A New Expression for the Coulomb Potential Corresponding to the Product of Two Exponential Functions Based on the Properties of the Integral Representations of the Bessel Functions

Abstract

The calculation of the Coulomb Potential corresponding to the product of two Exponential Type Functions, inherently has numerical challenges that must be resolved. In order to address these problems, in this paper it is presented a new partition of the Coulomb Potential. The proposed partition involves two terms. One of the terms is a one-dimensional integral, which allows geometrical and statistical interpretations. The other term is proportional to a Modified Bessel Function and it is obtained from a two-step procedure. As a first step, a Non-Rational Function is used for approximating one of the two integrals involved. Then, the remaining improper integral can be identified with an integral representation of an appropriate Modified Bessel Function. The existence of such a Non-Rational Approximant is proved and its numerical performance is shown through some examples.