Dielectric response of the charged Bose gas in the random-phase approximation

A closed form for the dielectric function of a charged Bose gas is found in the random-phase approximation. This dielectric constant is used to explore the quasiparticle energy spectrum (i.e., the allowed modes of oscillation of the gas), the damping of these quasiparticles, and the long-ranged form of the electrostatic potential around a test charge in the gas. We do this around the three temperature regions of interest; namely, $T=0\mathrm{}$, $T=\infty$, and $T=T_c$, where $T_c$ is the transition temperature of the gas.