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We show how to express a conformal map Φ of a general two-connected domain in the plane, such that neither boundary component is a point, onto a representative domain of the form \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal{A}_r = \left\{ {z:\left| {z + 1/z} \right| < 2r} \right\}$$\end{document}, where r > 1 is a constant. The domain\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal{A}_r$$\end{document} has the virtue of having an explicit algebraic Bergman kernel function, and we shall explain why it is the best analogue of the unit disc in the two-connected setting. The map Φ will be given as a simple and explicit algebraic function of an Ahlfors map of the domain associated to a specially chosen point. It will follow that the conformal map Φ can be found by solving the same extremal problem that determines a Riemann map in the simply connected case. In the last section, we show how these results can be used to give formulae for the Bergman kernel in two-connected domains.
Computational Methods and Function Theory – Springer Journals
Published: Apr 1, 2009
Keywords: Ahlfors map; Bergman kernel; Szegő kernel; 30C35
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