has to check that the resulting function f does indeed have all the required properties. Uniformization theorem The Riemann mapping theorem can be generalized to the context of Riemann surfaces: If U is a non-empty simply-connected open subset of a Riemann surface, then U is biholomorphic to one of the following: the Riemann sphere, C or D. This is known as the uniformization theorem. Smooth Riemann mapping theorem In the case of a simply connected bounded domain with smooth boundary, the Riemann mapping function and all its derivatives extend by continuity to the closure of the domain. This can be proved using regularity properties of solutions of the Dirichlet boundary value problem, which follow either from the theory of Sobolev spaces for planar domains or from classical potential theory. Other methods for proving the smooth Riemann mapping theorem include the theory of kernel functions or the Beltrami equation. Algorithms Computational conformal mapping is prominently featured in problems of applied analysis and mathematical physics, as well as in engineering disciplines, such as image processing. In the early 1980s an elementary algorithm for computing conformal maps was discovered. Given points in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve with This algorithm converges for Jordan regions in the sense of uniformly close boundaries. There are corresponding uniform estimates on the closed region and the closed disc for the mapping functions and their inverses. Improved estimates are obtained if the data points lie on a curve or a K-quasicircle. The algorithm was discovered as an approximate method for conformal welding; however, it can also be viewed as a discretization of the Loewner differential equation. The following is known about numerically approximating the conformal mapping between two planar domains. Positive results: There is an algorithm A that computes the uniformizing map in the following sense. Let be a bounded simply-connected domain, and ∂Ω is provided to A by an oracle representing it in a pixelated sense (i.e., if the screen is divided to pixels, the oracle can say whether each pixel belongs to the boundary or not). Then A computes the absolute values of the uniformizing map with precision in space bounded by and time , where C depends only on the diameter of and Furthermore, the algorithm computes the value of φ(w) with precision as long as Moreover, A queries