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

$F_j(x) = \frac{1}{\Gamma(j+1)} \int_0^\infty \frac{t^j}{e^{t-x} + 1}\,dt, \qquad (j > 0)$

This equals

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

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

Its derivative is

$\frac{dF_{j}(x)}{dx} = F_{j-1}(x) ,$

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

Special values

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

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

See also

• Incomplete Fermi–Dirac integral
• Gamma function
• Polylogarithm

References

• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "3.411.3.". In Zwillinger, Daniel; Moll, Victor Hugo. Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. p. 355. ISBN 0-12-384933-0. LCCN 2014010276. ISBN 978-0-12-384933-5.

External links

• GNU Scientific Library - Reference Manual
• Fermi-Dirac integral calculator for iPhone/iPad
• Notes on Fermi-Dirac Integrals