# ellipsoidal

(0.000 seconds)

## 1—10 of 14 matching pages

##### 1: 29.11 Lamé Wave Equation
###### §29.11 Lamé Wave Equation
The Lamé (or ellipsoidal) wave equation is given by …
##### 3: 29.18 Mathematical Applications
###### §29.18(ii) Ellipsoidal Coordinates
The wave equation (29.18.1), when transformed to ellipsoidal coordinates $\alpha,\beta,\gamma$: …
29.18.10 $u(\alpha,\beta,\gamma)=u_{1}(\alpha)u_{2}(\beta)u_{3}(\gamma),$
##### 4: 23.21 Physical Applications
###### §23.21(iii) Ellipsoidal Coordinates
Ellipsoidal coordinates $(\xi,\eta,\zeta)$ may be defined as the three roots $\rho$ of the equation …The Laplacian operator $\nabla^{2}$1.5(ii)) is given by
23.21.2 $(\eta-\zeta)(\zeta-\xi)(\xi-\eta)\nabla^{2}=(\zeta-\eta)f(\xi)f^{\prime}(\xi)% \frac{\partial}{\partial\xi}+(\xi-\zeta)f(\eta)f^{\prime}(\eta)\frac{\partial}% {\partial\eta}+(\eta-\xi)f(\zeta)f^{\prime}(\zeta)\frac{\partial}{\partial% \zeta},$
##### 5: 19.15 Advantages of Symmetry
These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)). For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). For example, the computation of depolarization factors for solid ellipsoids is simplified considerably; compare (19.33.7) with Cronemeyer (1991). …
##### 6: 30.14 Wave Equation in Oblate Spheroidal Coordinates
The coordinate surfaces $\xi=\mbox{const}.$ are oblate ellipsoids of revolution with focal circle $z=0$, $x^{2}+y^{2}=c^{2}$. …
###### §30.14(v) The Interior Dirichlet Problem for Oblate Ellipsoids
Equation (30.13.7) for $\xi\leq\xi_{0}$ together with the boundary condition $w=0$ on the ellipsoid given by $\xi=\xi_{0}$, poses an eigenvalue problem with $\kappa^{2}$ as spectral parameter. …
##### 7: 30.13 Wave Equation in Prolate Spheroidal Coordinates
The coordinate surfaces $\xi=\mbox{const}.$ are prolate ellipsoids of revolution with foci at $x=y=0$, $z=\pm c$. …
###### §30.13(v) The Interior Dirichlet Problem for Prolate Ellipsoids
Equation (30.13.7) for $\xi\leq\xi_{0}$, and subject to the boundary condition $w=0$ on the ellipsoid given by $\xi=\xi_{0}$, poses an eigenvalue problem with $\kappa^{2}$ as spectral parameter. …For the Dirichlet boundary-value problem of the region $\xi_{1}\leq\xi\leq\xi_{2}$ between two ellipsoids, the eigenvalues are determined from …
##### 8: 19.37 Tables
Here $\sigma^{2}=\tfrac{2}{3}((\ln a)^{2}+(\ln b)^{2}+(\ln c)^{2})$, $\cos\left(3\gamma\right)=(4/\sigma^{3})(\ln a)(\ln b)(\ln c)$, and $a,b,c$ are semiaxes of an ellipsoid with the same volume as the unit sphere. …
##### 9: Bibliography H
• F. E. Harris (2002) Analytic evaluation of two-center STO electron repulsion integrals via ellipsoidal expansion. Internat. J. Quantum Chem. 88 (6), pp. 701–734.
• E. W. Hobson (1931) The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, London-New York.
• ##### 10: Bibliography M
• J. Meixner (1944) Die Laméschen Wellenfunktionen des Drehellipsoids. Forschungsbericht No. 1952 ZWB (German).
• H. J. W. Müller (1966a) Asymptotic expansions of ellipsoidal wave functions and their characteristic numbers. Math. Nachr. 31, pp. 89–101.
• H. J. W. Müller (1966b) Asymptotic expansions of ellipsoidal wave functions in terms of Hermite functions. Math. Nachr. 32, pp. 49–62.
• H. J. W. Müller (1966c) On asymptotic expansions of ellipsoidal wave functions. Math. Nachr. 32, pp. 157–172.