Uniform approximation as a numerical tool for constructing conformal maps

摘要

Let f0 be the conformal mapping of a given simply connected region R onto a disk D. It is known that f0 is the unique minimizer of φ(f)=‖f‖∞, where f varies in a set H1 of holomorphic functions defined on R and ‖ ‖∞ is the uniform (Chebyshev) norm. In this paper approximations to f0 are computed numerically with the aid of approximation and optimization techniques. The essential feature is the incorporation of the geometry of R into the numerical part of this problem. Several numerical examples are given, with figures. In addition, it has been shown in another paper by one of the authors that by changing ‖ ‖∞ slightly, one can also produce conformal mappings of R onto prescribed regions other than D. An example of this kind involving a triangle is also given.