DLMF: Untitled Document

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Fox (1960, Table 3) tabulates $2\pi^{1/2}x^{1/4}\*\exp(\tfrac{2}{3}x^{3/2})\*\operatorname{Ai}\left(x\right)$ , $2\pi^{1/2}x^{-1/4}\*\exp(\tfrac{2}{3}x^{3/2})\*\operatorname{Ai}'\left(x\right)$ , $\pi^{1/2}x^{1/4}\*\exp(-\tfrac{2}{3}x^{3/2})\*\operatorname{Bi}\left(x\right)$ , and $\pi^{1/2}x^{-1/4}\*\exp(-\tfrac{2}{3}x^{3/2})\*\operatorname{Bi}'\left(x\right)$ for $\tfrac{3}{2}x^{-3/2}=0(.001)0.05$ , together with similar auxiliary functions for negative values of $x$ . Precision is 10D.

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Zhang and Jin (1996, p. 337) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

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Miller (1946) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ . Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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Sherry (1959) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; 20S.

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Gil et al. (2003c) tabulates the only positive zero of $\operatorname{Gi}'\left(z\right)$ , the first 10 negative real zeros of $\operatorname{Gi}\left(z\right)$ and $\operatorname{Gi}'\left(z\right)$ , and the first 10 complex zeros of $\operatorname{Gi}\left(z\right)$ , $\operatorname{Gi}'\left(z\right)$ , $\operatorname{Hi}\left(z\right)$ , and $\operatorname{Hi}'\left(z\right)$ . Precision is 11 or 12S.

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I. C. Tang (1969) Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.

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External Links:: ISSN 0044-2267, Document, MathReview (B. C. Carlson), ZentralBlatt
Referenced by:: §22.11.
Encodings:: BibTeX

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N. M. Temme (1996a) Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters. Methods Appl. Anal. 3 (3), pp. 335–344.

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External Links:: ISSN 1073-2772, FullText, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §8.12, §8.2(i), §8.2(ii), §8.6(ii).
Encodings:: BibTeX

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I. Thompson (2013) Algorithm 926: incomplete gamma functions with negative arguments. ACM Trans. Math. Software 39 (2), pp. Art. 14, 9.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: 7th item, §8.28(ii).
Encodings:: BibTeX

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See accompanying text — Figure 25.11.1: Hurwitz zeta function $\zeta\left(x,a\right)$ , $a$ = 0. …8, 1, $-20\leq x\leq 10$ . … Magnify

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25.11.30

\zeta\left(s,a\right)=\frac{\Gamma\left(1-s\right)}{2\pi i}\int_{-\infty}^{(0+% )}\frac{e^{az}z^{s-1}}{1-e^{z}}\,\mathrm{d}z,

s\neq 1

,

\Re a>0

,

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $a$ : real or complex parameter, $s$ : complex variable and $z$ : complex variable
Keywords:: contour integral, integral representation
Source:: Erdélyi et al. (1953a, (1.10.5), p. 25); with $z=-t$
Proof sketch:: This contour integral uses Hankel’s contour Figure 5.9.1. Assume $\Re{s}>1$ , collapse the integration path onto the negative real axis, apply (25.11.25), followed by analytic continuation for all $s\neq 1$ , $\Re{a}>0$ , since the integrand is analytic in $z$ , the convergence at the ends of the path is exponential for all $s\neq-1$ and the Hankel contour can be kept well clear of the singularity at the origin in the $z$ -plane.
Referenced by:: §25.11(i)
Permalink:: http://dlmf.nist.gov/25.11.E30
Encodings:: pMML, png, TeX
See also:: Annotations for §25.11(vii), §25.11 and Ch.25

►where the integration contour is a loop around the negative real axis as described for (25.5.20). … ►

25.11.32

\int_{0}^{a}x^{n}\psi\left(x\right)\,\mathrm{d}x=(-1)^{n-1}\zeta'\left(-n% \right)+(-1)^{n}H_{n}\frac{B_{n+1}}{n+1}-\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{% 0.0pt}{}{n}{k}H_{k}\frac{B_{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}^{n}(-1)^{k}% \genfrac{(}{)}{0.0pt}{}{n}{k}\zeta'\left(-k,a\right)a^{n-k},

n=1,2,\dots

,

\Re a>0

,

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $H_{\NVar{n}}$ : harmonic number, $\int$ : integral, $\Re$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $a$ : real or complex parameter and $h(n)$ : harmonic number
Keywords:: definite integral, integral representation
Source:: Adamchik (1998, (17), p. 197)
Referenced by:: Erratum (V1.1.4) for Notation
Permalink:: http://dlmf.nist.gov/25.11.E32
Encodings:: pMML, png, TeX
Notation (effective with 1.1.4):: The notation previously used for the harmonic number $h(n)$ (resp. $h(k)$ ) has been replaced to be $H_{n}$ (resp. $H_{k}$ ).
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.11(viii), §25.11 and Ch.25

