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1: 3.9 Acceleration of Convergence

… ►

§3.9(i) Sequence Transformations

►All sequences (series) in this section are sequences (series) of real or complex numbers. … ►, a sequence for which … ►

§3.9(iv) Shanks’ Transformation

… ►Sequences that are accelerated by Levin’s transformation include logarithmically convergent sequences, i. …

2: 3.12 Mathematical Constants

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3.12.1 $$\pi =\mathrm{3.14159\hspace{0.33em}26535\hspace{0.33em}89793\hspace{0.33em}23846}\mathrm{\dots }$$

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Defines:

$\pi$: the ratio of the circumference of a circle to its diameter

Notes:

For more digits see OEIS Sequence A000796; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/3.12.E1

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Annotations for §3.12 and Ch.3

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3.12.3 $$\mathrm{e}=\mathrm{2.71828\hspace{0.33em}18284\hspace{0.33em}59045\hspace{0.33em}23536}\mathrm{\dots },$$

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Symbols:

$\mathrm{e}$: base of natural logarithm

Notes:

For more digits see OEIS Sequence A001113; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/3.12.E3

Encodings:

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Annotations for §3.12 and Ch.3

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3.12.4 $$\gamma =\mathrm{0.57721\hspace{0.33em}56649\hspace{0.33em}01532\hspace{0.33em}86060}\mathrm{\dots },$$

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Symbols:

$\gamma$: Euler’s constant

Notes:

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

Permalink:

http://dlmf.nist.gov/3.12.E4

Encodings:

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Annotations for §3.12 and Ch.3

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3: 1.17 Integral and Series Representations of the Dirac Delta

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§1.17(i) Delta Sequences

… ►for a suitably chosen sequence of functions ${\delta }_n\left(x\right)$, $n=1,2,\mathrm{\dots }$. Such a sequence is called a delta sequence and we write, symbolically, ►

1.17.4 $$\underset{n\to \mathrm{\infty }}{lim}{\delta }_n\left(x\right)=\delta \left(x\right),$$ $x\in ℝ$.

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), ${\delta }_n\left(x\right)$: Dirac delta sequence, $\in$: element of, $ℝ$: real line and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.17.E4

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Annotations for §1.17(i), §1.17 and Ch.1

►An example of a delta sequence is provided by …

4: 26.22 Software

… ►

On-Line Encyclopedia of Integer Sequences (website).

…

5: 3.6 Linear Difference Equations

… ►Many special functions satisfy second-order recurrence relations, or difference equations, of the form … ►

3.6.4 $$w_n/g_n\to 0,$$ $n\to \mathrm{\infty }$.

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Symbols:

$w_n$: sequence and $g_n$: solution

Permalink:

http://dlmf.nist.gov/3.6.E4

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Annotations for §3.6(ii), §3.6 and Ch.3

… ►At the same time we construct a sequence $e_n$, $n=0,1,\mathrm{\dots }$, defined by ►

3.6.8 $$a_n{e}_n=c_n{e}_{n-1}-d_n{p}_n,$$

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Defines:

$e_n$: sequence (locally)

Symbols:

$p_n$: solutions, $a_n$: coefficient, $c_n$: coefficient and $d_n$: coefficient

Permalink:

http://dlmf.nist.gov/3.6.E8

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Annotations for §3.6(v), §3.6 and Ch.3

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3.6.10 $$p_{n+1}{w}_n=p_n{w}_{n+1}+e_n,$$

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Symbols:

$w_n$: sequence, $p_n$: solutions and $e_n$: sequence

Permalink:

http://dlmf.nist.gov/3.6.E10

Encodings:

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Annotations for §3.6(v), §3.6 and Ch.3

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6: 2.10 Sums and Sequences

§2.10 Sums and Sequences

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§2.10(i) Euler–Maclaurin Formula

… ► … ►

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

… ►See also Flajolet and Odlyzko (1990).

7: 2.1 Definitions and Elementary Properties

… ►Let ${\varphi }_s(x)$, $s=0,1,2,\mathrm{\dots }$, be a sequence of functions defined in $𝐗$ such that for each $s$ ►

2.1.19 $${\varphi }_{s+1}(x)=o\left({\varphi }_s(x)\right),$$ $x\to c$ in $𝐗$,

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Symbols:

$o\left(x\right)$: order less than, $𝐗$: point set, $c$: limit point and ${\varphi }_s(x)$: sequence of functions

Permalink:

http://dlmf.nist.gov/2.1.E19

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Annotations for §2.1(v), §2.1 and Ch.2

… ►Then $\{{\varphi }_s(x)\}$ is an asymptotic sequence or scale. Suppose also that $f(x)$ and $f_s(x)$ satisfy ►

2.1.20 $$f(x)=\sum _{s=0}^{n-1}{f}_s(x)+O\left({\varphi }_n(x)\right),$$ $x\to c$ in $𝐗$,

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Symbols:

$O\left(x\right)$: order not exceeding, $𝐗$: point set, $c$: limit point, ${\varphi }_s(x)$: sequence of functions, $f(x)$: function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/2.1.E20

Encodings:

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See also:

Annotations for §2.1(v), §2.1 and Ch.2

…

8: 1.9 Calculus of a Complex Variable

… ►

§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

… ► …

9: 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. …

10: Guide to Searching the DLMF

… ►

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. …