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1: Software Index

… ►

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

…

2: 22.5 Special Values

… ►

§22.5(ii) Limiting Values of $k$

►If $k\to 0+$, then $K\to \pi /2$ and $K^{\prime }\to \mathrm{\infty }$; if $k\to 1-$, then $K\to \mathrm{\infty }$ and $K^{\prime }\to \pi /2$. … ►

Table 22.5.3: Limiting forms of Jacobian elliptic functions as $k\to 0$.

► ►►

$\mathrm{sn}(z,k)$ $\to$	$\sin z$	$\mathrm{cd}(z,k)$ $\to$	$\cos z$	$\mathrm{dc}(z,k)$ $\to$	$\sec z$	$\mathrm{ns}(z,k)$ $\to$	$\csc z$
…

►

… ►Expansions for $K,K^{\prime }$ as $k\to 0$ or $1$ are given in §§19.5, 19.12. … ►

3: 33.21 Asymptotic Approximations for Large $|r|$

… ►We indicate here how to obtain the limiting forms of $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, and $c(ϵ,\mathrm{\ell };r)$ as $r\to \pm \mathrm{\infty }$, with $ϵ$ and $\mathrm{\ell }$ fixed, in the following cases: ►

When $r\to \pm \mathrm{\infty }$ with $ϵ>0$, Equations (33.16.4)–(33.16.7) are combined with (33.10.1).

►

When $r\to \pm \mathrm{\infty }$ with , Equations (33.16.10)–(33.16.13) are combined with

33.21.1

${\zeta }_{\mathrm{\ell }}(\nu ,r)\sim {\mathrm{e}}^{-r/\nu }{(2r/\nu )}^{\nu },$

${\xi }_{\mathrm{\ell }}(\nu ,r)\sim {\mathrm{e}}^{r/\nu }{(2r/\nu )}^{-\nu }$ , $r\to \mathrm{\infty }$,

ⓘ

Symbols:

$\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function

Permalink:

http://dlmf.nist.gov/33.21.E1

Encodings:

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See also:

Annotations for §33.21(i), §33.21 and Ch.33

33.21.2

${\zeta }_{\mathrm{\ell }}(-\nu ,r)\sim {\mathrm{e}}^{r/\nu }{(-2r/\nu )}^{-\nu },$

${\xi }_{\mathrm{\ell }}(-\nu ,r)\sim {\mathrm{e}}^{-r/\nu }{(-2r/\nu )}^{\nu },$ $r\to -\mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function

Permalink:

http://dlmf.nist.gov/33.21.E2

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §33.21(i), §33.21 and Ch.33

Corresponding approximations for $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$ as $r\to \mathrm{\infty }$ can be obtained via (33.16.17), and as $r\to -\mathrm{\infty }$ via (33.16.18).

►

When $r\to \pm \mathrm{\infty }$ with $ϵ=0$, combine (33.20.1), (33.20.2) with §§10.7(ii), 10.30(ii).

… ►For asymptotic expansions of $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$ as $r\to \pm \mathrm{\infty }$ with $ϵ$ and $\mathrm{\ell }$ fixed, see Curtis (1964a, §6).

4: 24.12 Zeros

… ►and as $n\to \mathrm{\infty }$ with $m(\ge 1)$ fixed, … ►When $n$ is odd $x_1^{(n)}=\frac{1}{2}$, $x_2^{(n)}=1$ $(n\ge 3)$, and as $n\to \mathrm{\infty }$ with $m(\ge 1)$ fixed, … ►

$x_{2m}^{(n)}\to m.$

… ►When $n(\ge 2)$ is even $y_1^{(n)}=1$, and as $n\to \mathrm{\infty }$ with $m(\ge 1)$ fixed, … ►and as $n\to \mathrm{\infty }$ with $m(\ge 1)$ fixed, …

5: 28.17 Stability as $x\to \pm \mathrm{\infty }$

§28.17 Stability as $x\to \pm \mathrm{\infty }$

►If all solutions of (28.2.1) are bounded when $x\to \pm \mathrm{\infty }$ along the real axis, then the corresponding pair of parameters $(a,q)$ is called stable. … ►For example, as $x\to +\mathrm{\infty }$ one of the solutions ${\mathrm{me}}_{\nu }(x,q)$ and ${\mathrm{me}}_{\nu }(-x,q)$ tends to $0$ and the other is unbounded (compare Figure 28.13.5). …

6: 2.2 Transcendental Equations

… ►

2.2.1 $$f(x)\sim x,$$ $x\to \mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: asymptotic equality and $f(x)$: function

Referenced by:

§2.2

Permalink:

http://dlmf.nist.gov/2.2.E1

Encodings:

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See also:

Annotations for §2.2 and Ch.2

… ►

2.2.2 $$x(y)\sim y,$$ $y\to \mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: asymptotic equality and $y$: root

Referenced by:

§2.2

Permalink:

http://dlmf.nist.gov/2.2.E2

Encodings:

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See also:

Annotations for §2.2 and Ch.2

… ►

2.2.4 $$t=y^{\frac{1}{2}}\left(1+o\left(1\right)\right),$$ $y\to \mathrm{\infty }$.

