 Solution

An electrostatic plasma wave has $\delta \mathbf{E}=-i\mathbf{k}\varphi$. In the absence of externally imposed charge density, the Gauss law for the magnetized cold-plasma dielectric tensor is $\mathbf{k}\cdot \mathbf{ϵ}\cdot \delta \mathbf{E}=0$. Consequently,The antisymmetric terms of the magnetized cold-plasma dielectric tensor cancel in this scalar contraction. For oblique propagation this is the leading longitudinal condition in the large-refractive index approximation; a small transverse electric field is generally required by the full Maxwell equations at finite $n$. For parallel propagation an exactly longitudinal solution exists: $P=0$, hence ${\omega }^2\simeq {\omega }_{pe}^2$. It coincides with the $\theta \to 0$ limit of the finite-coefficient plasma wave resonance condition in c. At exactly $\theta =0$, $P$ factors out of the transverse dispersion relation, leaving the independent plasma oscillation rather than an isolated divergence of the transverse branches.