Conformal Mapping of Nonsmooth Domains via the Kerzman–Stein Integral Equation

Abstract

The connection between the Riemann map and the Szego kernel is classical. But the fact that there is an efficient numerical procedure, based on the Kerzman᎐Stein integral equation, for computing the Szego kernel of a smoothly bounded domain in the plane is more recent. In this paper it is shown how to extend the results of Kerzman and Stein to certain nonsmooth domains. This provides a new method for numerically computing the Szego kernel, and hence the Riemann map, of these domains. ᮊ

Holomorphic reproducing kernels for piecewise-smooth planar domains

“…That modification depends on both the angles and the curvature at the corners, whereas our construction only depends on the angles at the corners and it applies to a larger class of domains. We also point out work of Thomas in [10]. He showed that the Nyström solution of Eq.…”

Untitled

“…In particular, Riemann_Map() works for domains with piecewise smooth boundary. For a justification of this point, see [Thomas 1996]. …”

Numerical Conformal Mapping for Exterior Regions via the Kerzman-Stein Kernel

“…Kerzman and Trummer [10] have derived an integral equation of the second kind, whose unique solution is the Szegö kernel. This integral equation is now called the Kerzman-Stein integral equation by some authors, see [3, p. 108], [17,13], or the integral equation of Kerzman and Trummer [7,p. 560].…”