asymptotic expansions for small parameters

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21—30 of 41 matching pages

21: 33.12 Asymptotic Expansions for Large $\eta$

§33.12 Asymptotic Expansions for Large $\eta$

… ►For asymptotic expansions of

F_{\ell}\left(\eta,\rho\right)

and

G_{\ell}\left(\eta,\rho\right)

when

\eta\to\pm\infty

see Temme (2015, Chapter 31). ►

§33.12(ii) Uniform Expansions

… ►The first set is in terms of Airy functions and the expansions are uniform for fixed

\ell

and

\delta\leq z<\infty

, where

\delta

is an arbitrary small positive constant. … ►

22: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$

§8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$

… ►

§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

… ►

Symmetric Case

… ►

General Case

… ►

Inverse Function

…

23: 13.19 Asymptotic Expansions for Large Argument

§13.19 Asymptotic Expansions for Large Argument

… ►Again,

\delta

denotes an arbitrary small positive constant. … ►Error bounds and exponentially-improved expansions are derivable by combining §§13.7(ii) and 13.7(iii) with (13.14.2) and (13.14.3). … ►For an asymptotic expansion of

W_{\kappa,\mu}\left(z\right)

z\to\infty

that is valid in the sector

|\operatorname{ph}z|\leq\pi-\delta

and where the real parameters

\kappa

\mu

are subject to the growth conditions

\kappa=o\left(z\right)

\mu=o\left(\sqrt{z}\right)

, see Wong (1973a).

24: 2.4 Contour Integrals

… ►

§2.4(i) Watson’s Lemma

… ►Then … ►For examples see Olver (1997b, pp. 315–320). ►

§2.4(iii) Laplace’s Method

… ►In consequence, the asymptotic expansion obtained from (2.4.14) is no longer null. …

25: 30.16 Methods of Computation

… ►For small

|\gamma^{2}|

we can use the power-series expansion (30.3.8). …If

|\gamma^{2}|

is large we can use the asymptotic expansions in §30.9. … ►If

|\gamma^{2}|

is large, then we can use the asymptotic expansions referred to in §30.9 to approximate

\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)

. … ►A fourth method, based on the expansion (30.8.1), is as follows. … ►

30.16.8

a_{n,k}^{m}(\gamma^{2})=\lim_{d\to\infty}e_{k+p,d},

ⓘ

Symbols:: $m$ : nonnegative integer, $n\geq m$ : integer degree, $d$ : dimension, $\gamma^{2}$ : real parameter and $a^{m}_{n,k}(\gamma^{2})$ : coefficients
Permalink:: http://dlmf.nist.gov/30.16.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §30.16(ii), §30.16 and Ch.30

…

26: 28.34 Methods of Computation

… ►

(a)

Summation of the power series in §§28.6(i) and 28.15(i) when $\left|q\right|$ is small.

►

(b)

Use of asymptotic expansions and approximations for large $q$ (§§28.8(i), 28.16). See also Zhang and Jin (1996, pp. 482–485).

… ►

(a)

Summation of the power series in §§28.6(ii) and 28.15(ii) when $\left|q\right|$ is small.

►

(b)

Use of asymptotic expansions and approximations for large $q$ (§§28.8(ii)–28.8(iv)).

… ►

(c)

Use of asymptotic expansions for large $z$ or large $q$ . See §§28.25 and 28.26.

27: 16.11 Asymptotic Expansions

§16.11 Asymptotic Expansions

►

§16.11(i) Formal Series

… ►

§16.11(ii) Expansions for Large Variable

… ►

§16.11(iii) Expansions for Large Parameters

… ►Asymptotic expansions for the polynomials

{{}_{p+2}F_{q}}\left(-r,r+a_{0},\mathbf{a};\mathbf{b};z\right)

r\to\infty

through integer values are given in Fields and Luke (1963b, a) and Fields (1965).

28: 16.5 Integral Representations and Integrals

… ►

16.5.1

\left({\textstyle\ifrac{\prod\limits_{k=1}^{p}\Gamma\left(a_{k}\right)}{\prod% \limits_{k=1}^{q}\Gamma\left(b_{k}\right)}}\right){{}_{p}F_{q}}\left({a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\frac{1}{2\pi\mathrm{i}}\int_{L}% \left({\textstyle\ifrac{\prod\limits_{k=1}^{p}\Gamma\left(a_{k}+s\right)}{% \prod\limits_{k=1}^{q}\Gamma\left(b_{k}+s\right)}}\right)\Gamma\left(-s\right)% (-z)^{s}\,\mathrm{d}s,

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §13.6(vi), §16.11(i), §16.5, §16.5
Permalink:: http://dlmf.nist.gov/16.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

… ►In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as

z\to 0

in the sector

|\operatorname{ph}\left(-z\right)|\leq(p+1-q-\delta)\pi/2

, where

\delta

is an arbitrary small positive constant. … ►

16.5.2

{{}_{p+1}F_{q+1}}\left({a_{0},\dots,a_{p}\atop b_{0},\dots,b_{q}};z\right)=% \frac{\Gamma\left(b_{0}\right)}{\Gamma\left(a_{0}\right)\Gamma\left(b_{0}-a_{0% }\right)}\int_{0}^{1}t^{a_{0}-1}(1-t)^{b_{0}-a_{0}-1}{{}_{p}F_{q}}\left({a_{1}% ,\dots,a_{p}\atop b_{1},\dots,b_{q}};zt\right)\,\mathrm{d}t,

