The linearized Vlasov equation for a plasma is solved by means of integration along the particle trajectories assuming an oscillatory electric field of variable amplitude. A procedure is established which gives exactly the Landau–Jackson dispersion equation, showing that if the assumption of the constancy of amplitude of the electric field is removed, then O'Neil's method of integration of the nonlinear Vlasov equation yields the correct result in the linear limit. A simple application of the present formalism allows a determination of the earliest time at which the Landau damping law becomes valid for an arbitrary stationary distribution (previous work is limited to the case of a Maxwellian distribution) and initial perturbation. The minimum value of this time as a function of wave number is calculated explicitly for a Maxwellian stationary distribution.