@Article{         10.2307/2373917,
  Author        = "Risch, Robert H.",
  Abstract      = {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).^\ast(1) An algorithm is given for telling when two elementary expressions define the same function. (2) A characterization is derived of those differential equations having elementary solutions. (3) The four basic functions of elementary calculus-exp, log, tan, tan-1-are shown to be "irredundant." (4) A characterization is given of elementary functions possessing elementary inverses.},
  date-added    = "2023-02-03 07:17:32 +0100",
  date-modified = "2023-02-03 07:17:32 +0100",
  ISSN          = "00029327, 10806377",
  Journal       = "American Journal of Mathematics",
  Number        = "4",
  Pages         = "743--759",
  Publisher     = "Johns Hopkins University Press",
  Title         = "Algebraic Properties of the Elementary Functions of Analysis",
  URL           = "http://www.jstor.org/stable/2373917",
  URLDate       = "2023-02-03",
  Volume        = "101",
  Year          = "1979",
  bdsk-url-1    = "http://www.jstor.org/stable/2373917",
  File          = "Algebraic Properties of the Elementary Functions of Analysis - risch1979 - b.pdf"
}

@Article{ 10.2307/2373917, Author = "Risch, Robert H.", Abstract = {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).^\ast(1) An algorithm is given for telling when two elementary expressions define the same function. (2) A characterization is derived of those differential equations having elementary solutions. (3) The four basic functions of elementary calculus-exp, log, tan, tan-1-are shown to be "irredundant." (4) A characterization is given of elementary functions possessing elementary inverses.}, date-added = "2023-02-03 07:17:32 +0100", date-modified = "2023-02-03 07:17:32 +0100", ISSN = "00029327, 10806377", Journal = "American Journal of Mathematics", Number = "4", Pages = "743--759", Publisher = "Johns Hopkins University Press", Title = "Algebraic Properties of the Elementary Functions of Analysis", URL = "http://www.jstor.org/stable/2373917", URLDate = "2023-02-03", Volume = "101", Year = "1979", bdsk-url-1 = "http://www.jstor.org/stable/2373917", File = "Algebraic Properties of the Elementary Functions of Analysis - risch1979 - b.pdf" }