of revolution

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1: 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. …

2: 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. … …

3: 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)). …

4: 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. …

5: 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. …

6: 1.6 Vectors and Vector-Valued Functions

… ►For a surface of revolution, $y=f(x)$, $x\in [a,b]$, about the $x$-axis, …

7: Bibliography M

… ►

Die Laméschen Wellenfunktionen des Drehellipsoids. Forschungsbericht No. 1952 ZWB (German).

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Notes:

English translation: Lamé wave functions of the ellipsoid of revolution appeared as NACA Technical Memorandum No. 1224, 1949

External Links:

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Referenced by:

§30.9(iii).

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