This tag is for questions relating to the Fermi-Dirac Integrals named after Enrico Fermi and Paul Dirac. Fermi–Dirac integrals arise in calculating pressure and density in degenerate matter, such as neutron stars; they also occur in the electronic density of semiconductors.
Fermi–Dirac integrals arise in calculating pressure and density in degenerate matter, such as neutron stars; they also occur in the electronic density of semiconductors.
Generalized Fermi-Dirac integrals: For an index $~j~$, Fermi–Dirac integral is defined by $$ F_j(x)=\frac{1}{\Gamma(j+1)}\int_0^{\infty}\frac{t^j}{e^{t-x}+1}~dt~,\qquad (j>-1)$$
$$~~~~~~~~~~~~~~~~~~~~~~~~=-\text{Li}_{j+1}(-e^x)\qquad \text{Where $~\text{Li}_s~$ is the polylogarithm}$$
The Fermi-Dirac integral function appears in a variety of areas in physics. I Among these we find transport phenomena in degenerate systems, thermionic emission, and astrophysics. It has also been found to be important in the operation of semiconductor lasers and in the description of several simplifying schemes such as the free and independent electron gas model in two and three dimensions.
For further reference see this paper named as "Notes on Fermi-Dirac Integrals " by Raseong Kim and Mark Lundstrom, and
$2.~$ Journal of Applied Physics 63, 2848 (1988); https://doi.org/10.1063/1.340957
