.世界杯决赛的时间_『网址:687.vii』日本2014年世界杯成绩_b5p6v3_2022年11月29日5时20分49秒_ewk244wui

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21: 26.8 Set Partitions: Stirling Numbers

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§26.8(vii) Asymptotic Approximations

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26.8.40 $$s(n+1,k+1)\sim {(-1)}^{n-k}\frac{n!}{k!}{(\gamma +\ln n)}^k,$$ $n\to \mathrm{\infty }$,

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Symbols:

$\gamma$: Euler’s constant, $s(n,k)$: Stirling number of the first kind, $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.8.E40

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See also:

Annotations for §26.8(vii), §26.8 and Ch.26

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26.8.41 $$s(n+k,k)\sim \frac{{(-1)}^n}{2^{n}n!}{k}^{2n},$$ $k\to \mathrm{\infty }$,

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Symbols:

$s(n,k)$: Stirling number of the first kind, $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.8.E41

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See also:

Annotations for §26.8(vii), §26.8 and Ch.26

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26.8.42 $$S(n,k)\sim \frac{k^n}{k!},$$ $n\to \mathrm{\infty }$,

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Symbols:

$S(n,k)$: Stirling number of the second kind, $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.8.E42

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See also:

Annotations for §26.8(vii), §26.8 and Ch.26

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26.8.43 $$S(n+k,k)\sim \frac{k^{2n}}{2^{n}n!},$$ $k\to \mathrm{\infty }$,

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Symbols:

$S(n,k)$: Stirling number of the second kind, $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§26.8(vii)

Permalink:

http://dlmf.nist.gov/26.8.E43

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See also:

Annotations for §26.8(vii), §26.8 and Ch.26

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22: Bibliography C

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A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

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External Links:

ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.

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External Links:

ISSN 0010-4655, Document, MathReview

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Permutation symmetry for theta functions. J. Math. Anal. Appl. 378 (1), pp. 42–48.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Mohammad Ahmad), ZentralBlatt

Referenced by:

§20.11(v), §20.7(i), §20.7(i), §20.7(ii), §20.7(ii), §20.7(ii), §20.7(iii), §20.7(iii), §20.7(vii), §20.7(vii), Profile Bille C. Carlson.

Encodings:

BibTeX

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Optimized fast Hankel transform filters. Geophysical Prospecting 38 (5), pp. 545–568.

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External Links:

Document

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Computation of Hankel transforms. SIAM Rev. 14 (2), pp. 278–285.

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External Links:

ISSN 1095-7200, Document, MathReview (R. D. Riess), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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23: 9.18 Tables

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National Bureau of Standards (1958) tabulates $A_0(x)\equiv \pi \mathrm{Hi}\left(-x\right)$ and $-A_0^{\prime }(x)\equiv \pi {\mathrm{Hi}}^{\prime }\left(-x\right)$ for $x=0(.01)1(.02)5(.05)11$ and $1/x=0.01(.01)0.1$; ${\int }_0^x{A}_0(t)dt$ for $x=0.5,1(1)11$. Precision is 8D.

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

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§9.18(vii) Generalized Airy Functions

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24: Bibliography H

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Fast Hankel transform algorithm. IEEE Trans. Acoust. Speech Signal Process. 32 (3), pp. 666–671.

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External Links:

ISSN 0096-3518, FullText, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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A Course of Pure Mathematics. 10th edition, Cambridge University Press.

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Notes:

Numerous reprintings exist, including the Centenary Edition with Foreword by T. W. Körner (Cambridge Univerity Press, 2008).

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.4(ii), §1.4(iii), §1.4(iv), §1.4(v), §1.4(vi), §1.4(vii), §4.4(iii), §4.6(i), §4.6(ii).

Encodings:

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Applied and Computational Complex Analysis. Vol. 1: Power Series—Integration—Conformal Mapping—Location of Zeros. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York.

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Notes:

Reprinted in 1988.

External Links:

ISBN 0-471-37244-7, MathReview (M. Marden), ZentralBlatt

Referenced by:

§1.10(vii), §1.9(vi), Ch.3.

Encodings:

BibTeX

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Coulomb Wave Functions. In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.), pp. 408–465.

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External Links:

MathReview (G. Källén)

Referenced by:

§33.2(iii), §33.23(vii), §33.23(vii), §33.24, §33.5(ii), §33.7, Ch.33.

Encodings:

BibTeX

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Analytical structure and properties of Coulomb wave functions for real and complex energies. Ann. Physics 155 (2), pp. 461–493.

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External Links:

ISSN 0003-4916, Document, MathReview

Referenced by:

§33.10(ii), §33.13, §33.22(vii).

