Euler–Mellin ingtegrals

Euler–Mellin integrals are multidimensional Mellin transforms of functions of the form $1/f^t$, where $f^t$ denotes the product of polynomials in complex powers, and are closely related to $A$-hypergeometric Euler type integrals. They converge and define analytic functions $M_{f}(z, t)$ in tube domains which can be given in terms of the Newton polyhedra of $f$, and the polynomials themselves are assumed to be quasi-elliptic in the sense of the definition given in [Ermolaeva–Tsikh, 1996]. According to the result of [Berkesch–Forsgard–Passare, 2014], the functions $M_{f}(z, t)$ admit a meromorphic continuation. In the talk, we discuss the details of this meromorphic continuation and alternative representations for the functions $M_{f}(z, t)$.