Conformal mapping of circular arc polygons
SIAM Journal on Scientific and Statistical Computing - Papers from the Second Conference on Parallel Processing for Scientific Computin
Numerical conformal mapping of circular arc polygons
Journal of Computational and Applied Mathematics - Special issue on computational complex analysis
A Multipole Method for Schwarz--Christoffel Mapping of Polygons with Thousands of Sides
SIAM Journal on Scientific Computing
Functions of a Complex Variable: Theory and Technique (Classics in Applied Mathematics)
Functions of a Complex Variable: Theory and Technique (Classics in Applied Mathematics)
A modified Schwarz-Christoffel mapping for regions with piecewise smooth boundaries
Journal of Computational and Applied Mathematics
Schwarz-Christoffel Mappings for Nonpolygonal Regions
SIAM Journal on Scientific Computing
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The Schwarz-Christoffel mapping from the upper half-plane to a polygonal region in the complex plane is an integral of a product with several factors, where each factor corresponds to a certain vertex in the polygon. Different modifications of the Schwarz-Christoffel mapping in which factors are replaced with the so-called curve factors to achieve polygons with rounded corners are known since long times. Among other requisites, the arguments of a curve factor and its correspondent scl factor must be equal outside some closed interval on the real axis. In this paper, the term approximate curve factor is defined such that many of the already known curve factors are included as special cases. Additionally, by alleviating the requisite on the argument from exact to asymptotic equality, new types of curve factors are introduced. While traditional curve factors have a C^1 regularity, C^~ regular approximate curve factors can be constructed, resulting in smooth boundary curves when used in conformal mappings. Applications include modelling of wave scattering in waveguides. When using approximate curve factors in modified Schwarz-Christoffel mappings, numerical conformal mappings can be constructed that preserve two important properties in the waveguides. First, the direction of the boundary curve can be well controlled, especially towards infinity, where the application requires two straight parallel walls. Second, a smooth (C^~) boundary curve can be achieved.