New functional expansions for the Fermi-Dirac functions

Abstract

By expressing the Fermi-Dirac functions ${F}_{\ensuremath{\alpha}}(z)$ as a contour integral, we can, after some manipulations, apply the theorem of residues to obtain several high-precision functional series expansions which represent these functions in wide regions of the complex $z$ plane, and which analytically continue the well-known power series expansions valid for $|z| <1$. We also evaluate the respective Fermi-Dirac functions ${F}_{\ensuremath{\alpha}}(z)$ to an accuracy of nine decimal places for $\ensuremath{\alpha}=\frac{1}{2}, \frac{3}{2}, \mathrm{and} \frac{5}{2}$ and positive real $z$ (the parameter $\ensuremath{\alpha}$ commonly found in the literature corresponds to $\ensuremath{\alpha}\ensuremath{-}1$ in our notation). In particular for $\ensuremath{\alpha}=\frac{3}{2} \mathrm{and} \frac{5}{2}$ (both cases of great interest for applications) elementary functions are found, which represent a very good approximation to the respective ${F}_{\ensuremath{\alpha}}(z)$ functions in wide intervals of the $z$ variable.