Anharmonic Solutions to the Riccati equation and elliptic modular functions

TL;DR

This paper explores algebraic solutions to the Riccati equation within rational and elliptic function fields, advancing understanding of their structure and properties.

Contribution

It introduces new methods for solving the Riccati equation algebraically over complex rational and elliptic function fields.

Findings

Characterization of algebraic solutions over $ ext{C}(t)$

Identification of solutions over elliptic function fields

Development of solution techniques for Riccati equations

Abstract

We study algebraic solutions of the Riccati equation over the field of rational functions $\mathbb C(t)$, and over the elliptic function field $\mathbb C(\wp,\wp^\prime)$.