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In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f from U onto the open unit disk
This mapping is known as a Riemann mapping.
Intuitively, the condition that U be simply connected means that U does not contain any “holes”. The fact that f is biholomorphic implies that it is a conformal map and therefore angle-preserving. Intuitively, such a map preserves the shape of any sufficiently small figure, while possibly rotating and scaling it.
Henri Poincaré proved that the map f is essentially unique: if z0 is an element of U and φ is an arbitrary angle, then there exists precisely one f as above such that f = 0 and such that the argument of the derivative of f at the point z0 is equal to φ. This is an easy consequence of the Schwarz lemma.