Complexity of the Classical Kernel Functions of Potential Theory

We show that the Bergman, Szegő, and Poisson kernels associated to an n-connected domain in the plane are not genuine functions of two complex variables. Rather, they are all given by elementary rational combinations of n+ 1 holomorphic functions of one complex variable and their conjugates. Moreover, all three kernel functions are composed of the same basic n+ 1 functions. Our results can be interpreted as saying that the kernel functions are simpler than one might expect. We also prove, however, 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. We also characterize those domains in the plane that have rational Szegő kernel functions.