Theorem

Riemann Mapping Theorem

Analysis

Any simply connected open subset of the complex plane that is not the whole plane can be conformally mapped, in a bijective and angle-preserving way, onto the open unit disk. Stated by Bernhard Riemann, it is a foundational result of complex analysis and conformal geometry.

Facts

Statement

The Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f from U onto the open unit disk D. 1

Proof Year

1900 1

Classification

Statement Form

Existence Theorem 1

Connections

Sources

1. Riemann Mapping Theorem (Wikipedia)

Wikimedia Foundation

lead section, first paragraph
the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from U onto the open unit disk D
History section, Osgood proof sentence
The first rigorous proof of the theorem was given by William Fogg Osgood in 1900.

Switch Atlas

The Inner World

The Civil World

The Natural World

Riemann Mapping Theorem

Associated With

Has Statement Form

In Branch

Proved By