Comparison of the Analytical Formulations and Variable Transformations for Near-Singularity Quadrature Involving the Static Potentials

In this work, we first investigate the problem of the near-singularity and shape-dependence in the analytical methods. It is found that the analytical methods are not immune to the two problems, due to the subtractive cancellation and round-off error. In order to increase the stability of the analytical formulations, an investigation is performed to find alternative formulations for the extreme cases. On the other hand, we propose a new variable transformation which can achieve machine-precision accuracy for the extreme case with a fast convergence rate. Some numerical results are presented to compare the modified analytical formulations and the singularity cancellation methods.