  A dynamical coupled-channel model is presented for investigating the nucleon
resonances in the meson production reactions induced by pions and photons. The
model is based on an energy-independent Hamiltonian which is derived from a set
of Lagrangians by using a unitary transformation method. By applying the
projection operator techniques,we derive a set of coupled-channel equations
which satisfy the unitarity conditions within the channel space spanned by the
considered two-particle meson-baryon states and the three-particle $\pi\pi N$
state. We present and explain in detail a numerical method based on a
spline-function expansion for solving the resulting coupled-channel equations
which contain logarithmically divergent one-particle-exchange driving terms
resulted from the $\pi\pi N$ unitarity cut. We show that this driving term can
generate rapidly varying structure in the reaction amplitudes associated with
the unstable particle channels. It also has large effects in determining the
two-pion production cross sections. Our results indicate that cautions must be
taken to interpret the $N^*$ parameters extracted from using models which do
not include $\pi\pi N$ cut effects.
