The solution of the problem of integration in finite terms
Robert H. Risch · Bulletin of the American Mathematical Society · 1970
Introduction.The problem of integration in finite terms asks for an algorithm for deciding whether an elementary function has an elementary indefinite integral and for finding the integral if it does."Elementary" is used here to denote those functions built up from the rational functions using only exponentiation, logarithms, trigonometric, inverse trigonometric and algebraic operations.This vaguely worded question has several precise, but inequivalent formulations.The writer has devised an algorithm which solves the classical problem of Liouville.A complete account is planned for a future publication.The present note is intended to indicate some of the ideas and techniques involved.Basic notions.We will deal exclusively with differential fields 3D of characteristic zero; a differential field being a field endowed with a unary operation ' which satisfies the sum and product rule for derivatives.3D has a differential subfield K t called the constant field of 3D.It consists of all a G 3D such that a! = 0.If 3) is a differential subfield of $, then ^ (and any /G*?) is said to be elementary over 3D iff 3F = 3D(0i, • • • , 0 n ) where each 0 t -satisfies at least one of the following conditions:(1(1), ( 2) and ( 3) are all the operations needed since we get the trigonometric and inverse trigonometric operations by adjoining V(-l) toK.The following basic result gives us the form assumed by elementary AMS Subject Classifications.