Infinite Product Representation for the Szego Kernel for an Annulus

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.