asymptotic expansions

SearchAdvancedHelp

(0.008 seconds)

21—30 of 164 matching pages

21: 2.3 Integrals of a Real Variable

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

As $t\to a+$

2.3.14

$$p(t)\sim p(a)+\sum _{s=0}^{\mathrm{\infty }}{p}_s{(t-a)}^{s+\mu },$$

$$q(t)\sim \sum _{s=0}^{\mathrm{\infty }}{q}_s{(t-a)}^{s+\lambda -1},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: left endpoint, $p(t)$: real function, $p_s$: coefficients, $q(t)$: function, $q_s$: coefficients, $\lambda$: positive constant and $\mu$: positive constant

Referenced by:

item (b)

Permalink:

http://dlmf.nist.gov/2.3.E14

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §2.3(iii), §2.3 and Ch.2

and the expansion for $p(t)$ is differentiable. Again $\lambda$ and $\mu$ are positive constants. Also $p_0>0$ (consistent with (a)).

… ►Then …

22: 2.9 Difference Equations

… ►

$f(n)\sim {\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{\displaystyle \frac{f_s}{n^s}},$

►

$g(n)\sim {\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{\displaystyle \frac{g_s}{n^s}}$ , $n\to \mathrm{\infty }$,

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

2.9.9 $$w_j(n)\sim {\rho }^n\exp \left({(-1)}^j\kappa \sqrt{n}\right)n^{\alpha }\sum _{s=0}^{\mathrm{\infty }}{(-1)}^{js}\frac{c_s}{n^{s/2}},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\exp z$: exponential function, $\rho$: roots, $\kappa$ , $c$: constant and $w_j(n)$: solutions

Permalink:

http://dlmf.nist.gov/2.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §2.9(ii), §2.9 and Ch.2

… ►

2.9.12 $$w_j(n)\sim {\rho }^n{n}^{{\alpha }_j}\sum _{s=0}^{\mathrm{\infty }}\frac{a_{s,j}}{n^s},$$ $n\to \mathrm{\infty }$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\rho$: roots and $w_j(n)$: solutions

Referenced by:

§2.9(ii)

Permalink:

http://dlmf.nist.gov/2.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.9(ii), §2.9 and Ch.2

…

23: 29.7 Asymptotic Expansions

§29.7 Asymptotic Expansions

►

§29.7(i) Eigenvalues

… ►

29.7.1 $$a_{\nu }^m\left(k^2\right)\sim p\kappa -{\tau }_0-{\tau }_1{\kappa }^{-1}-{\tau }_2{\kappa }^{-2}-\mathrm{\cdots },$$

ⓘ

Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $\sim$: Poincaré asymptotic expansion, $m$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, $\kappa$ and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

… ►

§29.7(ii) Lamé Functions

… ►In Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$ and ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$. …

24: 10.41 Asymptotic Expansions for Large Order

§10.41 Asymptotic Expansions for Large Order

►

§10.41(i) Asymptotic Forms

… ► … ►

§10.41(v) Double Asymptotic Properties (Continued)

… ►

25: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10 Uniform Asymptotic Expansions for Large Parameter

… ►Lastly, the function $g(\mu )$ in (12.10.3) and (12.10.4) has the asymptotic expansion: ►

12.10.14 $$g(\mu )\sim h(\mu )\left(1+\frac{1}{2}\sum _{s=1}^{\mathrm{\infty }}\frac{{\gamma }_s}{{(\frac{1}{2}{\mu }^2)}^s}\right),$$

ⓘ

Defines:

$g(\mu )$: expansion (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $h(\mu )$: expansion and ${\gamma }_s$: coefficients

Referenced by:

§12.10(vii)

Permalink:

http://dlmf.nist.gov/12.10.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(ii), §12.10 and Ch.12

… ► … ►

26: 12.14 The Function $W(a,x)$

… ►

§12.14(viii) Asymptotic Expansions for Large Variable

… ►

§12.14(ix) Uniform Asymptotic Expansions for Large Parameter

… ►The function $l(\mu )$ has the asymptotic expansion ►

12.14.29 $$l(\mu )\sim \frac{2^{\frac{1}{4}}}{{\mu }^{\frac{1}{2}}}\sum _{s=0}^{\mathrm{\infty }}\frac{l_s}{{\mu }^{4s}},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $l(\mu )$ and $l_s$: coefficient

Permalink:

http://dlmf.nist.gov/12.14.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

… ► …

27: 2.1 Definitions and Elementary Properties

… ►

§2.1(iii) Asymptotic Expansions

… ►Symbolically, … ►

§2.1(iv) Uniform Asymptotic Expansions

… ►

2.1.18 $$f(u,x)\sim \sum _{s=0}^{\mathrm{\infty }}{a}_s(u)x^{-s}$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $u$: parameter (or set), $f(u,x)$: function and $a_s(u)$: coefficients

