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The periodic, multiply-connected Schwarz-Christoffel mapping

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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).

This talk is part of the Waves Group (DAMTP) series.

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