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21: 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. …

22: 29.17 Other Solutions

… ►Lamé–Wangerin functions are solutions of (29.2.1) with the property that ${(\mathrm{sn}(z,k))}^{1/2}w(z)$ is bounded on the line segment from $\mathrm{i}{K}^{\prime }$ to $2K+\mathrm{i}{K}^{\prime }$. …

23: 34.8 Approximations for Large Parameters

… ►For approximations for the $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.

24: Bibliography I

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Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

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External Links:

ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(v).

Encodings:

BibTeX

… ►

Bound on the extreme zeros of orthogonal polynomials. Proc. Amer. Math. Soc. 115 (1), pp. 131–140.

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External Links:

ISSN 0002-9939, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(18.16.12), (18.16.13).

Encodings:

BibTeX

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25: 7.8 Inequalities

…

26: Mathematical Introduction

… ► ►►►►

$ℂ$	complex plane (excluding infinity).
…
$inf$	greatest lower bound (infimum).
$sup$	least upper bound (supremum).
…

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27: 11.2 Definitions

… ►(11.2.11) applies when $x$ is bounded, and (11.2.12) applies when $x$ is bounded away from the origin. … ►(11.2.13) applies when $0\le \mathrm{ph}z\le \pi$ and $|z|$ is bounded. (11.2.14) applies when $-\pi \le \mathrm{ph}z\le 0$ and $|z|$ is bounded. (11.2.15) applies when $\left|\mathrm{ph}z\right|\le \pi$ and $z$ is bounded away from the origin. … ►(11.2.16) applies when $|\mathrm{ph}z|\le \frac{1}{2}\pi$ with $|z|$ bounded. …

28: 1.4 Calculus of One Variable

… ►A generalization of the Riemann integral is the Stieltjes integral ${\int }_a^{b}f(x)d\alpha (x)$, where $\alpha (x)$ is a nondecreasing function on the closure of $(a,b)$, which may be bounded, or unbounded, and $d\alpha (x)$ is the Stieltjes measure. … ►A more general concept of integrability of a function on a bounded or unbounded interval is Lebesgue integrability, which allows discussion of functions which may not be well defined everywhere (especially on sets of measure zero) for $x\in ℝ$. … ►

Functions of Bounded Variation

… ►If , then $f(x)$ is of bounded variation on $(a,b)$. … …

29: 18.16 Zeros

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Other Bounds

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18.16.12 $$(n+2)x_{n,1}\ge {\left(n-1-\sqrt{n^2+(n+2)(\alpha +1)}\right)}^2-1,$$

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Symbols:

$n$: nonnegative integer and $x$: real variable

Referenced by:

§18.16(iv), Erratum (V1.0.5) for Equations (18.16.12), (18.16.13), Erratum (V1.2.0) for Equations (18.16.12), (18.16.13)

Permalink:

http://dlmf.nist.gov/18.16.E12

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This bound, taken from Dimitrov and Nikolov (2010), replaces the weaker bound $x_{n,1}>2n+\alpha -2-{(1+4(n-1)(n+\alpha -1))}^{\frac{1}{2}}$, which had been taken from Ismail and Li (1992).

See also:

Annotations for §18.16(iv), §18.16(iv), §18.16 and Ch.18

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18.16.13 $$(n+2)x_{n,n}\le {\left(n-1+\sqrt{n^2+(n+2)(\alpha +1)}\right)}^2-1.$$

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Symbols:

$n$: nonnegative integer and $x$: real variable

Referenced by:

§18.16(iv), Erratum (V1.0.5) for Equations (18.16.12), (18.16.13), Erratum (V1.2.0) for Equations (18.16.12), (18.16.13)

Permalink:

http://dlmf.nist.gov/18.16.E13

Encodings:

pMML, png, TeX

Modification (effective with 1.0.5):

This bound, taken from Dimitrov and Nikolov (2010), replaces the weaker bound , which had been taken from Ismail and Li (1992).

See also:

Annotations for §18.16(iv), §18.16(iv), §18.16 and Ch.18

… ►For an error bound for the first approximation yielded by this expansion see Olver (1997b, p. 408). …

30: 22.19 Physical Applications

… ►The bounded $(-\pi \le \theta \le \pi )$ oscillatory solution of (22.19.1) is traditionally written … ►This formulation gives the bounded and unbounded solutions from the same formula (22.19.3), for $k\ge 1$ and $k\le 1$, respectively. … ►For $\beta$ real and positive, three of the four possible combinations of signs give rise to bounded oscillatory motions. … ►There is bounded oscillatory motion near $x=0$, with period $4K\left(k\right)/\sqrt{1-\eta }$, and modulus $k=1/\sqrt{{\eta }^{-1}-1}$, for initial displacements with $|a|\le \sqrt{1/\beta }$. … ►For an initial displacement with , bounded oscillations take place near one of the two points of stable equilibrium $x=\pm \sqrt{1/\beta }$. …