Generalization of Risch's Algorithm to Special Functions

TL;DR

This paper reviews Risch's algorithm and recent advances, highlighting its applications in symbolic integration, special functions, and differential algebra, aiming to inform both mathematicians and physicists.

Contribution

It extends Risch's algorithm to handle special functions, providing a comprehensive overview and emphasizing its relevance in physics and computer algebra.

Findings

Enhanced algorithms for indefinite and definite integration

Methods for computing relations among integrals and identities for special functions

Increased awareness of computer algebra tools in the physics community

Abstract

Symbolic integration deals with the evaluation of integrals in closed form. We present an overview of Risch's algorithm including recent developments. The algorithms discussed are suited for both indefinite and definite integration. They can also be used to compute linear relations among integrals and to find identities for special functions given by parameter integrals. The aim of this presentation is twofold: to introduce the reader to some basic ideas of differential algebra in the context of integration and to raise awareness in the physics community of computer algebra algorithms for indefinite and definite integration.