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Perturbing Rational Harmonic Functions by Poles

  • Olivier Sète
  • Robert LuceEmail author
  • Jörg Liesen
Article

Abstract

We study how adding certain poles to rational harmonic functions of the form \(R(z)-\overline{z}\), with \(R(z)\) rational and of degree \(d\ge 2\), affects the number of zeros of the resulting functions. Our results are motivated by and generalize a construction of Rhie derived in the context of gravitational microlensing (arXiv:astro-ph/0305166). Of particular interest is the construction and the behavior of rational functions \(R(z)\) that are extremal in the sense that \(R(z)-\overline{z}\) has the maximal possible number of \(5(d-1)\) zeros.

Keywords

Complex valued harmonic function Rational function  Gravitational lensing 

Mathematics Subject Classification

30D05 31A05 85A04 

Notes

Acknowledgments

We would like to thank Elias Wegert for comments on the manuscript and creating the color scheme used in the illustrations. We are grateful to the anonymous referee for carefully reading the manuscript and for giving us many useful suggestions.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.BerlinGermany

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