 Normalized complete Fermi-Dirac integral

The convention $F_j(\eta )=\Gamma (j+1{)}^{-1}{\int }_0^{\infty }{x}^j/[1+e^{x-\eta }]dx$ is useful for parabolic-band carrier densities. In three dimensions, a spin-degenerate single valley has $n=2[m{k}_{B}T/(2\pi {\hslash }^2){]}^{3/2}{F}_{1/2}[(\mu -E_c)/(k_{B}T)]$. Other conventions omit the gamma normalization, so prefactors must be stated.