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# Riemann’s removable singularity theorem

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 $z\in V$.

Related:

Pole, EssentialSingularity, Meromorphic, RiemannsTheoremOnIsolatedSingularities

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

30D30*no label found*

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