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1: 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. …

2: 9.13 Generalized Airy Functions

… ►Swanson and Headley (1967) define independent solutions

A_{n}\left(z\right)

and

B_{n}\left(z\right)

of (9.13.1) by … ►Properties of

A_{n}\left(z\right)

and

B_{n}\left(z\right)

follow from the corresponding properties of the modified Bessel functions. … ►The distribution in

\mathbb{C}

and asymptotic properties of the zeros of

A_{n}\left(z\right)

A_{n}'\left(z\right)

B_{n}\left(z\right)

, and

B_{n}'\left(z\right)

are investigated in Swanson and Headley (1967) and Headley and Barwell (1975). … ►Their relations to the functions

A_{n}\left(z\right)

and

B_{n}\left(z\right)

are given by … ►The

A_{k}\left(z,p\right)

are related by …

3: Bibliography L

… ►

S. Lai and Y. Chiu (1992) Exact computation of the

9

j

symbols. Comput. Phys. Comm. 70 (3), pp. 544–556.

ⓘ

Notes:: Double-precision Fortran
External Links:: ISSN 0010-4655, Document, Code, MathReview
Referenced by:: 7th item.
Encodings:: BibTeX

… ►

E. M. Lifshitz and L. P. Pitaevskiĭ (1980) Statistical Physics, Part 2: Theory of the Condensed State. Pergamon Press, Oxford.

ⓘ

Notes:: Course of Theoretical Physics , Vol. 9. Translated from the Russian by J. B. Sykes and M. J. Kearsley.
External Links:: ISBN 0-08-023073-3; 0-08-023072-5, MathReview
Referenced by:: §25.17.
Encodings:: BibTeX

… ►

Lord Kelvin (1905) Deep water ship-waves. Phil. Mag. 9, pp. 733–757.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §36.13.
Encodings:: BibTeX

… ►

D. W. Lozier and F. W. J. Olver (1994) Numerical Evaluation of Special Functions. In Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics (Vancouver, BC, 1993), Proc. Sympos. Appl. Math., Vol. 48, pp. 79–125.

ⓘ

Notes:: An updated version exists online.
External Links:: FullText, MathReview (John P. Coleman), ZentralBlatt
Referenced by:: Ch.3.
Encodings:: BibTeX

… ►

Y. L. Luke (1971a) Miniaturized tables of Bessel functions. II. Math. Comp. 25 (116), pp. 789–795 and D14–E13.

ⓘ

Notes:: For Corrigendum see same journal 26(120), pp. A1–A7
External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

…

4: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►See Calkin et al. (2007), and Lehmer (1941, pp. 5–83). …

5: Bibliography D

… ►

S. D. Daymond (1955) The principal frequencies of vibrating systems with elliptic boundaries. Quart. J. Mech. Appl. Math. 8 (3), pp. 361–372.

ⓘ

External Links:: Document, MathReview (M. A. Hyman), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

… ►

K. Dilcher (1987b) Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.

ⓘ

External Links:: ISSN 0022-314X, Document, MathReview (T. Metsänkylä)
Referenced by:: §24.12(iii).
Encodings:: BibTeX

… ►

P. G. L. Dirichlet (1849) Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1849, pp. 69–83 (German).

ⓘ

Notes:: Reprinted in G. Lejeune Dirichlet’s Werke, Band II, pp. 49–66, Reimer, Berlin, 1897 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.
External Links:: ISBN 0-8284-0225-6, Link, MathReview
Referenced by:: §27.11.
Encodings:: BibTeX

… ►

G. Doetsch (1955) Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).

ⓘ

External Links:: MathReview (I. I. Hirschman), ZentralBlatt
Referenced by:: §2.5(i).
Encodings:: BibTeX

… ►

L. Durand (1978) Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.

ⓘ

External Links:: Document, MathReview (R. C. Varma), ZentralBlatt
Referenced by:: §18.17(iii).
Encodings:: BibTeX

…

6: 28.6 Expansions for Small $q$

… ►

28.6.2

a_{1}\left(q\right)=1+q-\tfrac{1}{8}q^{2}-\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^% {4}+\tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}+\tfrac{55}{94\;37184}q^{7% }-\tfrac{83}{353\;89440}q^{8}+\cdots,

ⓘ

Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(i), §28.6 and Ch.28

►

28.6.3

b_{1}\left(q\right)=1-q-\tfrac{1}{8}q^{2}+\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^% {4}-\tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}-\tfrac{55}{94\;37184}q^{7% }-\tfrac{83}{353\;89440}q^{8}+\cdots,

ⓘ

Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(i), §28.6 and Ch.28

… ►Leading terms of the of the power series for

m=7,8,9,\dots

are: ►

28.6.14

\rselection{a_{m}\left(q\right)\\ b_{m}\left(q\right)}=m^{2}+\frac{1}{2(m^{2}-1)}q^{2}+\frac{5m^{2}+7}{32(m^{2}-% 1)^{3}(m^{2}-4)}q^{4}+\frac{9m^{4}+58m^{2}+29}{64(m^{2}-1)^{5}(m^{2}-4)(m^{2}-% 9)}q^{6}+\cdots.

