Editing Apps/Wong1973Potential (section)

====Attempt #2====
Now, drawing upon the Key Relation,
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{{ Math/EQ_Toroidal02 }}
</div>
and setting n = m = 0 while adopting the association, <math>x \rightarrow \cosh\eta</math>, we obtain,
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  <td align="right">
<math>Q_{-1 / 2}(\coth\eta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\pi^{3/2}}{\sqrt{2\pi}}~ (\sinh\eta)^{1 / 2} P_{-1 / 2}(\cosh\eta)
</math>
  </td>
</tr>

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  <td align="right">
<math>\Rightarrow ~~~ P_{-1 / 2}(\cosh\eta)</math>
  </td>
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<math>=</math>
  </td>
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<math>
\frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} Q_{-1 / 2}(\coth\eta) \, .
</math>
  </td>
</tr>
</table>
We therefore can write,

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  <td align="right">
<math>\biggl(\frac{a}{GM} \biggr) \Phi_{\mathrm{W}0} (\eta,\theta) \biggr|_\mathrm{Exterior} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{\sqrt{2} D_0\cdot C_0}{\pi} 
\biggl[ \frac{(\cosh\eta - \cos\theta)^{1 / 2}}{ (\sinh\eta)^{1 / 2}}\biggr]~Q_{- 1 / 2}(\coth\eta)
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
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<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{\sqrt{2} D_0\cdot C_0}{\pi} 
\biggl[ \frac{(\cosh\eta - \cos\theta)^{1 / 2}}{ (\sinh\eta)^{1 / 2}}\biggr]~k K(k) 
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{2 D_0\cdot C_0}{\pi} 
\biggl[ \frac{\cosh\eta - \cos\theta}{ \sinh\eta + \cosh\eta}\biggr]^{1 / 2}~K(k) 
</math>
  </td>
</tr>
</table>
where,
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<math>k\equiv \biggl[  \frac{2}{1+\coth\eta } \biggr]^{1 / 2} = \biggl[  \frac{2\sinh\eta}{\sinh\eta+\cosh\eta } \biggr]^{1 / 2}\, .</math>
</div>

Now, converting back to cylindrical coordinates, we recognize that,
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<math>k^2</math>
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<math>=</math>
  </td>
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<math>
\frac{4a\varpi}{(\varpi + a)^2 + z^2 } \, ,
</math>
  </td>
</tr>
</table>
and,
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<tr>
  <td align="right">
<math>\frac{\cosh\eta - \cos\theta}{ \sinh\eta + \cosh\eta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{a\sinh\eta}{\varpi} \biggl[ \frac{ \{ [(\varpi - a)^2 + z^2]\cdot [(\varpi + a)^2 + z^2] \}^{1 / 2} }{ \varpi^2 + a^2 + z^2 +2a\varpi }  \biggr] </math>
  </td>
</tr>

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

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
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<math>\biggl[ \frac{ 2a^2}{ (\varpi + a)^2 + z^2 }  \biggr] \, . </math>
  </td>
</tr>
</table>
Hence,

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

<tr>
  <td align="right">
<math>\biggl(\frac{a}{GM} \biggr) \Phi_{\mathrm{W}0} (\varpi,z) \biggr|_\mathrm{Exterior} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{2 D_0\cdot C_0}{\pi} 
\biggl[ \frac{ 2a^2}{ (\varpi + a)^2 + z^2 }  \biggr]^{1 / 2}~K(k) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
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<math>
- \biggl[ \frac{2^{3/2} D_0\cdot C_0}{\pi} \biggr]
 \frac{ aK(k)  }{[ (\varpi + a)^2 + z^2]^{1 / 2} }  
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
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<math>\approx</math>
  </td>
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<math>
- \biggl[ \frac{2^{3/2} }{\pi} \biggr] \biggl[ \frac{2^{3/2} }{3\pi^2} \cdot \frac{\sinh^3\eta_0}{\cosh\eta_0} \biggr] \biggl[\biggl( \frac{3 \pi^2}{2^2} \biggr) \frac{1}{\cosh^2\eta_0}\biggr]
 \frac{ aK(k)  }{[ (\varpi + a)^2 + z^2]^{1 / 2} }  
</math>
  </td>
</tr>

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&nbsp;
  </td>
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<math>\approx</math>
  </td>
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<math>- \frac{2}{\pi} \cdot 
\frac{ aK(k)  }{[ (\varpi + a)^2 + z^2]^{1 / 2} }  \, ,
</math>
  </td>
</tr>
</table>
where, in the next-to-last step, we have inserted, from above, the expression for the product of <math>D_0</math> and <math>C_0</math> in the limit of <math>\eta_0 \rightarrow \infty</math>.  This is precisely the expression for the ''Gravitational Potential in the Thin Ring Approximation, <math>\Phi_\mathrm{TR}(\varpi,z)</math>, that we have presented elsewhere.