Study of a Class of Regularizations of 1/|x| Using Gaussian Integrals

This paper presents a comprehensive study of the functions $$ V_m^p(x) = \frac{pe^{x^p}}{\Gamma(m+1)} \int_x^\infty (t^p-x^p)^me^{-t^p} dt $$ for x > 0, m > -1, and p > 0. For large x these functions approximate x ^1-p . The case p=2 is of particular importance because the functions $V_m^2(x) \approx 1/x$ can be regarded as one-dimensional regularizations of the Coulomb potential 1/|x| which are finite at the origin for $m > - {\textstyle \frac{1}{2}}$.% \half$. The limiting behavior and monotonicity properties of these functions are discussed in terms of their dependence on m and p as well as x. Several classes of inequalities, some of which provide tight bounds, are established. Some differential equations and recursion relations satisfied by these functions are given. The recursion relations give rise to two classes of polynomials, one of which is related to confluent hypergeometric functions. Finally, it is shown that, for integer m, the function $1/V_m^2(x)$ is convex in x and this implies an analogue of the triangle inequality. Some comments are made about the range of p and m to which this convexity result can be extended and several related questions are raised.