ⓘ

Symbols:

$o\left(x\right)$: order less than and $y$: root

Permalink:

http://dlmf.nist.gov/2.2.E4

Encodings:

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See also:

Annotations for §2.2, §2.2 and Ch.2

… ►

2.2.6 $$t=y^{\frac{1}{2}}\left(1+\frac{1}{4}{y}^{-1}\ln y+o\left(y^{-1}\right)\right),$$ $y\to \mathrm{\infty }$.

ⓘ

Symbols:

$o\left(x\right)$: order less than, $\ln z$: principal branch of logarithm function and $y$: root

Permalink:

http://dlmf.nist.gov/2.2.E6

Encodings:

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See also:

Annotations for §2.2, §2.2 and Ch.2

… ►

2.2.7 $$f(x)\sim x+f_0+f_1{x}^{-1}+f_2{x}^{-2}+\mathrm{\cdots },$$ $x\to \mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $f(x)$: function and $f_s$: coefficients

Permalink:

http://dlmf.nist.gov/2.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.2, §2.2 and Ch.2

…

7: 2.1 Definitions and Elementary Properties

… ►As $x\to c$ in $𝐗$ …The symbol $O$ can also apply to the whole set $𝐗$, and not just as $x\to c$. … ►as $x\to \mathrm{\infty }$ in an unbounded set $𝐗$ in $ℝ$ or $ℂ$. … ►is bounded as $x\to \mathrm{\infty }$ in $𝐗$, uniformly for $u\in 𝐔$. …as $x\to \mathrm{\infty }$ in $𝐗$, uniformly with respect to $u\in 𝐔$. …

8: 10.30 Limiting Forms

… ►

§10.30(i) $z\to 0$

►When $\nu$ is fixed and $z\to 0$, …For $K_{\nu }\left(x\right)$, when $\nu$ is purely imaginary and $x\to 0+$, see (10.45.2) and (10.45.7). ►

§10.30(ii) $z\to \mathrm{\infty }$

►When $\nu$ is fixed and $z\to \mathrm{\infty }$, …

9: 18.24 Hahn Class: Asymptotic Approximations

… ►When the parameters $\alpha$ and $\beta$ are fixed and the ratio $n/N=c$ is a constant in the interval (0,1), uniform asymptotic formulas (as $n\to \mathrm{\infty }$ ) of the Hahn polynomials $Q_n(z;\alpha ,\beta ,N)$ can be found in Lin and Wong (2013) for $z$ in three overlapping regions, which together cover the entire complex plane. … ►With $x=\lambda N$ and $\nu =n/N$, Li and Wong (2000) gives an asymptotic expansion for $K_n(x;p,N)$ as $n\to \mathrm{\infty }$, that holds uniformly for $\lambda$ and $\nu$ in compact subintervals of $(0,1)$. … ►For two asymptotic expansions of $M_n(nx;\beta ,c)$ as $n\to \mathrm{\infty }$, with $\beta$ and $c$ fixed, see Jin and Wong (1998) and Wang and Wong (2011). … ►Dunster (2001b) provides various asymptotic expansions for $C_n(x;a)$ as $n\to \mathrm{\infty }$, in terms of elementary functions or in terms of Bessel functions. … ►For an asymptotic expansion of $P_n^{(\lambda )}(nx;\varphi )$ as $n\to \mathrm{\infty }$, with $\varphi$ fixed, see Li and Wong (2001). …

10: 24.11 Asymptotic Approximations

… ►As $n\to \mathrm{\infty }$ ►

24.11.1 $${(-1)}^{n+1}{B}_{2n}\sim \frac{2(2n)!}{{(2\pi )}^{2n}},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

►

24.11.2 $${(-1)}^{n+1}{B}_{2n}\sim 4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Permalink:

http://dlmf.nist.gov/24.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

►

24.11.3 $${(-1)}^n{E}_{2n}\sim \frac{2^{2n+2}(2n)!}{{\pi }^{2n+1}},$$

ⓘ

Symbols:

$E_n$: Euler numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

Permalink:

http://dlmf.nist.gov/24.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

►

24.11.4 $${(-1)}^n{E}_{2n}\sim 8\sqrt{\frac{n}{\pi }}{\left(\frac{4n}{\pi \mathrm{e}}\right)}^{2n}.$$

ⓘ

Symbols:

$E_n$: Euler numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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