Riemann mapping theorem

Let U be a simply connected open proper subsetMathworldPlanetmathPlanetmath of , and let aU. There is a unique analytic functionMathworldPlanetmath f:U such that

  1. 1.

    f(a)=0, and f(a) is real and positive;

  2. 2.

    f is injectivePlanetmathPlanetmath;

  3. 3.


Remark. As a consequence of this theorem, any two simply connected regions, none of which is the whole plane, are conformally equivalent.

Title Riemann mapping theoremMathworldPlanetmath
Canonical name RiemannMappingTheorem
Date of creation 2013-03-22 13:15:03
Last modified on 2013-03-22 13:15:03
Owner Koro (127)
Last modified by Koro (127)
Numerical id 5
Author Koro (127)
Entry type Theorem
Classification msc 30A99
Related topic ConformalRadius