Description
The Risch algorithm is used to integrate elementary functions. These are functions obtained by composing exponentials, logarithms, radicals, trigonometric functions, and the four arithmetic 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; as a computer program it was finally 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 αi and elementary functions ui and v the solution is of the form
Risch developed a method that allows one to consider only a finite set of elementary functions of Liouville's form.
The intuition for the Risch algorithm comes from the behavior of the exponential and logarithm functions under differentiation. For the function f eg, where f and g are differentiable functions, we have
so if eg were in the result of an indefinite integration, it should be expected to be inside the integral. Also, as
then if (ln g)n were in the result of an integration, then only a few powers of the logarithm should be expected.
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