International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 350326, 20 pages
The Schwarz-Christoffel Conformal Mapping for “Polygons” with Infinitely Many Sides
Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Avenue Vicuña Makenna 4860, 7820436 Macul, Santiago, Chile
Received 22 January 2008; Revised 15 May 2008; Accepted 1 July 2008
Academic Editor: Vladimir Mityushev
Copyright © 2008 Gonzalo Riera et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The classical Schwarz-Christoffel formula gives conformal mappings of the upper half-plane onto domains whose boundaries consist of a finite number
of line segments. In this paper, we explore extensions to boundary curves
which in one sense or another are made up of infinitely many line segments,
with specific attention to the “infinite staircase” and to the Koch snowflake,
for both of which we develop explicit formulas for the mapping function and
explain how one can use standard mathematical software to generate corresponding
graphics. We also discuss a number of open questions suggested
by these considerations, some of which are related to differentials on hyperelliptic
surfaces of infinite genus.