Sectionally analytic solutions of the Vlasov-Poisson equations

We investigate the solutions of the kinetic Vlasov-Poisson equations, which govern a plasma made of electrons and one species of mobile ions, in one rectangular dimension. We present a new formulation of Poisson's equation as an integral inverse problem. We prove inversion formulas which allow us to write the solution of this problem in such a way that the energy distribution of either of the particle species is related, in a straightforward way, to the energy distribution of the other species. We show that these distributions are retrieved from the boundary values of suitable sectionally analytic functions. These latter functions are the extension of the particle distributions into their respective complex energy domains