Dielectric function with exact exchange contribution in the electron liquid. II. Analytical expression

The first-order, in terms of electron-interaction in the perturbation theory, of the proper linear response function $\mathrm{\Pi }(\mathbf{k},\omega )$ gives rise to the exchange-contribution to the dielectric function $\epsilon ({\bf k} , \omega)$intheelectronliquid\mathrm{.}Itsimaginarypart,$Im \Pi_1 ({\bf k}, \omega)$,iscalculatedexactly\mathrm{.}Ananalyticalexpressionfor$Im \Pi_1 ({\bf k}, \omega)$ is derived which after refinement has a quite simple form.