Fast computation of analytic capacity

TL;DR

This paper introduces a boundary integral equation method for efficiently computing the analytic capacity of compact sets in the complex plane, applicable to smooth and piecewise smooth boundaries, and verifies some theoretical properties.

Contribution

It presents a novel fast computational approach using the Kerzman--Stein integral equation for analytic capacity, including for slit sets, with numerical validation.

Findings

Efficient computation of analytic capacity for various sets.

Recovery of known exact results.

Support for the subadditivity conjecture.

Abstract

A boundary integral equation method is presented for fast computation of the analytic capacities of compact sets in the complex plane. The method is based on using the Kerzman--Stein integral equation to compute the Szeg\"o kernel and then the value of the Ahlfors map at the point at infinity. The proposed method can be used for domains with smooth and piecewise smooth boundaries. When combined with conformal mappings, the method can be used for compact slit sets. Several numerical examples are presented to demonstrate the efficiency of the proposed method. We recover some known exact results and corroborate the conjectural subadditivity property of analytic capacity.