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21: 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.

22: 35 Functions of Matrix Argument

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23: 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. …

24: Bibliography Y

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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

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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

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25: 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 $$5\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n}>{(-1)}^{n+1}{B}_{2n}>4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$

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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

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

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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

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

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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

26: 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 $(p-1)p^{\mathrm{\ell }-1}|2n$.

27: 24.4 Basic Properties

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

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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

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

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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

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

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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

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

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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

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

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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

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28: Joyce E. Conlon

… ►She occupied various positions providing support for high performance scientific computing. …

29: Marjorie A. McClain

… ► 1956 in Ithaca, New York) is a mathematician in the Applied and Computational Mathematics Division of NIST where she has provided support for mathematical software libraries and assisted with numerical computing projects since 1979. …

30: Software Index

… ►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►

This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

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These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

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