.世界杯预测神_『网址:687.vii』世界杯预选赛完整视频_b5p6v3_y0aqso2ym

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11: 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:

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See also:

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

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

ⓘ

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:

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See also:

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

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12: 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:

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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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13: 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

… ►

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

… ►

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

14: 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:

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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:

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See also:

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

15: 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:

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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:

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See also:

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

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

ⓘ

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:

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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:

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See also:

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

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16: 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

…

17: 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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18: Bibliography T

… ►

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

… ►

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

… ►

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

ⓘ

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

… ►

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

ⓘ

External Links:

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

Referenced by:

§3.2(iii), §3.2(vii).

Encodings:

BibTeX

…

19: 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; .

ⓘ

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

►

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; .

ⓘ

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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20: 12.10 Uniform Asymptotic Expansions for Large Parameter

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

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

ⓘ

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

ⓘ

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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