annulus

An annulus is the region bounded between two (usually concentric) circles.

An open annulus is a domain in the complex plane of the form

$A=A_{w}(r,R)=\{z\in\mathbb{C}:r<|z-w|<R\},$

where $w$ is an arbitrary complex number, and $r$ and $R$ are real numbers with $0<r<R$ . 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 $R=\infty$ . (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

$\overline{A}=\overline{A}_{w}(r,R)=\{z\in\mathbb{C}:r\leq|z-w|\leq R\},$

where $w\in\mathbb{C}$ , and $r$ and $R$ are real numbers with $0<r<R$ .

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