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Euler--Mellin integrals and A-hypergeometric functions

We consider integrals that generalize both the Mellin transforms of rational functions of the form 1/f and the 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 the 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.