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##### 1: 34.2 Definition: $\mathit{3j}$ Symbol
###### §34.2 Definition: $\mathit{3j}$ Symbol
The quantities $j_{1},j_{2},j_{3}$ in the $\mathit{3j}$ symbol are called angular momenta. …They therefore satisfy the triangle conditions …where $r,s,t$ is any permutation of $1,2,3$. The corresponding projective quantum numbers $m_{1},m_{2},m_{3}$ are given by …
##### 2: 4.17 Special Values and Limits
4.17.1 $\lim_{z\to 0}\frac{\sin z}{z}=1,$
4.17.2 $\lim_{z\to 0}\frac{\tan z}{z}=1.$
4.17.3 $\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}.$
##### 3: 34.3 Basic Properties: $\mathit{3j}$ Symbol
###### §34.3 Basic Properties: $\mathit{3j}$ Symbol
When any one of $j_{1},j_{2},j_{3}$ is equal to $0,\tfrac{1}{2}$, or $1$, the $\mathit{3j}$ symbol has a simple algebraic form. …For these and other results, and also cases in which any one of $j_{1},j_{2},j_{3}$ is $\frac{3}{2}$ or $2$, see Edmonds (1974, pp. 125–127). …
###### §34.3(ii) Symmetry
Even permutations of columns of a $\mathit{3j}$ symbol leave it unchanged; odd permutations of columns produce a phase factor $(-1)^{j_{1}+j_{2}+j_{3}}$, for example, …
##### 4: 34.4 Definition: $\mathit{6j}$ Symbol
###### §34.4 Definition: $\mathit{6j}$ Symbol
The $\mathit{6j}$ symbol is defined by the following double sum of products of $\mathit{3j}$ symbols: …where the summation is taken over all admissible values of the $m$’s and $m^{\prime}$’s for each of the four $\mathit{3j}$ symbols; compare (34.2.2) and (34.2.3). Except in degenerate cases the combination of the triangle inequalities for the four $\mathit{3j}$ symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths $j_{1},j_{2},j_{3},l_{1},l_{2},l_{3}$; see Figure 34.4.1. … where ${{}_{4}F_{3}}$ is defined as in §16.2. …
##### 5: 34.1 Special Notation
The main functions treated in this chapter are the Wigner $\mathit{3j},\mathit{6j},\mathit{9j}$ symbols, respectively,
$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix},$
An often used alternative to the $\mathit{3j}$ 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 $\mathit{3j}$, $\mathit{6j}$, $\mathit{9j}$ symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).
##### 6: 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. … and the symbol $o\left(1\right)$ denotes a quantity that tends to zero as the parameters tend to infinity, as in §2.1(i). … Uniform approximations in terms of Airy functions for the $\mathit{3j}$ and $\mathit{6j}$ symbols are given in Schulten and Gordon (1975b). 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.
##### 7: 34.5 Basic Properties: $\mathit{6j}$ Symbol
If any lower argument in a $\mathit{6j}$ symbol is $0$, $\tfrac{1}{2}$, or $1$, then the $\mathit{6j}$ symbol has a simple algebraic form. …
34.5.6 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}+1\end{Bmatrix}=(-1)^{J}\left(\frac{(J-2j_{2}-1)(J-2j_{2})(J-2j% _{3}+1)(J-2j_{3}+2)}{(2j_{2}+1)(2j_{2}+2)(2j_{2}+3)(2j_{3}-1)2j_{3}(2j_{3}+1)}% \right)^{\frac{1}{2}},$
34.5.9 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\begin{Bmatrix}j_{1}&\frac{1}{2}(j_{2}+l_{2}+j_% {3}-l_{3})&\frac{1}{2}(j_{2}-l_{2}+j_{3}+l_{3})\\ l_{1}&\frac{1}{2}(j_{2}+l_{2}-j_{3}+l_{3})&\frac{1}{2}(-j_{2}+l_{2}+j_{3}+l_{3% })\end{Bmatrix},$
34.5.14 $\sum_{j_{3}}(2j_{3}+1)(2l_{3}+1)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l^{\prime}_{3}\end{Bmatrix}=\delta_{l_{3},l^{\prime}_{3}}.$
34.5.16 $(-1)^{j_{1}+j_{2}+j_{3}+j_{1}^{\prime}+j_{2}^{\prime}+l_{1}+l_{2}}\begin{% Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}\begin{Bmatrix}j_{1}^{\prime}&j_{2}^{\prime}&j_{% 3}\\ l_{1}&l_{2}&l_{3}^{\prime}\end{Bmatrix}=\sum_{j}(-1)^{l_{3}+l_{3}^{\prime}+j}(% 2j+1)\begin{Bmatrix}j_{1}&j_{1}^{\prime}&j\\ j_{2}^{\prime}&j_{2}&j_{3}\end{Bmatrix}\begin{Bmatrix}l_{3}&l_{3}^{\prime}&j\\ j_{1}^{\prime}&j_{1}&l_{2}\end{Bmatrix}\begin{Bmatrix}l_{3}&l_{3}^{\prime}&j\\ j_{2}^{\prime}&j_{2}&l_{1}\end{Bmatrix}.$
##### 8: 22.9 Cyclic Identities
Throughout this subsection $m$ and $p$ are positive integers with $1\leq m\leq p$. … In this subsection $1\leq m\leq p$ and $1\leq n\leq p$. … These identities are cyclic in the sense that each of the indices $m,n$ in the first product of, for example, the form $s_{m,p}^{(4)}s_{n,p}^{(4)}$ are simultaneously permuted in the cyclic order: $m\to m+1\to m+2\to\cdots p\to 1\to 2\to\cdots m-1$; $n\to n+1\to n+2\to\cdots p\to 1\to 2\to\cdots n-1$. …
###### §22.9(iii) Typical Identities of Rank 3
22.9.23 $s_{1,3}^{(4)}d_{1,3}^{(4)}c_{2,3}^{(4)}c_{3,3}^{(4)}+s_{2,3}^{(4)}d_{2,3}^{(4)% }c_{3,3}^{(4)}c_{1,3}^{(4)}+s_{3,3}^{(4)}d_{3,3}^{(4)}c_{1,3}^{(4)}c_{2,3}^{(4% )}=\frac{\kappa^{2}}{1-\kappa^{2}}\left(s_{1,3}^{(4)}d_{1,3}^{(4)}+s_{2,3}^{(4% )}d_{2,3}^{(4)}+s_{2,3}^{(4)}d_{2,3}^{(4)}\right).$
##### 9: 34.10 Zeros
###### §34.10 Zeros
In a $\mathit{3j}$ 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 $\mathit{3j}$ symbol is zero. 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. …
##### 10: 4.43 Cubic Equations
$A=\left(-\tfrac{4}{3}p\right)^{1/2},$
$B=\left(\tfrac{4}{3}p\right)^{1/2}.$
• (a)

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

• (b)

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

• (c)

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