Annulus (mathematics)

In mathematics, an annulus (the Latin word for "little ring", with plural annuli) is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. The adjective form is annular (for example, an annular eclipse).

The open annulus is topologically equivalent to both the open cylinder $$S^1 \times (0,1)$$ and the punctured plane.

The area of such an annulus is given by the difference in the areas of a circle of radius R and one of radius r:
 * $$A = \pi(R^2 - r^2)\,.$$

Interestingly, the area of an annulus can also be obtained by multiplying pi by the square of half of the length of the longest interval that can lie completely inside the annulus. 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. Given the above formula for area, half of the length of the interval will actually form a right angle, along with radius r, to form diagonal R.

This result can be obtained via calculus by dividing the annulus up into an infinite number of annuli of infinitesimal width $$d\rho$$ and area $$2\pi\rho\, d\rho$$ ( = circumference &times; width) and then integrating from $$\rho = r$$ to $$\rho = R$$:
 * $$A = \int_r^R 2\pi\rho\, d\rho = \pi(R^2-r^2).$$

Complex structure
In complex analysis an annulus ann(a; r, R) in the complex plane is an open region defined by:


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

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.