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1: 34.2 Definition: $\mathit{3}j$ Symbol

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

►The quantities $j_1,j_2,j_3$ in the $\mathit{3}j$ symbol are called angular momenta. …They therefore satisfy the triangle conditions …where $r,s,t$ is any permutation of $1,2,3$. The corresponding projective quantum numbers $m_1,m_2,m_3$ are given by …

2: 26.10 Integer Partitions: Other Restrictions

… ► $p({𝒟}^{\prime }3,n)$ 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 $\{2,3,4,\mathrm{\dots }\}$ is denoted by $T$. … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►Note that $p({𝒟}^{\prime }3,n)\le p(𝒟3,n)$, with strict inequality for $n\ge 9$. It is known that for $k>3$, $p(𝒟k,n)\ge p(\in A_{1,k+3},n)$, with strict inequality for $n$ sufficiently large, provided that $k=2^m-1,m=3,4,5$, or $k\ge 32$; see Yee (2004). …

3: 9.10 Integrals

… ►

9.10.6 $${\int }_{-\mathrm{\infty }}^x\mathrm{Ai}\left(t\right)dt={\pi }^{-1/2}{(-x)}^{-3/4}\cos \left(\frac{2}{3}{(-x)}^{3/2}+\frac{1}{4}\pi \right)+O\left({|x|}^{-9/4}\right),$$ $x\to -\mathrm{\infty }$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $x$: real variable

Proof sketch:

Integrate the leading terms of (9.7.5) and use (9.10.11)

Permalink:

http://dlmf.nist.gov/9.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(ii), §9.10 and Ch.9

►

9.10.7 $${\int }_{-\mathrm{\infty }}^x\mathrm{Bi}\left(t\right)dt={\pi }^{-1/2}{(-x)}^{-3/4}\sin \left(\frac{2}{3}{(-x)}^{3/2}+\frac{1}{4}\pi \right)+O\left({|x|}^{-9/4}\right),$$ $x\to -\mathrm{\infty }$.

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $x$: real variable

Proof sketch:

Integrate the leading terms of (9.7.7) and use (9.10.12)

Referenced by:

§9.10(ii)

Permalink:

http://dlmf.nist.gov/9.10.E7

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See also:

Annotations for §9.10(ii), §9.10 and Ch.9

… ►

${\displaystyle {\int }_0^{\mathrm{\infty }}}\mathrm{Ai}\left(t\right)dt=\frac{1}{3}$ ,

►

${\displaystyle {\int }_{-\mathrm{\infty }}^0}\mathrm{Ai}\left(t\right)dt=\frac{2}{3}$ ,

… ►For further integrals, including the Airy transform, see §9.11(iv), Widder (1979), Prudnikov et al. (1990, §1.8.1), Prudnikov et al. (1992a, pp. 405–413), Prudnikov et al. (1992b, §4.3.25), Vallée and Soares (2010, Chapters 3, 4).

4: Publications

… ►

Toward a Revised NBS Handbook of Mathematical Functions, Technical Report NISTIR 6072 (September 1997), National Institute of Standards and Technology.

… ►

Using Numerical Grid Generation to Facilitate 3D Visualization of Complicated Mathematical Functions, Technical Report NISTIR 6413 (November 1999), National Institute of Standards and Technology.

… ►

Handbook of Mathematical Functions, in A Century of Excellence in Measurements Standards and Technology (D. R. Lide, ed.), CRC Press, pp. 135–139.

►

The NIST Digital Library of Mathematical Functions Project, Annals of Mathematics and Artificial Intelligence—Special Issue on Mathematical Knowledge Management, Vol. 38, Nos. 1–3, pp. 105–119.

►

Technical Aspects of the Digital Library of Mathematical Functions, Annals of Mathematics and Artificial Intelligence—Special Issue on Mathematical Knowledge Management, Vol. 38, Nos. 1–3, pp. 121–136.

…

5: 22.9 Cyclic Identities

… ► … ►

22.9.10 $$d_{1,3}^{(2)}{d}_{2,3}^{(2)}+d_{2,3}^{(2)}{d}_{3,3}^{(2)}+d_{3,3}^{(2)}{d}_{1,3}^{(2)}=d_{1,3}^{(4)}{d}_{2,3}^{(4)}+d_{2,3}^{(4)}{d}_{3,3}^{(4)}+d_{3,3}^{(4)}{d}_{1,3}^{(4)}=\kappa (\kappa +2).$$

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Symbols:

$d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Referenced by:

§22.9(ii), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10)

Permalink:

http://dlmf.nist.gov/22.9.E10

Encodings:

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Correction (effective with 1.0.25):

Originally this equation was written incorrectly for all $d_{m,p}^{(2)}$ and $d_{m,p}^{(4)}$ with $p=2$. It has been corrected by writing $p=3$ throughout.

