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31: 10.17 Asymptotic Expansions for Large Argument

§10.17 Asymptotic Expansions for Large Argument

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§10.17(ii) Asymptotic Expansions of Derivatives

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§10.17(iii) Error Bounds for Real Argument and Order

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§10.17(v) Exponentially-Improved Expansions

… ►For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).

32: 10.19 Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

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§10.19(i) Asymptotic Forms

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§10.19(ii) Debye’s Expansions

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§10.19(iii) Transition Region

… ►See also §10.20(i).

33: 13.20 Uniform Asymptotic Approximations for Large $\mu$

§13.20 Uniform Asymptotic Approximations for Large $\mu$

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§13.20(i) Large $\mu$, Fixed $\kappa$

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§13.20(v) Large $\mu$, Other Expansions

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34: 10.20 Uniform Asymptotic Expansions for Large Order

§10.20 Uniform Asymptotic Expansions for Large Order

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10.20.5 $$Y_{\nu }\left(\nu z\right)\sim -{\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Bi}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Bi}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{B_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

A&S Ref:

9.3.36

Permalink:

http://dlmf.nist.gov/10.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

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§10.20(iii) Double Asymptotic Properties

►For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of $z$ see §10.41(v).

35: 12.9 Asymptotic Expansions for Large Variable

§12.9 Asymptotic Expansions for Large Variable

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§12.9(ii) Bounds and Re-Expansions for the Remainder Terms

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36: 8.25 Methods of Computation

… ►Although the series expansions in §§8.7, 8.19(iv), and 8.21(vi) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. For large $|z|$ the corresponding asymptotic expansions (generally divergent) are used instead. …

37: 13.22 Zeros

… ►Asymptotic approximations to the zeros when the parameters $\kappa$ and/or $\mu$ are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. For example, if $\mu (\ge 0)$ is fixed and $\kappa (>0)$ is large, then the $r$th positive zero ${\varphi }_r$ of $M_{\kappa ,\mu }\left(z\right)$ is given by …

38: 30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

… ►The cases of large $m$, and of large $m$ and large $|\gamma |$, are studied in Abramowitz (1949). …The behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ for complex ${\gamma }^2$ and large $|{\lambda }_n^m\left({\gamma }^2\right)|$ is investigated in Hunter and Guerrieri (1982).

39: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

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§8.11(i) Large $z$, Fixed $a$

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§8.11(ii) Large $a$, Fixed $z$

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§8.11(iii) Large $a$, Fixed $z/a$

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40: 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)). … ►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 $\alpha$). …