asymptotic estimate

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11: 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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12: 8.18 Asymptotic Expansions of $I_x(a,b)$

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8.18.3 $$I_x(a,b)=\frac{\mathrm{\Gamma }\left(a+b\right)}{\mathrm{\Gamma }\left(a\right)}\left(\sum _{k=0}^{n-1}{d}_k{F}_k+O\left(a^{-n}\right)F_0\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $n$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable, $F_k$: functions and $d_k$: coefficients

Referenced by:

Erratum (V1.0.14) for Equation (8.18.3)

Permalink:

http://dlmf.nist.gov/8.18.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Previously this equation appeared without the order estimate as $I_x(a,b)\sim \frac{\mathrm{\Gamma }\left(a+b\right)}{\mathrm{\Gamma }\left(a\right)}{\sum }_{k=0}^{\mathrm{\infty }}{d}_k{F}_k$. The range of $x$ was extended to include $1$.

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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13: 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. … ►shows that this direct estimate is correct to almost 3D. …

14: 2.5 Mellin Transform Methods

§2.5 Mellin Transform Methods

… ►This is allowable in view of the asymptotic formula … ►

§2.5(ii) Extensions

… ►(The last order estimate follows from the Riemann–Lebesgue lemma, §1.8(i).) The asymptotic expansion of $I(x)$ is then obtained from (2.5.29). …

15: Bibliography Q

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Sharp estimates for complete elliptic integrals. SIAM J. Math. Anal. 27 (3), pp. 823–834.

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External Links:

ISSN 0036-1410, Document, MathReview (G. D. Anderson), ZentralBlatt

Referenced by:

§19.9(i), §19.9(i).

Encodings:

BibTeX

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Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.

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External Links:

ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§2.4(vi).

Encodings:

BibTeX

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Asymptotic expansion of the Krawtchouk polynomials and their zeros. Comput. Methods Funct. Theory 4 (1), pp. 189–226.

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External Links:

ISSN 1617-9447, FullText, MathReview (C. M. Joshi), ZentralBlatt (N. M. Temme)

Referenced by:

§18.24.

Encodings:

BibTeX

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16: 2.3 Integrals of a Real Variable

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

§2.3(vi) Asymptotics of Mellin Transforms

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17: Errata

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8.18.3 $$I_x(a,b)=\frac{\mathrm{\Gamma }\left(a+b\right)}{\mathrm{\Gamma }\left(a\right)}\left(\sum _{k=0}^{n-1}{d}_k{F}_k+O\left(a^{-n}\right)F_0\right)$$

The range of $x$ was extended to include $1$. Previously this equation appeared without the order estimate as $I_x(a,b)\sim \frac{\mathrm{\Gamma }\left(a+b\right)}{\mathrm{\Gamma }\left(a\right)}{\sum }_{k=0}^{\mathrm{\infty }}{d}_k{F}_k$.

Reported 2016-08-30 by Xinrong Ma.

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18: 16.5 Integral Representations and Integrals

… ►In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as $z\to 0$ in the sector $|\mathrm{ph}\left(-z\right)|\le (p+1-q-\delta )\pi /2$, where $\delta$ is an arbitrary small positive constant. …

19: 2.6 Distributional Methods

… ►For rigorous derivations of these results and also order estimates for ${\delta }_n(x)$, see Wong (1979) and Wong (1989, Chapter 6).

20: 11.6 Asymptotic Expansions

§11.6 Asymptotic Expansions

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§11.6(i) Large $|z|$, Fixed $\nu$

… ►More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). … ►and for an estimate of the relative error in this approximation see Watson (1944, p. 336).