{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:12:37Z","timestamp":1776791557900,"version":"3.51.2"},"reference-count":27,"publisher":"American Mathematical Society (AMS)","issue":"271","license":[{"start":{"date-parts":[[2011,2,9]],"date-time":"2011-02-09T00:00:00Z","timestamp":1297209600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Box integrals\u2014expectations\n \n \n \n \n \n \u27e8\n \n <\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \n r<\/mml:mi>\n \n \u2192\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \n \n |<\/mml:mo>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msup>\n \n \u27e9\n \n <\/mml:mo>\n <\/mml:mrow>\n \\langle |\\vec r|^s \\rangle<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n or\n \n \n \n \n \n \u27e8\n \n <\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \n r<\/mml:mi>\n \n \u2192\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \n \u2212\n \n <\/mml:mo>\n \n \n q<\/mml:mi>\n \n \u2192\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \n \n |<\/mml:mo>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msup>\n \n \u27e9\n \n <\/mml:mo>\n <\/mml:mrow>\n \\langle |\\vec r - \\vec q|^s \\rangle<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n over the unit\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -cube\u2014have over three decades been occasionally given closed forms for isolated\n \n \n \n \n n<\/mml:mi>\n ,<\/mml:mo>\n s<\/mml:mi>\n <\/mml:mrow>\n n, s<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . By employing experimental mathematics together with a new, global analytic strategy, we prove that for each of\n \n \n \n \n n<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n 3<\/mml:mn>\n ,<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n n = 1,2,3,4<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n dimensions the box integrals are for any integer\n \n \n \n s<\/mml:mi>\n s<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n hypergeometrically closed (\u201chyperclosed\u201d) in an explicit sense we clarify herein. For\n \n \n \n \n n<\/mml:mi>\n =<\/mml:mo>\n 5<\/mml:mn>\n <\/mml:mrow>\n n = 5<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n dimensions, such a complete hyperclosure proof is blocked by a single, unresolved integral we call\n \n \n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n 5<\/mml:mn>\n <\/mml:msub>\n {\\mathcal K}_5<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ; although we do prove that all but a finite set of (\n \n \n \n \n n<\/mml:mi>\n =<\/mml:mo>\n 5<\/mml:mn>\n <\/mml:mrow>\n n = 5<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ) cases enjoy hyperclosure. We supply a compendium of exemplary closed forms that arise naturally from the theory.\n <\/p>","DOI":"10.1090\/s0025-5718-10-02338-0","type":"journal-article","created":{"date-parts":[[2010,4,15]],"date-time":"2010-04-15T09:01:02Z","timestamp":1271322062000},"page":"1839-1866","source":"Crossref","is-referenced-by-count":16,"title":["Advances in the theory of box integrals"],"prefix":"10.1090","volume":"79","author":[{"given":"D.","family":"Bailey","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J.","family":"Borwein","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R.","family":"Crandall","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2010,2,9]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1137\/0130003","article-title":"Concerning \u222b\u00b9\u2080^{\u2026}\u222b\u00b9\u2080(\ud835\udc65\u00b2\u2081+^{\u2026}+\ud835\udc65\u00b2_{\ud835\udc58})^{1\/2}\ud835\udc51\ud835\udc65\u2081^{\u2026}\ud835\udc51\ud835\udc65_{\ud835\udc58} and a Taylor series method","volume":"30","author":"Anderssen, R. S.","year":"1976","journal-title":"SIAM J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1399","issn-type":"print"},{"key":"2","unstructured":"M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, NBS (now NIST), 1965. See also http:\/\/dlmf.nist.gov\/."},{"key":"3","unstructured":"David H. Bailey and Jonathan M. Borwein, \u201cHighly parallel, high-precision numerical integration,\u201d Int. Journal of Computational Science and Engineering, accepted Jan. 2008. Available at http:\/\/crd.lbl.gov\/\u02dcdhbailey\/dhbpapers\/quadparallel.pdf."},{"key":"4","doi-asserted-by":"crossref","unstructured":"David H. Bailey and Jonathan M. Borwein, \u201cEffective error bounds for Euler-Maclaurin-based quadrature schemes,\u201d 20th Annual HPCS Proceedings, IEEE CD, May 2006. 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Mehoke, \u201cOn the fractal distribution of brain synapses,\u201d manuscript, 2009."},{"issue":"225","key":"15","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1090\/S0025-5718-99-00995-3","article-title":"Analysis of PSLQ, an integer relation finding algorithm","volume":"68","author":"Ferguson, Helaman R. P.","year":"1999","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"16","series-title":"Advanced Lectures in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-322-92918-1","volume-title":"Hypergeometric summation","author":"Koepf, Wolfram","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/3528069503"},{"key":"17","unstructured":"G. Lamb,\u201cAn evaluation of the integral \ud835\udca6\u2080,\u201d \u201cA method of evaluating the integral \ud835\udca6\u2081,\u201d and \u201cAn approach to evaluating the integral \ud835\udca6\u2083,\u201d preprint collection, 2006."},{"key":"18","unstructured":"G. Lamb,\u201cAn evaluation of the integral \ud835\udca6\u2086,\u201d and \u201cAn approach to evaluating the integral \ud835\udca6\u2084,\u201d preprint collection, 2009."},{"key":"19","isbn-type":"print","volume-title":"Polylogarithms and associated functions","author":"Lewin, Leonard","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/0444005501"},{"key":"20","doi-asserted-by":"crossref","unstructured":"Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart, LINPACK User\u2019s Guide, SIAM, Philadelphia, 1979.","DOI":"10.1137\/1.9781611971811"},{"key":"21","unstructured":"J. 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Trott, Private communication, 2005."},{"key":"26","unstructured":"M. Trott, \u201cThe area of a random triangle,\u201d Mathematica Journal, 7 (1998), 189\u2013198."},{"key":"27","unstructured":"E. Weisstein, \u201cHypercube line picking.\u201d Available at http:\/\/mathworld.wolfram.com\/ HypercubeLinePicking.html."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2010-79-271\/S0025-5718-10-02338-0\/S0025-5718-10-02338-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2010-79-271\/S0025-5718-10-02338-0\/S0025-5718-10-02338-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:29:52Z","timestamp":1776788992000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2010-79-271\/S0025-5718-10-02338-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,2,9]]},"references-count":27,"journal-issue":{"issue":"271","published-print":{"date-parts":[[2010,7]]}},"alternative-id":["S0025-5718-10-02338-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-10-02338-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2010,2,9]]}}}