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Computing Conformal Maps of Finitely Connected Domains onto Canonical Slit Domains

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Abstract

We continue the research initiated in Andreev et al. (Computing conformal maps onto circular domains, 2010, submitted) on the computability of conformal mappings of multiply connected domains by showing that the conformal maps of a finitely connected domain onto the canonical slit domains can be computed uniformly from the domain and its boundary. Along the way, we demonstrate the computability of finding analytic extensions of harmonic functions and solutions to Neuman problems. These results on conformal mapping then follow easily from M. Schiffer’s constructions (Dirichlet Principle, Conformal Mapping and Minimal Surfaces, Interscience, New York, 1950).

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Authors and Affiliations

  1. Department of Mathematics, Lamar University, Beaumont, TX, 77710, USA

    Valentin V. Andreev & Timothy H. McNicholl

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  1. Valentin V. Andreev
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  2. Timothy H. McNicholl
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Correspondence to Timothy H. McNicholl.

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Authors’ note: It is our understanding that R. Rettinger has obtained some of these results by different methods.

The results of this paper were presented at the Sixth International Conference on Computability and Complexity in Analysis, Ljubljana, Slovenia, 2009. A preliminary version appeared in the proceedings of this conference.

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Andreev, V.V., McNicholl, T.H. Computing Conformal Maps of Finitely Connected Domains onto Canonical Slit Domains. Theory Comput Syst 50, 354–369 (2012). https://doi.org/10.1007/s00224-010-9305-4

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  • DOI: https://doi.org/10.1007/s00224-010-9305-4

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