Annulus

A ring-shaped region between two concentric circles, $r<|z-z_0|<R$ .

An annulus centered at $z_0$ is the region strictly between two concentric circles, $r<|z-z_0|<R$ , for radii $0\le r<R\le\infty$ . It is a domain (open and connected) but never simply connected, since a loop circling $z_0$ inside the annulus cannot shrink to a point without crossing the missing inner disk; a Laurent series is exactly the tool for representing a function analytic throughout an annulus.

Examples

$1<|z|<3$ is an annulus of inner radius $1$ and outer radius $3$ .
Taking $r=0$ gives a punctured disk $0<|z-z_0|<R$ , the special case of an annulus with no inner circle at all.