RISCH ALGORITHM

The 'Risch algorithm', named after Robert H. Risch, is an algorithm for the calculus operation of indefinite integration (i.e. finding antiderivatives). The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch, who developed the algorithm in 1968, called it a decision procedure, because it is a method for deciding ''if'' a function has a simple-looking function as an indefinite integral; and also, if it does, determining it. The Risch-Norman algorithm, a faster but less powerful technique, was developed in 1976.
The Risch algorithm is used to integrate elementary functions. These are functions obtained by composing exponentials, logarithms, radicals, and the four operations (+ − × ÷). Laplace solved this problem for the case of rational functions, as he showed that the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions. The algorithm suggested by Laplace is usually described in calculus textbooks but was only implemented in the 1960s.
Liouville formulated the problem solved by the Risch algorithm. Liouville proved by analytical means that if there is an elementary solution ''g'' to the equation ''g'' ′ = ''f'' then for constants α_''i'' and elementary functions ''u_i'' and ''v'' the solution is of the form
:
Risch developed a method for finding a finite set of elementary functions to consider.
The intuition for the Risch algorithm comes from the behavior of the exponential and logarithm functions under differentiation. For the function ''f'' e^''g'', where ''f'' and ''g'' are differentiable functions, we have
: $(f\; cdot\; e^g)\text{'}\; =\; (f^prime\; +\; fcdot\; g^prime)cdot\; e^g,$
so if e^''g'' were in the result of an indefinite integration, it should be expected to be inside the integral. Also, as
:
then if ln^''n''''g'' were in the result of an integration, then only a few powers of the logarithm should be expected.
Transforming the Risch decision procedure into an algorithm that can be executed by a computer is a complex task that requires the use of heuristics and many refinements.

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References

See also

References

★ The Problem of Integration in Finite Terms, R. H. Risch, , , Transactions of the American Mathematical Society, 1969 [1]

★ Integration in finite terms, Maxwell Rosenlicht, , , American Mathematical Monthly, 1972

★ Algorithms for Computer Algebra, Geddes, Czapor, Labahn, , , Kluwer Academic Publishers, 1992, ISBN 0-7923-9259-0

★ Symbolic Integration I, Manuel Bronstein, , , Springer, 2005, ISBN 3-540-21493-3

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★ MathWorld entry on the Risch Algorithm

See also

★ Lists of integrals

★ Liouville's theorem (complex analysis)