Invariants of Superalgebras as Complex Integrals

In [A. Berele, Computing super matrix invariants, {\it Advances in Applied Math. \bf48} (2012), 273–289.] we defined integrals that approximated the Poincaré series of the invariants and concomitants of the general linear Lie supergroup or superalgebra. Budzik suggested in [K. Budzik, Supergroup Invariants and the Brane/Negative Brane Expansion, (preprint) arXiv:2509.20451] a way to adapt this method to get the exact Poincaré series. The purpose of this paper is to prove that Budzik’s ideas are correct. As a consequence we prove that the Poincaré series are rational functions.