= Solution
Take electrostatic perturbations proportional to $e^{i(kz-\omega t)}$, with <Ion>[ions] immobile on the fast <electron> <time scale>. Linearized Vlasov gives $f_1=-i(qE_1/m)(\partial f_0/\partial v_z)/(\omega-kv_z)$. Integrating over transverse velocities gives $F_0(v_z)=n_0\sqrt{m/(2\pi k_BT)}e^{-mv_z^2/(2k_BT)}$. Combine the perturbed <charge density> with <Poisson equation>[Poisson's equation]:
$$
1+\frac{e^2}{\epsilon_0mk}\int\frac{F_0'(v_z)}{\omega-kv_z}dv_z=0.
$$
Since $F_0'=-mv_zF_0/(k_BT)$, this is precisely
$$
\boxed{1-\frac{2n_0e^2}{\sqrt\pi\epsilon_0mk}\left(\frac m{2k_BT}\right)^{3/2}\int\frac{v_ze^{-mv_z^2/(2k_BT)}}{\omega-kv_z}dv_z=0.}
$$
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.
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