DLMF: Untitled Document

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G. A. Kalugin, D. J. Jeffrey, and R. M. Corless (2012) Bernstein, Pick, Poisson and related integral expressions for Lambert

W

. Integral Transforms Spec. Funct. 23 (11), pp. 817–829.

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External Links:: ISSN 1065-2469, Document, Link, MathReview (Ross C. McPhedran), ZentralBlatt
Referenced by:: §4.13.
Encodings:: BibTeX

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E. L. Kaplan (1948) Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.

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External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §19.12.
Encodings:: BibTeX

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R. P. Kerr (1963) Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11 (5), pp. 237–238.

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External Links:: Document, MathReview (C. Gilbert), ZentralBlatt
Referenced by:: §31.17(ii).
Encodings:: BibTeX

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K. S. Kölbig (1968) Algorithm 327: Dilogarithm [S22]. Comm. ACM 11 (4), pp. 270–271.

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Notes:: Includes Algol program, maximum accuracy 12D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §25.21(v).
Encodings:: BibTeX

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T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:: ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §15.14, §15.14.
Encodings:: BibTeX

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Table 4.17.1: Trigonometric functions: values at multiples of

\frac{1}{12}\pi

.

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$\theta$	$\sin\theta$	$\cos\theta$	$\tan\theta$	$\csc\theta$	$\sec\theta$	$\cot\theta$
…
$11\pi/12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$-\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$-(2-\sqrt{3})$	$\sqrt{2}(\sqrt{3}+1)$	$-\sqrt{2}(\sqrt{3}-1)$	$-(2+\sqrt{3})$
…

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Table 26.12.1: Plane partitions.

► ►►►►►

$n$	$\operatorname{pp}\left(n\right)$	$n$	$\operatorname{pp}\left(n\right)$	$n$	$\operatorname{pp}\left(n\right)$
…
4	13	21	1 18794	38	1324 41995
…
11	859	28	24 83234	45	17740 79109
12	1479	29	37 59612	46	25435 35902
…

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26.12.26

\operatorname{pp}\left(n\right)\sim\frac{\left(\zeta\left(3\right)\right)^{7/3% 6}}{2^{11/36}(3\pi)^{1/2}n^{25/36}}\exp\left(3\left(\zeta\left(3\right)\right)% ^{1/3}\left(\tfrac{1}{2}n\right)^{2/3}+\zeta'\left(-1\right)\right),

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\sim$ : asymptotic equality, $\exp\NVar{z}$ : exponential function, $\operatorname{pp}\left(\NVar{n}\right)$ : number of plane partitions of $n$ , $n$ : nonnegative integer and $\pi$ : plane partition
Referenced by:: §26.12(iv), Erratum (V1.0.5) for Equation (26.12.26)
Permalink:: http://dlmf.nist.gov/26.12.E26
Encodings:: pMML, png, TeX
Errata (effective with 1.0.5):: The original statement of this equation, $\operatorname{pp}\left(n\right)\sim\left(\frac{\zeta\left(3\right)}{2^{11}n^{2% 5}}\right)^{1/36}\*\exp\left(3\left(\frac{\zeta\left(3\right)n^{2}}{4}\right)^% {1/3}+\zeta'\left(-1\right)\right)$ , has been corrected.
Reported 2011-09-05 by Suresh Govindarajan
See also:: Annotations for §26.12(iv), §26.12 and Ch.26

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34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\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}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

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Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
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., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►

34.6.2

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.

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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.
Referenced by:: §34.13
Permalink:: http://dlmf.nist.gov/34.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

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W. Gautschi (1966) Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.

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External Links:: ISSN 0001-0782, Document
Referenced by:: §33.26(ii).
Encodings:: BibTeX

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W. Gautschi (1969) Algorithm 363: Complex error function. Comm. ACM 12 (11), pp. 635.

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Notes:: Includes Algol program, maximum accuracy 10S. For certification see same journal v. 15 (1972), pp. 465–466
External Links:: ISSN 0001-0782, Document
Referenced by:: §7.25(iii).
Encodings:: BibTeX

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A. Gervois and H. Navelet (1984) Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities. J. Math. Phys. 25 (11), pp. 3350–3356.

