# paraboloid of revolution

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##### 1: 12.17 Physical Applications
In a similar manner coordinates of the paraboloid of revolution transform the Helmholtz equation into equations related to the differential equations considered in this chapter. … …
##### 2: 13.28 Physical Applications
###### §13.28(i) Exact Solutions of the Wave Equation
The reduced wave equation $\nabla^{2}w=k^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi\eta}\cos\phi$, $y=2\sqrt{\xi\eta}\sin\phi$, $z=\xi-\eta$, can be solved via separation of variables $w=f_{1}(\xi)f_{2}(\eta)e^{\mathrm{i}p\phi}$, where …
##### 3: 28.31 Equations of Whittaker–Hill and Ince
###### §28.31(iii) Paraboloidal Wave Functions
With (28.31.10) and (28.31.11), …are called paraboloidal wave functions. … More important are the double orthogonality relations for $p_{1}\neq p_{2}$ or $m_{1}\neq m_{2}$ or both, given by …
##### 4: Sidebar 5.SB1: Gamma & Digamma Phase Plots
In the upper half of the image, the poles of $\Gamma\left(z\right)$ are clearly visible at negative integer values of $z$: the phase changes by $2\pi$ around each pole, showing a full revolution of the color wheel. …
##### 5: Bibliography U
• K. M. Urwin (1964) Integral equations for paraboloidal wave functions. I. Quart. J. Math. Oxford Ser. (2) 15, pp. 309–315.
• K. M. Urwin (1965) Integral equations for the paraboloidal wave functions. II. Quart. J. Math. Oxford Ser. (2) 16, pp. 257–262.
• ##### 6: 28.32 Mathematical Applications
###### §28.32(ii) Paraboloidal Coordinates
The general paraboloidal coordinate system is linked with Cartesian coordinates via …
##### 7: 14.31 Other Applications
The conical functions $\mathsf{P}^{m}_{-\frac{1}{2}+i\tau}\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). …
##### 8: 30.14 Wave Equation in Oblate Spheroidal Coordinates
The coordinate surfaces $\xi=\mbox{const}.$ are oblate ellipsoids of revolution with focal circle $z=0$, $x^{2}+y^{2}=c^{2}$. The coordinate surfaces $\eta=\mbox{const}.$ are halves of one-sheeted hyperboloids of revolution with the same focal circle. …
##### 9: 30.13 Wave Equation in Prolate Spheroidal Coordinates
The coordinate surfaces $\xi=\mbox{const}.$ are prolate ellipsoids of revolution with foci at $x=y=0$, $z=\pm c$. The coordinate surfaces $\eta=\mbox{const}.$ are sheets of two-sheeted hyperboloids of revolution with the same foci. …
##### 10: 1.6 Vectors and Vector-Valued Functions
For a surface of revolution, $y=f(x)$, $x\in[a,b]$, about the $x$-axis, …