large variable and/or large parameter

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31: 10.72 Mathematical Applications

… ►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)). …

32: 2.8 Differential Equations with a Parameter

… ►

2.8.1 $$d^{2}w/{dz}^2=\left(u^{2}f(z)+g(z)\right)w,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $f(x)$: function, $g(x)$: function, $w$: solution and $u$: large real or complex parameter

Referenced by:

§2.8(i), §2.8(v)

Permalink:

http://dlmf.nist.gov/2.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(i), §2.8 and Ch.2

►in which $u$ is a real or complex parameter, and asymptotic solutions are needed for large $|u|$ that are uniform with respect to $z$ in a point set $𝐃$ in $ℝ$ or $ℂ$. … ►

2.8.3 $$\frac{d^{2}W}{{d\xi }^2}=\left(u^2{\dot{z}}^{2}f(z)+\psi (\xi )\right)W,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $f(x)$: function, $u$: large real or complex parameter, $W$: change of variable, $\dot{z}$: derivative of $z$ with respect to $\xi$ and $\psi (\xi )$: function

Permalink:

http://dlmf.nist.gov/2.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(i), §2.8 and Ch.2

… ►

2.8.8 $$d^{2}W/{d\xi }^2=\left(u^2{\xi }^m+\psi (\xi )\right)W,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $u$: large real or complex parameter, $W$: change of variable and $\psi (\xi )$: function

Referenced by:

§2.8(i), §2.8(iv)

Permalink:

http://dlmf.nist.gov/2.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(i), §2.8 and Ch.2

… ►

2.8.9 $$\frac{d^{2}W}{{d\xi }^2}=\left(\frac{u^2}{\xi }+\frac{\rho }{{\xi }^2}\right)W,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $u$: large real or complex parameter, $W$: change of variable and $\rho$: limit

Permalink:

http://dlmf.nist.gov/2.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(i), §2.8 and Ch.2

…

33: 36.12 Uniform Approximation of Integrals

… ►where $k$ is a large real parameter and $𝐲=\{y_1,y_2,\mathrm{\dots }\}$ is a set of additional (nonasymptotic) parameters. …

34: 28.31 Equations of Whittaker–Hill and Ince

… ►When , we substitute … ►When $p$ is a nonnegative integer, the parameter $\eta$ can be chosen so that solutions of (28.31.3) are trigonometric polynomials, called Ince polynomials. … ►

Asymptotic Behavior

… ►All other periodic solutions behave as multiples of $\exp \left(\frac{1}{4}\xi \cos \left(2z\right)\right){(\cos z)}^{-p-2}$. … ►All other periodic solutions behave as multiples of $\exp \left(-\frac{1}{4}\xi \cos \left(2z\right)\right){\left(\cos z\right)}^p$.

35: 33.18 Limiting Forms for Large $\mathrm{\ell }$

§33.18 Limiting Forms for Large $\mathrm{\ell }$

… ►

$f(ϵ,\mathrm{\ell };r)\sim {\displaystyle \frac{{(2r)}^{\mathrm{\ell }+1}}{(2\mathrm{\ell }+1)!}},$

►

$h(ϵ,\mathrm{\ell };r)\sim {\displaystyle \frac{(2\mathrm{\ell })!}{\pi {(2r)}^{\mathrm{\ell }}}}.$

36: 33.11 Asymptotic Expansions for Large $\rho$

§33.11 Asymptotic Expansions for Large $\rho$

►For large $\rho$, with $\mathrm{\ell }$ and $\eta$ fixed, ►

33.11.1 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\sim {\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k{\left(b\right)}_k}{k!{(\pm 2\mathrm{i}\rho )}^k},$$

ⓘ

Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $a$: coefficient and $b$: coefficient

A&S Ref:

14.5.9

Referenced by:

Erratum (V1.0.19) for Equation (33.11.1), Erratum (V1.0.22) for Equation (33.11.1)

Permalink:

http://dlmf.nist.gov/33.11.E1

Encodings:

pMML, png, TeX

Errata (effective with 1.0.22):

Previously this formula was expressed as an equality. Since this formula expresses an asymptotic expansion, it has been corrected by using instead an $\sim$ relation.

Reported 2019-01-29 by Gergő Nemes

Errata (effective with 1.0.19):

Originally the factor in the denominator on the right-hand side was written incorrectly as ${(\mp 2\mathrm{i}\rho )}^k$. This has been corrected to ${(\pm 2\mathrm{i}\rho )}^k$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.11 and Ch.33

… ►

33.11.7 $$g(\eta ,\rho )\widehat{f}(\eta ,\rho )-f(\eta ,\rho )\widehat{g}(\eta ,\rho )=1.$$

ⓘ

Symbols:

$\rho$: nonnegative real variable, $\eta$: real parameter, $f(\eta ,\rho )$: function, $g(\eta ,\rho )$: function, $\widehat{f}(\eta ,\rho )$: function and $\widehat{g}(\eta ,\rho )$: function

A&S Ref:

14.5.8

Referenced by:

§33.11, Erratum (V1.0.22) for Equations (33.11.2)–(33.11.7), Erratum (V1.0.22) for Equations (33.11.2)–(33.11.7)

Permalink:

http://dlmf.nist.gov/33.11.E7

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.22):

The arguments of some of the functions in this equation were included to improve clarity of the presentation.

See also:

Annotations for §33.11 and Ch.33

►Here $f_0=1$, $g_0=0$, ${\widehat{f}}_0=0$, ${\widehat{g}}_0=1-(\eta /\rho )$, and for $k=0,1,2,\mathrm{\dots }$, …

37: 14.32 Methods of Computation

… ►

Application of the uniform asymptotic expansions for large values of the parameters given in §§14.15 and 14.20(vii)–14.20(ix).

…

38: 18.15 Asymptotic Approximations

… ►These approximations apply when the parameters are large, namely $\alpha$ and $\beta$ (subject to restrictions) in the case of Jacobi polynomials, $\lambda$ in the case of ultraspherical polynomials, and $|\alpha |+|x|$ in the case of Laguerre polynomials. …

39: 13.29 Methods of Computation

… ►For large values of the parameters $a$ and $b$ the approximations in §13.8 are available. …

40: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

§33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

►

§33.10(i) Large $\rho$

… ►

33.10.2 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\sim \exp \left(\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )\right),$$

ⓘ

Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\sim$: asymptotic equality, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.6.5

Permalink:

http://dlmf.nist.gov/33.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.10(i), §33.10 and Ch.33

… ►

§33.10(ii) Large Positive $\eta$

… ►

§33.10(iii) Large Negative $\eta$

…