---
tag: 0AJ0YVK743
kind: definition
name: Annulus
msc: [30A99]
defines: 0CW52NV5TC
uses: [0CTE7A31XQ, 0KW2X0TFBX]
sources:
  - url: https://en.wikipedia.org/wiki/Annulus_(mathematics)
    locator: Complex structure
---

## 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|0CTE7A31XQ]]. Here $|z - c|$ is the [[modulus|0KW2X0TFBX]] of $z - c$.

## Remarks

Each annulus is a [[domain|097KNJFJ7V]].
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|06WHRARZKY]].

## 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 $2$.