essential singularity

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

If $a$ is an essential singularity of $f$, then the image of any punctured neighborhood of $a$ under $f$ is dense in $ℂ$ (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 $ℂ$, 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