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 …
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