Asymptotic expansion of a class of integral transforms with algebraically dominated kernels

A technique is developed here which yields the asymptotic expansion, in the two limits λ → 0+ and λ → +∞, for a class of functions defined by integrals I(λ) = ∝0∞ h(λt)f(t) dt where h(t) and f(t) are algebraically dominated near both 0+ and +∞. The integral I(λ) is first expressed as a contour integral of Barnes type through the Parseval theorem for Mellin transforms. The integrand is found to involve M[h; z] and M[f, 1 - z], the Mellin transforms of h and f evaluated at z and 1 - z respectively. Under rather general conditions, these Mellin transforms can be analytically continued to meromorphic functions in a suitable half plane, with poles that can be located and classified. The desired asymptotic expansions of I(λ) are then obtained by systematically moving the contour of its Barnes integral representation to the right for λ → +∞ and to the left for λ → 0+. Each term in either of these expansions is just the residue corresponding to a pole of M[h; z]M[f; 1 - z]. Under somewhat stronger conditions, the expansions obtained originally for real λ are extended to complex λ and shown valid in appropriate sectors. Several examples are considered to illustrate the range of these results. © 1971.