Take electrostatic perturbations proportional to ei(kz−ωt), with ions immobile on the fast electrontime scale. Linearized Vlasov gives f1=−i(qE1/m)(∂f0/∂vz)/(ω−kvz). Integrating over transverse velocities gives F0(vz)=n0m/(2πkBT)e−mvz2/(2kBT). Combine the perturbed charge density with Poisson's equation:
The pole selects particles moving at phase speed. A causal initial-value prescription specifies how the contour passes it and produces resonant energy exchange, including Landau damping for a decreasing Maxwellian. A bare real improper integral without that prescription is incomplete. The unclosed brace in the printed Maxwellian exponent is typographical; the normalized Gaussian above fixes its meaning.
