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3j,6j,9j symbols

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11: 34.10 Zeros

§34.10 Zeros

… ►Similarly the

\mathit{6j}

symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four

\mathit{3j}

symbols in the summation. …However, the

\mathit{3j}

and

\mathit{6j}

symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. Such zeros are called nontrivial zeros. ►For further information, including examples of nontrivial zeros and extensions to

\mathit{9j}

symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).

12: 34.13 Methods of Computation

… ►Methods of computation for

\mathit{3j}

and

\mathit{6j}

symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). …

13: 16.7 Relations to Other Functions

… ►For

\mathit{3j}

,

\mathit{6j}

,

\mathit{9j}

symbols see Chapter 34. …

14: 16.4 Argument Unity

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

\mathit{3j}

symbols (Chapter 34). … ► …

15: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

►For large values of the parameters in the

\mathit{3j}

,

\mathit{6j}

, and

\mathit{9j}

symbols, different asymptotic forms are obtained depending on which parameters are large. … ►For approximations for the

\mathit{3j}

,

\mathit{6j}

, and

\mathit{9j}

symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.

16: 10.59 Integrals

… ►

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}

ⓘ

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

…

17: 10.54 Integral Representations

… ►

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.

ⓘ

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

►

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.

ⓘ

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

… ►

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.

ⓘ

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

…

18: 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

…

19: Software Index

… ► ►►►

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

…

20: 10.60 Sums

… ►

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).

ⓘ

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

… ►

10.60.11

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

ⓘ

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

… ►

10.60.12

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

ⓘ

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

►

10.60.13

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

ⓘ

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

►

10.60.14

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

ⓘ

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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