Annulus (mathematics)

In mathematics, an annulus (the Latin word for "little ring" is anulus, with plural anuli) is a ring-shaped object — a region bounded by two concentric circles. The adjectival form is annular (as in annular eclipse).

The open annulus is topologically equivalent to both the open cylinder $S 1 \times (0,1)$ and the punctured plane. Informally, it has the shape of a hardware washer.

The area of an annulus is the difference in the areas of the larger circle of radius $R$ and the smaller one of radius $r$ :

A=\pi R^{2}-\pi r^{2}=\pi \left(R^{2}-r^{2}\right).

The area of an annulus is determined by the length of the longest line segment within the annulus, which is the chord tangent to the inner circle, $2 d$ in the accompanying diagram. This can be shown using the Pythagorean theorem, since this line is tangent to the smaller circle and perpendicular to its radius at that point, so $d$ and $r$ are sides of a right-angled triangle with hypotenuse $R$ , and the area of the annulus is given by

A=\pi \left(R^{2}-r^{2}\right)=\pi d^{2}.

The area can also be obtained via calculus by dividing the annulus up into an infinite number of annuli of infinitesimal width $dρ$ and area $2π ρ dρ$ and then integrating from $ρ = r$ to $ρ = R$ :

A=\int _{r}^{R}\!\!2\pi \rho \,d\rho =\pi \left(R^{2}-r^{2}\right).

The area of an annulus sector of angle $θ$ , with $θ$ measured in radians, is given by

A={\frac {\theta }{2}}\left(R^{2}-r^{2}\right).

Complex structure

In complex analysis an annulus $ann(a; r, R)$ in the complex plane is an open region defined as

r<|z-a|<R.

If $r$ is $0$ , the region is known as the punctured disk of radius $R$ around the point $a$ .

As a subset of the complex plane, an annulus can be considered as a Riemann surface. The complex structure of an annulus depends only on the ratio $r / R$ . Each annulus $ann(a; r, R)$ can be holomorphically mapped to a standard one centered at the origin and with outer radius 1 by the map

z\mapsto {\frac {z-a}{R}}.

The inner radius is then $r / R < 1$ .

The Hadamard three-circle theorem is a statement about the maximum value a holomorphic function may take inside an annulus.

External links

Annulus definition and properties With interactive animation
Area of an annulus, formula With interactive animation

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Annulus (mathematics)

Complex structure

See also

External links