asymptotic behavior of coefficients

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1: 30.8 Expansions in Series of Ferrers Functions

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30.8.6 $$a_{n,k}^{-m}({\gamma }^2)=\frac{(n-m)!(n+m+2k)!}{(n+m)!(n-m+2k)!}{a}_{n,k}^m({\gamma }^2).$$

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Symbols:

$!$: factorial (as in $n!$), $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients

Referenced by:

§30.11(i)

Permalink:

http://dlmf.nist.gov/30.8.E6

Encodings:

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See also:

Annotations for §30.8(i), §30.8 and Ch.30

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2: Bibliography W

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On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.

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External Links:

ISSN 1044-677X, FullText

Referenced by:

§28.4(vii).

Encodings:

BibTeX

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3: Bibliography J

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Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

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External Links:

MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§5.10, §5.10.

Encodings:

BibTeX

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4: 2.10 Sums and Sequences

… ►What is the asymptotic behavior of $f_n$ as $n\to \mathrm{\infty }$ or $n\to -\mathrm{\infty }$? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion? … ►

The coefficients in the Laurent expansion

2.10.27 $$g(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{g}_n{z}^n,$$ ,

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Symbols:

$r$: radious of annulus, $g(z)$: comparison function and $g_n$: coefficients

Permalink:

http://dlmf.nist.gov/2.10.E27

Encodings:

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See also:

Annotations for §2.10(iv), §2.10 and Ch.2

have known asymptotic behavior as $n\to \pm \mathrm{\infty }$.

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5: 1.8 Fourier Series

… ►The series (1.8.1) is called the Fourier series of $f(x)$, and $a_n,b_n$ are the Fourier coefficients of $f(x)$. … ►

Asymptotic Estimates of Coefficients

… ►If $f(x)$ and $g(x)$ are continuous, have the same period and same Fourier coefficients, then $f(x)=g(x)$ for all $x$. ►

Lebesgue Constants

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6: 28.4 Fourier Series

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§28.4(vi) Behavior for Small $q$

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7: 2.4 Contour Integrals

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§2.4(i) Watson’s Lemma

… ►with known asymptotic behavior as $t\to +\mathrm{\infty }$. …For examples see Olver (1997b, pp. 315–320). … ►For integral representations of the $b_{2s}$ and their asymptotic behavior as $s\to \mathrm{\infty }$ see Boyd (1995). … ►For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). …

8: 8.12 Uniform Asymptotic Expansions for Large Parameter

§8.12 Uniform Asymptotic Expansions for Large Parameter

… ►With $\mu =\lambda -1$, the coefficients $c_k(\eta )$ are given by …where $g_k$, $k=0,1,2,\mathrm{\dots }$, are the coefficients that appear in the asymptotic expansion (5.11.3) of $\mathrm{\Gamma }\left(z\right)$. … ►For the asymptotic behavior of $c_k(\eta )$ as $k\to \mathrm{\infty }$ see Dunster et al. (1998) and Olde Daalhuis (1998c). … ►

Inverse Function

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9: Bibliography N

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Tables of Lagrangian Interpolation Coefficients. Columbia University Press, New York.

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Notes:

Technical Director: Arnold N. Lowan

External Links:

MathReview (W. Feller), ZentralBlatt

Referenced by:

§3.3(ii).

Encodings:

BibTeX

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Asymptotic behavior of the Lerch transcendent function. J. Approx. Theory 170, pp. 21–31.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Gabriel Alvarez)

Referenced by:

§25.14(ii).

Encodings:

BibTeX

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On an asymptotic expansion of the Kontorovich-Lebedev transform. Applicable Anal. 39 (4), pp. 249–263.

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External Links:

ISSN 0003-6811, Document, MathReview (Aloknath Chakrabarti), ZentralBlatt

Referenced by:

§10.43(v).

Encodings:

BibTeX

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On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

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External Links:

ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt

Referenced by:

§11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).

Encodings:

BibTeX

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The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

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Notes:

In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.

External Links:

ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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10: 30.9 Asymptotic Approximations and Expansions

§30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

… ►The asymptotic behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ and $a_{n,k}^m({\gamma }^2)$ as $n\to \mathrm{\infty }$ in descending powers of $2n+1$ is derived in Meixner (1944). …The asymptotic behavior of ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ and ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$ as $x\to \pm 1$ is given in Erdélyi et al. (1955, p. 151). …