Simplicity of the Bergman, Szego and Poisson kernel functions

We announce a proof that the Bergman, Szeg\H o, and Poisson kernels associated to a finitely connected domain in the plane are simple in the sense that they are not genuine functions of two variables. They are all composed of {\it finitely many holomorphic functions of one variable}. We can also prove that the kernels cannot be too simple by showing that the only finitely connected domains in the plane whose Bergman or Poisson kernels are rational functions are the simply connected domains which can be mapped onto the unit disc by a rational biholomorphic mapping. This leads to a proof that the classical Green’s function associated to a finitely connected domain in the plane is one half the logarithm of a real-valued rational function if and only if the domain is simply connected and there is a rational biholomorphic map of the domain onto the unit disc.

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Published 1 January 1995