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Schwarz-Christoffel mapping of multiply connected domains

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Abstract

A Schwarz-Christoffel mapping formula is established for polygonal domains of finite connectivitym≥2 thereby extending the results of Christoffel (1867) and Schwarz (1869) form=1 and Komatu (1945),m=2. A formula forf, the conformal map of the exterior ofm bounded disks to the exterior ofm bounded disjoint polygons, is derived. The derivation characterizes the global preSchwarzianf″ (z)/f′ (z) on the Riemann sphere in terms of its singularities on the sphere and its values on them boundary circles via the reflection principle and then identifies a singularity function with the same boundary behavior. The singularity function is constructed by a “method of images” infinite sequence of iterations of reflecting prevertex singularities from them boundary circles to the whole sphere.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Wichita State University, 67260-0033, Wichita, KS, USA

    T. K. Delillo & A. R. Elcrat

  2. Department of Mathematics, CB 3250, University of North Carolina, 27599-3250, Chapel Hill, NC, USA

    J. A. Pfaltzgraff

Authors
  1. T. K. Delillo
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  2. A. R. Elcrat
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  3. J. A. Pfaltzgraff
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Correspondence to T. K. Delillo.

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Delillo, T.K., Elcrat, A.R. & Pfaltzgraff, J.A. Schwarz-Christoffel mapping of multiply connected domains. J. Anal. Math. 94, 17–47 (2004). https://doi.org/10.1007/BF02789040

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  • DOI: https://doi.org/10.1007/BF02789040

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