ⓘ

Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $m$ : integer and $q=h^{2}$ : parameter
A&S Ref:: 20.2.26
Referenced by:: §28.6(i), §28.6(i), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)
Permalink:: http://dlmf.nist.gov/28.6.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(i), §28.6 and Ch.28

… ►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. …

7: Bibliography I

… ►

E. L. Ince (1940b) Further investigations into the periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 83–99.

ⓘ

External Links:: MathReview (H. Bateman), ZentralBlatt
Referenced by:: §29.1, §29.17(ii), §29.3(iii), §29.6(i), §29.6(ii), §29.6(iii), §29.6(iv), Ch.29, Notations E, Notations E, Notations E, Notations E.
Encodings:: BibTeX

… ►

A. Iserles, S. P. Nørsett, and S. Olver (2006) Highly Oscillatory Quadrature: The Story So Far. In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.), pp. 97–118.

ⓘ

Notes:: Proceedings of ENuMath, Santiago de Compostela (2005)
External Links:: ISBN 3-540-34287-7, MathReview, ZentralBlatt
Referenced by:: §3.5(vii).
Encodings:: BibTeX

…

8: 27.2 Functions

… ►

27.2.9

d\left(n\right)=\sum_{d\mathbin{|}n}1

ⓘ

Symbols:: $d_{\NVar{k}}\left(\NVar{n}\right)$ : divisor function, $d$ : positive integer and $n$ : positive integer
A&S Ref:: 24.3.3 I.A
Permalink:: http://dlmf.nist.gov/27.2.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §27.2(i), §27.2 and Ch.27

… ►It is the special case

k=2

of the function

d_{k}\left(n\right)

that counts the number of ways of expressing

n

as the product of

k

factors, with the order of factors taken into account. …Note that

\sigma_{0}\left(n\right)=d\left(n\right)

. … ►Table 27.2.2 tabulates the Euler totient function

\phi\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

. ►

Table 27.2.1: Primes.

► ►►►

$n$	$p_{n}$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
…
3	5	41	83	137	191	241	307	367	431	487
…

► …

9: Bibliography E

… ►

Á. Elbert (2001) Some recent results on the zeros of Bessel functions and orthogonal polynomials. J. Comput. Appl. Math. 133 (1-2), pp. 65–83.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (N. Hayek Calil), ZentralBlatt
Referenced by:: §10.21(iv).
Encodings:: BibTeX

… ►

A. Erdélyi (1942b) The Fuchsian equation of second order with four singularities. Duke Math. J. 9 (1), pp. 48–58.

ⓘ

External Links:: Document, MathReview (R. E. Langer), ZentralBlatt
Referenced by:: §31.11(i).
Encodings:: BibTeX

… ►

D. Erricolo and G. Carluccio (2013) Algorithm 934: Fortran 90 subroutines to compute Mathieu functions for complex values of the parameter. ACM Trans. Math. Softw. 40 (1), pp. 8:1–8:19.

ⓘ

External Links:: ISSN 0098-3500, FullText, Document, MathReview Entry, ZentralBlatt
Referenced by:: 4th item, §28.36(iii).
Encodings:: BibTeX

… ►

T. Estermann (1959) On the representations of a number as a sum of three squares. Proc. London Math. Soc. (3) 9, pp. 575–594.

ⓘ

External Links:: Document, MathReview (S. Chowla), ZentralBlatt
Referenced by:: §20.12(i).
Encodings:: BibTeX

… ►

J. A. Ewell (1990) A new series representation for

\zeta(3)

. Amer. Math. Monthly 97 (3), pp. 219–220.

ⓘ

External Links:: ISSN 0002-9890, FullText, MathReview (Christian Radoux), ZentralBlatt
Referenced by:: (25.8.10).
Encodings:: BibTeX

…

10: 3.9 Acceleration of Convergence

… ►

Table 3.9.1: Shanks’ transformation for

s_{n}=\sum_{j=1}^{n}(-1)^{j+1}j^{-2}

► ►►►►►►

$n$	$t_{n,2}$	$t_{n,4}$	$t_{n,6}$	$t_{n,8}$	$t_{n,10}$
…
3	0.82300 13550 14	0.82247 78118 35	0.82246 72851 83	0.82246 70397 56	0.82246 70335 90
4	0.82221 76684 88	0.82246 28314 41	0.82246 69467 93	0.82246 70314 36	0.82246 70333 75
…
8	0.82243 73137 33	0.82246 67719 32	0.82246 70301 49	0.82246 70333 73	0.82246 70334 23
9	0.82248 70624 89	0.82246 71865 91	0.82246 70351 34	0.82246 70334 48	0.82246 70334 24
…

► …