Symbolic Integration of Differential Forms: From Abel to Zeilberger

TL;DR

This paper reviews the evolution of symbolic integration of differential forms, highlighting classical and modern techniques, and introduces algorithmic methods for integrating closed rational forms and parameterized integrals.

Contribution

It extends classical integration methods by developing algorithmic approaches for closed rational forms and parameterized integrals, unifying algebraic and transcendental cases.

Findings

Algorithmic methods for integrating closed rational forms.

Unified framework for algebraic and transcendental integrals.

Extension of Hermite reduction and Liouville's theorem.

Abstract

This paper focuses on symbolic integration of differential forms, with a particular emphasis on historical and modern developments, from Abel's addition theorems for Abelian integrals to Zeilberger's creative telescoping for parameterized integrals. It explores closed rational $p$-forms and provides algorithmic approaches for their integration, extending classical results like Hermite reduction and Liouville's theorem. The integration of closed differential forms with parameters is further examined through telescopers, offering a unified framework for handling both algebraic and transcendental cases.