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21—30 of 47 matching pages

21: Preface

… ►All of the mathematical information contained in the Handbook is also contained in the DLMF, along with additional features such as more graphics, expanded tables, and higher members of some families of formulas; in consequence, in the Handbook there are occasional gaps in the numbering sequences of equations, tables, and figures. …

22: 5.2 Definitions

… ►

5.2.3 $$\gamma =\underset{n\to \mathrm{\infty }}{lim}\left(1+\frac{1}{2}+\frac{1}{3}+\mathrm{\cdots }+\frac{1}{n}-\ln n\right)=\mathrm{0.57721\hspace{0.33em}56649\hspace{0.33em}01532\hspace{0.33em}86060}\mathrm{\dots }.$$

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Defines:

$\gamma$: Euler’s constant

Symbols:

$\ln z$: principal branch of logarithm function and $n$: nonnegative integer

A&S Ref:

6.1.3 (where the 10D value is given, and Table 1.1 where the 24D value is given.)

Notes:

For more digits see OEIS Sequence A001620; see also Sloane (2003).

Referenced by:

§4.4(iii)

Permalink:

http://dlmf.nist.gov/5.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(ii), §5.2 and Ch.5

…

23: 5.17 Barnes’ $G$-Function (Double Gamma Function)

… ►

5.17.6 $$A={\mathrm{e}}^C=\mathrm{1.28242\hspace{0.33em}71291\hspace{0.33em}00622\hspace{0.33em}63687}\mathrm{\dots },$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $A$: Glaisher’s constant and $C$: log of Glaisher’s Constant

Notes:

For more digits see OEIS Sequence A074962; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.17.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

…

24: 1.16 Distributions

… ►A sequence $\{{\varphi }_n\}$ of test functions converges to a test function $\varphi$ if the support of every ${\varphi }_n$ is contained in a fixed compact set $K$ and as $n\to \mathrm{\infty }$ the sequence $\{{\varphi }_n^{(k)}\}$ converges uniformly on $K$ to ${\varphi }^{(k)}$ for $k=0,1,2,\mathrm{\dots }$. … ►We say that a sequence of distributions $\{{\mathrm{\Lambda }}_n\}$ converges to a distribution $\mathrm{\Lambda }$ in ${𝒟}^{\ast }$ if … ►A sequence $\{{\varphi }_n\}$ of functions in $𝒯$ is said to converge to a function $\varphi \in 𝒯$ as $n\to \mathrm{\infty }$ if the sequence $\{{\varphi }_n^{(k)}\}$ converges uniformly to ${\varphi }^{(k)}$ on every finite interval and if the constants $c_{k,N}$ in the inequalities … ►A sequence of tempered distributions ${\mathrm{\Lambda }}_n$ converges to $\mathrm{\Lambda }$ in ${𝒯}^{\ast }$ if … ►A sequence $\{{\varphi }_m\}$ of functions in ${𝒟}_n$ converges to a function $\varphi \in {𝒟}_n$ if the supports of ${\varphi }_m$ lie in a fixed compact subset $K$ of ${ℝ}^n$ and ${\varphi }_m^{(k)}$ converges uniformly to ${\varphi }^{(k)}$ in $K$ for every multi-index $k=(k_1,k_2,\mathrm{\dots },k_n)$. …

25: 4.4 Special Values and Limits

… ►

4.4.19 $$\underset{n\to \mathrm{\infty }}{lim}\left(\left(\sum _{k=1}^n\frac{1}{k}\right)-\ln n\right)=\gamma =\mathrm{0.57721\; 56649\; 01532\; 86060}\mathrm{\dots },$$

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Symbols:

$\gamma$: Euler’s constant, $\ln z$: principal branch of logarithm function, $k$: integer and $n$: integer

A&S Ref:

4.1.32 (with 10D value)

Notes:

For more digits see OEIS Sequence A001620; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/4.4.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(iii), §4.4 and Ch.4

…

26: 4.45 Methods of Computation

… ►Beginning with $x_0=x$, generate the sequence …

27: Bibliography C

… ►

Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Trans. Math. Software 9 (2), pp. 242–245.

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Notes:

Double-precision Fortran, maximum accuracy 24S.

External Links:

ISSN 0098-3500, Document, GAMS (module 597; package TOMS), ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

… ►

Department of Computer Science, University of Victoria, Canada.

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Notes:

Permutations, set and integer partitions, sequences, graphs, and trees.

External Links:

Combinatorial Object Server

Referenced by:

1st item.

Encodings:

BibTeX

… ►

A sequence of series for the Lambert $W$ function. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), pp. 197–204.

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External Links:

Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(4.13.4_1), (4.13.4_2), (4.13.5_2), §4.13.

Encodings:

BibTeX

…

28: 1.10 Functions of a Complex Variable

… ►The convergence of the infinite product is uniform if the sequence of partial products converges uniformly. … ►If $\{z_n\}$ is a sequence such that ${\sum }_{n=1}^{\mathrm{\infty }}\left|z_n^{-2}\right|$ is convergent, then … ►If $\{a_n\}$ and $\{z_n\}$ are sequences such that $z_m\ne z_n$ ($m\ne n$) and ${\sum }_{n=1}^{\mathrm{\infty }}\left|a_n{z}_n^{-2}\right|$ is convergent, then …

29: 18.40 Methods of Computation

… ►These quadrature weights and abscissas will then allow construction of a convergent sequence of approximations to $w(x)$, as will be considered in the following paragraphs. …

30: 19.30 Lengths of Plane Curves

… ►

19.30.13 $$P=4\sqrt{2a^2}{R}_F(0,1,2)=\sqrt{2a^2}\times \mathrm{5.24411\hspace{0.33em}51}\mathrm{\dots }=4aK\left(1/\sqrt{2}\right)=a\times \mathrm{7.41629\hspace{0.33em}87}\mathrm{\dots }.$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind and $P$: length

Notes:

For more digits see OEIS Sequence A064853; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/19.30.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.30(iii), §19.30 and Ch.19

…