Electrostatics and Riemann Surfaces

TL;DR

This paper explores the application of geometry and complex analysis to derive electric fields on complex surfaces, providing visualizations of elliptic functions and connecting these methods to Riemann surface theory for educational purposes.

Contribution

It offers a simplified derivation of electric fields on topologically complex surfaces and links complex analysis with Riemann surfaces, serving as an educational reference.

Findings

Visualization of elliptic functions with complex arguments

Connection between complex analysis and Riemann surfaces

Educational resource for advanced topics in geometry and physics

Abstract

Using techniques from geometry and complex analysis in their simplest form, we present a derivation of electric fields on surfaces with non-trivial topology. A byproduct of this analysis is an intuitive visualization of elliptic functions when their argument is complex-valued. The underlying connections between these techniques and the theory of Riemann surfaces are also explained. Our goal is to provide students and instructors a quick reference article for an extraordinary topic that is not included in the standard books.