In mathematics, the incomplete Fermi-Dirac integral, named after Enrico Fermi and Paul Dirac, for an index
is given by 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 upper incomplete gamma function.
is the complete Fermi-Dirac integral.
The closed form of the function exists for