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11: Bibliography D

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

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External Links:

Document, MathReview (M. A. Hyman), ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.

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External Links:

ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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History of the Theory of Numbers (3 volumes). Carnegie Institution of Washington, Washington, D.C..

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Notes:

Reprinted by Chelsea Publishing Co., New York, 1966.

External Links:

MathReview, ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

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Algorithm 708: Significant digit computation of the incomplete beta function ratios. ACM Trans. Math. Software 18 (3), pp. 360–373.

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Notes:

Single-precision Fortran, maximum accuracy 14D.

External Links:

ISSN 0098-3500, Document, GAMS (module 708; package TOMS), ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.

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External Links:

ISSN 0377-9017, Document, MathReview, ZentralBlatt

Referenced by:

§14.17(iv).

Encodings:

BibTeX

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12: Bibliography S

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Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$. II. Math. Comp. 30 (134), pp. 337–360.

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Notes:

For corrigendum see same journal v. 30 (1976), no. 136, p. 900

External Links:

FullText, Document, MathReview (R. P. Brent), ZentralBlatt

Referenced by:

§27.12.

Encodings:

BibTeX

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Recursive evaluation of $3j$- and $6j$- coefficients. Comput. Phys. Comm. 11 (2), pp. 269–278.

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Notes:

Double-precision Fortran provided in 3 parts, $j_1$-recursion for $3j$-coefficients, $m_2$-recursion for $3j$-coefficients, and $j_1$-recursion for $6j$-coefficients

External Links:

Document, Code 1, Code 2, Code 3

Referenced by:

8th item.

Encodings:

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Exact recursive evaluation of $3j$- and $6j$-coefficients for quantum-mechanical coupling of angular momenta. J. Mathematical Phys. 16 (10), pp. 1961–1970.

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External Links:

Document, MathReview (J.-L. Calais)

Referenced by:

§34.13, §34.3(iii).

Encodings:

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Semiclassical approximations to $3j$- and $6j$-coefficients for quantum-mechanical coupling of angular momenta. J. Mathematical Phys. 16 (10), pp. 1971–1988.

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External Links:

Document, MathReview (J.-L. Calais)

Referenced by:

§34.8.

Encodings:

BibTeX

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Arbitrary $3n-j$ symbols for $\mathrm{SU}(2)$ . Comput. Phys. Comm. 1 (3), pp. 207–215.

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Notes:

Single-precision Fortran

External Links:

Document, Code

Referenced by:

9th item.

Encodings:

BibTeX

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13: Bibliography B

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A generalisation of the Legendre polynomial. Proc. London Math. Soc. (2) 3 (3), pp. 111–123.

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Notes:

Reviewed in JFM 36.0512.01

External Links:

Document, ZentralBlatt

Referenced by:

§18.12.

Encodings:

BibTeX

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Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.

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External Links:

ISSN 0022-4715, Document, MathReview, ZentralBlatt

Referenced by:

§17.17.

Encodings:

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

Encodings:

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Integrable Hamiltonian systems and the Painlevé property. Phys. Rev. A (3) 25 (3), pp. 1257–1264.

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External Links:

ISSN 0556-2791, Document, MathReview

Referenced by:

§32.16.

Encodings:

BibTeX

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Asymptotic Expansions for the Coefficient Functions Associated with Linear Second-order Differential Equations: The Simple Pole Case. In Asymptotic and Computational Analysis (Winnipeg, MB, 1989), R. Wong (Ed.), Lecture Notes in Pure and Applied Mathematics, Vol. 124, pp. 53–73.

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External Links:

MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.8(iv).

