Editing Apps/Wong1973Potential (section)

===Summary===

<table border="1" align="center" width="85%" cellpadding="8">
<tr>
  <th align="center">
Summary of Relevant Toroidal Function Expressions<br />
(see also an [[Appendix/SpecialFunctions#Toroidal_Function_Evaluations|associated ''Special Functions'' appendix]])
  </th>
</tr>
<tr>
  <td align="left">
<table border="0" cellpadding="5" align="center">
<tr>
  <th align="left" colspan="3">Function Expression</th>
  <th align="center" colspan="1"><sup>&dagger;</sup>Case A</th>
  <th align="center" colspan="1">&nbsp; &nbsp; &nbsp;</th>
  <th align="center" colspan="1"><sup>&dagger;</sup>Case B</th>
</tr>

<tr>
  <td align="right">
<math>~Q^0_{-\frac{1}{2}}(z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
k~K( k )  \, ;
</math>
  </td>
  <td align="right"><math>~1.419337751 \times 10^0</math></td>
  <td align="center" colspan="1">&nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>~5.54081487 \times 10^{-1}</math></td>
</tr>

<tr>
  <td align="right">
<math>~Q^0_{+\frac{1}{2}}(z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
z k~K( k ) ~-~ [2(z+1)]^{1 / 2} E(k ) \, ;
</math>
  </td>
  <td align="right"><math>~1.426580119 \times 10^{-1}</math></td>
  <td align="center" colspan="1">&nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>~8.614495 \times 10^{-3}</math></td>
</tr>

<tr>
  <td align="right">
<math>~Q^0_{+\frac{3}{2}}(z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{3} \biggl[ 4zQ^0_{+\frac{1}{2}}(z) - Q^0_{-\frac{1}{2}}(z) \biggr] 
</math>
  </td>
  <td align="right"><math>~2.143519083 \times 10^{-2}</math></td>
  <td align="center" colspan="1">&nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>~1.99242 \times 10^{-4}</math></td>
</tr>

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

<tr>
  <td align="right">
<math>~Q^2_{-\frac{1}{2}}(z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{ 4 z E(k) - (z-1) K(k) }{ [2^{3} (z-1) (z^2-1) ]^{1 / 2}} \, ;
</math>
  </td>
  <td align="right"><math>~1.246521876 \times 10^{0}</math></td>
  <td align="center" colspan="1">&nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>~4.171704 \times 10^{-1}</math></td>
</tr>

<tr>
  <td align="right">
<math>~Q_{+ \frac{1}{2}}^{2}(z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\frac{1}{2^2}
\biggl\{ z k~K ( k )  ~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k)\biggr\} \, ;
</math>
  </td>
  <td align="right"><math>~5.80241772 \times 10^{-1}</math></td>
  <td align="center" colspan="1">&nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>~3.236720 \times 10^{-2}</math></td>
</tr>

<tr>
  <td align="right">
<math>~ Q^2_{+\frac{3}{2}}(z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
5Q^2_{-\frac{1}{2}}(z) - 4zQ^2_{+\frac{1}{2}}(z)
</math>
  </td>
  <td align="right"><math>~1.98094951 \times 10^{-1}</math></td>
  <td align="center" colspan="1">&nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>~1.758501 \times 10^{-3}</math></td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[ 2^3(z-1) (z^2-1) ]^{- 1 / 2} \biggl\{
\biggl[ 2^{3 / 2}z^2[ (z-1) (z^2-1) ]^{1 / 2}  k 
~-~ 5(z-1) \biggr] K(k)  
</math>
  </td>
  <td align="right">&nbsp;</td>
  <td align="center" colspan="1">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">&nbsp;</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~ \biggl[ 20 z  
~-~4z(z^2+3) \biggr] E(k) \biggr\} \, ;
</math>
  </td>
  <td align="right">&nbsp;</td>
  <td align="center" colspan="1">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">&nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>~C_0(z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{1}{2}Q^0_{+\frac{1}{2}}(z) Q_{- \frac{1}{2}}^2(z) 
~+~\tfrac{3}{2} Q^0_{- \frac{1}{2}}(z)~Q^2_{+ \frac{1}{2}}(z) \, ,
</math>
  </td>
  <td align="right"><math>~1.324251744 \times 10^{0}</math></td>
  <td align="center" colspan="1">&nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>~2.869795 \times 10^{-2}</math></td>
</tr>

