Riemann mapping theorem

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The Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number plane C (or even of the compact complex number sphere C U {∞}) which is different from C (and C U {∞}) and D = { z in C : |z| < 1 } denotes the open disk, then there exists a bijective holomorphic conformal map f : U -> D. 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.

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.

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.

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.