Analytic solution of the system of integro-differential equations for the plasma model in an external field

We study a system of two integro-differential equations that arises as the result of linearization of Boltzmann–Maxwell's kinetic equations, where the collision integral is chosen in the Bhatnagar–Gross–Krook approximation, and the unperturbed state of the plasma is characterized by the Fermi–Dirac distribution. The unknown functions are the linear parts of the perturbations of the distribution function of the charged particles and the electric field strength in plasma. In the paper, an analytical representation for the general solution of this system is found. When deriving this representation, some new results were applied to Fourier transforms of distributions (generalized functions).