A pair of optimal inequalities related to the error function

TL;DR

This paper introduces optimal inequalities for the error function, providing tight bounds that are useful across probability and physics applications.

Contribution

It presents new upper and lower bounds for the error function that are optimal within a specific class of functions, advancing mathematical approximation techniques.

Findings

Derived bounds are proven to be optimal within their class

Bounds improve accuracy of error function estimates

Applicable in probability and physics contexts

Abstract

The Error Function \begin{eqnarray} V(x) & \equiv & \sqrt{\pi} e^{x^2} [1 - \hbox{erf}(x)] \\ & = & \int_0^\infty \frac{ e^{-u} }{\sqrt{x^2 + u}} du = 2 e^{x^2}\int_x^\infty e^{-t^2} dt \nonumber \end{eqnarray} arises in many contexts, from probability to mathematical physics. We give estimates for the Error Function from above and below which are optimal within a certain class of functions.