 Solution

A plasma wave resonance has $n^2\to \infty$. At finite coefficients of the magnetized cold-plasma dielectric tensor, its generic condition is the vanishing of the coefficient of $n^4$:A plasma wave cutoff has $n^2=0$, so the constant term of the dispersion relation vanishes:Both statements are branch conditions: check the undivided dispersion relation at special angles and simultaneous zero factors. In particular, the parallel plasma oscillation at $P=0$ is a longitudinal branch with arbitrary $k$ in the cold limit, while the transverse parallel branches have plasma wave cutoffs at $R=0$ or $L=0$. Poles of the magnetized cold-plasma dielectric tensor can give additional limiting plasma wave resonances, such as the parallel cold-plasma right-handed wave at $\omega ={\Omega }_e$; these are not obtained by assuming finite $S,P,g$.