ABSTRACT
Bjo¨rn Gustafsson [Gu1] discovered the connection between the Schottky double of a domain and the property of being an area quadrature domain. In this last chapter, we will describe just enough of this theory to be able to prove that the Bergman kernel K(z, w) associated to an area quadrature domain without cusps in the boundary is particularly simple. It is a rational combination of z and the Schwarz function S(z) in the sense that K(z, w) is a rational function of z, S(z), w¯, and S(w). Consequently, since S(z) = z¯ on the boundary, the Bergman kernel is rational in z, z¯, w, and w¯ when z and w are boundary points, z 6= w. This is another instance of our claim that area quadrature domains share many properties with the unit disc. We will later combine this result with the improved Riemann mapping theorem that says that bounded finitely connected domains are biholomorphic to area quadrature domains, and use the transformation formula for the Bergman kernels to see that the Bergman kernel is a rational combination of three functions of one complex variable, in general.
