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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Implicitly elementary integrals


Author: Robert H. Risch
Journal: Proc. Amer. Math. Soc. 57 (1976), 1-7
MSC: Primary 12H05
MathSciNet review: 0409427
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Abstract: The explicitly elementary functions of complex variables $ {z_1}, \ldots ,{z_n}$ are those functions built up from $ {\mathbf{C}}({z_1}, \ldots ,{z_n})$ 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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DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0409427-1
PII: S 0002-9939(1976)0409427-1
Article copyright: © Copyright 1976 American Mathematical Society