Implicitly elementary integrals
Robert H. Risch
Proc. Amer. Math. Soc. 57 (1976), 1-7
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Abstract: The explicitly elementary functions of complex variables are those functions built up from by exponentiation, taking logarithms, and algebraic operations. The implicitly elementary functions are obtained by solving, via the implicit function theorem, for some of the variables in terms of the others, in systems of equations formed by setting a set of explicitly elementary functions equal to 0. Here we prove a 1923 conjecture of J. F. Ritt to the effect that if the indefinite integral of an explicitly elementary function is implicitly elementary, then it is explicitly elementary. The method features a geometrization of the concepts involved.
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