.世界杯小组赛抽签出线_『网址:687.vii』14年阿根廷世界杯成绩_b5p6v3_2022年11月29日9时32分58秒_iwusqq8gw

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11: Bibliography W

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The Algebraic Eigenvalue Problem. Monographs on Numerical Analysis. Oxford Science Publications, The Clarendon Press, Oxford University Press, Oxford.

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Notes:

Paperback reprint of original edition, published in 1965 by Clarendon Press, Oxford.

External Links:

ISBN 0-19-853418-3, MathReview, ZentralBlatt

Referenced by:

§3.2(v), §3.2(vi), §3.2(vii).

Encodings:

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A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.

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External Links:

Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§13.23(iv).

Encodings:

BibTeX

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Computation with Recurrence Relations. Pitman, Boston, MA.

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External Links:

ISBN 0-273-08508-5, MathReview (S. Conde), ZentralBlatt

Referenced by:

§13.29(iv), §16.25, §2.9(iii), §3.10(iii), §3.6(iii), §3.6(v), §3.6(vii).

Encodings:

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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:

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Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.

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External Links:

ISSN 0029-599X, Document, MathReview (A.J. Rodrigues), ZentralBlatt

Referenced by:

§10.74(vii), §3.5(vii).

Encodings:

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12: 34.9 Graphical Method

… ►For an account of this method see Brink and Satchler (1993, Chapter VII). For specific examples of the graphical method of representing sums involving the $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

13: Bibliography M

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Asymptotic expansions for the zeros of certain special functions. J. Comput. Appl. Math. 145 (2), pp. 261–267.

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External Links:

ISSN 0377-0427, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.70, §11.4(vii), §6.13.

Encodings:

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Contribution no. 14. The Riemann zeta function. BIT 5, pp. 138–141.

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Referenced by:

§25.21(ii).

Encodings:

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On the $G$-function. VII, VIII. Nederl. Akad. Wetensch., Proc. 49, pp. 1063–1072, 1165–1175 = Indagationes Math. 8, 661–670, 713–723 (1946).

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§16.11(ii).

Encodings:

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Jacobian Elliptic Function Tables. Dover Publications Inc., New York.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§22.20(vii).

Encodings:

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Stirling numbers of the second kind. Duke Math. J. 25 (1), pp. 29–43.

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External Links:

Document, MathReview (A. Sade), ZentralBlatt

Referenced by:

§26.1, §26.8(vii), Notations S.

Encodings:

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14: 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:

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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:

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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:

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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:

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15: 9.18 Tables

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Corless et al. (1992) gives the real and imaginary parts of ${\beta }_k$ for $k=1(1)13$; 14S.

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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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16: 10 Bessel Functions

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17: 18.16 Zeros

… ►For an asymptotic expansion of $x_{n,m}$ as $n\to \mathrm{\infty }$ that applies uniformly for $m=1,2,\mathrm{\dots },\lfloor \frac{1}{2}n\rfloor$, see Olver (1959, §14(i)). … ►

§18.16(vii) Discriminants

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18.16.19 $$\mathrm{Disc}\left(P_n^{(\alpha ,\beta )}\right)=2^{-n(n-1)}\prod _{j=1}^n{j}^{j-2n+2}{(j+\alpha )}^{j-1}{(j+\beta )}^{j-1}{(n+j+\alpha +\beta )}^{n-j}.$$

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Symbols:

$\mathrm{Disc}\left(x\right)$: discriminant function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial and $n$: nonnegative integer

Proved:

Ismail (2009, (3.4.16))(proved)

Referenced by:

Erratum (V1.2.0) Section 18.16

Permalink:

http://dlmf.nist.gov/18.16.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(vii), §18.16(vii), §18.16 and Ch.18

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18.16.20 $$\mathrm{Disc}\left(L_n^{(\alpha )}\right)=\prod _{j=1}^n{j}^{j-2n+2}{(j+\alpha )}^{j-1}.$$

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Symbols:

$\mathrm{Disc}\left(x\right)$: discriminant function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial and $n$: nonnegative integer

Proved:

Ismail (2009, (3.4.15))(proved)

Permalink:

http://dlmf.nist.gov/18.16.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(vii), §18.16(vii), §18.16 and Ch.18

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18.16.21 $$\mathrm{Disc}\left(H_n\right)=2^{\frac{3}{2}n(n-1)}\prod _{j=1}^n{j}^j.$$

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Symbols:

$\mathrm{Disc}\left(x\right)$: discriminant function, $H_n\left(x\right)$: Hermite polynomial and $n$: nonnegative integer

Proved:

Ismail (2009, (3.4.14))(proved)

Referenced by:

Erratum (V1.2.0) Section 18.16

Permalink:

http://dlmf.nist.gov/18.16.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(vii), §18.16(vii), §18.16 and Ch.18

18: 3.6 Linear Difference Equations

… ►In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome. … ►

Table 3.6.1: Weber function $w_n={𝐄}_n\left(1\right)$ computed by Olver’s algorithm.

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$n$	$p_n$		$e_n$		$e_n/(p_n{p}_{n+1})$		$w_n$
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14	$\mathrm{0.39792\hspace{0.33em}611}$	$\mathrm{\times 10¹⁴}$	$\mathrm{0.19555\hspace{0.33em}304}$	$\mathrm{\times 10¹³}$	$\mathrm{0.44167\hspace{0.33em}174}$	$\mathrm{\times 10⁻¹⁶}$	$0.327\underset{¯}{\mathrm{92\hspace{0.33em}861}}$	$\mathrm{\times 10⁻²}$
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§3.6(vii) Linear Difference Equations of Other Orders

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3.6.17 $$a_n{w}_{n+1}-b_n{w}_n=d_n.$$

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Symbols:

$w_n$: sequence, $a_n$: coefficient, $b_n$: coefficient and $d_n$: coefficient

Permalink:

http://dlmf.nist.gov/3.6.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §3.6(vii), §3.6 and Ch.3

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3.6.18 $$a_{n,k}{w}_{n+k}+a_{n,k-1}{w}_{n+k-1}+\mathrm{\cdots }+a_{n,0}{w}_n=d_n,$$

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Symbols:

$w_n$: sequence, $a_n$: coefficient and $d_n$: coefficient

Permalink:

http://dlmf.nist.gov/3.6.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §3.6(vii), §3.6 and Ch.3

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19: 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:

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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:

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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:

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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:

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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:

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20: Software Index

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	Open Source													With Book						Commercial
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7.25(vii) $ℱ\left(z\right)$, $G\left(z\right)$, $z\in ℂ$																					✓	✓	✓
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8.28(vii) $E_p\left(z\right)$, $z,p\in ℂ$	✓											✓	✓								✓	✓	✓
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11 Struve and Related Functions
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14 Legendre and Related Functions
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25.21(vii) Fermi–Dirac, Bose–Einstein					✓									✓			✓					✓
…

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