How constant shifts affect the zeros of certain rational harmonic functions

Abstract

We study the effect of constant shifts on the zeros of rational harmomic functions $f(z) = r(z) - \conj{z}$. 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.

26 pages, 9 figures

References in corpus (2)

• Perturbing rational harmonic functions by poles
• The maximum number of zeros of $r(z)-\overline{z}$ revisited

Cited by in corpus (4)

• A Newton method for harmonic mappings in the plane
• The index of singular zeros of harmonic mappings of anti-analytic degree one
• Multiplane gravitational lenses with an abundance of images
• The transport of images method: computing all zeros of harmonic mappings by continuation