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
This equals
where is the polylogarithm.
Its derivative is
and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for appears in the literature, for instance some authors omit the factor. The definition used here matches that in the .The closed form of the function exists for j = 0:
