Integrability in Finite Terms And Actions of Lie Groups

TL;DR

This paper discusses Liouville's Theorem on the non-elementary nature of indefinite integrals of elementary functions, exploring the role of differential Galois groups and Lie groups in the proof.

Contribution

It provides a detailed analysis of Liouville's Theorem using differential Galois groups and Lie group theory, extending the understanding of integrability in finite terms.

Findings

Differential Galois group of an extension does not determine elementary integrability.

Liouville's Theorem can be proved via algebraic and transcendental Galois groups.

Lie groups play a key role in understanding the structure of integrable functions.

Abstract

According to Liouville's Theorem, an indefinite integral of an elementary function is usually not an elementary function. In this notes, we discuss that statement and a proof of this result. The differential Galois group of the extension obtained by adjoining an integral does not determine whether the integral is an elementary function or not. Nevertheless, Liouville's Theorem can be proved using differential Galois groups. The first step towards such a proof was suggested by Abel. This step is related to algebraic extensions and their finite Galois groups. A significant part of this notes is dedicated to a second step, which deals with pure transcendent extensions and their Galois groups which are connected Lie groups. The idea of the proof goes back to J.Liouville and J.Ritt.