论文标题
六倍积分的挑战：对两个立方体之间牛顿电位的封闭式评估
The Challenge of Sixfold Integrals: The Closed-Form Evaluation of Newton Potentials between Two Cubes
论文作者
论文摘要
在基本封闭形式下，在数学和物理学文献的各个地方都采用了明确评估的弱构成六倍积分的挑战。它创造了一些引人注目的结果，具有任意性的光环，并且由于hackbusch而引起了一个复杂的一般程序。这些分散的实例主要是通过连续整合来解决问题的方向，同时跟踪在中级阶段产生的原始灌木丛。 在本文中，我们基于内核的拉普拉斯变换，提出了一种更容易，更短的方法。我们清楚地表明了通过使用有理多项式计算的显式算法获得的结果结构。该方法将单个积分的评估扩展到更高的维度。除其他示例外，我们很容易复制福恩伯格（Fornberg）的Treefethen两根立方体问题的惊人封闭式解决方案和Waldvogel的对称公式，以获得矩形Cuboid的牛顿潜力。
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.
