The map f is essentially unique: if z_{0} 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(z_{0}) = 0 and arg f '(z_{0}) = 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.
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