Computational Methods and Function Theory 9 (2009), No. 1, 323--334
Copyright Heldermann Verlag 2009
Steven R. Bell
bell@math.purdue.edu , Purdue University, Mathematics Department, West Lafayette, IN 47907, U.S.A.
Ersin Deger
ersin.deger@uc.edu , University of Cincinnati, Mathematics Department, Cincinnati, OH 45221, U.S.A.
Thomas Tegtmeyer
thomas.tegtmeyer@trinity.edu , Trinity University, Mathematics Department, San Antonio, TX 78212, U.S.A.
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 Ar={z:|z+1/z|<2r}, where r>1 is a constant. The domain Ar 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.
Keywords: Ahlfors map, Bergman kernel, Szegö kernel.
MSC 2000: 30C35.
[FullText-pdf (262 K)] [FullText-ps (448 K)]