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| [[Image:Toroidal coordinates.png|thumb|350px|right|Illustration of toroidal coordinates, which are obtained by rotating a two-dimensional [[bipolar coordinates|bipolar coordinate system]] about the axis separating its two foci. The foci are located at a distance 1 from the vertical ''z''-axis. The red sphere is the σ = 30° isosurface, the blue torus is the τ = 0.5 isosurface, and the yellow half-plane is the φ = 60° isosurface. The green half-plane marks the ''x''-''z'' plane, from which φ is measured. The black point is located at the intersection of the red, blue and yellow isosurfaces, at Cartesian coordinates roughly (0.996, −1.725, 1.911).]]
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| '''Toroidal coordinates''' are a three-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] that results from rotating the two-dimensional [[bipolar coordinates|bipolar coordinate system]] about the axis that separates its two foci. Thus, the two [[Focus (geometry)|foci]] <math>F_1</math> and <math>F_2</math> in [[bipolar coordinates]] become a ring of radius <math>a</math> in the <math>xy</math> plane of the toroidal coordinate system; the <math>z</math>-axis is the axis of rotation. The focal ring is also known as the reference circle.
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| ==Definition==
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| The most common definition of toroidal coordinates <math>(\sigma, \tau, \phi)</math> is
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| :<math>
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| x = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma} \cos \phi
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| </math>
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| :<math>
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| y = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma} \sin \phi
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| </math>
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| :<math>
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| z = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma}
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| </math>
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| where the <math>\sigma</math> coordinate of a point <math>P</math> equals the angle <math>F_{1} P F_{2}</math> and the <math>\tau</math> coordinate equals the [[natural logarithm]] of the ratio of the distances <math>d_{1}</math> and <math>d_{2}</math> to opposite sides of the focal ring
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| :<math>
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| \tau = \ln \frac{d_{1}}{d_{2}}.
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| </math>
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| The coordinate ranges are <math>-\pi<\sigma\le\pi</math> and <math>\tau\ge 0</math> and <math>0\le\phi < 2\pi.</math>
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| ===Coordinate surfaces===
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| [[Image:Apollonian circles.svg||thumb|right|350px|Rotating this two-dimensional [[bipolar coordinates|bipolar coordinate system]] about the vertical axis produces the three-dimensional toroidal coordinate system above. A circle on the vertical axis becomes the red [[sphere]], whereas a circle on the horizontal axis becomes the blue [[torus]].]]
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| Surfaces of constant <math>\sigma</math> correspond to spheres of different radii
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| :<math>
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| \left( x^{2} + y^{2} \right) +
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| \left( z - a \cot \sigma \right)^{2} = \frac{a^{2}}{\sin^{2} \sigma}
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| </math>
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| that all pass through the focal ring but are not concentric. The surfaces of constant <math>\tau</math> are non-intersecting tori of different radii
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| :<math>
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| z^{2} +
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| \left( \sqrt{x^{2} + y^{2}} - a \coth \tau \right)^{2} = \frac{a^{2}}{\sinh^{2} \tau}
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| </math>
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| that surround the focal ring. The centers of the constant-<math>\sigma</math> spheres lie along the <math>z</math>-axis, whereas the constant-<math>\tau</math> tori are centered in the <math>xy</math> plane.
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| ===Inverse transformation===
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| The (σ, τ, φ) coordinates may be calculated from the Cartesian coordinates (''x'', ''y'', ''z'') as follows. The azimuthal angle φ is given by the formula
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| :<math>
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| \tan \phi = \frac{y}{x}
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| </math>
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| The cylindrical radius ρ of the point P is given by
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| :<math>
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| \rho^{2} = x^{2} + y^{2}
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| </math>
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| and its distances to the foci in the plane defined by φ is given by
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| :<math>
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| d_{1}^{2} = (\rho + a)^{2} + z^{2}
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| </math>
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| :<math>
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| d_{2}^{2} = (\rho - a)^{2} + z^{2}
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| </math>
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| [[Image:Bipolar coordinates.png|thumb|right|350px|Geometric interpretation of the coordinates σ and τ of a point '''P'''. Observed in the plane of constant azimuthal angle φ, toroidal coordinates are equivalent to [[bipolar coordinates]]. The angle σ is formed by the two foci in this plane and '''P''', whereas τ is the logarithm of the ratio of distances to the foci. The corresponding circles of constant σ and τ are shown in red and blue, respectively, and meet at right angles (magenta box); they are orthogonal.]]
