Editing 2DStructure/ToroidalGreenFunction (section)

=By The Way &hellip;=

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<li>[http://user.xmission.com/~rimrock/Documents/The%20Charged%20Bowl%20in%20Toroidal%20Coordinates.pdf P. Lucht (2016)], ''The Charged Bowl in Toroidal Coordinates'' &#8212; Note from Tohline:  On 3 July 2018 I stumbled on this article by Phil Lucht, who lists his affiliation as Rimrock Digital Technology (rimrock at emission.com), Salt Lake City, Utah. As I have done over the past approximately half a year, Lucht appears to have spent quite a bit of time investigating how toroidal coordinates might be used to solve a few specific potential problems.  He draws on many of the same, rich technical resource publications as I have done and, at one point, explicitly expresses amazement at the extensive amount of useful material that can be found in [<b>[[User:Tohline/Appendix/References#MF53|<font color="red">MF53</font>]]</b>].  Anyone digging into this topic would be well-advised to read Lucht's '''Overview and Summary''' &#8212; especially including the subsection titled, ''Legendre Functions'' &#8212; and to recognize that his document contains a large number of technically detailed appendices. 
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To provide one point of comparison with [<b>[[User:Tohline/Appendix/References#MF53|<font color="red">MF53</font>]]</b>], Lucht derived an analytic expression for the potential outside of a torus (with toroidal-coordinate radius, <math>~\eta_0</math>) whose surface is held at uniform potential, <math>~\Phi_0</math>.  This is in contrast to Wong's (1973) derivation &#8212; which we have verified, above &#8212; of the potential outside (as well as inside) of a torus having uniform charge/mass.  For his particular example problem, Lucht obtains, 
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<math>~\Phi(\eta,\theta)</math>
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<math>~=</math>
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<math>~
\Phi_0 \frac{\sqrt{2}}{\pi} \sqrt{\cosh\eta - \cos\theta} \sum_{n=0}^\infty \epsilon_n P_{n-\frac{1}{2}}(\cosh\eta)
\Biggl[ \frac{Q_{n-\frac{1}{2}}(\cosh\eta_0)}{P_{n-\frac{1}{2}}(\cosh\eta_0)}  \Biggr] \cos(n\theta) \, ,
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[http://user.xmission.com/~rimrock/Documents/The%20Charged%20Bowl%20in%20Toroidal%20Coordinates.pdf Lucht (2016)], p. 68, Eq. (10.1.11)
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then he points out that the associated expression for the potential, which appears as equation (10.3.80) on p. 1304 of [<b>[[User:Tohline/Appendix/References#MF53|<font color="red">MF53</font>]]</b>], is missing the Neumann factor, <math>~\epsilon_n</math>.
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