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) := ∊ [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.

Keywords:

• Box integral
• trilogarithm
• arctan integral