# Higher-dimensional box integrals

Borwein, Jonathan M. and Chan, O-Yeat and Crandall, R. E. (2010) Higher-dimensional box integrals. Experimental Mathematics, 19 (4). pp. 431-446.

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Herein, with the aid of substantial symbolic computation, we solve previously open problems in the theory of n-dimensional box integrals $B_n(s) := <|r|^s;r\in[0,1]^n$ In particular, we resolve an elusive integral called $K_5$ 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.