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

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

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

where $w\in ℂ$, 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 $D_0(r,1)$.