03-22 19:14:12
Last modified on 2013-03-22 19:14:12
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Theorem
Classification msc 30B10
Related topic CoefficientsOfLaurentSeries
Related topic UniquenessOfFourierExpansion
Related topic UniquenessOfDigitalRepresentation