Editing Appendix/Mathematics/ToroidalFunctions (section)

=Just Compare Green Functions=

==First Dabbling==

According to [http://adsabs.harvard.edu/abs/1999ApJ...527...86C CT99], the Green's function written in toroidal coordinates is,
<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=-\infty}^{\infty} e^{im(\phi - \phi^')}Q_{m- 1 / 2}(\chi) \, ,
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </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(\phi - \phi^')] Q_{m- 1 / 2}(\chi) \, ,
</math>
  </td>
</tr>
</table>
where,<br />
<div align="center"><math>~\chi \equiv \frac{\varpi^2 + (\varpi^')^2 + (z - z^')^2}{2\varpi \varpi^'} \, ,</math>
</div>
and, <math>~Q_{m - 1 / 2}</math> is the zero-order, half-(odd)integer degree, Llegendre function of the second kind &#8212; also referred to as a ''toroidal'' function of zeroth order.  

From the [[Apps/DysonWongTori#Introducing_Toroidal_Coordinates|Dyson-Wong Toroid]] chapter,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a \sinh\eta }{(\cosh\eta - \cos\theta)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \, .</math>
  </td>
</tr>
</table>
</div>
<span id="FirstDab">Hence, we can rewrite the Green's function as,</span>
<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} 
\biggl[  \frac{a \sinh\eta^' }{(\cosh\eta^' - \cos\theta^')} \frac{a \sinh\eta }{(\cosh\eta - \cos\theta)}\biggr]^{- 1 / 2}
\sum_{m=0}^{\infty} \epsilon_m \cos[m(\phi - \phi^')] Q_{m- 1 / 2}(\chi) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{\pi a} [\sinh\eta^' \sinh\eta]^{- 1 / 2}
\biggl[  (\cosh\eta^' - \cos\theta^') (\cosh\eta - \cos\theta)\biggr]^{1 / 2}
\sum_{m=0}^{\infty} \epsilon_m \cos[m(\phi - \phi^')] Q_{m- 1 / 2}(\chi) 
</math>
  </td>
</tr>
</table>
<!--
Hence, a valid expression for the gravitational potential is,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(\varpi,\phi,z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-G \int \frac{1}{|\vec{x}^{~'} - \vec{x}|} ~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-G \int \biggl\{ 
\frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=-\infty}^{\infty} e^{im(\phi - \phi^')}Q_{m- 1 / 2}(\chi)
\biggr\}~
\rho(\varpi^',\phi^',z^') \varpi^' d\varpi^' d\phi^' dz^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{\pi \sqrt{\varpi}} \int \rho(\varpi^',\phi^',z^') \sqrt{\varpi^'} d\varpi^' d\phi^' dz^'  
\sum_{m=0}^{\infty} \epsilon_m \cos[m(\phi - \phi^')] Q_{m- 1 / 2}(\chi) \, ,
</math>
  </td>
</tr>
</table>
where, <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0=1</math> and <math>~\epsilon_m = 2</math> for <math>~m\ge 1</math>.  
-->
----
Wong (1973) states that in toroidal coordinates the Green's function is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{|~\vec{r} - {\vec{r}}^{~'} ~|} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<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}\, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.53)
  </td>
</tr>
</table>
</div>
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>,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\tan\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{y}{x} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_1^2 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[(x^2 + y^2)^{1 / 2} + a]^2 + z^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r_2^2 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[(x^2 + y^2)^{1 / 2} - a]^2 + z^2 \, ,</math>
  </td>
</tr>
</table>
</div>
and <math>~\theta</math> has the same sign as <math>~z</math>.

