# proof of extended Liouville’s theorem

This is a proof of the second, more general, form of Liouville’s theorem given in the parent (http://planetmath.org/LiouvillesTheorem2) article.

Let $f:\u2102\to \u2102$ be a holomorphic function^{} such that

$$ |

for some $c\in \mathbb{R}$ and for $z\in \u2102$ with $|z|$ sufficiently large. Consider

$$g(z)=\{\begin{array}{cc}\frac{f(z)-f(0)}{z}\hfill & z\ne 0\hfill \\ {f}^{\prime}(0)\hfill & z=0\hfill \end{array}$$ |

Since $f$ is holomorphic, $g$ is as well, and by the bound on $f$, we have

$$ |

again for $|z|$ sufficiently large.

By induction^{}, $g$ is a polynomial of degree at most $n-1$, and thus $f$ is a polynomial of degree at most $n$.

Title | proof of extended Liouville’s theorem |
---|---|

Canonical name | ProofOfExtendedLiouvillesTheorem |

Date of creation | 2013-03-22 16:18:31 |

Last modified on | 2013-03-22 16:18:31 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 8 |

Author | rm50 (10146) |

Entry type | Proof |

Classification | msc 30D20 |