.世界杯买球怎么买_『wn4.com_』2018世界杯一共多少支球队_w6n2c9o_2022年11月29日6时48分9秒_ok040qquy_gov_hk

Did you mean .世界杯买球怎么买_『welcom_』2018世界杯一共多少支球队_w6n2c9o_2022年11月29日6时48分9秒_ok040qquy_gov_hk ?

(0.006 seconds)

1—10 of 799 matching pages

2: 34.6 Definition: $\mathit{9j}$ Symbol

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

►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ 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

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ 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{9j}

symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

3: 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. …

4: 26.9 Integer Partitions: Restricted Number and Part Size

… ►

p_{k}\left(n\right)

denotes the number of partitions of

n

into at most

k

parts. See Table 26.9.1. … ►It follows that

p_{k}\left(n\right)

also equals the number of partitions of

n

into parts that are less than or equal to

k

. ►

p_{k}\left(\leq m,n\right)

is the number of partitions of

n

into at most

k

parts, each less than or equal to

m

. …

5: 28.6 Expansions for Small $q$

… ►Leading terms of the power series for

a_{m}\left(q\right)

and

b_{m}\left(q\right)

for

m\leq 6

are: … ►The coefficients of the power series of

a_{2n}\left(q\right)

b_{2n}\left(q\right)

and also

a_{2n+1}\left(q\right)

b_{2n+1}\left(q\right)

are the same until the terms in

q^{2n-2}

and

q^{2n}

, respectively. … ►Numerical values of the radii of convergence

\rho_{n}^{(j)}

of the power series (28.6.1)–(28.6.14) for

n=0,1,\dots,9

are given in Table 28.6.1. Here

j=1

for

a_{2n}\left(q\right)

j=2

for

b_{2n+2}\left(q\right)

, and

j=3

for

a_{2n+1}\left(q\right)

and

b_{2n+1}\left(q\right)

. … ►

§28.6(ii) Functions $\operatorname{ce}_{n}$ and $\operatorname{se}_{n}$

…

7: 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},

… ►

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}.

►An often used alternative to the

\mathit{3j}

symbol is the Clebsch–Gordan coefficient …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).

9: 26.16 Multiset Permutations

… ►

\mathfrak{S}_{S}

denotes the set of permutations of

S

for all distinct orderings of the

a_{1}+a_{2}+\cdots+a_{n}

integers. The number of elements in

\mathfrak{S}_{S}

is the multinomial coefficient (§26.4)

\genfrac{(}{)}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}

. … ►Thus

\mathop{\mathrm{inv}}(351322453154)=4+8+0+3+1+1+2+3+1+0+1=24

, and

\mathop{\mathrm{maj}}(351322453154)=2+4+8+9+11=34.

►The $q$ -multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by …and again with

S=\{1^{a_{1}},2^{a_{2}},\ldots,n^{a_{n}}\}

we have …

Abramowitz and Stegun (1964, Chapter 9) tabulates $j_{n,m}$ , $J_{n}'\left(j_{n,m}\right)$ , ${j^{\prime}_{n,m}}$ , $J_{n}\left({j^{\prime}_{n,m}}\right)$ , $n=0(1)8$ , $m=1(1)20$ , 5D (10D for $n=0$ ), $y_{n,m}$ , $Y_{n}'\left(y_{n,m}\right)$ , ${y^{\prime}_{n,m}}$ , $Y_{n}\left({y^{\prime}_{n,m}}\right)$ , $n=0(1)8$ , $m=1(1)20$ , 5D (8D for $n=0$ ), $J_{0}\left(j_{0,m}x\right)$ , $m=1(1)5$ , $x=0(.02)1$ , 5D. Also included are the first 5 zeros of the functions $xJ_{1}\left(x\right)-\lambda J_{0}\left(x\right)$ , $J_{1}\left(x\right)-\lambda xJ_{0}\left(x\right)$ , $J_{0}\left(x\right)Y_{0}\left(\lambda x\right)-Y_{0}\left(x\right)J_{0}\left(% \lambda x\right)$ , $J_{1}\left(x\right)Y_{1}\left(\lambda x\right)-Y_{1}\left(x\right)J_{1}\left(% \lambda x\right)$ , $J_{1}\left(x\right)Y_{0}\left(\lambda x\right)-Y_{1}\left(x\right)J_{0}\left(% \lambda x\right)$ for various values of $\lambda$ and $\lambda^{-1}$ in the interval $[0,1]$ , 4–8D.

