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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>-1)

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. Differing notation for $F_{j}$ appears in the literature, for instance some authors omit the factor $1/\Gamma (j+1)$ . The definition used here matches that in the NIST DLMF.

Special values

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

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

For x = 0, the result reduces to

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

where $\eta$ is the Dirichlet eta function.

See also

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 (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.

External links

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.

Quelle: https://en.wikipedia.org/wiki/Complete_Fermi%E2%80%93Dirac_integral