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21: 33.21 Asymptotic Approximations for Large $|r|$

§33.21 Asymptotic Approximations for Large $|r|$

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§33.21(i) Limiting Forms

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§33.21(ii) Asymptotic Expansions

►For asymptotic expansions of $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$ as $r\to \pm \mathrm{\infty }$ with $ϵ$ and $\mathrm{\ell }$ fixed, see Curtis (1964a, §6).

22: Bibliography W

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An asymptotic expansion of $W_{k,m}(z)$ with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.

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External Links:

FullText, Document, MathReview (J. A. Donaldson), ZentralBlatt

Referenced by:

§13.19.

Encodings:

BibTeX

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On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

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External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

Encodings:

BibTeX

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Error bounds for asymptotic expansions of Hankel transforms. SIAM J. Math. Anal. 7 (6), pp. 799–808.

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External Links:

ISSN 1095-7154, Document, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§10.22(v).

Encodings:

BibTeX

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Asymptotic expansions of the Kontorovich-Lebedev transform. Appl. Anal. 12 (3), pp. 161–172.

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External Links:

ISSN 0003-6811, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§10.43(v).

Encodings:

BibTeX

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Applications of some recent results in asymptotic expansions. Congr. Numer. 37, pp. 145–182.

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External Links:

ISSN 0384-9864, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§19.27(vi).

Encodings:

BibTeX

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23: 28.26 Asymptotic Approximations for Large $q$

§28.26 Asymptotic Approximations for Large $q$

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28.26.4 $${\mathrm{Fc}}_m(z,h)\sim 1+\frac{s}{8h{\cosh }^{2}z}+\frac{1}{2^{11}{h}^2}\left(\frac{s^4+86s^2+105}{{\cosh }^{4}z}-\frac{s^4+22s^2+57}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(-\frac{s^5+14s^3+33s}{{\cosh }^{2}z}-\frac{2s^5+124s^3+1122s}{{\cosh }^{4}z}+\frac{3s^5+290s^3+1627s}{{\cosh }^{6}z}\right)+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.9 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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28.26.5 $${\mathrm{Gc}}_m(z,h)\sim \frac{\sinh z}{{\cosh }^{2}z}\left(\frac{s^2+3}{2^{5}h}+\frac{1}{2^9{h}^2}\left(s^3+3s+\frac{4s^3+44s}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(5s^4+34s^2+9-\frac{s^6-47s^4+667s^2+2835}{12{\cosh }^{2}z}+\frac{s^6+505s^4+12139s^2+10395}{12{\cosh }^{4}z}\right)\right)+\mathrm{\cdots }.$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.10 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E5

Encodings:

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See also:

Annotations for §28.26(i), §28.26 and Ch.28

►The asymptotic expansions of ${\mathrm{Fs}}_m(z,h)$ and ${\mathrm{Gs}}_m(z,h)$ in the same circumstances are also given by the right-hand sides of (28.26.4) and (28.26.5), respectively. … ►

§28.26(ii) Uniform Approximations

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24: 13.8 Asymptotic Approximations for Large Parameters

§13.8 Asymptotic Approximations for Large Parameters

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§13.8(ii) Large $b$ and $z$, Fixed $a$ and $b/z$

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§13.8(iii) Large $a$

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§13.8(iv) Large $a$ and $b$

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25: 8.12 Uniform Asymptotic Expansions for Large Parameter

§8.12 Uniform Asymptotic Expansions for Large Parameter

… ►where $g_k$, $k=0,1,2,\mathrm{\dots }$, are the coefficients that appear in the asymptotic expansion (5.11.3) of $\mathrm{\Gamma }\left(z\right)$. … ►Lastly, a uniform approximation for $\mathrm{\Gamma }(a,ax)$ for large $a$, with error bounds, can be found in Dunster (1996a). … ►

Inverse Function

►For asymptotic expansions, as $a\to \mathrm{\infty }$, of the inverse function $x=x(a,q)$ that satisfies the equation …

26: 2.4 Contour Integrals

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§2.4(i) Watson’s Lemma

… ►For examples and extensions (including uniformity and loop integrals) see Olver (1997b, Chapter 4), Wong (1989, Chapter 1), and Temme (1985). … ►Furthermore, as $t\to 0+$, $q(t)$ has the expansion (2.3.7). … ►For examples see Olver (1997b, pp. 315–320). ►

§2.4(iii) Laplace’s Method

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27: 2.3 Integrals of a Real Variable

… ►Then … ►For the Fourier integral … ►

§2.3(ii) Watson’s Lemma

… ►Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … ►

§2.3(iii) Laplace’s Method

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28: 10.57 Uniform Asymptotic Expansions for Large Order

§10.57 Uniform Asymptotic Expansions for Large Order

►Asymptotic expansions for ${𝗃}_n\left((n+\frac{1}{2})z\right)$, ${𝗒}_n\left((n+\frac{1}{2})z\right)$, ${𝗁}_n^{(1)}\left((n+\frac{1}{2})z\right)$, ${𝗁}_n^{(2)}\left((n+\frac{1}{2})z\right)$, ${𝗂}_n^{(1)}\left((n+\frac{1}{2})z\right)$, and ${𝗄}_n\left((n+\frac{1}{2})z\right)$ as $n\to \mathrm{\infty }$ that are uniform with respect to $z$ can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). …

29: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

§8.20 Asymptotic Expansions of $E_p\left(z\right)$

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§8.20(i) Large $z$

… ►Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). ►For an exponentially-improved asymptotic expansion of $E_p\left(z\right)$ see §2.11(iii). ►

§8.20(ii) Large $p$

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30: 2.2 Transcendental Equations

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2.2.6 $$t=y^{\frac{1}{2}}\left(1+\frac{1}{4}{y}^{-1}\ln y+o\left(y^{-1}\right)\right),$$ $y\to \mathrm{\infty }$.

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Symbols:

$o\left(x\right)$: order less than, $\ln z$: principal branch of logarithm function and $y$: root

Permalink:

http://dlmf.nist.gov/2.2.E6

Encodings:

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Annotations for §2.2, §2.2 and Ch.2

►An important case is the reversion of asymptotic expansions for zeros of special functions. … ►

2.2.7 $$f(x)\sim x+f_0+f_1{x}^{-1}+f_2{x}^{-2}+\mathrm{\cdots },$$ $x\to \mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $f(x)$: function and $f_s$: coefficients

Permalink:

http://dlmf.nist.gov/2.2.E7

Encodings:

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See also:

Annotations for §2.2, §2.2 and Ch.2

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2.2.8 $$x\sim y-F_0-F_1{y}^{-1}-F_2{y}^{-2}-\mathrm{\cdots },$$ $y\to \mathrm{\infty }$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $y$: root and $F_s$: coefficients

Permalink:

http://dlmf.nist.gov/2.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §2.2, §2.2 and Ch.2

►where $F_0=f_0$ and $s{F}_s$ ($s\ge 1$) is the coefficient of $x^{-1}$ in the asymptotic expansion of ${(f(x))}^s$ (Lagrange’s formula for the reversion of series). …