COMPLEXITY OF THE CLASSICAL KERNEL FUNCTIONS OF POTENTIAL-THEORY

We show that the Bergman, Szego, and Poisson kernels associated to an n-connected domain in the plane are not genuine functions of two compl ex variables. Rather, they are all given by elementary rational combin ations of n + 1 holomorphic functions of one complex variable and thei r conjugates. Moreover, adl 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 p rove, however, that the kernels cannot be too simple by showing that t he only finitely connected domains in the plane whose Bergman or Poiss on kernels are rational functions are the simply connected domains whi ch can be mapped onto the unit disc by a rational biholomorphic mappin g. This leads to a proof that the classical Green's function associate d to a finitely connected domain in the plane is one half the logarith m of a real valued rational function if and only if the domain is simp ly connected and there is a rational biholomorphic map of the domain o nto the unit disc. We also characterize those domains in the plane tha t have rational Szego kernel functions.