# Riemann’s removable singularity theorem

Let $U\subset \u2102$ 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

$$\underset{z\to a}{lim}(z-a)f(z)=0.$$ |

In particular, $a$ is a removable singularity of $f$ if $f$ is http://planetmath.org/node/Bounded^{}bounded near $a$, i.e. if there is a punctured neighborhood $V$ of $a$ and a real number $M>0$ such that $$ for all $z\in V$.

Title | Riemann’s removable singularity theorem |
---|---|

Canonical name | RiemannsRemovableSingularityTheorem |

Date of creation | 2013-03-22 13:33:00 |

Last modified on | 2013-03-22 13:33:00 |

Owner | pbruin (1001) |

Last modified by | pbruin (1001) |

Numerical id | 4 |

Author | pbruin (1001) |

Entry type | Theorem |

Classification | msc 30D30 |

Related topic | Pole |

Related topic | EssentialSingularity |

Related topic | Meromorphic |

Related topic | RiemannsTheoremOnIsolatedSingularities |