The role of Bell polynomials in integration

摘要

It is shown that, in the evaluation of certain integrals, the answer will be a simple multiple of a Bell polynomial. Integrals of the form In,α,β≔∫0π/2lnn(sinαθcosβθ)dθ, where n is a nonnegative integer, are provided as examples. We focus in particular on the integrals In≔∫0π/2lnnsinθdθ, which have been frequently discussed in the past, following Euler's investigation of I1. It is also shown that in certain related but more complicated cases of integration, the answer will appear as a linear combination of Bell polynomials. As examples in this connection, the integrals Jn≔∫0∞(lnnt/(et+1))dt are evaluated. The expression of an integral in terms of Bell polynomials provides an apparently new connection between analysis and combinatorics. Because of the close links with combinatorics, it is possible to estimate the length of this expression, by means of an upper bound on the number of terms which arise. In fact, we are able to state the precise number of terms which arise in the expression for In,α,β, for general n. For the integrals Jn, this is also possible, but it is complicated by the fact that terms will cancel in the linear combination of Bell polynomials. We also subject the answers for In,α,β and for Jn to a certain modification (which depends on a well-known connection between the Riemann zeta function and Bernoulli numbers); this causes a significant and predictable reduction in the number of terms. It is further shown that the evaluation of the related integrals In∗≔∫0π/2lnn(2sinθ)dθ is possible, in terms of a Bell polynomial whose first argument is zero. For a general value of n, this causes a drastic and quantifiable reduction in the number of terms in the answer, compared to that in In, both before and after modification (for example, the initial numbers of terms for I16 and I16∗ are 231 and 55, respectively, whereas after modification, the numbers are 93 and 17, respectively). It is thus possible to provide, for general n, a combinatorial explanation of an observation which has been made previously when n satisfies 1⩽n⩽4.