How Constant Shifts Affect the Zeros of Certain Rational Harmonic Functions

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Abstract

We study the effect of constant shifts on the zeros of rational harmonic functions \(f(z) = r(z) - \overline{z}\). In particular, we characterize how shifting through the caustics of f changes the number of zeros and their respective orientations. This also yields insight into the nature of the singular zeros of f. Our results have applications in gravitational lensing theory, where certain such functions f represent gravitational point-mass lenses, and a constant shift can be interpreted as the position of the light source of the lens.

Keywords

Rational harmonic functions Gravitational lensing Critical curve and caustic Cusp and fold points Singular zeros 

Mathematics Subject Classification

30D05 31A05 85A04 

Notes

Acknowledgements

We thank Seung-Yeop Lee for sending us a PDF file of [23]. We also thank an anonymous referee for helpful comments.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.TU Berlin, Institute of MathematicsBerlinGermany

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