Asymptotic expansions of the dielectric responce functions of relativistic plasma

We carry out a detailed analytical investigation of the dielectric response function (longitudinal dielectric permeability) ε$_{1}$(k,ω) of a Maxwellian relativistic plasma. Correct expansions for this function are obtained in the regions ω « kc and ω ~ kc, which are used to obtain asymptotic expressions for ω$_{1}$(k,ω). We give the regions of ω and k for which these expressions are meaningful, and derive formulae for the dispersion and damping decrement of Langmuir waves.