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1: 34.4 Definition: $\mathit{6}j$ Symbol

§34.4 Definition: $\mathit{6}j$ Symbol

►The $\mathit{6}j$ symbol is defined by the following double sum of products of $\mathit{3}j$ symbols: … ►The $\mathit{6}j$ symbol can be expressed as the finite sum … ►where ${}_4{}^{}F_3^{}$ is defined as in §16.2. ►For alternative expressions for the $\mathit{6}j$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${}_4{}^{}F_3^{}$ of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).

2: 34.2 Definition: $\mathit{3}j$ Symbol

§34.2 Definition: $\mathit{3}j$ Symbol

►The quantities $j_1,j_2,j_3$ in the $\mathit{3}j$ symbol are called angular momenta. …They therefore satisfy the triangle conditions …The corresponding projective quantum numbers $m_1,m_2,m_3$ are given by … ►When both conditions are satisfied the $\mathit{3}j$ symbol can be expressed as the finite sum …

3: 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. … ►Biedenharn and Louck (1981) give tables of algebraic expressions for Clebsch–Gordan coefficients and $\mathit{6}j$ symbols, together with a bibliography of tables produced prior to 1975. … 270–289; similar tables for the $\mathit{6}j$ symbols are given on pp. …

4: 18.41 Tables

§18.41 Tables

… ►For $P_n\left(x\right)$ ($={𝖯}_n\left(x\right)$) see §14.33. ►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 $n=0(1)12$. … ►See also Abramowitz and Stegun (1964, Tables 25.4, 25.9, and 25.10). ►

§18.41(iii) Other Tables

…

5: 25.19 Tables

§25.19 Tables

►

Abramowitz and Stegun (1964) tabulates: $\zeta \left(n\right)$, $n=2,3,4,\mathrm{\dots }$, 20D (p. 811); ${\mathrm{Li}}_2\left(1-x\right)$, $x=0(.01)0.5$, 9D (p. 1005); $f(\theta )$, $\theta ={15}^{\circ }(1^{\circ }){30}^{\circ }(2^{\circ }){90}^{\circ }(5^{\circ }){180}^{\circ }$, $f(\theta )+\theta \ln \theta$, $\theta =0(1^{\circ }){15}^{\circ }$, 6D (p. 1006). Here $f(\theta )$ denotes Clausen’s integral, given by the right-hand side of (25.12.9).

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Morris (1979) tabulates ${\mathrm{Li}}_2\left(x\right)$ (§25.12(i)) for $\pm x=0.02(.02)1(.1)6$ to 30D.

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Cloutman (1989) tabulates $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

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Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta \left(s\right)$ for both real and complex $s$. §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta (s,a)$, and §22.17 lists tables for some Dirichlet $L$-functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

6: 26.2 Basic Definitions

… ►If, for example, a permutation of the integers 1 through 6 is denoted by $256413$, then the cycles are $\left(1,2,5\right)$, $\left(3,6\right)$, and $\left(4\right)$. …The function $\sigma$ also interchanges 3 and 6, and sends 4 to itself. … ►As an example, $\{1,3,4\}$, $\{2,6\}$, $\{5\}$ is a partition of $\{1,2,3,4,5,6\}$. … ►See Table 26.2.1 for $n=0(1)50$. For the actual partitions ($\pi$) for $n=1(1)5$ see Table 26.4.1. …

7: 35.11 Tables

§35.11 Tables

►Tables of zonal polynomials are given in James (1964) for $|\kappa |\le 6$, Parkhurst and James (1974) for $|\kappa |\le 12$, and Muirhead (1982, p. 238) for $|\kappa |\le 5$. Each table expresses the zonal polynomials as linear combinations of monomial symmetric functions.

8: 27.2 Functions

… ►Tables of primes (§27.21) reveal great irregularity in their distribution. …It can be expressed as a sum over all primes $p\le x$: … ►

§27.2(ii) Tables

►Table 27.2.1 lists the first 100 prime numbers $p_n$. Table 27.2.2 tabulates the Euler totient function $\varphi \left(n\right)$, the divisor function $d\left(n\right)$ ($={\sigma }_0(n)$), and the sum of the divisors $\sigma (n)$ ($={\sigma }_1(n)$), for $n=1(1)52$. …

9: 4.17 Special Values and Limits

… ►

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

► ►►►►►►

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
…
$\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$	$\sqrt{2}(\sqrt{3}+1)$	$\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$
$\pi /6$	$\frac{1}{2}$	$\frac{1}{2}\sqrt{3}$	$\frac{1}{3}\sqrt{3}$	$2$	$\frac{2}{3}\sqrt{3}$	$\sqrt{3}$
…
$5\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$	$\sqrt{2}(\sqrt{3}-1)$	$\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$
…
$5\pi /6$	$\frac{1}{2}$	$-\frac{1}{2}\sqrt{3}$	$-\frac{1}{3}\sqrt{3}$	$2$	$-\frac{2}{3}\sqrt{3}$	$-\sqrt{3}$
…

►

…

10: 8.26 Tables

§8.26 Tables

… ►

Pearson (1965) tabulates the function $I(u,p)$ ($=P(p+1,u)$) for $p=-1(.05)0(.1)5(.2)50$, $u=0(.1)u_p$ to 7D, where $I(u,u_p)$ rounds off to 1 to 7D; also $I(u,p)$ for $p=-0.75(.01)-1$, $u=0(.1)6$ to 5D.

►

Zhang and Jin (1996, Table 3.8) tabulates $\gamma (a,x)$ for $a=0.5,1,3,5,10,25,50,100$, $x=0(.1)1(1)3,5(5)30,50,100$ to 8D or 8S.

… ►

Zhang and Jin (1996, Table 3.9) tabulates $I_x(a,b)$ for $x=0(.05)1$, $a=0.5,1,3,5,10$, $b=1,10$ to 8D.

… ►

Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.