# Complete Fermi–Dirac integral

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

${\displaystyle F_{j}(x)={\frac {1}{\Gamma (j+1)}}\int _{0}^{\infty }{\frac {t^{j}}{e^{t-x}+1}}\,dt,\qquad (j>-1)}$

This equals

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

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

Its derivative is

${\displaystyle {\frac {dF_{j}(x)}{dx}}=F_{j-1}(x),}$

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

## Special values

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

${\displaystyle F_{0}(x)=\ln(1+\exp(x)).}$