On composite conformal mapping of an annulus to a plane with two holes

TL;DR

This paper investigates a composite conformal mapping from an annulus to a plane with two holes, demonstrating its effectiveness under certain geometric conditions and providing specific examples of such mappings.

Contribution

It introduces a new composite conformal mapping technique for annuli to regions with two holes, with analysis of its accuracy based on geometric parameters.

Findings

Mapping is effective when holes are far apart or one is small.

Examples include bilinear-hypotrochoids and bilinear-Schwarz-Christoffel mappings.

The method provides a practical approach for complex domain transformations.

Abstract

In the article we consider the composite conformal map which maps annulus to infinite region with symmetric hole and nearly circular hole. It is shown that such transformation is good if the distance between centers of holes are large or radius of circular hole is small. Examples for bilinear-hypotrochoids mapping and bilinear-Schwarz-Christoffel mapping are present.