Editing Appendix/Ramblings/Bordeaux (section)

====Leading (n  = 0) Term====
=====Wong's Expression=====
Now, from our [[Apps/Wong1973Potential#Attempt_.232|separate derivation]] we have,

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

<tr>
  <td align="right">
<math>~P_{-1 / 2}(\cosh\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} Q_{-1 / 2}(\coth\eta) \, .
</math>
  </td>
</tr>
</table>
<span id="KeyEquation">And if we make the function-argument substitution,</span> <math>z \rightarrow \coth\eta</math>, in the "[[Appendix/SpecialFunctions#Analytic_Expressions_.26_Plots|Key Equation]],"

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

<tr>
<td align="right">
[[Image:LSUkey.png|25px|link=Appendix/SpecialFunctions#Toroidal_Function_Evaluations]]
</td> 
  <td align="right">
<math>~Q_{-\frac{1}{2}}(z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sqrt{ \frac{2}{z+1} } ~K\biggl( \sqrt{ \frac{2}{z+1}} \biggr) 
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="4">
[https://books.google.com/books?id=MtU8uP7XMvoC&printsec=frontcover&dq=Abramowitz+and+stegun&hl=en&sa=X&ved=0ahUKEwialra5xNbaAhWKna0KHcLAASAQ6AEILDAA#v=onepage&q=Abramowitz%20and%20stegun&f=false Abramowitz &amp; Stegun (1995)], p. 337, eq. (8.13.3)
  </td>
</table>
we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_{-1 / 2}(\cosh\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k) \, ,
</math>
  </td>
</tr>
</table>

where, from above, we recognize that,
<div align="center">
<math>~
k \equiv \biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2}  
=
\biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} \, .
</math>
</div>

So, the leading (n = 0) term gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~C_0(\cosh\eta_0)P_{-\frac{1}{2}}(\cosh\eta) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-D_0~C_0(\cosh\eta_0)
\biggl[ \frac{a \sinh\eta}{\varpi} \biggr]^{1 / 2} ~\frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{D_0~C_0(\cosh\eta_0)}{\pi}
\biggl[ \frac{2a }{\varpi} \biggr]^{1 / 2} ~ k \boldsymbol{K}(k) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- C_0(\cosh\eta_0) \cdot \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] 
\frac{a}{ [ (\varpi + a)^2 + (z - Z_0)^2 ]^{1 / 2} } \cdot  \boldsymbol{K}(k) \, .
</math>
  </td>
</tr>
</table>

=====Thin-Ring Evaluation of C<sub>0</sub>=====
In an [[Apps/Wong1973Potential#Thin_Ring_Approximation|accompanying discussion of the thin-ring approximation]], we showed that as <math>~\cosh\eta_0 \rightarrow \infty</math>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~C_0(x)\biggr|_{x\rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{3 \pi^2}{2^2} \biggr) \frac{1}{\cosh^2\eta_0} \, .
</math>
  </td>
</tr>
</table>
Hence, in this limit we can write,

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

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\eta,\theta)\biggr|_\mathrm{thin-ring}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{2 }{\pi}  \cancelto{1}{\biggl[\frac{\sinh\eta_0}{\cosh\eta_0}\biggr]^3 } 
\frac{a}{ [ (\varpi + a)^2 + (z - Z_0)^2 ]^{1 / 2} } \cdot  \boldsymbol{K}(k) \, .
</math>
  </td>
</tr>
</table>

=====More General Evaluation of C<sub>0</sub>=====

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
<font color="red">NOTE of CAUTION:</font>  In our [[#KeyEquation|above evaluation of the toroidal function]], <math>~Q_{-\frac{1}{2}}(z)</math>, we appropriately associated the function argument, <math>~z</math>, with the hyperbolic-cotangent of <math>~\eta</math>; that is, we made the substitution, <math>~z \rightarrow \coth\eta</math>.  Here, as we assess the behavior of, and evaluate, the leading coefficient, <math>~C_0</math>, an alternate substitution is appropriate, namely, <math>~z_0 \rightarrow \cosh\eta_0</math>; we affix the subscript zero to this function argument in an effort to minimize possible confusion with the argument, <math>~z</math>.
</td></tr></table>

