Complete Fermi–Dirac integral

In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j  is defined by

This equals

where ${\displaystyle \operatorname {Li} _{s}(z)}$ is the polylogarithm.

Its derivative is

and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j.

The closed form of the function exists for j = 0:

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