The Schottky–Klein prime function: a theoretical and computational tool for applications

This paper surveys the important role, in a variety of applied mathematical contexts, played by the so-called Schottky-Klein prime function. While it is a classical special function, introduced by 19th century investigators, its theoretical significance for applications has only been realized in the last decade or so, especially with respect to solving problems defined in multiply connected, or “holey”, domains. It is shown here that, in terms of it, many well-known results pertaining only to the simply connected case (no holes) can be generalized, in a natural way, to the multiply connected case thereby contextualizing those well-known results within a more general framework of much broader applicability. Given the wide-ranging usefulness of the Schottky-Klein prime function it is important to be able to compute it efficiently. Here we introduce both a new theoretical formulation for its computation, as well as two distinct numerical methods to implement the construction. The combination of these theoretical and computational developments renders the Schottky-Klein prime function a powerful new tool in applied mathematics.