Drawing from our [[Appendix/SpecialFunctions#Analytic_Expressions_.26_Plots|accompanying tabulation of ''Toroidal Function Evaluations'']], we have more generally,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~2C_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ Q_{+\frac{1}{2}}(\cosh \eta_0) \biggr]
\biggl[ Q_{ - \frac{1}{2}}^2(\cosh \eta_0) \biggr]
+ 
3 \biggl[ Q_{ - \frac{1}{2}}(\cosh \eta_0) \biggr]
\biggl[ Q^2_{ + \frac{1}{2}}(\cosh \eta_0) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \cosh\eta_0 ~k_0~K(k_0) ~-~ [2(\cosh\eta_0+1)]^{1 / 2} E(k_0) \biggr]
\times \biggl\{ \frac{ 4 \cosh\eta_0 ~E(k_0) - (\cosh\eta_0-1) K(k_0) }{ [2^{3} (\cosh\eta_0+1) (\cosh\eta_0-1)^{2} ]^{1 / 2}} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 
\frac{3}{2^2} \biggl[ k_0 ~K ( k_0) \biggr]
\times \biggl\{ \cosh\eta_0~ k_0~K ( k_0 )  
~-~(\cosh^2\eta_0+3) \biggl[ \frac{2}{(\cosh\eta_0 - 1)(\cosh^2\eta_0 -1)} \biggr]^{1 / 2} E(k_0)
 \biggr\} \, ,
</math>
  </td>
</tr>
</table>
<span id="FirstEvaluations">where,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k_0</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2}{\cosh\eta_0+1}\biggr]^{1 / 2} ~~~\Rightarrow ~~~ (\cosh\eta_0 + 1) = \frac{2}{k_0^2} \, .</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="10">
<tr><td align="left">
Looking back at our [[Apps/Wong1973Potential#Exterior_Solution_.28n_.3D_0.29|previous numerical evaluation]] of <math>~C_0(\cosh\eta_0)</math> when <math>~z_0 = \cosh\eta_0 = 3 ~~\Rightarrow ~~~ k_0 = 2^{-1 / 2}</math>, we see that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
[[Appendix/SpecialFunctions#Toroidal_Function_Evaluations|Appendix Expression:]] <math>~Q_{-\tfrac{1}{2}}(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~k_0 K(k_0)</math>
  </td>
</tr>

<tr>
  <td align="right">
Hence [[Appendix/SpecialFunctions#Comparison_with_Table_IX_from_MF53|MF53 value]], <math>~Q_{-\tfrac{1}{2}}(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1.311028777 ~~~\Rightarrow ~~~ K(k_0) = 1.854074677</math>
  </td>
</tr>

<tr>
  <td align="right">
[[Appendix/SpecialFunctions#Toroidal_Function_Evaluations|Appendix Expression:]] <math>~Q_{+\tfrac{1}{2}}(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z~k_0 K(k_0) - [2(z+1)]^{1 / 2} E(k_0)</math>
  </td>
</tr>

<tr>
  <td align="right">
Hence [[Appendix/SpecialFunctions#Comparison_with_Table_IX_from_MF53|MF53 value]], <math>~Q_{+\tfrac{1}{2}}(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.1128885424 ~~~\Rightarrow~~~ E(k_0) = 1.350643881</math>
  </td>
</tr>

<tr>
  <td align="right">
[[Appendix/SpecialFunctions#Toroidal_Function_Evaluations|Appendix Expression:]] <math>~Q^2_{-\tfrac{1}{2}}(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[2^3(z-1)(z^2-1)]^{-1 / 2} [4zE(k_0) - (z-1)K(k_0)]</math>
  </td>
</tr>

<tr>
  <td align="right">
Hence, <math>~Q^2_{-\tfrac{1}{2}}(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1.104816977</math>, which matches  [[Appendix/SpecialFunctions#Comparison_with_Table_IX_from_MF53|MF53 value]]
  </td>
</tr>

<tr>
  <td align="right">
[[Appendix/Mathematics/ToroidalSynopsis01#Evaluating_Q2.CE.BD|Additional derivation:]] <math>~Q^2_{+\tfrac{1}{2}}(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\frac{1}{2^2}\biggl\{ z k_0~K ( k_0 )  
~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
Hence, <math>~Q^2_{+\tfrac{1}{2}}(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.449302588</math>
  </td>
</tr>
</table>

