发表机构
Delft University of Technology（代尔夫特理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对二维矩形中由点源和点汇产生的电流密度计算，本文利用与高斯多项式和雅可比theta函数相关的$q$-分析，克服了双傅里叶级数收敛慢的障碍，给出了电势和电流密度大小的精确且高效的解析公式。
AI 中文摘要
在$D=2$维欧几里得空间的矩形中，当相反的点电荷分别置于源点$s$和汇点$d$时，电流密度被计算为满足泊松方程（麦克斯韦第一定律）的电势的梯度。尽管二维矩形中电流密度的计算是一个经典问题，但若干障碍（例如双傅里叶级数的极慢收敛）最终通过$q$-分析得到了缓解，该分析与高斯多项式和雅可比theta函数相关，这些构成了任何椭圆函数的基本构建模块。我们提出了矩形中电势和电流密度大小的精确且计算上非常高效的解析公式。
英文摘要
The current density in a rectangle in the $D=2$ dimensional Euclidean space, where opposite point charges are placed in a source $s$ and a destination $d$, is computed as the gradient of a potential that obeys the Poisson equation (first law of Maxwell). Although the computation of the current density in a 2D rectangle is a classical problem, several barriers (e.g. very slow convergence of double Fourier series) were eventually alleviated by a $q$-analysis, related to Gaussian polynomials and Jacobi's theta-functions, that form the basic building blocks for any elliptic function. We present an exact and computationally very efficient analytic formula of the potential and the magnitude of the current density in a rectangle.
CommentsOur exact result did not appear before in the literature (to the best of my knowledge)
