asymptotic approximations and expansions for large %7Cr%7C

SearchAdvancedHelp

(0.003 seconds)

1—10 of 232 matching pages

1: 14.26 Uniform Asymptotic Expansions

§14.26 Uniform Asymptotic Expansions

…

2: 29.16 Asymptotic Expansions

§29.16 Asymptotic Expansions

… ►The approximations for Lamé polynomials hold uniformly on the rectangle $0\le \mathrm{\Re }z\le K$, $0\le \mathrm{\Im }z\le K^{\prime }$, when $nk$ and $n{k}^{\prime }$ assume large real values. …

3: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

… ►

§28.8(ii) Sips’ Expansions

… ►

Barrett’s Expansions

… ►

Dunster’s Approximations

… ►

4: 28.16 Asymptotic Expansions for Large $q$

§28.16 Asymptotic Expansions for Large $q$

… ►

28.16.1 $${\lambda }_{\nu }\left(h^2\right)\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

ⓘ

Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.16 and Ch.28

…

5: 10.41 Asymptotic Expansions for Large Order

§10.41 Asymptotic Expansions for Large Order

… ► … ►

§10.41(iv) Double Asymptotic Properties

… ►

§10.41(v) Double Asymptotic Properties (Continued)

… ►

6: 16.22 Asymptotic Expansions

§16.22 Asymptotic Expansions

►Asymptotic expansions of $G_{p,q}^{m,n}(z;𝐚;𝐛)$ for large $z$ are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). For asymptotic expansions of Meijer $G$-functions with large parameters see Fields (1973, 1983).

7: 12.16 Mathematical Applications

… ► …

8: 32.12 Asymptotic Approximations for Complex Variables

§32.12 Asymptotic Approximations for Complex Variables

… ►See Boutroux (1913), Novokshënov (1990), Kapaev (1991), Joshi and Kruskal (1992), Kitaev (1994), Its and Kapaev (2003), and Fokas et al. (2006, Chapter 7). … ►

9: Bibliography O

… ►

On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

ⓘ

External Links:

ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

►

Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.

ⓘ

External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

►

Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.

ⓘ

External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

ⓘ

External Links:

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

Referenced by:

§15.12(iii).

Encodings:

BibTeX

… ►

Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.

ⓘ

External Links:

Document, MathReview (F. Oberhettinger), ZentralBlatt

Referenced by:

§10.19(iii), §10.21(vii).

Encodings:

BibTeX

…

10: 2.11 Remainder Terms; Stokes Phenomenon

… ►

§2.11(i) Numerical Use of Asymptotic Expansions

… ► … ►

§2.11(iii) Exponentially-Improved Expansions

… ►For another approach see Paris (2001a, b). ►

§2.11(vi) Direct Numerical Transformations

…