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| The coordinate τ equals the [[natural logarithm]] of the focal distances
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| :<math>
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| \tau = \ln \frac{d_{1}}{d_{2}}
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| </math>
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| whereas the coordinate σ equals the angle between the rays to the foci, which may be determined from the [[law of cosines]]
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| :<math>
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| \cos \sigma = -\frac{4a^{2} - d_{1}^{2} - d_{2}^{2}}{2 d_{1} d_{2}}
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| </math>
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| where the sign of σ is determined by whether the coordinate surface sphere is above or below the ''x''-''y'' plane.
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| ===Scale factors===
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| The scale factors for the toroidal coordinates <math>\sigma</math> and <math>\tau</math> are equal
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| :<math>
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| h_\sigma = h_\tau = \frac{a}{\cosh \tau - \cos\sigma}
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| </math>
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| whereas the azimuthal scale factor equals
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| :<math>
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| h_\phi = \frac{a \sinh \tau}{\cosh \tau - \cos\sigma}
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| </math>
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| Thus, the infinitesimal volume element equals
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| :<math>
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| dV = \frac{a^3 \sinh \tau}{\left( \cosh \tau - \cos\sigma \right)^3} \, d\sigma \, d\tau \, d\phi
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| </math>
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| and the Laplacian is given by
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| :<math>
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| \begin{align}
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| \nabla^2 \Phi =
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| \frac{\left( \cosh \tau - \cos\sigma \right)^{3}}{a^{2}\sinh \tau}
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| & \left[
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| \sinh \tau
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| \frac{\partial}{\partial \sigma}
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| \left( \frac{1}{\cosh \tau - \cos\sigma}
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| \frac{\partial \Phi}{\partial \sigma}
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| \right) \right. \\[8pt]
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| & {} \quad +
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| \left. \frac{\partial}{\partial \tau}
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| \left( \frac{\sinh \tau}{\cosh \tau - \cos\sigma}
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| \frac{\partial \Phi}{\partial \tau}
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| \right) +
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| \frac{1}{\sinh \tau \left( \cosh \tau - \cos\sigma \right)}
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| \frac{\partial^2 \Phi}{\partial \phi^2}
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| \right]
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| \end{align}
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| </math>
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| Other differential operators such as <math>\nabla \cdot \mathbf{F}</math>
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| and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\sigma, \tau, \phi)</math> by substituting
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| the scale factors into the general formulae
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| found in [[orthogonal coordinates]].
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| ==Toroidal harmonics==
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| ===Standard separation ===
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| The 3-variable [[Laplace equation]]
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| :<math>\nabla^2\Phi=0</math>
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| admits solution via [[separation of variables]] in toroidal coordinates. Making the substitution
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| :<math>
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| V=U\sqrt{\cosh\tau-\cos\sigma}
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| </math>
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| A separable equation is then obtained. A particular solution obtained by [[separation of variables]] is:
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| :<math>V= \sqrt{\cosh\tau-\cos\sigma}\,\,S_\nu(\sigma)T_{\mu\nu}(\tau)\Phi_\mu(\phi)\,</math>
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| where each function is a linear combination of:
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| :<math>
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| S_\nu(\sigma)=e^{i\nu\sigma}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\nu\sigma}
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| </math>
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| :<math>
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| T_{\mu\nu}(\tau)=P_{\nu-1/2}^\mu(\cosh\tau)\,\,\,\,\mathrm{and}\,\,\,\,Q_{\nu-1/2}^\mu(\cosh\tau)
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| </math>
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| :<math>
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| \Phi_\mu(\phi)=e^{i\mu\phi}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\mu\phi}
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| </math>
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| Where P and Q are [[associated Legendre functions]] of the first and second kind. These Legendre functions are often referred to as toroidal harmonics.
