Algebraic Properties of the Elementary Functions of Analysis
Robert H. Risch · American Journal of Mathematics · 1979
The elementary functions of a complex variable z are those functions built up from the rational functions of z by exponentiation, taking logarithms, and algebraic operations. The purpose of this paper is first, to prove a structure theorem which shows that if an algebraic relation holds among a set of elementary functions, then they must satisfy an algebraic relation of a special kind. Then we make four applications of this theorem, obtaining both new and old results which are described here briefly (and imprecisely). (1) An algorithm is given for telling when two elementary expressions define the same function. (2) A characterization is derived of those ordinary differential equations having elementary solutions. (3) The four basic functions of elementary calculus-exp, log, tan, tan-' -are shown to be irredundant. (4) A characterization is given of elementary functions possessing elemen- tary inverses.