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11: 34.10 Zeros

§34.10 Zeros

►In a $\mathit{3}j$ symbol, if the three angular momenta $j_1,j_2,j_3$ do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the $\mathit{3}j$ symbol is zero. Similarly the $\mathit{6}j$ symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four $\mathit{3}j$ symbols in the summation. …However, the $\mathit{3}j$ and $\mathit{6}j$ symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. …

12: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

►For large values of the parameters in the $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols, different asymptotic forms are obtained depending on which parameters are large. … ►

34.8.1 $j_1,j_2,j_3\gg l_3\gg 1$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $o\left(x\right)$: order less than, $\sin z$: sine function, : $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $l,l_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §34.8 and Ch.34

… ►Uniform approximations in terms of Airy functions for the $\mathit{3}j$ and $\mathit{6}j$ symbols are given in Schulten and Gordon (1975b). For approximations for the $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.

13: 22.9 Cyclic Identities

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22.9.10 $$d_{1,3}^{(2)}{d}_{2,3}^{(2)}+d_{2,3}^{(2)}{d}_{3,3}^{(2)}+d_{3,3}^{(2)}{d}_{1,3}^{(2)}=d_{1,3}^{(4)}{d}_{2,3}^{(4)}+d_{2,3}^{(4)}{d}_{3,3}^{(4)}+d_{3,3}^{(4)}{d}_{1,3}^{(4)}=\kappa (\kappa +2).$$

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Symbols:

$d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Referenced by:

§22.9(ii), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10)

Permalink:

http://dlmf.nist.gov/22.9.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.0.25):

Originally this equation was written incorrectly for all $d_{m,p}^{(2)}$ and $d_{m,p}^{(4)}$ with $p=2$. It has been corrected by writing $p=3$ throughout.

Suggested 2019-11-07 by Juan Miguel Nieto

See also:

Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

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§22.9(iii) Typical Identities of Rank 3

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22.9.13 $$s_{1,3}^{(4)}{s}_{2,3}^{(4)}{s}_{3,3}^{(4)}=-\frac{1}{1-{\kappa }^2}\left(s_{1,3}^{(4)}+s_{2,3}^{(4)}+s_{3,3}^{(4)}\right),$$

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Symbols:

$s_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Permalink:

http://dlmf.nist.gov/22.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

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22.9.22 $$s_{1,3}^{(2)}{c}_{1,3}^{(2)}{d}_{2,3}^{(2)}{d}_{3,3}^{(2)}+s_{2,3}^{(2)}{c}_{2,3}^{(2)}{d}_{3,3}^{(2)}{d}_{1,3}^{(2)}+s_{3,3}^{(2)}{c}_{3,3}^{(2)}{d}_{1,3}^{(2)}{d}_{2,3}^{(2)}=\frac{{\kappa }^2+k^2-1}{1-{\kappa }^2}\left(s_{1,3}^{(2)}{c}_{1,3}^{(2)}+s_{2,3}^{(2)}{c}_{2,3}^{(2)}+s_{3,3}^{(2)}{c}_{3,3}^{(2)}\right),$$

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Symbols:

$k$: modulus, $s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity, $d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Permalink:

http://dlmf.nist.gov/22.9.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

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22.9.23 $$s_{1,3}^{(4)}{d}_{1,3}^{(4)}{c}_{2,3}^{(4)}{c}_{3,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}{c}_{3,3}^{(4)}{c}_{1,3}^{(4)}+s_{3,3}^{(4)}{d}_{3,3}^{(4)}{c}_{1,3}^{(4)}{c}_{2,3}^{(4)}=\frac{{\kappa }^2}{1-{\kappa }^2}\left(s_{1,3}^{(4)}{d}_{1,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}\right).$$

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Symbols:

$s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity, $d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Permalink:

http://dlmf.nist.gov/22.9.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

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14: 3 Numerical Methods

Chapter 3 Numerical Methods

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15: 4.43 Cubic Equations

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$A={\left(-\frac{4}{3}p\right)}^{1/2},$

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$B={\left(\frac{4}{3}p\right)}^{1/2}.$

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$A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=4q/A^3$, when $4p^3+27q^2\le 0$.

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$A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=-4q/A^3$, when , , and $4p^3+27q^2>0$.

