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
- [math]\displaystyle{ F_j(x) = \frac{1}{\Gamma(j+1)} \int_0^\infty \frac{t^j}{e^{t-x} + 1}\,dt, \qquad (j \gt -1) }[/math]
This equals
- [math]\displaystyle{ -\operatorname{Li}_{j+1}(-e^x), }[/math]
where [math]\displaystyle{ \operatorname{Li}_{s}(z) }[/math] is the polylogarithm.
Its derivative is
- [math]\displaystyle{ \frac{dF_{j}(x)}{dx} = F_{j-1}(x) , }[/math]
and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for [math]\displaystyle{ F_j }[/math] appears in the literature, for instance some authors omit the factor [math]\displaystyle{ 1/\Gamma(j+1) }[/math]. The definition used here matches that in the NIST DLMF.
Special values
The closed form of the function exists for j = 0:
- [math]\displaystyle{ F_0(x) = \ln(1+\exp(x)). }[/math]
For x = 0, the result reduces to
[math]\displaystyle{ F_j(0) = \eta(j+1), }[/math]
where [math]\displaystyle{ \eta }[/math] is the Dirichlet eta function.
See also
References
- "3.411.3." (in English). Table of Integrals, Series, and Products (8 ed.). Academic Press, Inc.. 2015. p. 355. ISBN:978-0-12-384933-5. 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.
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