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

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.