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It was observed recently that the multidimensional Mellin--Barnes integrals(Gustafson's integrals) arise naturally in studies of the $SL(2,R)$ spin chainmodels. We extend this analysis to the noncompact $SL(2,\mathbb{C})$ spinmagnets and obtain integrals which generalize Gustafson's integrals to thecomplex case.

Gustafson integrals for $SL(2,\mathbb{C})$ spin magnet