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The first-order, in terms of electron-interaction in the perturbation theory,of the proper linear response function $\Pi ({\bf k}, \omega )$ gives rise tothe exchange-contribution to the dielectric function $\epsilon ({\bf k} ,\omega)$ in the electron liquid. Its imaginary part, $Im \Pi_1 ({\bf k},\omega)$, is calculated exactly. An analytical expression for $Im \Pi_1 ({\bfk}, \omega)$ is derived which after refinement has a quite simple form.

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