angle

SearchAdvancedHelp

(0.000 seconds)

21—27 of 27 matching pages

21: 3.2 Linear Algebra

… ►Because $\left|{𝐲}^{\mathrm{T}}𝐱\right|=\left|\cos \theta \right|$, where $\theta$ is the angle between ${𝐲}^{\mathrm{T}}$ and $𝐱$ we always have $\kappa (\lambda )\ge 1$. …

22: 10.41 Asymptotic Expansions for Large Order

… ►The curve $E_{1}B{E}_2$ in the $z$-plane is the upper boundary of the domain $𝐊$ depicted in Figure 10.20.3 and rotated through an angle $-\frac{1}{2}\pi$. …

23: 22.19 Physical Applications

… ►The angle $\alpha =\pi$ is a separatrix, separating oscillatory and unbounded motion. …

24: 14.15 Uniform Asymptotic Approximations

… ►

14.15.11 $${𝖯}_{\nu }^{-\mu }\left(\cos \theta \right)=\frac{1}{{\nu }^{\mu }}{\left(\frac{\theta }{\sin \theta }\right)}^{1/2}\left(J_{\mu }\left(\left(\nu +\frac{1}{2}\right)\theta \right)+O\left(\frac{1}{\nu }\right)\mathrm{env}{J}_{\mu }\left(\left(\nu +\frac{1}{2}\right)\theta \right)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, : Ferrers function of the first kind, $\cos z$: cosine function, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$ , $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and $\theta$: angle

Permalink:

http://dlmf.nist.gov/14.15.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(iii), §14.15 and Ch.14

►

14.15.12 $${𝖰}_{\nu }^{-\mu }\left(\cos \theta \right)=-\frac{\pi }{2{\nu }^{\mu }}{\left(\frac{\theta }{\sin \theta }\right)}^{1/2}\left(Y_{\mu }\left(\left(\nu +\frac{1}{2}\right)\theta \right)+O\left(\frac{1}{\nu }\right)\mathrm{env}{Y}_{\mu }\left(\left(\nu +\frac{1}{2}\right)\theta \right)\right),$$

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and $\theta$: angle

Permalink:

http://dlmf.nist.gov/14.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(iii), §14.15 and Ch.14

…

25: 14.30 Spherical and Spheroidal Harmonics

… ►With $l$ and $m$ integers such that $|m|\le l$, and $\theta$ and $\varphi$ angles such that $0\le \theta \le \pi$, $0\le \varphi \le 2\pi$, …

26: 2.11 Remainder Terms; Stokes Phenomenon

… ►Following §2.4(iv), we rotate the integration path through an angle $-\theta$, which is valid by analytic continuation when . …

27: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►This is accomplished by the variable change $x\to x{\mathrm{e}}^{\mathrm{i}\theta }$, in $ℒ$, which rotates the continuous spectrum ${𝝈}_c\to {𝝈}_c{\mathrm{e}}^{-2\mathrm{i}\theta }$ and the branch cut of (1.18.66) into the lower half complex plain by the angle $-2\theta$, with respect to the unmoved branch point at $\lambda =0$; thus, providing access to resonances on the higher Riemann sheet should $\theta$ be large enough to expose them. …