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# annulus

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 word 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 $D_{0}(r,1)$.

## Mathematics Subject Classification

30-00*no label found*

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