Riemann Mapping Theorem
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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. Howev...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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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