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1: 34.4 Definition: $\mathit{6j}$ Symbol

… ►

34.4.1

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\sum_{m_{r}m^{\prime}_{s}}(-1)^{l_{1}+m^{\prime% }_{1}+l_{2}+m^{\prime}_{2}+l_{3}+m^{\prime}_{3}}\*\begin{pmatrix}j_{1}&j_{2}&j% _{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&l_{2}&l_{3}\\ m_{1}&m^{\prime}_{2}&-m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&j_{2}&l_% {3}\\ -m^{\prime}_{1}&m_{2}&m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&j_% {3}\\ m^{\prime}_{1}&-m^{\prime}_{2}&m_{3}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol
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., $l,l_{r}$ : non-negative integers or non-negative integers plus one half. and $r$ : nonnegative integer
Referenced by:: §34.10, §34.4, §34.9
Permalink:: http://dlmf.nist.gov/34.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.4 and Ch.34

… ►Except in degenerate cases the combination of the triangle inequalities for the four

\mathit{3j}

symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths

j_{1},j_{2},j_{3},l_{1},l_{2},l_{3}

; see Figure 34.4.1. … ►

34.4.2

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})% \Delta(l_{1}j_{2}l_{3})\Delta(l_{1}l_{2}j_{3})\*\sum_{s}\frac{(-1)^{s}(s+1)!}{% (s-j_{1}-j_{2}-j_{3})!(s-j_{1}-l_{2}-l_{3})!(s-l_{1}-j_{2}-l_{3})!(s-l_{1}-l_{% 2}-j_{3})!}\*\frac{1}{(j_{1}+j_{2}+l_{1}+l_{2}-s)!(j_{2}+j_{3}+l_{2}+l_{3}-s)!% (j_{3}+j_{1}+l_{3}+l_{1}-s)!},

ⓘ

Symbols:: $!$ : factorial (as in $n!$ ), $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half. and $\Delta(j_{1}j_{2}j_{3})$ : product
Referenced by:: Erratum (V1.0.7) for Equation (34.4.2)
Permalink:: http://dlmf.nist.gov/34.4.E2
Encodings:: pMML, png, TeX
Errata (effective with 1.0.7):: Originally the factor $\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})\Delta(l_{1}j_{2}l_{3})\Delta(l_% {1}l_{2}j_{3})$ was missing in this equation.
Reported 2012-12-31 by Yu Lin
See also:: Annotations for §34.4 and Ch.34

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34.4.3

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}={(-1)^{j_{1}+j_{3}+l_{1}+l_{3}}}\frac{\Delta(j_% {1}j_{2}j_{3})\Delta(j_{2}l_{1}l_{3})(j_{1}-j_{2}+l_{1}+l_{2})!(-j_{2}+j_{3}+l% _{2}+l_{3})!(j_{1}+j_{3}+l_{1}+l_{3}+1)!}{\Delta(j_{1}l_{2}l_{3})\Delta(j_{3}l% _{1}l_{2})(j_{1}-j_{2}+j_{3})!(-j_{2}+l_{1}+l_{3})!(j_{1}+l_{2}+l_{3}+1)!(j_{3% }+l_{1}+l_{2}+1)!}\*{{}_{4}F_{3}}\left({-j_{1}+j_{2}-j_{3},j_{2}-l_{1}-l_{3},-% j_{1}-l_{2}-l_{3}-1,-j_{3}-l_{1}-l_{2}-1\atop-j_{1}+j_{2}-l_{1}-l_{2},j_{2}-j_% {3}-l_{2}-l_{3},-j_{1}-j_{3}-l_{1}-l_{3}-1};1\right),

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $!$ : factorial (as in $n!$ ), $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half. and $\Delta(j_{1}j_{2}j_{3})$ : product
Referenced by:: §18.38(iii)
Permalink:: http://dlmf.nist.gov/34.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §34.4 and Ch.34

…

2: 34.5 Basic Properties: $\mathit{6j}$ Symbol

… ►

34.5.9

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\begin{Bmatrix}j_{1}&\frac{1}{2}(j_{2}+l_{2}+j_% {3}-l_{3})&\frac{1}{2}(j_{2}-l_{2}+j_{3}+l_{3})\\ l_{1}&\frac{1}{2}(j_{2}+l_{2}-j_{3}+l_{3})&\frac{1}{2}(-j_{2}+l_{2}+j_{3}+l_{3% })\end{Bmatrix},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.5(ii)
Permalink:: http://dlmf.nist.gov/34.5.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(ii), §34.5 and Ch.34

►

34.5.10

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\begin{Bmatrix}\frac{1}{2}(j_{2}+l_{2}+j_{3}-l_% {3})&\frac{1}{2}(j_{1}-l_{1}+j_{3}+l_{3})&\frac{1}{2}(j_{1}+l_{1}+j_{2}-l_{2})% \\ \frac{1}{2}(j_{2}+l_{2}-j_{3}+l_{3})&\frac{1}{2}(-j_{1}+l_{1}+j_{3}+l_{3})&% \frac{1}{2}(j_{1}+l_{1}-j_{2}+l_{2})\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.5(ii)
Permalink:: http://dlmf.nist.gov/34.5.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(ii), §34.5 and Ch.34

