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## 1—10 of 16 matching pages

##### 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 $n=\rho-p+\alpha,$
2.11.12 $F_{n+p}\left(z\right)=\frac{e^{-z}}{2\pi}\int_{0}^{\infty}\exp\left(-\rho\left% (te^{i\theta}-\ln t\right)\right)\frac{t^{\alpha-1}}{1+t}\,\mathrm{d}t.$
2.11.14 $a_{2}(\theta,\alpha)=\frac{1}{12}(6\alpha^{2}-6\alpha+1)-\frac{\alpha}{1+e^{i% \theta}}+\frac{1}{(1+e^{i\theta})^{2}}.$
2.11.16 $c(\theta)=\sqrt{2(1+e^{i\theta}+i(\theta-\pi))},$
2.11.18 $h_{0}(\theta,\alpha)=\frac{e^{i\alpha(\pi-\theta)}}{1+e^{-i\theta}}-\frac{i}{c% (\theta)}.$
##### 3: 1.5 Calculus of Two or More Variables
With $0\leq r<\infty$, $0\leq\phi\leq 2\pi$, … With $0\leq r<\infty$, $0\leq\phi\leq 2\pi$, $-\infty, … With $0\leq\rho<\infty$, $0\leq\phi\leq 2\pi$, $0\leq\theta\leq\pi$, …
1.5.39 $\frac{\partial(x,y)}{\partial(r,\phi)}=r\quad\text{(polar coordinates)}.$
1.5.41 $\frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)}=\rho^{2}\sin\theta\quad% \text{(spherical coordinates)}.$
##### 4: 22.18 Mathematical Applications
In polar coordinates, $x=r\cos\phi$, $y=r\sin\phi$, the lemniscate is given by $r^{2}=\cos\left(2\phi\right)$, $0\leq\phi\leq 2\pi$. …
##### 5: 1.9 Calculus of a Complex Variable
where
1.9.4 $r=(x^{2}+y^{2})^{1/2},$
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
$\epsilon=E/(Z_{1}^{2}Z_{2}^{2}m{c}^{2}{\alpha}^{2}/2).$
For $Z_{1}Z_{2}=-1$ and $m=m_{e}$, the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, $a_{0}=\hbar/(m_{e}c\alpha)$, 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 $\frac{1}{k!}\,f^{(k)}(x_{0})=\frac{1}{2\pi r^{k}}\int_{0}^{2\pi}f(x_{0}+re^{i% \theta})e^{-ik\theta}\,\mathrm{d}\theta.$
3.4.19 $\frac{1}{k!}=\frac{1}{2\pi r^{k}}\int_{0}^{2\pi}e^{r\cos\theta}\cos\left(r\sin% \theta-k\theta\right)\,\mathrm{d}\theta.$
##### 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. …