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1: 34.6 Definition: $\mathit{9j}$ Symbol

§34.6 Definition: $\mathit{9j}$ Symbol

►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►

34.6.2

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.13
Permalink:: http://dlmf.nist.gov/34.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►The

\mathit{9j}

symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

3: Staff

… ►

Frank W. J. Olver, University of Maryland and NIST, Chaps. 1, 2, 4, 9, 10

►

Richard B. Paris, University of Abertay, Chaps. 8, 11

… ►

Hans Volkmer, University of Wisconsin, Milwaukee, Chaps. 29, 30

… ►

Richard B. Paris, University of Abertay Dundee, for Chaps. 8, 11 (deceased)

… ►

Hans Volkmer, University of Wisconsin–Milwaukee, for Chaps. 29, 30

…

B. R. Miller and A. Youssef (2003) Technical Aspects of the Digital Library of Mathematical Functions, Annals of Mathematics and Artificial Intelligence—Special Issue on Mathematical Knowledge Management, Vol. 38, Nos. 1–3, pp. 121–136.

►

B. V. Saunders and Q. Wang (2005) Boundary/Contour Fitted Grid Generation for Effective Visualizations in a Digital Library of Mathematical Functions, Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, June 11–18, 2005. pp. 61–71.

►

Q. Wang and B. V. Saunders (2005) Web-Based 3D Visualization in a Digital Library of Mathematical Functions, Proceedings of the Web3D Symposium, Bangor, UK, March 29–April 1, 2005.

►

B. V. Saunders and Q. Wang (2006) From B-Spline Mesh Generation to Effective Visualizations for the NIST Digital Library of Mathematical Functions, in Curve and Surface Design, Proceedings of the Sixth International Conference on Curves and Surfaces, Avignon, France June 29–July 5, 2006, pp. 235–243.

…

5: 26.2 Basic Definitions

… ►

Table 26.2.1: Partitions

p\left(n\right)

► ►►►►►►

$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
…
6	11	23	1255	40	37338
…
9	30	26	2436	43	63261
…
11	56	28	3718	45	89134
12	77	29	4565	46	1 05558
…

►

6: 34.7 Basic Properties: $\mathit{9j}$ Symbol

§34.7 Basic Properties: $\mathit{9j}$ Symbol

… ►

§34.7(ii) Symmetry

… ►

§34.7(iv) Orthogonality

… ►

§34.7(vi) Sums

… ►It constitutes an addition theorem for the

\mathit{9j}

symbol. …

7: 26.9 Integer Partitions: Restricted Number and Part Size

… ►

Table 26.9.1: Partitions

p_{k}\left(n\right)

► ►►►►►►

$n$	$k$
$n$	0	1	2	3	4	5	6	7	8	9	10
…
6	0	1	4	7	9	10	11	11	11	11	11
7	0	1	4	8	11	13	14	15	15	15	15
…
9	0	1	5	12	18	23	26	28	29	30	30
…

► …

8: 11 Struve and Related Functions

Chapter 11 Struve and Related Functions

…

9: 29 Lamé Functions

Chapter 29 Lamé Functions

…

10: 34.9 Graphical Method

§34.9 Graphical Method

… ►For specific examples of the graphical method of representing sums involving the

\mathit{3j},\mathit{6j}

, and

\mathit{9j}

symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

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1—10 of 221 matching pages

1: 34.6 Definition: $\mathit{9j}$ Symbol

§34.6 Definition: $\mathit{9j}$ Symbol

2: 10 Bessel Functions

3: Staff

4: Publications

5: 26.2 Basic Definitions

6: 34.7 Basic Properties: $\mathit{9j}$ Symbol

§34.7 Basic Properties: $\mathit{9j}$ Symbol

§34.7(ii) Symmetry

§34.7(iv) Orthogonality

§34.7(vi) Sums

7: 26.9 Integer Partitions: Restricted Number and Part Size

8: 11 Struve and Related Functions

Chapter 11 Struve and Related Functions

9: 29 Lamé Functions

Chapter 29 Lamé Functions

10: 34.9 Graphical Method

§34.9 Graphical Method

$n$	$k$
$n$	0	1	2	3	4	5	6	7	8	9	10
…
6	0	1	4	7	9	10	11	11	11	11	11
7	0	1	4	8	11	13	14	15	15	15	15
…
9	0	1	5	12	18	23	26	28	29	30	30
…

$n$	$k$
$n$	0	1	2	3	4	5	6	7	8	9	10
…
6	0	1	4	7	9	10	11	11	11	11	11
7	0	1	4	8	11	13	14	15	15	15	15
…
9	0	1	5	12	18	23	26	28	29	30	30
…

.世界杯体彩串买『网址:mxsty.cc』.陪看世界杯女孩.m6q3s2-2022年11月29日7时0分9秒.字符>wcamcs

1—10 of 221 matching pages

1: 34.6 Definition: 9 ⁢ j Symbol

§34.6 Definition: 9 ⁢ j Symbol

2: 10 Bessel Functions

3: Staff

4: Publications

5: 26.2 Basic Definitions

6: 34.7 Basic Properties: 9 ⁢ j Symbol

§34.7 Basic Properties: 9 ⁢ j Symbol

§34.7(ii) Symmetry

§34.7(iv) Orthogonality

§34.7(vi) Sums

7: 26.9 Integer Partitions: Restricted Number and Part Size

8: 11 Struve and Related Functions

Chapter 11 Struve and Related Functions

9: 29 Lamé Functions

Chapter 29 Lamé Functions

10: 34.9 Graphical Method

§34.9 Graphical Method

1: 34.6 Definition: $\mathit{9j}$ Symbol

§34.6 Definition: $\mathit{9j}$ Symbol

6: 34.7 Basic Properties: $\mathit{9j}$ Symbol

§34.7 Basic Properties: $\mathit{9j}$ Symbol

$n$	$k$
$n$	0	1	2	3	4	5	6	7	8	9	10
…
6	0	1	4	7	9	10	11	11	11	11	11
7	0	1	4	8	11	13	14	15	15	15	15
…
9	0	1	5	12	18	23	26	28	29	30	30
…