Smooth Riemann Mapping Theorem
In the case of a simply connected bounded domain with smooth boundary, the Riemann mapping function and all its derivatives extend by continuity to the closure of the domain. This can be proved using regularity properties of solutions of the Dirichlet boundary value problem, which follow either from the theory of Sobolev spaces for planar domains or from classical potential theory. Other methods for proving the smooth Riemann mapping theorem include the theory of kernel functions or the Beltrami equation.
Read more about this topic: Riemann Mapping Theorem
Famous quotes containing the words smooth and/or theorem:
“We are born with luck
which is to say with gold in our mouth.
As new and smooth as a grape,
as pure as a pond in Alaska,
as good as the stem of a green bean
we are born and that ought to be enough....”
—Anne Sexton (1928–1974)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)