.2018世界杯cctv加转播『wn4.com』世界杯买独赢什么意思.w6n2c9o.2022年11月29日23时16分3秒.o08igewoe

SearchAdvancedHelp

(0.004 seconds)

21—30 of 590 matching pages

21: 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

… ►

Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.

ⓘ

External Links:

ISSN 1364-5021, Document, MathReview (Donald A. Lutz), ZentralBlatt

Referenced by:

§2.11(v), §2.7(ii).

Encodings:

BibTeX

… ►

Error bounds for stationary phase approximations. SIAM J. Math. Anal. 5 (1), pp. 19–29.

ⓘ

Notes:

See also Olver (1997b)

External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

BibTeX

… ►

Numerical solution of Riemann-Hilbert problems: Painlevé II. Found. Comput. Math. 11 (2), pp. 153–179.

ⓘ

External Links:

ISSN 1615-3375, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§32.17, §32.17.

Encodings:

BibTeX

… ►

Algorithm 22: Riccati-Bessel functions of first and second kind. Comm. ACM 3 (11), pp. 600–601.

ⓘ

Notes:

Includes Algol code, minimum accuracy 9S. For certification see same journal 13(1970), p. 448.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

Encodings:

BibTeX

…

22: Bibliography G

… ►

Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.

ⓘ

External Links:

ISSN 0001-0782, Document

Referenced by:

§33.26(ii).

Encodings:

BibTeX

►

Algorithm 363: Complex error function. Comm. ACM 12 (11), pp. 635.

ⓘ

Notes:

Includes Algol program, maximum accuracy 10S. For certification see same journal v. 15 (1972), pp. 465–466

External Links:

ISSN 0001-0782, Document

Referenced by:

§7.25(iii).

Encodings:

BibTeX

… ►

Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities. J. Math. Phys. 25 (11), pp. 3350–3356.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

… ►

Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

ⓘ

External Links:

FullText, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

Encodings:

BibTeX

… ►

Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).

ⓘ

Notes:

In Russian. English translation: Differential Equations 11(11), pp. 285–287

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.7(iii), §32.7(vi).

Encodings:

BibTeX

…

23: Bibliography D

… ►

Sur une classe de nombres rationnels réductibles aux nombres de Bernoulli. Bull. Sci. Math. (2) 28, pp. 29–32 (French).

ⓘ

External Links:

ZentralBlatt

Referenced by:

§24.1.

Encodings:

BibTeX

… ►

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.

ⓘ

External Links:

ISSN 0010-3640, Document, MathReview (D. S. Lubinsky), ZentralBlatt

Referenced by:

§18.32.

Encodings:

BibTeX

… ►

Algorithm 322. F-distribution. Comm. ACM 11 (2), pp. 116–117.

ⓘ

Notes:

Includes Algol program, maximum accuracy 10S. For certification see same journal 12(1969), p. 39

External Links:

ISSN 0001-0782, Document

Referenced by:

§8.28(iv).

Encodings:

BibTeX

… ►

Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).

ⓘ

Notes:

With an appendix by I. M. Krichever. In Russian. English translation: Russian Math. Surveys 36(1981), no. 2, pp. 11–92 (1982)

External Links:

ISSN 0042-1316, Document, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§21.1, §21.6(ii), §21.7(i), Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2, Ch.21, Notations T.

Encodings:

BibTeX

… ►

The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.

ⓘ

External Links:

ISSN 0003-9519, MathReview (Willard Parker), ZentralBlatt

Referenced by:

§8.17(i).

Encodings:

BibTeX

…

24: 18.8 Differential Equations

… ►

Table 18.8.1: Classical OP’s: differential equations $A(x)f^{\prime \prime }(x)+B(x)f^{\prime }(x)+C(x)f(x)+{\lambda }_{n}f(x)=0$.

► ►►►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	${\lambda }_n$
…
3	${(\sin x)}^{\alpha +\frac{1}{2}}{P}_n^{(\alpha ,\alpha )}\left(\cos x\right)$	$1$	$0$	$(\frac{1}{4}-{\alpha }^2)/{\sin }^{2}x$	${(n+\alpha +\frac{1}{2})}^2$
…
6	$U_n\left(x\right)$	$1-x^2$	$-3x$	$0$	$n(n+2)$
…
11	${\mathrm{e}}^{-n^{-1}x}{x}^{\mathrm{\ell }+1}{L}_{n-\mathrm{\ell }-1}^{(2\mathrm{\ell }+1)}\left(2n^{-1}x\right)$	$1$	$0$	$\frac{2}{x}-\frac{\mathrm{\ell }(\mathrm{\ell }+1)}{x^2}$	$-\frac{1}{n^2}$
…

►

►Item 11 of Table 18.8.1 yields (18.39.36) for $Z=1$.

25: Bibliography H

… ►

On the distribution of the zeros of generalized Airy functions. Math. Comp. 29 (131), pp. 863–877.

ⓘ

External Links:

FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.13(i).

Encodings:

BibTeX

… ►

Inverse virial symmetry of diatomic potential curves. J. Chem. Phys. 109 (1), pp. 11–19.

