error term

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1: 8.27 Approximations

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Luke (1975, §4.3) gives Padé approximation methods, combined with a detailed analysis of the error terms, valid for real and complex variables except on the negative real $z$-axis. See also Temme (1994b, §3).

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2: 3.5 Quadrature

… ►In particular, when $k=\mathrm{\infty }$ the error term is an exponentially-small function of $1/h$, and in these circumstances the composite trapezoidal rule is exceptionally efficient. … ►Simpson’s rule can be regarded as a combination of two trapezoidal rules, one with step size $h$ and one with step size $h/2$ to refine the error term. … ►As in Simpson’s rule, by combining the rule for $h$ with that for $h/2$, the first error term $c_1{h}^2$ in (3.5.9) can be eliminated. … ►An interpolatory quadrature rule … ►Equation (3.5.36), without the error term, becomes …

3: 5.19 Mathematical Applications

… ►By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of $f(z)$ for large $|z|$, or small $|z|$, can be obtained complete with an integral representation of the error term. …

4: 9.7 Asymptotic Expansions

… ►In (9.7.5) and (9.7.6) the $n$th error term, that is, the error on truncating the expansion at $n$ terms, is bounded in magnitude by the first neglected term and has the same sign, provided that the following term is of opposite sign, that is, if $n\ge 0$ for (9.7.5) and $n\ge 1$ for (9.7.6). ►In (9.7.7) and (9.7.8) the $n$th error term is bounded in magnitude by the first neglected term multiplied by $\chi (n+\sigma )+1$ where $\sigma =\frac{1}{6}$ for (9.7.7) and $\sigma =0$ for (9.7.8), provided that $n\ge 0$ in the first case and $n\ge 1$ in the second case. ►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. … ►The $n$th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by …

5: Bibliography W

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Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials. Anal. Appl. (Singap.) 1 (2), pp. 213–241.

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External Links:

ISSN 0219-5305, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.15(i), §18.15(i).

Encodings:

BibTeX

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Explicit error terms for asymptotic expansions of Mellin convolutions. J. Math. Anal. Appl. 72 (2), pp. 740–756.

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External Links:

ISSN 0022-247X, Document, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.6(ii), §2.6(iv).

Encodings:

BibTeX

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6: 12.9 Asymptotic Expansions for Large Variable

… ►Bounds and re-expansions for the error term in (12.9.1) can be obtained by use of (12.7.14) and §§13.7(ii), 13.7(iii). …

7: 25.10 Zeros

… ►The error term $R(t)$ can be expressed as an asymptotic series that begins …

8: 27.11 Asymptotic Formulas: Partial Sums

… ►Dirichlet’s divisor problem (unsolved as of 2022) is to determine the least number ${\theta }_0$ such that the error term in (27.11.2) is $O\left(x^{\theta }\right)$ for all $\theta >{\theta }_0$. … ►The error terms given here are not necessarily the best known. …

9: 2.3 Integrals of a Real Variable

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2.3.3 $${\sigma }_n=\underset{(0,\mathrm{\infty })}{sup}(t^{-1}\ln |q^{(n)}(t)/q^{(n)}(0)|)$$

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Defines:

${\sigma }_n$ (locally)

Symbols:

$\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $sup$: least upper bound (supremum), $q(t)$: infinitely differentiable function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/2.3.E3

Encodings:

pMML, png, TeX

See also:

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

►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 … ►For other estimates of the error term see Lyness (1971). …

10: 18.15 Asymptotic Approximations

… ►When $\alpha ,\beta \in (-\frac{1}{2},\frac{1}{2})$, the error term in (18.15.1) is less than twice the first neglected term in absolute value, in which one has to take $\cos {\theta }_{n,m,\mathrm{\ell }}=1$. … ►For higher coefficients see Baratella and Gatteschi (1988), and for another estimate of the error term in a related expansion see Wong and Zhao (2003). … ►For a bound on the error term in (18.15.10) see Szegő (1975, Theorem 8.21.11). … ►For an error bound for the first term in the Airy-function expansions see Olver (1997b, p. 403). …