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2010 Higher-Dimensional Box Integrals
Jonathan M. Borwein, O-Yeat Chan, R. E. Crandall
Experiment. Math. 19(4): 431-446 (2010).


Herein, with the aid of substantial symbolic computation, we solve previously open problems in the theory of $n$-dimensional box integrals $B_n(s) := \langle|\vec{r}|^s\rangle, \vec{r} ∈ [0, 1]^n$. In particular, we resolve an elusive integral called $\mathcal{K}_5$ that previously acted as a “blockade” against closed-form evaluation in $n = 5$ dimensions. In consequence, we now know that $B_n$(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 $B_6$(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.


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Jonathan M. Borwein. O-Yeat Chan. R. E. Crandall. "Higher-Dimensional Box Integrals." Experiment. Math. 19 (4) 431 - 446, 2010.


Published: 2010
First available in Project Euclid: 4 October 2011

zbMATH: 1251.33002
MathSciNet: MR2778656

Keywords: arctan integral , Box integral , trilogarithm

Rights: Copyright © 2010 A K Peters, Ltd.


Vol.19 • No. 4 • 2010
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