Indian Journal of Science and Technology
Year: 2015, Volume: 8, Issue: 32, Pages: 1-8
Ali W. K. Sangawi2,3, Kashif Nazar4,5 and Ali H. M. Murid1,4*
1 UTM Centre for Industrial and Applied Mathematics (UTM-CIAM), Ibnu Sina Institute of Science and Industrial Research, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
2 Department of Mathematics, School of Science, Faculty of Science and Science Education, University of Sulaimani, 46001 Sulaimani, Kurdistan, Iraq
3 Department of Computer, College of Basic Education, Charmo University, Chamchamal, 46001 Sulaimani, Kurdistan, Iraq
4 Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
5 Department of Mathematics, COMSATS Institute of Information Technology, P.O. Box 54000 Defence Road Off Raiwind Road Lahore Pakistan
The Ahlfors map of an n-connected region is a n-to-one map from the region onto the unit disk. The Ahlfors map being n-toone map has n zeros. Previously, the exact zeros of the Ahlfors map are known only for the annulus region. The zeros of the Ahlfors map for general bounded doubly connected regions has been unknown for many years. This paper presents a numerical method for computing the zeros of the Ahlfors map of any bounded doubly connected region. The method depends on the values of Szego kernel, its derivative and the derivative of boundary correspondence function of the Ahlfors map. The Ahlfors map and Szego kernel are both classically related to each other. Ahlfors map can be computed using Szego kernel without relying on the zeros of Ahlfors map. The Szego kernel is a solution of a Fredholm integral equation of the second kind with the Kerzman-Stein kernel. The numerical examples presented here prove the effectiveness of the proposed method.
Keywords: Ahlfors Map, Generalized Neumann Kernel, Neumann Kernel, Szego Kernel
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