  For coupled-channel resonance scattering we derive a model with a closed form
solution for the $T$-matrix that satisfies unitarity and analyticity. The
two-channel case is handled explicitly for an arbitrary number of resonances.
The method focuses on the expansion of the transition matrix elements,
$\Gamma(s)$, in known analytical functions. The appropriate hadronic form
factors and the related energy shifts can be determined from the scattering
data. The differences between this method and the $K$-matrix and the
Breit-Wigner approximation are illustrated in the case of the $S_{11}$
resonances $S_{11}(1535)$ and $S_{11}(1650)$.
