Perturbing Rational Harmonic Functions by Poles

We study how adding certain poles to rational harmonic functions of the form R ( z ) - z ¯ , with R ( z ) rational and of degree d ≥ 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...

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Bibliographic Details
Published in:Computational methods and function theory 2015-03, Vol.15 (1), p.9-35
Main Authors: Sète, Olivier, Luce, Robert, Liesen, Jörg
Format: Article
Language:English
Subjects:
Analysis
Computational Mathematics and Numerical Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
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Summary:We study how adding certain poles to rational harmonic functions of the form R ( z ) - z ¯ , with R ( z ) rational and of degree d ≥ 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 ) - z ¯ has the maximal possible number of 5 ( d - 1 ) zeros.
ISSN:1617-9447
2195-3724
DOI:10.1007/s40315-014-0083-x