Note that:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cosh\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl[ e^\eta + e^{-\eta} \biggr] = ~\frac{1}{2}\biggl[ \frac{r_1}{r_2} + \frac{r_2}{r_1}\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl[ \frac{\sqrt{[ \varpi + a]^2 + z^2}}{\sqrt{[ \varpi - a]^2 + z^2}} 
+ \frac{\sqrt{[ \varpi - a]^2 + z^2}}{\sqrt{[ \varpi + a]^2 + z^2}}\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 4\cosh^2\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ \frac{ [( \varpi + a)^2 + z^2]^{1 / 2} }{ [( \varpi - a)^2 + z^2]^{1 / 2}} 
+ \frac{[( \varpi - a)^2 + z^2]^{1 / 2}}{ [( \varpi + a)^2 + z^2]^{1 / 2} }\biggr\}^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 4\cosh^2\eta~  [( \varpi + a)^2 + z^2] [( \varpi - a)^2 + z^2]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{  [( \varpi + a)^2 + z^2] 
+ [( \varpi - a)^2 + z^2] \biggr\}^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 4\cosh^2\eta~  [ \varpi^2 + 2a\varpi + a^2 + z^2] [ \varpi^2 -2a\varpi + a^2 + z^2]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4 [ \varpi^2 + a^2 + z^2]^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
\frac{ [ \varpi^2 + a^2 + z^2]^2}{\cosh^2\eta}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[ \varpi^4 -2a\varpi^3 + a^2\varpi^2 + \varpi^2 z^2 + 2a\varpi^3 - 4a^2\varpi^2 +2a^3\varpi + 2a\varpi z^2 + a^2\varpi^2 - 2a^3\varpi + a^4 + a^2z^2 + \varpi^2 z^2 -2a\varpi z^2 +a^2z^2 + z^4  ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[ \varpi^4   +  2\varpi^2 z^2 - 2a^2\varpi^2 + a^4   + 2a^2z^2 + z^4  ]
</math>
  </td>
</tr>
</table>
</div>

==Second Dabbling==

If <math>~\eta = \ln(r_1/r_2)</math>, as defined by Wong, then we can show that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\coth\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ \varpi^2 + a^2 + z^2 }{ 2a\varpi  } \, .
</math>
  </td>
</tr>
</table>
</div>
This expression matches the expression for <math>~\Chi</math>, as defined in the CT99; see, for example, [[2DStructure/ToroidalCoordinates#Statement_of_the_Problem|our accompanying discussion]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Chi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ \varpi^2 + R_*^2 + (Z_* -Z)^2 }{ 2R_*\varpi  } \, .
</math>
  </td>
</tr>
</table>
</div>

It is the context of the CT99 derivation that we state,
<div align="center">
<math>Q_{-1/2}(\Chi)= Q_{-1/2}(\coth\eta) = \mu K(\mu) \, ,</math>
</div>
where the argument of the elliptic integral is related to <math>~\Chi</math> via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{2}{1+\Chi}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{4R_*\varpi}{(R_* + \varpi)^2 + (Z_* - Z)^2} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>


<table border="1" align="center" cellpadding="15" width="85%">
<tr>
  <td align="left">
In a [[Appendix/Mathematics/ToroidalSynopsis01#SummaryTable|separate discussion]], we have derived expressions for the quantities, <math>~\Chi</math> and <math>~\mu^2</math>, in terms of Wong's toroidal coordinates.
  </td>
</tr>
</table>


We note that, while the argument of the CT99 toroidal function is <math>~\coth\eta</math>, the argument of Wong's toroidal function is <math>~\cosh\eta</math>.  It should be useful to keep in mind, therefore, that you can move back and forth between <math>~\coth\eta</math> and <math>~\cosh\eta</math> via the mapping (in either direction), 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ x  }{ \sqrt{x^2-1}  } \, .</math>
  </td>
</tr>
</table>
</div>

