Perturbing Rational Harmonic Functions by Poles

@article{Ste2014PerturbingRH,
title={Perturbing Rational Harmonic Functions by Poles},
author={Olivier S{\e}te and Robert Luce and J{\"o}rg Liesen},
journal={Computational Methods and Function Theory},
year={2014},
volume={15},
pages={9-35}
}`
• Published 2014
• Mathematics, Physics
• Computational Methods and Function Theory
We study how adding certain poles to rational harmonic functions of the form $$R(z)-\overline{z}$$R(z)-z¯, with $$R(z)$$R(z) rational and of degree $$d\ge 2$$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 $$R(z)$$R(z) that are extremal in the sense that $$R… Expand 12 Citations Figures from this paper The Maximum Number of Zeros of$$r(z) - \overline{z}$$r(z)-z¯ Revisited • Mathematics • 2017 Generalizing several previous results in the literature on rational harmonic functions, we derive bounds on the maximum number of zeros of functions$$f(z) = \frac{p(z)}{q(z)} -Expand
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References

SHOWING 1-10 OF 27 REFERENCES
Counting Zeros of Harmonic Rational Functions and Its Application to Gravitational Lensing
• Mathematics, Physics
• 2012
General Relativity gives that finitely many point masses between an observer and a light source create many images of the light source. Positions of these images are solutions of $r(z)=\bar{z},$Expand
Sharp parameter bounds for certain maximal point lenses
• Physics, Mathematics
• 2014
Starting from an $$n$$n-point circular gravitational lens having $$3n+1$$3n+1 images, Rhie (ArXiv Astrophysics e-prints, 2003) used a perturbation argument to construct an $$(n+1)$$(n+1)-point lensExpand
On the number of zeros of certain rational harmonic functions
• Mathematics, Physics
• 2004
Extending a result of Khavinson and Swiatek (2003) we show that the rational harmonic function r(z) - z, where r(z) is a rational function of degree n > 1, has no more than 5n - 5 complex zeros.Expand
When is the Adjoint of a Matrix a Low Degree Rational Function in the Matrix?
• J. Liesen
• Mathematics, Computer Science
• SIAM J. Matrix Anal. Appl.
• 2007
It is shown that unless the eigenvalues of A lie on a single circle in the complex plane, the ratio of the normal degree and the McMillan degree of $A$ is bounded by a small constant that depends neither on the number nor on the distribution of the eigens of £A. Expand
The valence of harmonic polynomials
The paper gives an upper bound for the valence of harmonic polynomials. An example is given to show that this bound is sharp. Interest in harmonic mappings in the complex plane has increased due toExpand
From the Fundamental Theorem of Algebra to Astrophysics: a \Harmonious" Journey
The fundamental theorem of algebra (FTA) tells us that every complex polynomial of degree n has precisely n complex roots. The first published proofs (J. d’Alembert in 1746 and C. F. Gauss 1799) ofExpand
Phase Plots of Complex Functions: a Journey in Illustration
A universality theorem for Riemann's Zeta function in the language of phase plots is reformulate in the form of a theorem based on color--coding the points on the unit circle. Expand
Local behavior of harmonic mappings
• Mathematics
• 2000
The argument principle is an important and useful result for meromorphic functions on domains in the plane. Duren, Hengartner and Laugesen have obtained an argument principle for harmonic mappings onExpand
Sharp bounds for the valence of certain harmonic polynomials
In Khavinson and Światek (2002) it was proved that harmonic polynomials z - p(z), where p is a holomorphic polynomial of degree n > 1, have at most 3n - 2 complex zeros. We show that this bound isExpand
n-point Gravitational Lenses with 5(n-1) Images
It has been conjectured (astro-ph/0103463) that a gravitational lens consisting of n point masses can not produce more than 5(n-1) images as is known to be the case for n = 2 and 3. The reasoning isExpand