.世界杯1比1_『网址:687.vii』activ足球世界杯录像_b5p6v3_wka0kcuei.com

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11: 28.4 Fourier Series

… ►For $n=0,1,2,3,\mathrm{\dots }$, … ►For fixed $s=1,2,3,\mathrm{\dots }$ and fixed $m=1,2,3,\mathrm{\dots }$, … ►

§28.4(vii) Asymptotic Forms for Large $m$

… ►

28.4.25 $$\frac{A_{2m+1}^{2n+1}(q)}{A_1^{2n+1}(q)}=\frac{{(-1)}^{m+1}}{{\left({\left(\frac{1}{2}\right)}_{m+1}\right)}^2}{\left(\frac{q}{4}\right)}^{m+1}\frac{2\left(1+O\left(m^{-1}\right)\right)}{w_{\text{II}}^{\prime }(\frac{1}{2}\pi ;a_{2n+1}\left(q\right),q)},$$

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Symbols:

$O\left(x\right)$: order not exceeding, $a_n\left(q\right)$: eigenvalues of Mathieu equation, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $q=h^2$: parameter, $n$: integer, $w(z)$: Mathieu’s equation solution and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E25

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Annotations for §28.4(vii), §28.4 and Ch.28

… ►For the basic solutions $w_{\text{I}}$ and $w_{\text{II}}$ see §28.2(ii).

12: 13.2 Definitions and Basic Properties

… ►Similarly, when $a-b+1=-n$, $n=0,1,2,\mathrm{\dots }$, … ►When $b=n+1$, $n=0,1,2,\mathrm{\dots }$, and $a\ne 0,-1,-2,\mathrm{\dots }$, … ►When $b=n+1$, $n=0,1,2,\mathrm{\dots }$, and $a=-m$, $m=0,1,2,\mathrm{\dots }$, … ►Except when $a=0,-1,\mathrm{\dots }$ (polynomial cases), … ►

§13.2(vii) Connection Formulas

…

13: 24.4 Basic Properties

… ►

24.4.7 $$\sum _{k=1}^m{k}^n=\frac{B_{n+1}\left(m+1\right)-B_{n+1}}{n+1},$$

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $k$: integer, $m$: integer and $n$: integer

A&S Ref:

23.1.4

Referenced by:

§24.17(iii)

Permalink:

http://dlmf.nist.gov/24.4.E7

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Annotations for §24.4(iii), §24.4 and Ch.24

… ►

§24.4(vii) Derivatives

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24.4.34 $$\frac{d}{dx}{B}_n\left(x\right)=n{B}_{n-1}\left(x\right),$$ $n=1,2,\mathrm{\dots }$,

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: integer and $x$: real or complex

A&S Ref:

23.1.5

Referenced by:

§24.13(i), (25.11.7)

Permalink:

http://dlmf.nist.gov/24.4.E34

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Annotations for §24.4(vii), §24.4 and Ch.24

►

24.4.35 $$\frac{d}{dx}{E}_n\left(x\right)=n{E}_{n-1}\left(x\right),$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: integer and $x$: real or complex

A&S Ref:

23.1.5

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.4.E35

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Annotations for §24.4(vii), §24.4 and Ch.24

… ►Let $P(x)$ denote any polynomial in $x$, and after expanding set ${(B(x))}^n=B_n\left(x\right)$ and ${(E(x))}^n=E_n\left(x\right)$. …

14: 12.14 The Function $W(a,x)$

… ►Here $w_1(a,x)$ and $w_2(a,x)$ are the even and odd solutions of (12.2.3): … ►

§12.14(vii) Relations to Other Functions

… ►For real $\mu$ and $t$ oscillations occur outside the $t$-interval $[-1,1]$. … ►uniformly for $t\in [-1+\delta ,1-\delta ]$, with $\eta$ given by (12.10.23) and ${\widetilde{𝒜}}_s(t)$ given by (12.10.24). … ►uniformly for $t\in [-1+\delta ,\mathrm{\infty })$, with $\zeta$, $\varphi (\zeta )$, $A_s(\zeta )$, and $B_s(\zeta )$ as in §12.10(vii). …

