We introduce the Mellin transform of a certain beta function. It arises in the study of a class of Laplace-type integrals with exponential phase function. An asymptotic expansion in terms of digamma functions is established for this Mellin transform, which exposes its logarithmic asymptotic character, and shows that it has the properties of an asymptotic scale. As a consequence, if the Mellin transform of the amplitude can be continued meromorphically, the initial Laplacian integral admits a logarithmic expansion. If the amplitude is even exponential, an additional logarithmic-exponential term appears.
Henrik Kaiser (Sat,) studied this question.