# Liouville’s theorem

A bounded entire function^{} is constant. That is, a bounded complex function $f:\u2102\to \u2102$ which is holomorphic on the entire complex plane^{} is always a constant function.

More generally, any holomorphic function $f:\u2102\to \u2102$ which satisfies a polynomial^{} bound condition of the form

$$ |

for some $c\in \mathbb{R}$, $n\in \mathbb{Z}$, and all $z\in \u2102$ with $|z|$ sufficiently large is necessarily equal to a polynomial function.

Liouville’s theorem is a vivid example of how stringent the holomorphicity condition on a complex function really is. One has only to compare the theorem to the corresponding statement for real functions (namely, that a bounded differentiable^{} real function is constant, a patently false statement) to see how much stronger the complex differentiability condition is compared to real differentiability.

Applications of Liouville’s theorem include proofs of the fundamental theorem of algebra^{} and of the partial fraction decomposition theorem for rational functions^{}.

Title | Liouville’s theorem |
---|---|

Canonical name | LiouvillesTheorem |

Date of creation | 2013-03-22 12:04:31 |

Last modified on | 2013-03-22 12:04:31 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 8 |

Author | djao (24) |

Entry type | Theorem |

Classification | msc 30D20 |