.阿根廷世界杯预选赛_『网址:687.vii』世界杯塞尔维亚对哥_b5p6v3_2022年11月28日21时21分47秒_6smwawga4

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21: 18.17 Integrals

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§18.17(vii) Mellin Transforms

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18.17.36 $${\int }_{-1}^1{(1-x)}^{z-1}{(1+x)}^{\beta }{P}_n^{(\alpha ,\beta )}\left(x\right)dx=\frac{2^{\beta +z}\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(1+\beta +n\right){\left(1+\alpha -z\right)}_n}{n!\mathrm{\Gamma }\left(1+\beta +z+n\right)},$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

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18.17.38 $${\int }_0^1{P}_{2n}\left(x\right)x^{z-1}dx=\frac{{(-1)}^n{\left(\frac{1}{2}-\frac{1}{2}z\right)}_n}{2{\left(\frac{1}{2}z\right)}_{n+1}},$$ $\mathrm{\Re }z>0$,

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

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18.17.39 $${\int }_0^1{P}_{2n+1}\left(x\right)x^{z-1}dx=\frac{{(-1)}^n{\left(1-\frac{1}{2}z\right)}_n}{2{\left(\frac{1}{2}+\frac{1}{2}z\right)}_{n+1}},$$ $\mathrm{\Re }z>-1$.

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

… ►For further integrals, see Apelblat (1983, pp. 189–204), Erdélyi et al. (1954a, pp. 38–39, 94–95, 170–176, 259–261, 324), Erdélyi et al. (1954b, pp. 42–44, 271–294), Gradshteyn and Ryzhik (2000, pp. 788–806), Gröbner and Hofreiter (1950, pp. 23–30), Marichev (1983, pp. 216–247), Oberhettinger (1972, pp. 64–67), Oberhettinger (1974, pp. 83–92), Oberhettinger (1990, pp. 44–47 and 152–154), Oberhettinger and Badii (1973, pp. 103–112), Prudnikov et al. (1986b, pp. 420–617), Prudnikov et al. (1992a, pp. 419–476), and Prudnikov et al. (1992b, pp. 280–308).

22: 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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

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

Encodings:

pMML, png, TeX

See also:

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

23: Bibliography J

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Fast Hankel transforms. Geophysical Prospecting 27 (4), pp. 876–901.

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External Links:

Document

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Numerical stability in evaluating continued fractions. Math. Comp. 28 (127), pp. 795–810.

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External Links:

FullText, MathReview (N. N. Abdelmalek), ZentralBlatt

Referenced by:

§3.10(iii).

Encodings:

BibTeX

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Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, Vol. 11, Addison-Wesley Publishing Co., Reading, MA.

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External Links:

ISBN 0-201-13510-8, MathReview (E. Frank), ZentralBlatt

Referenced by:

§1.12(ii), §1.12(iii), §1.12(iv), §1.12(v), §13.17, §13.17, §13.5, §13.5, §4.25, §5.10.

Encodings:

BibTeX

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Calculus of Finite Differences. Hungarian Agent Eggenberger Book-Shop, Budapest.

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Notes:

Third edition published in 1965 by Chelsea Publishing Co., New York, and American Mathematical Society, AMS Chelsea Book Series, Providence, RI

External Links:

MathReview (W. E. Milne), ZentralBlatt

Referenced by:

§26.1, §26.1, §26.8(vii), §5.16, Notations S.

Encodings:

BibTeX

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Memoire sur l’itération des fonctions rationnelles. J. Math. Pures Appl. 8 (1), pp. 47–245 (French).

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External Links:

FullText, ZentralBlatt

Referenced by:

§3.8(viii).

Encodings:

BibTeX

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:

BibTeX

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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: 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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25.21(vii) Fermi–Dirac, Bose–Einstein					✓									✓			✓					✓
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28 Mathieu Functions and Hill’s Equation
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26: 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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27: 8.28 Software

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

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28: 10.77 Software

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

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29: 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:

BibTeX

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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:

BibTeX

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An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.

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External Links:

ISBN 0-471-58884-9, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii).

Encodings:

BibTeX

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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:

BibTeX

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Asymptotic development of the Stirling numbers of the first kind. J. London Math. Soc. 33, pp. 133–146.

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External Links:

Document, MathReview (N. J. Fine), ZentralBlatt

Referenced by:

§26.1, §26.8(vii).

Encodings:

BibTeX

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30: 3.2 Linear Algebra

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3.2.2

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Permalink:

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

Encodings:

pMML, png, TeX

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

Encodings:

pMML, png, TeX

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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