We show that in the presence of free charge carriers the definition of thefrequency-dependent dielectric permittivity requires additional regularization.As an example, the dielectric permittivity of the Drude model is considered andits time-dependent counterpart is derived and analyzed. The respective electricdisplacement cannot be represented in terms of the standard Fourier integral.The regularization procedure allowing to circumvent these difficulties issuggested. For purpose of comparison it is shown that the frequency-dependentdielectric permittivity of insulators satisfies all rigorous mathematicalcriteria. This permits us to conclude that in the presence of free chargecarriers the concept of dielectric permittivity is not as well defined as forinsulators and we make a link to widely discussed puzzles in the theory ofthermal Casimir force which might be caused by the use of such kindpermittivities.
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