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21—30 of 140 matching pages

21: 26.8 Set Partitions: Stirling Numbers

… ►

Table 26.8.1: Stirling numbers of the first kind

s\left(n,k\right)

► ►►►►►

$n$	$k$
$n$	…
4	$0$	$-6$	$11$	$-6$	$1$
5	$0$	$24$	$-50$	$35$	$-10$	$1$
…
8	$0$	$-5040$	$13068$	$-13132$	$6769$	$-1960$	$322$	$-28$	$1$
…

► ►

Table 26.8.2: Stirling numbers of the second kind

S\left(n,k\right)

► ►►►

$n$	$k$
$n$	…
8	0	1	127	966	1701	1050	266	28	1
…

► …

22: 6.19 Tables

… ►

•

Abramowitz and Stegun (1964, Chapter 5) includes $x^{-1}\operatorname{Si}\left(x\right)$ , $-x^{-2}\operatorname{Cin}\left(x\right)$ , $x^{-1}\operatorname{Ein}\left(x\right)$ , $-x^{-1}\operatorname{Ein}\left(-x\right)$ , $x=0(.01)0.5$ ; $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=0.5(.01)2$ ; $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $xe^{-x}\operatorname{Ei}\left(x\right)$ , $xe^{x}E_{1}\left(x\right)$ , $x=2(.1)10$ ; $x\mathrm{f}\left(x\right)$ , $x^{2}\mathrm{g}\left(x\right)$ , $xe^{-x}\operatorname{Ei}\left(x\right)$ , $xe^{x}E_{1}\left(x\right)$ , $x^{-1}=0(.005)0.1$ ; $\operatorname{Si}\left(\pi x\right)$ , $\operatorname{Cin}\left(\pi x\right)$ , $x=0(.1)10$ . Accuracy varies but is within the range 8S–11S.

… ►

•

Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_{1}\left(z\right)$ , $\pm x=0.5,1,3,5,10,15,20,50,100$ , $y=0(.5)1(1)5(5)30,50,100$ , 8S.

23: 34.7 Basic Properties: $\mathit{9j}$ Symbol

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34.7.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{13}\\ j_{31}&j_{31}&0\end{Bmatrix}=\frac{(-1)^{j_{12}+j_{21}+j_{13}+j_{31}}}{((2j_{1% 3}+1)(2j_{31}+1))^{\frac{1}{2}}}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{22}&j_{21}&j_{31}\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.
Permalink:: http://dlmf.nist.gov/34.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.7(i), §34.7 and Ch.34

… ►

34.7.2

\sum_{j_{12}\,j_{34}}(2j_{12}+1)(2j_{34}+1)(2j_{13}+1)(2j_{24}+1)\begin{% Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j_{13}&j_{24}&j\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j^{\prime}_{13}&j^{\prime}_{24}&j\end{Bmatrix}=\delta_{j_{13},j^{\prime}_{13}}% \delta_{j_{24},j^{\prime}_{24}}.

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.7(iv), §34.7 and Ch.34

… ►

34.7.3

\sum_{j_{13}\,j_{24}}(-1)^{2j_{2}+j_{24}+j_{23}-j_{34}}(2j_{13}+1)(2j_{24}+1)% \begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j_{13}&j_{24}&j\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{3}&j_{13}\\ j_{4}&j_{2}&j_{24}\\ j_{14}&j_{23}&j\end{Bmatrix}=\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{4}&j_{3}&j_{34}\\ j_{14}&j_{23}&j\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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.9
Permalink:: http://dlmf.nist.gov/34.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §34.7(vi), §34.7 and Ch.34

… ►

34.7.4

\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{m_{r1},m_{r2},r=1,2,3}\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}.

ⓘ

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, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $r$ : nonnegative integer
Referenced by:: Erratum (V1.0.10) for Equation (34.7.4)
Permalink:: http://dlmf.nist.gov/34.7.E4
Encodings:: pMML, png, TeX
Errata (effective with 1.0.10):: Originally the third $\mathit{3j}$ symbol in the summation was written incorrectly as $\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}.$
Reported 2015-01-19 by Yan-Rui Liu
See also:: Annotations for §34.7(vi), §34.7 and Ch.34

►

34.7.5

\sum_{j^{\prime}}(2j^{\prime}+1)\begin{Bmatrix}j_{11}&j_{12}&j^{\prime}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}\begin{Bmatrix}j_{11}&j_{12}&j^{\prime}\\ j_{23}&j_{33}&j\end{Bmatrix}={(-1)^{2j}}\begin{Bmatrix}j_{21}&j_{22}&j_{23}\\ j_{12}&j&j_{32}\end{Bmatrix}\begin{Bmatrix}j_{31}&j_{32}&j_{33}\\ j&j_{11}&j_{21}\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.
Permalink:: http://dlmf.nist.gov/34.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §34.7(vi), §34.7 and Ch.34

24: 5.22 Tables

… ►Abramowitz and Stegun (1964, Chapter 6) tabulates

\Gamma\left(x\right)

\ln\Gamma\left(x\right)

\psi\left(x\right)

, and

\psi'\left(x\right)

for

x=1(.005)2

to 10D;

