An extension of an asymptotic result of Tricomi concerning a definite integral

TL;DR

This paper extends Tricomi's asymptotic analysis of a specific integral involving the error function and explores an additional integral related to airfoil theory, providing numerical validation of the expansions.

Contribution

It introduces a new asymptotic expansion for Tricomi's integral as the parameter grows large and extends the analysis to an integral relevant in airfoil theory.

Findings

Derived an asymptotic expansion for the integral as m approaches infinity.

Validated the expansion with numerical results showing high accuracy.

Extended the analysis to a related integral in aerodynamics.

Abstract

We consider the expansion of an integral considered by F.G. Tricomi given by \[\int_{-\infty}^\infty x e^{-x^2}(\frac{1}{2}+\frac{1}{2}\mbox{erf}\,x)^{m} dx\] as $m\to\infty$. The procedure involves a suitable change of variable and the inversion of the complementary error function $\mbox{erfc}\,x$. Numerical results are presented to demonstrate the accuracy of the expansion. A second part examines an extension of an integral arising in airfoil theory.