An exponentially terminated Poincaré expansion

Abstract

We point out an example of an asymptotic expansion whose form is consistent with the definition due to Poincaré, but is abruptly terminated by an exponentially small order. This example is a hankel transform. First, we employ the classical Mellin Transform Method and show that it fails to establish the exponential smallness of the error term which is apparent in numerical calculations. Finally, we use a technique of translating the contour of integration of the function to show the exponential dependence of the error term of its resulting asymptotic expansion.