Permalink:

http://dlmf.nist.gov/2.1.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §2.1(iv), §2.1 and Ch.2

… ►

§2.1(v) Generalized Asymptotic Expansions

…

28: 8.11 Asymptotic Approximations and Expansions

… ►For an exponentially-improved asymptotic expansion (§2.11(iii)) see Olver (1991a). … ►With $x=1$, an asymptotic expansion of $e_n(nx)/{\mathrm{e}}^{nx}$ follows from (8.11.14) and (8.11.16). … ►

8.11.16 $$S_n(1)-\frac{1}{2}\frac{n!{\mathrm{e}}^n}{n^n}\sim -\frac{2}{3}+\frac{4}{135}{n}^{-1}-\frac{8}{2835}{n}^{-2}-\frac{16}{8505}{n}^{-3}+\mathrm{\dots },$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $S_n(x)$: function

Referenced by:

§8.11(v), §8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(v), §8.11 and Ch.8

►

8.11.17 $$S_n(-1)\sim -\frac{1}{2}+\frac{1}{8}{n}^{-1}+\frac{1}{32}{n}^{-2}-\frac{1}{128}{n}^{-3}-\frac{13}{512}{n}^{-4}+\mathrm{\dots }.$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $n$: nonnegative integer and $S_n(x)$: function

Referenced by:

§8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(v), §8.11 and Ch.8

… ►

8.11.18 $$S_n(x)\sim \sum _{k=0}^{\mathrm{\infty }}{d}_k(x)n^{-k},$$ $n\to \mathrm{\infty }$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $x$: real variable, $k$: nonnegative integer, $n$: nonnegative integer, $S_n(x)$: function and $d_k(x)$: coefficients

Referenced by:

§8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(v), §8.11 and Ch.8

…

29: 8.12 Uniform Asymptotic Expansions for Large Parameter

§8.12 Uniform Asymptotic Expansions for Large Parameter

… ►

8.12.7 $$S(a,\eta )\sim \frac{{\mathrm{e}}^{-\frac{1}{2}a{\eta }^2}}{\sqrt{2\pi a}}\sum _{k=0}^{\mathrm{\infty }}{c}_k(\eta )a^{-k},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $a$: parameter, $k$: nonnegative integer, $\eta (\lambda )$ , $S(a,\eta )$: function and $c_k(\eta )$: coefficients

Referenced by:

§8.12

Permalink:

http://dlmf.nist.gov/8.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

►

8.12.8 $$T(a,\eta )\sim \frac{{\mathrm{e}}^{\frac{1}{2}a{\eta }^2}}{\sqrt{2\pi a}}\sum _{k=0}^{\mathrm{\infty }}{c}_k(\eta ){(-a)}^{-k},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $a$: parameter, $k$: nonnegative integer, $\eta (\lambda )$ , $T(a,\eta )$: function and $c_k(\eta )$: coefficients

Permalink:

http://dlmf.nist.gov/8.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

… ►

8.12.15 $$Q(a,a)\sim \frac{1}{2}+\frac{1}{\sqrt{2\pi a}}\sum _{k=0}^{\mathrm{\infty }}{c}_k(0)a^{-k},$$ $|\mathrm{ph}a|\le \pi -\delta$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $Q(a,z)$: normalized incomplete gamma function, $\mathrm{ph}$: phase, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant and $c_k(\eta )$: coefficients

Permalink:

http://dlmf.nist.gov/8.12.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

… ►

8.12.22 $$x(a,\frac{1}{2})\sim a-\frac{1}{3}+\frac{8}{405}{a}^{-1}+\frac{184}{25515}{a}^{-2}+\frac{2248}{\mathrm{34\hspace{0.33em}44525}}{a}^{-3}+\mathrm{\cdots },$$ $a\to \mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.12.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12, §8.12 and Ch.8

30: 7.13 Zeros

… ►As $n\to \mathrm{\infty }$ ►

$x_n\sim \lambda -\frac{1}{4}\mu {\lambda }^{-1}+\frac{1}{16}(1-\mu +\frac{1}{2}{\mu }^2){\lambda }^{-3}-\mathrm{\cdots },$

… ►

$x_n\sim \lambda +{\displaystyle \frac{\alpha (\alpha \pi -4)}{8\pi {\lambda }^3}}+\mathrm{\cdots },$

►

$y_n\sim {\displaystyle \frac{\alpha }{2\lambda }}+\mathrm{\cdots }$ ,

… ►