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| Toroidal harmonics have many interesting properties. If you make a variable substitution <math>\,\!1<z=\cosh\eta\,</math> then, for instance, with vanishing order (the convention is to not write the order when it vanishes) and <math>\,\!n=0</math>
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| :<math>Q_{-\frac12}(z)=\sqrt{\frac{2}{1+z}}K\left(\sqrt{\frac{2}{1+z}}\right)</math>
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| and
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| :<math>P_{-\frac12}(z)=\frac{2}{\pi}\sqrt{\frac{2}{1+z}}K \left( \sqrt{\frac{z-1}{z+1}} \right)</math>
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| where <math>\,\!K</math> and <math>\,\!E</math> are the complete [[elliptic integrals]] of the [[Elliptic integral#Complete elliptic integral of the first kind|first]] and [[Elliptic integral#Complete elliptic integral of the second kind|second]] kind respectively. The rest of the toroidal harmonics can be obtained, for instance, in terms of the complete elliptic integrals, by using recurrence relations for associated Legendre functions.
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| The classic applications of toroidal coordinates are in solving [[partial differential equations]],
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| e.g., [[Laplace's equation]] for which toroidal coordinates allow a [[separation of variables]] or the [[Helmholtz equation]], for which toroidal coordinates do not allow a separation of variables. Typical examples would be the [[electric potential]] and [[electric field]] of a conducting torus, or in the degenerate case, a conducting ring.
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| ===An alternative separation===
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| Alternatively, a different substitution may be made (Andrews 2006)
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| :<math>
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| V=\frac{U}{\sqrt{\rho}}
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| </math>
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| where
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| :<math>
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| \rho=\sqrt{x^2+y^2}=\frac{\cosh\tau-\cos\sigma}{a\sinh\tau}.
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| </math>
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| Again, a separable equation is obtained. A particular solution obtained by [[separation of variables]] is then:
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| :<math>V= \frac{a}{\rho}\,\,S_\nu(\sigma)T_{\mu\nu}(\tau)\Phi_\mu(\phi)\,</math>
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| where each function is a linear combination of:
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| :<math> | |
| S_\nu(\sigma)=e^{i\nu\sigma}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\nu\sigma}
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| </math>
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| :<math>
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| T_{\mu\nu}(\tau)=P_{\mu-1/2}^\nu(\coth\tau)\,\,\,\,\mathrm{and}\,\,\,\,Q_{\mu-1/2}^\nu(\coth\tau)
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| </math>
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| :<math>
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| \Phi_\mu(\phi)=e^{i\mu\phi}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\mu\phi}.
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| </math>
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| Note that although the toroidal harmonics are used again for the ''T'' function, the argument is <math>\coth\tau</math> rather than <math>\cosh\tau</math> and the <math>\mu</math> and <math>\nu</math> indices are exchanged. This method is useful for situations in which the boundary conditions are independent of the spherical angle <math>\theta</math>, such as the charged ring, an infinite half plane, or two parallel planes. For identities relating the toroidal harmonics with argument hyperbolic
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| cosine with those of argument hyperbolic cotangent, see the [[Whipple formulae]].
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| ==References==
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| *Byerly, W E. (1893) ''[http://www.archive.org/details/elemtreatfour00byerrich An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics]'' Ginn & co. pp. 264–266
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| *{{cite book | author = Arfken G | year = 1970 | title = Mathematical Methods for Physicists | edition = 2nd | publisher = Academic Press | location = Orlando, FL | pages = 112–115}}
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| *{{cite journal |last=Andrews |first=Mark |year=2006 |title=Alternative separation of Laplace's equation in toroidal coordinates and its application to electrostatics |journal=Journal of Electrostatics |volume=64|pages=664–672 |doi=10.1016/j.elstat.2005.11.005 |issue=10}}
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| ==Bibliography==
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| *{{cite book | author = Morse P M, Feshbach H | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw–Hill | location = New York | page = 666}}
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| *{{cite book | author = Korn G A, Korn T M |year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | page = 182 | lccn = 5914456}}
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| *{{cite book | author = Margenau H, Murphy G M | year = 1956 | title = The Mathematics of Physics and Chemistry | publisher = D. van Nostrand | location = New York| pages = 190–192 | lccn = 5510911 }}
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| *{{cite book | author = Moon P H, Spencer D E | year = 1988 | chapter = Toroidal Coordinates (''η'', ''θ'', ''ψ'') | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = 2nd ed., 3rd revised printing | publisher = Springer Verlag | location = New York | isbn = 0-387-02732-7 | pages = 112–115 (Section IV, E4Ry)}}
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| ==External links==
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| *[http://mathworld.wolfram.com/ToroidalCoordinates.html MathWorld description of toroidal coordinates]
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| {{Orthogonal coordinate systems}}
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| [[Category:Coordinate systems]]
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