Encyclopedia > Riemann mapping theorem

  Article Content

Riemann mapping theorem

The Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number plane C (or even of the compact complex number sphere C U {∞}) which is different from C (and C U {∞}),, then there exists a bijective holomorphic conformal map f : U -> D, where D = { z in C : |z| < 1 } denotes the open disk. Intuitively, the condition that U be simply connected means that U does not contain any "holes"; the conformality of f means that f maintains the shape of small figures.

The map f is essentially unique: if z0 is an element of U and φ in (-π, π] is an arbitrary angle, then there exists precisely one f as above with the additional properties f(z0) = 0 and arg f '(z0) = φ.

As a corollary, any two such simply connected open sets (which are different from C and C U {∞}) can be conformally mapped into each other.

The theorem was proved by Bernhard Riemann in 1851, but his proof depended on a statement in the calculus of variations which was only later proven by David Hilbert.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article

... 13112 are male, and 13338 are female. The population density of the community is 18 inhabitants per km². External links Ludvika (http://www.ludvika.se) - ...

This page was created in 25.9 ms