In mathematics, the incomplete Fermi-Dirac integral, named after Enrico Fermi and Paul Dirac, for an index
and parameter
is given by

| This article relies largely or entirely on a single source. (December 2020) |
Its derivative is

and this derivative relationship may be used to find the value of the incomplete Fermi-Dirac integral for non-positive indices
.[1]
This is an alternate definition of the incomplete polylogarithm, since:

Which can be used to prove the identity:

where
is the gamma function and
is the upper incomplete gamma function. Since
, it follows that:

where
is the complete Fermi-Dirac integral.