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Solutions of the Riemann–Hilbert–Poincaré problem and the Robin problem for the inhomogeneous Cauchy–Riemann equation

Published online by Cambridge University Press:  13 March 2009

Alip Mohammed  and
M. W. Wong
Alip Mohammed
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada (mwwong@mathstat.yorku.ca)
M. W. Wong
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada (mwwong@mathstat.yorku.ca)
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Abstract

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The Riemann–Hilbert–Poincaré problem with general coefficient for the inhomogeneous Cauchy–Riemann equation on the unit disc is studied using Fourier analysis. It is shown that the problem is well posed only if the coeffcient is holomorphic. If the coefficient has a pole, then the problem is transformed into a system of linear equations and a finite number of boundary conditions are imposed in order to find a unique and explicit solution. In the case when the coefficient has an essential singularity, it is shown that the problem is well posed only for the Robin boundary condition.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 139 , Issue 1 , February 2009 , pp. 157 - 181
Copyright
Copyright © Royal Society of Edinburgh 2009