  We investigate determinants of Koszul complexes of holomorphic functions of a
commuting tuple of bounded operators acting on a Hilbert space. Our main result
shows that the analytic joint torsion, which compares two such determinants,
can be computed by a local formula which involves a tame symbol of the involved
holomorphic functions. As an application we are able to extend the classical
tame symbol of meromorphic functions on a Riemann surface to the more involved
setting of transversal functions on a complex analytic curve. This follows by
spelling out our main result in the case of Toeplitz operators acting on the
Hardy space over the polydisc.
