Exactification of the Poincaré asymptotic expansion of the Hankel transform of entire exponential type functions using the distribution theory approach

Abstract

An asymptotic evaluation of the Hankel transform, ∫₀^∞ f(x) J_ν(λx) dx, of entire exponential type function, f(x), of type τ using the distribution theory approach due to McClure and Wong is done in this work. We show that the results from distribution theory is a special case that agrees with a generalized result which was obtained by performing a shifting of contour integration in the complex plane. It is also shown that the exponentially small terms being recovered from the distribution theory approach cancel each other and thus no longer contribute to the resulting expansion provided that the condition λ > τ is satisfied. Then, the resulting expansion terminates to a polynomial of inverse powers of the asymptotic parameter λ as λ → ∞.