On the singularity of the static Green's function and the derivations of equations associated with potentials

The singularity of the static Green's function incurs mathematical difficulty. It is pointed out that this singularity is unnecessarily complicated and can be removed by a physically meaningful assumption which regularizes the static Green's function without substantially affecting the electromagnetic theory. Further, this regularization smooths the electric field in the close proximity of the source and leads to that the electrostatic force due to a charged particle exerted on itself is zero. Thereby, the Poisson equation of the regularized static Green's function can be obtained in a simple manner. Then, the wave equations of the electric scalar potential and the magnetic vector potential are derived in a new approach. Furthermore, we derive the Lorentz gauge, rather than assume it.