radius of convergence of a complex function

Let f be an analytic functionMathworldPlanetmath defined in a disk of radius R about a point z0. Then the radius of convergenceMathworldPlanetmath of the Taylor seriesMathworldPlanetmath of f about z0 is at least R.

For example, the function a(z)=1/(1-z)2 is analytic inside the disk |z|<1. Hence its the radius of covergence of its Taylor series about 0 is at least 1. By direct examination of the Taylor series we can see that its radius of convergence is, in fact, equal to 1.

Colloquially, this theorem is stated in the sometimes imprecise but memorable form “The radius of convergence of the Taylor series is the distance to the nearest singularity.”

Title radius of convergence of a complex function
Canonical name RadiusOfConvergenceOfAComplexFunction
Date of creation 2013-03-22 14:40:33
Last modified on 2013-03-22 14:40:33
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Theorem
Classification msc 30B10