TY - JOUR
T1 - The challenge of sixfold integrals
T2 - the closed-form evaluation of Newton potentials between two cubes
AU - Bornemann, Folkmar
N1 - Publisher Copyright:
© 2022 The Author(s).
PY - 2022/6/29
Y1 - 2022/6/29
N2 - 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.
AB - 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.
KW - Error function
KW - Gravitational and electrostatic forces
KW - Integration in finite terms
KW - Laplace transform
KW - Newton potential
UR - http://www.scopus.com/inward/record.url?scp=85134056137&partnerID=8YFLogxK
U2 - 10.1098/rspa.2022.0254
DO - 10.1098/rspa.2022.0254
M3 - Article
AN - SCOPUS:85134056137
SN - 1364-5021
VL - 478
JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
IS - 2262
M1 - 20220254
ER -