asymptotic approximations for large parameters

SearchAdvancedHelp

(0.004 seconds)

21—30 of 51 matching pages

21: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

§28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

… ►

28.25.1 $${\mathrm{M}}_{\nu }^{(3,4)}(z,h)\sim \frac{{\mathrm{e}}^{\pm \mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi \right)}}{{\left(\pi h(\cosh z+1)\right)}^{\frac{1}{2}}}\sum _{m=0}^{\mathrm{\infty }}\frac{D_m^{\pm }}{{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)}^m},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: 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

… ►

28.25.3 $$(m+1)D_{m+1}^{\pm }+\left({(m+\frac{1}{2})}^2\pm (m+\frac{1}{4})8\mathrm{i}h+2h^2-a\right)D_m^{\pm }\pm (m-\frac{1}{2})\left(8\mathrm{i}hm\right)D_{m-1}^{\pm }=0,$$ $m\ge 0$.

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit, $m$: integer, $h$: parameter, $a$: parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.2 (for $+$) 20.9.4 (for $-$)

Permalink:

http://dlmf.nist.gov/28.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

… ►

28.25.4 $$\mathrm{\Re }z\to +\mathrm{\infty },$$ $-\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le 2\pi -\delta$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: 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

… ►

28.25.5 $$\mathrm{\Re }z\to +\mathrm{\infty },$$ $-2\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le \pi -\delta$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $h$: parameter, $z$: complex variable and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/28.25.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

…

22: Bibliography W

… ►

Asymptotic approximations for certain $6$-$j$ and $9$-$j$ symbols. J. Phys. A 32 (39), pp. 6901–6902.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§34.8, §34.8.

Encodings:

BibTeX

… ►

Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions. In Algorithms for Approximation, A. Iske and J. Levesley (Eds.), pp. 331–348.

ⓘ

External Links:

ISBN 3-540-33283-9, Document, MathReview Entry, ZentralBlatt

Referenced by:

§2.10(i).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§13.19.

Encodings:

BibTeX

►

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

ⓘ

External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

Encodings:

BibTeX

… ►

Error bounds for asymptotic approximations of special functions. Ann. Numer. Math. 2 (1-4), pp. 181–197.

ⓘ

Notes:

Special functions (Torino, 1993)

External Links:

ISSN 1021-2655, MathReview (Andrei Martínez Finkelshtein), ZentralBlatt

Referenced by:

§10.21(vi).

Encodings:

BibTeX

…

23: 28.20 Definitions and Basic Properties

… ►

28.20.1 $$w^{\prime \prime }-\left(a-2q\cosh \left(2z\right)\right)w=0,$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

A&S Ref:

20.1.2 (in slightly different form) 20.8.6

Referenced by:

§28.20(iii), §28.32(i), 1st item, 2nd item, item (b), §28.8(iv), §28.8(iv)

Permalink:

http://dlmf.nist.gov/28.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(i), §28.20 and Ch.28

… ►

28.20.2 $$({\zeta }^2-1)w^{\prime \prime }+\zeta w^{\prime }+\left(4q{\zeta }^2-2q-a\right)w=0,$$ $\zeta =\cosh z$.

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

Referenced by:

§28.20(iii)

Permalink:

http://dlmf.nist.gov/28.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(i), §28.20 and Ch.28

… ►

§28.20(iii) Solutions ${\mathrm{M}}_{\nu }^{(j)}$

… ►

28.20.8 $$h=\sqrt{q}(>0).$$

ⓘ

Symbols:

$q=h^2$: parameter and $h$: parameter

Permalink:

http://dlmf.nist.gov/28.20.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

►Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ${\zeta }^{1/2}{\mathrm{e}}^{\pm 2\mathrm{i}h\zeta }$ as $\zeta \to \mathrm{\infty }$ in the respective sectors $|\mathrm{ph}\left(\mp \mathrm{i}\zeta \right)|\le \frac{3}{2}\pi -\delta$, $\delta$ being an arbitrary small positive constant. …