Suggested 2019-11-07 by Juan Miguel Nieto

See also:

Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

… ►

§22.9(iii) Typical Identities of Rank 3

… ►

22.9.22 $$s_{1,3}^{(2)}{c}_{1,3}^{(2)}{d}_{2,3}^{(2)}{d}_{3,3}^{(2)}+s_{2,3}^{(2)}{c}_{2,3}^{(2)}{d}_{3,3}^{(2)}{d}_{1,3}^{(2)}+s_{3,3}^{(2)}{c}_{3,3}^{(2)}{d}_{1,3}^{(2)}{d}_{2,3}^{(2)}=\frac{{\kappa }^2+k^2-1}{1-{\kappa }^2}\left(s_{1,3}^{(2)}{c}_{1,3}^{(2)}+s_{2,3}^{(2)}{c}_{2,3}^{(2)}+s_{3,3}^{(2)}{c}_{3,3}^{(2)}\right),$$

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Symbols:

$k$: modulus, $s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity, $d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Permalink:

http://dlmf.nist.gov/22.9.E22

Encodings:

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See also:

Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

►

22.9.23 $$s_{1,3}^{(4)}{d}_{1,3}^{(4)}{c}_{2,3}^{(4)}{c}_{3,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}{c}_{3,3}^{(4)}{c}_{1,3}^{(4)}+s_{3,3}^{(4)}{d}_{3,3}^{(4)}{c}_{1,3}^{(4)}{c}_{2,3}^{(4)}=\frac{{\kappa }^2}{1-{\kappa }^2}\left(s_{1,3}^{(4)}{d}_{1,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}\right).$$

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Symbols:

$s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity, $d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Permalink:

http://dlmf.nist.gov/22.9.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

…

6: 1.11 Zeros of Polynomials

… ► $f(z)=z^3-6z^2+6z-2$, $g(w)=w^3-6w-6$, $A=3\sqrt[3]{4}$, $B=3\sqrt[3]{2}$. Roots of $f(z)=0$ are $2+\sqrt[3]{4}+\sqrt[3]{2}$, $2+\sqrt[3]{4}\rho +\sqrt[3]{2}{\rho }^2$, $2+\sqrt[3]{4}{\rho }^2+\sqrt[3]{2}\rho$. … ►Set $z=w-\frac{1}{4}a$ to reduce $f(z)=z^4+a{z}^3+b{z}^2+cz+d$ to … ►Resolvent cubic is $z^3+12z^2+20z+9=0$ with roots ${\theta }_1=-1$, ${\theta }_2=-\frac{1}{2}(11+\sqrt{85})$, ${\theta }_3=-\frac{1}{2}(11-\sqrt{85})$, and $\sqrt{-{\theta }_1}=1$, $\sqrt{-{\theta }_2}=\frac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-{\theta }_3}=\frac{1}{2}(\sqrt{17}-\sqrt{5})$. So $2{\alpha }_1=1+\sqrt{17}$, $2{\alpha }_2=1-\sqrt{17}$, $2{\alpha }_3=-1+\sqrt{5}$, $2{\alpha }_4=-1-\sqrt{5}$, and the roots of $f(z)=0$ are $\frac{1}{2}(3\pm \sqrt{17})$, $\frac{1}{2}(1\pm \sqrt{5})$. …

7: Bibliography L

… ►

Démonstration d’un Théoréme d’Arithmétique. Nouveau Mém. Acad. Roy. Sci. Berlin, pp. 123–133 (French).

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Notes:

Reprinted in Oeuvres, Vol. 3, pp. 189–201, Gauthier-Villars, Paris, 1867

External Links:

FullText

Referenced by:

§27.13(iii).

Encodings:

BibTeX

… ►

Quantum Mechanics: Non-relativistic Theory. Pergamon Press Ltd., Oxford.

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Notes:

Revised second edition. Course of Theoretical Physics, Vol. 3. Translated from the Russian by J. B. Sykes and J. S. Bell

External Links:

MathReview, ZentralBlatt

Referenced by:

§9.16.

Encodings:

BibTeX

… ►

Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§4.13, 2nd item, §4.48(iv).

Encodings:

BibTeX

… ►

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

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Notes:

Corrigendum, ibid. 15(3) (1989), 287

External Links:

ISSN 0098-3500, Document, MathReview (T. Feagin), ZentralBlatt

Referenced by:

§3.8(iii).

Encodings:

BibTeX

… ►

Ramanujan’s function $\tau (n)$ . Duke Math. J. 10 (3), pp. 483–492.

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External Links:

Document, MathReview (H. S. Zuckerman), ZentralBlatt

Referenced by:

§27.20, §27.20, §27.21.

