asymptotic behavior of coefficients

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1: 30.8 Expansions in Series of Ferrers Functions

… ►

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}).

ⓘ

Symbols:: $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter and $a^{m}_{n,k}(\gamma^{2})$ : coefficients
Referenced by:: §30.11(i)
Permalink:: http://dlmf.nist.gov/30.8.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §30.8(i), §30.8 and Ch.30

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2: Bibliography W

… ►

G. Wolf (2008) 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

…

W. B. Jones and W. Van Assche (1998) 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

…

4: 2.10 Sums and Sequences

… ►What is the asymptotic behavior of

f_{n}

n\to\infty

n\to-\infty

? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion? … ►

(c)

The coefficients in the Laurent expansion

2.10.27

g(z)=\sum_{n=-\infty}^{\infty}g_{n}z^{n},

0<|z|<r

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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:: pMML, png, TeX
See also:: Annotations for §2.10(iv), §2.10 and Ch.2

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

…

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

…

6: 28.4 Fourier Series

… ►

§28.4(vi) Behavior for Small $q$

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7: 2.4 Contour Integrals

… ►

§2.4(i) Watson’s Lemma

… ►with known asymptotic behavior as

t\to+\infty

. …For examples see Olver (1997b, pp. 315–320). … ►For integral representations of the

b_{2s}

and their asymptotic behavior as

s\to\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,\dots

, are the coefficients that appear in the asymptotic expansion (5.11.3) of

\Gamma\left(z\right)

. … ►For the asymptotic behavior of

c_{k}(\eta)

k\to\infty

see Dunster et al. (1998) and Olde Daalhuis (1998c). … ►

Inverse Function

…

9: 30.9 Asymptotic Approximations and Expansions

§30.9 Asymptotic Approximations and Expansions

►

§30.9(i) Prolate Spheroidal Wave Functions

… ►The asymptotic behavior of

\lambda^{m}_{n}\left(\gamma^{2}\right)

and

a^{m}_{n,k}(\gamma^{2})

n\to\infty

in descending powers of

2n+1

is derived in Meixner (1944). …The asymptotic behavior of

\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)

and

\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)

x\to\pm 1

is given in Erdélyi et al. (1955, p. 151). …

10: 3.6 Linear Difference Equations

… ►

3.6.1

a_{n}w_{n+1}-b_{n}w_{n}+c_{n}w_{n-1}=d_{n},

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Defines:: $a_{n}$ : coefficient (locally), $b_{n}$ : coefficient (locally), $c_{n}$ : coefficient (locally) and $d_{n}$ : coefficient (locally)
Symbols:: $w_{n}$ : sequence
Referenced by:: §3.6(iv), §3.6(v)
Permalink:: http://dlmf.nist.gov/3.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §3.6(i), §3.6 and Ch.3

… ►The values of

w_{N}

and

w_{N+1}

needed to begin the backward recursion may be available, for example, from asymptotic expansions (§2.9). … ►A new problem arises, however, if, as

n\to\infty

, the asymptotic behavior of

w_{n}

is intermediate to those of two independent solutions

f_{n}

and

g_{n}

of the corresponding inhomogeneous equation (the complementary functions). … ►Thus the asymptotic behavior of the particular solution

\mathbf{E}_{n}\left(1\right)

is intermediate to those of the complementary functions

J_{n}\left(1\right)

and

Y_{n}\left(1\right)

; moreover, the conditions for Olver’s algorithm are satisfied. … ►Here

\ell\in[0,k]

, and its actual value depends on the asymptotic behavior of the wanted solution in relation to those of the other solutions. …