Hand-to-hand combat with thousand-digit integrals

In this paper we describe numerical investigations of definite integrals that arise by considering the moments of multi-step uniform random walks in the plane, together with a closely related class of integrals involving the elliptic functions K, K′, E and E′. We find that in many cases such integrals can be “experimentally” evaluated in closed form or that intriguing linear relations exist within a class of similar integrals. Discovering these identities and relations often requires the evaluation of integrals to extreme precision, combined with large-scale runs of the “PSLQ” integer relation algorithm. This paper presents details of the techniques used in these calculations and mentions some of the many difficulties that can arise.

Research highlights
► In this study, we present an instance of a novel application of state-of-the-art computing: we have discovered some new mathematical results related to certain classes of definite integrals, by first calculating their numerical values to very high numerical precision (typically 1,000 decimal digits or more) and then applying what is known as the PSLQ integer relation algorithm to identify these numerical values in terms of relatively simple mathematical constants and functions.
► The integrals we have addressed derive from the study of random walks, which is a staple of elementary probability courses, and are not significantly more complicated than many studied by college students in freshman calculus courses.
► The techniques we employ have very broad applicability to other problems in pure mathematics and mathematical physics.