Picard’s theorem

Let $f$ be an holomorphic function with an essential singularity at $w\in ℂ$. Then there is a number $z_0\in ℂ$ such that the image of any neighborhood of $w$ by $f$ contains $ℂ-\{z_0\}$. In other words, $f$ assumes every complex value, with the possible exception of $z_0$, in any neighborhood of $w$.

Remark. Little Picard theorem follows as a corollary: Given a nonconstant entire function $f$, if it is a polynomial, it assumes every value in $ℂ$ as a consequence of the fundamental theorem of algebra. 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