  In the coherent electron cooling, the modern hadron beam cooling technique,
each hadron receives an individual kick from the electric field of the
amplified electron density perturbation created in the modulator by this hadron
in a co-propagating electron beam. We developed a method for computing the
dynamics of these density perturbations in an infinite electron plasma with any
equilibrium velocity distribution -- a possible model for the modulator. We
derived analytical expressions for the dynamics of the density perturbations in
the Fourier-Laplace domain for a variety of 1D, 2D, and 3D equilibrium
distributions of the electron beam. To obtain the space-time dynamics, we
employed the fast Fourier transform (FFT) algorithm. We also found an
analytical solution in the space-time domain for the 1D Cauchy equilibrium
distribution, which serves as a benchmark for our general approach based on
numerical evaluation of the integral transforms and as a fast alternative to
the numerical computations. We tested the method for various distributions and
initial conditions.
