DLMF: Untitled Document

… ►

M. Abramowitz (1949) 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

… ►

S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

ⓘ

External Links:: ISSN 0024-1318, MathReview (James M. Horner)
Referenced by:: §13.9(i).
Encodings:: BibTeX

… ►

D. E. Amos (1983a) Algorithm 609. A portable FORTRAN subroutine for the Bickley functions

{\rm Ki}_{n}(x)

. ACM Trans. Math. Software 9 (4), pp. 480–493.

ⓘ

Notes:: Single-precision Fortran, maximum accuracy 18S.
External Links:: ISSN 0098-3500, Document, GAMS (module 609; package TOMS), MathReview (Marietta J. Tretter), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

D. E. Amos (1983c) Uniform asymptotic expansions for exponential integrals

E_{n}(x)

and Bickley functions

{\rm{K}i}_{n}(x)

. ACM Trans. Math. Software 9 (4), pp. 467–479.

ⓘ

External Links:: ISSN 0098-3500, Document, MathReview (Marietta J. Tretter), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

… ►

V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).

ⓘ

Notes:: Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. In Russian. English translation: Russian Math. Surveys, 29(1974), no. 2, pp. 10–50.
External Links:: Document, MathReview (G. R. Belickii), ZentralBlatt
Referenced by:: §36.2(i).
Encodings:: BibTeX

…

… ►

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

.

► ►►►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	$\lambda_{n}$
…
9	${\mathrm{e}}^{-\frac{1}{2}x^{2}}x^{\alpha+\frac{1}{2}}L^{(\alpha)}_{n}\left(x^% {2}\right)$	$1$	$0$	$-x^{2}+(\frac{1}{4}-\alpha^{2})x^{-2}$	$4n+2\alpha+2$
…
11	${\mathrm{e}}^{-n^{-1}x}x^{\ell+1}L^{(2\ell+1)}_{n-\ell-1}\left(2n^{-1}x\right)$	$1$	$0$	$\frac{2}{x}-\frac{\ell(\ell+1)}{x^{2}}$	$-\frac{1}{n^{2}}$
12	$H_{n}\left(x\right)$	$1$	$-2x$	$0$	$2n$
…

► ►Item 11 of Table 18.8.1 yields (18.39.36) for

Z=1

.

… ►

B_{2n+1}=0

,

… ►

24.2.4

B_{n}=B_{n}\left(0\right),

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials and $n$ : integer
A&S Ref:: 23.1.2
Referenced by:: (25.11.7)
Permalink:: http://dlmf.nist.gov/24.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.2(i), §24.2 and Ch.24

… ►

Table 24.2.4: Euler numbers

E_{n}

.

► ►►

$n$	$E_{n}$
…

► ►

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}

.

► ►►

	$k$
…

► ►

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}

.

► ►►

	$k$
…

►

… ►

26.12.9

\left(\prod_{h=1}^{r}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right)^{2};

ⓘ

Symbols:: $r$ : positive integer, $s$ : positive integer, $t$ : positive integer, $h$ : positive integer and $j$ : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §26.12(i), §26.12 and Ch.26

… ►

26.12.10

\left(\prod_{h=1}^{r}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right)\*\left(\prod_% {h=1}^{r+1}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right);

ⓘ

Symbols:: $r$ : positive integer, $s$ : positive integer, $t$ : positive integer, $h$ : positive integer and $j$ : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §26.12(i), §26.12 and Ch.26

… ►

26.12.11

\left(\prod_{h=1}^{r+1}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right)\*\left(% \prod_{h=1}^{r}\prod_{j=1}^{s+1}\frac{h+j+t-1}{h+j-1}\right).

