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

SearchAdvancedHelp

(0.005 seconds)

11—20 of 232 matching pages

11: 10.70 Zeros

§10.70 Zeros

►Asymptotic approximations for large zeros are as follows. … ►

$$\text{zeros of }{\mathrm{ber}}_{\nu }x\sim \sqrt{2}(t-f(t)),$$ $t=(m-\frac{1}{2}\nu -\frac{3}{8})\pi$,

►

$$\text{zeros of }{\mathrm{bei}}_{\nu }x\sim \sqrt{2}(t-f(t)),$$ $t=(m-\frac{1}{2}\nu +\frac{1}{8})\pi$,

►

$$\text{zeros of }{\ker }_{\nu }x\sim \sqrt{2}(t+f(-t)),$$ $t=(m-\frac{1}{2}\nu -\frac{5}{8})\pi$,

…

12: 30.9 Asymptotic Approximations and Expansions

§30.9 Asymptotic Approximations and Expansions

… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). … ►

§30.9(iii) Other Approximations and Expansions

… ►

13: Bibliography T

… ►

Uniform asymptotic expansions of confluent hypergeometric functions. J. Inst. Math. Appl. 22 (2), pp. 215–223.

ⓘ

External Links:

ISSN 0020-2932, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.8(ii), §13.8(ii).

Encodings:

BibTeX

… ►

The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

… ►

Laguerre polynomials: Asymptotics for large degree. Technical report Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.

ⓘ

External Links:

FullText

Referenced by:

§2.4(vi).

Encodings:

BibTeX

… ►

Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a, b and z. Indagationes Mathematicae.

ⓘ

External Links:

ISSN 0019-3577, Document, Link

Referenced by:

§13.8(iv).

Encodings:

BibTeX

… ►

Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

ⓘ

External Links:

ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.8(iv).

Encodings:

BibTeX

…

14: 35.10 Methods of Computation

… ►For large $\Vert 𝐓\Vert$ the asymptotic approximations referred to in §35.7(iv) are available. … ►Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …

15: Bibliography N

… ►

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

ⓘ

External Links:

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

Referenced by:

§10.43(v).

Encodings:

BibTeX

… ►

Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.

ⓘ

External Links:

ISSN 1452-8630, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§5.11(ii), §5.11(ii).

Encodings:

BibTeX

… ►

Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions. Acta Appl. Math. 150, pp. 141–177.

ⓘ

External Links:

ISSN 0167-8019, Document, MathReview Entry, ZentralBlatt

Referenced by:

(9.7.16), (9.7.17), §9.7(iii), §9.7(iv).

Encodings:

BibTeX

►

Error bounds for the large-argument asymptotic expansions of the Lommel and allied functions. Stud. Appl. Math. 140 (4), pp. 508–541.

ⓘ

External Links:

ISSN 0022-2526, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§11.11(i), §11.11(i).

Encodings:

BibTeX

… ►

Mathematical Approximation of Special Functions. Nova Science Publishers Inc., Commack, NY.

ⓘ

Notes:

Ten papers on Chebyshev expansions

External Links:

ISBN 1-56072-052-2, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.76(ii), §10.76(ii), §10.76(ii), §10.76(iii), 2nd item.

Encodings:

BibTeX

…

16: 19.38 Approximations

§19.38 Approximations

… ►Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. Cody (1965b) gives Chebyshev-series expansions (§3.11(ii)) with maximum precision 25D. ►Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). …The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for $\varphi$ near $\pi /2$ with the improvements made in the 1970 reference. …

17: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

… ►Ismail (1986) gives asymptotic expansions as $n\to \mathrm{\infty }$, with $x$ and other parameters fixed, for continuous $q$-ultraspherical, big and little $q$-Jacobi, and Askey–Wilson polynomials. These asymptotic expansions are in fact convergent expansions. … ►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). …

18: 2.1 Definitions and Elementary Properties

… ►means that for each $n$, the difference between $f(x)$ and the $n$th partial sum on the right-hand side is $O\left({(x-c)}^n\right)$ as $x\to c$ in $𝐗$. … ►Some asymptotic approximations are expressed in terms of two or more Poincaré asymptotic expansions. …For an example see (2.8.15). … ►

§2.1(iv) Uniform Asymptotic Expansions

… ►

§2.1(v) Generalized Asymptotic Expansions

…

19: 10.69 Uniform Asymptotic Expansions for Large Order

§10.69 Uniform Asymptotic Expansions for Large Order

… ►All fractional powers take their principal values. ►All four expansions also enjoy the same kind of double asymptotic property described in §10.41(iv). … ►

20: 18.24 Hahn Class: Asymptotic Approximations

… ►With $x=\lambda N$ and $\nu =n/N$, Li and Wong (2000) gives an asymptotic expansion for $K_n(x;p,N)$ as $n\to \mathrm{\infty }$, that holds uniformly for $\lambda$ and $\nu$ in compact subintervals of $(0,1)$. … ►With $\mu =N/n$ and $x$ fixed, Qiu and Wong (2004) gives an asymptotic expansion for $K_n(x;p,N)$ as $n\to \mathrm{\infty }$, that holds uniformly for $\mu \in [1,\mathrm{\infty })$. … ►For two asymptotic expansions of $M_n(nx;\beta ,c)$ as $n\to \mathrm{\infty }$, with $\beta$ and $c$ fixed, see Jin and Wong (1998) and Wang and Wong (2011). … ►Dunster (2001b) provides various asymptotic expansions for $C_n(x;a)$ as $n\to \mathrm{\infty }$, in terms of elementary functions or in terms of Bessel functions. … ►For an asymptotic expansion of $P_n^{(\lambda )}(nx;\varphi )$ as $n\to \mathrm{\infty }$, with $\varphi$ fixed, see Li and Wong (2001). …