## Smoothness to the boundary of conformal maps.(English)Zbl 0626.30005

A self contained and elementary proof of the following classical result is given. Theorem. Let $$\Omega\subseteq {\mathbb{C}}$$ be a bounded simply connected domain with $$C^{\infty}$$ smooth boundary. Let $$f: \Omega\to \Delta$$ be a one-to-one conformal map of $$\Omega$$ onto the unit disc, as provided by the Riemann mapping theorem. Then f continues up to the boundary to be a diffeomorphism between the closure of $$\Omega$$ and the closure of $$\Delta$$.
A brief history of the theorem is given, together with applications to the Dirichlet problem.

### MSC:

 30C35 General theory of conformal mappings 30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable 30C40 Kernel functions in one complex variable and applications

### Keywords:

boundary regularity; Riemann mapping theorem
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