Uniformizing the boundaries of multiply connected quadrature domains using Fuchsian groups

This paper proposes a new method for uniformizing the boundaries of multiply connected quadrature domains by means of conformal maps that are automorphic functions invariant under the action of a Schottky group that is a Fuchsian group of the second kind. The conformal maps are expressed as ratios of Schottky-Klein prime functions. The result is significant because a classical infinite product formula for the Schottky-Klein prime function associated with Fuchsian groups of the second kind is known to be convergent.