The Schwarz problem for analytic functions in torus-related domains

The Cauchy kernel is one of the two significant tools for solving the Riemann boundary value problem for analytic functions. For poly-domains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in poly-domains. This s...

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Bibliographic Details
Published in:Applicable analysis Vol. 85; no. 9; pp. 1079 - 1101
Main Authors: Begehr, Heinrich, Mohammed, Alip
Format: Journal Article
Language:English
Published: Taylor & Francis Group 01.09.2006
Subjects:
Holomorphic functions
Mathematics Subject Classifications: 32A26
Poly-domain
Schwarz problem
ISSN:0003-6811, 1563-504X
Online Access:Get full text
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Summary:The Cauchy kernel is one of the two significant tools for solving the Riemann boundary value problem for analytic functions. For poly-domains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in poly-domains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. It is also decisive for the establishment of connection between the Riemann-Hilbert problem and the Riemann problem. Thus, not only the Schwarz problem for holomorphic functions in poly-domains is solved, but also the basis is established for solving some other problems. The boundary values of functions, holomorphic in poly-domains, are classified in the Wiener algebra. The general integral representation formulas for these functions, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary poly-domains is given. At the end, well-posed formulations of the torus-related problems are considered. †Dedicated to Professor Wei Lin on the occasion of his 70th birthday.
ISSN:0003-6811
1563-504X
DOI:10.1080/00036810600835128