Complete Fermi–Dirac integral

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

F_j(x) = \frac{1}{\Gamma(j+1)} \int_0^\infty \frac{t^j}{e^{t-x} + 1}\,dt.

This equals

-\operatorname{Li}_{j+1}(-e^x),

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

Special values[edit]

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

F_0(x) = \ln(1+\exp(x)).\,

Compare this with the value of the polylogarithm at s=1:

\operatorname{Li}_{1}(z)=-\log(1-z).

See also[edit]

External links[edit]