The Schottky-Klein prime function on the Schottky double of planar domains
This article should be seen as either a review or a summary of the author's achievements in the field of conformal mappings. It provides an overview of the properties and uses of the Schottky-Klein prime function (which can be seen as a generalization of the elementary function \(\omega (\zeta,z)=\zeta-z\) that is suitable for compact Riemann surfaces) on the Schottky double of multiply connected planar domains (that is a special compact Riemann surface with genus one less than the connectivity of the planar domain, which can be constructed by identifying the boundaries of the given domain with the one obtained by reflection). The use of the Schottky-Klein prime function makes it possible to write down explicit expressions (in terms of these functions) for the conformal mappings from a multiply connected circular domain to canonical multiply connected slit domains. As a corollary, a new geometrical interpretation of a recently derived formula for multiply connected Schwarz-Christoffel mappings is given. It is postulated that this geometric step-by-step procedure provides a better understanding and somehow simplifies the proof of the mentioned (very complicated and technical) formula. The formula itself is given in [\textit{D. Crowdy}, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 461, 2653--2678 (2005; Zbl 1186.30005)]. The article is very clear and pleasant to read.
scientific article; zbMATH DE number 5840922
scientific article