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11: Bibliography W

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Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Reports 10 (5-6), pp. 189–371.

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External Links:

Document

Referenced by:

§2.11(vi), §3.9(v), §3.9(vi).

Encodings:

BibTeX

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Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.

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External Links:

Document

Referenced by:

§2.11(vi), §2.11(vi).

Encodings:

BibTeX

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Sequence Transformations and their Applications. Mathematics in Science and Engineering, Vol. 154, Academic Press Inc., New York.

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External Links:

ISBN 0-12-757940-0, MathReview (Jan Zítko), ZentralBlatt

Referenced by:

§3.9(vi).

Encodings:

BibTeX

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12: 8.18 Asymptotic Expansions of $I_x(a,b)$

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General Case

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8.18.14 $$I_x(a,b)\sim Q(b,a\zeta )-\frac{{(2\pi b)}^{-1/2}}{{\mathrm{\Gamma }}^{\ast }\left(b\right)}{\left(\frac{x}{x_0}\right)}^a{\left(\frac{1-x}{1-x_0}\right)}^b\sum _{k=0}^{\mathrm{\infty }}\frac{h_k(\zeta ,\mu )}{a^k},$$

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Symbols:

$I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $Q(a,z)$: normalized incomplete gamma function, ${\mathrm{\Gamma }}^{\ast }\left(z\right)$: scaled gamma function, $k$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable, $\mu$ , $\zeta$: variable and $h_k(\zeta ,\mu )$: coefficients

Permalink:

http://dlmf.nist.gov/8.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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13: 2.10 Sums and Sequences

§2.10 Sums and Sequences

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§2.10(i) Euler–Maclaurin Formula

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§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method

… ►See also Flajolet and Odlyzko (1990).

14: 1.9 Calculus of a Complex Variable

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§1.9(v) Infinite Sequences and Series

… ►This sequence converges pointwise to a function $f(z)$ if … ►

§1.9(vii) Inversion of Limits

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Double Sequences and Series

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15: 3.8 Nonlinear Equations

… ►Let $z_1,z_2,\mathrm{\dots }$ be a sequence of approximations to a root, or fixed point, $\zeta$. …for all $n$ sufficiently large, where $A$ and $p$ are independent of $n$, then the sequence is said to have convergence of the $p$th order. … ►For real functions $f(x)$ the sequence of approximations to a real zero $\xi$ will always converge (and converge quadratically) if either: … ►Starting this iteration in the neighborhood of one of the four zeros $\pm 1,\pm \mathrm{i}$, sequences $\{z_n\}$ are generated that converge to these zeros. For an arbitrary starting point $z_0\in ℂ$, convergence cannot be predicted, and the boundary of the set of points $z_0$ that generate a sequence converging to a particular zero has a very complicated structure. …

16: Guide to Searching the DLMF

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any double-quoted sequence of textual words and numbers.

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any LaTeX-like sequence of terms, excluding phrases.

… ►If you do not want a term or a sequence of terms in your query to undergo math processing, you should quote them as a phrase. …

17: 6.13 Zeros

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6.13.1 $$x_0=\mathrm{0.37250\hspace{0.33em}74107\hspace{0.33em}81366\hspace{0.33em}63446\hspace{0.33em}19918\hspace{0.33em}66580}\mathrm{\dots }.$$

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Symbols:

$x$: real variable

Notes:

For more digits see OEIS Sequence A091723; see also Sloane (2003).

Referenced by:

§6.13

Permalink:

http://dlmf.nist.gov/6.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.13 and Ch.6

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18: 8.25 Methods of Computation

… ►Expansions involving incomplete gamma functions often require the generation of sequences $P(a+n,x)$, $Q(a+n,x)$, or ${\gamma }^{\ast }(a+n,x)$ for fixed $a$ and $n=0,1,2,\mathrm{\dots }$. …

19: 5.4 Special Values and Extrema

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5.4.6 $$\mathrm{\Gamma }\left(\frac{1}{2}\right)={\pi }^{1/2}=\mathrm{1.77245\hspace{0.33em}38509\hspace{0.33em}05516\hspace{0.33em}02729}\mathrm{\dots },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $\pi$: the ratio of the circumference of a circle to its diameter

A&S Ref:

6.1.8

Notes:

For more digits see OEIS Sequence A002161; see also Sloane (2003).

Referenced by:

§4.10(ii), §5.4(i)

Permalink:

http://dlmf.nist.gov/5.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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5.4.7 $$\mathrm{\Gamma }\left(\frac{1}{3}\right)=\mathrm{2.67893\hspace{0.33em}85347\hspace{0.33em}07747\hspace{0.33em}63365}\mathrm{\dots },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.11 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A073005; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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5.4.8 $$\mathrm{\Gamma }\left(\frac{2}{3}\right)=\mathrm{1.35411\hspace{0.33em}79394\hspace{0.33em}26400\hspace{0.33em}41694}\mathrm{\dots },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.13 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A073006; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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5.4.9 $$\mathrm{\Gamma }\left(\frac{1}{4}\right)=\mathrm{3.62560\hspace{0.33em}99082\hspace{0.33em}21908\hspace{0.33em}31193}\mathrm{\dots },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.10 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A068466; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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5.4.10 $$\mathrm{\Gamma }\left(\frac{3}{4}\right)=\mathrm{1.22541\hspace{0.33em}67024\hspace{0.33em}65177\hspace{0.33em}64512}\mathrm{\dots }.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.14 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A068465; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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20: 17.12 Bailey Pairs

… ►A sequence of pairs of rational functions of several variables $({\alpha }_n,{\beta }_n)$, $n=0,1,2,\mathrm{\dots }$, is called a Bailey pair provided that for each $n\geqq 0$ … ►When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. …