asymptotic estimate

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1: Bibliography T

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Asymptotic estimates for Laguerre polynomials. Z. Angew. Math. Phys. 41 (1), pp. 114–126.

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External Links:

ISSN 0044-2275, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§18.11(i), §18.16(vi).

Encodings:

BibTeX

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Asymptotic estimates of Stirling numbers. Stud. Appl. Math. 89 (3), pp. 233–243.

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External Links:

ISSN 0022-2526, FullText, MathReview (E. Rodney Canfield), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

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2: 27.2 Functions

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27.2.3 $$\pi \left(x\right)\sim \frac{x}{\ln x}.$$

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Symbols:

$\sim$: asymptotic equality, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

§25.16(i), §27.11

Permalink:

http://dlmf.nist.gov/27.2.E3

Encodings:

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Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

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3: Bibliography L

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An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

BibTeX

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4: 1.8 Fourier Series

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Asymptotic Estimates of Coefficients

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5: 26.8 Set Partitions: Stirling Numbers

… ►For asymptotic estimates for generalized Stirling numbers see Chelluri et al. (2000). …

6: Bibliography C

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

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7: 27.12 Asymptotic Formulas: Primes

… ►The best available asymptotic error estimate (2009) appears in Korobov (1958) and Vinogradov (1958): there exists a positive constant $d$ such that …

8: 5.4 Special Values and Extrema

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5.4.20 $$x_n=-n+\frac{1}{\pi }\arctan \left(\frac{\pi }{\ln n}\right)+O\left(\frac{1}{n{(\ln n)}^2}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $x$: real variable

A&S Ref:

6.3.20 (The error estimate has been improved.)

Referenced by:

§5.4(iii)

Permalink:

http://dlmf.nist.gov/5.4.E20

Encodings:

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See also:

Annotations for §5.4(iii), §5.4 and Ch.5

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9: 35.9 Applications

… ►For applications of the integral representation (35.5.3) see McFarland and Richards (2001, 2002) (statistical estimation of misclassification probabilities for discriminating between multivariate normal populations). The asymptotic approximations of §35.7(iv) are applied in numerous statistical contexts in Butler and Wood (2002). ►In chemistry, Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix argument, and develop asymptotic approximations for these density functions. …

10: 13.29 Methods of Computation

… ►For large $|z|$ the asymptotic expansions of §13.7 should be used instead. … ►In the sector the integration has to be towards the origin, with starting values computed from asymptotic expansions (§§13.7 and 13.19). … ►and estimate … ►and estimate ►

13.29.8 $$w(n)\sim \frac{\sqrt{\pi }{\mathrm{e}}^{\frac{1}{2}z}{z}^{\frac{1}{4}(4a-2b+1)}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a+1-b\right)}{n}^{\frac{1}{4}(4a-2b-3)}{\mathrm{e}}^{-2\sqrt{nz}},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.29.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

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