  The exchange contribution, $\Pi_1 ({\bf k}, 0)$, to the static dielectric
function in the electron liquid is evaluated exactly. Expression for it is
derived analytically in terms of one quadrature. The expression, as presented
in Eq. (3) in the Introduction, turns out to be very simple. A fully explicit
expression (with no more integral in it) for $\Pi_1 ({\bf k}, 0)$ is further
developed in terms of series. Equation (3) is proved to be equal to the
expression obtained before under some mathematical assumption by Engel and
Vosko, thus in the meanwhile putting the latter on a rigorous basis. The
expansions of $\Pi_1 ({\bf k}, 0)$ at the wavectors of $k=0$, $k=2k_F$, and at
limiting large $k$ are derived. The results all verify those obtained by Engel
and Vosko.
