The Cauchy Integral Formula, Quadrature Domains, and Riemann Mapping Theorems

Abstract

It is well known that a domain in the plane is a quadrature domain with respect to area measure if and only if the function z extends meromorphically to the double, and it is a quadrature domain with respect to boundary arc length measure if and only if the complex unit tangent vector function T(z) extends meromorphically to the double. By applying the Cauchy integral formula to $\bar{z}$, we will shed light on the density of area quadrature domains among smooth domains with real analytic boundary. By extending $\bar{z}$ and T(z) and applying the Cauchy integral formula to the Szegő kernel, we will obtain conformal mappings to nearby arc length quadrature domains and even domains that are like the unit disc in that they are simultaneously area and arc length quadrature domains. These “double quadrature domains” can be thought of as analogs of the unit disc in the multiply connected setting and the mappings so obtained as generalized Riemann mappings. The main theorems of this paper are not new, but the methods used in their proofs are new and more constructive than previous methods. The new computational methods give rise to numerical methods for computing generalized Riemann maps to nearby quadrature domains.