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A Survey on the Maximal Number of Solutions of Equations Related to Gravitational Lensing

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Complex Analysis and Dynamical Systems

Part of the book series: Trends in Mathematics ((TM))

Abstract

This paper is a survey of what is known about the maximal number of solutions of the equation \(f(z) = \bar {z},\) in particular when f is the Cauchy transform of a compactly supported positive measure. When f is a rational function, the number of solutions of this equation is equal to the number of images seen by an observer of a single light source deflected by a gravitational lens (such as a galaxy). We will discuss what is known in the context of harmonic polynomials, rational functions, polynomials in z and \(\bar {z}\) (but not harmonic!) and even transcendental functions that arise in situations involving continuous mass distributions for different shapes. In particular, we discuss an example related to the lens equation for a limaçon-shaped gravitational lens.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of South Florida, Tampa, FL, 33620, USA

    Catherine Bénéteau & Nicole Hudson

Authors
  1. Catherine Bénéteau
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  2. Nicole Hudson
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Corresponding author

Correspondence to Catherine Bénéteau .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel

    Mark Agranovsky

  2. Department of Mathematics, Holon Institute of Technology, Holon, Israel

    Anatoly Golberg

  3. Department of Mathematics, ORT Braude College, Karmiel, Israel

    Fiana Jacobzon

  4. Department of Mathematics, Holon Institute of Technology, Holon, Israel

    David Shoikhet

  5. Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel

    Lawrence Zalcman

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Bénéteau, C., Hudson, N. (2018). A Survey on the Maximal Number of Solutions of Equations Related to Gravitational Lensing. In: Agranovsky, M., Golberg, A., Jacobzon, F., Shoikhet, D., Zalcman, L. (eds) Complex Analysis and Dynamical Systems. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-70154-7_2

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