ⓘ

Symbols:: $r$ : positive integer, $s$ : positive integer, $t$ : positive integer, $h$ : positive integer and $j$ : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §26.12(i), §26.12 and Ch.26

… ►The notation

\sum_{\pi\subseteq B(r,s,t)}

denotes the sum over all plane partitions contained in

B(r,s,t)

, and

|\pi|

denotes the number of elements in

\pi

. … ►where

\sigma_{2}(j)

is the sum of the squares of the divisors of

j

. …

… ►

C. Eckart (1930) The penetration of a potential barrier by electrons. Phys. Rev. 35 (11), pp. 1303–1309.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §15.18.
Encodings:: BibTeX

… ►

F. H. L. Essler, H. Frahm, A. R. Its, and V. E. Korepin (1996) Painlevé transcendent describes quantum correlation function of the

X X Z

antiferromagnet away from the free-fermion point. J. Phys. A 29 (17), pp. 5619–5626.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview (Estelle L. Basor), ZentralBlatt
Referenced by:: §32.16.
Encodings:: BibTeX

… ►

L. Euler (1768) Institutiones Calculi Integralis. Opera Omnia (1), Vol. 11, pp. 110–113.

ⓘ

Referenced by:: §25.12(i).
Encodings:: BibTeX

… ►

W. N. Everitt, L. L. Littlejohn, and R. Wellman (2004) The Sobolev orthogonality and spectral analysis of the Laguerre polynomials

\{L^{-k}_{n}\}

for positive integers

k

. J. Comput. Appl. Math. 171 (1-2), pp. 199–234.

ⓘ

External Links:: ISSN 0377-0427, Document, Link, MathReview (Khélifa Trimèche)
Referenced by:: §18.36(v).
Encodings:: BibTeX

… ►

W. N. Everitt (2008) Note on the

X_{1}

-Laguerre orthogonal polynomials.

ⓘ

Notes:: arXiv:0811.3559v2
Referenced by:: §18.36(vi).
Encodings:: BibTeX

…

… ►

Table 4.17.1: Trigonometric functions: values at multiples of

\frac{1}{12}\pi

.

► ►►►

$\theta$	$\sin\theta$	$\cos\theta$	$\tan\theta$	$\csc\theta$	$\sec\theta$	$\cot\theta$
…
$11\pi/12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$-\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$-(2-\sqrt{3})$	$\sqrt{2}(\sqrt{3}+1)$	$-\sqrt{2}(\sqrt{3}-1)$	$-(2+\sqrt{3})$
…

► ►

4.17.1

\lim_{z\to 0}\frac{\sin z}{z}=1,

ⓘ

Symbols:: $\sin\NVar{z}$ : sine function and $z$ : complex variable
A&S Ref:: 4.3.74
Permalink:: http://dlmf.nist.gov/4.17.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §4.17 and Ch.4

►

4.17.2

\lim_{z\to 0}\frac{\tan z}{z}=1.

ⓘ

Symbols:: $\tan\NVar{z}$ : tangent function and $z$ : complex variable
A&S Ref:: 4.3.75
Permalink:: http://dlmf.nist.gov/4.17.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §4.17 and Ch.4

►

4.17.3

\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}.

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.17.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.17 and Ch.4

… ►

G. A. Kalugin, D. J. Jeffrey, and R. M. Corless (2012) Bernstein, Pick, Poisson and related integral expressions for Lambert

W

. Integral Transforms Spec. Funct. 23 (11), pp. 817–829.

ⓘ

External Links:: ISSN 1065-2469, Document, Link, MathReview (Ross C. McPhedran), ZentralBlatt
Referenced by:: §4.13.
Encodings:: BibTeX

… ►

E. L. Kaplan (1948) Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.

ⓘ

External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §19.12.
Encodings:: BibTeX

… ►

S. K. Kim (1972) The asymptotic expansion of a hypergeometric function

{}_{2}F_{2}(1,\,\alpha;\,\rho_{1},\,\rho_{2};\,z)

. Math. Comp. 26 (120), pp. 963.

ⓘ

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §16.11(ii), §16.11(ii).
Encodings:: BibTeX

… ►

Y. S. Kim, A. K. Rathie, and R. B. Paris (2013) An extension of Saalschütz’s summation theorem for the series

{}_{r+3}F_{r+2}

. Integral Transforms Spec. Funct. 24 (11), pp. 916–921.

