The Schottky-Klein prime function: A theoretical and computational tool for applications

Abstract

This article surveys the important role, in a variety of applied mathematical contexts, played by the socalled Schottky-Klein (S-K) 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 wideranging usefulness of the S-K prime function it is important to be able to compute it efficiently. Here we introduce botha 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 S-K prime function a powerful new tool in applied mathematics. The authors 2016.