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## 1—10 of 69 matching pages

##### 1: 12.8 Recurrence Relations and Derivatives
12.8.5 $zV\left(a,z\right)-V\left(a+1,z\right)+(a-\tfrac{1}{2})V\left(a-1,z\right)=0,$
12.8.6 $V'\left(a,z\right)-\tfrac{1}{2}zV\left(a,z\right)-(a-\tfrac{1}{2})V\left(a-1,z% \right)=0,$
12.8.7 $V'\left(a,z\right)+\tfrac{1}{2}zV\left(a,z\right)-V\left(a+1,z\right)=0,$
12.8.8 $2V'\left(a,z\right)-V\left(a+1,z\right)-(a-\tfrac{1}{2})V\left(a-1,z\right)=0.$
##### 2: 7.19 Voigt Functions Figure 7.19.2: Voigt function 𝖵 ⁡ ( x , t ) , t = 0.1 , 2.5 , 5 , 10 . Magnify
$\lim_{t\to 0}\mathsf{V}\left(x,t\right)=\frac{x}{1+x^{2}}.$
$\mathsf{V}\left(-x,t\right)=-\mathsf{V}\left(x,t\right).$
$-1\leq\mathsf{V}\left(x,t\right)\leq 1.$
7.19.9 $\mathsf{U}\left(x,t\right)=1-x\mathsf{V}\left(x,t\right)-2t\frac{\partial% \mathsf{V}\left(x,t\right)}{\partial x}.$
##### 3: 5.19 Mathematical Applications
The volume $V$ and surface area $S$ of the $n$-dimensional sphere of radius $r$ are given by
$V=\frac{\pi^{\frac{1}{2}n}r^{n}}{\Gamma\left(\frac{1}{2}n+1\right)},$
$S=\frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\Gamma\left(\frac{1}{2}n\right)}=\frac{n}{% r}V.$
##### 4: 18.39 Physical Applications
Consider, for example, the one-dimensional form of this equation for a particle of mass $m$ with potential energy $V(x)$:
18.39.1 $\left(\frac{-\hbar^{2}}{2m}\frac{{\partial}^{2}}{{\partial x}^{2}}+V(x)\right)% \psi(x,t)=i\hbar\frac{\partial}{\partial t}\psi(x,t),$
##### 5: 12.3 Graphics Figure 12.3.2: V ⁡ ( a , x ) , a = 0. … Magnify Figure 12.3.4: V ⁡ ( a , x ) , a = − 0.5 , − 2 , − 3.5 , − 5 . Magnify Figure 12.3.8: V ⁡ ( a , x ) , − 2.5 ≤ a ≤ 2.5 , − 2.5 ≤ x ≤ 2.5 . Magnify 3D Help
##### 6: 36.13 Kelvin’s Ship-Wave Pattern
A ship moving with constant speed $V$ on deep water generates a surface gravity wave. …
36.13.2 $\rho=\ifrac{gr}{V^{2}}.$
36.13.6 $\omega(\mathbf{k})=\sqrt{gk}+\mathbf{V}\cdot\mathbf{k}.$
Here $k=|\mathbf{k}|$, and $\mathbf{V}$ is the ship velocity (so that $\mathrm{V}=|\mathbf{V}|$). …
##### 7: 9.1 Special Notation
Other notations that have been used are as follows: $\operatorname{Ai}\left(-x\right)$ and $\operatorname{Bi}\left(-x\right)$ for $\operatorname{Ai}\left(x\right)$ and $\operatorname{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\operatorname{Ai}\left(x\right)$ and $\operatorname{Bi}\left(x\right)$); $U(x)=\sqrt{\pi}\operatorname{Bi}\left(x\right)$, $V(x)=\sqrt{\pi}\operatorname{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-\ifrac{1}{3}}\pi\operatorname{Ai}\left(-3^{-\ifrac{1}{3}}x\right)$ (Szegő (1967, §1.81)); $e_{0}(x)=\pi\operatorname{Hi}(-x)$, $\widetilde{e}_{0}(x)=-\pi\operatorname{Gi}(-x)$ (Tumarkin (1959)).
##### 8: 13.28 Physical Applications
$f_{1}(\xi)=\xi^{-\frac{1}{2}}V_{\kappa,\frac{1}{2}p}^{(1)}(2\mathrm{i}k\xi)$ ,
$f_{2}(\eta)=\eta^{-\frac{1}{2}}V_{\kappa,\frac{1}{2}p}^{(2)}(-2\mathrm{i}k\eta)$ ,
and $V^{(j)}_{\kappa,\mu}(z)$, $j=1,2$, denotes any pair of solutions of Whittaker’s equation (13.14.1). …
##### 9: 19.33 Triaxial Ellipsoids
The surface area of an ellipsoid with semiaxes $a,b,c$, and volume $V=4\pi abc/3$ is given by The potential is …and the electric capacity $C=Q/V(0)$ is given by … Let a homogeneous magnetic ellipsoid with semiaxes $a,b,c$, volume $V=4\pi abc/3$, and susceptibility $\chi$ be placed in a previously uniform magnetic field $H$ parallel to the principal axis with semiaxis $c$. …
##### 10: 28.32 Mathematical Applications
then becomes
28.32.3 $\frac{{\partial}^{2}V}{{\partial\xi}^{2}}+\frac{{\partial}^{2}V}{{\partial\eta% }^{2}}+\frac{1}{2}c^{2}k^{2}(\cosh\left(2\xi\right)-\cos\left(2\eta\right))V=0.$
The separated solutions $V(\xi,\eta)=v(\xi)w(\eta)$ can be obtained from the modified Mathieu’s equation (28.20.1) for $v$ and from Mathieu’s equation (28.2.1) for $w$, where $a$ is the separation constant and $q=\tfrac{1}{4}c^{2}k^{2}$. …
28.32.8 $\nabla^{2}V+k^{2}V=0$