Annulus
From Freepedia
In mathematics, an annulus (the Latin word for "little ring", with plural annuli, more at Wiktionary:anus) 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 the open cylinder <math>S^1 \times (0,1)</math>.
The area of such a annulus is given by the difference in the areas of a circle of radius R and one of radius r:
- <math>A = \pi(R^2 - r^2)\,</math>
This result can be obtained via calculus by dividing the annulus up into an infinite number of annuli of infinitesimal width <math>d\rho</math> and area <math>2\pi\rho d\rho</math> ( = circumference × width) and then integrating from <math>\rho = r</math> to <math>\rho = R</math>:
- <math>A = \int_r^R 2\pi\rho\, d\rho = \pi(R^2-r^2).</math>
Complex structure
In complex analysis an annulus ann(a; r, R) in the complex plane is an open region defined by:
- <math> r < |z-a| < R.\,</math>
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
- <math>z \mapsto \frac{z-a}{R}.</math>
The inner radius is then r/R < 1.
