Editing Apps/Wong1973Potential (section)

==Focus on Exterior Solution==
In an [[Appendix/Ramblings/Dyson1893Part1#External_Potential_in_Terms_of_Angle_.CF.88|accompanying discussion]], we provide details related to Dyson's (1893a) derivation of the potential exterior to an anchor ring (torus).  His derived expression for the potential was written in terms of complete elliptic integrals of the first and second kind whose arguments, most naturally, were,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{R_1-R}{R_1+R} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tanh\biggl(\frac{\eta}{2}\biggr) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1-e^{-\eta}}{1 + e^{-\eta}} \, . </math>
  </td>
</tr>
</table>

(See [[2DStructure/ToroidalGreenFunction#Appendix_B:_Elliptic_Integrals|Appendix B]] for examples of alternate parameter definitions.)  Drawing from our [[Appendix/SpecialFunctions#Toroidal_Function_Evaluations|accompanying table of ''Toroidal Function Evaluations'']], let's nudge Wong's analytic ''exterior'' solution into a similar form.

The first (n = 0) term in Wong's series summation is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_{\mathrm{W}0} (\eta,\theta)\biggr|_\mathrm{exterior} </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
(\cosh\eta - \cos\theta)^{1 / 2} ~C_0(\cosh\eta_0)\biggl[ \frac{\pi}{2} \cdot \cosh\frac{\eta}{2} \biggr]^{-1} K\biggl(\tanh\frac{\eta}{2} \biggr) 
</math>
  </td>
</tr>
</table>

The second (n = 1) term in the series is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_{\mathrm{W}1}(\eta,\theta)\biggr|_\mathrm{exterior} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-2D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\cos(\theta) C_1(\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>~
-2D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\cos(\theta) C_1(\cosh\eta_0) \biggl[\frac{2}{\pi}~e^{\eta/2}\biggr]~ E( \sqrt{1-e^{-2\eta}} ) 
</math>
  </td>
</tr>
</table>

The third (n = 2) term in the series is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_{\mathrm{W}2}(\eta,\theta)\biggr|_\mathrm{exterior} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-2D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\cos(2\theta) C_2(\cosh\eta_0)P_{+\frac{3}{2}}(\cosh\eta) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-2D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\cos(2\theta) C_2(\cosh\eta_0) \biggl\{
4 \cosh\eta \biggl[ \frac{n-1}{2n-1} \biggr] P_{n-\frac{3}{2}}(\cosh\eta) - \biggl[ \frac{2n-3}{2n-1}\biggr]P_{n-\frac{5}{2}}(\cosh\eta)
\biggr\}_{n=2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-2D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\cos(2\theta) C_2(\cosh\eta_0) \biggl\{
4 \cosh\eta \biggl[ \frac{1}{3} \biggr] P_{+\frac{1}{2}}(\cosh\eta) - \biggl[ \frac{1}{3}\biggr]P_{-\frac{1}{2}}(\cosh\eta)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\biggl( \frac{2}{3} \biggr)D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\cos(2\theta) C_2(\cosh\eta_0) \biggl\{
4 \cosh\eta ~ \biggl[\frac{2}{\pi}~e^{\eta/2}\biggr]~ E( \sqrt{1-e^{-2\eta}} )  
- \biggl[ \frac{\pi}{2} \cdot \cosh\frac{\eta}{2} \biggr]^{-1} K\biggl(\tanh\frac{\eta}{2} \biggr)
\biggr\}
</math>
  </td>
</tr>
</table>

Again referencing our [[2DStructure/ToroidalGreenFunction#Appendix_B:_Elliptic_Integrals|accompanying Appendix B]], we can make the following self-consistent parameter associations:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sqrt{1 - e^{-2\eta}}</math>
  </td>
  <td align="center">
<math>~\leftrightarrow</math>
  </td>
  <td align="left">
<math>~k</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\tanh\biggl(\frac{\eta}{2}\biggr)</math>
  </td>
  <td align="center">
<math>~\leftrightarrow</math>
  </td>
  <td align="left">
<math>~k_1 = \mu \, .</math>
  </td>
</tr>
</table>
Combining the definition of <math>~D_0</math>, as provided above, with our [[Apps/DysonWongTori#DensityFormula|expression for the torus density]], <math>~\rho_0</math>, we also have,
<div align="center">
<math>D_0 = \frac{2^{5 / 2}}{3} \cdot \frac{a^3\rho_0}{M} \, .</math>
</div>
Appreciating, as well, that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\cosh\eta - \cos\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2a^2}{RR_1} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~e^{\eta/2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{R_1}{R}\biggr)^{1 / 2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\cosh \frac{\eta}{2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R_1 + R}{2(RR_1)^{1 / 2}} \, ,</math>
  </td>
</tr>
</table>
we have,

