# annulus

An *open annulus* is a domain in the complex plane^{} of the form

$$ |

where $w$ is an arbitrary complex number^{}, and $r$ and $R$ are real numbers with $$. Such a set is often called an *annular region*.

It should be noted that the annulus usually refers to an open annulus.

More generally, one can allow $r=0$ or $R=\mathrm{\infty}$. (This makes sense for the purposes of the bound on $|z-w|$ above.) This would make an annulus include the cases of a punctured disc, and some unbounded domains.

Analogously, a *closed annulus* is a set of the form

$$\overline{A}={\overline{A}}_{w}(r,R)=\{z\in \u2102:r\le |z-w|\le R\},$$ |

where $w\in \u2102$, and $r$ and $R$ are real numbers with $$.

One can show that two annuli ${D}_{w}(r,R)$ and ${D}_{{w}^{\prime}}({r}^{\prime},{R}^{\prime})$ are conformally equivalent if and only if $R/r={R}^{\prime}/{r}^{\prime}$. More generally, the complement of any closed disk in an open disk is conformally equivalent to precisely one annulus of the form ${D}_{0}(r,1)$.

Title | annulus |
---|---|

Canonical name | Annulus1 |

Date of creation | 2013-03-22 13:34:52 |

Last modified on | 2013-03-22 13:34:52 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 7 |

Author | Wkbj79 (1863) |

Entry type | Definition |

Classification | msc 30-00 |

Synonym | open annulus |

Synonym | annular region |

Related topic | Annulus |

Defines | closed annulus |