Einstein summation convention for vectors

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1: 1.6 Vectors and Vector-Valued Functions

§1.6 Vectors and Vector-Valued Functions

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Unit Vectors

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Einstein Summation Convention

►Much vector algebra involves summation over suffices of products of vector components. … ► …

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

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

… ►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)

3: 4.44 Other Applications

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

4: 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)). …

5: 29.19 Physical Applications

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

6: 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). …

7: 27.17 Other Applications

… ►Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.

8: 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. …

9: Bibliography B

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Integral representations and summations of the modified Struve function. Acta Math. Hungar. 141 (3), pp. 254–281.

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External Links:

ISSN 0236-5294, Document, FullText, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§11.7(v), §11.7(v).

Encodings:

BibTeX

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Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.

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External Links:

Document, MathReview (T. M. Apostol), ZentralBlatt

Referenced by:

§24.16(ii).

Encodings:

BibTeX

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Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§24.16(iii), §24.8(ii).

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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The Borel transform and its use in the summation of asymptotic expansions. Stud. Appl. Math. 105 (2), pp. 83–113.

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External Links:

ISSN 0022-2526, Document, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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10: 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–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 … ►

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

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