Hence, for example, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cosh^2\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\coth^2\eta \biggl[ \coth^2\eta-1  \biggr]^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{ \varpi^2 + a^2 + z^2 }{ 2a\varpi  }  \biggr]^{2} \biggl\{ \biggl[\frac{ \varpi^2 + a^2 + z^2 }{ 2a\varpi  }  \biggr]^{2}-1  \biggr\}^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{\varpi^2 + a^2 + z^2}{\cosh\eta} \biggr]^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4a^2\varpi^2  \biggl\{ \biggl[\frac{ \varpi^2 + a^2 + z^2 }{ 2a\varpi  }  \biggr]^{2}-1  \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  [ \varpi^2 + a^2 + z^2 ]^{2} - 4a^2\varpi^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  \biggl[ ( \varpi^2 + a^2 + z^2 ) + 2a \varpi \biggr] \biggl[ ( \varpi^2 + a^2 + z^2 ) - 2a \varpi \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ [ (\varpi + a)^2 + z^2 ] [ (\varpi - a)^2 + z^2 ] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\cosh\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl\{ \frac{  [ \varpi^2 + a^2 + z^2 ]^2  }{ [ (\varpi + a)^2 + z^2 ] [ (\varpi - a)^2 + z^2 ] } \biggr\}^{1 / 2} \, .</math>
  </td>
</tr>
</table>
</div>

So, according to Wong, for <math>~\eta^' > \eta</math>, the ''axisymmetric'' potential is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~U(\eta^',\theta^')\biggr|_\mathrm{ axi }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\rho_0 a^2 (\cosh \eta^' - \cos \theta^')^{1 / 2} \sum\limits_n \epsilon_n \int_{\eta_0}^\infty d\eta 
\int_{-\pi}^{\pi} \biggl[\frac{\cos[n(\theta - \theta^')]}{(\cosh\eta - \cos\theta)^{5 / 2}} \biggr]d\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sinh\eta ~\begin{cases}P_{n-1 / 2}(\cosh\eta) ~Q_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P_{n-1 / 2}(\cosh\eta^') ~Q_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta
\end{cases}\, 
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.55)
  </td>
</tr>

<tr>
  <td align="right">
<math>~U(\eta^',\theta^')\biggr|_\mathrm{ axi }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\rho_0 a^2 (\cosh \eta^' - \cos \theta^')^{1 / 2} \sum\limits_n \epsilon_n \int_{\eta_0}^\infty  
\int_{-\pi}^{\pi} \biggl[\frac{\cos[n(\theta - \theta^')]}{(\cosh\eta - \cos\theta)^{5 / 2}} \biggr] \sinh\eta ~P_{n-1 / 2}(\cosh\eta) ~Q_{n-1 / 2}(\cosh\eta^') ~d\theta ~d\eta
</math>
  </td>
</tr>
</table>
</div>

Drawing from equations (2.7), (2.17) and (2.18) of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], we see that the volume,</span> <math>~V</math>, of a torus that is bounded by surface <math>~\eta_s</math> is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{V}{a^3} = \frac{1}{a^3} \iiint\limits_{\eta_s}d^3 r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint\limits_{\eta_s} \biggl[ \frac{\sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta~ d\psi~ = \frac{2\pi^2\cosh{\eta_s}}{\sinh^3\eta_s} \, .</math>
  </td>
</tr>
</table>
</div>
This means that, in toroidal coordinates, just the integration over the azimuthal angle,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{a^3} \iiint\limits_{\eta_s}d^3 r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi \iint\limits \biggl[ \frac{\sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta  \, .</math>
  </td>
</tr>
</table>
</div>
To be compared with the same expression in cylindrical coordinates,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{a^3} \iiint\limits_{\eta_s}d^3 r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2\pi}{a^3} \iint\limits \varpi ~d\varpi ~dz  \, .</math>
  </td>
</tr>
</table>
</div>
This means that the coordinate mapping is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\varpi ~d\varpi ~dz}{a^3} </math>
  </td>
  <td align="center">
<math>~\leftrightarrow</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta  \, .</math>
  </td>
</tr>
</table>
</div>

