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Riemann's removable singularity theorem
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(Theorem)
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Let $U\subset\mathbb{C}$ be a domain, $a\in U$ , and let $f:U\setminus\{a\}$ be holomorphic. Then $a$ is a removable singularity of $f$ if and only if $$ \lim_{z\to a}(z-a)f(z)=0. $$
In particular, $a$ is a removable singularity of $f$ if $f$ is bounded near $a$ , i.e. if there is a punctured neighborhood $V$ of $a$ and a real number $M>0$ such that $|f(z)|<M$ for all $z\in V$ .
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"Riemann's removable singularity theorem" is owned by pbruin.
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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 3599 times total.
Classification:
| AMS MSC: | 30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory) |
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Pending Errata and Addenda
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