The approximate conformal mapping of a disk onto domain with an acute angle

TL;DR

This paper presents a boundary curve reparametrization method combining Fredholm equations and spline interpolation to approximate conformal mappings from a disk to domains with acute angles, including multiply connected cases.

Contribution

It introduces a novel approach for approximate conformal mapping of domains with boundary angle points using boundary reparametrization, Fredholm equations, and spline interpolation.

Findings

Method effectively constructs approximate conformal maps with boundary angles.

Applicable to multiply connected domains with boundary angle points.

Demonstrated through several example mappings.

Abstract

The method of boundary curve reparametrization is applied to construction of the approximate analytical conformal mapping of the unit disk onto an arbitrary given finite domain with a boundary smooth at every point but fininte number of acute angle points. The method is based on both the Fredholm equation solution and spline-interpolation. This approach consists of approximate solution of a linear system with unknown Fourier coefficients and construction of correction splines. The approximate mapping function has the form of a Cauchy integral. The method presentation is supported by demonstration of some examples. This method is applicable to the case of multiply connected domains with boundary angle points.