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Mathematics of Computation

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Advances in the theory of box integrals

Authors: D. H. Bailey, J. M. Borwein and R. E. Crandall
Journal: Math. Comp. 79 (2010), 1839-1866
MSC (2010): Primary 11Y60; Secondary 28-04
Published electronically: February 9, 2010
MathSciNet review: 2630017
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Abstract: Box integrals--expectations $ \langle \vert\vec r\vert^s \rangle$ or $ \langle \vert\vec r - \vec q\vert^s \rangle$ over the unit $ n$-cube--have over three decades been occasionally given closed forms for isolated $ n, s$. By employing experimental mathematics together with a new, global analytic strategy, we prove that for each of $ n = 1,2,3,4$ dimensions the box integrals are for any integer $ s$ hypergeometrically closed (``hyperclosed'') in an explicit sense we clarify herein. For $ n = 5$ dimensions, such a complete hyperclosure proof is blocked by a single, unresolved integral we call $ {\mathcal K}_5$; although we do prove that all but a finite set of ($ n = 5$) cases enjoy hyperclosure. We supply a compendium of exemplary closed forms that arise naturally from the theory.

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Additional Information

D. H. Bailey
Affiliation: Lawrence Berkeley National Laboratory, Berkeley, California 94720

J. M. Borwein
Affiliation: School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia and Faculty of Computer Science, Dalhousie University, Halifax, NS, B3H 2W5, Canada

R. E. Crandall
Affiliation: Center for Advanced Computation, Reed College, Portland, Oregon

Received by editor(s): March 3, 2009
Received by editor(s) in revised form: August 13, 2009
Published electronically: February 9, 2010
Additional Notes: The first author was supported in part by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC02-05CH11231.
The second author was supported in part by ARC, NSERC and the Canada Research Chair Programme.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.