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Euler-Mellin integrals and $$A$$-hypergeometric functions. (English) Zbl 1290.32008
Summary: 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.

##### MSC:
 32A60 Zero sets of holomorphic functions of several complex variables 33C60 Hypergeometric integrals and functions defined by them ($$E$$, $$G$$, $$H$$ and $$I$$ functions)
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