PlanetMath: Riemann's removable singularity theorem

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Riemann's removable singularity theorem

(Theorem)

Let $U\subset\mathbb{C}$ be a domain, $a\in U$ , and let $f:U\setminus\{a\}$ be holomorphic. Then is a removable singularity of if and only if

$\displaystyle \lim_{z\to a}(z-a)f(z)=0.$

In particular, is a removable singularity of if is bounded near , i.e. if there is a punctured neighborhood of and a real number such that $\vert f(z)\vert<M$ for all $z\in V$ .

"Riemann's removable singularity theorem" is owned by pbruin.

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See Also: pole, essential singularity, meromorphic

Attachments:: proof of Riemann's removable singularity theorem (Proof) by pbruin
Riemann's removable singularity theorem in several variables (Theorem) by jirka

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Cross-references: real number, neighborhood, near, removable singularity, holomorphic, domain
There are 2 references to this entry.

This is version 1 of Riemann's removable singularity theorem, born on 2003-04-05.
Object id is 4152, canonical name is RiemannsRemovableSingularityTheorem.
Accessed 2808 times total.

Classification:

AMS MSC:

30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory)

Pending Errata and Addenda

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