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The theorem was stated (under the assumption that the boundary of 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”. [2]
In mathematics, the measurable Riemann mapping theorem is a theorem proved in 1960 by Lars Ahlfors and Lipman Bers in complex analysis and geometric function theory.Contrary to its name, it is not a direct generalization of the Riemann mapping theorem, but instead a result concerning quasiconformal mappings and solutions of the Beltrami equation.
Although Riemann's mapping theorem demonstrates the existence of a mapping function, it does not actually exhibit this function. An example is given below. An example is given below. In the above figure, consider D 1 {\displaystyle D_{1}} and D 2 {\displaystyle D_{2}} as two simply connected regions different from C {\displaystyle \mathbb {C} } .
The theorem generalizes the Riemann mapping theorem from conformal to quasiconformal homeomorphisms, and is stated as follows. Suppose that D is a simply connected domain in C that is not equal to C , and suppose that μ : D → C is Lebesgue measurable and satisfies ‖ μ ‖ ∞ < 1 {\displaystyle \|\mu \|_{\infty }<1} .
Riemann mapping theorem (complex analysis) Riemann series theorem (mathematical series) Riemann's existence theorem (algebraic geometry) Riemann's theorem on removable singularities (complex analysis) Riemann–Roch theorem (Riemann surfaces, algebraic curves) Riemann–Roch theorem for smooth manifolds (differential topology)
In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon.Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard Riemann in 1851); the Schwarz–Christoffel formula provides an explicit construction.
In mathematics, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply ...
The Riemann mapping theorem, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of admits a bijective conformal map to the open unit disk in . Informally, this means that any blob can be transformed into a perfect circle by some conformal map.