asymptotic behavior for large variable

SearchAdvancedHelp

(0.002 seconds)

1—10 of 14 matching pages

1: 28.31 Equations of Whittaker–Hill and Ince

… ►

Asymptotic Behavior

…

2: 30.11 Radial Spheroidal Wave Functions

… ►

§30.11(iii) Asymptotic Behavior

…

3: 15.12 Asymptotic Approximations

… ►

§15.12(i) Large Variable

►For the asymptotic behavior of $𝐅(a,b;c;z)$ as $z\to \mathrm{\infty }$ with $a$, $b$, $c$ fixed, combine (15.2.2) with (15.8.2) or (15.8.8). ►

§15.12(ii) Large $c$

… ►As $\lambda \to \mathrm{\infty }$, … ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).

4: 30.9 Asymptotic Approximations and Expansions

§30.9 Asymptotic Approximations and Expansions

►

§30.9(i) Prolate Spheroidal Wave Functions

… ►The asymptotic behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ and $a_{n,k}^m({\gamma }^2)$ as $n\to \mathrm{\infty }$ in descending powers of $2n+1$ is derived in Meixner (1944). …The behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ for complex ${\gamma }^2$ and large $|{\lambda }_n^m\left({\gamma }^2\right)|$ is investigated in Hunter and Guerrieri (1982).

5: 25.11 Hurwitz Zeta Function

… ►

§25.11(xii) $a$-Asymptotic Behavior

… ►

25.11.41 $$\zeta (s,a+1)=\zeta \left(s\right)-s\zeta \left(s+1\right)a+O\left(a^2\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $a$: real or complex parameter and $s$: complex variable

Keywords:

asymptotic approximation

Source:

Apostol (1952, (15), p. 6)

Permalink:

http://dlmf.nist.gov/25.11.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xii), §25.11 and Ch.25

… ►

25.11.42 $$\zeta (s,\alpha +\mathrm{i}\beta )\to 0,$$

ⓘ

Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\mathrm{i}$: imaginary unit and $s$: complex variable

Keywords:

asymptotic approximation

Source:

Apostol (1952, (16), p. 6)

Permalink:

http://dlmf.nist.gov/25.11.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xii), §25.11 and Ch.25

… ►As $a\to \mathrm{\infty }$ in the sector , with $s(\ne 1)$ and $\delta$ fixed, we have the asymptotic expansion … ► …

6: 18.15 Asymptotic Approximations

§18.15 Asymptotic Approximations

►

§18.15(i) Jacobi

… ►For large $\beta$, fixed $\alpha$, and $0\le n/\beta \le c$, Dunster (1999) gives asymptotic expansions of $P_n^{(\alpha ,\beta )}\left(z\right)$ that are uniform in unbounded complex $z$-domains containing $z=\pm 1$. …This reference also supplies asymptotic expansions of $P_n^{(\alpha ,\beta )}\left(z\right)$ for large $n$, fixed $\alpha$, and $0\le \beta /n\le c$. … ►The asymptotic behavior of the classical OP’s as $x\to \pm \mathrm{\infty }$ with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. …

7: 19.12 Asymptotic Approximations

§19.12 Asymptotic Approximations

►With $\psi \left(x\right)$ denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of $K\left(k\right)$ and $E\left(k\right)$ near the singularity at $k=1$ is given by the following convergent series: … ►For the asymptotic behavior of $F(\varphi ,k)$ and $E(\varphi ,k)$ as $\varphi \to \frac{1}{2}\pi -$ and $k\to 1-$ see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007). … ►Asymptotic approximations for $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$, with different variables, are given in Karp et al. (2007). They are useful primarily when $(1-k)/(1-\sin \varphi )$ is either small or large compared with 1. …

8: 2.4 Contour Integrals

… ►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 …with known asymptotic behavior as $t\to +\mathrm{\infty }$. …If this integral converges uniformly at each limit for all sufficiently large $t$, then by the Riemann–Lebesgue lemma (§1.8(i)) … ►For integral representations of the $b_{2s}$ and their asymptotic behavior as $s\to \mathrm{\infty }$ see Boyd (1995). …

9: 8.13 Zeros

… ►When $-5\le a\le 4$ the behavior of the $x$-zeros as functions of $a$ can be seen by taking the slice ${\gamma }^{\ast }(a,x)=0$ of the surface depicted in Figure 8.3.6. … ►For asymptotic approximations for $x_{+}(a)$ and $x_{-}(a)$ as $a\to -\mathrm{\infty }$ see Tricomi (1950b), with corrections by Kölbig (1972b). For more accurate asymptotic approximations see Thompson (2012). … ►For information on the distribution and computation of zeros of $\gamma (a,\lambda a)$ and $\mathrm{\Gamma }(a,\lambda a)$ in the complex $\lambda$-plane for large values of the positive real parameter $a$ see Temme (1995a). … ►Approximations to $a_n^{\ast }$, $x_n^{\ast }$ for large $n$ can be found in Kölbig (1970). …

10: Bibliography B

… ►

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

… ►

Coulomb functions for large charges and small velocities. Phys. Rev. (2) 97 (2), pp. 542–554.

ⓘ

External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§33.5(iv).

Encodings:

BibTeX

… ►

Asymptotic behavior of the Pollaczek polynomials and their zeros. Stud. Appl. Math. 96, pp. 307–338.

ⓘ

External Links:

ISSN 0022-2526, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§18.35(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

… ►

The behavior at unit argument of the hypergeometric function ${}_3{}^{}F_2^{}$ . SIAM J. Math. Anal. 18 (5), pp. 1227–1234.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (K. M. Saksena), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

…