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

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

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.

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

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

For x = 0, the result reduces to

${\displaystyle F_{j}(0)=\eta (j+1),}$

where ${\displaystyle \eta }$ is the Dirichlet eta function.

• Incomplete Fermi–Dirac integral
• Gamma function
• Polylogarithm

• 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 (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. p. 355. ISBN 978-0-12-384933-5. LCCN 2014010276. ISBN 978-0-12-384933-5.
• R.B.Dingle (1957). Fermi-Dirac Integrals. Appl.Sci.Res. B6. pp. 225–239.

• GNU Scientific Library - Reference Manual
• Fermi-Dirac integral calculator for iPhone/iPad
• Notes on Fermi-Dirac Integrals
• Section in NIST Digital Library of Mathematical Functions
• npplus: Python package that provides (among others) Fermi-Dirac integrals and inverses for several common orders.
• Wolfram's MathWorld: Definition given by Wolfram's MathWorld.

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