<tr>
  <td align="right">
<math>~C_1(z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{3}{2} Q^0_{+\frac{3}{2}}(z) Q_{+ \frac{1}{2}}^2(z) 
+ \tfrac{1}{2} Q^0_{+ \frac{1}{2}}(z)~Q^2_{+ \frac{3}{2}}(z) \, ;
</math>
  </td>
  <td align="right"><math>~3.278631 \times 10^{-2}</math></td>
  <td align="center" colspan="1">&nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>~1.724765 \times 10^{-5}</math></td>
</tr>
</table>

  </td>
</tr>
<tr>
  <td align="left">
<sup>&dagger;</sup>''Example'' values are given here in an effort to illustrate agreement with &#8212; and partial extension of &#8212; toroidal function evaluations that we have [[Appendix/SpecialFunctions#Comparison_with_Table_IX_from_MF53|tabulated elsewhere for comparison with MF53]].  Tabulated function values are for the argument:
* '''Case A:''' &nbsp; <math>~z = 13/5 = 2.6</math>, in which case, <math>~k = \sqrt{2/(z+1)} = \sqrt{5}/3</math>, <math>~K(k) = 1.904241417</math>, and <math>~E(k) = 1.322119966</math>. 
* '''Case B:''' &nbsp; <math>~k = \sin(\pi/9)</math>, in which case, <math>~z = (2/k^2 - 1) = 16.09726435</math>, <math>~K(k) = 1.62002589</math>, and <math>~E(k) = 1.52379921</math>. 

See, also, a separate [[2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals|table giving example evaluations of elliptic integrals]], and other [[2DStructure/ToroidalGreenFunction#Appendix_B:_Elliptic_Integrals|useful elliptic integral expressions/relations]].
  </td>
</tr>
</table>


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

<tr>
  <td align="right">
<math>~D_0 \equiv \frac{2^{3 / 2}}{3\pi^2} \biggl[\frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2^{3 / 2}}{3\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] \, ,</math>
  </td>
</tr>
</table>


From [[#ThreeTermsAdded|above]], when added together, the three leading terms in Wong's expression for the exterior potential give,

<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(z) 
-~\biggl( \frac{2}{3} \biggr)
C_2(z)~\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(z) + \frac{2}{3} \cdot C_2(z) \biggr]\cos(\theta)  
+~\biggl( \frac{2}{3} \biggr)\biggl( \frac{4c^2 }{RR_1} \biggr)
C_2(z)~\cos(2\theta)  
+~\biggl( \frac{2}{3} \biggr)
C_2(z)~\cos (3\theta )       
\biggr\} 
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{E(k)}{R}  \biggr]   \biggl\{
~ \frac{\epsilon^4}{2}\biggl[ \frac{\pi^2}{2^2(1-\epsilon^2)} \biggr] \frac{2^{4}}{3\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr]  \cos\theta 
~+~ C_2(z) \cdot \frac{2^{5}}{3^2\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr]  \biggl[ \cos(\theta)  
~+~\biggl( \frac{4c^2 }{RR_1} \biggr)~\cos(2\theta)  
~+~\cos (3\theta )  \biggr]     
\biggr\} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{E(k)}{R}  \biggr]   \biggl\{
~ \frac{2\epsilon^2}{3} (1 + \epsilon^2)^{3 / 2} (1-\epsilon^2)^{-1}  \cos\theta 
~+~\frac{2^{5}}{3^2\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] C_2(z)  \biggl[ \cos(\theta)  
~+~\biggl( \frac{4c^2 }{RR_1} \biggr)~\cos(2\theta)  
~+~\cos (3\theta )  \biggr]     
\biggr\} \, .
</math>
  </td>
</tr>
</table>