Vol. 3, No. 4, 2010

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ISSN: 1944-4184 (e-only)
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Visual representation of the Riemann and Ahlfors maps via the Kerzman–Stein equation

Michael Bolt, Sarah Snoeyink and Ethan Van Andel

Vol. 3 (2010), No. 4, 405–420
Abstract

The Szegő kernel serves as one of the canonical functions associated to a region in the complex plane, and from it one can compute the Riemann (or Ahlfors) map, the essentially unique conformal transformation of the region to the unit disc. We provide an elementary description of the method that Kerzman and Stein used to compute the Szegő kernel, and subsequently, the Riemann and Ahlfors maps. A description, too, is provided for a new tool that generates visual representations of these functions and is included with the open-source computer algebra system Sage.

Keywords
Riemann map, Ahlfors map, Szegő kernel, Kerzman–Stein
Mathematical Subject Classification 2000
Primary: 30C30
Secondary: 65E05
Milestones
Received: 23 July 2010
Revised: 16 November 2010
Accepted: 20 November 2010
Published: 6 January 2011

Proposed: Michael Dorff
Communicated by Michael Dorff
Authors
Michael Bolt
Department of Mathematics and Statistics
Calvin College
3201 Burton St., S.E.
Grand Rapids, MI 49546
United States
http://www.calvin.edu/~mdb7/
Sarah Snoeyink
Department of Mathematics
University of Colorado at Boulder
Campus Box 395
Boulder, CO 80309
United States
Ethan Van Andel
Department of Mathematics and Statistics
Calvin College
3201 Burton St., S.E.
Grand Rapids, MI 49546
United States