Infinite Product Representation for the Szego Kernel for an Annulus

TL;DR

This paper derives an infinite product and bilateral series representation for the Szego kernel of an annulus, revealing its zeros and connections to special functions, with numerical validation.

Contribution

It introduces a novel infinite product representation of the Szego kernel for an annulus using Ramanujan's sum, enhancing understanding and computation.

Findings

Infinite product representation clearly shows the zeros of the Szego kernel.

Connections established between the Szego kernel, gamma function, and Jacobi theta function.

Numerical comparisons validate the effectiveness of the new representations.

Abstract

The Szego kernel has many applications to problems in conformal mapping and satisfies the Kerzman-Stein integral equation. The Szego kernel for an annulus can be expressed as a bilateral series. In this paper, we show how to represent the Szego kernel for an annulus as a basic bilateral series and an infinite product representation through the application of the Ramanujan's sum. The infinite product clearly exhibits the zero of the Szego kernel for an annulus. Its connection with basic gamma function and modified Jacobi theta function is also presented. The results are extended to the Szego kernel for general annulus and weighted Szego kernel. Numerical comparisons on computing the Szego kernel for an annulus based on the bilateral series and the infinite product are also presented.