Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S16 / 3 / 2 / Solution

When $|k{v}_z/\omega |\ll 1$ over the populated velocities, expand the denominator in powers of $k{v}_z/\omega$. Odd Gaussian moments vanish. The first two nonzero terms use $⟨v_z^2⟩=v_{th}^2$ and $⟨v_z^4⟩=3v_{th}^4$, yielding $1-{\omega }_p^2/{\omega }^2-3{\omega }_p^2{k}^2{v}_{th}^2/{\omega }^4+\cdots =0$. Solving consistently to order $k^2$ givesThis is a long-wavelength/high-phase-speed approximation, not an exact Maxwellian dispersion relation at arbitrary $k$.