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1: DLMF Project News

error generating summary

2: Foreword

… ►However, we have also seen the birth of a new age of computing technology, which has not only changed how we utilize special functions, but also how we communicate technical information. …

3: Frank W. J. Olver

… ►Having witnessed the birth of the computer age firsthand (as a colleague of Alan Turing at NPL, for example), Olver is also well known for his contributions to the development and analysis of numerical methods for computing special functions. … ►He continued his editing work until the time of his death on April 22, 2013 at age 88.

4: Publications

… ►

A Special Functions Handbook for the Digital Age, Notices of the American Mathematical Society 58, 7 (2011), pp. 905–911.

…

5: 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. … ►

6: 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:

pMML, pMML, png, png, TeX, TeX

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).

7: 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, …

8: 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). …

9: 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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

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

…

10: 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 $u\in 𝐔$. …