Abstract
The Riemann mapping to the complement in a disk of a finite union of disjoint disks bounded by horocycles has a Schwarzian derivative in the form of a simple rational function R = R[{z k}, {r k}](z) with two accessory parameters z k, r k for each vertex ωk. It is shown that if the prevertices z k are presupposed (while the ωk are undetermined), there exists a unique set of values {r k} for which R is the Schwarzian derivative of such a horocyclic mapping. These values depend on the combinatorial structure of the adjacencies of horocycles. An algorithm is developed for calculating the correspondence, and numerical examples are presented.
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Porter, R.M. Numerical Calculation of Conformal Mapping to a Disk Minus Finitely Many Horocycles. Comput. Methods Funct. Theory 5, 471–488 (2006). https://doi.org/10.1007/BF03321111
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DOI: https://doi.org/10.1007/BF03321111