Fermi–Dirac integral

Remarks

The integral converges for $j > -1$: near $t = 0$ the integrand is bounded by $t^j$, and for large $t$ it decays like $t^j e^{x - t}$.

For $x < 0$, expanding $\frac{1}{e^{t-x} + 1} = \sum_{k \ge 1} (-1)^{k-1} e^{-k(t - x)}$ and integrating term by term gives $$F_j(x) = \sum_{k=1}^{\infty} (-1)^{k-1} \frac{e^{kx}}{k^{j+1}} = -\operatorname{Li}_{j+1}(-e^x),$$ where $\operatorname{Li}_{j+1}$ is the polylogarithm, and the identity holds for all real $x$ by analytic continuation. It also gives $F_j'(x) = F_{j-1}(x)$ for $j > 0$.

Example: for $j = 0$, $\int_0^\infty \frac{dt}{e^{t-x} + 1} = \log(1 + e^x)$, which matches $-\operatorname{Li}_1(-e^x) = \log(1 + e^x)$.

Non-example: for $j = -1$ the integrand behaves like $\frac{1}{t (e^{-x} + 1)}$ near $t = 0$, so the integral diverges; this is why $j > -1$ is required, and $F_{-1}$ is defined instead through the polylogarithm form.

Some authors omit the factor $1/\Gamma(j+1)$; the normalized version above is the one with $F_j(x) \sim e^x$ as $x \to -\infty$.