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21: Bibliography F

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J. P. M. Flude (1998) The Edmonds asymptotic formulas for the

3j

and

6j

symbols. J. Math. Phys. 39 (7), pp. 3906–3915.

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External Links:: ISSN 0022-2488, Document, MathReview (A. J. Coleman), ZentralBlatt
Referenced by:: §34.8.
Encodings:: BibTeX

…

22: 34.7 Basic Properties: $\mathit{9j}$ Symbol

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34.7.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{13}\\ j_{31}&j_{31}&0\end{Bmatrix}=\frac{(-1)^{j_{12}+j_{21}+j_{13}+j_{31}}}{((2j_{1% 3}+1)(2j_{31}+1))^{\frac{1}{2}}}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{22}&j_{21}&j_{31}\end{Bmatrix}.

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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.
Permalink:: http://dlmf.nist.gov/34.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.7(i), §34.7 and Ch.34

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34.7.5

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

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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.
Permalink:: http://dlmf.nist.gov/34.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §34.7(vi), §34.7 and Ch.34

23: 10.59 Integrals

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10.59.1

\int_{-\infty}^{\infty}e^{ibt}\mathsf{j}_{n}\left(t\right)\,\mathrm{d}t=\begin% {cases}\pi i^{n}P_{n}\left(b\right),&-1<b<1,\\ \frac{1}{2}\pi(\pm\mathrm{i})^{n},&b=\pm 1,\\ 0,&\pm b>1,\end{cases}

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Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind and $n$ : integer
A&S Ref:: 11.4.26
Referenced by:: §10.59
Permalink:: http://dlmf.nist.gov/10.59.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.59 and Ch.10

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24: 34.3 Basic Properties: $\mathit{3j}$ Symbol

… ►Similar conventions apply to all subsequent summations in this chapter. …

25: 10.54 Integral Representations

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10.54.1

\mathsf{j}_{n}\left(z\right)=\frac{z^{n}}{2^{n+1}n!}\int_{0}^{\pi}\cos\left(z% \cos\theta\right)(\sin\theta)^{2n+1}\,\mathrm{d}\theta.

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.13
Referenced by:: §10.54
Permalink:: http://dlmf.nist.gov/10.54.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.54 and Ch.10

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10.54.2

\mathsf{j}_{n}\left(z\right)=\frac{(-i)^{n}}{2}\int_{0}^{\pi}e^{iz\cos\theta}P% _{n}\left(\cos\theta\right)\sin\theta\,\mathrm{d}\theta.

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Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.14
Permalink:: http://dlmf.nist.gov/10.54.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.54 and Ch.10

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10.54.4

\mathsf{j}_{n}\left(z\right)=\frac{(-i)^{n+1}}{2\pi}\int_{i\infty}^{(-1+,1+)}e% ^{izt}Q_{n}\left(t\right)\,\mathrm{d}t,

|\operatorname{ph}z|<\tfrac{1}{2}\pi.

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, $\operatorname{ph}$ : phase, $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.54.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §10.54 and Ch.10

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26: 16.4 Argument Unity

… ►See Raynal (1979) for a statement in terms of

\mathit{3j}

symbols (Chapter 34). … ►These series contain

\mathit{6j}

symbols as special cases when the parameters are integers; compare §34.4. …

27: Software Index

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	Open Source													With Book						Commercial
…
34 3j, 6j, 9j Symbols
…

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28: 10.60 Sums

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10.60.2

\frac{\sin w}{w}=\sum_{n=0}^{\infty}(2n+1)\mathsf{j}_{n}\left(v\right)\mathsf{% j}_{n}\left(u\right)P_{n}\left(\cos\alpha\right).

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Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind and $n$ : integer
A&S Ref:: 10.1.45
Referenced by:: §10.60(iii)
Permalink:: http://dlmf.nist.gov/10.60.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.60(i), §10.60 and Ch.10

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10.60.11

\sum_{n=0}^{\infty}{\mathsf{j}_{n}}^{2}\left(z\right)=\frac{\operatorname{Si}% \left(2z\right)}{2z}.

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Symbols:: $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.52
Referenced by:: §10.60(iii), Ch.10
Permalink:: http://dlmf.nist.gov/10.60.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §10.60(iii), §10.60 and Ch.10

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10.60.12

\sum_{n=0}^{\infty}(2n+1){\mathsf{j}_{n}}^{2}\left(z\right)=1,

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Symbols:: $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.50
Referenced by:: §10.60(iii)
Permalink:: http://dlmf.nist.gov/10.60.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §10.60(iii), §10.60 and Ch.10

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10.60.13

\sum_{n=0}^{\infty}(-1)^{n}(2n+1){\mathsf{j}_{n}}^{2}\left(z\right)=\frac{\sin% \left(2z\right)}{2z},

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Symbols:: $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.51
Referenced by:: §10.60(iii)
Permalink:: http://dlmf.nist.gov/10.60.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §10.60(iii), §10.60 and Ch.10

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10.60.14

\sum_{n=0}^{\infty}(2n+1)(\mathsf{j}_{n}'\left(z\right))^{2}=\tfrac{1}{3}.

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Symbols:: $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
Referenced by:: §10.60(iii)
Permalink:: http://dlmf.nist.gov/10.60.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §10.60(iii), §10.60 and Ch.10

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29: Bibliography

… ►

H. Appel (1968) Numerical Tables for Angular Correlation Computations in

\alpha

-,

\beta

- and

\gamma

-Spectroscopy:

3j

-,

6j

-,

9j

-Symbols, F- and

\Gamma

-Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.

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External Links:: ISBN 978-3-540-04218-1
Referenced by:: §34.14.
Encodings:: BibTeX

…

30: Bibliography G

… ►

P. Gianni, M. Seppälä, R. Silhol, and B. Trager (1998) Riemann surfaces, plane algebraic curves and their period matrices. J. Symbolic Comput. 26 (6), pp. 789–803.

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External Links:: ISSN 0747-7171, Document, MathReview (Gabino González Díez), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

J. N. Ginocchio (1991) A new identity for some six-

j

symbols. J. Math. Phys. 32 (6), pp. 1430–1432.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview
Referenced by:: §34.5(vi).
Encodings:: BibTeX

…

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