Encodings:

BibTeX

…

8: 34.3 Basic Properties: $\mathit{3}j$ Symbol

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

… ►

§34.3(ii) Symmetry

… ►

§34.3(iv) Orthogonality

… ►

§34.3(vi) Sums

… ►

9: 23.22 Methods of Computation

… ►Suppose that the invariants $g_2=c$, $g_3=d$, are given, for example in the differential equation (23.3.10) or via coefficients of an elliptic curve (§23.20(ii)). … ►

In the general case, given by $cd\ne 0$, we compute the roots $\alpha$, $\beta$, $\gamma$, say, of the cubic equation $4t^3-ct-d=0$; see §1.11(iii). These roots are necessarily distinct and represent $e_1$, $e_2$, $e_3$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is $c^3-27d^2>0$, then $e_1$, $e_2$, $e_3$ can be identified via (23.5.1), and $k^2$, $k_{}^{\prime }{}_{}{}^2$ obtained from (23.6.16).

If , or $c$ and $d$ are not both real, then we label $\alpha$, $\beta$, $\gamma$ so that the triangle with vertices $\alpha$, $\beta$, $\gamma$ is positively oriented and $[\alpha ,\gamma ]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha$, $\beta$, $\gamma$ are collinear, then we label them so that $\beta$ is on the line segment $(\alpha ,\gamma )$. In consequence, $k^2=(\beta -\gamma )/(\alpha -\gamma )$, $k_{}^{\prime }{}_{}{}^2=(\alpha -\beta )/(\alpha -\gamma )$ satisfy $\mathrm{\Im }{k}^2\ge 0\ge \mathrm{\Im }k_{}^{\prime }{}_{}{}^2$ (with strict inequality unless $\alpha$, $\beta$, $\gamma$ are collinear); also $|k^2|$, $|k_{}^{\prime }{}_{}{}^2|\le 1$.

Finally, on taking the principal square roots of $k^2$ and $k_{}^{\prime }{}_{}{}^2$ we obtain values for $k$ and $k^{\prime }$ that lie in the 1st and 4th quadrants, respectively, and $2{\omega }_1$, $2{\omega }_3$ are given by

23.22.1 $$2{\omega }_{1}M(1,k^{\prime })=-2\mathrm{i}{\omega }_{3}M(1,k)=\frac{\pi }{3}\sqrt{\frac{c(2+k^{2}k_{}^{\prime }{}_{}{}^2)(k_{}^{\prime }{}_{}{}^2-k^2)}{d(1-k^{2}k_{}^{\prime }{}_{}{}^2)}},$$

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

Proof sketch:

Combine (23.6.16), (23.6.17), (22.20.6), (23.3.5)–(23.3.7), and use analytical continuation.

Referenced by:

§23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E1

Encodings:

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See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 2 possible choices of the square root.

►

If $d=0$, then

23.22.2 $$2{\omega }_1=-2\mathrm{i}{\omega }_3=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{4}\right)\right)}^2}{2\sqrt{\pi }{c}^{1/4}}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§22.20(iv), §23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E2

Encodings:

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See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

There are 4 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 4 rotations of a square lattice. The lemniscatic case occurs when $c>0$ and ${\omega }_1>0$.

►

If $c=0$, then

23.22.3 $$2{\omega }_1=2{\mathrm{e}}^{-\pi \mathrm{i}/3}{\omega }_3=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{3}\right)\right)}^3}{2\pi d^{1/6}}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E3

Encodings:

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See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

There are 6 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 6 rotations of a lattice of equilateral triangles. The equianharmonic case occurs when $d>0$ and ${\omega }_1>0$.

… ►Assume $c=g_2=-4(3-2\mathrm{i})$ and $d=g_3=4(4-2\mathrm{i})$. …

10: Bibliography

… ►

Algorithm 556: Exponential integrals. ACM Trans. Math. Software 6 (3), pp. 420–428.

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Notes:

Single-precision Fortran, maximum accuracy 14S. For a Remark see same journal v. 9 (1983), no. 4, p. 525.

External Links:

ISSN 0098-3500, Document, Remark,GAMS (module 556; package TOMS), ZentralBlatt

Referenced by:

1st item, 1st item.

Encodings:

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Computation of exponential integrals. ACM Trans. Math. Software 6 (3), pp. 365–377.

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External Links:

ISSN 0098-3500, Document, MathReview (M. M. Chawla), ZentralBlatt

Referenced by:

§8.25(v).

Encodings:

BibTeX

… ►

Normal forms of functions near degenerate critical points, the Weyl groups $A_k,D_k,E_k$ and Lagrangian singularities. Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).

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Notes:

In Russian. English translation: Functional Anal. Appl., 6(1973), pp. 254–272.

External Links:

MathReview (G. R. Belickii), ZentralBlatt

Referenced by:

§36.2(i).

Encodings:

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… ►

Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 30(1975), no. 5, pp. 1–75.

External Links:

Document, MathReview (D. Siersma), ZentralBlatt

Referenced by:

§36.2(i), Ch.36.

Encodings:

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… ►

Chiral-symmetry breaking in QCD. I. The infrared domain. Phys. Rev. D (3) 37 (8), pp. 2290–2295.

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External Links:

ISSN 0556-2821, Document, MathReview

Referenced by:

§15.18.

Encodings:

BibTeX

…