Encodings:

BibTeX

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14: 25.11 Hurwitz Zeta Function

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Figure 25.11.1: Hurwitz zeta function $\zeta (x,a)$, $a$ = 0. 3, 0. … Magnify

… ►where $h,k$ are integers with $1\le h\le k$ and $n=1,2,3,\mathrm{\dots }$. … ►

25.11.23 $${\zeta }^{\prime }(1-2n,\frac{1}{3})=-\frac{\pi (9^n-1)B_{2n}}{8n\sqrt{3}(3^{2n-1}-1)}-\frac{B_{2n}\ln 3}{4n\cdot 3^{2n-1}}-\frac{{(-1)}^n{\psi }^{(2n-1)}\left(\frac{1}{3}\right)}{2\sqrt{3}{(6\pi )}^{2n-1}}-\frac{\left(3^{2n-1}-1\right){\zeta }^{\prime }\left(1-2n\right)}{2\cdot 3^{2n-1}},$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Keywords:

derivative

Source:

Miller and Adamchik (1998, (18), p. 205)

Permalink:

http://dlmf.nist.gov/25.11.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

… ►For further sums see Prudnikov et al. (1990, pp. 396–397) and Hansen (1975, pp. 358–360). … ►

25.11.45 $${\zeta }^{\prime }(-2,a)-\frac{1}{12}a+\frac{1}{9}{a}^3-\left(\frac{1}{6}a-\frac{1}{2}{a}^2+\frac{1}{3}{a}^3\right)\ln a\sim \sum _{k=1}^{\mathrm{\infty }}\frac{2B_{2k+2}}{(2k+2)(2k+1)2k(2k-1)}{a}^{-(2k-1)}.$$

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\sim$: Poincaré asymptotic expansion, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $a$: real or complex parameter

Keywords:

asymptotic approximation

Source:

Elizalde (1986, (19), p. 350)

Permalink:

http://dlmf.nist.gov/25.11.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xii), §25.11 and Ch.25

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15: 1.9 Calculus of a Complex Variable

… ►The cross ratio of $z_1,z_2,z_3,z_4\in ℂ\cup \{\mathrm{\infty }\}$ is defined by ►

1.9.45 $$\frac{(z_1-z_2)(z_3-z_4)}{(z_1-z_4)(z_3-z_2)},$$

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Symbols:

$z$: variable

Permalink:

http://dlmf.nist.gov/1.9.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(iv), §1.9(iv), §1.9 and Ch.1

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$b_2=(a_1^2-a_0{a}_2)/a_0^3,$

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1.9.57 $$\ln f(z)=q_{1}z+q_2{z}^2+q_3{z}^3+\mathrm{\cdots },$$

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Symbols:

$\ln z$: principal branch of logarithm function, $z$: variable and $q_j$: coefficients

Referenced by:

§1.9(vi)

Permalink:

http://dlmf.nist.gov/1.9.E57

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

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$q_3=(3a_3-3a_1{a}_2+a_1^3)/3,$

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16: 16.8 Differential Equations

… ►For further information see Hille (1976, pp. 360–370). …

17: 34.11 Higher-Order $3nj$ Symbols

§34.11 Higher-Order $3nj$ Symbols

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18: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

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$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
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$\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$	$\sqrt{2}(\sqrt{3}+1)$	$\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$
$\pi /6$	$\frac{1}{2}$	$\frac{1}{2}\sqrt{3}$	$\frac{1}{3}\sqrt{3}$	$2$	$\frac{2}{3}\sqrt{3}$	$\sqrt{3}$
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$\pi /3$	$\frac{1}{2}\sqrt{3}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{2}{3}\sqrt{3}$	$2$	$\frac{1}{3}\sqrt{3}$
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$2\pi /3$	$\frac{1}{2}\sqrt{3}$	$-\frac{1}{2}$	$-\sqrt{3}$	$\frac{2}{3}\sqrt{3}$	$-2$	$-\frac{1}{3}\sqrt{3}$
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$5\pi /6$	$\frac{1}{2}$	$-\frac{1}{2}\sqrt{3}$	$-\frac{1}{3}\sqrt{3}$	$2$	$-\frac{2}{3}\sqrt{3}$	$-\sqrt{3}$
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19: 34.12 Physical Applications

§34.12 Physical Applications

►The angular momentum coupling coefficients ($\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols) are essential in the fields of nuclear, atomic, and molecular physics. …$\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).

20: 34.3 Basic Properties: $\mathit{3}j$ Symbol

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

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§34.3(ii) Symmetry

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§34.3(iv) Orthogonality

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§34.3(vi) Sums

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