existence of power series

In this entry we shall demonstrate the logical equivalence of the holomorphic and analyticPlanetmathPlanetmath concepts. As is the case with so many basic results in complex analysis, the proof of these facts hinges on the Cauchy integral theorem, and the Cauchy integral formulaPlanetmathPlanetmath.

Holomorphic implies analytic.

Theorem 1

Let UC be an open domain that contains the origin, and let f:UC, be a function such that the complex derivativeMathworldPlanetmath


exists for all zU. Then, there exists a power seriesMathworldPlanetmath representation


for a sufficiently small radius of convergenceMathworldPlanetmath R>0.

Note: it is just as easy to show the existence of a power series representation around every basepoint in z0U; one need only consider the holomorphic function f(z-z0).

Proof. Choose an R>0 sufficiently small so that the disk zR is contained in U. By the Cauchy integral formula we have that


where, as usual, the integration contour is oriented counterclockwise. For every ζ of modulus R, we can expand the integrand as a geometric power series in z, namely


The circle of radius R is a compact set; hence f(ζ) is boundedPlanetmathPlanetmathPlanetmath on it; and hence, the power series above converges uniformly with respect to ζ. Consequently, the order of the infiniteMathworldPlanetmath summation and the integration operationsMathworldPlanetmath can be interchanged. Hence,




as desired. QED

Analytic implies holomorphic.

Theorem 2



be a power series, converging in D=Dϵ(0), the open disk of radius ϵ>0 about the origin. Then the complex derivative


exists for all zD, i.e. the function f:DC is holomorphic.

Note: this theorem generalizes immediately to shifted power series in z-z0,z0.

Proof. For every z0D, the function f(z) can be recast as a power series centered at z0. Hence, without loss of generality it suffices to prove the theorem for z=0. The power series


convergesPlanetmathPlanetmath, and equals (f(ζ)-f(0))/ζ for ζ0. Consequently, the complex derivative f(0) exists; indeed it is equal to a1. QED

Title existence of power series
Canonical name ExistenceOfPowerSeries
Date of creation 2013-03-22 12:56:27
Last modified on 2013-03-22 12:56:27
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 5
Author rmilson (146)
Entry type Result
Classification msc 30B10