Editing 2DStructure/ToroidalGreenFunction (section)

===As Presented in Cohl &amp; Tohline (1999)===

This last, compact Green's function expression &#8212; which we have derived, here, from [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] published Green's function by drawing strategically upon a variety of special function relations &#8212; precisely matches the "compact cylindrical Green's function expression" that has been derived independently by [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)] via a less tortuous route, namely,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) 
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="2">&nbsp;</td>
  <td align="left" colspan="1">
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], p. 88, Eq. (17)
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Chi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{(\varpi^')^2 + \varpi^2 + (z^' - z)^2}{2\varpi^' \varpi}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^'} \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="2">
&nbsp; &nbsp;
  </td>
  <td align="left" colspan="3">
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], p. 88, Eq. (16)
  </td>
</tr>
</table>


'''Note from J. E. Tohline (June, 2018)''':  &nbsp;This is the first time that I have been able to formally demonstrate to myself that these two separately derived Green's function expressions are identical.  See, however, the earlier identification of ''new'' addition theorems in association with equations (49) and (50) of [http://adsabs.harvard.edu/abs/2000AN....321..363C Cohl et al. (2000)].