.02年世界杯吉祥物_『网址:687.vii』世界杯假球黑幕视频_b5p6v3_gqyiq6mwc

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

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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

22: 1.10 Functions of a Complex Variable

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§1.10(vii) Inverse Functions

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1.10.12 $$f(z)=w$$

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Symbols:

$z$: variable and $w$: variable

Referenced by:

§1.10(vii), §1.10(vii)

Permalink:

http://dlmf.nist.gov/1.10.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(vii), §1.10(vii), §1.10 and Ch.1

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1.10.13 $$F(w)=z_0+\sum _{n=1}^{\mathrm{\infty }}{F}_n{(w-w_0)}^n$$

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Defines:

$F(w)$: inverse function of $f(z)$ (locally)

Symbols:

$z$: variable, $w$: variable, $n$: nonnegative integer and $F_n$: residue

Referenced by:

§1.10(vii)

Permalink:

http://dlmf.nist.gov/1.10.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(vii), §1.10(vii), §1.10 and Ch.1

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1.10.14 $$g(F(w))=g(z_0)+\sum _{n=1}^{\mathrm{\infty }}{G}_n{(w-w_0)}^n,$$

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Symbols:

$z$: variable, $w$: variable, $n$: nonnegative integer, $F(w)$: inverse function of $f(z)$ and $G_n$: residue

Referenced by:

§1.10(vii)

Permalink:

http://dlmf.nist.gov/1.10.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(vii), §1.10(vii), §1.10 and Ch.1

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1.10.16 $$F(w)=z_0+\sum _{n=1}^{\mathrm{\infty }}{F}_n{(w-w_0)}^{n/\mu }$$

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Symbols:

$z$: variable, $w$: variable, $n$: nonnegative integer, $F(w)$: inverse function of $f(z)$ , $F_n$: residue and $\mu$: parameter

Referenced by:

§1.10(vii)

Permalink:

http://dlmf.nist.gov/1.10.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(vii), §1.10(vii), §1.10 and Ch.1

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23: Bibliography Q

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“Best possible” upper and lower bounds for the zeros of the Bessel function $J_{\nu }(x)$ . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.

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External Links:

ISSN 0002-9947, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

BibTeX

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24: 32.7 Bäcklund Transformations

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§32.7(vii) Sixth Painlevé Equation

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32.7.33 $$z_1=1/z_0,$$

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Symbols:

$z$: real

Permalink:

http://dlmf.nist.gov/32.7.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §32.7(vii), §32.7 and Ch.32

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32.7.34 $$z_2=1-z_0,$$

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Symbols:

$z$: real

Permalink:

http://dlmf.nist.gov/32.7.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §32.7(vii), §32.7 and Ch.32

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32.7.35 $$z_3=1/z_0,$$

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Symbols:

$z$: real

Permalink:

http://dlmf.nist.gov/32.7.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §32.7(vii), §32.7 and Ch.32

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32.7.40 $${𝒮}_2:w_2(z_2)=1-w_0(z_0),$$

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Symbols:

$z$: real and ${𝒮}_j$: transformation

Permalink:

http://dlmf.nist.gov/32.7.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §32.7(vii), §32.7 and Ch.32

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25: Bibliography T

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Asymptotic estimates of Stirling numbers. Stud. Appl. Math. 89 (3), pp. 233–243.

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External Links:

ISSN 0022-2526, FullText, MathReview (E. Rodney Canfield), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

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Asymptotic Methods for Integrals. Series in Analysis, Vol. 6, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.

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External Links:

ISBN 978-981-4612-15-9, MathReview Entry, ZentralBlatt

Referenced by:

§12.9(i), §12.9(i), (13.8.13), (13.8.14), (13.8.15), (13.8.16), §13.8(ii), §13.8(ii), §13.8(iii), §13.8(iii), §14.15(i), §14.15(i), §14.15(iii), §14.15(iii), §15.12(iii), §15.12(iii), §18.15(vi), §18.15(vi), §2.4(v), §2.4(v), §2.4(vi), §2.4(vi), §26.8(vii), §26.8(vii), §33.12(i), §33.12(i), §33.12(ii), §33.12(ii), Profile Nico M. Temme.

Encodings:

BibTeX

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High Speed Numerical Integration of Fermi Dirac Integrals. Master’s Thesis, Naval Postgraduate School, Monterey, CA.

