The Complex Scaling Method (CSM) provides scattering wave functions which
regularize resonances and suggest a resolution of the identity in terms of such
resonances, completed by the bound states and a smoothed continuum. But, in the
case of inelastic scattering with many channels, the existence of such a
resolution under complex scaling is still debated. Taking advantage of results
obtained earlier for the two channel case, this paper proposes a representation
in which the convergence of a resolution of the identity can be more easily
tested. The representation is valid for any finite number of coupled channels
for inelastic scattering without rearrangement.