Editing Apps/Wong1973Potential (section)

====W====

At first glance, the expression for <math>\Phi_\mathrm{W}</math> is much more complex than the expression for <math>\Phi_\mathrm{CT}</math>.  It not only requires a double integration, but also an infinite-series summation over the integer index, <math>n</math>; and the radial integrand contains a product of two (relatively unfamiliar) associated Legendre functions of various half-integer orders (toroidal functions).  Developing a numerical algorithm to evaluate <math>\Phi_\mathrm{W}</math> to a certain accuracy &#8212; presumably set by the number of terms included in the summation over <math>~n</math> &#8212; would certainly be a more challenging task than developing a numerical algorithm to evaluate <math>\Phi_\mathrm{CT}</math>.

=====Analytic Integration Over the Angular Coordinate=====
However, focusing only on the integration over the angular coordinate, we see that the integrand in the expression for <math>\Phi_\mathrm{W}</math> is significantly less imposing than the one that appears in the expression for <math>\Phi_\mathrm{CT}</math>.  {{ Wong73 }} was able to evaluate this definite integral in closed form, analytically.  While Wong does not record the detailed steps that he used to evaluate this definite integral, he does indicate that he received guidance from Volume I of [https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erd&eacute;lyi's (1953)] ''Higher Transcendental Functions''.  We therefore presume that he adopted the line of reasoning that we have [[#A.3|detailed in the Appendix, below, in deriving the expression labeled]] <font color="green" size="+1">&#x2462;</font>.  Wong recognized, what we have explicitly demonstrated, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\int_{-\pi}^\pi d\theta^' (\cosh\eta^' - \cos\theta^')^{- 5 / 2} \cos[n(\theta - \theta^')]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="right">
<math>2\cos(n\theta)
\int_0^\pi \frac{ \cos(n\theta^')~d\theta^' }{ (\cosh\eta^' - \cos \theta^')^{\frac{5}{2}} }
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{8\sqrt{2}}{3} \biggl[ \frac{\cos(n\theta)}{\sinh^2\eta^'} \biggr] Q^2_{n- \frac{1}{2}} (\cosh\eta^') \,.
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="5">
{{ Wong73 }}, p. 293, Eq. (2.56)
  </td>
</tr>
</table>

<table border="1" cellpadding="10" align="center" width="90%"><tr><td align="left">
<font color="red">'''CAUTION:'''</font>&nbsp; It is important to appreciate that, in this expression as well as in the expressions to follow, the term, <math>Q^2_{n-\frac{1}{2}}(z)</math>, is ''not'' the square of the zero-order toroidal function, <math>Q^0_{n - \frac{1}{2}}(z)</math>, but is instead the toroidal function of order two.  In an [[Appendix/Mathematics/ToroidalSynopsis01#Evaluating_Q2.CE.BD|accompanying discussion]] we present an analytic expression for <math>Q^2_{-\frac{1}{2}}(z)</math> &#8212; and a separate analytic expression for <math>Q^1_{-\frac{1}{2}}(z)</math> &#8212; in terms of complete elliptic integrals, as well as a recurrence relation that can be used to generate analytic expressions for all other order-two (and all other order-one) toroidal functions that have higher half-integer degrees, <math>n-\tfrac{1}{2}</math> for <math>n \ge 1</math>.
</td></tr></table>


Hence, Wong was able to simplify the expression for <math>\Phi_\mathrm{W}</math> to one that &#8212; albeit, in addition to an infinite summation over the index, <math>n</math> &#8212; only requires integration over the radial coordinate, <math>\eta^'</math>.  Specifically, he obtained, 

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

<tr>
  <td align="right">
<math>\Phi_\mathrm{W}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 2G \rho_0 a^2 (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n 
\int_{\eta_0}^\infty d\eta^' ~\sinh\eta^'~P_{n-1 / 2}(\cosh\eta_<) ~Q_{n-1 / 2}(\cosh\eta_>)\biggl\{
\frac{8\sqrt{2}}{3} \biggl[ \frac{\cos(n\theta)}{\sinh^2\eta^'} \biggr] Q^2_{n- \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{16\sqrt{2}}{3}  \biggr) G \rho_0 a^2 (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta)
\int_{\eta_0}^\infty d\eta^' \Biggl[ \frac{Q^2_{n- \frac{1}{2}} (\cosh\eta^')}{\sinh\eta^'} \Biggr] ~P_{n-1 / 2}(\cosh\eta_<) ~Q_{n-1 / 2}(\cosh\eta_>)
\, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
{{ Wong73 }}, p. 294, Eq. (2.57)
  </td>
</tr>
</table>

=====Analytic Integration Over the Radial Coordinate=====

Now, in considering how to handle integration over the radial coordinate, <math>\eta^'</math>, let's examine, first, the case where <math>\eta^' \ge \eta_0 > \eta</math>, that is, the potential is being evaluated at a location that is entirely outside of the torus.  In this case,

