Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. R. Anderssen, R. Brent, D. Daley, and P. Moran, ``Concerning $ \int_0^1 \cdots \int_0^1 (x_1^2 + \cdots x_n^2)^{\frac 12} dx_1 \cdots dx_n$ and a Taylor series method,'' SIAM J. Applied Math., 30 (1976), 22-30. MR 0394974 (52:15773)
  • 2. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, NBS (now NIST), 1965. See also
  • 3. David H. Bailey and Jonathan M. Borwein, ``Highly parallel, high-precision numerical integration,'' Int. Journal of Computational Science and Engineering, accepted Jan. 2008. Available at˜dhbailey/dhbpapers/quadparallel.pdf.
  • 4. David H. Bailey and Jonathan M. Borwein, ``Effective error bounds for Euler-Maclaurin-based quadrature schemes,'' 20th Annual HPCS Proceedings, IEEE CD, May 2006. Available at˜dhbailey/dhbpapers/em-error.pdf.
  • 5. D.H. Bailey, J.M. Borwein and R.E. Crandall, ``Box integrals,'' Journal of Computational and Applied Mathematics, 206 (2007), 196-208. [D-drive 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), 12271-12302. Available at˜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), 481-509. Available at 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), 1719-1736. MR 1836930 (2002b:11174)
  • 9. David H. Bailey, Xiaoye S. Li and Karthik Jeyabalan, ``A comparison of three high-precision quadrature schemes,'' Experimental Mathematics, 14 (2005), 317-329. Available at˜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˜dhbailey/dhbpapers/arprec.pdf.
  • 11. Jonathan M. Borwein and David H. Bailey, Mathematics by Experiment, AK Peters, 2003. Second edition, 2008. See also MR 2473161 (2010c:00001)
  • 12. J. M. Borwein and B. Salvy, ``A proof of a recursion for Bessel moments,'' Exp. Mathematics, vol. 17 (2008), 223-230. MR 2433887
  • 13. Chaunming Zong, The Cube-A 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), 351-369. 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 $ {\mathcal K}_0$,'' ``A method of evaluating the integral $ {\mathcal K}_1$,'' and ``An approach to evaluating the integral $ {\mathcal K}_3$,'' preprint collection, 2006.
  • 18. G. Lamb,``An evaluation of the integral $ {\mathcal K}_6$,'' and ``An approach to evaluating the integral $ {\mathcal K}_4$,'' 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 5-cube,'' 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), 19-29. MR 0243741 (39:5062)
  • 24. Hidetosi Takahasi and Masatake Mori, ``Quadrature formulas obtained by variable transformation,'' Numerische Mathematik, 21 (1973), 206-219. MR 0331738 (48:10070)
  • 25. M. Trott, Private communication, 2005.
  • 26. M. Trott, ``The area of a random triangle,'' Mathematica Journal, 7 (1998), 189-198.
  • 27. E. Weisstein, ``Hypercube line picking.'' Available at HypercubeLinePicking.html.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11Y60, 28-04

Retrieve articles in all journals with MSC (2010): 11Y60, 28-04

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.

American Mathematical Society