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There is an ever increasing interest in the development of plasmonic 2D nanomaterials, with widespread applications in optoelectronics, high resolution microscopy, imaging and sensing, among others. With the current ability of ultrathin noble metal film deposition down to a few monolayers in thickness, there is a need for an analytical expression of the thickness dependent complex dielectric function for predicting optical properties for arbitrary thicknesses. The free and bound electron contributions to the dielectric function are dealt with independently, since their influences affect separate wavelengths ranges. The former is dealt within the Drude model framework for large wavelengths with appropriately addressed damping constant and plasma frequency parameters to account for thickness dependence. Applying our previously developed method, we determine these parameters for specific film thicknesses, based on refractive index experimental values for Ag and Au thin films. Fitting separately each one of these parameters allowed us to find an analytical expression for their dependence on arbitrary film thickness and consequently for the free electron contribution. Concerning bound electrons, it is seen that its contribution for small wavelengths is the same for all analyzed thicknesses and may be set equal to the bulk bound contribution. Taking all these facts into account, the complex dielectric function can be rewritten analytically, in terms of the bulk dielectric function plus corrective film thickness dependent terms. In particular, the fitting process for the damping constant allows us to determine that the electron scattering at the film boundary is mainly diffusive (inelastic) for both silver and gold thin films. It is also shown that, in accordance with theoretical studies, plasma frequency shows a red shift as the film thickness decreases.