asymptotic behavior

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21: Bibliography B

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Asymptotic behavior of the Pollaczek polynomials and their zeros. Stud. Appl. Math. 96, pp. 307–338.

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External Links:

ISSN 0022-2526, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§18.35(iii).

Encodings:

BibTeX

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22: 2.7 Differential Equations

… ►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: …

23: 18.15 Asymptotic Approximations

… ►The asymptotic behavior of the classical OP’s as $x\to \pm \mathrm{\infty }$ with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. …

24: 3.5 Quadrature

… ►In practical applications the weight function $w(x)$ is chosen to simulate the asymptotic behavior of the integrand as the endpoints are approached. …

25: Frank W. J. Olver

… ►He is particularly known for his extensive work in the study of the asymptotic solution of differential equations, i. …, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e. … ►His well-known book, Asymptotics and Special Functions, was reprinted in the AKP Classics Series by AK Peters, Wellesley, Massachusetts, in 1997. … ►In 1989 the conference “Asymptotic and Computational Analysis” was held in Winnipeg, Canada, in honor of Olver’s 65th birthday, with Proceedings published by Marcel Dekker in 1990. … ►

2 Asymptotic Approximations

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26: 28.4 Fourier Series

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§28.4(vi) Behavior for Small $q$

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27: 14.8 Behavior at Singularities

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14.8.16 $${𝑸}_{-n-(1/2)}^{\mu }\left(x\right)\sim \frac{{\pi }^{1/2}\mathrm{\Gamma }\left(\mu +n+\frac{1}{2}\right)}{n!\mathrm{\Gamma }\left(\mu -n+\frac{1}{2}\right){(2x)}^{n+(1/2)}},$$ $n=1,2,3,\mathrm{\dots }$, $\mu -n+\frac{1}{2}\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $x$: real variable, $n$: nonnegative integer and $\mu$: general order

Referenced by:

§14.21(iii), §14.8(iii)

Permalink:

http://dlmf.nist.gov/14.8.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(iii), §14.8 and Ch.14

28: 2.3 Integrals of a Real Variable

… ►Then … ►For the Fourier integral … ►Other types of singular behavior in the integrand can be treated in an analogous manner. … ►Then … ►

§2.3(vi) Asymptotics of Mellin Transforms

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29: 27.11 Asymptotic Formulas: Partial Sums

§27.11 Asymptotic Formulas: Partial Sums

►The behavior of a number-theoretic function $f(n)$ for large $n$ is often difficult to determine because the function values can fluctuate considerably as $n$ increases. It is more fruitful to study partial sums and seek asymptotic formulas of the form … ►Equations (27.11.3)–(27.11.11) list further asymptotic formulas related to some of the functions listed in §27.2. … ►The prime number theorem for arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if $\left(h,k\right)=1$, then the number of primes $p\le x$ with $p\equiv h(modk)$ is asymptotic to $x/(\varphi \left(k\right)\ln x)$ as $x\to \mathrm{\infty }$.

30: 19.20 Special Cases

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19.20.19 $$R_D(x,y,z)\sim 3x^{-1/2}{y}^{-1/2}{z}^{-1/2},$$ $z/\sqrt{xy}\to 0$.

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables and $\sim$: asymptotic equality

Referenced by:

§19.20(iv), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.20.E19

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

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