large c

SearchAdvancedHelp

(0.003 seconds)

31—40 of 82 matching pages

31: 33.21 Asymptotic Approximations for Large $|r|$

§33.21 Asymptotic Approximations for Large $|r|$

►

§33.21(i) Limiting Forms

►We indicate here how to obtain the limiting forms of $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, and $c(ϵ,\mathrm{\ell };r)$ as $r\to \pm \mathrm{\infty }$, with $ϵ$ and $\mathrm{\ell }$ fixed, in the following cases: … ►

When $r\to \pm \mathrm{\infty }$ with $ϵ=0$, combine (33.20.1), (33.20.2) with §§10.7(ii), 10.30(ii).

►

§33.21(ii) Asymptotic Expansions

…

32: 2.1 Definitions and Elementary Properties

… ►Let $𝐗$ be a point set with a limit point $c$. … ►If $\sum a_s{x}^{-s}$ converges for all sufficiently large $|x|$, then it is automatically the asymptotic expansion of its sum as $x\to \mathrm{\infty }$ in $ℂ$. ►If $c$ is a finite limit point of $𝐗$, then … ►Similarly for finite limit point $c$ in place of $\mathrm{\infty }$. … ►where $c$ is a finite, or infinite, limit point of $𝐗$. …

33: 2.4 Contour Integrals

… ►Except that $\lambda$ is now permitted to be complex, with $\mathrm{\Re }\lambda >0$, we assume the same conditions on $q(t)$ and also that the Laplace transform in (2.3.8) converges for all sufficiently large values of $\mathrm{\Re }z$. Then … ►For large $t$, the asymptotic expansion of $q(t)$ may be obtained from (2.4.3) by Haar’s method. This depends on the availability of a comparison function $F(z)$ for $Q(z)$ that has an inverse transform …If this integral converges uniformly at each limit for all sufficiently large $t$, then by the Riemann–Lebesgue lemma (§1.8(i)) … ►in which $z$ is a large real or complex parameter, $p(\alpha ,t)$ and $q(\alpha ,t)$ are analytic functions of $t$ and continuous in $t$ and a second parameter $\alpha$. …

34: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10 Uniform Asymptotic Expansions for Large Parameter

… ►

§12.10(vi) Modifications of Expansions in Elementary Functions

… ► … ►

Modified Expansions

… ►

35: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

§13.21 Uniform Asymptotic Approximations for Large $\kappa$

►

§13.21(i) Large $\kappa$, Fixed $\mu$

… ►

§13.21(ii) Large $\kappa$, $0\le \mu \le (1-\delta )\kappa$

… ►

§13.21(iv) Large $\kappa$, Other Expansions

… ►

36: 3.6 Linear Difference Equations

… ►Stability can be restored, however, by backward recursion, provided that $c_n\ne 0$, $\forall n$: starting from $w_N$ and $w_{N+1}$, with $N$ large, equation (3.6.3) is applied to generate in succession $w_{N-1},w_{N-2},\mathrm{\dots },w_0$. …

37: 11.6 Asymptotic Expansions

… ►

§11.6(i) Large $|z|$, Fixed $\nu$

… ►

§11.6(ii) Large $|\nu |$, Fixed $z$

… ►

§11.6(iii) Large $|\nu |$, Fixed $z/\nu$

… ►Here …These and higher coefficients $c_k(\lambda )$ can be computed via the representations in Nemes (2015b). …

38: 1.17 Integral and Series Representations of the Dirac Delta

… ►More generally, assume $\varphi (x)$ is piecewise continuous (§1.4(ii)) when $x\in [-c,c]$ for any finite positive real value of $c$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$. …

39: Bibliography K

… ►

A proof of the $q$-Macdonald-Morris conjecture for $B{C}_n$ . Mem. Amer. Math. Soc. 108 (516), pp. vi+80.

ⓘ

External Links:

ISSN 0065-9266, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§17.14.

Encodings:

BibTeX

… ►

On the evaluation of the Gauss hypergeometric function. C. R. Acad. Bulgare Sci. 45 (6), pp. 35–36.

ⓘ

External Links:

ISSN 0861-1459, MathReview, ZentralBlatt

Referenced by:

§15.15, §15.19(i).

Encodings:

BibTeX

… ►

Quantum Groups. Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York.

ⓘ

External Links:

ISBN 0-387-94370-6, MathReview (Yu. N. Bespalov), ZentralBlatt

Referenced by:

§17.17.

Encodings:

BibTeX

… ►

Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters. J. B. Wolters, Groningen.

ⓘ

External Links:

MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

… ►

Askey-Wilson Polynomials for Root Systems of Type $BC$ . In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, pp. 189–204.

ⓘ

External Links:

FullText, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

…

40: 28.35 Tables

… ►

National Bureau of Standards (1967) includes the eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$ for $n=0(1)3$ with $q=0(.2)20(.5)37(1)100$, and $n=4(1)15$ with $q=0(2)100$; Fourier coefficients for ${\mathrm{ce}}_n(x,q)$ and ${\mathrm{se}}_n(x,q)$ for $n=0(1)15$, $n=1(1)15$, respectively, and various values of $q$ in the interval $[0,100]$; joining factors $g_{e,n}(\sqrt{q})$, $f_{e,n}(\sqrt{q})$ for $n=0(1)15$ with $q=0(.5\text{ to }10)100$ (but in a different notation). Also, eigenvalues for large values of $q$. Precision is generally 8D.

►

Stratton et al. (1941) includes $b_n$, $b_n^{\prime }$, and the corresponding Fourier coefficients for ${\mathrm{Se}}_n(c,x)$ and ${\mathrm{So}}_n(c,x)$ for $n=0$ or $1(1)4$, $c=0(.1\text{or}.2)4.5$. Precision is mostly 5S. Notation: $c=2\sqrt{q}$, $b_n=a_n+2q$, $b_n^{\prime }=b_n+2q$, and for ${\mathrm{Se}}_n(c,x)$, ${\mathrm{So}}_n(c,x)$ see §28.1.

…