Complete Fermi–Dirac integral

In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j  is defined by


 * $$F_j(x) = \frac{1}{\Gamma(j+1)} \int_0^\infty \frac{t^j}{e^{t-x} + 1}\,dt, \qquad (j > -1)$$

This equals
 * $$-\operatorname{Li}_{j+1}(-e^x),$$

where $$\operatorname{Li}_{s}(z)$$ is the polylogarithm.

Its derivative is
 * $$\frac{dF_{j}(x)}{dx} = F_{j-1}(x), $$

and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for $$F_j$$ appears in the literature, for instance some authors omit the factor $$1/\Gamma(j+1)$$. The definition used here matches that in the NIST DLMF.

Special values
The closed form of the function exists for j = 0:


 * $$F_0(x) = \ln(1+\exp(x)).$$

For x = 0, the result reduces to

$$ F_j(0) = \eta(j+1), $$

where $$\eta$$ is the Dirichlet eta function.