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1: 33.11 Asymptotic Expansions for Large $\rho$

§33.11 Asymptotic Expansions for Large $\rho$

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

2: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

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

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§33.10(i) Large $\rho$

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§33.10(ii) Large Positive $\eta$

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§33.10(iii) Large Negative $\eta$

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3: 2.11 Remainder Terms; Stokes Phenomenon

… ►For large $\rho$ the integrand has a saddle point at $t={\mathrm{e}}^{-\mathrm{i}\theta }$. … ►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. …

4: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

§33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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§33.5(iv) Large $\mathrm{\ell }$

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5: 33.12 Asymptotic Expansions for Large $\eta$

§33.12 Asymptotic Expansions for Large $\eta$

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6: 10.21 Zeros

… ►With $a=(t+\frac{1}{2}\nu -\frac{1}{4})\pi$, the right-hand side is the asymptotic expansion of ${\rho }_{\nu }(t)$ for large $t$. …

7: 19.36 Methods of Computation

… ►Descending Gauss transformations of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). This method loses significant figures in $\rho$ if ${\alpha }^2$ and $k^2$ are nearly equal unless they are given exact values—as they can be for tables. …

8: 2.8 Differential Equations with a Parameter

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

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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

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9: 10.72 Mathematical Applications

… ►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 $z\to z_0$. …

10: 33.9 Expansions in Series of Bessel Functions

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§33.9(i) Spherical Bessel Functions

… ►The series (33.9.1) converges for all finite values of $\eta$ and $\rho$. ►

§33.9(ii) Bessel Functions and Modified Bessel Functions

… ►With $t=2\left|\eta \right|\rho$, … ►Next, as $\eta \to +\mathrm{\infty }$ with $\rho$ ($>0$) fixed, …