For the bump-in-tail instability, the bump must show up in the 1-dimensional distribution function F₀ (after integrating out the electron velocity components orthogonal to the wave vector v).

Consider an arbitrary, isotropic, 3-dimensional distribution function f₀ = f₀(|v|) for electron velocities—one that might even have an isotropic bump at large |v|. Show that this distribution is stable against the growth of Langmuir waves (i.e., it produces ω_i < 0 for all wave vectors k).