# How constant shifts affect the zeros of certain rational harmonic functions

## TL;DR

This paper investigates how constant shifts influence the zeros of rational harmonic functions, providing insights into their behavior and applications in gravitational lensing, especially regarding zero count and orientation changes.

## Contribution

It characterizes the zero behavior of rational harmonic functions under constant shifts, linking mathematical properties to gravitational lensing phenomena.

## Key findings

- Shifting through caustics alters zero count and orientation.
- Insights into singular zeros of rational harmonic functions.
- Applications to gravitational lensing models.

## 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.