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

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§13.28(i) Exact Solutions of the Wave Equation

►The reduced wave equation ${\nabla }^{2}w=k^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi \eta }\cos \varphi$, $y=2\sqrt{\xi \eta }\sin \varphi$, $z=\xi -\eta$, can be solved via separation of variables $w=f_1(\xi )f_2(\eta ){\mathrm{e}}^{\mathrm{i}p\varphi }$, where …

3: Sidebar 5.SB1: Gamma & Digamma Phase Plots

… ►In the upper half of the image, the poles of $\mathrm{\Gamma }\left(z\right)$ 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. …

4: 28.31 Equations of Whittaker–Hill and Ince

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§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 $p_1\ne p_2$ or $m_1\ne m_2$ or both, given by … ►

Asymptotic Behavior

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5: Bibliography U

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Integral equations for paraboloidal wave functions. I. Quart. J. Math. Oxford Ser. (2) 15, pp. 309–315.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.31(iii), §28.31(iii).

Encodings:

BibTeX

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Integral equations for the paraboloidal wave functions. II. Quart. J. Math. Oxford Ser. (2) 16, pp. 257–262.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.31(iii), §28.31(iii).

Encodings:

BibTeX

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6: 28.32 Mathematical Applications

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§28.32(ii) Paraboloidal Coordinates

►The general paraboloidal coordinate system is linked with Cartesian coordinates via …

7: 14.31 Other Applications

… ►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ 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 $\xi =\text{const}.$ are oblate ellipsoids of revolution with focal circle $z=0$, $x^2+y^2=c^2$. The coordinate surfaces $\eta =\text{const}.$ 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 $\xi =\text{const}.$ are prolate ellipsoids of revolution with foci at $x=y=0$, $z=\pm c$. The coordinate surfaces $\eta =\text{const}.$ 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, …