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25.11.34

n\int_{0}^{a}\zeta'\left(1-n,x\right)\,\mathrm{d}x=\zeta'\left(-n,a\right)-% \zeta'\left(-n\right)+\frac{B_{n+1}-B_{n+1}\left(a\right)}{n(n+1)},

n=1,2,\dots

,

\Re a>0

.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $n$ : nonnegative integer, $x$ : real variable and $a$ : real or complex parameter
Keywords:: definite integral, integral representation
Source:: Adamchik (1998, (15), p. 196)
Permalink:: http://dlmf.nist.gov/25.11.E34
Encodings:: pMML, png, TeX
See also:: Annotations for §25.11(viii), §25.11 and Ch.25

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34.3.1

\begin{pmatrix}j&j&0\\ m&-m&0\end{pmatrix}=\frac{(-1)^{j-m}}{(2j+1)^{\frac{1}{2}}},

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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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.3.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(i), §34.3 and Ch.34

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34.3.2

\begin{pmatrix}j&j&1\\ m&-m&0\end{pmatrix}=(-1)^{j-m}\frac{2m}{\left(2j(2j+1)(2j+2)\right)^{\frac{1}{% 2}}},

j\geq\tfrac{1}{2}

,

ⓘ

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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.3.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(i), §34.3 and Ch.34

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34.3.3

\begin{pmatrix}j&j&1\\ m&-m-1&1\end{pmatrix}=(-1)^{j-m}\left(\frac{2(j-m)(j+m+1)}{2j(2j+1)(2j+2)}% \right)^{\frac{1}{2}},

j\geq\tfrac{1}{2}

.

ⓘ

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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.3.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(i), §34.3 and Ch.34

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34.3.4

J=j_{1}+j_{2}+j_{3}.

ⓘ

Defines:: $J$ : sum (locally)
Symbols:: $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.5(i)
Permalink:: http://dlmf.nist.gov/34.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(i), §34.3 and Ch.34

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34.3.8

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\begin{pmatrix}j_{2}&j_{3}&j_{1}\\ m_{2}&m_{3}&m_{1}\end{pmatrix}=\begin{pmatrix}j_{3}&j_{1}&j_{2}\\ m_{3}&m_{1}&m_{2}\end{pmatrix},

ⓘ

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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.3(ii)
Permalink:: http://dlmf.nist.gov/34.3.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(ii), §34.3 and Ch.34

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$2j_{1},2j_{2},2j_{3},2l_{1},2l_{2},2l_{3}$	nonnegative integers.
…

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34.1.1

\left(j_{1}\;m_{1}\;j_{2}\;m_{2}|j_{1}\;j_{2}\;j_{3}\,\,m_{3}\right)=(-1)^{j_{% 1}-j_{2}+m_{3}}(2j_{3}+1)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&-m_{3}\end{pmatrix};

ⓘ

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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Sources:: Edmonds (1974, p. 46, Eq. (3.7.3)); Rotenberg et al. (1959, p. 1, Eq. (1.1a))
Permalink:: http://dlmf.nist.gov/34.1.E1
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.15):: A wording change reflects that the Clebsch–Gordan coefficients are an alternative formulation of angular momentum problems, rather than alternative notation for the $\mathit{3j}$ . The sign of $m_{3}$ was changed for clarity.
See also:: Annotations for §34.1 and Ch.34

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25.12.2

\operatorname{Li}_{2}\left(z\right)=-\int_{0}^{z}t^{-1}\ln\left(1-t\right)\,% \mathrm{d}t,

z\in\mathbb{C}\setminus(1,\infty)

.

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Symbols:: $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\in$ : element of, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, $x$ : real variable and $z$ : complex variable
Keywords:: definite integral, integral representation
Source:: Maximon (2003, (2.1), p. 2807)
A&S Ref:: 27.7.1 (is a modified form with $z=1-x$ )
Referenced by:: §25.12(i)
Permalink:: http://dlmf.nist.gov/25.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §25.12(i), §25.12 and Ch.25

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25.12.9

\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}{n^{2}}=-\int_{0}^{\theta}% \ln\left(2\sin\left(\tfrac{1}{2}x\right)\right)\,\mathrm{d}x.

