Conjugate Function Method and Conformal Mappings in Multiply Connected Domains

TL;DR

This paper extends the conjugate function method to compute conformal mappings in multiply connected domains, addressing the construction of conjugate domains while preserving key properties, with implementation and examples provided.

Contribution

It generalizes the conjugate function method for multiply connected domains, a significant extension from simply and doubly connected cases.

Findings

Method preserves reciprocal relation of moduli

Algorithm successfully applied to various examples

Implementation demonstrates practical effectiveness

Abstract

The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and doubly connected domains. In this paper the conjugate function method is generalized for multiply connected domains. The key challenge addressed here is the construction of the conjugate domain and the associated conjugate problem. All variants of the method preserve the so-called reciprocal relation of the moduli. An implementation of the algorithm, along with several examples and illustrations are given.