# sphero-conal coordinates

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## 11—20 of 40 matching pages

##### 11: 14.19 Toroidal (or Ring) Functions
###### §14.19(i) Introduction
This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates $(\eta,\theta,\phi)$, which are related to Cartesian coordinates $(x,y,z)$ by …
##### 12: 31.17 Physical Applications
Introduce elliptic coordinates $z_{1}$ and $z_{2}$ on $S_{2}$. Then
31.17.2 $\frac{x_{s}^{2}}{z_{k}}+\frac{x_{t}^{2}}{z_{k}-1}+\frac{x_{u}^{2}}{z_{k}-a}=0,$ $k=1,2$,
##### 13: Howard S. Cohl
Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …
##### 14: 36.5 Stokes Sets
For $z\neq 0$, the Stokes set is expressed in terms of scaled coordinates
36.5.7 $X=\dfrac{9}{20}+20u^{4}-\frac{Y^{2}}{20u^{2}}+6u^{2}\operatorname{sign}\left(z% \right),$
36.5.10 $160u^{6}+40u^{4}=Y^{2}.$
With coordinates
36.5.17 $Y_{\mathrm{S}}(X)=Y(u,|X|),$
##### 15: 1.5 Calculus of Two or More Variables
###### Spherical Coordinates
For applications and other coordinate systems see §§12.17, 14.19(i), 14.30(iv), 28.32, 29.18, 30.13, 30.14. …
19.26.3 $\sqrt{z}=\frac{\xi\zeta^{\prime}+\eta^{\prime}\zeta-\xi\eta^{\prime}}{\sqrt{% \xi\eta\zeta^{\prime}}+\sqrt{\xi^{\prime}\eta^{\prime}\zeta}},$
where …
19.26.14 $(p-y)R_{C}\left(x,p\right)+(q-y)R_{C}\left(x,q\right)=(\eta-\xi)R_{C}\left(\xi% ,\eta\right),$ $x\geq 0$, $y\geq 0$; $p,q\in\mathbb{R}\setminus\{0\}$,
19.26.24 $z=(\xi\zeta+\eta\zeta-\xi\eta)^{2}/(4\xi\eta\zeta),$ $(\xi,\eta,\zeta)=(x+\lambda,y+\lambda,z+\lambda)$,
##### 18: 22.19 Physical Applications
where $V(x)$ is the potential energy, and $x(t)$ is the coordinate as a function of time $t$. …
22.19.5 $V(x)=\pm\tfrac{1}{2}x^{2}\pm\tfrac{1}{4}\beta x^{4}$
22.19.6 $x(t)=a\operatorname{cn}\left(t\sqrt{1+2\eta},k\right).$
22.19.8 $x(t)=a\operatorname{dn}\left(t\sqrt{\eta},k\right).$
##### 19: 30.2 Differential Equations
In applications involving prolate spheroidal coordinates $\gamma^{2}$ is positive, in applications involving oblate spheroidal coordinates $\gamma^{2}$ is negative; see §§30.13, 30.14. …
##### 20: 23.20 Mathematical Applications
or equivalently, on replacing $x$ by $x/z$ and $y$ by $y/z$ (projective coordinates), into the form
23.20.2 $C:y^{2}z=x^{3}+axz^{2}+bz^{3},$
Let $T$ denote the set of points on $C$ that are of finite order (that is, those points $P$ for which there exists a positive integer $n$ with $nP=o$), and let $I,K$ be the sets of points with integer and rational coordinates, respectively. …