  Deriving analytical expressions of dielectric permittivities is required for
numerical and physical modeling of optical systems and the soar of
non-hermitian photonics motivates their prolongation in the complex plane.
Analytical models are based on the association of microscopic models to
describe macroscopic effects. However, the question is to know whether the
resulting Debye Drude Lorentz models are not too restrictive. Here we show that
the permittivity must be treated as a meromorphic transfer function that
complies with the requirements of complex analysis. This function can be
naturally expanded on a set of complex singularities. This singularity
expansion of the dielectric permittivity allows us to derive a generalized
expression of the Debye Drude Lorentz model that complies with the requirements
of complex analysis and the constraints of physical systems. We show that the
complex singularities and other parameters of this generalized expression can
be retrieved from experimental data acquired along the real frequency axis. The
accuracy of this expression is assessed for a wide range of materials including
metals, 2D materials and dielectrics, and we show how the distribution of the
retrieved poles helps in characterizing the materials.
