Secondary Schottky–Klein prime functions associated with multiply connected planar domains

In recent years, a general mathematical framework for solving applied problems in multiply connected domains has been developed based on use of the Schottky–Klein (S–K) prime function of an underlying compact Riemann surface known as the Schottky double of the domain. In this paper, we describe additional function-theoretic objects that are naturally associated with planar multiply connected domains and which we refer to as secondary S–K prime functions. The basic idea develops, and extends, an observation of Burnside dating back to 1892. Applications of the new functions to represent conformal slit maps of mixed type that have been a topic of recent interest in the literature are given. Other possible applications are also surveyed.