On computability of multiple integrals by means of a sum of local residues

We consider $n$-fold integrals of meromorphic differential $n$-forms on an $n$-dimensional complex manifold and study the problem of computability of such integrals by means of local (Grothendieck) residues of these forms. This problem is relevant in various fields of theoretical physics (in superstring theory for study of periods of Calabi–Yau manifolds, in particle physics for computation of anomalous magnetic moments of muons). The obtained theorems refine earlier results of A.K. Tsikh and A.P. Yuzhakov.