The main results of the paper are two effective versions of the Riemann mapping theorem. The first, uniform version is based on the constructive proof of the Riemann mapping theorem by Cheng and by ...Bishop and Bridges and formulated in the computability framework developed by Kreitz and Weihrauch. It states which topological information precisely one needs about a nonempty, proper, open, connected, and simply connected subset of the complex plane in order to compute a description of a holomorphic bijection from the unit disk onto this set, and vice versa, which topological information about the set can be obtained from a description of a holomorphic bijection. The second version, which is derived from the first by considering the sets and the functions with computable descriptions, characterizes the subsets of the complex plane for which there exists a computable holomorphic bijection from the unit disk. This gives a partial answer to a problem posed by Pour-El and Richards. We also show that this class of sets is strictly larger than a class of sets considered by Zhou, which solves an open problem posed by him. In preparation, recursively enumerable open subsets and closed subsets of Euclidean spaces are considered and several effective results in complex analysis are proved.
Let Ω be a bounded convex domain and let ω be a finite Blaschke product of order N = 1, 2, ... It is known that the elliptic differential equation admits a one-to-one solution normalized by ƒ(0) = 0, ...ƒz(0) > 0 and maps the open unit disc onto a convex (n + 2)-gon whose vertices belong to ∂Ω. In this article it is shown that this solution is unique.
Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapid ...introduction to techniques in complex dynamics. Its central concern is the structure of an infinitely renormalizable quadratic polynomial f(z) = z2 + c. As discovered by Feigenbaum, such a mapping exhibits a repetition of form at infinitely many scales. Drawing on universal estimates in hyperbolic geometry, this work gives an analysis of the limiting forms that can occur and develops a rigidity criterion for the polynomial f. This criterion supports general conjectures about the behavior of rational maps and the structure of the Mandelbrot set. The course of the main argument entails many facets of modern complex dynamics. Included are foundational results in geometric function theory, quasiconformal mappings, and hyperbolic geometry. Most of the tools are discussed in the setting of general polynomials and rational maps.
We analyze an only recently identified Riemann autograph that had been in the possession of Georg August Thieme (1831–1910). These are the notes which Bernhard Riemann (1826–1866) made during his ...conversation with Thieme in order to explain the central points of his theory of functions of a complex variable, in particular his version of the Dirichlet principle. Copyright 1999 Academic Press.
Wir analysieren ein erst kürzlich identifiziertes Riemann-Autograph aus dem Besitz von Georg August Thieme (1831–1910). Es handelt sich um Notizen, mittels derer Bernhard Riemann (1826–1866) im Gespräch mit Thieme die zentralen Punkte seiner Theorie der Funktionen einer komplexen Veränderlichen erläutert hat, insbesondere seine Version des Dirichlet-Prinzips. Copyright 1999 Academic Press.
MSC 1991 subject classification: 01A55; 30-03; 31-03; 01A72.
We give a simple proof of a Riemann mapping theorem for domains in a Stein manifold with a spherical boundary. Those boundaries are real hypersurfaces which are locally CR equivalent to the sphere ...inside complex space. The main observation is that for higher dimensional Stein manifolds, the fundamental group of its boundary coincides with that of its interior.
In this article we give a description of some particular aspects of Schwarz-Christoffel transformations either from a theoretical point of view or related to numerical approach. Furthermore some ...interesting geometrical properties of these transforms are shown and displayed by means of a graphical interface. In particular, an approximation of the identity in the unit disk of
by means of a sequence of Schwarz-Christoffel transformations is exhibited.
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