24: 10.72 Mathematical Applications

… ►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. …where $z$ is a real or complex variable and $u$ is a large real or complex parameter. … ►In regions in which (10.72.1) has a simple turning point $z_0$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_0$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)). … ►If $f(z)$ has a double zero $z_0$, or more generally $z_0$ is a zero of order $m$, $m=2,3,4,\mathrm{\dots }$, then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order $1/(m+2)$. … ►Then for large $u$ asymptotic approximations of the solutions $w$ can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on $u$ and $\alpha$). …

25: 12.16 Mathematical Applications

… ►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). … ►In Brazel et al. (1992) exponential asymptotics are considered in connection with an eigenvalue problem involving PCFs. ►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …

26: 13.29 Methods of Computation

… ►Although the Maclaurin series expansion (13.2.2) converges for all finite values of $z$, it is cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. For large $|z|$ the asymptotic expansions of §13.7 should be used instead. …For large values of the parameters $a$ and $b$ the approximations in §13.8 are available. … ►In the sector the integration has to be towards the origin, with starting values computed from asymptotic expansions (§§13.7 and 13.19). … ►

13.29.8 $$w(n)\sim \frac{\sqrt{\pi }{\mathrm{e}}^{\frac{1}{2}z}{z}^{\frac{1}{4}(4a-2b+1)}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a+1-b\right)}{n}^{\frac{1}{4}(4a-2b-3)}{\mathrm{e}}^{-2\sqrt{nz}},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.29.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

…

27: 30.9 Asymptotic Approximations and Expansions

§30.9 Asymptotic Approximations and Expansions

►

§30.9(i) Prolate Spheroidal Wave Functions

… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►

§30.9(iii) Other Approximations and Expansions

… ►

28: 8.13 Zeros

… ►

8.13.1 .

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.13(i), §8.13 and Ch.8

… ►For asymptotic approximations for $x_{+}(a)$ and $x_{-}(a)$ as $a\to -\mathrm{\infty }$ see Tricomi (1950b), with corrections by Kölbig (1972b). For more accurate asymptotic approximations see Thompson (2012). … ►For information on the distribution and computation of zeros of $\gamma (a,\lambda a)$ and $\mathrm{\Gamma }(a,\lambda a)$ in the complex $\lambda$-plane for large values of the positive real parameter $a$ see Temme (1995a). … ►Approximations to $a_n^{\ast }$, $x_n^{\ast }$ for large $n$ can be found in Kölbig (1970). …

29: 33.21 Asymptotic Approximations for Large $|r|$

§33.21 Asymptotic Approximations for Large $|r|$

►

§33.21(i) Limiting Forms

… ►

When $r\to \pm \mathrm{\infty }$ with , Equations (33.16.10)–(33.16.13) are combined with

33.21.1

${\zeta }_{\mathrm{\ell }}(\nu ,r)\sim {\mathrm{e}}^{-r/\nu }{(2r/\nu )}^{\nu },$

${\xi }_{\mathrm{\ell }}(\nu ,r)\sim {\mathrm{e}}^{r/\nu }{(2r/\nu )}^{-\nu }$ , $r\to \mathrm{\infty }$,

ⓘ

Symbols:

$\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function

Permalink:

http://dlmf.nist.gov/33.21.E1

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §33.21(i), §33.21 and Ch.33

33.21.2

${\zeta }_{\mathrm{\ell }}(-\nu ,r)\sim {\mathrm{e}}^{r/\nu }{(-2r/\nu )}^{-\nu },$

${\xi }_{\mathrm{\ell }}(-\nu ,r)\sim {\mathrm{e}}^{-r/\nu }{(-2r/\nu )}^{\nu },$ $r\to -\mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function

Permalink:

http://dlmf.nist.gov/33.21.E2

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §33.21(i), §33.21 and Ch.33

Corresponding approximations for $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$ as $r\to \mathrm{\infty }$ can be obtained via (33.16.17), and as $r\to -\mathrm{\infty }$ via (33.16.18).

… ►

§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).

30: 10.41 Asymptotic Expansions for Large Order

§10.41 Asymptotic Expansions for Large Order

►

§10.41(i) Asymptotic Forms

… ►

§10.41(iv) Double Asymptotic Properties

… ►

§10.41(v) Double Asymptotic Properties (Continued)

… ►