  Smooth, highly accurate analytical representations of Fermi-Dirac (FD)
integral combinations important in free-energy density functional calculations
are presented. Specific forms include those that occur in the local density
approximation (LDA), generalized gradient approximation (GGA), and fourth-order
gradient expansion of the non-interacting free energy as well as in the LDA and
second-order gradient expansion for exchange. By construction, all the
representations and their derivatives of any order are continuous on the full
domains of their independent variables. The same type of technique provides an
analytical representation of the function inverse to the FD integral of order
$1/2$. It plays an important role in physical problems related to the electron
gas at finite temperature. From direct evaluation, the quality of these
improved representations is shown to be substantially superior to existing
ones, many of which were developed before the era of large-scale computation or
early in the era.
