Abstract

Schwarz-Christoffel (S-C) formulae provide conformal mappings from circular domains to polygonal domains. Consequently, they are important tools for applied mathematicians seeking to solve problems involving complicated geometries. We present a major extension to previous S-C mappings by permitting the target domain to consist of a periodic array of polygons. Moreover, by employing the transcendental Schottky-Klein prime function, the new S-C formula is valid for any number of polygons in each period window. We demonstrate several examples of the new mapping for a range of connectivities and show that, when coupled with the “new calculus of two-dimensional vortex dynamics”, the new S-C mapping is a powerful tool for the study of fluid flows in periodic domains. Finally, we formulate the accessory parameter problem of determining the pre-vertices and conformal geometry in the canonical domain in order to achieve the desired target domain. This work is in collaboration with Prof. Darren Crowdy (Imperial).