… ►

•

Makinouchi (1966) tabulates all values of $j_{\nu,m}$ and $y_{\nu,m}$ in the interval $(0,100)$ , with at least 29S. These are for $\nu=0(1)5$ , 10, 20; $\nu=\tfrac{3}{2}$ , $\tfrac{5}{2}$ ; $\nu=m/n$ with $m=1(1)n-1$ and $n=3(1)8$ , except for $\nu=\tfrac{1}{2}$ .

… ►

•

Abramowitz and Stegun (1964, Chapter 11) tabulates $\int_{0}^{x}J_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{0}^{x}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)10$ , 10D; $\int_{0}^{x}t^{-1}(1-J_{0}\left(t\right))\,\mathrm{d}t$ , $\int_{x}^{\infty}t^{-1}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)5$ , 8D.

… ►

•

Leung and Ghaderpanah (1979), tabulates all zeros of the principal value of $K_{n}\left(z\right)$ , for $n=2(1)10$ , 29S.

… ►

•

Abramowitz and Stegun (1964, Chapter 11) tabulates $e^{-x}\int_{0}^{x}I_{0}\left(t\right)\,\mathrm{d}t$ , $e^{x}\int_{x}^{\infty}K_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)10$ , 7D; $e^{-x}\int_{0}^{x}t^{-1}(I_{0}\left(t\right)-1)\,\mathrm{d}t$ , $xe^{x}\int_{x}^{\infty}t^{-1}K_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)5$ , 6D.

…

.世界杯买球怎么买_『wn4.com_』2018世界杯一共多少支球队_w6n2c9o_2022年11月29日6时48分9秒_ok040qquy_gov_hk

1—10 of 799 matching pages

1: 19.2 Definitions

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

2: 34.6 Definition: $\mathit{9j}$ Symbol

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

3: 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

4: 26.9 Integer Partitions: Restricted Number and Part Size

5: 28.6 Expansions for Small $q$

§28.6(ii) Functions $\operatorname{ce}_{n}$ and $\operatorname{se}_{n}$

6: 10 Bessel Functions

7: 34.1 Special Notation

8: 27.2 Functions

9: 26.16 Multiset Permutations

10: 10.75 Tables

.世界杯 买球怎么买_『wn4.com_』2018世界杯一共多少支球队_w6n2c9o_2022年11月29日6时48分9秒_ok040qquy_gov_hk

1—10 of 799 matching pages

1: 19.2 Definitions

§19.2(iv) A Related Function: R C ⁡ ( x , y )

2: 34.6 Definition: 9 ⁢ j Symbol

§34.6 Definition: 9 ⁢ j Symbol

3: 34.7 Basic Properties: 9 ⁢ j Symbol

§34.7 Basic Properties: 9 ⁢ j Symbol

§34.7(ii) Symmetry

§34.7(iv) Orthogonality

§34.7(vi) Sums

4: 26.9 Integer Partitions: Restricted Number and Part Size

5: 28.6 Expansions for Small q

§28.6(ii) Functions ce n and se n

6: 10 Bessel Functions

7: 34.1 Special Notation

8: 27.2 Functions

9: 26.16 Multiset Permutations

10: 10.75 Tables

.世界杯买球怎么买_『wn4.com_』2018世界杯一共多少支球队_w6n2c9o_2022年11月29日6时48分9秒_ok040qquy_gov_hk

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

2: 34.6 Definition: $\mathit{9j}$ Symbol

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

3: 34.7 Basic Properties: $\mathit{9j}$ Symbol

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

5: 28.6 Expansions for Small $q$

§28.6(ii) Functions $\operatorname{ce}_{n}$ and $\operatorname{se}_{n}$