Picard’s theorem

Let f be an holomorphic functionMathworldPlanetmath with an essential singularityMathworldPlanetmath at w. Then there is a number z0 such that the image of any neighborhood of w by f contains -{z0}. In other words, f assumes every complex value, with the possible exception of z0, in any neighborhood of w.

Remark. Little Picard theorem follows as a corollary: Given a nonconstant entire functionMathworldPlanetmath f, if it is a polynomialMathworldPlanetmathPlanetmathPlanetmath, it assumes every value in as a consequence of the fundamental theorem of algebraMathworldPlanetmath. If f is not a polynomial, then g(z)=f(1/z) has an essential singularity at 0; Picard’s theorem implies that g (and thus f) assumes every complex value, with one possible exception.

Title Picard’s theorem
Canonical name PicardsTheorem
Date of creation 2013-03-22 13:15:23
Last modified on 2013-03-22 13:15:23
Owner Koro (127)
Last modified by Koro (127)
Numerical id 9
Author Koro (127)
Entry type Theorem
Classification msc 32H25
Synonym great Picard theorem
Related topic EssentialSingularity
Related topic CasoratiWeierstrassTheorem
Related topic ProofOfCasoratiWeierstrassTheorem