Higher-Dimensional Box Integrals

Herein, with the aid of substantial symbolic computation, we solve previously open problems in the theory of n-dimensional box integrals Bn(s) := 〈|~r|〉; ~r ∈ [0, 1]. In particular we resolve an elusive integral called K5 that previously acted as a blockade against closed-form evaluation in n = 5 dimensions. In consequence we now know that Bn(integer) can be given a closed form for n = 1, 2, 3, 4, 5. We also nd the general residue at the pole at s = −n, this leading to new relations and de nite integrals for example, we are able to give the rst nontrivial closed forms for 6-dimensional box integrals and to show hyperclosure of B6(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 simpli cation methods. ∗Centre for Computer Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, Callaghan, New South Wales, 2308, Australia [email protected]. Research supported by the Australian Research Council †School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, New South Wales, 2308, Australia [email protected] ‡Center for Advanced Computation, Reed College, Portland OR [email protected]