# essential singularity

Let $U\subset \u2102$ be a domain, $a\in U$, and let $f:U\setminus \{a\}\to \u2102$ be holomorphic. If the Laurent series^{} expansion of $f(z)$ around $a$ contains infinitely many terms with negative powers of $z-a$, then $a$ is said to be an *essential singularity ^{}* of $f$. Any singularity of $f$ is a removable singularity

^{}, a pole or an essential singularity.

If $a$ is an essential singularity of $f$, then the image of any punctured neighborhood^{} of $a$ under $f$ is dense in $\u2102$ (the Casorati-Weierstrass theorem). In fact, an even stronger statement is true: according to Picard’s theorem, the image of any punctured neighborhood of $a$ is $\u2102$, with the possible exception of a single point.

Title | essential singularity |
---|---|

Canonical name | EssentialSingularity |

Date of creation | 2013-03-22 13:32:10 |

Last modified on | 2013-03-22 13:32:10 |

Owner | pbruin (1001) |

Last modified by | pbruin (1001) |

Numerical id | 7 |

Author | pbruin (1001) |

Entry type | Definition |

Classification | msc 30D30 |

Related topic | LaurentSeries |

Related topic | Pole |

Related topic | RemovableSingularity |

Related topic | PicardsTheorem |

Related topic | RiemannsRemovableSingularityTheorem |