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34.5.11

{(2j_{1}+1)\left((J_{3}+J_{2}-J_{1})(L_{3}+L_{2}-J_{1})-2(J_{3}L_{3}+J_{2}L_{2% }-J_{1}L_{1})\right)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}}\\ =j_{1}E(j_{1}+1)\begin{Bmatrix}j_{1}+1&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}+(j_{1}+1)E(j_{1})\begin{Bmatrix}j_{1}-1&j_{2}&j% _{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half., $J_{r}$ , $L_{r}$ and $E(j)$
Permalink:: http://dlmf.nist.gov/34.5.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(iii), §34.5 and Ch.34

… ►

L_{r}=l_{r}(l_{r}+1),

… ►

34.5.16

(-1)^{j_{1}+j_{2}+j_{3}+j_{1}^{\prime}+j_{2}^{\prime}+l_{1}+l_{2}}\begin{% Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}\begin{Bmatrix}j_{1}^{\prime}&j_{2}^{\prime}&j_{% 3}\\ l_{1}&l_{2}&l_{3}^{\prime}\end{Bmatrix}=\sum_{j}(-1)^{l_{3}+l_{3}^{\prime}+j}(% 2j+1)\begin{Bmatrix}j_{1}&j_{1}^{\prime}&j\\ j_{2}^{\prime}&j_{2}&j_{3}\end{Bmatrix}\begin{Bmatrix}l_{3}&l_{3}^{\prime}&j\\ j_{1}^{\prime}&j_{1}&l_{2}\end{Bmatrix}\begin{Bmatrix}l_{3}&l_{3}^{\prime}&j\\ j_{2}^{\prime}&j_{2}&l_{1}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.5(vi)
Permalink:: http://dlmf.nist.gov/34.5.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(vi), §34.5 and Ch.34

…

3: B. L. J. Braaksma

Profile
Boele L. J. Braaksma

… ►Boele L. J. Braaksma (b. …

4: 25.15 Dirichlet $L$ -functions

§25.15 Dirichlet $L$ -functions

►

§25.15(i) Definitions and Basic Properties

►The notation

L\left(s,\chi\right)

was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series … … ►

§25.15(ii) Zeros

…

5: 20 Theta Functions

…

6: 14.30 Spherical and Spheroidal Harmonics

… ►With

l

and

m

integers such that

|m|\leq l

, and

\theta

and

\phi

angles such that

0\leq\theta\leq\pi

0\leq\phi\leq 2\pi

, …

Y_{l}^{m}\left(\theta,\phi\right)

are known as surface harmonics of the first kind: tesseral for

|m|<l

and sectorial for

|m|=l

. Sometimes

Y_{{l},{m}}\left(\theta,\phi\right)

is denoted by

i^{-l}\mathfrak{D}_{lm}(\theta,\phi)

; also the definition of

Y_{{l},{m}}\left(\theta,\phi\right)

can differ from (14.30.1), for example, by inclusion of a factor

(-1)^{m}

. … ►Here, in spherical coordinates,

\mathrm{L}^{2}

is the squared angular momentum operator: …and

\mathrm{L}_{z}

is the

z

component of the angular momentum operator …

7: 34.8 Approximations for Large Parameters

… ►

34.8.1

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{2}&j_{1}&l_{3}\end{Bmatrix}=(-1)^{j_{1}+j_{2}+j_{3}+l_{3}}\*\left(\frac{4}{% \pi(2j_{1}+1)(2j_{2}+1)(2l_{3}+1)\sin\theta}\right)^{\frac{1}{2}}\*\left(\cos% \left((l_{3}+\tfrac{1}{2})\theta-\tfrac{1}{4}\pi\right)+o\left(1\right)\right),

j_{1},j_{2},j_{3}\gg l_{3}\gg 1

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $o\left(\NVar{x}\right)$ : order less than, $\sin\NVar{z}$ : sine function, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.8 and Ch.34

…

8: 18.39 Applications in the Physical Sciences

… ►where

\mathrm{L}^{2}

is the (squared) angular momentum operator (14.30.12). The eigenfunctions of

\mathrm{L}^{2}

are the spherical harmonics

Y_{{l},{m_{l}}}\left(\theta,\phi\right)

with eigenvalues

\hbar^{2}l(l+1)

, each with degeneracy

2l+1

m_{l}=-l,-l+1,\dots,l

. … ►The functions

\psi_{p,l}(r)

satisfy the equation, … ►The radial Coulomb wave functions

R_{n,l}(r)

, solutions of … ►These, taken together with the infinite sets of bound states for each

l

, form complete sets. …

9: 22 Jacobian Elliptic Functions

…

10: Peter L. Walker

Profile
Peter L. Walker

… ►Peter L. Walker (b. …