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11: 34.13 Methods of Computation

… ►Methods of computation for

\mathit{3j}

and

\mathit{6j}

symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). ►For

\mathit{9j}

symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989). …

12: 34.9 Graphical Method

§34.9 Graphical Method

… ►For specific examples of the graphical method of representing sums involving the

\mathit{3j},\mathit{6j}

, and

\mathit{9j}

symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

13: 34.10 Zeros

… ►Such zeros are called nontrivial zeros. ►For further information, including examples of nontrivial zeros and extensions to

\mathit{9j}

symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).

14: 34.7 Basic Properties: $\mathit{9j}$ Symbol

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

… ►

§34.7(ii) Symmetry

… ►

§34.7(iv) Orthogonality

… ►

§34.7(vi) Sums

… ►It constitutes an addition theorem for the

\mathit{9j}

symbol. …

15: 19.9 Inequalities

… ►Further inequalities for

K\left(k\right)

and

E\left(k\right)

can be found in Alzer and Qiu (2004), Anderson et al. (1992a, b, 1997), and Qiu and Vamanamurthy (1996). … ►Even for the extremely eccentric ellipse with

a=99

and

b=1

, this is correct within 0. … ►Sharper inequalities for

F\left(\phi,k\right)

are: … ►Inequalities for both

F\left(\phi,k\right)

and

E\left(\phi,k\right)

involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). … ►Other inequalities for

F\left(\phi,k\right)

can be obtained from inequalities for

R_{F}\left(x,y,z\right)

given in Carlson (1966, (2.15)) and Carlson (1970) via (19.25.5).

16: 19.5 Maclaurin and Related Expansions

… ►where

{{}_{2}F_{1}}

is the Gauss hypergeometric function (§§15.1 and 15.2(i)). …where

{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)

is an Appell function (§16.13). … ►Coefficients of terms up to

\lambda^{49}

are given in Lee (1990), along with tables of fractional errors in

K\left(k\right)

and

E\left(k\right)

0.1\leq k^{2}\leq 0.9999

, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). … ►Series expansions of

F\left(\phi,k\right)

and

E\left(\phi,k\right)

are surveyed and improved in Van de Vel (1969), and the case of

F\left(\phi,k\right)

is summarized in Gautschi (1975, §1.3.2). For series expansions of

\Pi\left(\phi,\alpha^{2},k\right)

when

|\alpha^{2}|<1

see Erdélyi et al. (1953b, §13.6(9)). …

17: 24.16 Generalizations

… ►For

\ell=0,1,2,\dotsc

, Bernoulli and Euler polynomials of order $\ell$ are defined respectively by … ►

24.16.2

\left(\frac{2}{e^{t}+1}\right)^{\ell}e^{xt}=\sum_{n=0}^{\infty}E^{(\ell)}_{n}% \left(x\right)\frac{t^{n}}{n!},

|t|<\pi

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $E^{(\NVar{\ell})}_{\NVar{n}}\left(\NVar{x}\right)$ : generalized Euler polynomials, $\ell$ : integer, $n$ : integer, $t$ : real or complex and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.16.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

►When

x=0

they reduce to the Bernoulli and Euler numbers of order $\ell$ : … ►

E^{(\ell)}_{n}=E^{(\ell)}_{n}\left(0\right).

… ►

24.16.13

E_{n}\left(x\right)=-\frac{2^{1-n}}{n+1}B_{n+1,\chi_{4}}(2x-1).

ⓘ

Symbols:: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $n$ : integer, $x$ : real or complex, $\chi(a)$ : Dirichlet character and $B_{n,\chi}$ : Generalized Bernoulli numbers
Referenced by:: §24.16(ii)
Permalink:: http://dlmf.nist.gov/24.16.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §24.16(ii), §24.16 and Ch.24

…

18: 34.1 Special Notation

… ► ►►

$2j_{1},2j_{2},2j_{3},2l_{1},2l_{2},2l_{3}$	nonnegative integers.
…

►The main functions treated in this chapter are the Wigner

\mathit{3j},\mathit{6j},\mathit{9j}

symbols, respectively, … ►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).

