  The electron self-energy for long-range Coulomb interactions plays a crucial
role in understanding the many-body physics of interacting electron systems
(e.g. in metals and semiconductors), and has been studied extensively for
decades. In fact, it is among the oldest and the most-investigated many body
problems in physics. However, there is a lack of an analytical expression for
the self-energy $Re \Sigma^{(R)}( \varepsilon,T)$ when energy $\varepsilon$ and
temperature $k_{B} T$ are arbitrary with respect to each other (while both
being still small compared with the Fermi energy). We revisit this problem and
calculate analytically the self-energy on the mass shell for a two-dimensional
electron system with Coulomb interactions in the high density limit $r_s \ll
1$, for temperature $ r_s^{3/2} \ll k_{B} T/ E_F \ll r_s$ and energy $r_s^{3/2}
\ll |\varepsilon |/E_F \ll r_s$. We provide the exact high-density analytical
expressions for the real and imaginary parts of the electron self-energy with
arbitrary value of $\varepsilon /k_{B} T$, to the leading order in the
dimensionless Coulomb coupling constant $r_s$, and to several higher than
leading orders in $k_{B} T/r_s E_F$ and $\varepsilon /r_s E_F$. We also obtain
the asymptotic behavior of the self-energy in the regimes $|\varepsilon | \ll
k_{B} T$ and $|\varepsilon | \gg k_{B} T$. The higher-order terms have subtle
and highly non-trivial compound logarithmic contributions from both
$\varepsilon $ and $T$, explaining why they have never before been calculated
in spite of the importance of the subject matter.
