### The problem of integration in finite terms

Robert H. Risch
1969 Transactions of the American Mathematical Society
This paper deals with the problem of telling whether a given elementary function, in the sense of analysis, has an elementary indefinite integral. In §1 of this work, we give a precise definition of the elementary functions and develop the theory of integration of functions of a single variai ¿. By using functions of a complex, rather than a real variable, we can limit ourselves to exponentiation, taking logs, and algebraic operations in defining the elementary functions, since sin, tan"1,
more » ... ce sin, tan"1, etc., can be expressed in terms of these three. Following Ostrowski [9], we use the concept of a differential field. We strengthen the classical Liouville theorem and derive a number of consequences. §2 uses the terminology of mathematical logic to discuss formulations of the problem of integration in finite terms. §3 (the major part of this paper) uses the previously developed theory to give an algorithm for determining the elementary integrability of those elementary functions which can be built up (roughly speaking) using only the rational operations, exponentiation and taking logarithms; however, if these exponentiations and logarithms can be replaced by adjoining constants and performing algebraic operations, the algorithm, as it is presented here, cannot be applied. The reader need only be familiar with some standard facts from algebra and complex analysis in order to understand this paper. Some basic results from differential algebra are used, but they are explicitly stated and references are given for their proofs. 1. Liouville theory of elementary functions. A field 3> with a unary operation d/dz is said to be a differential field iff for any a, b in S>: (1) |(û+é)^a+|è ®_ íz^ = aíb+bTza-