 9958 Mary Beth Ruskai, Elisabeth Werner
 Study of a Class of Regularizations \\ of
$1/x$ using Gaussian Integrals
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Feb 21, 99

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Abstract. This paper presents a comprehensive study of the functions
$ V_m^p(x) = \frac{pe^{x^p}}{\Gamma(m+1)}
\int_x^\infty (t^px^p)^me^{t^p} dt $ for $x > 0$,
$m > 1$ and $p > 0$. For large $x$ these functions approximate
$x^{1p}$. The case $p=2$ is of particular importance because
the functions $V_m^2(x) \approx 1/x$ can be regarded as
onedimensional regularizations of the Coulomb potential $1/x$
which are finite at the origin for $m >  \half$.
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
open questions are raised.
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