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, $n$ : nonnegative integer, $x$ : real variable and $\theta$ : phase
Keywords:: definite integral, infinite series
Source:: Maximon (2003, (8.7), (8.8), p. 2813)
A&S Ref:: 27.8.6 (integration of first series)
Referenced by:: 1st item
Permalink:: http://dlmf.nist.gov/25.12.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §25.12(i), §25.12 and Ch.25

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►

…

… ►and the second order term in (1.5.18) is positive definite (negative definite), that is, …

… ►A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative. … ►A similar relation holds for the changes in sign of the coefficients of

f(-z)

, and hence for the number of negative zeros of

f(z)

. … ►Both polynomials have one change of sign; hence for each polynomial there is one positive zero, one negative zero, and six complex zeros. … ►Resolvent cubic is

z^{3}+12z^{2}+20z+9=0

with roots

\theta_{1}=-1

,

\theta_{2}=-\tfrac{1}{2}(11+\sqrt{85})

,

\theta_{3}=-\tfrac{1}{2}(11-\sqrt{85})

, and

\sqrt{-\theta_{1}}=1

,

\sqrt{-\theta_{2}}=\tfrac{1}{2}(\sqrt{17}+\sqrt{5})

,

\sqrt{-\theta_{3}}=\tfrac{1}{2}(\sqrt{17}-\sqrt{5})

. … ►with real coefficients, is called stable if the real parts of all the zeros are strictly negative. …

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34.2.1

|j_{r}-j_{s}|\leq j_{t}\leq j_{r}+j_{s},

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Symbols:: $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r\in 1,2,3$ , $t\in 1,2,3$ and $s$ : nonnegative integer
Referenced by:: §34.10, §34.2, §34.3(iv)
Permalink:: http://dlmf.nist.gov/34.2.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.2 and Ch.34

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34.2.2

m_{r}=-j_{r},-j_{r}+1,\dots,j_{r}-1,j_{r},

r=1,2,3

,

ⓘ

Symbols:: $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $r\in 1,2,3$
Referenced by:: §34.4
Permalink:: http://dlmf.nist.gov/34.2.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.2 and Ch.34

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34.2.4

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

ⓘ

Defines:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol
Symbols:: $!$ : factorial (as in $n!$ ), $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $\Delta(j_{1}j_{2}j_{3})$ : product and $s$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §34.2 and Ch.34

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34.2.5

\Delta(j_{1}j_{2}j_{3})=\left(\frac{(j_{1}+j_{2}-j_{3})!(j_{1}-j_{2}+j_{3})!(-% j_{1}+j_{2}+j_{3})!}{(j_{1}+j_{2}+j_{3}+1)!}\right)^{\frac{1}{2}},

ⓘ

Defines:: $\Delta(j_{1}j_{2}j_{3})$ : product (locally)
Symbols:: $!$ : factorial (as in $n!$ ) and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §34.2 and Ch.34

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34.2.6

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{j_{2}-m_{1}+m_{3}}}\frac{(j_{1}+j_{2}+m_% {3})!(j_{2}+j_{3}-m_{1})!}{\Delta(j_{1}j_{2}j_{3})(j_{1}+j_{2}+j_{3}+1)!}\left% (\frac{(j_{1}+m_{1})!(j_{3}-m_{3})!}{(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})% !(j_{3}+m_{3})!}\right)^{\frac{1}{2}}\*{{{}_{3}F_{2}}\left(-j_{1}-j_{2}-j_{3}-% 1,-j_{1}+m_{1},-j_{3}-m_{3};-j_{1}-j_{2}-m_{3},-j_{2}-j_{3}+m_{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 $n!$ ), $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $\Delta(j_{1}j_{2}j_{3})$ : product
Referenced by:: §18.38(iii)
Permalink:: http://dlmf.nist.gov/34.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §34.2 and Ch.34

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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

… ►

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

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negative%20definite

11—20 of 211 matching pages

11: 9.18 Tables

12: Bibliography T

13: 25.11 Hurwitz Zeta Function

14: 34.3 Basic Properties: $\mathit{3j}$ Symbol

15: 34.1 Special Notation

16: 25.12 Polylogarithms

17: 1.5 Calculus of Two or More Variables

18: 1.11 Zeros of Polynomials

19: 34.2 Definition: $\mathit{3j}$ Symbol

20: 34.4 Definition: $\mathit{6j}$ Symbol