Dielectric Tensor in Vlasov Plasmas near Cyclotron Harmonics

The relativistic expression for the dielectric tensor obtained by Trubnikov is simplified in the very weakly relativistic limit at and near electron cyclotron harmonics. Wavenumbers parallel to magnetic field are included, leading to relativistic damping when this wavenumber is minute and to cyclotron damping when it is sufficiently large. The transition to the nonrelativistic Z function is shown and the regions of validity of the various functions are indicated. Collisional damping is neglected. The dielectric elements given here are also applicable to cases of complex ω and real k. An example of such a situation arises in Alouette cyclotron harmonic reception when one is concerned with an initial time value problem. For this application the analytic continuation of a complicated function is provided, and the tracks where it is real for complex ω are investigated.