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11: 7.8 Inequalities

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7.8.5 $x\ge 0$.

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Symbols:

$𝖬\left(x\right)$: Mills’ ratio and $x$: real variable

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.8 and Ch.7

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12: 5.11 Asymptotic Expansions

… ►uniformly for bounded real values of $x$. ►

§5.11(ii) Error Bounds and Exponential Improvement

… ►For error bounds for (5.11.8) and an exponentially-improved extension, see Nemes (2013b). … ►For further information see Olver (1997b, pp. 293–295), and for other error bounds see Whittaker and Watson (1927, §12.33), Spira (1971), and Schäfke and Finsterer (1990). … ►For realistic error bounds in (5.11.14) see Frenzen (1987a, 1992). …

13: Bibliography I

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

Encodings:

BibTeX

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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

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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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14: 2.3 Integrals of a Real Variable

… ►is finite and bounded for $n=0,1,2,\mathrm{\dots }$, then the $n$th error term (that is, the difference between the integral and $n$th partial sum in (2.3.2)) is bounded in absolute value by $|q^{(n)}(0)/(x^n(x-{\sigma }_n))|$ when $x$ exceeds both $0$ and ${\sigma }_n$. … ►In both cases the $n$th error term is bounded in absolute value by $x^{-n}{𝒱}_{a,b}\left(q^{(n-1)}(t)\right)$, where the variational operator ${𝒱}_{a,b}$ is defined by ►

2.3.6 $${𝒱}_{a,b}\left(f(t)\right)={\int }_a^b\left|f^{\prime }(t)\right|dt;$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, ${𝒱}_{a,b}\left(f\right)$: total variation, $a$: left endpoint and $b$: right endpoint

Referenced by:

§10.17(iv), Erratum (V1.1.1) for Equation (2.3.6)

Permalink:

http://dlmf.nist.gov/2.3.E6

Encodings:

pMML, png, TeX

Correction (effective with 1.1.1):

The integrand was corrected so that the absolute value does not include the differential.

Suggested 2021-02-08 by Juan Luis Varona

See also:

Annotations for §2.3(i), §2.3 and Ch.2

… ►For error bounds for Watson’s lemma and Laplace’s method see Boyd (1993) and Olver (1997b, Chapter 3). … ►For proofs of the results of this subsection, error bounds, and an example, see Olver (1974). …

15: 9.7 Asymptotic Expansions

… ►

§9.7(iii) Error Bounds for Real Variables

… ►In (9.7.9)–(9.7.12) the $n$th error term in each infinite series is bounded in magnitude by the first neglected term and has the same sign, provided that the following term in the series is of opposite sign. … ►

§9.7(iv) Error Bounds for Complex Variables

►The $n$th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by … ►Corresponding bounds for the errors in (9.7.7) to (9.7.14) may be obtained by use of these results and those of §9.2(v) and their differentiated forms. …

16: Bibliography Q

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A new lower bound in the second Kershaw’s double inequality. J. Comput. Appl. Math. 214 (2), pp. 610–616.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

§5.6(i), §5.6(i).

Encodings:

BibTeX

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“Best possible” upper and lower bounds for the zeros of the Bessel function $J_{\nu }(x)$ . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.

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External Links:

ISSN 0002-9947, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

BibTeX

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17: Bibliography F

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

BibTeX

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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:

BibTeX

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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:

BibTeX

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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:

BibTeX

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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:

BibTeX

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18: 20 Theta Functions

Chapter 20 Theta Functions

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19: Bibliography J

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Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35 (2), pp. 475–501.

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External Links:

FullText, MathReview (I. Olkin), ZentralBlatt

Referenced by:

§35.11, §35.4(i), §35.4(ii), §35.8(i), §35.8(ii), §35.8(iv), §35.9, Ch.35.

Encodings:

BibTeX

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Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.

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Notes:

ISSN 1389-2355

External Links:

FullText

Referenced by:

§7.8, §7.8.

Encodings:

BibTeX

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20: 2.7 Differential Equations

… ►For error bounds for (2.7.14) see Olver (1997b, Chapter 7). … ►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: … ►provided that . …and $𝒱$ denotes the variational operator (§2.3(i)). … ►Assuming also , we have …