Complexity of the classical kernel functions of potential theory

Details Complexity of the classical kernel functions of potential theory Complexity of the classical kernel functions of potential theory

1993-09-30

arxiv.org/abs/math/9309201v1

Mathematics

Complex Variables

Scientific paper

We show that the Bergman, Szego, and Poisson kernels associated to a finitely connected domain in the plane are all composed of finitely many easily computed functions of one variable. The new formulas give rise to new methods for computing the Bergman and Szeg\H o kernels in which all integrals used in the computations are line integrals; at no point is an integral with respect to area measure required. The results mentioned so far can be interpreted as saying that the kernel functions are simpler than one might expect. However, we also prove that the kernels cannot be too simple by showing that the only finitely connected domains in the plane whose Bergman or Szeg\H o kernels are rational functions are the obvious ones. This leads to a proof that the classical Green's function associated to a finitely connected domain in the plane is the logarithm of a rational function if and only if the domain is simply connected and rationally equivalent to the unit disc.

Affiliated with

Also associated with

No associations

Rating

Complexity of the classical kernel functions of potential theory does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Complexity of the classical kernel functions of potential theory, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Complexity of the classical kernel functions of potential theory will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-50388