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# essential singularity

Let $U\subset\mathbb{C}$ be a domain, $a\in U$, and let $f:U\setminus\{a\}\to\mathbb{C}$ 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 $\mathbb{C}$ (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 $\mathbb{C}$, with the possible exception of a single point.

Related:

LaurentSeries, Pole, RemovableSingularity, PicardsTheorem, RiemannsRemovableSingularityTheorem

Type of Math Object:

Definition

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Reference

## Mathematics Subject Classification

30D30*no label found*

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