In mathematics, the incomplete Fermi-Dirac integral, named after Enrico Fermi and Paul Dirac, for an index formula_1 and parameter formula_2 is given by
formula_3
Its derivative is
formula_4
and this derivative relationship may be used to find the value of the incomplete Fermi-Dirac integral for non-positive indices formula_1.
This is an alternate definition of the incomplete polylogarithm, since:
formula_6
Which can be used to prove the identity:
formula_7
where formula_8 is the gamma function and formula_9 is the upper incomplete gamma function. Since formula_10, it follows that:
formula_11
where formula_12 is the complete Fermi-Dirac integral.
Special values.
The closed form of the function exists for formula_13: 
formula_14
References.
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