\Re b_{0}>\Re a_{0}>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Keywords:: Mellin transform
Referenced by:: §16.5, §16.5
Permalink:: http://dlmf.nist.gov/16.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

►

16.5.3

{{}_{p+1}F_{q}}\left({a_{0},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\frac% {1}{\Gamma\left(a_{0}\right)}\int_{0}^{\infty}{\mathrm{e}}^{-t}t^{a_{0}-1}{{}_% {p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\cdots,b_{q}};zt\right)\,\mathrm{% d}t,

\Re z<1

\Re a_{0}>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Keywords:: Laplace transform, Mellin transform
Referenced by:: §16.5
Permalink:: http://dlmf.nist.gov/16.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

… ►Lastly, the restrictions on the parameters can be eased by replacing the integration paths with loop contours; see Luke (1969a, §3.6). …

29: 28.25 Asymptotic Expansions for Large $\Re z$

§28.25 Asymptotic Expansions for Large $\Re z$

… ►

28.25.1

{\operatorname{M}^{(3,4)}_{\nu}}\left(z,h\right)\sim\frac{e^{\pm\mathrm{i}% \left(2h\cosh z-\left(\frac{1}{2}\nu+\frac{1}{4}\right)\pi\right)}}{\left(\pi h% (\cosh z+1)\right)^{\frac{1}{2}}}\*\sum_{m=0}^{\infty}\dfrac{D^{\pm}_{m}}{% \left(\mp 4\mathrm{i}h(\cosh z+1)\right)^{m}},

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable, $\nu$ : complex parameter and $D_{m}^{\pm}$ : coefficients
A&S Ref:: 20.9.1 (for 3 and for integer $\nu$ ) 20.9.3 (for 4 and for integer $\nu$ )
Referenced by:: §28.25
Permalink:: http://dlmf.nist.gov/28.25.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.25 and Ch.28

… ►The expansion (28.25.1) is valid for

{\operatorname{M}^{(3)}_{\nu}}\left(z,h\right)

when ►

28.25.4

\Re z\to+\infty,

-\pi+\delta\leq\operatorname{ph}h+\Im z\leq 2\pi-\delta

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\Im$ : imaginary part, $\operatorname{ph}$ : phase, $\Re$ : real part, $h$ : parameter, $z$ : complex variable and $\delta$ : arbitrary small positive constant
Permalink:: http://dlmf.nist.gov/28.25.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §28.25 and Ch.28

… ►where

\delta

again denotes an arbitrary small positive constant. …

30: 18.24 Hahn Class: Asymptotic Approximations

… ►For two asymptotic expansions of

M_{n}\left(nx;\beta,c\right)

n\to\infty

, with

\beta

and

c

fixed, see Jin and Wong (1998) and Wang and Wong (2011). The first expansion holds uniformly for

\delta\leq x\leq 1+\delta

, and the second for

1-\delta\leq x\leq 1+\delta^{-1}

\delta

being an arbitrary small positive constant. … ►Dunster (2001b) provides various asymptotic expansions for

C_{n}\left(x;a\right)

n\to\infty

, in terms of elementary functions or in terms of Bessel functions. Taken together, these expansions are uniformly valid for

-\infty<x<\infty

and for

a

in unbounded intervals—each of which contains

[0,(1-\delta)n]

, where

\delta

again denotes an arbitrary small positive constant. … ►For an asymptotic expansion of

P^{(\lambda)}_{n}\left(nx;\phi\right)

n\to\infty

, with

\phi

fixed, see Li and Wong (2001). …

asymptotic expansions for small parameters

21—30 of 41 matching pages

21: 33.12 Asymptotic Expansions for Large η

§33.12 Asymptotic Expansions for Large η

§33.12(ii) Uniform Expansions

22: 8.18 Asymptotic Expansions of I x ⁡ ( a , b )

§8.18 Asymptotic Expansions of I x ⁡ ( a , b )

§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

Symmetric Case

General Case

Inverse Function

23: 13.19 Asymptotic Expansions for Large Argument

§13.19 Asymptotic Expansions for Large Argument

24: 2.4 Contour Integrals

§2.4(i) Watson’s Lemma

§2.4(iii) Laplace’s Method

25: 30.16 Methods of Computation

26: 28.34 Methods of Computation

27: 16.11 Asymptotic Expansions

§16.11 Asymptotic Expansions

§16.11(i) Formal Series

§16.11(ii) Expansions for Large Variable

§16.11(iii) Expansions for Large Parameters

28: 16.5 Integral Representations and Integrals

29: 28.25 Asymptotic Expansions for Large ℜ ⁡ z

§28.25 Asymptotic Expansions for Large ℜ ⁡ z

30: 18.24 Hahn Class: Asymptotic Approximations

21: 33.12 Asymptotic Expansions for Large $\eta$

§33.12 Asymptotic Expansions for Large $\eta$

22: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$

§8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$

29: 28.25 Asymptotic Expansions for Large $\Re z$

§28.25 Asymptotic Expansions for Large $\Re z$