Semi-analytical description of the modulator section of the coherent electron cooling

We discuss the theoretical description of the modulator section of the coherent electron cooling (CeC) [1], the modern realization of the stochastic electron cooling, where the electron beam serves as a modulator and a kicker, i.e., it records the information about the hadron beam via electron density perturbations resulting from the shielding of the hadrons and then accelerates or decelerates hadrons by its electric field with respect to their velocities. To analyze the performance of the CeC shielding of a hadron in an electron beam should be computed with high precision. We propose a solution of this problem via Fourier and Laplace transforms for 1D, 2D and 3D plasmas. In some cases there are fully analytical solutions, which gave an opportunity to test semi-analytical ones involving numerical evaluations of the inverse integral transforms. Having its own practical value this solution will also serve as a testing ground for our general solution via numerical treatment of the integral equations applicable for the realistic case of the finite beam [3].