Schwarz-Christoffel mapping of multiply connected domains
DeLillo, Thomas K. Elcrat, Alan R. Pfaltzgraff, J. A.
DeLillo, Thomas K.
Elcrat, Alan R.
Pfaltzgraff, J. A.
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2004-12-01
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Delillo, T., A. Elcrat, et al. (2004). "Schwarz-Christoffel mapping of multiply connected domains." Journal d'Analyse Mathématique 94(1): 17-47.
Abstract
A Schwarz-Christoffel mapping formula is established for polygonal
domains of finite connectivity m _> 2 thereby extending the results of Christoffel
(1867) and Schwarz (1869) for m = 1 and Komatu (1945), m = 2. A formula
for f, the conformal map of the exterior of m bounded disks to the exterior of
m bounded disjoint polygons, is derived. The derivation characterizes the global
preSchwarzian f" (z) If' (z) on the Riemann sphere in terms of its singularities
on the sphere and its values on the m 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 the m boundary circles to the
whole sphere.
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Springer Verlag
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Journal d’Analyse Mathematique;v.94 no.1
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0021-7670
1565-8538
1565-8538