In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j is defined by This equals where
Its derivative is and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j.
appears in the literature, for instance some authors omit the factor
The definition used here matches that in the NIST DLMF.
The closed form of the function exists for j = 0: For x = 0, the result reduces to