  The on-shell self-energy of the homogeneous electron gas in second order of
exchange, $\Sigma_{2{\rm x}}= {\rm Re} \Sigma_{2{\rm x}}(k_{\rm F},k_{\rm
F}^2/2)$, is given by a certain integral. This integral is treated here in a
similar way as Onsager, Mittag, and Stephen [Ann. Physik (Leipzig) {\bf 18}, 71
(1966)] have obtained their famous analytical expression $e_{2{\rm x}}={1/6}\ln
2- 3\frac{\zeta(3)}{(2\pi)^2}$ (in atomic units) for the correlation energy in
second order of exchange. Here it is shown that the result for the
corresponding on-shell self-energy is $\Sigma_{2{\rm x}}=e_{2{\rm x}}$. The
off-shell self-energy $\Sigma_{2{\rm x}}(k,\omega)$ correctly yields $2e_{2{\rm
x}}$ (the potential component of $e_{2{\rm x}}$) through the Galitskii-Migdal
formula. The quantities $e_{2{\rm x}}$ and $\Sigma_{2{\rm x}}$ appear in the
high-density limit of the Hugenholtz-van Hove (Luttinger-Ward) theorem.
