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This paper presents a comprehensive study of a class of functions whichapproximate 1/|x| for large x but which are finite at the origin. Thesefunctions arise naturally in the study of atoms in strong magnetic fields wherethe so-called "Landau states" give rise to Gaussian integrals. Generalizationsin which e^{-x^2} is replaced by e^{-x^p} are also considered and approximatex^{1-p} for large x.  The limiting behavior and monotonicity properties of these functions arediscussed in terms of parameters which arise in the approximations as well asx. Several classes of inequalities, some of which provide tight bounds, areestablished. Some differential equations and recursion relations satisfied bythese functions are given. The recursion relations give rise to two classes ofpolynomials, one of which is related to confluent hypergeometric functions.Finally, it is shown that the inverse of the approximating function is convexand this implies an analogue of the triangle inequality. Some comments are madeabout possible extensions and several open questions are raised.

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