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External Links:

FullText

Referenced by:

§25.21(vii).

Encodings:

BibTeX

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The Theory of Functions. 2nd edition, Oxford University Press, Oxford.

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External Links:

ZentralBlatt

Referenced by:

§1.10(ix), §1.10(v), §1.8(i), §1.8(ii), §1.8(iii), §1.9(vii).

Encodings:

BibTeX

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Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-361-7, MathReview (Moody T. Chu), ZentralBlatt

Referenced by:

§3.2(iii), §3.2(vii).

Encodings:

BibTeX

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26: 8.17 Incomplete Beta Functions

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8.17.5 $$I_x(m,n-m+1)=\sum _{j=m}^n\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)x^j{(1-x)}^{n-j},$$ $m,n$ positive integers; .

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Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

6.6.4 (Relation to the Binomial Expansion)

Referenced by:

§8.17(i), §8.17(vii), Erratum (V1.0.5) for Subsection 8.17(i)

Permalink:

http://dlmf.nist.gov/8.17.E5

Encodings:

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See also:

Annotations for §8.17(i), §8.17 and Ch.8

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§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

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8.17.24 $$I_x(m,n)={(1-x)}^n\sum _{j=m}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{n+j-1}{j}\right)x^j,$$ $m,n$ positive integers; .

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Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

26.5.26 (The upper limit of summation has been corrected.)

Referenced by:

§8.17(i), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/8.17.E24

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation is the same as Equation (26.5.26) of Abramowitz and Stegun (1964), except that the upper limit of the summation has been corrected to $\mathrm{\infty }$.

Suggested 2011-03-23 by Stephen Bourn

See also:

Annotations for §8.17(vii), §8.17 and Ch.8

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27: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►These cases are treated in §§12.10(vii)–12.10(viii). … ►

12.10.7 $$\xi =\frac{1}{2}t\sqrt{t^2-1}-\frac{1}{2}\ln \left(t+\sqrt{t^2-1}\right).$$

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Defines:

$\xi$ (locally)

Symbols:

$\ln z$: principal branch of logarithm function

Referenced by:

§12.10(vi), §12.10(vii), §12.14(ix)

Permalink:

http://dlmf.nist.gov/12.10.E7

Encodings:

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See also:

Annotations for §12.10(ii), §12.10 and Ch.12

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12.10.23 $$\eta =\frac{1}{2}\arccos t-\frac{1}{2}t\sqrt{1-t^2},$$

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Defines:

$\eta$ (locally)

Symbols:

$\arccos z$: arccosine function

Referenced by:

§12.10(vii), §12.14(ix)

Permalink:

http://dlmf.nist.gov/12.10.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(iv), §12.10 and Ch.12

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§12.10(vii) Negative $a$, . Expansions in Terms of Airy Functions

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12.10.40 $$\varphi (\zeta )={\left(\frac{\zeta }{t^2-1}\right)}^{\frac{1}{4}}.$$

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Defines:

$\varphi (\zeta )$: function (locally)

Symbols:

$\zeta$: change of variable

Referenced by:

§12.10(vii)

Permalink:

http://dlmf.nist.gov/12.10.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vii), §12.10 and Ch.12

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28: 1.15 Summability Methods

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1.15.50 $${𝐼}^{\alpha }f(x)=\sum _{k=0}^{\mathrm{\infty }}\frac{k!}{\mathrm{\Gamma }\left(k+\alpha +1\right)}{a}_k{x}^{k+\alpha }.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: integer and ${𝐼}^{\alpha }$: fractional integral

Referenced by:

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

Permalink:

http://dlmf.nist.gov/1.15.E50

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(vi), §1.15 and Ch.1

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§1.15(vii) Fractional Derivatives

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1.15.51 $${𝐷}^{\alpha }f(x)=\frac{d^n}{{dx}^n}{𝐼}^{n-\alpha }f(x),$$

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Defines:

${𝐷}^{\alpha }$: fractional derivative (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and ${𝐼}^{\alpha }$: fractional integral

Referenced by:

§1.15(vii)

Permalink:

http://dlmf.nist.gov/1.15.E51

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(vii), §1.15 and Ch.1

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1.15.52 $${𝐷}^k{𝐼}^{\alpha }={𝐷}^n{𝐼}^{\alpha +n-k},$$ $k=1,2,\mathrm{\dots },n$.

