Asymptotic solution of a class of inhomogeneous integral equations

The purpose of this paper is to derive the asymptotic solutions to a class of inhomogeneous integral equations which reduce to algebraic equations when a parameter η goes to zero (the kernel becoming proportional to a Dirac δ-function). This class includes the integral equations obtained from the system of Vlasov and Poisson equations for the Fourier transform in space and the Laplace transform in time of the electric potential, when the equilibrium magnetic field is uniform and the equilibrium plasma density depends on ηx, with the coordinate z being the direction of the magnetic field. In this case the inhomogeneous term is given by the initial conditions and possibly by sources, and the Laplace transform variable ω is the eigenvalue parameter. (orig.)