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11: 4.29 Graphics

… ►The conformal mapping $w=\sinh z$ is obtainable from Figure 4.15.7 by rotating both the $w$-plane and the $z$-plane through an angle $\frac{1}{2}\pi$, compare (4.28.8). …

12: 7.20 Mathematical Applications

… ►Furthermore, because $dy/dx=\tan \left(\frac{1}{2}\pi t^2\right)$, the angle between the $x$-axis and the tangent to the spiral at $P(t)$ is given by $\frac{1}{2}\pi t^2$. …

13: 22.18 Mathematical Applications

… ►In polar coordinates, $x=r\cos \varphi$, $y=r\sin \varphi$, the lemniscate is given by $r^2=\cos \left(2\varphi \right)$, $0\le \varphi \le 2\pi$. … ►Discussion of parametrization of the angles of spherical trigonometry in terms of Jacobian elliptic functions is given in Greenhill (1959, p. 131) and Lawden (1989, §4.4). …

14: 2.4 Contour Integrals

… ►in which $a$ is finite, $b$ is finite or infinite, and $\omega$ is the angle of slope of $𝒫$ at $a$, that is, $lim(\mathrm{ph}\left(t-a\right))$ as $t\to a$ along $𝒫$. … ►

$z$ ranges along a ray or over an annular sector ${\theta }_1\le \theta \le {\theta }_2$, $|z|\ge Z$, where $\theta =\mathrm{ph}z$, , and $Z>0$. $I(z)$ converges at $b$ absolutely and uniformly with respect to $z$.

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2.4.13 $$|\theta +\mu \omega +\mathrm{ph}{p}_0|\le \frac{1}{2}\pi .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\omega$: angle, $p_s$: coefficients, ${\theta }_j$: angles and $\mu$: positive constant

Permalink:

http://dlmf.nist.gov/2.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(iii), §2.4 and Ch.2

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15: 36.13 Kelvin’s Ship-Wave Pattern

… ►This is the angle of the familiar V-shaped wake. …

16: 6.12 Asymptotic Expansions

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17: 1.5 Calculus of Two or More Variables

… ►With , $0\le \varphi \le 2\pi$, … ►With , $0\le \varphi \le 2\pi$, , … ►

1.5.39 $$\frac{\partial (x,y)}{\partial (r,\varphi )}=r\text{(polar coordinates)}.$$

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Symbols:

$(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $r$: radius and $\varphi$: angle

Permalink:

http://dlmf.nist.gov/1.5.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(vi), §1.5(vi), §1.5 and Ch.1

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18: 7.12 Asymptotic Expansions

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19: Bibliography K

… ►

A Fortran computer program for calculating the prolate and oblate angle functions of the first kind and their first and second derivatives. NRL Report No. 7161 Naval Res. Lab.  Washingtion, D.C..

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Referenced by:

§30.18(iii).

Encodings:

BibTeX

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20: 2.1 Definitions and Elementary Properties

… ►For (2.1.14) $𝐗$ can be the positive real axis or any unbounded sector in $ℂ$ of finite angle. …