----

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ C_0(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2}~Q_{+\frac{1}{2}}(3) \cdot Q_{- \frac{1}{2}}^2(3) 
+ \frac{3}{2}~ Q_{- \frac{1}{2}}(3)\cdot Q^2_{+ \frac{1}{2}}(3) 
=
0.945933522 \, .
</math>
  </td>
</tr>
</table>

</td></tr>
</table>


Attempting to simplify this expression, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~2C_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \cosh\eta_0 ~k_0~K(k_0) ~-~ \biggl(\frac{2}{k_0}\biggr) E(k_0) \biggr\}
\times \biggl\{ \frac{ 4 \cosh\eta_0 ~E(k_0) - (\cosh\eta_0-1) K(k_0) }{ [2^{2} k_0^{-1} (\cosh\eta_0-1) ]} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 
\frac{3}{2^2} \biggl[ k_0 ~K ( k_0) \biggr]
\times \biggl\{ \cosh\eta_0~ k_0~K ( k_0 )  
~-~(\cosh^2\eta_0+3) \biggl[ \frac{k_0^2}{(\cosh\eta_0 - 1)^2} \biggr]^{1 / 2} E(k_0)
 \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 2^3(\cosh\eta_0 - 1)C_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \cosh\eta_0 ~k_0^2~K(k_0) ~-~ 2 E(k_0) \biggr\}
\times \biggl\{ 4 \cosh\eta_0 ~E(k_0) - (\cosh\eta_0-1) K(k_0)  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 
3 k_0 ~K ( k_0) 
\times \biggl\{ \cosh\eta_0(\cosh\eta_0 - 1)~ k_0~K ( k_0 )  
~-~(\cosh^2\eta_0+3) k_0 E(k_0)
 \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~K(k_0)\cdot K(k_0) \biggl[ (\cosh\eta_0-1) \cdot \cosh\eta_0 ~k_0^2 + 3\cosh\eta_0~ (\cosh\eta_0~-1)k_0^2\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ K(k_0)\cdot E(k_0) \biggl[ 2^2 \cosh^2\eta_0 ~k_0^2  + 2(\cosh\eta_0 ~-1)  + 3k_0^2 (\cosh^2\eta_0 ~ + 3)\biggr] 
- E(k_0)\cdot E(k_0) \biggl[2^3\cosh\eta_0  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{ 2^3(\cosh\eta_0 - 1)}{k_0^2} \biggr] C_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~K(k_0)\cdot K(k_0) \biggl[ (\cosh\eta_0-1) \cdot \cosh\eta_0  + 3\cosh\eta_0~ (\cosh\eta_0~-1) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ K(k_0)\cdot E(k_0) \biggl[ 2^2 \cosh^2\eta_0   + \frac{2}{k_0^2}(\cosh\eta_0 ~-1)  + 3 (\cosh^2\eta_0 ~ + 3)\biggr] 
- E(k_0)\cdot E(k_0) \biggl[\frac{2^3\cosh\eta_0}{k_0^2}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (\cosh^2\eta_0 - 1) C_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
K(k_0)\cdot K(k_0) \biggl[ \cosh\eta_0(1 - \cosh\eta_0)  \biggr] 
+ 2K(k_0)\cdot E(k_0) \biggl[ \cosh^2\eta_0  + 1\biggr] 
- E(k_0)\cdot E(k_0) \biggl[ \cosh\eta_0(1 + \cosh\eta_0)  \biggr]
</math>
  </td>
</tr>
</table>

This last, simplifed expression gives, as above, <math>~C_0(3) = 0.945933523</math>.  <font color="red">TERRIFIC!</font>

Finally then, for any choice of <math>~\eta_0</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\eta,\theta)\biggr|_\mathrm{exterior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{2^{3} }{3\pi^3} 
\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] 
\frac{a}{ [ (\varpi + a)^2 + (z - Z_0)^2 ]^{1 / 2} } \cdot  \boldsymbol{K}(k) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \biggl\{
K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ] 
+ 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0  + 1 ] 
- E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0)  ]
\biggr\}  \, .
</math>
  </td>
</tr>
</table>