ⓘ

External Links:

Document

Referenced by:

§15.18.

Encodings:

BibTeX

… ►

Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems. J. Mathematical Phys. 11 (8), pp. 2501–2504.

ⓘ

External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(xiv).

Encodings:

BibTeX

… ►

Algorithm 395: Student’s t-distribution. Comm. ACM 13 (10), pp. 617–619.

ⓘ

Notes:

Includes Algol program, maximum accuracy 11S. For Remarks see ACM Trans. Math. Software 5(1979) and 7(1981)

External Links:

ISSN 0001-0782, Document, Remark 1, Remark 2

Referenced by:

§8.28(iv).

Encodings:

BibTeX

… ►

An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§24.6.

Encodings:

BibTeX

…

26: 18 Orthogonal Polynomials

…

27: 24.2 Definitions and Generating Functions

… ►

Table 24.2.5: Coefficients $b_{n,k}$ of the Bernoulli polynomials $B_n\left(x\right)={\sum }_{k=0}^n{b}_{n,k}{x}^k$.

► ►►►►►

	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
…
$11$	$0$	$\frac{5}{6}$	$0$	$-\frac{11}{2}$	$0$	$11$	$0$	$-11$	$0$	$\frac{55}{6}$	$-\frac{11}{2}$	$1$
$12$	$-\frac{691}{2730}$	$0$	$5$	$0$	$-\frac{33}{2}$	$0$	$22$	$0$	$-\frac{33}{2}$	$0$	$11$	$-6$	$1$
…

►

►

Table 24.2.6: Coefficients $e_{n,k}$ of the Euler polynomials $E_n\left(x\right)={\sum }_{k=0}^n{e}_{n,k}{x}^k$.

► ►►►►

	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
…
$11$	$\frac{691}{4}$	$0$	$-\frac{1705}{2}$	$0$	$\frac{2805}{4}$	$0$	$-231$	$0$	$\frac{165}{4}$	$0$	$-\frac{11}{2}$	$1$
…

►

28: Bibliography B

… ►

Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.

ⓘ

External Links:

FullText, MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§8.7.

Encodings:

BibTeX

… ►

Vortices in Ginzburg-Landau Equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 11–19.

ⓘ

External Links:

ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§17.17.

Encodings:

BibTeX

… ►

Rainbow over Woolsthorpe Manor. Notes and Records Roy. Soc. London 36 (1), pp. 3–11 (1 plate).

ⓘ

External Links:

ISSN 0035-9149, Document, MathReview

Referenced by:

Sidebar 9.SB1.

Encodings:

BibTeX

… ►

Algorithm 21: Bessel function for a set of integer orders. Comm. ACM 3 (11), pp. 600.

ⓘ

Notes:

Includes Algol code, minimum accuracy 5S. For certification see same journal 8(1965), p. 219.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

Encodings:

BibTeX

… ►

Stability of repulsive Bose-Einstein condensates in a periodic potential. Phys. Rev. E (3) 63 (036612), pp. 1–11.

ⓘ

External Links:

Document

Referenced by:

§29.19(i).

Encodings:

BibTeX

…

29: Bibliography F

… ►

Tables of Values of the Function $w(z)=e^{-z^2}(1+2i{\pi }^{-1/2}{\int }_0^z{e}^{t^2}𝑑t)$ for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

V. N. Faddeeva and N. M. Terent’ev (1954).

Encodings:

BibTeX

… ►

Polynomial relations in the Heisenberg algebra. J. Math. Phys. 35 (11), pp. 6144–6149.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (Shao-Ming Fei), ZentralBlatt

Referenced by:

§13.3(ii), §15.5(i), §16.3(i).

Encodings:

BibTeX

… ►

On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

Encodings:

BibTeX

… ►

Application of the $\tau$-function theory of Painlevé equations to random matrices: ${\mathrm{P}}_{\mathrm{VI}}$, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. 174, pp. 29–114.

ⓘ

External Links:

ISSN 0027-7630, MathReview, ZentralBlatt

Referenced by:

§32.10(vi), §32.6(v).

Encodings:

BibTeX

… ►

Algorithm 435: Modified incomplete gamma function. Comm. ACM 15 (11), pp. 993–995.

ⓘ

Notes:

Includes Fortran code. For a Remark see ACM Trans. Math. Software 4 (1978), no. 3, pp. 296–304.

External Links:

ISSN 0001-0782, Document, Remark

Referenced by:

§8.28(ii).

Encodings:

BibTeX

…

30: 5.10 Continued Fractions

… ►

5.10.1 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)+z-\left(z-\frac{1}{2}\right)\ln z-\frac{1}{2}\ln \left(2\pi \right)=\frac{a_0}{z+}\frac{a_1}{z+}\frac{a_2}{z+}\frac{a_3}{z+}\frac{a_4}{z+}\frac{a_5}{z+}\mathrm{\cdots },$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $z$: complex variable and $a_k$: coefficient

Referenced by:

Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.10.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.10 and Ch.5

… ►

$a_3=\frac{195}{371},$

… ►For exact values of $a_7$ to $a_{11}$ and 40S values of $a_0$ to $a_{40}$, see Char (1980). …