Computing the zeros of the Szego?kernel for doubly connected regions using conformal mapping

An explicit formula for the zero of the Szego center dot kernel for an annulus region is well-known. There exists a transformation formula for the Szego center dot kernel from a doubly connected region onto an annulus. Based on conformal mapping, we derive an analytical formula for the zeros of the Szego center dot kernel for a general doubly connected region with smooth boundaries. Special cases are the explicit formulas for the zeros of the Szego center dot kernel for doubly connected regions bounded by circles, limacons, ellipses, and ovals of Cassini. For a general doubly connected region with smooth boundaries, the zero of the Szego center dot kernel must be computed numerically. This paper describes the application of conformal mapping via integral equation with the generalized Neumann kernel for computing the zeros of the Szego center dot kernel for smooth doubly connected regions. Some numerical examples and comparisons are also presented. It is shown that the conformal mapping approach also yields good accuracy for a narrow region or region with boundaries that are close to each other.