# Box Integrals

Borwein, Jonathan M. and Bailey, David H. and Crandall, Richard E. (2007) Box Integrals. Journal of Computational and Applied Mathematics, 206 .

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By a box integral'' we mean here an expectation $\langle |\vec r - \vec q|^s \rangle$ where $\vec r$ runs over the unit $n$-cube, with $\vec q$ and $s$ fixed, explicitly: \begin{eqnarray*} &&\int_0^1 \cdots \int_0^1 \left((r_1 - q_1)^2 + \dots + (r_n-q_n)^2\right)^{s/2} \ dr_1 \cdots dr_n. \end{eqnarray*} The study of box integrals leads one naturally into several disparate fields of analysis. While previous studies have focused upon symbolic evaluation and asymptotic analysis of special cases (notably $s = 1$), we work herein more generally---in interdisciplinary fashion---developing results such as: (1) analytic continuation (in complex $s$), (2) relevant combinatorial identities, (3) rapidly converging series, (4) statistical inferences, (5) connections to mathematical physics, and (6) extreme-precision quadrature techniques appropriate for these integrals. These intuitions and results open up avenues of experimental mathematics, with a view to new conjectures and theorems on integrals of this type.