.世界杯2022赛程欧洲『网址:687.vii』02年世界杯巴西队.b5p6v3-1v5pbd

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21: 10.77 Software

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§10.77(vii) Bessel Functions–Complex Order and Real Argument

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22: 34.9 Graphical Method

… ►For an account of this method see Brink and Satchler (1993, Chapter VII). …

23: 34.13 Methods of Computation

… ►A review of methods of computation is given in Srinivasa Rao and Rajeswari (1993, Chapter VII, pp. 235–265). …

24: 10.74 Methods of Computation

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§10.74(vii) Integrals

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25: 33.23 Methods of Computation

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§33.23(vii) WKBJ Approximations

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26: 14.32 Methods of Computation

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Application of the uniform asymptotic expansions for large values of the parameters given in §§14.15 and 14.20(vii)–14.20(ix).

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27: 8.21 Generalized Sine and Cosine Integrals

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§8.21(vii) Auxiliary Functions

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8.21.18 $$f(a,z)=\mathrm{si}(a,z)\cos z-\mathrm{ci}(a,z)\sin z,$$

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Defines:

$f(a,z)$: auxiliary function (locally)

Symbols:

$\cos z$: cosine function, $\mathrm{ci}(a,z)$: generalized cosine integral, $\mathrm{si}(a,z)$: generalized sine integral, $\sin z$: sine function, $z$: complex variable and $a$: parameter

A&S Ref:

5.2.6 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.19 $$g(a,z)=\mathrm{si}(a,z)\sin z+\mathrm{ci}(a,z)\cos z.$$

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Defines:

$g(a,z)$: auxiliary function (locally)

Symbols:

$\cos z$: cosine function, $\mathrm{ci}(a,z)$: generalized cosine integral, $\mathrm{si}(a,z)$: generalized sine integral, $\sin z$: sine function, $z$: complex variable and $a$: parameter

A&S Ref:

5.2.7 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.22 $$f(a,z)={\int }_0^{\mathrm{\infty }}\frac{\sin t}{{(t+z)}^{1-a}}dt,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $z$: complex variable, $a$: parameter and $f(a,z)$: auxiliary function

A&S Ref:

5.2.12 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.23 $$g(a,z)={\int }_0^{\mathrm{\infty }}\frac{\cos t}{{(t+z)}^{1-a}}dt.$$

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable, $a$: parameter and $g(a,z)$: auxiliary function

A&S Ref:

5.2.13 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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28: 18.30 Associated OP’s

… ►For corresponding corecursive associated Jacobi polynomials, corecursive associated polynomials being discussed in §18.30(vii), see Letessier (1995). … ►

§18.30(vii) Corecursive and Associated Monic Orthogonal Polynomials

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18.30.27 $$x{\widehat{p}}_n(x;c)={\widehat{p}}_{n+1}(x;c)+{\alpha }_{n+c}{\widehat{p}}_n(x;c)+{\beta }_{n+c}{\widehat{p}}_{n-1}(x;c),$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$n$: nonnegative integer and $x$: real variable

Referenced by:

§18.30(vii), §18.30(vii), §18.30(vii)

Permalink:

http://dlmf.nist.gov/18.30.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(vii), §18.30(vii), §18.30 and Ch.18

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18.30.29 $${\widehat{p}}_n^{(0)}(x)={\widehat{p}}_{n-1}(x;1)$$

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Symbols:

$n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.30.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(vii), §18.30(vii), §18.30 and Ch.18

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18.30.30 $${\widehat{p}}_n^{(k)}(x)={\widehat{p}}_{n-1}(x;k+1).$$

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Symbols:

$k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.30.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(vii), §18.30(vii), §18.30 and Ch.18

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29: Bibliography I

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Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.

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External Links:

ISSN 0044-2267, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Highly Oscillatory Quadrature: The Story So Far. In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.), pp. 97–118.

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Notes:

Proceedings of ENuMath, Santiago de Compostela (2005)

External Links:

ISBN 3-540-34287-7, MathReview, ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

BibTeX

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$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.

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External Links:

ISSN 0002-9947, FullText, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.28(vii).

Encodings:

BibTeX

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Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

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Notes:

With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Corrected reprint of the 2005 original.

External Links:

ISBN 978-0-521-14347-9, MathReview Entry, ZentralBlatt

Referenced by:

§18.12, (18.15.4_5), (18.16.19), (18.16.20), (18.16.21), §18.17(ii), §18.18(vii), §18.19, (18.2.20), §18.2(vi), §18.2(xii), §18.2(xii), §18.2(x), §18.20(i), §18.20(ii), §18.21(ii), §18.22(i), §18.22(ii), §18.22(iii), §18.23, §18.25(i), §18.25(iii), §18.26(i), §18.26(iv), (18.27.4), §18.27(i), §18.27(ii), §18.27(iii), §18.27(v), §18.27(vi), §18.28(i), §18.28(ii), §18.28(iii), §18.28(v), §18.28(vi), §18.28(vii), §18.28(viii), §18.30(vi), §18.30(vii), §18.30(iii), §18.30(iv), §18.30, §18.30(vi), §18.34(i), §18.34(ii), §18.34(ii), §18.34(iii), (18.35.4), (18.35.5), (18.35.6), (18.35.6_2), (18.35.6_3), (18.35.6_4), (18.35.7), (18.35.8), §18.35(ii), §18.35(iii), §18.36(iii), §18.38(ii), §18.39(iv), §18.39(iv), §18.39(iv), (18.40.4), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.

Encodings:

BibTeX

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From Gauss to Painlevé: A Modern Theory of Special Functions. Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.

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External Links:

ISBN 3-528-06355-6, MathReview (Takeshi Sasaki), ZentralBlatt

Referenced by:

§32.2(vi), §32.7(vii).

Encodings:

BibTeX

30: 18.16 Zeros

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

ⓘ

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