Editing Riemann mapping theorem (section)

==History==
The theorem was stated (under the assumption that the [[boundary (topology)|boundary]] of <math>U</math> 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”.<ref>{{Citation | last=Ahlfors | first=Lars | author-link=Lars Ahlfors | title=Developments of the Theory of Conformal Mapping and Riemann Surfaces Through a Century | journal=Contributions to the Theory of Riemann Surfaces | editor1=L. Ahlfors | editor2=E. Calabi | editor3=M. Morse | editor4=L. Sario | editor5=D. Spencer | year=1953 | pages=3–4}}</ref> 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 <math>U</math> (namely, that it is a [[Jordan curve theorem|Jordan curve]]) which are not valid for simply connected [[Domain (mathematical analysis)|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 functions|Green's function]] on arbitrary simply connected domains other than <math>\mathbb{C}</math> itself; this established the Riemann mapping theorem.<ref>For the original paper, see {{harvnb|Osgood|1900}}. For accounts of the history, see {{harvnb|Walsh|1973|pp=270–271}}; {{harvnb|Gray|1994|pp=64–65}}; {{harvnb|Greene|Kim|2017|p=4}}. Also see {{harvnb|Carathéodory|1912|p=108|loc=footnote **}} (acknowledging that {{harvnb|Osgood|1900}} had already proven the Riemann mapping theorem).</ref>

[[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]].<ref>{{harvnb|Gray|1994|pp=78–80}}, citing {{harvnb|Carathéodory|1912}}</ref> His proof used Montel's concept of normal families, which became the standard method of proof in textbooks.<ref>{{harvnb|Greene|Kim|2017|p=1}}</ref> 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 (conformal mapping)|Carathéodory's theorem]]).<ref>{{harvnb|Gray|1994|pp=80–83}}</ref>

Carathéodory's proof used [[Riemann surface]]s 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.<ref>{{Cite web |title=What did Riemann Contribute to Mathematics? Geometry, Number Theory and Others |url=https://www.researchgate.net/publication/344401528}}</ref>