- Title
- Higher-dimensional box integrals
- Creator
- Borwein, Jonathan M.; Chan, O-Yeat; Crandall, R. E.
- Relation
- Experimental Mathematics Vol. 19, Issue 4, p. 431-446
- Publisher Link
- http://dx.doi.org/10.1080/10586458.2010.10390634
- Publisher
- A K Peters
- Resource Type
- journal article
- Date
- 2010
- Description
- Herein, with the aid of substantial symbolic computation, we solve previously open problems in the theory of n-dimensional box integrals Bn(s) := 〈|r̅|⁸〉, r̅ ∈ [0, 1]ⁿ. In particular, we resolve an elusive integral called K₅ that previously acted as a “blockade” against closed-form evaluation in n = 5 dimensions. In consequence, we now know that Bn(integer) can be given a closed form for n = 1,2,3,4,5. We also find the general residue at the pole at s = −n, this leading to new relations and definite integrals; for example, we are able to give the first nontrivial closed forms for six-dimensional box integrals and to show hyperclosure of B₆(even). The Clausen function and its generalizations play a central role in these higher-dimensional evaluations. Our results provide stringent test scenarios for symbolic-algebra simplification methods.
- Subject
- box integral; trilogarithm; arctan integral
- Identifier
- http://hdl.handle.net/1959.13/928063
- Identifier
- uon:10329
- Identifier
- ISSN:1058-6458
- Rights
- This is an electronic version of an article published in Experimental Mathematics Vol. 19, Issue 4, p. 431-446. Experimental Mathematics is available online at: http://www.tandfonline.com/openurl?genre=article&issn=1058-6458&volume=19&issue=4&spage=431
- Language
- eng
- Full Text
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