Annulus (mathematics)

From Wikipedia, the free encyclopedia

In mathematics, an annulus (the Latin word for "little ring", with plural annuli) is a ring-shaped object, especially 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(R^2 - r^2)\,.

The area of an annulus can be obtained from the length of the longest interval that can lie completely inside the annulus, 2*d in the accompanying diagram. This can be proven by the Pythagorean theorem; the length of the longest interval that can lie completely inside the annulus will be tangent to the smaller circle and form a right angle with its radius at that point. Therefore, d and r are the sides of a right angled triangle with hypotenuse R and the area 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

2 πρ 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 by:

r < |z-a| < R.\,

r

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.

Mathematics

Sunday, 1 November 2015