15: 3.6 Linear Difference Equations

… ►Miller (Bickley et al. (1952, pp. xvi–xvii)) that arbitrary “trial values” can be assigned to $w_N$ and $w_{N+1}$, for example, $1$ and $0$. … ►The Weber function ${𝐄}_n\left(1\right)$ satisfies …Thus the asymptotic behavior of the particular solution ${𝐄}_n\left(1\right)$ is intermediate to those of the complementary functions $J_n\left(1\right)$ and $Y_n\left(1\right)$; moreover, the conditions for Olver’s algorithm are satisfied. … ►The values of $w_n$ for $n=1,2,\mathrm{\dots },10$ are the wanted values of ${𝐄}_n\left(1\right)$. … ►

§3.6(vii) Linear Difference Equations of Other Orders

…

16: 18.17 Integrals

… ►For the beta function $\mathrm{B}(a,b)$ see §5.12, and for the confluent hypergeometric function ${}_1{}^{}F_1^{}$ see (16.2.1) and Chapter 13. … ►Formulas (18.17.21_2) and (18.17.21_3) are respectively the limit case $c\to \frac{1}{2}$ and the special case $c=1$ of (18.17.21_1). … ►For the confluent hypergeometric function ${}_1{}^{}F_1^{}$ see (16.2.1) and Chapter 13. … ►

§18.17(vii) Mellin Transforms

… ►For the hypergeometric function ${}_2{}^{}F_1^{}$ see §§15.1 and 15.2(i). …

17: 8.5 Confluent Hypergeometric Representations

… ►For the confluent hypergeometric functions $M$, $𝐌$, $U$, and the Whittaker functions $M_{\kappa ,\mu }$ and $W_{\kappa ,\mu }$, see §§13.2(i) and 13.14(i). ►

8.5.1 $$\gamma (a,z)=a^{-1}{z}^a{\mathrm{e}}^{-z}M(1,1+a,z)=a^{-1}{z}^{a}M(a,1+a,-z),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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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, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.12

Referenced by:

§8.11(ii), §8.14

Permalink:

http://dlmf.nist.gov/8.5.E1

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Annotations for §8.5 and Ch.8

… ►

8.5.3 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-z}U(1-a,1-a,z)=z^a{\mathrm{e}}^{-z}U(1,1+a,z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.14, §8.19(vi)

Permalink:

http://dlmf.nist.gov/8.5.E3

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Annotations for §8.5 and Ch.8

►

8.5.4 $$\gamma (a,z)=a^{-1}{z}^{\frac{1}{2}a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}z}{M}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.5

Permalink:

http://dlmf.nist.gov/8.5.E4

Encodings:

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See also:

Annotations for §8.5 and Ch.8

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8.5.5 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}a-\frac{1}{2}}{W}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.5.E5

Encodings:

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See also:

Annotations for §8.5 and Ch.8

18: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►These cases are treated in §§12.10(vii)–12.10(viii). … ►For $s=0,1,2$, … ►uniformly for $t\in [-1+\delta ,1-\delta ]$. … ►starting with ${𝖠}_0(\tau )=1$. … ►where $\varphi (\zeta )$ is as in (12.10.40), $u_k(t)$ is as in §12.10(ii), ${\alpha }_0=1$, and …

19: 26.8 Set Partitions: Stirling Numbers

… ►where the summation is over all nonnegative integers $c_1,c_2,\mathrm{\dots },c_k$ such that $c_1+c_2+\mathrm{\cdots }+c_k=n-k.$ … ►where ${\left(x\right)}_n$ is the Pochhammer symbol: $x(x+1)\mathrm{\cdots }(x+n-1)$. … ►For $n\ge 1$, … ►

§26.8(vii) Asymptotic Approximations

… ►For asymptotic approximations for $s(n+1,k+1)$ and $S(n,k)$ that apply uniformly for $1\le k\le n$ as $n\to \mathrm{\infty }$ see Temme (1993) and Temme (2015, Chapter 34). …

20: Bibliography I

… ►

The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

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External Links:

ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

BibTeX

… ►

The factorization method. Rev. Modern Phys. 23 (1), pp. 21–68.

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Referenced by:

§18.38(iii).

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

… ►

$q$-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

… ►

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