Editing Riemann mapping theorem (section)

== Sketch proof via Dirichlet problem ==
Given <math>U</math> and a point <math>z_0\in U</math>, we want to construct a function <math>f</math> which maps <math>U</math> to the unit disk and <math>z_0</math> to <math>0</math>. For this sketch, we will assume that ''U'' is bounded and its boundary is smooth, much like Riemann did. Write
:<math>f(z) = (z - z_0)e^{g(z)},</math>
where <math>g=u+iv</math> is some (to be determined) holomorphic function with real part <math>u</math> and imaginary part <math>v</math>. It is then clear that <math>z_0</math> is the only zero of <math>f</math>. We require <math>|f(z)|=1</math> for <math>z\in\partial U</math>, so we need 
:<math>u(z) = -\log|z - z_0|</math>
on the boundary. Since <math>u</math> is the real part of a holomorphic function, we know that <math>u</math> is necessarily a [[harmonic function]]; i.e., it satisfies [[Laplace's equation]].

The question then becomes: does a real-valued harmonic function <math>u</math> exist that is defined on all of <math>U</math> and has the given boundary condition? The positive answer is provided by the [[Dirichlet principle]]. Once the existence of <math>u</math> has been established, the [[Cauchy–Riemann equations]] for the holomorphic function <math>g</math> allow us to find <math>v</math> (this argument depends on the assumption that <math>U</math> be simply connected). Once <math>u</math> and <math>v</math> have been constructed, one has to check that the resulting function <math>f</math> does indeed have all the required properties.<ref>{{harvnb|Gamelin|2001|pages=390–407}}</ref>