Thermodynamics of the electron-positron plasma at very high temperatures

Thermodynamic properties of the electron-positron plasma (or gas) at high and very high temperatures are investigated. To achieve this goal we have derived a number of analytical formulas for the Fermi-Dirac distribution functions (or spectral functions) which can be applied to various Fermi gases in different cases. Almost all these formulas are represented in the form of series expansions. The coefficients in these expansions are the explicit and relatively simple functions of the $\frac{\mu}{T}$ ratio, where $T$ is the temperature and $\mu$ is the chemical potential of this Fermi system. Our new approach works very well for high temperature electron-positron plasma, which is in thermal equilibrium with the photon gas of annihilation $\gamma-$quanta, and for the model ultra-relativistic gas of fermions, where there is no radiation at all.