.1994年世界杯小组分组_『网址:68707.vip』2018世界杯直播咪咕_b5p6v3_2022年11月30日7时57分10秒_z9fzftt9n.cc

Did you mean .1994年世界杯小组分组_『网址:687.vii』2018世界杯直播咪咕_b5p6v3_2022年11月30日7时57分10秒_z9fzftt9n.cc ?

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1: Publications

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B. V. Saunders and Q. Wang (2005) Boundary/Contour Fitted Grid Generation for Effective Visualizations in a Digital Library of Mathematical Functions, Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, June 11–18, 2005. pp. 61–71.

►

Q. Wang and B. V. Saunders (2005) Web-Based 3D Visualization in a Digital Library of Mathematical Functions, Proceedings of the Web3D Symposium, Bangor, UK, March 29–April 1, 2005.

… ►

A. Youssef (2007) Methods of Relevance Ranking and Hit-content Generation in Math Search, Proceedings of Mathematical Knowledge Management (MKM2007), RISC, Hagenberg, Austria, June 27–30, 2007.

… ►

B. I. Schneider, B. R. Miller and B. V. Saunders (2018) NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53.

2: DLMF Project News

error generating summary

3: Antony Ross Barnett

… ► 2018) was Senior Research Fellow in the Mathematics Department, University of Waikato, Hamilton, New Zealand. …

4: Bibliography N

… ►

G. Nemes (2018) Error bounds for the large-argument asymptotic expansions of the Lommel and allied functions. Stud. Appl. Math. 140 (4), pp. 508–541.

ⓘ

External Links:: ISSN 0022-2526, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt
Referenced by:: §11.11(i), §11.11(i).
Encodings:: BibTeX

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E. Neuman (1969a) Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

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Notes:: Includes Algol program.
External Links:: MathReview
Referenced by:: §19.39(iii).
Encodings:: BibTeX

►

E. Neuman (1969b) On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(ii).
Encodings:: BibTeX

… ►

C. J. Noble and I. J. Thompson (1984) COULN, a program for evaluating negative energy Coulomb functions. Comput. Phys. Comm. 33 (4), pp. 413–419.

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Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: 2nd item, 7th item.
Encodings:: BibTeX

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V. Yu. Novokshënov (1985) The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

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Notes:: In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.
External Links:: ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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5: Bibliography K

… ►

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

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

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

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

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

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T. H. Koornwinder and M. Mazzocco (2018) Dualities in the

q

-Askey scheme and degenerate DAHA. Stud. Appl. Math. 141 (4), pp. 424–473.

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External Links:: ISSN 0022-2526, Document, Link, MathReview (Marzena Szajewska)
Referenced by:: (18.28.1_5).
Encodings:: BibTeX

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T. Koornwinder, A. Kostenko, and G. Teschl (2018) Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

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External Links:: ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)
Referenced by:: §18.14(i).
Encodings:: BibTeX

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6: 27.12 Asymptotic Formulas: Primes

… ►

p_{n}

is the

n

th prime, beginning with

p_{1}=2

. … ►

27.12.1

\lim_{n\to\infty}\frac{p_{n}}{n\ln n}=1,

ⓘ

Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : positive integer and $p,p_{1},\ldots$ : prime numbers
Permalink:: http://dlmf.nist.gov/27.12.E1
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.12 and Ch.27

►

27.12.2

p_{n}>n\ln n,

n=1,2,\dots

ⓘ

Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : positive integer and $p,p_{1},\ldots$ : prime numbers
Permalink:: http://dlmf.nist.gov/27.12.E2
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.12 and Ch.27

►

27.12.3

\pi\left(x\right)=\left\lfloor x\right\rfloor-1-\sum_{p_{j}\leq\sqrt{x}}\left% \lfloor\frac{x}{p_{j}}\right\rfloor+\sum_{r\geq 2}(-1)^{r}\*\sum_{p_{j_{1}}<p_% {j_{2}}<\cdots<p_{j_{r}}\leq\sqrt{x}}\left\lfloor\frac{x}{p_{j_{1}}p_{j_{2}}% \cdots p_{j_{r}}}\right\rfloor,

x\geq 1

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Symbols:: $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\pi\left(\NVar{x}\right)$ : number of primes not exceeding a number, $p,p_{1},\ldots$ : prime numbers and $x$ : real number
Permalink:: http://dlmf.nist.gov/27.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §27.12 and Ch.27

… ►The largest known prime (2018) is the Mersenne prime

2^{82,589,933}-1

. …

7: Bibliography I

… ►

IEEE (2018) IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017. The Institute of Electrical and Electronics Engineers, Inc..

