加拿大28久久预测零一预测【亚博官方qee9.com】▄3aT

SearchAdvancedHelp

(0.003 seconds)

21—30 of 717 matching pages

21: 15.12 Asymptotic Approximations

… ►Also let $a,b,z$ be real or complex and fixed, and at least one of the following conditions be satisfied: … ►

15.12.9 $${(z+1)}^{3\lambda /2}{(2\lambda )}^{c-1}𝐅(\genfrac{}{}{0pt}{}{a+\lambda ,b+2\lambda }{c};-z)={\lambda }^{-1/3}\left({\mathrm{e}}^{\pi \mathrm{i}(a-c+\lambda +(1/3))}\mathrm{Ai}\left({\mathrm{e}}^{-2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)+{\mathrm{e}}^{\pi \mathrm{i}(c-a-\lambda -(1/3))}\mathrm{Ai}\left({\mathrm{e}}^{2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)\right)\left(a_0(\zeta )+O({\lambda }^{-1})\right)+{\lambda }^{-2/3}\left({\mathrm{e}}^{\pi \mathrm{i}(a-c+\lambda +(2/3))}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{-2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)+{\mathrm{e}}^{\pi \mathrm{i}(c-a-\lambda -(2/3))}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)\right)\left(a_1(\zeta )+O({\lambda }^{-1})\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\zeta$: change of variable, $\beta$ and $a_j(\zeta )$: function

Referenced by:

§15.12(iii)

Permalink:

http://dlmf.nist.gov/15.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

… ►

15.12.11 $$\beta ={\left(-\frac{3}{2}\zeta +\frac{9}{4}\ln \left(\frac{2+{\mathrm{e}}^{\zeta }}{2+{\mathrm{e}}^{-\zeta }}\right)\right)}^{1/3},$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\ln z$: principal branch of logarithm function, $\zeta$: change of variable and $\beta$

Permalink:

http://dlmf.nist.gov/15.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

… ►By combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for $F(a+e_1\lambda ,b+e_2\lambda ;c+e_3\lambda ;z)$ can be obtained with $e_j=\pm 1$ or $0$, $j=1,2,3$. …For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).

22: 20.10 Integrals

… ►

20.10.2 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}({\theta }_3\left(0\right|\mathrm{i}{x}^2)-1)dx={\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E2

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_3\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_3\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.32) and (20.4.4).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

… ►

20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

…

23: Bibliography

… ►

Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.

ⓘ

External Links:

MathReview (J. G. van der Corput), ZentralBlatt

Referenced by:

§30.9(iii).

Encodings:

BibTeX

… ►

Theory of Incomplete Cylindrical Functions and Their Applications. Springer-Verlag, Berlin.

ⓘ

Notes:

Translated from the Russian by H. E. Fettis, J. W. Goresh and D. A. Lee, Die Grundlehren der mathematischen Wissenschaften, Band 160

External Links:

MathReview, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Study of nuclear structure by electromagnetic excitation with accelerated ions. Rev. Mod. Phys. 28, pp. 432–542.

ⓘ

External Links:

ISSN 0034-6861, Document, MathReview, ZentralBlatt

Referenced by:

§33.22(ii).

Encodings:

BibTeX

… ►

Computation of modified Bessel functions and their ratios. Math. Comp. 28 (125), pp. 239–251.

ⓘ

External Links:

ISSN 0025-5718, FullText, MathReview (L. Lorch), ZentralBlatt

Referenced by:

§10.74(v).

Encodings:

BibTeX

… ►

Sturm-Liouville Theory. Birkhäuser Verlag, Basel.

ⓘ

Notes:

Past and present, Including papers from the International Colloquium held at the University of Geneva, Geneva, September 15–19, 2003

External Links:

ISBN 978-3-7643-7066-4; 3-7643-7066-1, Document, Link, MathReview (Anton Zettl)

Referenced by:

§1.13(viii), §1.18(x).

Encodings:

BibTeX

…

24: 28.29 Definitions and Basic Properties

… ►If (28.29.1) has a periodic solution with minimum period $n\pi$, $n=3,4,\mathrm{\dots }$, then all solutions are periodic with period $n\pi$. … ►Its order of growth for $|\lambda |\to \mathrm{\infty }$ is exactly $\frac{1}{2}$; see Magnus and Winkler (1966, Chapter II, pp. 19–28). … ►For this purpose the discriminant can be expressed as an infinite determinant involving the Fourier coefficients of $Q(x)$; see Magnus and Winkler (1966, §2.3, pp. 28–36). … ►

28.29.17 $${\mu }_n,n=1,2,3,\mathrm{\dots },\text{ with }△({\mu }_n)=-2.$$

ⓘ

Defines:

${\mu }_n$: eigenvalues (locally)

Symbols:

$n$: integer

Referenced by:

§28.30(i)

Permalink:

http://dlmf.nist.gov/28.29.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(iii), §28.29 and Ch.28

… ►

28.29.18

ⓘ

Symbols:

${\lambda }_n$: eigenvalues and ${\mu }_n$: eigenvalues

Referenced by:

§28.30(i)

Permalink:

http://dlmf.nist.gov/28.29.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(iii), §28.29 and Ch.28

…

25: 26.12 Plane Partitions

… ►As an example, there are six plane partitions of 3: ►

$3$ ,

… ►It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point $(h,j,k)\in \pi$. … ►

26.12.2

ⓘ

Referenced by:

§26.12(i)

Permalink:

http://dlmf.nist.gov/26.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(i), §26.12 and Ch.26

… ►

26.12.18

ⓘ

Permalink:

http://dlmf.nist.gov/26.12.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(i), §26.12 and Ch.26

…

26: Software Index

… ► ►►►►

	Open Source													With Book						Commercial
…
28 Mathieu Functions and Hill’s Equation
…
34 3j, 6j, 9j Symbols
…

… ►

This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

… ►

A cross index of mathematical software in use at NIST.

