Perturbing rational harmonic functions by poles
Abstract
We study how adding certain poles to rational harmonic functions of the form $R(z)\bar{z}$, with $R(z)$ rational and of degree $d\geq 2$, 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 eprint 2003). Of particular interest is the construction and the behavior of rational functions $R(z)$ that are {\em extremal} in the sense that $R(z)\bar{z}$ has the maximal possible number of $5(d1)$ zeros.
 Publication:

arXiv eprints
 Pub Date:
 March 2014
 arXiv:
 arXiv:1403.0906
 Bibcode:
 2014arXiv1403.0906S
 Keywords:

 Mathematics  Complex Variables;
 Astrophysics  Astrophysics of Galaxies;
 30D05 31A05 85A04
 EPrint:
 Minor corrections, better color scheme for phase portraits