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21: 3.11 Approximation Techniques

… ►If $f$ is continuously differentiable on $[-1,1]$, then with … ► Moreover, the set of minimax approximations $p_0(x),p_1(x),p_2(x),\mathrm{\dots },p_n(x)$ requires the calculation and storage of $\frac{1}{2}(n+1)(n+2)$ coefficients, whereas the corresponding set of Chebyshev-series approximations requires only $n+1$ coefficients. … ►With $b_0=1$, the last $q$ equations give $b_1,\mathrm{\dots },b_q$ as the solution of a system of linear equations. … ►Here $x_j$, $j=1,2,\mathrm{\dots },J$, is a given set of distinct real points and $J\ge n+1$. … ►(3.11.29) is a system of $n+1$ linear equations for the coefficients $a_0,a_1,\mathrm{\dots },a_n$. …

22: 24.11 Asymptotic Approximations

… ►

24.11.1 $${(-1)}^{n+1}{B}_{2n}\sim \frac{2(2n)!}{{(2\pi )}^{2n}},$$

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Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

►

24.11.2 $${(-1)}^{n+1}{B}_{2n}\sim 4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$

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Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Permalink:

http://dlmf.nist.gov/24.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

►

24.11.3 $${(-1)}^n{E}_{2n}\sim \frac{2^{2n+2}(2n)!}{{\pi }^{2n+1}},$$

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Symbols:

$E_n$: Euler numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

Permalink:

http://dlmf.nist.gov/24.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

… ►

24.11.5

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $\sin z$: sine function, $n$: integer and $x$: real or complex

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

►

24.11.6

ⓘ

Symbols:

$E_n\left(x\right)$: Euler polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $\sin z$: sine function, $n$: integer and $x$: real or complex

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

…

23: Foreword

… ►In 1964 the National Institute of Standards and Technology published the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by Milton Abramowitz and Irene A. … ►Particular attention is called to the generous support of the National Science Foundation, which made possible the participation of experts from academia and research institutes worldwide. …

24: 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). …If you have trouble viewing the WebGL visualizations in your web browser, see x3dom.org or caniuse.com/webgl for information on WebGL browser support. … ►Please see caniuse.com/webgl or x3dom.org for information on WebGL browser support.

25: 35 Functions of Matrix Argument

…

26: Gloria Wiersma

… ► 1937 in Washington, DC) joined the NIST staff in 1973, where she occupied various positions providing support for the Physics Laboratory until 1993. …

27: Bibliography Y

… ►

Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State. American Journal of Physics 57 (1), pp. 85–86.

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External Links:

Document

Referenced by:

§1.18(viii).

Encodings:

BibTeX

… ►

Software for interval arithmetic: A reasonably portable package. ACM Trans. Math. Software 5 (1), pp. 50–63.

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External Links:

ISSN 0098-3500, Document

Referenced by:

§4.48(ii).

Encodings:

BibTeX

…

28: 24.9 Inequalities

… ►Except where otherwise noted, the inequalities in this section hold for $n=1,2,\mathrm{\dots }$. … ►(24.9.3)–(24.9.5) hold for $\frac{1}{2}>x>0$. … ►

24.9.6 $$5\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n}>{(-1)}^{n+1}{B}_{2n}>4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Referenced by:

§24.9, §24.9

Permalink:

http://dlmf.nist.gov/24.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

… ►

24.9.8 $$\frac{2(2n)!}{{(2\pi )}^{2n}}\frac{1}{1-2^{\beta -2n}}\ge {(-1)}^{n+1}{B}_{2n}\ge \frac{2(2n)!}{{(2\pi )}^{2n}}\frac{1}{1-2^{-2n}}$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: integer and $\beta$: quantity

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

… ►

24.9.10 $$\frac{4^{n+1}(2n)!}{{\pi }^{2n+1}}>{(-1)}^n{E}_{2n}>\frac{4^{n+1}(2n)!}{{\pi }^{2n+1}}\frac{1}{1+3^{-1-2n}}.$$

ⓘ

Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

A&S Ref:

23.1.15

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.9.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

29: 24.10 Arithmetic Properties

… ►where the summation is over all $p$ such that $p-1$ divides $2n$. …where $n\ge 2$, and $\mathrm{\ell }(\ge 1)$ is an arbitrary integer such that $(p-1)p^{\mathrm{\ell }}|2n$. … ►where $m\equiv n\cancel{\equiv }0(modp-1)$. … ►valid for fixed integers $\mathrm{\ell }(\ge 1)$, and for all $n(\ge 1)$ such that $2n\cancel{\equiv }0$ $(modp-1)$ and $p^{\mathrm{\ell }}|2n$. …valid for fixed integers $\mathrm{\ell }(\ge 1)$ and for all $n(\ge 1)$ such that $(p-1)p^{\mathrm{\ell }-1}|2n$.

30: 24.4 Basic Properties

… ►

24.4.7 $$\sum _{k=1}^m{k}^n=\frac{B_{n+1}\left(m+1\right)-B_{n+1}}{n+1},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $k$: integer, $m$: integer and $n$: integer

A&S Ref:

23.1.4

Referenced by:

§24.17(iii)

Permalink:

http://dlmf.nist.gov/24.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iii), §24.4 and Ch.24

… ►

24.4.11 $$\sum _{\begin{array}{c}k=1\\ \left(k,m\right)=1\end{array}}^m{k}^n=\frac{1}{n+1}\sum _{j=1}^{n+1}\left(\genfrac{}{}{0pt}{}{n+1}{j}\right)\left(\prod _{p|m}(1-p^{n-j})B_{n+1-j}\right)m^j.$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\left(m,n\right)$: greatest common divisor (gcd), $j$: integer, $k$: integer, $m$: integer, $n$: integer and $p$: prime

Referenced by:

§24.4(iii)

Permalink:

http://dlmf.nist.gov/24.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iii), §24.4 and Ch.24

… ►

24.4.26 $$E_n\left(0\right)=-E_n\left(1\right)=-\frac{2}{n+1}(2^{n+1}-1)B_{n+1},$$ $n>0$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials and $n$: integer

A&S Ref:

23.1.20

Referenced by:

§24.9, Erratum (V1.0.5) for Equation (24.4.26)

Permalink:

http://dlmf.nist.gov/24.4.E26

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

This equation is true only for $n>0$. Previously, $n=0$ was also allowed.

Reported 2012-05-14 by Vladimir Yurovsky

See also:

Annotations for §24.4(vi), §24.4 and Ch.24

… ►

24.4.30 $$E_{2n-1}\left(\frac{1}{3}\right)=-E_{2n-1}\left(\frac{2}{3}\right)=-\frac{(1-3^{1-2n})(2^{2n}-1)}{2n}{B}_{2n},$$ $n=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials and $n$: integer

A&S Ref:

23.1.22

Permalink:

http://dlmf.nist.gov/24.4.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(vi), §24.4 and Ch.24

… ►

24.4.33 $$E_{2n}\left(\frac{1}{6}\right)=E_{2n}\left(\frac{5}{6}\right)=\frac{1+3^{-2n}}{2^{2n+1}}{E}_{2n}.$$

ⓘ

Symbols:

$E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials and $n$: integer

Referenced by:

§24.4(vi)

Permalink:

http://dlmf.nist.gov/24.4.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(vi), §24.4 and Ch.24

…