Multidimensional Mellin transforms

The Mellin transforms figure prominently in the complex analysis due to being the most appropriate for using the theory of residues techniques. 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. Domains $\Theta$ and $U$ predetermine the asymptotics of functions. Moreover, the asymptotics of the original function $f(x)\in M_{\Theta}^{U}$ is defined by singularities of its Mellin transform $M[f](z)\in W_{U}^{\Theta}$. It is the fundamental correspondence which determines the scope of application for Mellin transforms. In my talk, I will speak about properties of the Mellin transform for rational functions with quasi-elliptic or hypoelliptic denominators and about using the inverse Mellin transform (Mellin–Barnes integral) as a tool of getting the analytic continuation for algebraic functions. I also will focus on the role of the Mellin transforms in the realization of residue currents.