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Mellin transforms of multivariate rational functions. (English) Zbl 1271.44001
If \(h\) is a locally integrable function on the positive real axis, the Mellin transform is defined by the formula \[ M_h(s)= \int^\infty_0 h(z)\,z^s{dz\over z}, \] where \(s= \sigma+ it\).
This transform is closely related to the Fourier-Laplace transform.
In the paper, the authors consider Mellin transforms of rational functions \(h= g/f\), where \(g\) and \(f\) are polynomials, among them in several variables.
It is proved that the polar set of such Mellin transforms consists of finitely many families of parallel hyperplanes, with all planes in each family being integral translates of a specific facial hyperplane of the Newton polytope of the denominator \(f\).
The Mellin transform is related to the so-called coamoeba and the theory of \(A\)-hypergeometric functions.

44A15 Special integral transforms (Legendre, Hilbert, etc.)
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
Full Text: DOI arXiv
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