Riemann mapping theorem

In complex analysis, the Riemann mapping theorem, named after Bernhard Riemann (pronounced REE mahn), states the following: if U is a simply connected open subset of the complex number plane C which is not all of C, then there exists a bijective holomorphic conformal map

Related Topics:
Complex analysis - Theorem - Bernhard Riemann - Simply connected - Complex number plane - Bijective - Holomorphic - Conformal map

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:f : U → D,

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where D = { z in C : |z| < 1 } denotes the open unit disk. Intuitively, the condition that U be simply connected means that U does not contain any "holes"; the conformality of f means that f maintains the shape of small figures.

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The map f is essentially unique: if z0 is an element of U and φ in (−π, π] is an arbitrary angle, then there exists precisely one f as above with the additional properties f(z0) = 0 and arg f '(z0) = φ.

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As a corollary, any two such simply connected open sets (which are different from C and C U {∞}) can be conformally mapped into each other.

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The theorem was proved by Bernhard Riemann in 1851, but his proof depended on a statement in the calculus of variations which was only later proven by David Hilbert.

Related Topics:
Bernhard Riemann - 1851 - Calculus of variations - David Hilbert

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