27: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at $\eta =0$, and the Maclaurin series expansion of $c_k(\eta )$ is given by …where $d_{0,0}=-\frac{1}{3}$, …and ${\alpha }_3,{\alpha }_4,\mathrm{\dots }$ are defined by …For numerical values of $d_{k,n}$ to 30D for $k=0(1)9$ and $n=0(1)N_k$, where $N_k=28-4\lfloor k/2\rfloor$, see DiDonato and Morris (1986). … ►A different type of uniform expansion with coefficients that do not possess a removable singularity at $z=a$ is given by …

28: Bibliography O

… ►

On the $\tau$-function of the Painlevé equations. Phys. D 2 (3), pp. 525–535.

ⓘ

External Links:

ISSN 0167-2789, Document, MathReview (D. A. Lutz), ZentralBlatt

Referenced by:

§32.6(i), §32.6(ii).

Encodings:

BibTeX

… ►

Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.

ⓘ

External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

… ►

On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.

ⓘ

External Links:

ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§2.7(ii).

Encodings:

BibTeX

… ►

Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank unity. Methods Appl. Anal. 4 (4), pp. 375–403.

ⓘ

External Links:

ISSN 1073-2772, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§2.7(ii).

Encodings:

BibTeX

… ►

Modified quotients of cylinder functions. Math. Tables Aids Comput. 10, pp. 27–28.

ⓘ

External Links:

ISSN 0891-6837, MathReview Entry, ZentralBlatt

Referenced by:

§10.29(i), §10.6(i).

Encodings:

BibTeX

…

29: 4.13 Lambert $W$-Function

… ► $W_0\left(z\right)$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },-{\mathrm{e}}^{-1}]$, real-valued when $z>-{\mathrm{e}}^{-1}$, and has a square root branch point at $z=-{\mathrm{e}}^{-1}$. …The other branches $W_k\left(z\right)$ are single-valued analytic functions on $ℂ\setminus (-\mathrm{\infty },0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. … ►

$$p_n(x)=(1+x)p_{n-1}^{\prime }(x)+(1-n(x+3))p_{n-1}(x),$$ $n=1,2,3,\mathrm{\dots }$.

… ►

4.13.7 $$c_0=1,c_1=1,c_2=\frac{1}{3},c_3=\frac{1}{36},c_4=-\frac{1}{270},$$

ⓘ

Symbols:

$c_{\mathrm{i}}$: coefficients

Permalink:

http://dlmf.nist.gov/4.13.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.13 and Ch.4

… ►For the foregoing results and further information see Borwein and Corless (1999), Corless et al. (1996), de Bruijn (1961, pp. 25–28), Olver (1997b, pp. 12–13), and Siewert and Burniston (1973). …

30: Bibliography G

… ►

Variable-precision recurrence coefficients for nonstandard orthogonal polynomials. Numer. Algorithms 52 (3), pp. 409–418.

ⓘ

External Links:

ISSN 1017-1398, Document, Link, MathReview Entry

Referenced by:

§18.39(iii), §18.40(ii).

Encodings:

BibTeX

… ►

Algorithm 819: AIZ, BIZ: two Fortran 77 routines for the computation of complex Airy functions. ACM Trans. Math. Software 28 (3), pp. 325–336.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 18S.

External Links:

ISSN 0098-3500, Document, GAMS (module 819; package TOMS), MathReview Entry, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

►

Algorithm 822: GIZ, HIZ: two Fortran 77 routines for the computation of complex Scorer functions. ACM Trans. Math. Software 28 (4), pp. 436–447.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 18S.

External Links:

ISSN 0098-3500, Document, GAMS (module 822; package TOMS), MathReview Entry, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).

ⓘ

Notes:

In Russian. English translation: Differential Equations 18(3), pp. 317–326.

External Links:

ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.10(iv), §32.10(v), §32.8(iii), §32.8(v), §32.9(ii).

Encodings:

BibTeX

… ►

Painlevé Differential Equations in the Complex Plane. Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin-New York.

ⓘ

External Links:

ISBN 3-11-017379-4, MathReview, ZentralBlatt

Referenced by:

§32.10(i), §32.10(iii), §32.10(iv), §32.10(v), §32.10(vi), §32.2(iv), §32.7(iii), §32.7(iv), §32.7(v), §32.7(vi), §32.7(vii), §32.8(ii), §32.8(iii), §32.8(iv), §32.8(v), §32.9(i), §32.9(ii), §32.9(iii).

Encodings:

BibTeX

…