This means that the CT99 axisymmetric potential is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(R_*,Z_*)\biggr|_\mathrm{ axi }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 2G  \iint \frac{Q_{-1/2}(\coth\eta)}{ (R_* \varpi)^{1 / 2}}  \rho(\varpi, Z) ~\varpi~d\varpi dZ</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 2Ga^3  \iint \frac{Q_{-1/2}(\coth\eta)}{ (R_* \varpi)^{1 / 2}}  \biggl[ \frac{\sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] \rho~d\eta~ d\theta</math>
  </td>
</tr>
</table>
</div>
Given that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{ 1 }{ \varpi }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(\cosh\eta - \cos\theta)}{a \sinh\eta} \, ,</math>
  </td>
</tr>
</table>
</div>
we therefore have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(R_*,Z_*)\biggr|_\mathrm{ axi }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 2Ga^2\biggl( \frac{a}{R_*}\biggr)^{1 / 2}  \iint Q_{-1/2}(\coth\eta)  \biggl[ \frac{\sinh\eta}{(\cosh\eta - \cos\theta)^5} \biggr]^{1 / 2} \rho~d\eta~ d\theta</math>
  </td>
</tr>
</table>
</div>

==Third Dabbling==

From our [[#FirstDab|"First Dabbling" expression]] for the Green's Function, we have,
<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 a} [\sinh\eta^' \sinh\eta]^{- 1 / 2}
\biggl[  (\cosh\eta^' - \cos\theta^') (\cosh\eta - \cos\theta)\biggr]^{1 / 2}
\sum_{m=0}^{\infty} \epsilon_m \cos[m(\phi - \phi^')] Q_{m- 1 / 2}(\chi) \, .
</math>
  </td>
</tr>
</table>

And from our [[Appendix/Mathematics/ToroidalConfusion#Joel.27s_Additional_Manipulations|examination of the summation expression]], we have found that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\sum_{n=0}^{\infty} \epsilon_n
Q_{n-\frac{1}{2}}\left(\cosh\xi\right) \cos\left(n
\phi\right) 

</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  \biggl[
\dfrac{ \pi/\sqrt{2} }{\left(\cosh\xi-\cos\phi\right)^{\frac{1}{2}}  }\biggr] \, .
</math>
  </td>
</tr>
</table>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
\sum_{m=0}^{\infty} \epsilon_m
Q_{m-\frac{1}{2}}
\left(\cosh\xi\right) \cos\left[m( \phi - \phi^')\right]

</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  \frac{\pi}{\sqrt{2}} 
\biggl\{ \cosh\xi-\cos\left[( \phi - \phi^')\right]\biggr\}^{ - \frac{1}{2}} \, .
</math>
  </td>
</tr>
</table>
</div>

How does this jive with [[Appendix/Mathematics/ToroidalSynopsis01#SummaryTable|our separate recognition]] that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } \biggr] \, ;
</math>
  </td>
</tr>
</table>
</div>
Is it completely legitimate to make the association, <math>~\chi \leftrightarrow \cosh\xi</math> &nbsp; &nbsp;?  If so, then the Green's function becomes,
<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}{a \sqrt{2}} [\sinh\eta^' \sinh\eta]^{- 1 / 2}
\biggl[  (\cosh\eta^' - \cos\theta^') (\cosh\eta - \cos\theta)\biggr]^{1 / 2}
\biggl\{ \biggl[ \frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } \biggr]-\cos\left[( \phi - \phi^')\right]\biggr\}^{ - \frac{1}{2}}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{a \sqrt{2}} 
\biggl[  \frac{ (\cosh\eta^' - \cos\theta^') (\cosh\eta - \cos\theta) }{ \sinh\eta^' \sinh\eta } \biggr]^{1 / 2}
\biggl[ \frac{ \sinh\eta \cdot \sinh\eta^' }{\cosh\eta \cdot \cosh\eta^' - \sinh\eta \cdot \sinh\eta^'\cos ( \phi - \phi^') - \cos(\theta^' - \theta)  } \biggr]^{ \frac{1}{2}}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{a \sqrt{2}} 
\biggl[ \frac{(\cosh\eta^' - \cos\theta^') (\cosh\eta - \cos\theta)  }{\cosh\eta \cdot \cosh\eta^' - \sinh\eta \cdot \sinh\eta^'\cos ( \phi - \phi^') - \cos(\theta^' - \theta)  } \biggr]^{ \frac{1}{2}}  \, .
</math>
  </td>
</tr>
</table>