# Casorati-Weierstrass theorem

Given a domain $U\subset\mathbb{C}$, $a\in U$, and $f:U\setminus\{a\}\to\mathbb{C}$ 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 $\mathbb{C}$. 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 CasoratiWeierstrassTheorem 2013-03-22 13:32:36 2013-03-22 13:32:36 PrimeFan (13766) PrimeFan (13766) 7 PrimeFan (13766) Theorem msc 30D30 Weierstrass-Casorati theorem PicardsTheorem