Finite Temperatur Dynamic Susceptibility of the Free Bose Gas

A detailed calculation of the real part of the finite temperature dynamic susceptibility of the free Bose gas is presented. After a short discussion on the different ways in which it can be calculated for temperatures above and below the Bose--Einstein transition temperature, its main properties and its evolution with $q$ and $T$ are analyzed. Finally, expressions for the lowest order energy weighted sum rules are also derived and studied.