Asymptotic expansion of a class of integral transforms via Mellin transforms

Asymptotic Solution to a Class of Nonlinear Volterra Integral Equations

“…We hope to accomplish this by displacing the contour in (2.2) to the right, picking up in this process contributions corresponding to each singularity in the analytic continuations of the functions M[h;z], M[g; 1 z] into the right half-plane. In [4], it is shown that in order to locate and classify these singularities one need only assume appropriate asymptotic forms for h as + oe and g as --* 0 +. Indeed we have the following two lemmas.…”

“…The basic method of analysis to be employed involves the use of some new results on the asymptotic analysis of a class of integral transforms developed by Handelsman and Lew in [4] and [5]. These results are themselves heavily dependent on the theory of the Mellin transform and are summarized in 2 below.…”

“…Although we shall make use of this, we note that in most instances more direct methods can also be used. [4] concerning the asymptotic behavior of functions I(t) defined by definite integrals of the form…”

Asymptotic expansions of modified Hankel transform

“…Our investigation is based on the method developed by Handelsman and Lew [5][6][7], which yields asymptotic expansions of integrals of the form…”

Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales

“…Their method, which is based on the Laplace transform, gives explicit asymptotic expansions for a limited class of exponentially small Hankel integrals. A powerful method of obtaining explicit asymptotic expansion of a Hankel integral is the Mellin-Barnes method [7], which is a modification of the classical Mellin transform method [6,17,18] based on the asymptotic expansion of the ratio of products of gamma functions [19][20][21][22][23]. This method is capable of obtaining explicit expansions where the method of Franzen and Wong can give only order estimates.…”