Generalization of Risch’s Algorithm to Special Functions

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 1 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.

Additive Decompositions in Primitive Extensions

“…The efficiency of the Risch algorithm is further improved by Rothstein [26], Davenport [11], Trager [29], Bronstein [6,7] etc. Some standard references on this topic are Bronstein's book [8] and Raab's survey [22] that gives an overview of the Risch algorithm and its recent developments.…”

Stability Problems in Symbolic Integration

“…The standard references on symbolic integration are firstly Bronstein's book [8] and some chapters in [50,12]. Recently, Raab gave an informative survey [37] on the Risch algorithm and its recent developments. Another significant current trend is the arithmetic studies of elementary integration initialized by Masser an Zannier in [51,10,29,30].…”

Stability Problems in Symbolic Integration

“…The standard references on symbolic integration are firstly Bronstein's book [8] and some chapters in [12,49]. Recently, Raab gave an informative survey [36] on the Risch algorithm and its recent developments. Another significant current trend is the arithmetic studies of elementary integration initialized by Masser an Zannier in [10,28,29,50].…”