Annulus

Definition

Let $c \in \mathbb{C}$ and let $r, R$ be such that $0 \le r < R \le \infty$. The annulus with center $c$, inner radius $r$ and outer radius $R$ is the subset $$A(c; r, R) = \{ z \in \mathbb{C} : r < |z - c| < R \}$$ of the complex plane. Here $|z - c|$ is the modulus of $z - c$.

Remarks

Each annulus is a domain. For $r = 0$, the annulus is the punctured disc $0 < |z - c| < R$. For $r = 0$ and $R = \infty$, it is $\mathbb{C} \setminus \{c\}$. For $R < \infty$, its area is $\pi (R^2 - r^2)$. Holomorphic functions on an annulus have Laurent expansions.

Explanations

L1 human

An annulus is the ring between two circles with the same center, without the two circles. For example, $1 < |z| < 2$ is the set of the points whose distance to $0$ is between $1$ and $2$.