# Runge’s theorem

Let $K$ be a compact subset of $\u2102$, and let $E$ be a subset of
${\u2102}_{\mathrm{\infty}}=\u2102\cup \{\mathrm{\infty}\}$ (the extended complex plane) which intersects every connected component^{} of ${\u2102}_{\mathrm{\infty}}-K$. If $f$ is an analytic function^{} in an open set containing $K$, given $\epsilon >0$, there is a rational function $R(z)$ whose only poles are in $E$, such that
$$ for all $z\in K$.

Title | Runge’s theorem |
---|---|

Canonical name | RungesTheorem |

Date of creation | 2013-03-22 13:15:12 |

Last modified on | 2013-03-22 13:15:12 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 6 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 30E10 |

Related topic | MergelyansTheorem |