ABSTRACT
Throughout this book, we have seen that there are beautiful formulas relating almost any two objects we have defined. In particular, the Bergman, Szego˝, and Poisson kernels are all closely connected. In this chapter, we address the question of just how complicated these kernels are. Are they true functions of two complex variables, or might they be comprised of more elementary functions of one complex variable? We begin by showing that the Szego˝ kernel associated to a smooth domain is a rational combination of more elementary functions. In the simply connected setting, we may use a Riemann map to map to the unit disc and use the transformation formula for the Szego˝ kernels to see that the Szego˝ kernel is a simple rational combination of the Riemann map and a square root of its derivative. Thus, this question is interesting only in the multiply connected setting.
