3j symbols

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21: Leonard C. Maximon

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34 3j, 6j, 9j Symbols

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22: Bibliography R

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C. C. J. Roothaan and S. Lai (1997) Calculation of

3n

j

symbols by Labarthe’s method. International Journal of Quantum Chemistry 63 (1), pp. 57–64.

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External Links:: Document
Referenced by:: §34.13.
Encodings:: BibTeX

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M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr. (1959) The

3

j

and

6

j

Symbols. The Technology Press, MIT, Cambridge, MA.

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External Links:: ISBN 0-262-18004-9
Referenced by:: (34.1.1), §34.1, §34.1, §34.14, Erratum (V1.0.14) for Section 34.1, Erratum (V1.0.15) for Section 34.1.
Encodings:: BibTeX

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23: 18.41 Tables

… ►The ranges of

x

are

0.2(.2)1

for

T_{n}\left(x\right)

and

U_{n}\left(x\right)

, and

0.5,1,3,5,10

for

L_{n}\left(x\right)

and

H_{n}\left(x\right)

. …

24: Bibliography M

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M. Micu (1968) Recursion relations for the

3

j

symbols. Nuclear Physics A 113 (1), pp. 215–220.

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External Links:: Document
Referenced by:: §34.3(iii).
Encodings:: BibTeX

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25: 16.7 Relations to Other Functions

… ►For

\mathit{3j}

\mathit{6j}

\mathit{9j}

symbols see Chapter 34. …

26: 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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27: Bibliography L

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S. Lai and Y. Chiu (1990) Exact computation of the

3

j

and

6

j

symbols. Comput. Phys. Comm. 61 (3), pp. 350–360.

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Notes:: A Fortran program is included.
External Links:: Document, ZentralBlatt
Referenced by:: §34.15.
Encodings:: BibTeX

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S. Lai and Y. Chiu (1992) Exact computation of the

9

j

symbols. Comput. Phys. Comm. 70 (3), pp. 544–556.

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Notes:: Double-precision Fortran
External Links:: ISSN 0010-4655, Document, Code, MathReview
Referenced by:: 7th item.
Encodings:: BibTeX

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J. H. Luscombe and M. Luban (1998) Simplified recursive algorithm for Wigner

3j

and

6j

symbols. Phys. Rev. E 57 (6), pp. 7274–7277.

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External Links:: Document
Referenced by:: §34.13, §34.3(iii).
Encodings:: BibTeX

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28: 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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29: 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

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30: Bibliography S

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J. Shapiro (1970) Arbitrary

3n-j

symbols for

\mathrm{SU}(2)

. Comput. Phys. Comm. 1 (3), pp. 207–215.

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Notes:: Single-precision Fortran
External Links:: Document, Code
Referenced by:: 9th item.
Encodings:: BibTeX

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