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31: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$, or $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$, to determine the scattering $S$-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). ►For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function $W_{-\eta ,\mathrm{\ell }+\frac{1}{2}}\left(2\rho \right)$. The functions ${\varphi }_{n,\mathrm{\ell }}(r)$ defined by (33.14.14) are the hydrogenic bound states in attractive Coulomb potentials; their polynomial components are often called associated Laguerre functions; see Christy and Duck (1961) and Bethe and Salpeter (1977). …

32: Bibliography F

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A uniform asymptotic expansion of the Jacobi polynomials with error bounds. Canad. J. Math. 37 (5), pp. 979–1007.

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External Links:

ISSN 0008-414X, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.15(i), §18.16(ii).

Encodings:

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Error bounds for asymptotic expansions of the ratio of two gamma functions. SIAM J. Math. Anal. 18 (3), pp. 890–896.

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External Links:

ISSN 0036-1410, Document, FullText, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§5.11(iii), §5.11(iii).

Encodings:

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Error bounds for a uniform asymptotic expansion of the Legendre function $Q_n^{-m}(\cosh z)$ . SIAM J. Math. Anal. 21 (2), pp. 523–535.

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External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.26.

Encodings:

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Error bounds for the asymptotic expansion of the ratio of two gamma functions with complex argument. SIAM J. Math. Anal. 23 (2), pp. 505–511.

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External Links:

ISSN 0036-1410, Document, FullText, MathReview (A. Reinfelds), ZentralBlatt

Referenced by:

§5.11(iii), §5.11(iii).

Encodings:

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33: 3.3 Interpolation

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3.3.12 $$c_n=\frac{1}{(n+1)!}\max \prod _{k=n_0}^{n_1}\left|t-k\right|,$$

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Defines:

$c_n$: bound coefficient (locally)

Symbols:

$!$: factorial (as in $n!$)

Referenced by:

§3.4(i)

Permalink:

http://dlmf.nist.gov/3.3.E12

Encodings:

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See also:

Annotations for §3.3(ii), §3.3 and Ch.3

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3.3.13 $$\left|R_{n,t}\right|\le c_n{h}^{n+1}\left|f^{(n+1)}(\xi )\right|.$$

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Symbols:

$f(z)$: function, $R_{n,t}(x)$: remainder and $c_n$: bound coefficient

Permalink:

http://dlmf.nist.gov/3.3.E13

Encodings:

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Annotations for §3.3(ii), §3.3 and Ch.3

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3.3.15 $$c_1=\frac{1}{8},$$ .

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Symbols:

$c_n$: bound coefficient

A&S Ref:

25.2.10 (modification of)

Permalink:

http://dlmf.nist.gov/3.3.E15

Encodings:

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Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3

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3.3.18 $$c_2=1/(9\sqrt{3})=0.0641\mathrm{\dots },$$ .

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Symbols:

$c_n$: bound coefficient

A&S Ref:

25.2.12 (modification of)

Permalink:

http://dlmf.nist.gov/3.3.E18

Encodings:

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See also:

Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3

… ►For interpolation of a bounded function $f$ on $ℝ$ the cardinal function of $f$ is defined by …

34: 30.15 Signal Analysis

… ►The maximum (or least upper bound) $\mathrm{B}$ of all numbers … ►

30.15.11 $$\arccos \sqrt{\mathrm{B}}+\arccos \sqrt{\alpha }=\arccos \sqrt{{\mathrm{\Lambda }}_0},$$

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Symbols:

$\arccos z$: arccosine function, $\mathrm{\Lambda }$ and $\mathrm{B}$: maximum bound

Permalink:

http://dlmf.nist.gov/30.15.E11

Encodings:

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See also:

Annotations for §30.15(v), §30.15 and Ch.30

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30.15.12 $$\mathrm{B}={\left(\sqrt{{\mathrm{\Lambda }}_0\alpha }+\sqrt{1-{\mathrm{\Lambda }}_0}\sqrt{1-\alpha }\right)}^2.$$

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Symbols:

$\mathrm{\Lambda }$ and $\mathrm{B}$: maximum bound

Permalink:

http://dlmf.nist.gov/30.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §30.15(v), §30.15 and Ch.30

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35: Bibliography S

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On Lindelöf’s error bound for Stirling’s series. J. Reine Angew. Math. 404, pp. 135–139.

