  We present a pedagogical study of the commonly employed Speed-Plot (SP) and
Time-delay (TD) methods for extracting the resonance parameters from the data
of two particle coupled-channels reactions. Within several exactly solvable
models, it is found that these two methods find poles on different Riemann
sheets and are not always valid. We then develop an analytic continuation
method for extracting the nucleon resonances within a dynamical coupled-channel
formulation of the $\pi N$ and $\gamma N$ reactions. The main focus is on
resolving the complications due to the coupling with the unstable $\pi \Delta,
\rho N, \sigma N$ channels which decay into $\pi \pi N$ states. By using the
results from the considered exactly solvable models, explicit numerical
procedures are presented and verified. As a first application of the developed
analytic continuation method, we present the nucleon resonances in the $S_{11}$
and $S_{31}$ partial waves extracted within a recently developed
coupled-channels model of $\pi N$ reactions. The results from this realistic
$\pi N$ model, which includes $\pi N$, $\eta N$, $\pi\Delta$, $\rho N$, and
$\sigma N$ channels, also show that the simple pole parametrization of the
resonant propagator using the poles extracted from SP and TD methods works
poorly.
