= Solution
When $|kv_z/\omega|\ll1$ over the populated velocities, expand the denominator in powers of $kv_z/\omega$. Odd Gaussian moments vanish. The first two nonzero terms use $\langle v_z^2\rangle=v_{th}^2$ and $\langle v_z^4\rangle=3v_{th}^4$, yielding $1-\omega_p^2/\omega^2-3\omega_p^2k^2v_{th}^2/\omega^4+\cdots=0$. Solving consistently to order $k^2$ gives
$$
\boxed{\omega^2\simeq\omega_p^2+3k^2v_{th}^2.}
$$
This is a long-wavelength/high-phase-speed approximation, not an exact Maxwellian <dispersion relation> at arbitrary $k$.
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