asymptotic expansions for large order

SearchAdvancedHelp

(0.009 seconds)

31—40 of 64 matching pages

31: 8.18 Asymptotic Expansions of $I_x(a,b)$

… ►

§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

… ►

Symmetric Case

… ►

General Case

… ►

Inverse Function

►For asymptotic expansions for large values of $a$ and/or $b$ of the $x$-solution of the equation …

32: Bibliography F

… ►

Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.

ⓘ

External Links:

ISSN 0219-5305, Document, Link, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii), §15.12(iii).

Encodings:

BibTeX

… ►

Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II. J. Math. Anal. Appl. 7 (3), pp. 440–451.

ⓘ

External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

►

Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. J. Math. Anal. Appl. 6 (3), pp. 394–403.

ⓘ

External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions of certain classes of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 4 (3), pp. 482–507.

ⓘ

External Links:

Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§16.22.

Encodings:

BibTeX

►

Uniform asymptotic expansions of a class of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (Roop N. Kesarwani), ZentralBlatt

Referenced by:

§16.22.

Encodings:

BibTeX

…

33: 2.8 Differential Equations with a Parameter

… ►in which $u$ is a real or complex parameter, and asymptotic solutions are needed for large $|u|$ that are uniform with respect to $z$ in a point set $𝐃$ in $ℝ$ or $ℂ$. … ►

§2.8(iv) Case III: Simple Pole

… ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). … ►In both cases uniform asymptotic approximations are obtained in terms of Bessel functions of order $1/(\lambda +2)$. … ►For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24. …

34: 28.4 Fourier Series

… ►

§28.4(vi) Behavior for Small $q$

… ►

28.4.21 $$A_{2s}^0(q)=\left(\frac{{(-1)}^{s}2}{{(s!)}^2}{\left(\frac{q}{4}\right)}^s+O\left(q^{s+2}\right)\right)A_0^0(q),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), $q=h^2$: parameter and $A_m(q)$: Fourier coefficient

A&S Ref:

20.2.29 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.4.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(vi), §28.4 and Ch.28

… ►

28.4.23 $$\begin{array}{c}{A}_{m-2s}^m(q)\\ B_{m-2s}^m(q)\end{array}\}=\left(\frac{(m-s-1)!}{s!(m-1)!}{\left(\frac{q}{4}\right)}^s+O\left(q^{s+1}\right)\right)\{\begin{array}{c}{A}_m^m(q),\\ B_m^m(q).\end{array}$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $A_m(q)$: Fourier coefficient and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(vi), §28.4 and Ch.28

►For further terms and expansions see Meixner and Schäfke (1954, p. 122) and McLachlan (1947, §3.33). ►

§28.4(vii) Asymptotic Forms for Large $m$

…

35: 2.1 Definitions and Elementary Properties

… ►

§2.1(i) Asymptotic and Order Symbols

… ►

§2.1(iii) Asymptotic Expansions

… ►Symbolically, … ►

§2.1(iv) Uniform Asymptotic Expansions

… ►

§2.1(v) Generalized Asymptotic Expansions

…

36: 13.8 Asymptotic Approximations for Large Parameters

§13.8 Asymptotic Approximations for Large Parameters

… ►For other asymptotic expansions for large $b$ and $z$ see López and Pagola (2010). … ►

§13.8(iii) Large $a$

… ► … ►

§13.8(iv) Large $a$ and $b$

…

37: 2.6 Distributional Methods

… ►

§2.6(ii) Stieltjes Transform

… ►To derive an asymptotic expansion of $𝒮f\left(z\right)$ for large values of $|z|$, with , we assume that $f(t)$ possesses an asymptotic expansion of the form … ►An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). ►

§2.6(iii) Fractional Integrals

… ►We now derive an asymptotic expansion of ${𝐼}^{\mu }f(x)$ for large positive values of $x$. …

38: 13.20 Uniform Asymptotic Approximations for Large $\mu$

§13.20 Uniform Asymptotic Approximations for Large $\mu$

►

§13.20(i) Large $\mu$, Fixed $\kappa$

… ► … ►

§13.20(v) Large $\mu$, Other Expansions

… ►

39: Bibliography B

… ►

Asymptotic expansions of the modified Bessel function of the third kind of imaginary order. SIAM J. Appl. Math. 15, pp. 1315–1323.

ⓘ

External Links:

ISSN 0036-1399, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.45.

Encodings:

BibTeX

… ►

Anharmonic oscillator. II. A study of perturbation theory in large order. Phys. Rev. D 7, pp. 1620–1636.

ⓘ

External Links:

Document

Referenced by:

§22.19(ii).

Encodings:

BibTeX

… ►

Unfolding the high orders of asymptotic expansions with coalescing saddles: Singularity theory, crossover and duality. Proc. Roy. Soc. London Ser. A 443, pp. 107–126.

ⓘ

External Links:

ISSN 0962-8444, FullText, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§36.12(i), §36.12(ii), §36.12(iii).

Encodings:

BibTeX

… ►

Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§31.13.

Encodings:

BibTeX

… ►

Asymptotic Expansions for the Coefficient Functions Associated with Linear Second-order Differential Equations: The Simple Pole Case. In Asymptotic and Computational Analysis (Winnipeg, MB, 1989), R. Wong (Ed.), Lecture Notes in Pure and Applied Mathematics, Vol. 124, pp. 53–73.

ⓘ

External Links:

MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.8(iv).

Encodings:

BibTeX

…

40: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

… ►where $\delta$ denotes an arbitrary small positive constant. … ►For an exponentially-improved asymptotic expansion (§2.11(iii)) see Olver (1991a). … ►With $x=1$, an asymptotic expansion of $e_n(nx)/{\mathrm{e}}^{nx}$ follows from (8.11.14) and (8.11.16). … ►