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

<tr>
  <td align="right">
<math>\Phi_\mathrm{W}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \biggl( \frac{16\sqrt{2}}{3}  \biggr) G \rho_0 a^2 (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) ~P_{n-1 / 2}(\cosh\eta) 
\int_{\eta_0}^\infty d\eta^' \Biggl[ \frac{Q^2_{n- \frac{1}{2}} (\cosh\eta^')}{\sinh\eta^'} \Biggr] ~Q_{n-1 / 2}(\cosh\eta^')
\, .
</math>
  </td>
</tr>
</table>

So we are interested in carrying out the "radial" integral,

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

<tr>
  <td align="right">
<math>
\int_{\eta_0}^\infty d\eta^' \sinh\eta^' \Biggl[ \frac{Q^2_{n- \frac{1}{2}} (\cosh\eta^')}{\sinh^2\eta^'} \Biggr] ~Q_{n-1 / 2}(\cosh\eta^')
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\int_{\cosh(\eta_0)}^{\cosh(\infty)} dt \Biggl[ \frac{Q^2_{n- \frac{1}{2}} (t)~Q_{n-\frac{1}{2}}(t)}{t^2 - 1} \Biggr]  \, ,
</math>
  </td>
</tr>
</table>
where we have made the association, <math>\cosh\eta^' \rightarrow t</math>, in which case, <math>dt = \sinh\eta^' d\eta^'</math>.  Following the line of reasoning that we have [[#A.4|detailed in the Appendix, below, in deriving the expression labeled]] <font color="green" size="+1">&#x2463;</font>, this integral can be evaluated in closed form to give,

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

<tr>
  <td align="right">
<math>
\int_a^b\biggl[  \frac{Q_{n - \frac{1}{2}}^2(t) ~Q_{n - \frac{1}{2}}(t) }{(t^2-1)}\biggr]~dt
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{4} \biggl[
(n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t) ~ Q_{n - \frac{1}{2}}(t)
-  (n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(t) Q_{n - \frac{1}{2}}^2(t) 
\biggr]_a^b \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Wong73 }}, Eq. (2.58)
  </td>
</tr>
</table>
Hence, we have,

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

<tr>
  <td align="right">
<math>\Phi_\mathrm{W}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \biggl( \frac{4\sqrt{2}}{3}  \biggr) G \rho_0 a^2 (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times P_{n-1 / 2}(\cosh\eta) 
\biggl[
(n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t) ~ Q_{n - \frac{1}{2}}(t)
-  (n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(t) Q_{n - \frac{1}{2}}^2(t) 
\biggr]_{t = \cosh \eta_0}^{t = \cosh\infty} \, .
</math>
  </td>
</tr>
</table>
Or, given that, in this uniform-density configuration, the [[Apps/DysonWongTori#DensityFormula|density is exactly the mass divided by the torus volume]], that is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\rho_0 = \frac{M}{V}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{M \sinh^3\eta_0}{2\pi^2 a^3 \cosh{\eta_0}} \, ,</math>
  </td>
</tr>
</table>
we have,

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

<tr>
  <td align="right">
<math>\Phi_\mathrm{W}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-  \frac{2\sqrt{2}~ GM}{3\pi^2 a}  \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times P_{n-1 / 2}(\cosh\eta) 
\biggl[
(n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t) ~ Q_{n - \frac{1}{2}}(t)
-  (n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(t) Q_{n - \frac{1}{2}}^2(t) 
\biggr]_{t = \cosh \eta_0}^{t = \cosh\infty} \, .
</math>
  </td>
</tr>
</table>

In a [[Appendix/Mathematics/ToroidalFunctions#Asymptotic_Behavior|separate chapter]], we have examined the asymptotic behavior of the toroidal functions, <math>Q_{n-\tfrac{1}{2}}(t)</math>, which is,

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

<tr>
  <td align="right">
<math>\lim_{\chi\rightarrow \infty} Q_{n-\frac{1}{2}}(t)</math>
  </td>
  <td align="center">
<math>\propto</math>
  </td>
  <td align="left">
<math>
\frac{1}{t^{n+\frac{1}{2}} } \, .
</math>
  </td>
</tr>

</table>
This means that, at the upper integration limit, these toroidal functions go to zero.  As a result, we can write,

<table border="1" align="center" cellpadding="8">
<tr><td align="center">Exterior Solution: &nbsp;<math>\eta_0 \ge \eta</math></td></tr>
<tr><td align="left">

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

<tr>
  <td align="right">
<math>\Phi_\mathrm{W}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-  \frac{2\sqrt{2}~ GM}{3\pi^2 a}  \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times P_{n-1 / 2}(\cosh\eta) 
\biggl[
(n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(\cosh \eta_0) Q_{n - \frac{1}{2}}^2(\cosh \eta_0) 
- (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(\cosh \eta_0) ~ Q_{n - \frac{1}{2}}(\cosh \eta_0)
\biggr] \, .
</math>
  </td>
</tr>
</table>

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

This expression matches the potential for the ''exterior region'' that is obtained by combining Eqs. (2.59), (2.61) and (2.63) from {{ Wong73 }}.