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

<tr>
  <td align="right">
<math>~\Phi_{\mathrm{W}0} (\eta,\theta)\biggr|_\mathrm{exterior} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl(\frac{2D_0}{\pi}\biggr)~C_0(\cosh\eta_0) \biggl[ \frac{2a^2}{RR_1} \biggr]^{1 / 2}
\biggl[ \frac{2(RR_1)^{1 / 2}}{R_1 + R}\biggr] K(\mu) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl(\frac{a}{\pi}\biggr) 2^{3 / 2} D_0 \cdot~C_0(\cosh\eta_0) \biggl[ \frac{2K(\mu)}{R_1 + R} \biggr] \, ;
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_{\mathrm{W}1}(\eta,\theta)\biggr|_\mathrm{exterior} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{4D_0}{\pi} \cdot ~ C_1(\cosh\eta_0) 
\biggl[ \frac{2a^2}{RR_1} \biggr]^{1 / 2}\biggl[\frac{R_1}{R}\biggr]^{1 / 2} ~\cos(\theta) ~ E( k ) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl( \frac{a}{\pi}\biggr) 2^{5 / 2}D_0 \cdot ~ C_1(\cosh\eta_0) 
~\frac{\cos(\theta)}{R} \biggl[
\frac{2E(\mu)}{1+\mu} - (1-\mu)K(\mu)
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl( \frac{a}{\pi}\biggr) 2^{5 / 2}D_0 \cdot ~ C_1(\cosh\eta_0) 
~\cos(\theta) \biggl[
\frac{E(\mu)(R_1 + R)}{R R_1}  - \frac{2K(\mu)}{R_1+R}
\biggr] \, ;
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_{\mathrm{W}2}(\eta,\theta)\biggr|_\mathrm{exterior} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\biggl( \frac{4}{3\pi} \biggr)D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\cos(2\theta) C_2(\cosh\eta_0) \biggl\{
4 \cosh\eta ~ \biggl[\frac{R_1}{R}\biggr]^{1 / 2}~ E( k )  
- \biggl[ \frac{2(RR_1)^{1 / 2}}{R_1 + R} \biggr] K(\mu)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\biggl( \frac{4}{3\pi} \biggr)D_0 \biggl[ \frac{2a^2}{RR_1} \biggr]^{1 / 2}
C_2(\cosh\eta_0)~\cos(2\theta)  \biggl\{
4 \biggl[ \frac{R_1^2 + R^2}{2RR_1} \biggr] ~ \biggl[\frac{R_1}{R}\biggr]^{1 / 2}\biggl[ \frac{2E(\mu)}{1+\mu} - (1-\mu)K(\mu) \biggr]  
- \biggl[ \frac{2(RR_1)^{1 / 2}}{R_1 + R} \biggr] K(\mu)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\biggl(\frac{a}{\pi}\biggr)\biggl( \frac{2^{5 / 2}}{3} \biggr)D_0 
C_2(\cosh\eta_0)~\cos(2\theta)  \biggl\{
4 \biggl[ \frac{R_1^2 + R^2}{2R^2R_1} \biggr] \biggl[ \frac{2E(\mu)}{1+\mu} - (1-\mu)K(\mu) \biggr]  
- \biggl[ \frac{2}{R_1 + R} \biggr] K(\mu)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\biggl(\frac{a}{\pi}\biggr)\biggl( \frac{2^{5 / 2}}{3} \biggr)D_0 
C_2(\cosh\eta_0)~\cos(2\theta)  \biggl\{
4 \biggl[ \frac{R_1^2 + R^2}{R^2R_1} \biggr] \biggl[ \frac{R_1+R}{2R_1}  \biggr]E(\mu)  
~-~4 \biggl[ \frac{R_1^2 + R^2}{2R^2R_1} \biggr] \biggl[\frac{2R}{R_1+R}\biggr]K(\mu)   
~-~ \biggl[ \frac{2}{R_1 + R} \biggr] K(\mu)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\biggl(\frac{a}{\pi}\biggr)\biggl( \frac{2^{7 / 2}}{3} \biggr)D_0 
C_2(\cosh\eta_0)~\cos(2\theta)  \biggl\{
\biggl[ \frac{R_1^2 + R^2}{RR_1} \biggr]  \frac{E(\mu)(R_1+R)}{RR_1}    
~-~\biggl[ \frac{R_1^2 + R^2}{RR_1} \biggr] \biggl[\frac{2K(\mu)}{R_1+R}\biggr]   
~-~ \biggl[ \frac{K(\mu)}{R_1 + R} \biggr] 
\biggr\}  \, .
</math>
  </td>
</tr>
</table>