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External Links:

ISSN 0075-4102, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§5.11(ii).

Encodings:

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Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros. Math. Comp. 70 (235), pp. 1205–1220.

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External Links:

ISSN 0025-5718, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.74(vi).

Encodings:

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Bounds for ratios of modified Bessel functions and associated Turán-type inequalities. J. Math. Anal. Appl. 374 (2), pp. 516–528.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Beyza Caliskan Aslan), ZentralBlatt

Referenced by:

§10.37, §10.37.

Encodings:

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The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions. Annals of Physics 97 (2), pp. 279–288.

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External Links:

Document, MathReview Entry

Referenced by:

§1.18(viii).

Encodings:

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Error bounds for asymptotic solutions of differential equations. I. The distinct eigenvalue case. J. Res. Nat. Bur. Standards Sect. B 70B, pp. 167–186.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.7(ii).

Encodings:

BibTeX

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36: 11.11 Asymptotic Expansions of Anger–Weber Functions

… ►For sharp error bounds and exponentially-improved extensions, see Nemes (2018). … ►If $z$ is fixed, and $\nu \to \mathrm{\infty }$ in $|\mathrm{ph}\nu |\le \pi$ in such a way that $\nu$ is bounded away from the set of all integers, then … ►uniformly for bounded complex values of $a$. … ►Error bounds for (11.11.8) and (11.11.10) are given in Meijer (1932) and Nemes (2014b, c). …

37: Bibliography L

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Bounds for modified Bessel functions. J. Comput. Appl. Math. 34 (3), pp. 263–267.

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External Links:

ISSN 0377-0427, Document, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.37.

Encodings:

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Bessel functions: Monotonicity and bounds. J. London Math. Soc. (2) 61 (1), pp. 197–215.

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External Links:

ISSN 0024-6107, Document, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§10.14, §10.15.

Encodings:

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Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0010-4655, Document, Code, ZentralBlatt

Referenced by:

4th item.

Encodings:

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Error bounds for asymptotic expansions of Laplace convolutions. SIAM J. Math. Anal. 25 (6), pp. 1537–1553.

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External Links:

ISSN 0036-1410, Document, MathReview (Walter Schempp), ZentralBlatt

Referenced by:

§2.6(iii).

Encodings:

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Comparison of a pair of upper bounds for a ratio of gamma functions. Math. Balkanica (N.S.) 16 (1-4), pp. 195–202.

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External Links:

ISSN 0205-3217, MathReview, ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

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38: 3.2 Linear Algebra

… ►Then we have the a posteriori error bound …

39: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Bounded and Unbounded Linear Operators

… ►A linear operator $T$ on $V$ is bounded with norm $\Vert T\Vert$ if … ►If $T$ is a bounded linear operator on $V$ then its adjoint is the bounded linear operator $T^{\ast }$ such that, for $v,w\in V$, … ►If $T$ is a bounded operator then its spectrum is a closed bounded subset of $ℂ$. If $T$ is self-adjoint (bounded or unbounded) then $\sigma (T)$ is a closed subset of $ℝ$ and the residual spectrum is empty. …

40: 2.11 Remainder Terms; Stokes Phenomenon

… ►When a rigorous bound or reliable estimate for the remainder term is unavailable, it is unsafe to judge the accuracy of an asymptotic expansion merely from the numerical rate of decrease of the terms at the point of truncation. …First, it is impossible to bound the tail by majorizing its terms. … ►However, regardless whether we can bound the remainder, the accuracy achievable by direct numerical summation of a divergent asymptotic series is always limited. … ►where $z=\rho {\mathrm{e}}^{\mathrm{i}\theta }$, and $|\alpha |$ is bounded as $n\to \mathrm{\infty }$. … ►For error bounds see Dunster (1996c). …