On the Construction of Multiply Connected Arc Integral Quadrature Domains

Abstract

In this paper, we construct an explicit parameterisation describing quadrature domains of finite area and arbitrary finite connectivity admitting quadrature identities with integrals along interior arcs. The formulation is in terms of conformal maps from circular domains, where these maps are expressed in terms of the associated Schottky–Klein prime function. An important role in the derivation is played by the Bergman kernel function. To illustrate the results, several examples are presented.