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External Links:: ISBN 978-1-5044-4602-0, Document
Referenced by:: §3.1(ii), §3.1(ii), Erratum (V1.0.25) for Section 3.1.
Encodings:: BibTeX

… ►

Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada (1993) The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of

J_{0}(z)-iJ_{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

…

8: 3.1 Arithmetics and Error Measures

… ►where

s

is equal to

1

0

, each

b_{j}

j\geq 1

, is either

0

1

b_{1}

is the most significant bit,

p

(

\in\mathbb{N}

) is the number of significant bits

b_{j}

b_{p-1}

is the least significant bit,

E

is an integer called the exponent,

b_{0}.b_{1}b_{2}\dots b_{p-1}

is the significand, and

f=.b_{1}b_{2}\dots b_{p-1}

is the fractional part. … ►with

b_{0}=1

and all allowable choices of

E

p

s

, and

b_{j}

. … ►Let

E_{{\rm min}}\leq E\leq E_{{\rm max}}

with

E_{{\rm min}}<0

and

E_{{\rm max}}>0

. … ►The respective machine precisions are

\frac{1}{2}\epsilon_{M}=0.596\times 10^{-7}

\frac{1}{2}\epsilon_{M}=0.111\times 10^{-15}

and

\frac{1}{2}\epsilon_{M}=0.963\times 10^{-34}

. … ►For interval arithmetic, one should refer to the IEEE Standards for Interval Arithmetic IEEE (2015, 2018). …

9: Bibliography R

… ►

M. Rahman (2001) The Associated Classical Orthogonal Polynomials. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 255–279.

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External Links:: MathReview (Wadim Zudilin), ZentralBlatt
Referenced by:: §18.30(viii).
Encodings:: BibTeX

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M. Reed and B. Simon (1980) Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis. Elsevier, New York.

ⓘ

Notes:: revised and expanded Edition from 1972,
External Links:: ISBN 10: 0-12-585050-6(vol. 1), MathReview Entry
Referenced by:: §1.18(x).
Encodings:: BibTeX

… ►

W. P. Reinhardt (2018) Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

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Notes:: arXiv:1812.02196
Referenced by:: §18.40(ii).
Encodings:: BibTeX

… ►

H. P. Robinson (1972) Roots of

\tan x=x

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Notes:: 10 typewritten pages deposited in the UMT file at Lawrence Berkeley Laboratory, University of California, Berkeley, CA.
Referenced by:: §4.46.
Encodings:: BibTeX

►

M. Robnik (1980) An extremum property of the

n

-dimensional sphere. J. Phys. A 13 (10), pp. L349–L351.

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External Links:: ISSN 0305-4470, Document, MathReview
Referenced by:: §5.19(iii).
Encodings:: BibTeX

…

10: Bibliography C

… ►

L. Carlitz (1960) Note on Nörlund’s polynomial

B_{n}^{(z)}

. Proc. Amer. Math. Soc. 11 (3), pp. 452–455.

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External Links:: FullText, MathReview (M. P. Drazin), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

… ►

P. A. Clarkson and K. Jordaan (2018) Properties of generalized Freud polynomials. J. Approx. Theory 225, pp. 148–175.

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External Links:: ISSN 0021-9045, Document, Link, MathReview (Maria das Neves Rebocho)
Referenced by:: §18.32.
Encodings:: BibTeX

►

P. A. Clarkson and M. D. Kruskal (1989) New similarity reductions of the Boussinesq equation. J. Math. Phys. 30 (10), pp. 2201–2213.

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External Links:: ISSN 0022-2488, Document, Link, MathReview (B. V. Baby), ZentralBlatt
Referenced by:: Profile Peter A. Clarkson.
Encodings:: BibTeX

… ►

F. Cooper, A. Khare, and A. Saxena (2006) Exact elliptic compactons in generalized Korteweg-de Vries equations. Complexity 11 (6), pp. 30–34.

ⓘ

External Links:: ISSN 1076-2787, Document, MathReview
Referenced by:: §19.6(i).
Encodings:: BibTeX

… ►

Cunningham Project (website)

ⓘ

Notes:: Seeks to factor numbers of the form $b^{n}\pm 1$ for $b=2,3,5,6,7,10,11,12$ .
External Links:: Home Page
Referenced by:: 3rd item.
Encodings:: BibTeX

…