Volume 36, pp. 195-223, 2009-2010.

Slit maps and Schwarz-Christoffel maps for multiply connected domains

Thomas K. DeLillo and Everett H. Kropf

Abstract

We review recent derivations of formulas for conformal maps from finitely connected domains with circular holes to canonical radial or circular slit domains. The formulas are infinite products based on simple reflection arguments. An earlier similar derivation of the Schwarz-Christoffel formula for the bounded multiply connected case and recent progress in its numerical implementation are also reviewed. We give some sample calculations with a reflection method and an estimate of its accuracy. We also discuss the relation of our approach to that of D. Crowdy and J. Marshall. In addition, a slit map calculation using Laurent series computed by the least squares method in place of the reflection method is given as an example of a possible direction for future improvements in the numerics.

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Key words

conformal mapping, Schwarz-Christoffel transformation, multiply connected domains, canonical slit domains, Schottky-Klein prime function

AMS subject classifications

30C30, 65E05

ETNA articles which cite this article

Vol. 39 (2012), pp. 286-297 Roman Czapla, Vladimir Mityushev, and Natalia Rylko: Conformal mapping of circular multiply connected domains onto slit domains
Vol. 45 (2016), pp. 1-15 Olivier Sète and Jörg Liesen: On conformal maps from multiply connected domains onto lemniscatic domains

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