Improving upon Landau and van Kampen-Case: Proper Asymptotics and Disappearance of Decaying Discrete Modes

Landau's approach (generalized by Jackson) to the initial value problem for the one-dimensional linear Vlasov--Poisson system shifts and deforms the Bromwich contour around the poles of the analytically-continued dielectric function. For an unstable equilibrium, this produces the growing but not the decaying discrete modes. However, in the van Kampen--Case construction, growing and decaying discrete modes occur together: an apparent contradiction. We present a more general, yet more transparent solution and show that the decaying discrete modes do not ultimately contribute; part of the continuum always exactly cancels the decaying discrete modes. We evaluate the Bromwich integral using properties of Cauchy-type integrals instead of deforming the contour and therefore avoid difficulties arising from the Landau--Jackson analytic continuation. The latter can result in divergences from incorrect asymptotic assumptions, where the initial condition plays an important role in the complex plane that we properly account for. We avoid complicated principal value integrals and singular eigenfunctions of van Kampen--Case; a straightforward Laurent series expansion is used instead. We show specific examples using equilibria and initial conditions with distinct, previously unseen properties.