Euler-Mellin integrals and A-hypergeometric functions

We consider integrals that generalize both Mellin transforms of rational functions of the form 1/f and classical Euler integrals. The domains of integration of our so-called Euler-Mellin integrals are naturally related to the coamoeba of f, and the components of the complement of the closure of this coamoeba give rise to a family of these integrals. After performing an explicit meromorphic continuation of Euler-Mellin integrals, we interpret them as A-hypergeometric functions and discuss their linear independence and relation to Mellin-Barnes integrals.