Recipes for Classes of Definite Integrals Involving Exponentials and Logarithms

What is it about?

There are many classes of definite integrals for which the corresponding indefinite integral cannot be expressed in closed form whereas the definite integral can be expressed (often in terms of special functions). A computer algebra system should be capable of recognizing a wide variety of definite integrals and, in order to achieve a broad coverage, it is desirable to encode this knowledge in programs which are more gen-eral than simple table look-up. By exploiting integral definitions of the various special functions of mathematics and by generalization and differentiation, we are able to derive closed-form solutions for broad classes of definite integrals. In this paper we treat integrals involving exponentials and logarithms. The resulting programs, based on pattern matching and differentiation, are very efficient. - Comment: this is the paper and presentation that pioneered symbolic integration of definite integrals based on special functions that has been emulated in all major computer algebra systems.

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Photo by Antoine Dautry on Unsplash

Photo by Antoine Dautry on Unsplash

Why is it important?

This approach was presented at a pivotal and historical "Computers and Mathematics" conference held at MIT in 1989. This approach which used pattern matching was controversial but turned out to be VERY successful.

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