Representation of the Szegö Kernel on an Annulus Using Infinite Products

Abstract

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

 Keywords: Szegö kernel, bilateral series, Kerzman-Stein integral equation, basic bilateral series, Ramanujan’s sum, basic gamma function, modified Jacobi theta function.