Generalizing Laplace's method by means of Mellin transforms

Abstract: A well-known procedure for the asymptotic evaluation of Laplace transforms is Laplace's method. Despite its wide applicability, however, it is easy to find relevant examples where the technique is infeasible, because the integrand admits no power series expansion near the critical point, e.g., due to exponential behaviour there. We circumvent this issue through an extension of the method of Mellin transforms, based on an integral representation for the kernel and a generalization of the Taylor expansion. Our main result then extends the Laplacian method in the sense that it provides asymptotic expansions for a wider scope of Laplace transforms in terms of known special functions, rather than only in powers of the asymptotic parameter. The results are illustrated in two examples.