Einstein summation convention for vectors

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11: 25.21 Software

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

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12: 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. …

13: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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External Links:

ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

Encodings:

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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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Multilateral summation theorems for ordinary and basic hypergeometric series in $\mathrm{U}(n)$ . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.

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External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§17.15.

Encodings:

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14: 34.3 Basic Properties: $\mathit{3}j$ Symbol

… ►In the summations (34.3.16)–(34.3.18) the summation variables range over all values that satisfy the conditions given in (34.2.1)–(34.2.3). Similar conventions apply to all subsequent summations in this chapter. …

15: 35.4 Partitions and Zonal Polynomials

… ►A partition $\kappa =(k_1,\mathrm{\dots },k_m)$ is a vector of nonnegative integers, listed in nonincreasing order. Also, $|\kappa |$ denotes $k_1+\mathrm{\cdots }+k_m$, the weight of $\kappa$; $\mathrm{\ell }(\kappa )$ denotes the number of nonzero $k_j$; $a+\kappa$ denotes the vector $(a+k_1,\mathrm{\dots },a+k_m)$. … ►

35.4.4

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Symbols:

$Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝟎$: $m\times m$ matrix of zeros and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

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35.4.5 $$Z_{\kappa }\left(𝐇𝐓{𝐇}^{-1}\right)=Z_{\kappa }\left(𝐓\right),$$ $𝐇\in 𝐎(m)$.

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Symbols:

$\in$: element of, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐓$: real symmetric $m\times m$ matrix, $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E5

Encodings:

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See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

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Summation

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16: Bibliography D

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Introduction to Vector Analysis. 5th edition, Allyn and Bacon Inc., Boston, MA.

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Notes:

Seventh edition printed by WCB/McGraw-Hill, 1995.

External Links:

ISBN 0-697-06356-9; 0-697-16099-8, MathReview (R. H. Bowman), ZentralBlatt

Referenced by:

§1.5(ii).

Encodings:

BibTeX

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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:

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Vector coupling coefficients as products of prime factors. Comput. Phys. Comm. 4 (2), pp. 268–274.

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Notes:

Single-precision Fortran

External Links:

Document, Code

Referenced by:

4th item.

Encodings:

BibTeX

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17: Bibliography L

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The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity. Methuen and Co., Ltd., London.

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Notes:

Translated from Das Relativitätsprinzip by W. Perrett and G. B. Jeffery. Reprinted by Dover Publications Inc., New York, 1952.

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.6(ii).

Encodings:

BibTeX

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VOIGTL – A fast subroutine for Voigt function evaluation on vector processors. Comput. Phys. Comm. 75 (1-2), pp. 135–142.

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Notes:

Single-precision Fortran, minimum accuracy 13D.

External Links:

Document, Code, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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18: 3.11 Approximation Techniques

… ►When $n>0$ and $0\le j\le n$, $0\le k\le n$, … ►Here the single prime on the summation symbol means that the first term is to be halved. … ►

Summation of Chebyshev Series: Clenshaw’s Algorithm

… ► … ►The pair of vectors $\{𝐟,𝐚\}$ …

19: 18.20 Hahn Class: Explicit Representations

… ►Here we use as convention for (16.2.1) with $b_q=-N$, $a_1=-n$, and $n=0,1,\mathrm{\dots },N$ that the summation on the right-hand side ends at $k=n$. …

20: 5.19 Mathematical Applications

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§5.19(i) Summation of Rational Functions

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