19: 9.4 Maclaurin Series

… ►

9.4.1

\operatorname{Ai}\left(z\right)=\operatorname{Ai}\left(0\right)\left(1+\frac{1% }{3!}z^{3}+\frac{1\cdot 4}{6!}z^{6}+\frac{1\cdot 4\cdot 7}{9!}z^{9}+\cdots% \right)+\operatorname{Ai}'\left(0\right)\left(z+\frac{2}{4!}z^{4}+\frac{2\cdot 5% }{7!}z^{7}+\frac{2\cdot 5\cdot 8}{10!}z^{10}+\cdots\right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $!$ : factorial (as in $n!$ ) and $z$ : complex variable
Source:: Olver (1997b, p. 54)
A&S Ref:: 10.4.2 (with 10.4.4 and 10.4.5)
Permalink:: http://dlmf.nist.gov/9.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §9.4 and Ch.9

►

9.4.2

\operatorname{Ai}'\left(z\right)=\operatorname{Ai}'\left(0\right)\left(1+\frac% {2}{3!}z^{3}+\frac{2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots% \right)+\operatorname{Ai}\left(0\right)\left(\frac{1}{2!}z^{2}+\frac{1\cdot 4}% {5!}z^{5}+\frac{1\cdot 4\cdot 7}{8!}z^{8}+\cdots\right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $!$ : factorial (as in $n!$ ) and $z$ : complex variable
Proof sketch:: Use methods of Olver (1997b, p. 54).
Permalink:: http://dlmf.nist.gov/9.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §9.4 and Ch.9

►

9.4.3

\operatorname{Bi}\left(z\right)=\operatorname{Bi}\left(0\right)\left(1+\frac{1% }{3!}z^{3}+\frac{1\cdot 4}{6!}z^{6}+\frac{1\cdot 4\cdot 7}{9!}z^{9}+\cdots% \right)+\operatorname{Bi}'\left(0\right)\left(z+\frac{2}{4!}z^{4}+\frac{2\cdot 5% }{7!}z^{7}+\frac{2\cdot 5\cdot 8}{10!}z^{10}+\cdots\right),

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $!$ : factorial (as in $n!$ ) and $z$ : complex variable
Proof sketch:: Use methods of Olver (1997b, p. 54).
A&S Ref:: 10.4.3 (with 10.4.4 and 10.4.5)
Referenced by:: (9.12.15)
Permalink:: http://dlmf.nist.gov/9.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §9.4 and Ch.9

►

9.4.4

\operatorname{Bi}'\left(z\right)=\operatorname{Bi}'\left(0\right)\left(1+\frac% {2}{3!}z^{3}+\frac{2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots% \right)+\operatorname{Bi}\left(0\right)\left(\frac{1}{2!}z^{2}+\frac{1\cdot 4}% {5!}z^{5}+\frac{1\cdot 4\cdot 7}{8!}z^{8}+\cdots\right).

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $!$ : factorial (as in $n!$ ) and $z$ : complex variable
Proof sketch:: Use methods of Olver (1997b, p. 54).
Permalink:: http://dlmf.nist.gov/9.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §9.4 and Ch.9

20: Software Index

… ► ►►►►►►►

	Open Source										With Book					Commercial
…
8.28(vi) $E_{p}\left(x\right)$ , $x\in\mathbb{R}$ , $p\in\mathbb{Z}$	✓		✓	✓		✓		✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓
8.28(vii) $E_{p}\left(z\right)$ , $z,p\in\mathbb{C}$	✓								✓	✓							✓	✓	✓
9 Airy and Related Functions
…
24.21(ii) $B_{n}$ , $B_{n}\left(x\right)$ , $E_{n}$ , $E_{n}\left(x\right)$	✓	✓			✓		✓		a	✓	✓			✓	✓		✓	✓	✓	Derive, MuPAD
…
34 3j, 6j, 9j Symbols
…

…