A quantum system is considered that can move in N two-body channels with the
potentials that may include the Coulomb interaction. For this system, the Jost matrix
is constructed in such a way that all its dependencies on the channel momenta and
Sommerfeld parameters are factorized in the form of explicit analytic expressions.
It is shown that the two remaining unknown matrices are single-valued analytic
functions of the energy and therefore can be expanded in the Taylor series near an
arbitrary point within the domain of their analyticity. It is derived a system of firstorder
differential equations whose solutions determine the expansion coefficients
of these series. Alternatively, the unknown expansion coefficients can be used as
fitting parameters for parametrizing experimental data similarly to the effective-range
expansion. Such a parametrization has the advantage of preserving proper analytic
structure of the Jost matrix and can be done not only near the threshold energies,
but around any collision or even complex energy. As soon as the parameters are
obtained, the Jost matrix (and therefore the S-matrix) is known analytically on all
sheets of the Riemann surface, and thus enables one to locate possible resonances.