An Inversion Theorem for the Singular Integral {P}oisson Equation

Abstract

The one dimensional Poisson equation governing the electric potential and the charge distributions in a plasma composed of electrons and one species of fully ionized ions is reduced to a singular integral equation. We prove an inversion theorem``which allow us to solve this equation in favour of the distribution function of one of the particle species, chosen according to need, once the electric potential and the distribution of the particle of the other species be given. At variance withprevious results, the unknown distribution function is determined over its whole energy range and it is written as the boundary value of a suitable sectionally analytic function. This fact allows us to extend the distribution function thus found over thewhole complex energy domai