Risch Algorithm
The Risch algorithm is a decision procedure for indefinite integration that determines whether a given integral is elementary, and if so,
returns a closed-form result for the integral. It builds a tower of logarithmic,
exponential, and algebraic extensions. The case of algebraic extensions is quite
complicated and is therefore not completely implemented in any computer algebra system.
Liouville's principle, which dates back to
the 19th century, is an important part of the Risch algorithm. There are extensions
to the Risch algorithm, notably by Cherry, to be able to handle some special functions.
SEE ALSO: Elementary Function,
Horowitz Reduction,
Indefinite
Integral,
Liouville's Principle
This entry contributed by Bhuvanesh
Bhatt
REFERENCES:
Bronstein, M. Symbolic
Integration I: Transcendental Functions. New York: Springer-Verlag, 1997.
Cherry, G. W. Algorithms for Integrating Elementary Functions in Terms of Logarithmic Integrals and Error Functions. Ph.D. thesis. University of Delaware,
1983.
Cherry, G. W. "Integration in Finite Terms with Special Functions: The
Logarithmic Integral." SIAM J. Computing 15, 1-12, 1986.
Cherry, G. W. "An Analysis of the Rational Exponential Integral."
SIAM J. Computing 18, 893-905, 1989.
Davenport, J. H. On
the Integration of Algebraic Functions. Berlin: Springer-Verlag, 1981.
Geddes, K. O.; Czapor, S. R.; and Labahn, G. "The Risch Integration Algorithm." Ch. 12 in Algorithms
for Computer Algebra. Amsterdam, Netherlands: Kluwer, pp. 511-573, 1992.
Risch, R. "On the Integration of Elementary Functions Which are Built Up using Algebraic Operations." Report SP-2801/002/00. Santa Monica, CA: Sys. Dev. Corp., 1968.
Risch, R. "The Problem of Integration in Finite Terms." Trans. Amer.
Math. Soc. 139, 167-189, 1969.
Risch, R. "The Solution of the Problem of Integration in Finite Terms."
Bull. Amer. Math. Soc., 1-76, 605-608, 1970.
Risch, R. "Algebraic Properties of Elementary Functions of Analysis." Amer.
J. Math. 101, 743-759, 1979.
Referenced on Wolfram|Alpha:
Risch Algorithm
CITE THIS AS:
Bhatt, Bhuvanesh. "Risch Algorithm." From MathWorld--A Wolfram Web Resource, created by Eric
W. Weisstein. http://mathworld.wolfram.com/RischAlgorithm.html
Contact the MathWorld Team
indefinite integrals
