Complex Structure
In complex analysis an annulus ann(a; r, R) in the complex plane is an open region defined by:
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
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.
Read more about this topic: Annulus (mathematics)
Famous quotes containing the words complex and/or structure:
“By “object” is meant some element in the complex whole that is defined in abstraction from the whole of which it is a distinction.”
—John Dewey (1859–1952)
“I’m a Sunday School teacher, and I’ve always known that the structure of law is founded on the Christian ethic that you shall love the Lord your God and your neighbor as yourself—a very high and perfect standard. We all know the fallibility of man, and the contentions in society, as described by Reinhold Niebuhr and many others, don’t permit us to achieve perfection.”
—Jimmy Carter (James Earl Carter, Jr.)