In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane which is not all of , then there exists a biholomorphic (bijective and holomorphic) mapping from onto the open unit disk
Intuitively, the condition that U be simply connected means that U does not contain any “holes”. The fact that f is biholomorphic implies that it is a conformal map and therefore angle-preserving. Intuitively, such a map preserves the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it.
Henri Poincaré proved that the map f is essentially unique: if z0 is an element of U and φ is an arbitrary angle, then there exists precisely one f as above with the additional properties that f maps z0 into 0 and that the argument of the derivative of f at the point z0 is equal to φ. This is an easy consequence of the Schwarz lemma.
As a corollary of the theorem, any two simply connected open subsets of the Riemann sphere (which each lack at least two points of the sphere) can be conformally mapped into each other (because conformal equivalence is an equivalence relation).
The theorem was stated (under the assumption that the boundary of U is piecewise smooth) by Bernhard Riemann in 1851 in his PhD thesis. Lars Ahlfors wrote once, concerning the original formulation of the theorem, that it was “ultimately formulated in terms which would defy any attempt of proof, even with modern methods”. Riemann's flawed proof depended on the Dirichlet principle (whose name was created by Riemann himself), which was considered sound at the time. However, Karl Weierstraß found that this principle was not universally valid. Later, David Hilbert was able to prove that, to a large extent, the Dirichlet principle is valid under the hypothesis that Riemann was working with. However, in order to be valid, the Dirichlet principle needs certain hypotheses concerning the boundary of U which are not valid for simply connected domains in general. Simply connected domains with arbitrary boundaries were first treated in 1900 by W. F. Osgood.
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