High Quality Content by WIKIPEDIA articles! In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane Bbb C which is not all of Bbb C, then there exists a biholomorphic (bijective and holomorphic) mapping f, from U, onto the open unit disk D={zin {Bbb C} :|z|<1}. 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 (but not reflecting) it.

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