Complete Fermi–Dirac integral

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

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

Incomplete Fermi–Dirac integral
Gamma function
Polylogarithm
Table of Integrals, Series, and Products, I.S. Gradshteyn, I.M. Ryzhik, 5th edition, p. 370, formula № 3.411.3.

External links