Journal Title
Title of Journal: Comput Methods Funct Theory
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Abbravation: Computational Methods and Function Theory
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Publisher
Springer-Verlag
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Authors: Jonathan Tsai
Publish Date: 2012/10/06
Volume: 12, Issue: 2, Pages: 687-706
Abstract
The classical SchwarzChristoffel formula allows one to compute the conformal map of the unit disc onto a domain bounded by a polygon An extension of this formula can be obtained that allows one to compute the conformal map of the unit disc onto a domain bounded by a “quadratic differential polygon” ie a closed Jordan curve that is the union of trajectory arcs of a certain quadratic differential and their end points Now suppose that R is a finite Riemann surface with boundary and π D → R is a universal covering map of R Let Qz dz 2 be a quadratic differential on R and let R be a subRiemann surface of R such that the boundary of R is the union of finitesided Qz dz 2 polygons If the lift of R D = π 1R is a simplyconnected domain then the Riemann mapping theorem ensures that there is a conformal map f of the unit disc onto D In this paper we will prove a generalization of the SchwarzChristoffel formula that allows us to compute f Then π ◦ f is a universal covering map of R
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