removable singularity

Let U be an open neighbourhood of a point a. We say that a function f:U\{a} has a removable singularityMathworldPlanetmath at a, if the complex derivativeMathworldPlanetmath f(z) exists for all za, and if f(z) is boundedPlanetmathPlanetmath near a.

Removable singularities can, as the name suggests, be removed.

Theorem 1

Suppose that f:U\{a}C has a removable singularity at a. Then, f(z) can be holomorphically extended to all of U, i.e. there exists a holomorphic g:UC such that g(z)=f(z) for all za.

Proof. Let C be a circle centered at a, oriented counterclockwise, and sufficiently small so that C and its interior are contained in U. For z in the interior of C, set


Since C is a compact set, the defining limit for the derivative


converges uniformly for ζC. Thanks to the uniform convergenceMathworldPlanetmath, the order of the derivative and the integral operations can be interchanged. Hence, we may deduce that g(z) exists for all z in the interior of C. Furthermore, by the Cauchy integral formulaPlanetmathPlanetmath we have that f(z)=g(z) for all za, and therefore g(z) furnishes us with the desired extension.

Title removable singularity
Canonical name RemovableSingularity
Date of creation 2013-03-22 12:56:01
Last modified on 2013-03-22 12:56:01
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 5
Author rmilson (146)
Entry type Definition
Classification msc 30E99
Related topic EssentialSingularity