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External Links:: ISSN 0022-2488, Document, MathReview (B. R. Bhonsle), ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

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H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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External Links:: FullText, MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

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V. I. Gromak (1975) Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).

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Notes:: In Russian. English translation: Differential Equations 11(11), pp. 285–287
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.7(iii), §32.7(vi).
Encodings:: BibTeX

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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}

.

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	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
…
$11$	$0$	$\tfrac{5}{6}$	$0$	$-\tfrac{11}{2}$	$0$	$11$	$0$	$-11$	$0$	$\tfrac{55}{6}$	$-\tfrac{11}{2}$	$1$
$12$	$-\tfrac{691}{2730}$	$0$	$5$	$0$	$-\tfrac{33}{2}$	$0$	$22$	$0$	$-\tfrac{33}{2}$	$0$	$11$	$-6$	$1$
…

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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}

.

► ►►►►

	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
…
$11$	$\tfrac{691}{4}$	$0$	$-\tfrac{1705}{2}$	$0$	$\tfrac{2805}{4}$	$0$	$-231$	$0$	$\tfrac{165}{4}$	$0$	$-\tfrac{11}{2}$	$1$
…

►

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R. Barakat (1961) Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.

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External Links:: FullText, MathReview (J. C. P. Miller), ZentralBlatt
Referenced by:: §8.7.
Encodings:: BibTeX

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B. C. Berndt, S. Bhargava, and F. G. Garvan (1995) Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.

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External Links:: ISSN 0002-9947, FullText, MathReview (J. Borwein), ZentralBlatt
Referenced by:: §15.8(v), §20.11(iii).
Encodings:: BibTeX

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F. Bethuel (1998) Vortices in Ginzburg-Landau Equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 11–19.

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External Links:: ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §17.17.
Encodings:: BibTeX

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A. Bhattacharyya and L. Shafai (1988) Theoretical and experimental investigation of the elliptical annual ring antenna. IEEE Trans. Antennas and Propagation 36 (11), pp. 1526–1530.

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External Links:: Document
Referenced by:: 5th item.
Encodings:: BibTeX

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R. L. Bishop (1981) Rainbow over Woolsthorpe Manor. Notes and Records Roy. Soc. London 36 (1), pp. 3–11 (1 plate).

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External Links:: ISSN 0035-9149, Document, MathReview
Referenced by:: Sidebar 9.SB1.
Encodings:: BibTeX

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V. N. Faddeyeva and N. M. Terent’ev (1961) Tables of Values of the Function

w(z)=e^{-z^{2}}(1+2i\pi^{-1/2}\int_{0}^{z}e^{t^{2}}dt)

for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.

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External Links:: MathReview, ZentralBlatt
Referenced by:: V. N. Faddeeva and N. M. Terent’ev (1954).
Encodings:: BibTeX

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N. Fleury and A. Turbiner (1994) Polynomial relations in the Heisenberg algebra. J. Math. Phys. 35 (11), pp. 6144–6149.

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External Links:: ISSN 0022-2488, Document, MathReview (Shao-Ming Fei), ZentralBlatt
Referenced by:: §13.3(ii), §15.5(i), §16.3(i).
Encodings:: BibTeX

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A. S. Fokas and M. J. Ablowitz (1982) On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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External Links:: ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.13(i), §32.7(i), §32.7(iii).
Encodings:: BibTeX

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P. J. Forrester and N. S. Witte (2004) Application of the

\tau

-function theory of Painlevé equations to random matrices:

\rm P_{\rm VI}

, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. 174, pp. 29–114.

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External Links:: ISSN 0027-7630, MathReview, ZentralBlatt
Referenced by:: §32.10(vi), §32.6(v).
Encodings:: BibTeX

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L. W. Fullerton (1972) Algorithm 435: Modified incomplete gamma function. Comm. ACM 15 (11), pp. 993–995.

