Resummation of the Poincaré asymptotic expansion of the Hankel integral by Borel summation and superasymptotic integration

Abstract

We subject the Poincaré asymptotic expansion of the Hankel integral to resummation through Borel transformation to determine an asymptotic expansion that can approximate the Hankel integral for small values of the asymptotic parameter. Two cases are considered where the odd and even values of the series are separated. In the resummation process, the integral after Borel summation happens to be divergent thus we perform superasymptotic integration. We find that both resummations show superiority than its Poincaré asymptotic expansion in approximating the value of the Hankel integral for small values of the parameter.