Encodings:

BibTeX

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25: 13.14 Definitions and Basic Properties

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13.14.4 $$M(a,b,z)={\mathrm{e}}^{\frac{1}{2}z}{z}^{-\frac{1}{2}b}{M}_{\frac{1}{2}b-a,\frac{1}{2}b-\frac{1}{2}}\left(z\right),$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Referenced by:

§10.16, (13.11.2), §13.11, §13.14(iii), §13.14(vii), §13.16(i), §13.18(i), §13.18(ii), §13.18(iii), §13.18(iv), §13.18(v), §13.23(iii), §13.25, §13.8(ii), §13.8(ii), §13.8(iii)

Permalink:

http://dlmf.nist.gov/13.14.E4

Encodings:

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See also:

Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

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§13.14(vii) Connection Formulas

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13.14.31 $$W_{\kappa ,\mu }\left(z\right)=W_{\kappa ,-\mu }\left(z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.14(iii), §13.14(vii)

Permalink:

http://dlmf.nist.gov/13.14.E31

Encodings:

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See also:

Annotations for §13.14(vii), §13.14 and Ch.13

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13.14.32 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left(z\right)=\frac{{\mathrm{e}}^{\pm (\kappa -\mu -\frac{1}{2})\pi \mathrm{i}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{W}_{\kappa ,\mu }\left(z\right)+\frac{{\mathrm{e}}^{\pm \kappa \pi \mathrm{i}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{W}_{-\kappa ,\mu }\left({\mathrm{e}}^{\pm \pi \mathrm{i}}z\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§13.14(vii)

Permalink:

http://dlmf.nist.gov/13.14.E32

Encodings:

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See also:

Annotations for §13.14(vii), §13.14 and Ch.13

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13.14.33 $$W_{\kappa ,\mu }\left(z\right)=\frac{\mathrm{\Gamma }\left(-2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{M}_{\kappa ,\mu }\left(z\right)+\frac{\mathrm{\Gamma }\left(2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{M}_{\kappa ,-\mu }\left(z\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.1.34

Referenced by:

§13.14(i), §13.14(vii)

Permalink:

http://dlmf.nist.gov/13.14.E33

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Annotations for §13.14(vii), §13.14 and Ch.13

26: 10.21 Zeros

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§10.21(vii) Asymptotic Expansions for Large Order

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10.21.24 $$\theta \left(-2^{\frac{1}{3}}\alpha \right)=\pi t,$$

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Defines:

$\alpha$: coefficients (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $\theta \left(z\right)$: Airy phase function

Permalink:

http://dlmf.nist.gov/10.21.E24

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See also:

Annotations for §10.21(vii), §10.21 and Ch.10

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10.21.32 $$j_{\nu ,m}\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k}{{\nu }^{2k/3}},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$ , $m$: integer, $k$: nonnegative integer, $\nu$: complex parameter and $\alpha$: coefficients

Referenced by:

§10.21(vii), §10.21(vii)

Permalink:

http://dlmf.nist.gov/10.21.E32

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Annotations for §10.21(vii), §10.21 and Ch.10

… ►(There is an inaccuracy in Figures 11 and 14 in this reference. … ►This information includes asymptotic approximations analogous to those given in §§10.21(vi), 10.21(vii), and 10.21(x). …

27: 7.25 Software

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§7.25(vii) $ℱ\left(z\right)$, $G\left(z\right)$, $z\in ℂ$

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28: 8.28 Software

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§8.28(vii) Generalized Exponential Integral for Complex Argument and/or Parameter

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29: 3.2 Linear Algebra

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3.2.2

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Permalink:

http://dlmf.nist.gov/3.2.E2

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See also:

Annotations for §3.2(i), §3.2 and Ch.3

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3.2.3

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Referenced by:

§3.2(i)

Permalink:

http://dlmf.nist.gov/3.2.E3

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See also:

Annotations for §3.2(i), §3.2 and Ch.3

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§3.2(vii) Computation of Eigenvalues

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30: 20.7 Identities

… ►The symmetry, applicable also to §§20.7(iii) and 20.7(vii), is obtained by modifying traditional theta functions in the manner recommended by Carlson (2011) and used for further purposes by Fukushima (2012). … ►

§20.7(vii) Derivatives of Ratios of Theta Functions

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20.7.25 $$\frac{d}{dz}\left(\frac{{\theta }_2\left(z\right|\tau )}{{\theta }_4\left(z\right|\tau )}\right)=-\frac{{\theta }_3^2\left(0\right|\tau ){\theta }_1\left(z\right|\tau ){\theta }_3\left(z\right|\tau )}{{\theta }_4^2\left(z\right|\tau )}.$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.7.E25

Encodings:

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See also:

Annotations for §20.7(vii), §20.7 and Ch.20

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