ⓘ

External Links:: ISSN 1065-2469, Document, FullText, MathReview (Christian Lavault), ZentralBlatt
Referenced by:: §16.4(ii), §16.4(ii).
Encodings:: BibTeX

… ►

K. S. Kölbig (1968) Algorithm 327: Dilogarithm [S22]. Comm. ACM 11 (4), pp. 270–271.

ⓘ

Notes:: Includes Algol program, maximum accuracy 12D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §25.21(v).
Encodings:: BibTeX

…

… ►

A. Leitner and J. Meixner (1960) Eine Verallgemeinerung der Sphäroidfunktionen. Arch. Math. 11, pp. 29–39.

ⓘ

External Links:: ISSN 0003-9268, Document, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §30.12.
Encodings:: BibTeX

… ►

M. Lerch (1887) Note sur la fonction

\mathfrak{K}(w,x,s)=\sum_{k=0}^{\infty}\frac{e^{2k\pi ix}}{(w+k)^{s}}

. Acta Math. 11 (1-4), pp. 19–24 (French).

ⓘ

External Links:: ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt
Referenced by:: §25.14(i), Notations K.
Encodings:: BibTeX

… ►

J. L. López, P. Pagola, and E. Pérez Sinusía (2013b) Asymptotics of the first Appell function

F_{1}

with large parameters. Integral Transforms Spec. Funct. 24 (9), pp. 715–733.

ⓘ

External Links:: ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §16.13, §16.13.
Encodings:: BibTeX

… ►

H. Lotsch and M. Gray (1964) Algorithm 244: Fresnel integrals. Comm. ACM 7 (11), pp. 660–661.

ⓘ

Notes:: Includes Algol program, maximum accuracy 12D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §7.25(iv).
Encodings:: BibTeX

… ►

N. A. Lukaševič (1967b) On the theory of Painlevé’s third equation. Differ. Uravn. 3 (11), pp. 1913–1923 (Russian).

ⓘ

Notes:: In Russian. English translation: Differential Equations 3(11), pp. 994–999
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.10(iii), §32.8(iii), §32.9(i), §32.9(ii).
Encodings:: BibTeX

…

… ►

K. Okamoto (1986) Studies on the Painlevé equations. III. Second and fourth Painlevé equations,

P_{{\rm II}}

and

P_{\mathrm{IV}}

. Math. Ann. 275 (2), pp. 221–255.

ⓘ

External Links:: ISSN 0025-5831, Document, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.10(ii), §32.10(iv), §32.6(iii), §32.6(v), §32.7(viii).
Encodings:: BibTeX

… ►

A. B. Olde Daalhuis (1998a) 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

… ►

F. W. J. Olver (1974) 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

… ►

S. Olver (2011) 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

… ►

H. Oser (1960) 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

…

… ►

Frank W. J. Olver, University of Maryland and NIST, Chaps. 1, 2, 4, 9, 10

►

Richard B. Paris, University of Abertay, Chaps. 8, 11

… ►

Hans Volkmer, University of Wisconsin, Milwaukee, Chaps. 29, 30

… ►

Richard B. Paris, University of Abertay Dundee, for Chaps. 8, 11 (deceased)

… ►

Hans Volkmer, University of Wisconsin–Milwaukee, for Chaps. 29, 30

…

.世界杯买球怎么买_『wn4.com_』2018世界杯一共多少支球队_w6n2c9o_2022年11月29日6时48分9秒_ok040qquy_gov_hk

11—20 of 799 matching pages

11: Bibliography

12: 18.8 Differential Equations

13: 24.2 Definitions and Generating Functions

14: 26.12 Plane Partitions

15: Bibliography E

16: 4.17 Special Values and Limits

17: Bibliography K

18: Bibliography L

19: Bibliography O

20: Staff

.世界杯 买球怎么买_『wn4.com_』2018世界杯一共多少支球队_w6n2c9o_2022年11月29日6时48分9秒_ok040qquy_gov_hk

11—20 of 799 matching pages

11: Bibliography

12: 18.8 Differential Equations

13: 24.2 Definitions and Generating Functions

14: 26.12 Plane Partitions

15: Bibliography E

16: 4.17 Special Values and Limits

17: Bibliography K

18: Bibliography L

19: Bibliography O

20: Staff

.世界杯买球怎么买_『wn4.com_』2018世界杯一共多少支球队_w6n2c9o_2022年11月29日6时48分9秒_ok040qquy_gov_hk