30-00-Annulus1.tex

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\pmtitle{annulus}
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\pmtype{Definition}
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\begin{document}
An \emph{annulus} is the region bounded between two (usually concentric) circles.

An \emph{open annulus} is a domain in the complex plane of the form
\[
A = A_w(r,R) = \{z \in \bbC : r < |z-w| < R\},
\]
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 \emph{annular region}.

It should be noted that the \PMlinkescapetext{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 \emph{closed annulus} is a set of the form
\[
\ov{A} = \ov{A}_w(r,R) = \{z \in \bbC : r \leq |z-w| \leq R\},
\]
where $w \in \bbC$, and $r$ and $R$ are real numbers with $0 < r < R$.

One can show that two annuli $D_w(r,R)$ and $D_{w'}(r',R')$ are conformally equivalent if and only if $R/r = R'/r'$.  More generally, the complement of any closed disk in an open disk is conformally equivalent to precisely one annulus of the form $D_0(r,1)$.
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