The Challenge of Sixfold Integrals: The Closed-Form Evaluation of Newton Potentials between Two Cubes

TL;DR

This paper introduces a simplified, algorithmic method using Laplace transforms to explicitly evaluate sixfold integrals for Newton potentials between two cubes, improving upon previous complex approaches.

Contribution

It presents a new, easier approach based on Laplace transforms for closed-form evaluation of sixfold integrals, extending to higher dimensions and reproducing known solutions.

Findings

Simplified explicit algorithm for potential integrals

Reproduces known closed-form solutions

Extends method to higher dimensions

Abstract

The challenge of explicitly evaluating, in elementary closed form, the weakly singular sixfold integrals for potentials and forces between two cubes has been taken up at various places in the mathematics and physics literature. It created some strikingly specific results, with an aura of arbitrariness, and a single intricate general procedure due to Hackbusch. Those scattered instances were mostly addressing the problem heads on, by successive integration while keeping track of a thicket of primitives generated at intermediate stages. In this paper we present a substantially easier and shorter approach, based on a Laplace transform of the kernel. We clearly exhibit the structure of the results as obtained by an explicit algorithm, just computing with rational polynomials. The method extends, up to the evaluation of single integrals, to higher dimensions. Among other examples, we easily reproduce Fornberg's startling closed form solution of Trefethen's two-cubes problem and Waldvogel's symmetric formula for the Newton potential of a rectangular cuboid.