Numerical Conformal Mapping to One-Tooth Gear-Shaped Domains and Applications

TL;DR

This paper explores numerical conformal mappings from simple domains to gear-shaped regions, focusing on the Schwarzian derivative, with applications including singular integral evaluation and mapping to complex gear-like shapes.

Contribution

It introduces numerical methods for conformal mapping to gear-shaped domains and demonstrates their applications in complex analysis and geometric mapping.

Findings

Developed numerical techniques for gear-shaped domain mappings

Applied methods to evaluate singular integrals and map to complex gear geometries

Enhanced understanding of conformal mappings involving Schwarzian derivatives

Abstract

We study conformal mappings from the unit disk (or a rectangle) to one-tooth gear-shaped planar domains from the point of view of the Schwarzian derivative, with emphasis on numerical considerations. Applications are given to evaluation of a singular integral, mapping to the complement of an annular rectangle, and symmetric multitooth domains.