Equation of state of a Fermi gas: Approximations for various degrees of relativism and degeneracy

We give a complete review of various expansions which may be used for quick and accurate computing of the thermodynamic functions of an ideal Fermi gas. We begin with the well-known expression for the equation of state of an ultrarelativistic Fermi gas, which is in thermal equilibrium with blackbody radiation. This elementary expression is valid when the fermion mass is completely neglected in comparison to the average fermion kinetic energy. After that we show how to take into account the fermion mass; in other words, how to find an analytic form of the equation of state which is valid for a moderate degree of relativism of fermions. Contrary to the known expressions, the obtained expansion over the mass powers, as well as the elementary equation of state of an ultrarelativistic gas, holds for both a cold degenerate and a hot nondegenerate gas. Then we present expressions efficient in various ranges of temperature and density and derive some new approximations. The boundaries of applicability of all the expansions are specified on the temperature-density plane.