\psi''\left(x\right)

and

\psi^{(3)}\left(x\right)

for

x=1(.01)2

to 10D;

\Gamma\left(n\right)

\ifrac{1}{\Gamma\left(n\right)}

\Gamma\left(n+\tfrac{1}{2}\right)

\psi\left(n\right)

\operatorname{log}_{10}\Gamma\left(n\right)

\operatorname{log}_{10}\Gamma\left(n+\tfrac{1}{3}\right)

\operatorname{log}_{10}\Gamma\left(n+\tfrac{1}{2}\right)

, and

\operatorname{log}_{10}\Gamma\left(n+\tfrac{2}{3}\right)

for

n=1(1)101

to 8–11S;

\Gamma\left(n+1\right)

for

n=100(100)1000

to 20S. Zhang and Jin (1996, pp. 67–69 and 72) tabulates

\Gamma\left(x\right)

\ifrac{1}{\Gamma\left(x\right)}

\Gamma\left(-x\right)

\ln\Gamma\left(x\right)

\psi\left(x\right)

\psi\left(-x\right)

\psi'\left(x\right)

, and

\psi'\left(-x\right)

for

x=0(.1)5

to 8D or 8S;

\Gamma\left(n+1\right)

for

n=0(1)100(10)250(50)500(100)3000

to 51S. …

25: Bibliography P

… ►

R. B. Paris and A. D. Wood (1995) Stokes phenomenon demystified. Bull. Inst. Math. Appl. 31 (1-2), pp. 21–28.

ⓘ

External Links:: ISSN 0905-5628, MathReview Entry, ZentralBlatt
Referenced by:: §2.11(iv).
Encodings:: BibTeX

… ►

J. B. Parkinson (1969) Optical properties of layer antiferromagnets with

\mathrm{K}_{2}\mathrm{NiF}_{4}

structure. J. Phys. C: Solid State Physics 2 (11), pp. 2012–2021.

ⓘ

External Links:: Document
Referenced by:: §19.35(ii).
Encodings:: BibTeX

… ►

R. Piessens and M. Branders (1984) Algorithm 28. Algorithm for the computation of Bessel function integrals. J. Comput. Appl. Math. 11 (1), pp. 119–137.

ⓘ

External Links:: ISSN 0377-0427, Document, ZentralBlatt
Referenced by:: §10.77(ix).
Encodings:: BibTeX

…

26: Simon Ruijsenaars

… ►

28 Mathieu Functions and Hill’s Equation

27: Bibliography L

… ►

D. Le (1985) An efficient derivative-free method for solving nonlinear equations. ACM Trans. Math. Software 11 (3), pp. 250–262.

ⓘ

Notes:: Corrigendum, ibid. 15(3) (1989), 287
External Links:: ISSN 0098-3500, Document, MathReview (T. Feagin), ZentralBlatt
Referenced by:: §3.8(iii).
Encodings:: BibTeX

… ►

A. Leitner and J. Meixner (1960) Eine Verallgemeinerung der Sphäroidfunktionen. Arch. Math. 11, pp. 29–39.

ⓘ

External Links:: ISSN 0003-9268, Document, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §30.12.
Encodings:: BibTeX

… ►

H. Lotsch and M. Gray (1964) Algorithm 244: Fresnel integrals. Comm. ACM 7 (11), pp. 660–661.

ⓘ

Notes:: Includes Algol program, maximum accuracy 12D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §7.25(iv).
Encodings:: BibTeX

… ►

N. A. Lukaševič (1967b) On the theory of Painlevé’s third equation. Differ. Uravn. 3 (11), pp. 1913–1923 (Russian).

ⓘ

Notes:: In Russian. English translation: Differential Equations 3(11), pp. 994–999
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.10(iii), §32.8(iii), §32.9(i), §32.9(ii).
Encodings:: BibTeX

… ►

Y. L. Luke (1977a) Algorithms for rational approximations for a confluent hypergeometric function. Utilitas Math. 11, pp. 123–151.

ⓘ

External Links:: MathReview (F. Gotze), ZentralBlatt
Referenced by:: §13.31(iii).
Encodings:: BibTeX

…

28: 28.29 Definitions and Basic Properties

… ►Its order of growth for

|\lambda|\to\infty

is exactly

\tfrac{1}{2}

; see Magnus and Winkler (1966, Chapter II, pp. 19–28). … ►For this purpose the discriminant can be expressed as an infinite determinant involving the Fourier coefficients of

Q(x)

; see Magnus and Winkler (1966, §2.3, pp. 28–36). …

29: 10 Bessel Functions

…

30: 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).

.世界杯中国足球队排名『网址:mxsty.cc』.2010世界杯经典画面-m4x5s2-2022年11月28日19时45分50秒ckoekkqwq

21—30 of 140 matching pages

21: 26.8 Set Partitions: Stirling Numbers

22: 6.19 Tables

23: 34.7 Basic Properties: 9 ⁢ j Symbol

24: 5.22 Tables

25: Bibliography P

26: Simon Ruijsenaars

27: Bibliography L

28: 28.29 Definitions and Basic Properties

29: 10 Bessel Functions

30: 34.9 Graphical Method

23: 34.7 Basic Properties: $\mathit{9j}$ Symbol