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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


On the boundary values of Riemann's mapping function

Author: R. J. V. Jackson
Journal: Trans. Amer. Math. Soc. 259 (1980), 281-297
MSC: Primary 30C20; Secondary 30C40
MathSciNet review: 561837
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Abstract: Classically, the calculus of variations is required to prove the existence of a biholomorphism from the unit disk to a given simply-connected, smooth domain in the complex plane. Here, the problem is reduced to the solution of an ordinary differential equation along the boundary of the domain. The sole coefficient in this equation is identified with the bounded term in the asymptotic expansion of the Bergman kernel function. It is shown that this coefficient can not depend upon any differential expression involving only the curvature function of the boundary.

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PII: S 0002-9947(1980)0561837-3
Keywords: Riemann's mapping function, Plateau's problem, ordinary differential equations, potential theory, Hilbert's transform, asymptotics of Bergman's kernel function
Article copyright: © Copyright 1980 American Mathematical Society