On the number of solutions of a transcendental equation arising in the theory of gravitational lensing

TL;DR

This paper proves that a specific transcendental equation in gravitational lensing theory has at most six solutions, and all numbers from one to six solutions are possible, clarifying the maximum number of images formed.

Contribution

It establishes an upper bound of six solutions for the lensing equation and demonstrates the realizability of all solution counts from one to six.

Findings

Maximum of six solutions for the lensing equation.

All solution counts from 1 to 6 are achievable.

Provides mathematical proof for the solution bounds.

Abstract

The equation in the title describes the number of bright images of a point source under lensing by an elliptic object with isothermal density. We prove that this equation has at most 6 solutions. Any number of solutions from 1 to 6 can actually occur.