Multidimensional Mellin Transforms: Fundamental Correspondence

There is a precise correspondence between individual terms in the asymptotic expansion of an original function and singularities of its Mellin transform. This general phenomenon is called the fundamental correspondence which determines the scope of application for Mellin fransforms. A pair of convex domains $\Theta, U \subset {\mathbb R}^n$ encodes isomorphic functional spaces $M_{\Theta}^{U}, W_{U}^{\Theta}$ which are transformed to each other by the direct and inverse Mellin transforms. The domains $\Theta$ and $U$ predetermine the asymptotic behaviour of functions in classes $M_{\Theta}^{U}$ and $W_{U}^{\Theta}$ respectively.
Mellin transforms figure prominently in the complex analysis due to being the most appropriate for using the residue theory techniques. In particular, the direct Mellin transform is used for computing the Bochner-Martinelli residue currents, and the inverse transform (the Mellin-Barnes integral representation) serves as a tool for analytic continuation of algebraic functions.
In my talk, I will focus on the properties of the multidimensional Mellin transform for rational functions with quasi-elliptic or hyperelliptic denominators.