Explicit evaluation of certain definite integrals involving powers of logarithms

Abstract

In recent years, considerable progress has been made in evaluating algorithmically, by symbolic computation on a computer, several classes of indefinite integrals. However, far fewer procedures seem to be available for the algorithmic computation of definite integrals. This is hardly surprising in view of the many different methods that are required to evaluate such integrals, some of them needing special tricks and a certain amount of experience. In this paper we report on a symbolic procedure that has been implemented in REDUCE for the evaluation of a class of definite integrals involving powers of exponentials and logarithms. In integral tables, these integrals can only be found for some particular values of the parameters. The symbolic procedure works uniformly for all values of the parameters (within the time and space limitations of the given machine). It is based on theoretical results derived in a series of recent papers, see (K61big, 1982, 1983a, b, 1985). These results show how the integrals considered can be represented as multiple sums involving Bernoulli, Euler, and Stirling numbers and the coefficients Ck that appear in the power series expansion of certain functions involving products and quotients of gamma functions. The symbolic determination of the coefficients Ck (depending on a number of parameters) and the simplification of the multiple sums can be totally automatised by using a computer algebra system. It is very tedious to carry these symbolic manipulations out by hand. Hence, the determination of the integrals involves, on the one hand, a mathemat.ical derivation that has been carried out by hand once and for all in the cited papers and, on the other hand, the evaluation of certain sums for specific parameter values by a symbolic procedure programmed in a computer algebra system. In section 2, we define the coefficients Ck and state a recurrence relation for the Ck suitable for symbolic evaluation. In section 3, a compilation of the sum representations of the integrals is given that can be read as symbolic procedures for the evaluation of the integrals. Some examples of the results of a REDUCE program evaluating these sums are provided in section 4.