The challenge of sixfold integrals: the closed-form evaluation of Newton potentials between two cubes

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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 head-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.

Original languageEnglish
Article number20220254
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume478
Issue number2262
DOIs
StatePublished - 29 Jun 2022

Keywords

  • Error function
  • Gravitational and electrostatic forces
  • Integration in finite terms
  • Laplace transform
  • Newton potential

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