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21: 16.13 Appell Functions

… ►For large parameter asymptotics see López et al. (2013a, b), and Ferreira et al. (2013a, b).

22: Bibliography T

… ►

On the computation of the incomplete gamma functions for large values of the parameters. In Algorithms for approximation (Shrivenham, 1985), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 10, pp. 479–489.

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External Links:

FullText, MathReview (G. Meinardus), ZentralBlatt

Referenced by:

§8.25(iii).

Encodings:

BibTeX

… ►

Computational aspects of incomplete gamma functions with large complex parameters. In Approximation and Computation. A Festschrift in Honor of Walter Gautschi, R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 551–562.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.11(iii), §8.20(ii), §8.25(i), §8.25(iii), 2nd item, 3rd item.

Encodings:

BibTeX

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Large parameter cases of the Gauss hypergeometric function. J. Comput. Appl. Math. 153 (1-2), pp. 441–462.

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External Links:

ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

… ►

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

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External Links:

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

Referenced by:

§13.8(iv).

Encodings:

BibTeX

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23: 2.3 Integrals of a Real Variable

… ►Then … ►Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … ►When $p(t)$ is real and $x$ is a large positive parameter, the main contribution to the integral … ►When the parameter $x$ is large the contributions from the real and imaginary parts of the integrand in … ► $k$ ($\le \mathrm{\infty }$) and $\lambda$ are positive constants, $\alpha$ is a variable parameter in an interval ${\alpha }_1\le \alpha \le {\alpha }_2$ with ${\alpha }_1\le 0$ and , and $x$ is a large positive parameter. …

24: 15.12 Asymptotic Approximations

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§15.12(iii) Other Large Parameters

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25: 2.4 Contour Integrals

… ►Then … ►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 $\alpha$. …

26: 25.11 Hurwitz Zeta Function

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§25.11(xii) $a$-Asymptotic Behavior

… ►Similarly, as $a\to \mathrm{\infty }$ in the sector , …

27: 14.32 Methods of Computation

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Application of the uniform asymptotic expansions for large values of the parameters given in §§14.15 and 14.20(vii)–14.20(ix).

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28: 2.8 Differential Equations with a Parameter

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2.8.1 $$d^{2}w/{dz}^2=\left(u^{2}f(z)+g(z)\right)w,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $f(x)$: function, $g(x)$: function, $w$: solution and $u$: large real or complex parameter

Referenced by:

§2.8(i), §2.8(v)

Permalink:

http://dlmf.nist.gov/2.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(i), §2.8 and Ch.2

►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 $ℂ$. … ►

2.8.3 $$\frac{d^{2}W}{{d\xi }^2}=\left(u^2{\dot{z}}^{2}f(z)+\psi (\xi )\right)W,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $f(x)$: function, $u$: large real or complex parameter, $W$: change of variable, $\dot{z}$: derivative of $z$ with respect to $\xi$ and $\psi (\xi )$: function

Permalink:

http://dlmf.nist.gov/2.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(i), §2.8 and Ch.2

… ►

2.8.8 $$d^{2}W/{d\xi }^2=\left(u^2{\xi }^m+\psi (\xi )\right)W,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $u$: large real or complex parameter, $W$: change of variable and $\psi (\xi )$: function

Referenced by:

§2.8(i), §2.8(iv)

Permalink:

http://dlmf.nist.gov/2.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(i), §2.8 and Ch.2

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2.8.9 $$\frac{d^{2}W}{{d\xi }^2}=\left(\frac{u^2}{\xi }+\frac{\rho }{{\xi }^2}\right)W,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $u$: large real or complex parameter, $W$: change of variable and $\rho$: limit

Permalink:

http://dlmf.nist.gov/2.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(i), §2.8 and Ch.2

…

29: 36.12 Uniform Approximation of Integrals

… ►where $k$ is a large real parameter and $𝐲=\{y_1,y_2,\mathrm{\dots }\}$ is a set of additional (nonasymptotic) parameters. …

30: 10.72 Mathematical Applications

… ►where $z$ is a real or complex variable and $u$ is a large real or complex parameter. …