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

An open annulus is a domain in the complex plane of the form $$ A = A_w(r,R) = \{z \in \bbC : r < |z-w| < R\}, $$ 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 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 $$ \ov{A} = \ov{A}_w(r,R) = \{z \in \bbC : r \leq |z-w| \leq R\}, $$ 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 $D_0(r,1)$ .