In mathematics, the incomplete Fermi-Dirac integral, named after Enrico Fermi and Paul Dirac, for an index and parameter is given by
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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.
The closed form of the function exists for : [1]