Advances in the theory of box integrals
Authors:
D. H. Bailey, J. M. Borwein and R. E. Crandall
Journal:
Math. Comp. 79 (2010), 18391866
MSC (2010):
Primary 11Y60; Secondary 2804
Published electronically:
February 9, 2010
MathSciNet review:
2630017
Fulltext PDF
Abstract 
References 
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Additional Information
Abstract: Box integralsexpectations or over the unit cubehave over three decades been occasionally given closed forms for isolated . By employing experimental mathematics together with a new, global analytic strategy, we prove that for each of dimensions the box integrals are for any integer hypergeometrically closed (``hyperclosed'') in an explicit sense we clarify herein. For dimensions, such a complete hyperclosure proof is blocked by a single, unresolved integral we call ; although we do prove that all but a finite set of () cases enjoy hyperclosure. We supply a compendium of exemplary closed forms that arise naturally from the theory.
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317–329. MR 2172710
(2006e:65047)
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David H. Bailey, Yozo Hida, Xiaoye S. Li and Brandon Thompson, ``ARPREC: An arbitrary precision computation package,'' September 2002. Available at http://crd.lbl.gov/˜dhbailey/dhbpapers/arprec.pdf.
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R.E. Crandall, C. Joe, and T. Mehoke, ``On the fractal distribution of brain synapses,'' manuscript, 2009.
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R. P. Ferguson, David
H. Bailey, and Steve
Arno, Analysis of PSLQ, an integer relation
finding algorithm, Math. Comp.
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Charles
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Computational Phys. 4 (1969), 19–29. MR 0243741
(39 #5062)
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Hidetosi
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M. Trott, ``The area of a random triangle,'' Mathematica Journal, 7 (1998), 189198.
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E. Weisstein, ``Hypercube line picking.'' Available at http://mathworld.wolfram.com/ HypercubeLinePicking.html.
 1.
 R. Anderssen, R. Brent, D. Daley, and P. Moran, ``Concerning and a Taylor series method,'' SIAM J. Applied Math., 30 (1976), 2230. MR 0394974 (52:15773)
 2.
 M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, NBS (now NIST), 1965. See also http://dlmf.nist.gov/.
 3.
 David H. Bailey and Jonathan M. Borwein, ``Highly parallel, highprecision numerical integration,'' Int. Journal of Computational Science and Engineering, accepted Jan. 2008. Available at http://crd.lbl.gov/˜dhbailey/dhbpapers/quadparallel.pdf.
 4.
 David H. Bailey and Jonathan M. Borwein, ``Effective error bounds for EulerMaclaurinbased quadrature schemes,'' 20th Annual HPCS Proceedings, IEEE CD, May 2006. Available at http://crd.lbl.gov/˜dhbailey/dhbpapers/emerror.pdf.
 5.
 D.H. Bailey, J.M. Borwein and R.E. Crandall, ``Box integrals,'' Journal of Computational and Applied Mathematics, 206 (2007), 196208. [Ddrive Preprint 320]. MR 2337437 (2008i:65036)
 6.
 D.H. Bailey, J.M. Borwein and R.E. Crandall, ``Integrals of the Ising class,'' J. Phys. A., 39 (2006), 1227112302. Available at http://crd.lbl.gov/˜dhbailey/ dhbpapers/Ising.pdf. MR 2261886 (2007j:33044)
 7.
 David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor, and Eric W. Weisstein, ``Ten problems in experimental mathematics,'' Amer. Mathematical Monthly, 113 (2006), 481509. Available at http://locutus.cs.dal.ca:8088/archive/00000316/ MR 2231135 (2007b:65001)
 8.
 David H. Bailey and David J. Broadhurst, "Parallel integer relation detection: Techniques and applications,'' Mathematics of Computation, 70 (Oct 2000), 17191736. MR 1836930 (2002b:11174)
 9.
 David H. Bailey, Xiaoye S. Li and Karthik Jeyabalan, ``A comparison of three highprecision quadrature schemes,'' Experimental Mathematics, 14 (2005), 317329. Available at http://crd.lbl.gov/˜dhbailey/dhbpapers/quadrature.pdf. MR 2172710 (2006e:65047)
 10.
 David H. Bailey, Yozo Hida, Xiaoye S. Li and Brandon Thompson, ``ARPREC: An arbitrary precision computation package,'' September 2002. Available at http://crd.lbl.gov/˜dhbailey/dhbpapers/arprec.pdf.
 11.
 Jonathan M. Borwein and David H. Bailey, Mathematics by Experiment, AK Peters, 2003. Second edition, 2008. See also http://www.experimentalmath.info. MR 2473161 (2010c:00001)
 12.
 J. M. Borwein and B. Salvy, ``A proof of a recursion for Bessel moments,'' Exp. Mathematics, vol. 17 (2008), 223230. MR 2433887
 13.
 Chaunming Zong, The CubeA Window to Convex and Discrete Geometry, Cambridge Tracts in Mathematics, 2006.
 14.
 R.E. Crandall, C. Joe, and T. Mehoke, ``On the fractal distribution of brain synapses,'' manuscript, 2009.
 15.
 Helaman R. P. Ferguson, David H. Bailey and Stephen Arno, ``Analysis of PSLQ, an integer relation finding algorithm,'' Mathematics of Computation, 68 (Jan 1999), 351369. MR 1489971 (99c:11157)
 16.
 Wolfram Koepf, Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities, American Mathematical Society, Providence, RI, 1998. MR 1644447 (2000c:33002)
 17.
 G. Lamb,``An evaluation of the integral ,'' ``A method of evaluating the integral ,'' and ``An approach to evaluating the integral ,'' preprint collection, 2006.
 18.
 G. Lamb,``An evaluation of the integral ,'' and ``An approach to evaluating the integral ,'' preprint collection, 2009.
 19.
 Leonard Lewin, Polylogarithms and Associated Functions, North Holland, 1981. MR 618278 (83b:33019)
 20.
 Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart, LINPACK User's Guide, SIAM, Philadelphia, 1979.
 21.
 J. Philip, ``The distance between two random points in a 4 and 5cube,'' preprint, 2008.
 22.
 D. Robbins, ``Average distance between two points in a box,'' Amer. Mathematical Monthly, 85 (1978), 278.
 23.
 Charles Schwartz, ``Numerical integration of analytic functions,'' Journal of Computational Physics, 4 (1969), 1929. MR 0243741 (39:5062)
 24.
 Hidetosi Takahasi and Masatake Mori, ``Quadrature formulas obtained by variable transformation,'' Numerische Mathematik, 21 (1973), 206219. MR 0331738 (48:10070)
 25.
 M. Trott, Private communication, 2005.
 26.
 M. Trott, ``The area of a random triangle,'' Mathematica Journal, 7 (1998), 189198.
 27.
 E. Weisstein, ``Hypercube line picking.'' Available at http://mathworld.wolfram.com/ HypercubeLinePicking.html.
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Additional Information
D. H. Bailey
Affiliation:
Lawrence Berkeley National Laboratory, Berkeley, California 94720
Email:
dhbailey@lbl.gov
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
Email:
jonathan.borwein@newcastle.edu.au, jborwein@cs.dal.ca
R. E. Crandall
Affiliation:
Center for Advanced Computation, Reed College, Portland, Oregon
Email:
crandall@reed.edu
DOI:
http://dx.doi.org/10.1090/S0025571810023380
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 DEAC0205CH11231.
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.
