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1: 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 …Then $\sum f_s(x)$ is a generalized asymptotic expansion of $f(x)$ with respect to the scale $\{{\varphi }_s(x)\}$. …

2: Bibliography G

… ►

Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2162), pp. 20130529, 16.

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External Links:

ISSN 1364-5021, Document, Link, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§10.22(v), §10.22(v).

Encodings:

BibTeX

…

3: 10.41 Asymptotic Expansions for Large Order

… ►This is because $A_k(\zeta )$ and ${\zeta }^{-\frac{1}{2}}{B}_k(\zeta ),k=0,1,\mathrm{\dots }$, do not form an asymptotic scale (§2.1(v)) as $\zeta \to +\mathrm{\infty }$; see Olver (1997b, pp. 422–425). …

4: 5.11 Asymptotic Expansions

… ►

5.11.3 $$\mathrm{\Gamma }\left(z\right)={\mathrm{e}}^{-z}{z}^z{\left(\frac{2\pi }{z}\right)}^{1/2}{\mathrm{\Gamma }}^{\ast }\left(z\right)\sim {\mathrm{e}}^{-z}{z}^z{\left(\frac{2\pi }{z}\right)}^{1/2}\sum _{k=0}^{\mathrm{\infty }}\frac{g_k}{z^k},$$

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Defines:

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

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $z$: complex variable and $g_k$: coefficients

A&S Ref:

6.1.37

Referenced by:

§10.19(i), §13.8(iv), §5.11(i), §5.11(i), §5.11(i), §5.11(ii), §5.11(ii), §5.21, §5.9(i), §8.11(ii), §8.12, (8.18.13), (8.18.13), §8.18(ii), §8.18(ii), Erratum (V1.1.7) for Additions, Erratum (V1.1.8) for Rearrangement

Permalink:

http://dlmf.nist.gov/5.11.E3

Encodings:

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Addition (effective with 1.1.7):

The definition of the scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ was inserted after the first equals sign.

Changes (effective with 1.0.11):

An unnecessary bracket around the sum was removed.

See also:

Annotations for §5.11(i), §5.11 and Ch.5

… ►The scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ is defined in (5.11.3) and its main property is ${\mathrm{\Gamma }}^{\ast }\left(z\right)\sim 1$ as $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$. …

5: 15.12 Asymptotic Approximations

… ►For the asymptotic behavior of $𝐅(a,b;c;z)$ as $z\to \mathrm{\infty }$ with $a$, $b$, $c$ fixed, combine (15.2.2) with (15.8.2) or (15.8.8). …

6: 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. …

7: 3.12 Mathematical Constants

… ►

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

Encodings:

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Annotations for §3.12 and Ch.3

… ►

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

… ►

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:

pMML, png, TeX

See also:

Annotations for §3.12 and Ch.3

…

8: 1.17 Integral and Series Representations of the Dirac Delta

… ►

§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 …

9: 26.22 Software

… ►

On-Line Encyclopedia of Integer Sequences (website).

…

10: 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

Encodings:

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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

Encodings:

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

… ►

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:

pMML, png, TeX

See also:

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

…