Asymptotic expansion of a class of integral transforms via Mellin transforms

An asymptotic expansion for large λ of functions I(λ) defined by definite integrals of the form $$I(\lambda ) = \mathop \smallint \limits_0^\infty h(\lambda t)f(t)dt$$ is obtained in the case where h(t)=O(exp(-βt ^p )) as t→∞ with β, ?>0. To obtain the expansion for such integral transforms, I(λ) is first represented as a contour integral involving M h; z], the Mellin transform of the kernel h(t) evaluated at z, and Mf; 1-z], the Mellin transform of the function f(t) evaluated at 1-z. By assuming a rather general asymptotic expansion for f(t) near t=0, it is shown that Mf; 1-z] can be continued into the right-half plane as a meromorphic function with poles that can be located and classified. The desired asymptotic expansion of I is then obtained by systematically moving the contour in its integral representation to the right. Each term in the expansion arises as a residue contribution corresponding to a pole of Mf; 1-z]. It is then shown how the expansion, originally found for large positive λ, can be extended to complex λ. Finally several examples are considered which illustrate the scope of our expansion theorems.