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$B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=-4q/B^3$, when $p>0$.

…

16: 34.14 Tables

§34.14 Tables

►Tables of exact values of the squares of the $\mathit{3}j$ and $\mathit{6}j$ symbols in which all parameters are $\le 8$ are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols on pp. … ►Tables of $\mathit{3}j$ and $\mathit{6}j$ symbols in which all parameters are $\le 17/2$ are given in Appel (1968) to 6D. …Other tabulations for $\mathit{3}j$ symbols are listed on pp. … ►In Varshalovich et al. (1988) algebraic expressions for the Clebsch–Gordan coefficients with all parameters $\le 5$ and numerical values for all parameters $\le 3$ are given on pp. …

17: 7.3 Graphics

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Figure 7.3.1: Complementary error functions $\mathrm{erfc}x$ and $\mathrm{erfc}\left(10x\right)$, $-3\le x\le 3$. Magnify

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Figure 7.3.5: $|\mathrm{erf}\left(x+\mathrm{i}y\right)|$, $-3\le x\le 3$, $-3\le y\le 3$. … Magnify 3D Help

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Figure 7.3.6: $|\mathrm{erfc}\left(x+\mathrm{i}y\right)|$, $-3\le x\le 3$, $-3\le y\le 3$. … Magnify 3D Help

18: 23.5 Special Lattices

… ►Then $\mathrm{\Delta }>0$ and the parallelogram with vertices at $0$, $2{\omega }_1$, $2{\omega }_1+2{\omega }_3$, $2{\omega }_3$ is a rectangle. … ►Also, $e_2$ and $g_3$ have opposite signs unless ${\omega }_3=\mathrm{i}{\omega }_1$, in which event both are zero. ►As functions of $\mathrm{\Im }{\omega }_3$, $e_1$ and $e_2$ are decreasing and $e_3$ is increasing. … ►The parallelogram $0$, $2{\omega }_1$, $2{\omega }_1+2{\omega }_3$, $2{\omega }_3$ is a square, and … ►The parallelogram $0$, $2{\omega }_1-2{\omega }_3$, $2{\omega }_1$, $2{\omega }_3$, is a rhombus: see Figure 23.5.1. …

19: 34 3j, 6j, 9j Symbols

Chapter 34 $\mathit{3}j,\mathit{6}j,\mathit{9}j$ Symbols

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20: 9.10 Integrals

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${\displaystyle {\int }_0^{\mathrm{\infty }}}\mathrm{Ai}\left(t\right)dt=\frac{1}{3}$ ,

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9.10.14 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Ai}\left(t\right)dt={\mathrm{e}}^{-p^3/3}\left(\frac{1}{3}-\frac{p{}_1{}^{}F_1^{}(\frac{1}{3};\frac{4}{3};\frac{1}{3}{p}^3)}{3^{4/3}\mathrm{\Gamma }\left(\frac{4}{3}\right)}+\frac{p^2{}_1{}^{}F_1^{}(\frac{2}{3};\frac{5}{3};\frac{1}{3}{p}^3)}{3^{5/3}\mathrm{\Gamma }\left(\frac{5}{3}\right)}\right),$$ $p\in ℂ$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(v), §9.10 and Ch.9

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9.10.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Ai}\left(-t\right)dt=\frac{1}{3}{\mathrm{e}}^{p^3/3}\left(\frac{\mathrm{\Gamma }(\frac{1}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}+\frac{\mathrm{\Gamma }(\frac{2}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\right),$$ $\mathrm{\Re }p>0$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(v), §9.10 and Ch.9

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9.10.16 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Bi}\left(-t\right)dt=\frac{1}{\sqrt{3}}{\mathrm{e}}^{p^3/3}\left(\frac{\mathrm{\Gamma }(\frac{2}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}-\frac{\mathrm{\Gamma }(\frac{1}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right),$$ $\mathrm{\Re }p>0$.

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(v), §9.10 and Ch.9

… ►For further integrals, including the Airy transform, see §9.11(iv), Widder (1979), Prudnikov et al. (1990, §1.8.1), Prudnikov et al. (1992a, pp. 405–413), Prudnikov et al. (1992b, §4.3.25), Vallée and Soares (2010, Chapters 3, 4).