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31: 15.12 Asymptotic Approximations

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§15.12(i) Large Variable

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§15.12(ii) Large $c$

… ►For large $b$ and $c$ with $c>b+1$ see López and Pagola (2011). ►

§15.12(iii) Other Large Parameters

… ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).

32: 2.10 Sums and Sequences

… ►for large $n$. … ►As a first estimate for large $n$ … ►(5.11.7) shows that the integrals around the large quarter circles vanish as $n\to \mathrm{\infty }$. Hence … ►

Example

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33: 10.72 Mathematical Applications

… ►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)). … ►In regions in which the function $f(z)$ has a simple pole at $z=z_0$ and ${(z-z_0)}^{2}g(z)$ is analytic at $z=z_0$ (the case $\lambda =-1$ in §10.72(i)), asymptotic expansions of the solutions $w$ of (10.72.1) for large $u$ can be constructed in terms of Bessel functions and modified Bessel functions of order $\pm \sqrt{1+4\rho }$, where $\rho$ is the limiting value of ${(z-z_0)}^{2}g(z)$ as $z\to z_0$. … ►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 $\alpha$). …

34: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

►For large values of the parameters in the $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols, different asymptotic forms are obtained depending on which parameters are large. … ►

34.8.1 $j_1,j_2,j_3\gg l_3\gg 1$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $o\left(x\right)$: order less than, $\sin z$: sine function, : $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $l,l_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §34.8 and Ch.34

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35: 13.9 Zeros

… ►For fixed $a,b\in ℂ$ the large $z$-zeros of $M(a,b,z)$ satisfy …where $n$ is a large positive integer, and the logarithm takes its principal value (§4.2(i)). … ►For fixed $b$ and $z$ in $ℂ$ the large $a$-zeros of $M(a,b,z)$ are given by …where $n$ is a large positive integer. … ►where $n$ is a large positive integer. …

36: 19.12 Asymptotic Approximations

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19.12.4 $$(1-{\alpha }^2)\mathrm{\Pi }({\alpha }^2,k)=\left(\ln \frac{4}{k^{\prime }}\right)\left(1+O\left(k_{}^{\prime }{}_{}{}^2\right)\right)-{\alpha }^2{R}_C(1,1-{\alpha }^2),$$ ,

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\ln z$: principal branch of logarithm function, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

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19.12.5 $$(1-{\alpha }^2)\mathrm{\Pi }({\alpha }^2,k)=\left(\ln \left(\frac{4}{k^{\prime }}\right)-R_C(1,1-{\alpha }^{-2})\right)\left(1+O\left(k_{}^{\prime }{}_{}{}^2\right)\right),$$ .

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\ln z$: principal branch of logarithm function, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

… ►They are useful primarily when $(1-k)/(1-\sin \varphi )$ is either small or large compared with 1. … ►

19.12.6 $$R_C(x,y)=\frac{\pi }{2\sqrt{y}}-\frac{\sqrt{x}}{y}\left(1+O\left(\sqrt{\frac{x}{y}}\right)\right),$$ $x/y\to 0$,

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\pi$: the ratio of the circumference of a circle to its diameter

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

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19.12.7 $$R_C(x,y)=\frac{1}{2\sqrt{x}}\left(\left(1+\frac{y}{2x}\right)\ln \left(\frac{4x}{y}\right)-\frac{y}{2x}\right)(1+O\left(y^2/x^2\right)),$$ $y/x\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\ln z$: principal branch of logarithm function

Referenced by:

§19.12, §19.2(iv)

Permalink:

http://dlmf.nist.gov/19.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

37: 32.11 Asymptotic Approximations for Real Variables

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32.11.1 $$w(x)=-\sqrt{\frac{1}{6}|x|}+d{|x|}^{-1/8}\sin \left(\varphi (x)-{\theta }_0\right)+o\left({|x|}^{-1/8}\right),$$ $x\to -\mathrm{\infty }$,

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Symbols:

$o\left(x\right)$: order less than, $\sin z$: sine function, $x$: real, $\varphi (z)$: function, $d$: constant and ${\theta }_0$: constant

Permalink:

http://dlmf.nist.gov/32.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(i), §32.11 and Ch.32

… ►If , then $w_k(x)$ exists for all sufficiently large $|x|$ as $x\to -\mathrm{\infty }$, and ►

32.11.6 $$w_k(x)=d{|x|}^{-1/4}\sin \left(\varphi (x)-{\theta }_0\right)+o\left({|x|}^{-1/4}\right),$$

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Symbols:

$o\left(x\right)$: order less than, $\sin z$: sine function, $x$: real, $k$: real, $\varphi (z)$: function, $d$: constant and ${\theta }_0$: constant

Permalink:

http://dlmf.nist.gov/32.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(ii), §32.11 and Ch.32

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32.11.19 $$w(x)=\sigma \sqrt{\frac{1}{2}x}+\sigma \rho {(2x)}^{-1/4}\cos \left(\psi (x)+\theta \right)+O\left(x^{-1}\right),$$ $x\to +\mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\cos z$: cosine function, $x$: real, $\sigma$: real constant, $\rho$: real constant, $\theta$: real constant and $\psi (z)$: function

Permalink:

http://dlmf.nist.gov/32.11.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(iii), §32.11 and Ch.32

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32.11.33 $$w_h(x)=-\frac{2}{3}x+\frac{4}{3}d\sqrt{3}\sin \left(\varphi (x)-{\theta }_0\right)+O\left(x^{-1}\right),$$ $x\to -\mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\sin z$: sine function, $x$: real, $\varphi (z)$: function, $d$: constant and ${\theta }_0$: constant

Permalink:

http://dlmf.nist.gov/32.11.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(v), §32.11 and Ch.32

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38: 18.39 Applications in the Physical Sciences

… ►A relativistic treatment becoming necessary as $Z$ becomes large as corrections to the non-relativistic Schrödinger picture are of approximate order ${\left(\alpha Z\right)}^2\sim {\left(Z/137\right)}^2$, $\alpha$ being the dimensionless fine structure constant $e^2/(4\pi {\epsilon }_0\mathrm{\hslash }c)$, where $c$ is the speed of light. …

39: 2.11 Remainder Terms; Stokes Phenomenon

… ►In order to guard against this kind of error remaining undetected, the wanted function may need to be computed by another method (preferably nonasymptotic) for the smallest value of the (large) asymptotic variable $x$ that is intended to be used. … ►Hence from §7.12(i) $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ is of the same exponentially-small order of magnitude as the contribution from the other terms in (2.11.15) when $\rho$ is large. …

40: 2.1 Definitions and Elementary Properties

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§2.1(i) Asymptotic and Order Symbols

… ►As $x\to c$ in $𝐗$ … ►

§2.1(ii) Integration and Differentiation

►Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. … ►If $\sum a_s{x}^{-s}$ converges for all sufficiently large $|x|$, then it is automatically the asymptotic expansion of its sum as $x\to \mathrm{\infty }$ in $ℂ$. …