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Symbols:

$k$: integer, $n$: nonnegative integer, ${𝐼}^{\alpha }$: fractional integral and ${𝐷}^{\alpha }$: fractional derivative

Referenced by:

§1.15(vii)

Permalink:

http://dlmf.nist.gov/1.15.E52

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(vii), §1.15 and Ch.1

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1.15.53 $${𝐷}^{\alpha }{𝐷}^{\beta }={𝐷}^{\alpha +\beta }.$$

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Symbols:

${𝐷}^{\alpha }$: fractional derivative

Permalink:

http://dlmf.nist.gov/1.15.E53

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(vii), §1.15 and Ch.1

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29: 11.10 Anger–Weber Functions

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§11.10(vii) Special Values

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11.10.25 ${𝐉}_{\nu }\left(0\right)={\displaystyle \frac{\sin \left(\pi \nu \right)}{\pi \nu }},$ ${𝐄}_{\nu }\left(0\right)={\displaystyle \frac{1-\cos \left(\pi \nu \right)}{\pi \nu }}.$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $\nu$: real or complex order

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E25

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §11.10(vii), §11.10 and Ch.11

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11.10.26 ${𝐄}_0\left(z\right)=-{𝐇}_0\left(z\right),$ ${𝐄}_1\left(z\right)={\displaystyle \frac{2}{\pi }}-{𝐇}_1\left(z\right).$

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E26

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §11.10(vii), §11.10 and Ch.11

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11.10.27 $${\frac{\partial }{\partial \nu }{𝐉}_{\nu }\left(z\right)|}_{\nu =0}=\frac{1}{2}\pi {𝐇}_0\left(z\right),$$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vii), §11.10 and Ch.11

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11.10.29 $${𝐉}_n\left(z\right)=J_n\left(z\right),$$ $n\in \mathbb{Z}$.

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\in$: element of, $\mathbb{Z}$: set of all integers, $z$: complex variable and $n$: integer order

A&S Ref:

12.3.2

Referenced by:

§11.10(vii), §11.11(ii)

Permalink:

http://dlmf.nist.gov/11.10.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vii), §11.10 and Ch.11

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30: 13.2 Definitions and Basic Properties

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§13.2(vii) Connection Formulas

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13.2.39 $$M(a,b,z)={\mathrm{e}}^{z}M(b-a,b,-z),$$

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Symbols:

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

A&S Ref:

13.1.27

Referenced by:

§13.12, §13.8(i), §13.9(ii), §18.17(vi)

Permalink:

http://dlmf.nist.gov/13.2.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(vii), §13.2(vii), §13.2 and Ch.13

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13.2.40 $$U(a,b,z)=z^{1-b}U(a-b+1,2-b,z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.1.29

Referenced by:

§13.8(iii)

Permalink:

http://dlmf.nist.gov/13.2.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(vii), §13.2(vii), §13.2 and Ch.13

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13.2.41 $$\frac{1}{\mathrm{\Gamma }\left(b\right)}M(a,b,z)=\frac{{\mathrm{e}}^{\mp a\pi \mathrm{i}}}{\mathrm{\Gamma }\left(b-a\right)}U(a,b,z)+\frac{{\mathrm{e}}^{\pm (b-a)\pi \mathrm{i}}}{\mathrm{\Gamma }\left(a\right)}{\mathrm{e}}^{z}U(b-a,b,{\mathrm{e}}^{\pm \pi \mathrm{i}}z).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer 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:

§13.12, §13.14(vii), §13.4(i), §13.7(ii)

Permalink:

http://dlmf.nist.gov/13.2.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(vii), §13.2(vii), §13.2 and Ch.13

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13.2.42 $$U(a,b,z)=\frac{\mathrm{\Gamma }\left(1-b\right)}{\mathrm{\Gamma }\left(a-b+1\right)}M(a,b,z)+\frac{\mathrm{\Gamma }\left(b-1\right)}{\mathrm{\Gamma }\left(a\right)}{z}^{1-b}M(a-b+1,2-b,z).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.1.3 (in different form)

Referenced by:

§13.14(vii), §13.2(ii), §13.6(v), §13.9(ii), §33.13

Permalink:

http://dlmf.nist.gov/13.2.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(vii), §13.2(vii), §13.2 and Ch.13