# 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 defined by

${\displaystyle F_{j}(x)={\frac {1}{\Gamma (j+1)}}\int _{0}^{\infty }{\frac {t^{j}}{e^{t-x}+1}}\,dt,\qquad (j>-1)}$

This equals

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

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

Its derivative is

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

and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for ${\displaystyle F_{j}}$ appears in the literature, for instance some authors omit the factor ${\displaystyle 1/\Gamma (j+1)}$. The definition used here matches that in the NIST DLMF.

## Special values

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

${\displaystyle F_{0}(x)=\ln(1+\exp(x)).}$

## 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.
• R.B.Dingle (1957). Fermi-Dirac Integrals. Appl.Sci.Res. B6. pp. 225–239.