What is it about?
Herein, it is shown that by exploiting integral definitions of well known special functions, through generalizations and differentiations, broad classes of definite integrals can be solved in closed form or in terms of special functions. This is especially useful when there is no closed form solution to the indefinite form of the integral. In this paper, three such classes of definite integrals are presented. Two of these classes incorporate and supercede all of Kolbig's integration formulae, including his formulation for the computation of Cauchy principal values. Also presented are the mathematical derivations that support the implementation of a third class which exploits the incomplete Gamma function. The resulting programs, based on pattern matching, differentiation, and occasionally limits, are very efficient.
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Why is it important?
This was the next development beyond the MIT Computers and Mathematics paper. This class of integrals and its exact analytical solutions provides handles about 75% of the integrals asked by users of Computer Algebra systems. It provides a very powerful first line of attack for symbolic integration.
Perspectives
Other classes were also added and this approach has become standard in prominent computer algebra systems like Maple, Mathematica and MuPAD.
Dr Tony Cyril Scott
RWTH-Aachen University
Read the Original
This page is a summary of: Evaluation of classes of definite integrals involving elementary functions via differentiation of special functions, Applicable Algebra in Engineering Communication and Computing, September 1990, Springer Science + Business Media,
DOI: 10.1007/bf01810298.
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