Research Article Received 10 September 2013 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/mma.3118 MOS subject classification: 78A25; 45K05 Admissible frequency domain response functions of dielectrics Michele Caputo a,b and Mauro Fabrizio c * † Communicated by J. Muñoz Rivera Few empirical formulae produced to represent the complex dielectric constant in the frequency domain are discussed here from the view point of the possibility of their representation in the time domain. Because a generic function of the frequency does not necessarily admit an inverse Fourier transform, it is not always possible to obtain the corresponding time domain representation of the constant defined in the time frequency. With the use of the fractional derivatives, we introduce a tentative model of the dielectric constant including a linear term, the complex dielectric constant with its loss, and the loss of the direct current electrical conductivity. With appropriate values of the parameters, the system reproduces in the frequency domain the K. Cole and R. Cole model, without the contribution of the loss of the direct cur- rent electrical conductivity, which is studied by an autonomous constitutive equation. Copyright © 2014 John Wiley & Sons, Ltd. Keywords: memory; dielectric constant; fractional derivatives; electrical conductivity 1. Introduction The study of the dielectric constant has been successfully made experimentally in the frequency domain (e.g., Cole and Cole [1], Davidson and Cole [2], Havriliak and Negami [2]) obtaining, in this domain, formulae relating the electric field and the displacement. Cametti [3] has contributed with a review of the dielectric constant particularly discussing the dielectric and conductometric properties of heterogeneous systems and illustrating the frequency domain formulae relating the electrical field to the reaction of the dielectric. In this note, we will attempt a discussion of the problem concerning the existence and bring out the time domain expression of other possible constitutive equations for the electrical field and the dielectric response. In the linear response theory, the macroscopic description of the constitutive properties of dielectric media is based on some characteristic functions, each of which determines the behavior of the material. If the frequency dependent electric field e.p/ acts upon the material, the frequency dependent electrical displacement d.p/ is given by a Laplace transform (LT) d.p/ D " 0 R.p/e.p/ (1.1) where R.p/ is the response function and the " 0 dielectric constant of free space, while p D˛ C i!.˛ > 0/ is the Laplace variable. Among these response functions, the Debye relaxation function has proven to be successful in describing the frequency dependence behavior of a large variety of materials. However, there are other materials as well, which show significant departure from this behavior. For this class of materials, a number of empirical expressions have been proposed, in order to take into account what was experimentally observed. In this note, we consider four models of dielectric and conductor coefficients: (a) The model originated from the Cole and Cole (1941) empirical formula without the effect of the direct electrical conductivity loss in the frequency domain. (b) The Davidson and Cole formula obtained generalizing the Cole and Cole formula but including a term describing the electrical conductivity loss, represented algebraically in the same form of the dielectric constant of (a). a Department of Physics, University of Rome La Sapienza, Rome, Italy b Department of Geology and Geophysics, Texas A&M University, College station, TX, USA c Department of Mathematics, University of Bologna, Bologna, Italy * Correspondence to: Mauro Fabrizio, Department of Mathematics, University of Bologna, Bologna, Italy. † E-mail: mauro.fabrizio@unibo.it Copyright © 2014 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2014