In complex analysis, the Riemann mapping theorem states that if U {\displaystyle U} is a non-empty simply connected open subset of the complex number plane C {\displaystyle \mathbb {C} } which is not all of C {\displaystyle \mathbb {C} } , then there exists a biholomorphic mapping f {\displaystyle f} (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from U {\displaystyle U} onto the open unit disk
D = { z ∈ C : | z | < 1 } . {\displaystyle D=\{z\in \mathbb {C} :|z|<1\}.}
This mapping is sometimes called the Riemann mapping from U {\displaystyle U} to the unit disk. Intuitively, the condition that U {\displaystyle U} be simply connected means that U {\displaystyle U} does not contain any "holes". The fact that f {\displaystyle f} is biholomorphic implies that it is a conformal map and therefore angle-preserving. Such a map may be interpreted as preserving the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it. Henri Poincaré proved that the map f {\displaystyle f} is unique up to rotation and recentering: if z 0 {\displaystyle z_{0}} is an element of U {\displaystyle U} and ϕ {\displaystyle \phi } is an arbitrary angle, then there exists precisely one f as above such that f ( z 0 ) = 0 {\displaystyle f(z_{0})=0} and such that the argument of the derivative of f {\displaystyle f} at the point z 0 {\displaystyle z_{0}} is equal to ϕ {\displaystyle \phi } . 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 both lack at least two points of the sphere can be conformally mapped into each other.
History The theorem was stated (under the assumption that the boundary of U {\displaystyle 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 (which was named 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 {\displaystyle U} (namely, that it is a Jordan curve) which are not valid for simply connected domains in general. The first rigorous proof of the theorem was given by William Fogg Osgood in 1900. He proved the existence of Green's function on arbitrary simply connected domains other than C {\displaystyle \mathbb {C} } itself; this established the Riemann mapping theorem. Constantin Carathéodory gave another proof of the theorem in 1912, which was the first to rely purely on the methods of function theory rather than potential theory. His proof used Montel's concept of normal families, which became the standard method of proof in textbooks. Carathéodory continued in 1913 by resolving the additional question of whether the Riemann mapping between the domains can be extended to a homeomorphism of the boundaries (see Carathéodory's theorem). Carathéodory's proof used Riemann surfaces and it was simplified by Paul Koebe two years later in a way that did not require them. Another proof, due to Lipót Fejér and to Frigyes Riesz, was published in 1922 and it was rather shorter than the previous ones. In this proof, like in Riemann's proof, the desired mapping was obtained as the solution of an extremal problem. The Fejér–Riesz proof was further simplified by Alexander Ostrowski and by Carathéodory.
Importance The following points detail the uniqueness and power of the Riemann mapping theorem:
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