Implicitly elementary integrals

Author:
Robert H. Risch

Journal:
Proc. Amer. Math. Soc. **57** (1976), 1-7

MSC:
Primary 12H05

MathSciNet review:
0409427

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Serge Lang,*Introduction to algebraic geometry*, Interscience Publishers, Inc., New York-London, 1958. MR**0100591****[2]**Shoshichi Kobayashi and Katsumi Nomizu,*Foundations of differential geometry. Vol I*, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR**0152974****[3]**J. F. Ritt,*On the integrals of elementary functions*, Trans. Amer. Math. Soc.**25**(1923), no. 2, 211–222. MR**1501240**, 10.1090/S0002-9947-1923-1501240-7**[4]**-,*Integration in finite terms. Liouville's theory of elementary methods*, Columbia Univ. Press, New York, 1948. MR**9**, 573.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
12H05

Retrieve articles in all journals with MSC: 12H05

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1976-0409427-1

Article copyright:
© Copyright 1976
American Mathematical Society