asymptotic forms

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21: 2.8 Differential Equations with a Parameter

… ►The form of the asymptotic expansion depends on the nature of the transition points in

\mathbf{D}

, that is, points at which

f(z)

has a zero or singularity. …

22: Bibliography L

… ►

J. N. Lyness (1971) Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature. Math. Comp. 25 (113), pp. 87–104.

ⓘ

External Links:: FullText, MathReview (J. Bryant), ZentralBlatt
Referenced by:: §2.3(iv).
Encodings:: BibTeX

…

23: 26.5 Lattice Paths: Catalan Numbers

… ►

§26.5(iv) Limiting Forms

►

26.5.6

C\left(n\right)\sim\frac{4^{n}}{\sqrt{\pi n^{3}}},

n\to\infty

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iv), §26.5 and Ch.26

…

24: 10.57 Uniform Asymptotic Expansions for Large Order

§10.57 Uniform Asymptotic Expansions for Large Order

►Asymptotic expansions for

\mathsf{j}_{n}\left((n+\tfrac{1}{2})z\right)

\mathsf{y}_{n}\left((n+\tfrac{1}{2})z\right)

{\mathsf{h}^{(1)}_{n}}\left((n+\tfrac{1}{2})z\right)

{\mathsf{h}^{(2)}_{n}}\left((n+\tfrac{1}{2})z\right)

{\mathsf{i}^{(1)}_{n}}\left((n+\tfrac{1}{2})z\right)

, and

\mathsf{k}_{n}\left((n+\tfrac{1}{2})z\right)

n\to\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). …

25: 2.9 Difference Equations

§2.9 Difference Equations

… ►For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a). ►

§2.9(ii) Coincident Characteristic Values

… ►For analogous results for difference equations of the form … ►

26: 18.34 Bessel Polynomials

… ►

18.34.8

\lim_{\alpha\to\infty}\frac{P^{(\alpha,a-\alpha-2)}_{n}\left(1+\alpha x\right)% }{P^{(\alpha,a-\alpha-2)}_{n}\left(1\right)}=y_{n}\left(x;a\right).

ⓘ

Symbols:: $y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Bessel polynomial, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.34(iii)
Permalink:: http://dlmf.nist.gov/18.34.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §18.34(iii), §18.34 and Ch.18

►In this limit the finite system of Jacobi polynomials

P^{(\alpha,\beta)}_{n}\left(x\right)

which is orthogonal on

(1,\infty)

(see §18.3) tends to the finite system of Romanovski–Bessel polynomials which is orthogonal on

(0,\infty)

(see (18.34.5_5)). ►For uniform asymptotic expansions of

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

n\to\infty

in terms of Airy functions (§9.2) see Wong and Zhang (1997) and Dunster (2001c). For uniform asymptotic expansions in terms of Hermite polynomials see López and Temme (1999b). …

27: 33.11 Asymptotic Expansions for Large $\rho$

§33.11 Asymptotic Expansions for Large $\rho$

…

28: 7.12 Asymptotic Expansions

§7.12 Asymptotic Expansions

►

§7.12(i) Complementary Error Function

… ►

§7.12(ii) Fresnel Integrals

►The asymptotic expansions of

C\left(z\right)

and

S\left(z\right)

are given by (7.5.3), (7.5.4), and … ►

§7.12(iii) Goodwin–Staton Integral

…

29: 28.8 Asymptotic Expansions for Large $q$

… ►

28.8.4

U_{m}(\xi)\sim D_{m}\left(\xi\right)-\frac{1}{2^{6}h}\left(D_{m+4}\left(\xi% \right)-4!\dbinom{m}{4}D_{m-4}\left(\xi\right)\right)+\frac{1}{2^{13}h^{2}}% \left(D_{m+8}\left(\xi\right)-2^{5}(m+2)D_{m+4}\left(\xi\right)+4!\,2^{5}(m-1)% \dbinom{m}{4}D_{m-4}\left(\xi\right)+8!\genfrac{(}{)}{0.0pt}{}{m}{8}D_{m-8}% \left(\xi\right)\right)+\cdots,

ⓘ

Defines:: $U_{m}(\xi)$ : function (locally)
Symbols:: $D_{\NVar{\nu}}\left(\NVar{z}\right)$ : parabolic cylinder function, $\sim$ : Poincaré asymptotic expansion, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $m$ : integer, $h$ : parameter and $\xi$ : variable
A&S Ref:: 20.9.17 (in different form)
Permalink:: http://dlmf.nist.gov/28.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(ii), §28.8 and Ch.28

… ►

28.8.6

\widehat{C}_{m}\sim\left(\frac{\pi h}{2(m!)^{2}}\right)^{\ifrac{1}{4}}\left(1+% \frac{2m+1}{8h}+\dfrac{m^{4}+2m^{3}+263m^{2}+262m+108}{2048h^{2}}+\cdots\right% )^{-\ifrac{1}{2}},

ⓘ

Defines:: $\widehat{C}_{m}$ : coefficient (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $m$ : integer and $h$ : parameter
A&S Ref:: 20.9.19 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.8.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(ii), §28.8 and Ch.28

►

28.8.7

\widehat{S}_{m}\sim\left(\frac{\pi h}{2(m!)^{2}}\right)^{\ifrac{1}{4}}\left(1-% \frac{2m+1}{8h}+\dfrac{m^{4}+2m^{3}-121m^{2}-122m-84}{2048h^{2}}+\cdots\right)% ^{-\ifrac{1}{2}}.

ⓘ

Defines:: $\widehat{S}_{m}$ : coefficient (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $m$ : integer and $h$ : parameter
A&S Ref:: 20.9.20 (in slightly different form)
Referenced by:: §28.8(ii), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)
Permalink:: http://dlmf.nist.gov/28.8.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(ii), §28.8 and Ch.28

… ►

28.8.11

P_{m}(x)\sim 1+\dfrac{s}{2^{3}h{\cos}^{2}x}+\dfrac{1}{h^{2}}\left(\dfrac{s^{4}% +86s^{2}+105}{2^{11}{\cos}^{4}x}-\dfrac{s^{4}+22s^{2}+57}{2^{11}{\cos}^{2}x}% \right)+\cdots,

ⓘ

Defines:: $P_{m}(x)$ (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion, $\cos\NVar{z}$ : cosine function, $m$ : integer, $h$ : parameter and $x$ : real variable
A&S Ref:: 20.9.14 (in different form)
Permalink:: http://dlmf.nist.gov/28.8.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(iii), §28.8 and Ch.28

►

28.8.12

Q_{m}(x)\sim\dfrac{\sin x}{{\cos}^{2}x}\left(\dfrac{1}{2^{5}h}(s^{2}+3)+\dfrac% {1}{2^{9}h^{2}}\left(s^{3}+3s+\dfrac{4s^{3}+44s}{{\cos}^{2}x}\right)\right)+\cdots.

ⓘ

Defines:: $Q_{m}(x)$ (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter and $x$ : real variable
A&S Ref:: 20.9.14 (in different form)
Permalink:: http://dlmf.nist.gov/28.8.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(iii), §28.8 and Ch.28

…

30: 8.20 Asymptotic Expansions of $E_{p}\left(z\right)$

§8.20 Asymptotic Expansions of $E_{p}\left(z\right)$

►

§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 $p$

…