Recognizing, again (as immediately above and as in our [[Appendix/Ramblings/Dyson1893Part1#EKrelation|separate summary of Dyson's results]]), that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{E(\mu)(R_1+R)}{RR_1} - \frac{2K(\mu)}{R_1+R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{E(k)}{R} \, ,</math>
  </td>
</tr>
</table>
where the alternate parameter, 
<div align="center">
<math>~k = \frac{2\sqrt{\mu}}{1+\mu} = \biggl[1 - \biggr(\frac{R}{R_1}\biggr)^2\biggr]^{1 / 2} 
= \biggl[ \frac{2}{\coth\eta+1}\biggr]^{1 / 2} = \sqrt{1-e^{-2\eta}} \, ,</math>
</div>
<span id="ThreeTermsAdded">these three terms added together give,</span>

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

<tr>
  <td align="right">
<math>~\frac{\pi}{a}\biggl[\Phi_{\mathrm{W}0} + \Phi_{\mathrm{W}1} + \Phi_{\mathrm{W}2}\biggr]_\mathrm{exterior} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2^{3 / 2} D_0 \biggl[ \frac{2K(\mu)}{R_1 + R} \biggr] \biggl\{ C_0(\cosh\eta_0) 
-~\biggl( \frac{2}{3} \biggr)
C_2(\cosh\eta_0)~\cos(2\theta)  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 2^{5 / 2}D_0 \biggl[ \frac{E(k)}{R}  \biggr]   \biggl\{
~ C_1(\cosh\eta_0)~\cos(\theta)  
+~\biggl( \frac{2}{3} \biggr)
C_2(\cosh\eta_0)~\cos(2\theta)  
\biggl[ \frac{R_1^2 + R^2}{RR_1} \biggr]      
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2^{3 / 2} D_0 \biggl[ \frac{2K(\mu)}{R_1 + R} \biggr] \biggl\{ C_0(\cosh\eta_0) 
-~\biggl( \frac{2}{3} \biggr)
C_2(\cosh\eta_0)~\cos(2\theta)  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 2^{5 / 2}D_0 \biggl[ \frac{E(k)}{R}  \biggr]   \biggl\{
~ C_1(\cosh\eta_0)~\cos(\theta)  
+~\biggl( \frac{2}{3} \biggr)
C_2(\cosh\eta_0)~\cos(2\theta)  
\biggl( \frac{4c^2 }{RR_1} \biggr)      
+~\biggl( \frac{4}{3} \biggr)
C_2(\cosh\eta_0)~\cos(2\theta)\cos\theta        
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2^{3 / 2} D_0 \biggl[ \frac{2K(\mu)}{R_1 + R} \biggr] \biggl\{ C_0(\cosh\eta_0) 
-~\biggl( \frac{2}{3} \biggr)
C_2(\cosh\eta_0)~\cos(2\theta)  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 2^{5 / 2}D_0 \biggl[ \frac{E(k)}{R}  \biggr]   \biggl\{
~ \biggl[ C_1(\cosh\eta_0) + \frac{2}{3} \cdot C_2(\cosh\eta_0) \biggr]\cos(\theta)  
+~\biggl( \frac{2}{3} \biggr)\biggl( \frac{4c^2 }{RR_1} \biggr)
C_2(\cosh\eta_0)~\cos(2\theta)  
+~\biggl( \frac{2}{3} \biggr)
C_2(\cosh\eta_0)~\cos (3\theta )       
\biggr\} 
</math>
  </td>
</tr>
</table>

<span id="Speculation">When</span> this is [[Appendix/Ramblings/Dyson1893Part1#Comparison|compared with Dyson's result]], it appears that,
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center">Speculation</th>
</tr>

<tr><td align="center">
<math>~C_0(\cosh\eta_0) = 2 - \frac{1}{2^3} \frac{a^2}{c^2} - \frac{3}{2^8} \frac{a^4}{c^4} + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, ;</math><br />

<math>~C_1(\cosh\eta_0) = \frac{1}{2^4} \frac{a^2}{c^2} + \frac{5}{2^8\cdot 3} \frac{a^4}{c^4}- \frac{2}{3}C_2(\cosh\eta_0) 
= \frac{1}{2^4} \frac{a^2}{c^2} + \frac{1}{2^7} \frac{a^4}{c^4}  + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)   \, ;</math><br />

<math>~C_2(\cosh\eta_0) = - \frac{1}{2^9} \frac{a^4}{c^4}  + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)  \, .</math>
</td></tr></table>