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Notes:: Includes Fortran code. For a Remark see ACM Trans. Math. Software 4 (1978), no. 3, pp. 296–304.
External Links:: ISSN 0001-0782, Document, Remark
Referenced by:: §8.28(ii).
Encodings:: BibTeX

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28.6.2

a_{1}\left(q\right)=1+q-\tfrac{1}{8}q^{2}-\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^% {4}+\tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}+\tfrac{55}{94\;37184}q^{7% }-\tfrac{83}{353\;89440}q^{8}+\cdots,

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Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(i), §28.6 and Ch.28

►

28.6.3

b_{1}\left(q\right)=1-q-\tfrac{1}{8}q^{2}+\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^% {4}-\tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}-\tfrac{55}{94\;37184}q^{7% }-\tfrac{83}{353\;89440}q^{8}+\cdots,

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Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(i), §28.6 and Ch.28

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28.6.10

a_{5}\left(q\right)=25+\tfrac{1}{48}q^{2}+\tfrac{11}{7\;74144}q^{4}+\tfrac{1}{% 1\;47456}q^{5}+\tfrac{37}{8918\;13888}q^{6}+\cdots,

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Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(i), §28.6 and Ch.28

►

28.6.11

b_{5}\left(q\right)=25+\tfrac{1}{48}q^{2}+\tfrac{11}{7\;74144}q^{4}-\tfrac{1}{% 1\;47456}q^{5}+\tfrac{37}{8918\;13888}q^{6}+\cdots,

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Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(i), §28.6 and Ch.28

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28.6.21

2^{\ifrac{1}{2}}\operatorname{ce}_{0}\left(z,q\right)=1-\tfrac{1}{2}q\cos 2z+% \tfrac{1}{32}q^{2}\left(\cos 4z-2\right)-\tfrac{1}{128}q^{3}\left(\tfrac{1}{9}% \cos 6z-11\cos 2z\right)+\cdots,

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $q=h^{2}$ : parameter and $z$ : complex variable
A&S Ref:: 20.2.27
Referenced by:: §28.6(ii)
Permalink:: http://dlmf.nist.gov/28.6.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(ii), §28.6 and Ch.28

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R. J. Wells (1999) Rapid approximation to the Voigt/Faddeeva function and its derivatives. J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.

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Notes:: Includes Fortran code in the appendices to compute the functions to an accuracy between $10^{-2}$ and $10^{-5}$ .
External Links:: ISSN 0022-4073, Document
Referenced by:: §7.21, §7.21.
Encodings:: BibTeX

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J. A. Wilson (1980) Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 11 (4), pp. 690–701.

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External Links:: ISSN 0036-1410, Document, MathReview (R. G. Buschman), ZentralBlatt
Referenced by:: §18.25(i), §18.25(i), §18.25(ii), §18.25(iii), §18.26(i), §18.26(ii), §18.38(iii).
Encodings:: BibTeX

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G. Wolf (2008) On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.

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External Links:: ISSN 1044-677X, FullText
Referenced by:: §28.4(vii).
Encodings:: BibTeX

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E. M. Wright (1935) The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. (2) 38, pp. 257–270.

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External Links:: Document, ZentralBlatt
Referenced by:: §10.46.
Encodings:: BibTeX

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E. M. Wright (1940b) The generalized Bessel function of order greater than one. Quart. J. Math., Oxford Ser. 11, pp. 36–48.

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External Links:: Document, MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §10.46.
Encodings:: BibTeX

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.世界杯足球决赛时间表_『网址:68707.vip』世界杯德国的球员名单_b5p6v3_2022年11月29日4时47分38秒_mgo6cgio4

11—20 of 144 matching pages

11: Bibliography K

12: 4.17 Special Values and Limits

13: 26.12 Plane Partitions

14: 34.6 Definition: $\mathit{9j}$ Symbol

15: Bibliography G

16: 24.2 Definitions and Generating Functions

17: Bibliography B

18: Bibliography F

19: 28.6 Expansions for Small $q$

20: Bibliography W