Liouville’s theorem on functions with elementary integrals

Abstract

Defining a function of one variable to be elementary if it has an explicit representation in terms of a finite number of algebraic operations, logarithms, and exponentials, Liouville’s theorem in its simplest case says that if an algebraic function has an elementaryintegral then the latter is itself an algebraicfunction plus a sum of constant multiples of logarithms of algebraic functions. Ostrowski has generalized Liouville’s results to widerclasses of meromorphic functions on regions of the complex plane and J. F. Ritt has given the classical account of the entire subject in his Integration in Finite Terms, Columbia University Press, 1948. In spite of the essentially algebraic nature of the problem, all proofs so far have been analytic. This paper gives a self contained purely algebraic exposition of the problem, making a few new points in addition to the resulting simplicity and generalization. © 1968 by Pacific Journal of Mathematics.