”lecain” logint issue Number 1206⤄360⤄1604 customer support ☎️"Number"

SearchAdvancedHelp

(0.004 seconds)

1—10 of 257 matching pages

1: 24.1 Special Notation

… ►

Bernoulli Numbers and Polynomials

►The origin of the notation $B_n$, $B_n\left(x\right)$, is not clear. … ►

Euler Numbers and Polynomials

… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations $E_n$, $E_n\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

2: 27.12 Asymptotic Formulas: Primes

… ►

Prime Number Theorem

… ►For the logarithmic integral $\mathrm{li}\left(x\right)$ see (6.2.8). … ► $\pi \left(x\right)-\mathrm{li}\left(x\right)$ changes sign infinitely often as $x\to \mathrm{\infty }$; see Littlewood (1914), Bays and Hudson (2000). … ►There are infinitely many Carmichael numbers.

3: 6.16 Mathematical Applications

… ►

§6.16(ii) Number-Theoretic Significance of $\mathrm{li}\left(x\right)$

►If we assume Riemann’s hypothesis that all nonreal zeros of $\zeta \left(s\right)$ have real part of $\frac{1}{2}$ (§25.10(i)), then ►

6.16.5 $$\mathrm{li}\left(x\right)-\pi (x)=O\left(\sqrt{x}\ln x\right),$$ $x\to \mathrm{\infty }$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $x$: real variable and $\pi (x)$: number of primes $\le x$

A&S Ref:

5.1.50

Permalink:

http://dlmf.nist.gov/6.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(ii), §6.16 and Ch.6

►where $\pi (x)$ is the number of primes less than or equal to $x$. … ►

►►

Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

4: 27.21 Tables

§27.21 Tables

… ►Bressoud and Wagon (2000, pp. 103–104) supplies tables and graphs that compare $\pi \left(x\right),x/\ln x$, and $\mathrm{li}\left(x\right)$. … ►Lehmer (1941) gives a comprehensive account of tables in the theory of numbers, including virtually every table published from 1918 to 1941. … ►No sequel to Lehmer (1941) exists to date, but many tables of functions of number theory are included in Unpublished Mathematical Tables (1944).

5: Bibliography L

… ►

A table of integrals involving the functions $E_n(x)$ .

ⓘ

Notes:

National Research Council of Canada, Division of Atomic Energy, Document no. MT-131 (NRC 1553)

Referenced by:

§8.19(x).

Encodings:

BibTeX

… ►

An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

ⓘ

External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

BibTeX

… ►

Guide to Tables in the Theory of Numbers. Bulletin of the National Research Council, No. 105, National Research Council, Washington, D.C..

ⓘ

Notes:

Reprinted in 1961.

External Links:

MathReview (R. D. Carmichael), ZentralBlatt

Referenced by:

§27.20, §27.20, §27.21, §27.21.

Encodings:

BibTeX

►

List of Prime Numbers from 1 to 10,006,721. Publ. No. 165, Carnegie Institution of Washington, Washington, D.C..

ⓘ

External Links:

ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

… ►

Highly accurate tables for elementary functions. BIT 35 (3), pp. 352–360.

ⓘ

External Links:

ISSN 0006-3835, Document, MathReview, ZentralBlatt

Referenced by:

§4.45(i), §4.46.

Encodings:

BibTeX

…

6: 8.19 Generalized Exponential Integral

… ►For $j=1,2,3,\mathrm{\dots }$, ►

8.19.15 $$\frac{{\partial }^j{E}_p\left(z\right)}{{\partial p}^j}={(-1)}^j{\int }_1^{\mathrm{\infty }}{(\ln t)}^j{t}^{-p}{\mathrm{e}}^{-zt}dt,$$ $\mathrm{\Re }z>0$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\mathrm{\Re }$: real part, $z$: complex variable, $p$: parameter and $j$: numbers

Permalink:

http://dlmf.nist.gov/8.19.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(v), §8.19(v), §8.19 and Ch.8

… ►For collections of integrals involving $E_p\left(z\right)$, especially for integer $p$, see Apelblat (1983, §§7.1–7.2) and LeCaine (1945). …

7: 6.1 Special Notation

… ►The main functions treated in this chapter are the exponential integrals $\mathrm{Ei}\left(x\right)$, $E_1\left(z\right)$, and $\mathrm{Ein}\left(z\right)$; the logarithmic integral $\mathrm{li}\left(x\right)$; the sine integrals $\mathrm{Si}\left(z\right)$ and $\mathrm{si}\left(z\right)$; the cosine integrals $\mathrm{Ci}\left(z\right)$ and $\mathrm{Cin}\left(z\right)$. …

8: Viewing DLMF Interactive 3D Graphics

… ►WebGL is supported in the current versions of most common web browsers. … ►1, some advanced features of X3DOM are currently not fully supported (see x3dom.org). Until these issues are resolved we cannot guarantee that the DLMF WebGL visualizations can be viewed in Internet Explorer. … ►After installing the viewer you must select Customize DLMF on the DLMF Menu bar and choose either VRML or X3D under “Visualization Format. … ►Please see caniuse.com/webgl or x3dom.org for information on WebGL browser support.

9: Need Help?

… ►

• DLMF pages look all messed up! See Browsers, or perhaps About MathML.

• I’m seeing things like or or ?; Or, My browser claims that I am missing fonts! See Special fonts for MathML.

• What’s this ‘MathML’, anyway? See About MathML.

…

10: 4.10 Integrals

… ►

4.10.7 $${⨍}_0^x\frac{dt}{\ln t}=\mathrm{li}\left(x\right),$$ $x>1$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

A&S Ref:

4.1.57

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

… ►For $\mathrm{li}\left(x\right)$ see §6.2(i). …