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Aside from what seems like a simple typo ($\Omega v_\parallel$ instead of $\Omega / v_\parallel$ ), I think there's an issue with expression of $k_{res}$ in terms of $r_L$ . In the text it was simply $r_L^{-1}$ , which can't be the case since there's also a pitch angle dependence. But also in the lectures we wrote $\frac{1}{r_L \mu}$ (as per my and Luis's notes), which also seems to miss the $\sin \theta$ from the definition of Larmor radius.
So, here I tried to write down the correct definition, with$\tan$ originating from $v_\perp / v_\parallel$