  We provide a self-consistent electromagnetic theory of the coupling between
dipole emitters and dissipative nanoresonators. The theory that relies on the
concept of quasi-normal modes with complex frequencies provides an accurate
closed-form expression for the electromagnetic local density of states (LDOS)
of any photonic or plasmonic resonator with strong radiation leakage,
absorption and material dispersion. It represents a powerful tool to calculate
and conceptualize the electromagnetic response of systems that are governed by
a small number of resonance modes. We use the formalism to revisit Purcell's
factor. The new formula substantially differs from the usual one; in
particular, it predicts that a spectral detuning between the emitter and the
resonance does not necessarily result in a Lorentzian response in the presence
of dissipation. Comparisons with fully-vectorial numerical calculations for
plasmonic nanoresonators made of gold nanorods evidence the high accuracy of
the predictions achieved by our semi-analytical treatment.
