Casorati-Weierstrass theorem

Given a domain $U\subset ℂ$, $a\in U$, and $f:U\setminus \{a\}\to ℂ$ being holomorphic, then $a$ is an essential singularity of $f$ if and only if the image of any punctured neighborhood of $a$ under $f$ is dense in $ℂ$. Put another way, a holomorphic function can come in an arbitrarily small neighborhood of its essential singularity arbitrarily close to any complex value.

Title	Casorati-Weierstrass theorem
Canonical name	CasoratiWeierstrassTheorem
Date of creation	2013-03-22 13:32:36
Last modified on	2013-03-22 13:32:36
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	7
Author	PrimeFan (13766)
Entry type	Theorem
Classification	msc 30D30
Synonym	Weierstrass-Casorati theorem
Related topic	PicardsTheorem