Higher-Dimensional Box Integrals

Abstract

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.