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.
