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1: 5.19 Mathematical Applications

… ►The volume $V$ and surface area $S$ of the $n$-dimensional sphere of radius $r$ are given by …

2: 2.11 Remainder Terms; Stokes Phenomenon

… ►

2.11.8 $$n=\rho -p+\alpha ,$$

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Symbols:

$\rho$: radius

Referenced by:

§2.11(iii), §2.11(iii)

Permalink:

http://dlmf.nist.gov/2.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

… ►

2.11.12 $$F_{n+p}\left(z\right)=\frac{{\mathrm{e}}^{-z}}{2\pi }{\int }_0^{\mathrm{\infty }}\exp \left(-\rho \left(t{\mathrm{e}}^{\mathrm{i}\theta }-\ln t\right)\right)\frac{t^{\alpha -1}}{1+t}dt.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\ln z$: principal branch of logarithm function, $F_p\left(z\right)$: terminant function, $\rho$: radius and $\theta$: radius

Permalink:

http://dlmf.nist.gov/2.11.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

… ►

2.11.14 $$a_2(\theta ,\alpha )=\frac{1}{12}(6{\alpha }^2-6\alpha +1)-\frac{\alpha }{1+{\mathrm{e}}^{\mathrm{i}\theta }}+\frac{1}{{(1+{\mathrm{e}}^{\mathrm{i}\theta })}^2}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\theta$: radius and $a_{2s}(\theta ,\alpha )$: coefficients

Referenced by:

§2.11(iii)

Permalink:

http://dlmf.nist.gov/2.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

… ►

2.11.16 $$c(\theta )=\sqrt{2(1+{\mathrm{e}}^{\mathrm{i}\theta }+\mathrm{i}(\theta -\pi ))},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\theta$: radius and $c(\theta )$

Permalink:

http://dlmf.nist.gov/2.11.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

… ►

2.11.18 $$h_0(\theta ,\alpha )=\frac{{\mathrm{e}}^{\mathrm{i}\alpha (\pi -\theta )}}{1+{\mathrm{e}}^{-\mathrm{i}\theta }}-\frac{\mathrm{i}}{c(\theta )}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\theta$: radius, $c(\theta )$ and $h_{2s}(\theta ,\alpha )$: functions

Permalink:

http://dlmf.nist.gov/2.11.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

…

3: 1.5 Calculus of Two or More Variables

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

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

… ►

1.5.41 $$\frac{\partial (x,y,z)}{\partial (\rho ,\theta ,\varphi )}={\rho }^2\sin \theta \text{(spherical coordinates)}.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\varphi$: longitude, $\rho$: radius and $\theta$: azimuth

Permalink:

http://dlmf.nist.gov/1.5.E41

Encodings:

pMML, png, TeX

See also:

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

…

4: 22.18 Mathematical Applications

… ►In polar coordinates, $x=r\cos \varphi$, $y=r\sin \varphi$, the lemniscate is given by $r^2=\cos \left(2\varphi \right)$, $0\le \varphi \le 2\pi$. … ►

22.18.4 $$l(r)=(1/\sqrt{2})\mathrm{arccn}(r,1/\sqrt{2}).$$

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Symbols:

$\mathrm{arccn}(x,k)$: inverse Jacobian elliptic function, $r$: radius and $l(r)$: arc length

Permalink:

http://dlmf.nist.gov/22.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.18(i), §22.18(i), §22.18 and Ch.22

… ►

22.18.5 $$r=\mathrm{cn}(\sqrt{2}l,1/\sqrt{2}),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $r$: radius and $l(r)$: arc length

Permalink:

http://dlmf.nist.gov/22.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.18(i), §22.18(i), §22.18 and Ch.22

…

5: 1.9 Calculus of a Complex Variable

… ►where ►

1.9.4 $$r={(x^2+y^2)}^{1/2},$$

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Symbols:

$r$: radius

A&S Ref:

3.7.3

Permalink:

http://dlmf.nist.gov/1.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

… ►If $h(w)$ is continuous on $\left|w\right|=R$, then with $z=r{\mathrm{e}}^{\mathrm{i}\theta }$ … ►

§1.9(vi) Power Series

… ►The circle $\left|z-z_0\right|=R$ is called the circle of convergence of the series, and $R$ is the radius of convergence. …

6: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►

$ϵ=E/(Z_1^2{Z}_2^{2}m{c}^2{\alpha }^2/2).$

►For $Z_1{Z}_2=-1$ and $m=m_{\mathrm{e}}$, the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, $a_0=\mathrm{\hslash }/(m_{\mathrm{e}}c\alpha )$, and to a multiple of the Rydberg constant, …

7: 3.4 Differentiation

… ►Taking $C$ to be a circle of radius $r$ centered at $x_0$, we obtain ►

3.4.18 $$\frac{1}{k!}{f}^{(k)}(x_0)=\frac{1}{2\pi r^k}{\int }_0^{2\pi }f(x_0+r{\mathrm{e}}^{\mathrm{i}\theta }){\mathrm{e}}^{-\mathrm{i}k\theta }d\theta .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral and $r$: radius

Referenced by:

§3.4(ii)

Permalink:

http://dlmf.nist.gov/3.4.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(ii), §3.4 and Ch.3

… ►

3.4.19 $$\frac{1}{k!}=\frac{1}{2\pi r^k}{\int }_0^{2\pi }{\mathrm{e}}^{r\cos \theta }\cos \left(r\sin \theta -k\theta \right)d\theta .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function and $r$: radius

Permalink:

http://dlmf.nist.gov/3.4.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(ii), §3.4(ii), §3.4 and Ch.3

…

8: 5.20 Physical Applications

… ►For $n$ charges free to move on a circular wire of radius $1$, …

9: 19.34 Mutual Inductance of Coaxial Circles

… ►The mutual inductance $M$ of two coaxial circles of radius $a$ and $b$ with centers at a distance $h$ apart is given in cgs units by …

10: 22.10 Maclaurin Series

… ►The radius of convergence is the distance to the origin from the nearest pole in the complex $k$-plane in the case of (22.10.4)–(22.10.6), or complex $k^{\prime }$-plane in the case of (22.10.7)–(22.10.9); see §22.17. …