  A two-channel problem is considered within a method based on first order
differential equations that are equivalent to the corresponding Schr\"odinger
equation but are more convenient for dealing with resonant phenomena. Using
these equations, it is possible to directly calculate the Jost matrix for
practically any complex value of the energy. The spectral points (bound and
resonant states) can therefore be located in a rigorous way, namely, as zeros
of the Jost matrix determinant. When calculating the Jost matrix, the
differential equations are solved and thus, at the same time, the wave function
is obtained with the correct asymptotic behavior that is embedded in the
solution analytically. The method offers very accurate way of calculating not
only total widths of resonances but their partial widths as well. For each pole
of the S-matrix, its residue can be calculated rather accurately, which makes
it possible to obtain the Mittag-Leffler type expansion of the S-matrix as a
sum of the singular terms (representing the resonances) and the background term
(contour integral). As an example, the two-channel model by Noro and Taylor is
considered. It is demonstrated how the contributions of individual resonance
poles to the scattering cross section can be analyzed using the Mittag-Leffler
expansion and the Argand plot technique. This example shows that even poles
situated far away from the physical real axis may give significant
contributions to the cross section.
