Editing Appendix/Mathematics/ToroidalSynopsis01 (section)

===Green's Function===
Wong (1973) points out that in toroidal coordinates the Green's function is,
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<math>~\frac{1}{|~\vec{r} - {\vec{r}}^{~'} ~|} </math>
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<math>~=</math>
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<math>~
\frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 }
\sum\limits_{m,n} (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
</math>
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&nbsp;
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&nbsp;
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<math>~
\times \cos[m(\psi - \psi^')][\cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta
\end{cases}\, ,
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[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.53)
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where, <math>~P^m_{n-1 / 2}, Q^m_{n-1 / 2}</math> are "<font color="darkgreen">Legendre functions of the first and second kind with order <math>~n - \tfrac{1}{2}</math> and degree <math>~m</math> (toroidal harmonics)</font>," and <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0 = 1</math> and <math>~\epsilon_m = 2</math> for all <math>~m \ge 1</math>.  According to [http://adsabs.harvard.edu/abs/1999ApJ...527...86C CT99], the Green's function written in toroidal coordinates is,
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<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}</math>
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<math>~=</math>
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<math>~ 
\frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) 
</math>
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&nbsp;
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<math>~=</math>
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<math>~ 
\frac{1}{a\pi} 
\biggl[  \frac{(\cosh\eta^' - \cos\theta^')}{\sinh\eta^' } \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2}
\sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) \, .
</math>
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Things to note:
<ol>
  <li>The argument of <math>~Q_{m - 1 / 2}</math> in the CT99 expression is very different from the argument of <math>~Q^m_{n - 1 / 2}</math> (or <math>~P^m_{n - 1 / 2}</math>) in Wong's expression.</li>
  <li>In both expressions, <math>~m</math> is the integer multiplying the azimuthal angle, <math>~\psi</math>, but in the CT99 expression this index serves as the ''subscript'' index of the function, <math>~Q</math>, whereas in Wong's expression it serves as the ''superscript'' index of both functions, <math>~Q</math> and <math>~P</math>.  In this context, note that,
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<math>~Q^m_{n-\frac{1}{2}}(\cosh\eta)</math>
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<math>~=</math>
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<math>~(-1)^m
\sqrt{\frac{\pi}{2}} ~\Gamma(m-n+\tfrac{1}{2}) \biggl[ \frac{1}{ \sqrt{\sinh\eta}} \biggr] P^{n}_{m - \frac{1}{2}} (\coth\eta) \, .
</math>
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  </li>
  <li>Wong's expression contains not only a summation over the index, <math>~m</math>, but also an explicit summation over the index, <math>~n</math>, which multiplies the "polar" angle, <math>~\theta</math>; no such additional summation appears in the CT99 expression, indicating that the summation over <math>~n</math> has implicitly already been completed.  In this context, note that the [[Appendix/Mathematics/ToroidalConfusion#Joel.27s_Additional_Manipulations|summation expression]] gives,

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<math>~
Q^{\mu}_{-\frac{1}{2}}\left(\cosh\xi\right)
+
2\sum_{n=1}^{\infty}
Q^{\mu}_{n-\frac{1}{2}}\left(\cosh\xi\right) \cos\left[ n (\theta - \theta^') \right] 
</math>
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<math>~=</math>
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<math>~ e^{\mu\pi i} \Gamma\left(\mu+ \tfrac{1}{2} \right) \biggl[
\dfrac{\left(\frac{1}{2}\pi\right)^{1/2}\left(\sinh\xi\right)^{\mu
}}{\left\{ \cosh\xi -\cos\left[ n (\theta - \theta^') \right] \right\}^{\mu+(1/2)}}\biggr]
\, ;
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or, specifically for the case of <math>~\mu = 0</math>,
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<math>~
\sum_{n=0}^{\infty} \epsilon_n
Q_{n-\frac{1}{2}}\left(\cosh\xi\right) \cos\left[ n(\theta - \theta^') \right] 
</math>
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<math>~=</math>
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<math>~ 
\dfrac{ \pi/\sqrt{2} }{\left[ \cosh\xi-\cos(\theta - \theta^') \right]^{\frac{1}{2}}  } \, .
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  </li>
  <li>Next thought &hellip;</li>
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