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11: Nico M. Temme

… ►

3 Numerical Methods

►

6 Exponential, Logarithmic, Sine, and Cosine Integrals

… ►

12 Parabolic Cylinder Functions

… ►In November 2015, Temme was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 3, 6, 7, and 12.

12: 16.24 Physical Applications

… ►

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

►The

\mathit{3j}

symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. They can be expressed as

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

functions with unit argument. The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner

\mathit{6j}

symbols. These are balanced

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

functions with unit argument. …

13: 34.5 Basic Properties: $\mathit{6j}$ Symbol

§34.5 Basic Properties: $\mathit{6j}$ Symbol

… ►

§34.5(ii) Symmetry

… ►

§34.5(iv) Orthogonality

… ►

§34.5(vi) Sums

… ►Equation (34.5.23) can be regarded as an alternative definition of the

\mathit{6j}

symbol. …

14: 34.12 Physical Applications

§34.12 Physical Applications

►The angular momentum coupling coefficients (

\mathit{3j}

\mathit{6j}

, and

\mathit{9j}

symbols) are essential in the fields of nuclear, atomic, and molecular physics. …

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

, and

\mathit{9j}

symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).

15: 26.10 Integer Partitions: Other Restrictions

… ►

p\left(\mathcal{D}^{\prime}3,n\right)

denotes the number of partitions of

n

into parts with difference at least 3, except that multiples of 3 must differ by at least 6. …The set

\{2,3,4,\ldots\}

is denoted by

T

. … ►where the sum is over nonnegative integer values of

k

for which

n-\frac{1}{2}(3k^{2}\pm k)\geq 0

. … ►Note that

p\left(\mathcal{D}^{\prime}3,n\right)\leq p\left(\mathcal{D}3,n\right)

, with strict inequality for

n\geq 9

. It is known that for

k>3

p\left(\mathcal{D}k,n\right)\geq p\left(\in\!A_{1,k+3},n\right)

, with strict inequality for

n

sufficiently large, provided that

k=2^{m}-1,m=3,4,5

, or

k\geq 32

; see Yee (2004). …

16: 24.2 Definitions and Generating Functions

… ►

Table 24.2.5: Coefficients

b_{n,k}

of the Bernoulli polynomials

B_{n}\left(x\right)=\sum_{k=0}^{n}b_{n,k}x^{k}

► ►►►►

	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
…
$12$	$-\tfrac{691}{2730}$	$0$	$5$	$0$	$-\tfrac{33}{2}$	$0$	$22$	$0$	$-\tfrac{33}{2}$	$0$	$11$	$-6$	$1$
…

► ►

Table 24.2.6: Coefficients

e_{n,k}

of the Euler polynomials

E_{n}\left(x\right)=\sum_{k=0}^{n}e_{n,k}x^{k}

► ►►►►►

	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
…
$6$	$0$	$-3$	$0$	$5$	$0$	$-3$	$1$
…
$12$	$0$	$2073$	$0$	$-3410$	$0$	$1683$	$0$	$-396$	$0$	$55$	$0$	$-6$	$1$
…

►

17: 26.9 Integer Partitions: Restricted Number and Part Size

… ►

…

Figure 26.9.1: Ferrers graph of the partition

7+4+3+3+2+1

… ►The conjugate to the example in Figure 26.9.1 is

6+5+4+2+1+1+1

. … ►Equations (26.9.2)–(26.9.3) are examples of closed forms that can be computed explicitly for any positive integer

k

. … ►

p_{3}\left(n\right)=1+\left\lfloor\frac{n^{2}+6n}{12}\right\rfloor.

… ►equivalently, partitions into at most

k

parts either have exactly

k

parts, in which case we can subtract one from each part, or they have strictly fewer than

k

parts. …

18: 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: … ►Higher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations: … ►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)

. … ►For

m=3,4,5,\dots

, … ►For the corresponding expansions of

\operatorname{se}_{m}\left(z,q\right)

for

m=3,4,5,\dots

change

\cos

\sin

everywhere in (28.6.26). …

19: 34.7 Basic Properties: $\mathit{9j}$ Symbol

… ►

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

20: 23.20 Mathematical Applications

… ►Points

P=(x,y)

on the curve can be parametrized by

x=\wp\left(z;g_{2},g_{3}\right)

2y=\wp'\left(z;g_{2},g_{3}\right)

, where

g_{2}=-4a

and

g_{3}=-4b

: in this case we write

P=P(z)

. … ►The geometric nature of this construction is illustrated in McKean and Moll (1999, §2.14), Koblitz (1993, §§6, 7), and Silverman and Tate (1992, Chapter 1, §§3, 4): each of these references makes a connection with the addition theorem (23.10.1). … ►

T

must have one of the forms

\mathbb{Z}/(n\mathbb{Z})

1\leq n\leq 10

n=12

, or

(\mathbb{Z}/(2\mathbb{Z}))\times(\mathbb{Z}/(2n\mathbb{Z}))

1\leq n\leq 4

. …The order of a point (if finite and not already determined) can have only the values 3, 5, 6, 7, 9, 10, or 12, and so can be found from

2P=-P

4P=-P

4P=-2P

8P=P

8P=-P

8P=-2P

, or

8P=-4P

. … ►See also Silverman and Tate (1992), Serre (1973, Part 2, Chapters 6, 7), Koblitz (1993), and Cornell et al. (1997). …

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11—20 of 743 matching pages

11: Nico M. Temme

12: 16.24 Physical Applications

§16.24(iii) 3 ⁢ j , 6 ⁢ j , and 9 ⁢ j Symbols

13: 34.5 Basic Properties: 6 ⁢ j Symbol

§34.5 Basic Properties: 6 ⁢ j Symbol

§34.5(ii) Symmetry

§34.5(iv) Orthogonality

§34.5(vi) Sums

14: 34.12 Physical Applications

§34.12 Physical Applications

15: 26.10 Integer Partitions: Other Restrictions

16: 24.2 Definitions and Generating Functions

17: 26.9 Integer Partitions: Restricted Number and Part Size

18: 28.6 Expansions for Small q

19: 34.7 Basic Properties: 9 ⁢ j Symbol

20: 23.20 Mathematical Applications

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

13: 34.5 Basic Properties: $\mathit{6j}$ Symbol

§34.5 Basic Properties: $\mathit{6j}$ Symbol

18: 28.6 Expansions for Small $q$

19: 34.7 Basic Properties: $\mathit{9j}$ Symbol