# Picard’s theorem

Let $f$ be an holomorphic function^{} with an essential singularity^{} at $w\in \u2102$. Then there is a number ${z}_{0}\in \u2102$ such that the image of any neighborhood of $w$ by $f$ contains $\u2102-\{{z}_{0}\}$. In other words, $f$ assumes every complex value, with the possible exception of ${z}_{0}$, in any neighborhood of $w$.

*Remark.* Little Picard theorem follows as a corollary:
Given a nonconstant entire function^{} $f$, if it is a polynomial^{}, it assumes every value in $\u2102$ as a consequence of the fundamental theorem of algebra^{}. If $f$ is not a polynomial, then $g(z)=f(1/z)$ has an essential singularity at $0$; Picard’s theorem implies that $g$ (and thus $f$) assumes every complex value, with one possible exception.

Title | Picard’s theorem |
---|---|

Canonical name | PicardsTheorem |

Date of creation | 2013-03-22 13:15:23 |

Last modified on | 2013-03-22 13:15:23 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 9 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 32H25 |

Synonym | great Picard theorem |

Related topic | EssentialSingularity |

Related topic | CasoratiWeierstrassTheorem |

Related topic | ProofOfCasoratiWeierstrassTheorem |