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, it is shown 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 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. It is shown 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 it is shown how the distribution of the retrieved poles helps in characterizing the materials.