Computational Methods and Function Theory 5 (2005), No. 2, 471--488
Copyright Heldermann Verlag 2005
R. Michael Porter
mike@math.cinvestav.mx , Departamento de Matemáticas, CINVESTAV-I.P.N, Apartado Postal 14-740, 07000 M\'exico, D.F., Mexico.
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[{zk},{rk}](z) with two accessory parameters zk, rk for each vertex wk. It is shown that if the prevertices zk are presupposed (while the wk are undetermined), there exists a unique set of values {rk} 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.
Keywords: Numerical conformal mapping, Schwarzian derivative, horocycle domain, cusp, Broyden's method.
MSC 2000: Primary 30C30; Secondary 30C20.
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