Hand-to-hand combat with thousand-digit integrals |
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Authors: | David H Bailey Jonathan M Borwein |
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Institution: | 1. School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454000, Henan, China;2. Division of Computing and Mathematics, University of Abertay, Dundee DD1 1HG, UK |
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Abstract: | 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. |
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