.%E5%9C%A8%E5%93%AA%E9%87%8C%E5%8F%AF%E4%BB%A5%E4%B9%B0%E4%B8%96%E7%95%8C%E6%9D%AF%E3%80%8Ewn4.com%E3%80%8F%E7%9C%8B%E4%B8%96%E7%95%8C%E6%9D%AF%E6%A0%87%E8%AF%AD.w6n2c9o.2022%E5%B9%B411%E6%9C%8830%E6%97%A57%E6%97%B645%E5%88%8649%E7%A7%92.gkswkagoe.gov.hk

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11: 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).

12: 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). …

13: 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. …

14: 16.24 Physical Applications

… ►

§16.24(iii) $\mathit{3j}$ , $\mathit{6j}$ , and $\mathit{9j}$ Symbols

… ►They can be expressed as

{{}_{3}F_{2}}

functions with unit argument. …These are balanced

{{}_{4}F_{3}}

functions with unit argument. Lastly, special cases of the

\mathit{9j}

symbols are

{{}_{5}F_{4}}

functions with unit argument. …

15: 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).

16: 7.20 Mathematical Applications

… ►For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951). … ►Let the set

\{x(t),y(t),t\}

be defined by

x(t)=C\left(t\right)

y(t)=S\left(t\right)

t\geq 0

. … ►

See accompanying text — Figure 7.20.1: Cornu’s spiral, formed from Fresnel integrals, is defined parametrically by $x=C\left(t\right)$ , $y=S\left(t\right)$ , $t\in[0,\infty)$ . Magnify

…

17: 26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

►

D(m,n)

is the number of paths from

(0,0)

(m,n)

that are composed of directed line segments of the form

(1,0)

(0,1)

, or

(1,1)

. … ►

26.6.12

C\left(n\right)=\sum_{k=1}^{n}N(n,k),

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $k$ : nonnegative integer, $n$ : nonnegative integer and $N(n,k)$ : Narayana number
Permalink:: http://dlmf.nist.gov/26.6.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(iv), §26.6 and Ch.26

►

26.6.13

M(n)=\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}C\left(n+1-k\right),

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : nonnegative integer, $n$ : nonnegative integer and $M(n)$ : Motzkin number
Permalink:: http://dlmf.nist.gov/26.6.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(iv), §26.6 and Ch.26

►

26.6.14

C\left(n\right)=\sum_{k=0}^{2n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{2n}{k}M(2n-k).

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : nonnegative integer, $n$ : nonnegative integer and $M(n)$ : Motzkin number
Permalink:: http://dlmf.nist.gov/26.6.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(iv), §26.6 and Ch.26

18: 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

19: 10 Bessel Functions

…

20: 23 Weierstrass Elliptic and Modular
Functions

…