Numerical conformal mapping

TL;DR

This paper discusses the theory and methods of conformal mapping in complex analysis, including mappings of simply connected, doubly-connected, and multiply connected domains to canonical forms.

Contribution

It provides an overview of classical conformal mapping results and the canonical domains used for different connectivity levels.

Findings

Conformal maps exist for all simply connected nonempty proper subsets of the complex plane.

Doubly-connected domains can be conformally mapped to circular annuli with a unique modulus.

Higher connectivity domains can be mapped to canonical domains with slits or smaller disks.

Abstract

Conformal mapping may be the best-known topic in complex analysis. Any simply connected nonempty domain $\Omega$ in the complex plane ${{\mathbb{C}}}$ (assuming $\Omega\ne {{\mathbb{C}}}$) can be mapped bijectively to the unit disk by an analytic function with nonvanishing derivative, as in Figure 1. If $\Omega$ is doubly-connected, it can be mapped to a circular annulus $1<|z|<R$ for some $R$, called the conformal modulus, which is uniquely determined by $\Omega$, as in Figure 2. If $\Omega$ has connectivity higher than $2$, it can be mapped onto various canonical domains such as a disk with exclusions in the form of slits or smaller disks, as in Figure 3.