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31: 22.20 Methods of Computation

… ►Then as $n\to \mathrm{\infty }$ sequences $\{a_n\}$, $\{b_n\}$ converge to a common limit $M=M(a_0,b_0)$, the arithmetic-geometric mean of $a_0,b_0$. …

32: 28.29 Definitions and Basic Properties

… ►To every equation (28.29.1), there belong two increasing infinite sequences of real eigenvalues: …

33: 30.15 Signal Analysis

… ►The sequence ${\varphi }_n$, $n=0,1,2,\mathrm{\dots }$ forms an orthonormal basis in the space of $\sigma$-bandlimited functions, and, after normalization, an orthonormal basis in $L^2(-\tau ,\tau )$. …

34: Bibliography S

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Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.

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External Links:

MathReview (J. Kuntzmann), ZentralBlatt

Referenced by:

§17.18.

Encodings:

BibTeX

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The On-Line Encyclopedia of Integer Sequences. Notices Amer. Math. Soc. 50 (8), pp. 912–915.

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External Links:

ISSN 0002-9920, FullText, Electronic Database, MathReview (Christian Krattenthaler), ZentralBlatt

Referenced by:

(19.20.2), (19.20.22), (19.30.13), (25.11.40), (3.12.1), (3.12.3), (3.12.4), §3.12, (4.2.11), (4.2.17), (4.2.18), (4.4.19), (5.17.6), (5.2.3), (5.4.10), (5.4.6), (5.4.7), (5.4.8), (5.4.9), (6.13.1).

Encodings:

BibTeX

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35: 18.14 Inequalities

… ►The successive maxima of $|H_n\left(x\right)|$ form a decreasing sequence for $x\le 0$, and an increasing sequence for $x\ge 0$. …

36: 19.20 Special Cases

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19.20.2 $${\int }_0^1\frac{dt}{\sqrt{1-t^4}}=R_F(0,1,2)=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{4}\right)\right)}^2}{4{(2\pi )}^{1/2}}=\mathrm{1.31102\hspace{0.33em}87771\hspace{0.33em}46059\hspace{0.33em}90523}\mathrm{\dots }.$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ and $\int$: integral

Notes:

For more digits see OEIS Sequence A085565; see also Sloane (2003).

Referenced by:

§19.20(i), §19.20(iv), §19.21(i)

Permalink:

http://dlmf.nist.gov/19.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(i), §19.20 and Ch.19

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19.20.22 $${\int }_0^1\frac{t^{2}dt}{\sqrt{1-t^4}}=\frac{1}{3}{R}_D(0,2,1)=\frac{{\left(\mathrm{\Gamma }\left(\frac{3}{4}\right)\right)}^2}{{(2\pi )}^{1/2}}=\mathrm{0.59907\hspace{0.33em}01173\hspace{0.33em}67796\hspace{0.33em}10371}\mathrm{\dots }.$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ and $\int$: integral

Notes:

For more digits see OEIS Sequence A085566; see also Sloane (2003).

Referenced by:

§19.20(iv), §19.21(i), Figure 19.3.6, Figure 19.3.6, Figure 19.3.6

Permalink:

http://dlmf.nist.gov/19.20.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

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37: Bibliography K

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Unimodal sequences show that Lambert $W$ is Bernstein. C. R. Math. Acad. Sci. Soc. R. Can. 33 (2), pp. 50–56.

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External Links:

ISSN 0706-1994, MathReview (Fana Tangara), ZentralBlatt

Referenced by:

§4.13.

Encodings:

BibTeX

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38: Bibliography L

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Monotonic sequences related to zeros of Bessel functions. Numer. Algorithms 49 (1-4), pp. 221–233.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

Encodings:

BibTeX

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39: 1.11 Zeros of Polynomials

… ►where the column vector $h_k^{(m)}$ consists of the first $k$ members of the sequence $a_m,a_{m-1},a_{m-2},\mathrm{\dots }$ with $a_j=0$ if or $j>n$. …

40: Bibliography R

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Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.

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Notes:

Software packages for hypergeometric and $q$-hypergeometric summation, sequences and power series, permutation groups, partition analysis, and difference equations.

External Links:

RISC Combinatorics Group

Referenced by:

6th item.

Encodings:

BibTeX

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