essential singularity

Let U be a domain, aU, and let f:U{a} be holomorphic. If the Laurent seriesMathworldPlanetmath expansion of f(z) around a contains infinitely many terms with negative powers of z-a, then a is said to be an essential singularityMathworldPlanetmath of f. Any singularity of f is a removable singularityMathworldPlanetmath, a pole or an essential singularity.

If a is an essential singularity of f, then the image of any punctured neighborhoodMathworldPlanetmath of a under f is dense in (the Casorati-Weierstrass theorem). In fact, an even stronger statement is true: according to Picard’s theorem, the image of any punctured neighborhood of a is , with the possible exception of a single point.

Title essential singularity
Canonical name EssentialSingularity
Date of creation 2013-03-22 13:32:10
Last modified on 2013-03-22 13:32:10
Owner pbruin (1001)
Last modified by pbruin (1001)
Numerical id 7
Author pbruin (1001)
Entry type Definition
Classification msc 30D30
Related topic LaurentSeries
Related topic Pole
Related topic RemovableSingularity
Related topic PicardsTheorem
Related topic RiemannsRemovableSingularityTheorem