In mathematics, the incomplete Fermi-Dirac integral, named after Enrico Fermi and Paul Dirac, for an index and parameter is given by
| This article does not cite any sources. (December 2020) |
Its derivative is
and this derivative relationship is used to define the incomplete Fermi-Dirac integral for non-positive indices .
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.