# Casorati-Weierstrass theorem

Given a domain $U\subset \u2102$, $a\in U$, and $f:U\setminus \{a\}\to \u2102$ being holomorphic, then $a$ is an essential singularity^{} of $f$ if and only if the image of any punctured neighborhood^{} of $a$ under $f$ is dense in $\u2102$. Put another way, a holomorphic function can come in an arbitrarily small neighborhood of its essential singularity arbitrarily close to any complex value.

Title | Casorati-Weierstrass theorem |
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Canonical name | CasoratiWeierstrassTheorem |

Date of creation | 2013-03-22 13:32:36 |

Last modified on | 2013-03-22 13:32:36 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 7 |

Author | PrimeFan (13766) |

Entry type | Theorem |

Classification | msc 30D30 |

Synonym | Weierstrass-Casorati theorem |

Related topic | PicardsTheorem |