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

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

►The $\mathit{9}j$ symbol may be defined either in terms of $\mathit{3}j$ symbols or equivalently in terms of $\mathit{6}j$ symbols: ►

34.6.1

ⓘ

Defines:

: $\mathit{9}j$ symbol

Symbols:

: $\mathit{3}j$ 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

ⓘ

Symbols:

: $\mathit{9}j$ symbol, : $\mathit{6}j$ 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 $\mathit{9}j$ symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

2: 34.7 Basic Properties: $\mathit{9}j$ Symbol

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

… ►

§34.7(ii) Symmetry

… ►Odd permutations of columns or rows introduce a phase factor ${(-1)}^R$, where $R$ is the sum of all arguments of the $\mathit{9}j$ symbol. … ►

§34.7(iv) Orthogonality

… ►

§34.7(vi) Sums

…

3: 12 Parabolic Cylinder Functions

Chapter 12 Parabolic Cylinder Functions

…

4: 4.17 Special Values and Limits

… ►

Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

► ►►►

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
…
$\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$	$\sqrt{2}(\sqrt{3}+1)$	$\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$
…

►

►

4.17.1 $$\underset{z\to 0}{lim}\frac{\sin z}{z}=1,$$

ⓘ

Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.74

Permalink:

http://dlmf.nist.gov/4.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

►

4.17.2 $$\underset{z\to 0}{lim}\frac{\tan z}{z}=1.$$

ⓘ

Symbols:

$\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.75

Permalink:

http://dlmf.nist.gov/4.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

►

4.17.3 $$\underset{z\to 0}{lim}\frac{1-\cos z}{z^2}=\frac{1}{2}.$$

ⓘ

Symbols:

$\cos z$: cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

5: 32.3 Graphics

… ►

►►

Figure 32.3.1: $w_k(x)$ for $-12\le x\le 1.33$ and $k=0.5$, $0.75$, $1$, $1.25$, and the parabola $6w^2+x=0$, shown in black. Magnify

… ►with $\beta =0$, $\alpha =2\nu +1$, and … ►

►►

Figure 32.3.7: $u_k(x;-\frac{1}{2})$ for $-12\le x\le 4$ with $k=\mathrm{0.33554\hspace{0.33em}691}$, $\mathrm{0.33554\hspace{0.33em}692}$. … Magnify

►

►►

Figure 32.3.8: $u_k(x;\frac{1}{2})$ for $-12\le x\le 4$ with $k=0.47442$, $0.47443$. …The curves $u^2+\frac{1}{3}x\pm \frac{1}{6}\sqrt{x^2+12}=0$ are shown in green and black, respectively. Magnify

…

6: 34.9 Graphical Method

§34.9 Graphical Method

… ►For specific examples of the graphical method of representing sums involving the $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

7: 10.48 Graphs

… ►

►►

Figure 10.48.1: ${𝗃}_n\left(x\right)$, $n=0(1)4$, $0\le x\le 12$. Magnify

►

►►

Figure 10.48.2: ${𝗒}_n\left(x\right)$, $n=0(1)4$, . Magnify

►

►►

Figure 10.48.3: ${𝗃}_5\left(x\right)$, ${𝗒}_5\left(x\right)$, $\sqrt{{𝗃}_5^2\left(x\right)+{𝗒}_5^2\left(x\right)}$, $0\le x\le 12$. Magnify

►

►►

Figure 10.48.4: ${𝗃}_5^{\prime }\left(x\right)$, ${𝗒}_5^{\prime }\left(x\right)$, $\sqrt{𝗃_5^{\prime }{}_{}{}^2\left(x\right)+𝗒_5^{\prime }{}_{}{}^2\left(x\right)}$, $0\le x\le 12$. Magnify

… ►

►►

Figure 10.48.6: ${𝗂}_1^{(1)}\left(x\right),{𝗂}_1^{(2)}\left(x\right),{𝗄}_1\left(x\right)$, $0\le x\le 4$. Magnify

…

8: 28.16 Asymptotic Expansions for Large $q$

… ►Let $s=2m+1$, $m=0,1,2,\mathrm{\dots }$, and $\nu$ be fixed with . … ►

28.16.1 $${\lambda }_{\nu }\left(h^2\right)\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

ⓘ

Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.16 and Ch.28

…

9: 34.1 Special Notation

… ►The main functions treated in this chapter are the Wigner $\mathit{3}j,\mathit{6}j,\mathit{9}j$ symbols, respectively, … ►

►An often used alternative to the $\mathit{3}j$ symbol is the Clebsch–Gordan coefficient …see Edmonds (1974, p. 46, Eq. (3.7.3)) and Rotenberg et al. (1959, p. 1, Eq. (1.1a)). For other notations for $\mathit{3}j$, $\mathit{6}j$, $\mathit{9}j$ symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).

10: 18.41 Tables

… ►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates $T_n\left(x\right)$, $U_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ for $n=0(1)12$. 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 $H_n\left(x\right)$. …