Fermi–Dirac integral

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1: 25.19 Tables

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Morris (1979) tabulates ${\mathrm{Li}}_2\left(x\right)$ (§25.12(i)) for $\pm x=0.02(.02)1(.1)6$ to 30D.

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Cloutman (1989) tabulates $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

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2: 25.20 Approximations

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Morris (1979) gives rational approximations for ${\mathrm{Li}}_2\left(x\right)$ (§25.12(i)) for $0.5\le x\le 1$. Precision is varied with a maximum of 24S.

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Antia (1993) gives minimax rational approximations for $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals and , with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$. For each $s$ there are three sets of approximations, with relative maximum errors ${10}^{-4},{10}^{-8},{10}^{-12}$.

3: 25.12 Polylogarithms

… ►The Fermi–Dirac and Bose–Einstein integrals are defined by ►

25.12.14 $$F_s(x)=\frac{1}{\mathrm{\Gamma }\left(s+1\right)}{\int }_0^{\mathrm{\infty }}\frac{t^s}{{\mathrm{e}}^{t-x}+1}dt,$$ $s>-1$,

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Defines:

$F_s(x)$: Fermi–Dirac integral (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $x$: real variable and $s$: complex variable

Keywords:

definition, Fermi–Diracintegral, improper integral, integral representation

Source:

Dingle (1957b, (1), p. 226)

Referenced by:

(25.12.16), 3rd item, 5th item

Permalink:

http://dlmf.nist.gov/25.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(iii), §25.12 and Ch.25

… ►In terms of polylogarithms ►

$F_s(x)=-{\mathrm{Li}}_{s+1}\left(-{\mathrm{e}}^x\right),$

… ►For a uniform asymptotic approximation for $F_s(x)$ see Temme and Olde Daalhuis (1990).

4: 25.18 Methods of Computation

… ►For dilogarithms and polylogarithms see Jacobs and Lambert (1972), Osácar et al. (1995), Spanier and Oldham (1987, pp. 231–232), and Zudilin (2007). ►For Fermi–Dirac and Bose–Einstein integrals see Cloutman (1989), Gautschi (1993), Mohankumar and Natarajan (1997), Natarajan and Mohankumar (1993), Paszkowski (1988, 1991), Pichon (1989), and Sagar (1991a, b). …

5: Bibliography P

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Evaluation of Fermi-Dirac Integral. In Nonlinear Numerical Methods and Rational Approximation (Wilrijk, 1987), A. Cuyt (Ed.), Mathematics and Its Applications, Vol. 43, pp. 435–444.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Evaluation of the Fermi-Dirac integral of half-integer order. Zastos. Mat. 21 (2), pp. 289–301.

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External Links:

ISSN 0044-1899, MathReview (Alan M. Cohen), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Numerical calculation of the generalized Fermi-Dirac integrals. Comput. Phys. Comm. 55 (2), pp. 127–136.

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External Links:

Document

Referenced by:

§25.18(i).

Encodings:

BibTeX

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6: Bibliography T

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Uniform asymptotic approximation of Fermi-Dirac integrals. J. Comput. Appl. Math. 31 (3), pp. 383–387.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§25.12(iii).

Encodings:

BibTeX

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High Speed Numerical Integration of Fermi Dirac Integrals. Master’s Thesis, Naval Postgraduate School, Monterey, CA.

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External Links:

FullText

Referenced by:

§25.21(vii).

Encodings:

BibTeX

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Rational Chebyshev approximation for the Fermi-Dirac integral $F_{-3/2}(x)$ . Solid–State Electronics 41 (5), pp. 771–773.

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External Links:

Document

Referenced by:

§25.21(vii).

Encodings:

BibTeX

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7: 25.21 Software

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§25.21(vii) Fermi–Dirac and Bose–Einstein Integrals

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8: Bibliography N

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On the numerical evaluation of the generalised Fermi-Dirac integrals. Comput. Phys. Comm. 76 (1), pp. 48–50.

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External Links:

ISSN 0010-4655, Document, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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9: Bibliography F

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Generalized Fermi-Dirac integrals—FD, FDG, FDH. Comput. Phys. Comm. 39 (2), pp. 181–185.

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External Links:

ISSN 0010-4655, Document, Code, MathReview

Referenced by:

4th item.

Encodings:

BibTeX

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10: Bibliography S

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A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals. Comput. Phys. Comm. 66 (2-3), pp. 271–275.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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On the evaluation of the Fermi-Dirac integrals. Astrophys. J. 376 (1, part 1), pp. 364–366.

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External Links:

ISSN 0004-637X, Document, MathReview (De Hui Chen)

Referenced by:

§25.18(i).

Encodings:

BibTeX

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