Exactifying the Hankel integral Poincaré asymptotic expansion by the distributional method and the role of rearrangement in the accuracy of the exactified expansion

Abstract

We use the distributional approach to determine the asymptotic expansion of the Hankel integral ∫₀^∞ Φ(x) x^–λ J_ν(bx) dx for arbitrarily large b. We find that the approach reproduces the known Poincaré asymptotic expansion plus the exactifying terms that are missed out by the Poincaré expansion. It is demonstrated that exactifying the Poincaré expansion does not necessarily produce an expansion that is superior to the Poincaré expansion in numerically approximating the Hankel integral. However, it is also demonstrated that a rearrangement of the exactified expansion may lead to an expansion that is more accurate than the Poincaré asymptotic expansion.