Recipes for the classical kernel functions associated to a multiply connected domain in the plane

Abstract

I have recently shown that the Bergman kernel associated to a finitely connected domain in the plane is given as an explicit rational combination of finitely many basic functions of one complex variable. In this paper, it is proved that all the basic functions and constants in the new formula for the Bergman kernel can be evaluated using onedimensional integrals and simple linear algebra. In fact, all integrals used in the computations are line integrals over boundary curves; at no point is an integral with respect to area measure required. From a theoretical perspective, these results lead to an understanding of the complexity of the Bergman kernel. From a practical point of view, they give an efficient method to numerically compute the Bergman kernel. Similar results are also proved for the Szego kernel function, the Poisson kernel, and the classical Green's function.