Fermi–Dirac integral

Summary

The complete Fermi–Dirac integral $F_j(x)$ is the integral $\frac{1}{\Gamma(j+1)} \int_0^\infty \frac{t^j}{e^{t-x} + 1} \, dt$. It arises in the statistical physics of fermions, such as electrons in metals and semiconductors, where $x$ plays the role of a reduced chemical potential. It is a polylogarithm in disguise: $F_j(x) = -\operatorname{Li}_{j+1}(-e^x)$.