Bose–Einstein integral

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1: 25.12 Polylogarithms

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§25.12(iii) Fermi–Dirac and Bose–Einstein Integrals

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

25.12.15 $$G_s(x)=\frac{1}{\mathrm{\Gamma }\left(s+1\right)}{\int }_0^{\mathrm{\infty }}\frac{t^s}{{\mathrm{e}}^{t-x}-1}dt,$$ $s>-1$, ; or $s>0$, $x\le 0$,

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Defines:

$G_s(x)$: Bose–Einstein 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:

Bose–Einsteinintegral, definition, improper integral, integral representation

Source:

Dingle (1957a, p. 240)

Referenced by:

(25.12.16)

Permalink:

http://dlmf.nist.gov/25.12.E15

Encodings:

pMML, png, TeX

See also:

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

… ►In terms of polylogarithms … ►

$G_s(x)={\mathrm{Li}}_{s+1}\left({\mathrm{e}}^x\right).$

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2: 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). …

3: 25.21 Software

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

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4: Bibliography G

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On the computation of generalized Fermi-Dirac and Bose-Einstein integrals. Comput. Phys. Comm. 74 (2), pp. 233–238.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§25.12(iii), §25.18(i), 5th item.

Encodings:

BibTeX

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5: Bibliography D

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The Bose-Einstein integrals ${ℬ}_p(\eta )={(p!)}^{-1}{\int }_0^{\mathrm{\infty }}{ϵ}^p{(e^{ϵ-\eta }-1)}^{-1}𝑑ϵ$ . Appl. Sci. Res. B. 6, pp. 240–244.

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External Links:

ISSN 0003-6994, MathReview Entry, ZentralBlatt

Referenced by:

(25.12.15), §25.12(iii), §25.12.

Encodings:

BibTeX

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6: Software Index

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	Open Source													With Book						Commercial
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6 Exponential, Logarithmic, Sine, and Cosine Integrals
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11.16(iii) Integrals of …														✓					✓		✓	✓	a
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11.16(vi) Integrals of …														✓							✓	✓	a
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19 Elliptic Integrals
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25.21(vii) Fermi–Dirac, Bose–Einstein					✓									✓			✓					✓
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7: Bibliography B

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A note on integrals involving parabolic cylinder functions. SIAM J. Appl. Math. 16 (1), pp. 71–74.

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External Links:

Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12.

Encodings:

BibTeX

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Numerical calculation of a generalized complete elliptic integral. Rev. Mod. Phys. 10, pp. 264–269.

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External Links:

Document, ZentralBlatt

Referenced by:

§19.2(iii).

Encodings:

BibTeX

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Normalization integrals of orthogonal Heun functions. J. Math. Phys. 38 (7), pp. 3692–3699.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§31.9(i).

Encodings:

BibTeX

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Stability of repulsive Bose-Einstein condensates in a periodic potential. Phys. Rev. E (3) 63 (036612), pp. 1–11.

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External Links:

Document

Referenced by:

§29.19(i).

Encodings:

BibTeX

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On a generalization of Barton’s integral and related integrals of complete elliptic integrals. Math. Proc. Cambridge Philos. Soc. 101 (1), pp. 1–5.

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External Links:

ISSN 0305-0041, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§19.13(i).

Encodings:

BibTeX

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