Einstein functions

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1: 4.44 Other Applications

… ►The Einstein functions and Planck’s radiation function are elementary combinations of exponentials, or exponentials and logarithms. …

2: 29.19 Physical Applications

… ►Bronski et al. (2001) uses Lamé functions in the theory of Bose–Einstein condensates. …

3: 25.17 Physical Applications

… ►See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). ►The zeta function arises in the calculation of the partition function of ideal quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the critical gas temperature and density for the Bose–Einstein condensation phase transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)). …

4: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate

Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate

… ►Jacobian elliptic functions arise as solutions to certain nonlinear Schrödinger equations, which model many types of wave propagation phenomena. Among these are the formation of vortex rings in Bose Einstein condensates. …For details see the NIST news item Decay of a dark soliton into vortex rings in a Bose–Einstein condensate. … ►Cornell, Watching Dark Solitons Decay into Vortex Rings in a Bose–Einstein Condensate, Phys. Rev. Lett. 86, 2926–2929 (2001)

5: William P. Reinhardt

… ►He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions. …

6: 25.18 Methods of Computation

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§25.18(i) Function Values and Derivatives

… ►For the Hurwitz zeta function $\zeta (s,a)$ see Spanier and Oldham (1987, p. 653) and Coffey (2009). ►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). ►

§25.18(ii) Zeros

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7: 25.12 Polylogarithms

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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 $s>0$, $x\le 0$,

ⓘ

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–Einstein integral, 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 …

8: 1.6 Vectors and Vector-Valued Functions

§1.6 Vectors and Vector-Valued Functions

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Einstein Summation Convention

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§1.6(iii) Vector-Valued Functions

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9: Bernard Deconinck

… ►He has worked on integrable systems, algorithms for computations with Riemann surfaces, Bose-Einstein condensates, and methods to investigate the stability of solutions of nonlinear wave equations. He is the coauthor of several Maple commands to work with Riemann surfaces and the command to compute multidimensional theta functions numerically. … ►

21 Multidimensional Theta Functions

►Deconinck served as a Validator for the original release and publication in May 2010 of the NIST Digital Library of Mathematical Functions and the NIST Handbook of Mathematical Functions. …

10: Software Index

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	Open Source													With Book						Commercial
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25.21(vii) Fermi–Dirac, Bose–Einstein					✓									✓			✓					✓
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …