Unperturbed-orbit integration and the 3D kinetic dispersion relation of the electron cyclotron drift instability

High-frequency instabilities in crossed-field ($\mathit{E}\times \mathit{B}$) plasmas are widely implicated in anomalous cross-field electron transport in Hall thrusters and related devices. Building on the fully kinetic 3D electrostatic dispersion relations reported by Ducrocq \emph{et al.} and later by Lafleur \emph{et al.}, we provide a concise, self-contained derivation of the key missing step: the magnetized-electron density perturbation $n_{e1}$ obtained from the linearized Vlasov equation via a retarded integration along unperturbed orbits, including finite-Larmor-radius effects and cyclotron harmonics. We collect the required mathematical identities in appendices and clarify the mapping between the Ducrocq Poisson-form and the Lafleur dielectric-form representations, including ion closures (cold-fluid versus kinetic Landau response). We conclude with a brief discussion of the assumptions and possible extensions toward more realistic configurations.