# Riemann mapping theorem

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## Riemann mapping theorem

[′rē‚män ′map·iŋ ‚thir·əm]
(mathematics)
Any simply connected domain in the plane with boundary containing more than one point can be conformally mapped onto the interior of the unit disk.
References in periodicals archive ?
Walsh proved the existence of such maps in 1956 and thus obtained a direct generalization of the Riemann mapping theorem to multiply connected domains.
Walsh's theorem is a direct generalization of the Riemann mapping theorem, and for n = 1, the two results are in fact equivalent.
Note that for n = 1, the lemniscatic domain L is the exterior of a disk with radius [mu] > 0, and Theorem 2.1 is equivalent to the classical Riemann mapping theorem.
For the modern mapping theory, which also considers dimensions n [greater than or equal to] 3, we do not have a Riemann mapping theorem and therefore it is natural to look for counterparts of the hyperbolic metric.
(b) the usual methods of function theory based on Cauchy's Formula, Morera's Theorem, Residue Theorem, The Residue Calculus and Consequences, Laurent Series, Schwarz's Lemma, Automorphisms of the Unit Disc, Riemann Mapping Theorem, and so forth, are not applicable in the higher-dimensional theory;
Probably the most important of these is that there exists no analogue of the Riemann mapping theorem when n > 2.
Among the topics are arithmetic and topology in the complex plane, holomorphic functions and differential forms, isolated singularities of holomorphic functions, harmonic functions, the Riemann mapping theorem and Dirichlet's problem, and the complex Fourier transform.
When talking about conformal mappings of a planar region onto another planar region a mathematician usually first thinks about complex analysis, the Riemann mapping theorem, and mapping with analytic functions.
* The Riemann Mapping Theorem. If D is a non-empty simply connected open subset of the complex plane C which is not all of C, then there exists a biholomorphic (bijective and holomorphic) mapping f from D onto the open unit disk U = {z [member of] C :[absolute value of z] <1} (Krantz, 1999, Section 6.4.3, p.
In the case that A is a simply connected domain in the plane which is not the whole plane, our argument yields the following generalization to the Riemann mapping theorem:
RIEMANN MAPPING THEOREM FOR RELATIVE CIRCLE DOMAINS.
INVERSE RIEMANN MAPPING THEOREM FOR RELATIVE CIRCLE DOMAINS.
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