Now, let's examine the potential evaluated at a radial location, <math>\eta</math>, that is positioned inside the surface of the torus.  That is, <math>\eta_0 < \eta \le \infty</math>.  We need to add expressions that have two different sets of integration limits as follows.  First, over the subregion, <math>\eta \le \eta^' \le \infty</math>, we use the same expression for <math>\Phi_\mathrm{W}</math> but employ the new limits, as indicated; and second, over the subregion <math>\eta_0 \le \eta^' \le \eta</math>, we swap the roles of <math>P_{n-\frac{1}{2}}</math> and <math>Q_{n-\frac{1}{2}}</math>.

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


<tr>
  <td align="right">
<math>\Phi_\mathrm{W}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\Phi_\mathrm{W}|_\mathrm{subregion1} + \Phi_\mathrm{W}|_\mathrm{subregion2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-  \frac{2\sqrt{2}~ GM}{3\pi^2 a}  \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times P_{n-1 / 2}(\cosh\eta) 
\biggl[
(n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t) ~ Q_{n - \frac{1}{2}}(t)
-  (n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(t) Q_{n - \frac{1}{2}}^2(t) 
\biggr]_{t = \cosh \eta}^{t = \cosh\infty} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
-  \frac{2\sqrt{2}~ GM}{3\pi^2 a}  \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times Q_{n-1 / 2}(\cosh\eta) 
\biggl[
(n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t) ~ P_{n - \frac{1}{2}}(t)
-  (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(t) Q_{n - \frac{1}{2}}^2(t) 
\biggr]_{t = \cosh \eta_0}^{t = \cosh\eta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
-  \frac{2\sqrt{2}~ GM}{3\pi^2 a}  \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times \biggl\{ P_{n-1 / 2}(\cosh\eta) 
\biggl[
(n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(\cosh\eta) Q_{n - \frac{1}{2}}^2(\cosh\eta) 
- (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(\cosh\eta) ~ Q_{n - \frac{1}{2}}(\cosh\eta)
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ Q_{n-1 / 2}(\cosh\eta) 
\biggl[
(n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(\cosh\eta) ~ P_{n - \frac{1}{2}}(\cosh\eta)
-  (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh\eta) Q_{n - \frac{1}{2}}^2(\cosh\eta) 
\biggr]
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-  \frac{2\sqrt{2}~ GM}{3\pi^2 a}  \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times \biggl\{ P_{n-1 / 2}(\cosh\eta) 
\biggl[
(n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(\cosh\eta) Q_{n - \frac{1}{2}}^2(\cosh\eta) 
\biggr]
- Q_{n-1 / 2}(\cosh\eta) 
\biggl[
(n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh\eta) Q_{n - \frac{1}{2}}^2(\cosh\eta) 
\biggr]
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-  \frac{2\sqrt{2}~ GM}{3\pi^2 a}  \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times \biggl\{  
Q_{n-1 / 2}(\cosh\eta) 
\biggl[
 (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh{\eta_0}) Q_{n - \frac{1}{2}}^2(\cosh{\eta_0}) 
- (n - \tfrac{3}{2}) ~ P_{n - \frac{1}{2}}(\cosh{\eta_0})~Q^2_{n + \frac{1}{2}}(\cosh{\eta_0})
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- (n+\tfrac{1}{2})Q_{n - \frac{1}{2}}^2(\cosh\eta)  \biggl[
Q_{n-1 / 2}(\cosh\eta) P_{n+\frac{1}{2}}(\cosh\eta) 
- Q_{n+\frac{1}{2}}(\cosh\eta) P_{n-1 / 2}(\cosh\eta)
\biggr]
\biggr\}
</math>
  </td>
</tr>
</table>

Drawing on the identified "Key Equation" expression,

{{ Math/EQ_Toroidal06 }}

and adopting the associations, <math>z = \xi = \cosh\eta</math> and <math>\ell \rightarrow (n-\tfrac{1}{2})</math>, we recognize that, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>(n+\tfrac{1}{2})[P_{n+\frac{1}{2}}(\cosh\eta) Q_{n-\frac{1}{2}} (\cosh\eta) - P_{n-\frac{1}{2}}(\cosh\eta)Q_{n+\frac{1}{2}}(\cosh\eta)] </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 \, .
</math>
  </td>
</tr>
</table>

Hence, we can write,

<table border="1" align="center" cellpadding="8">
<tr><td align="center">Interior Solution: &nbsp;<math>\eta \ge \eta_0 </math></td></tr>
<tr><td align="left">

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

<tr>
  <td align="right">
<math>\Phi_\mathrm{W}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-  \frac{2\sqrt{2}~ GM}{3\pi^2 a}  \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times \biggl\{  
Q_{n-1 / 2}(\cosh\eta) 
\biggl[
 (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh{\eta_0}) Q_{n - \frac{1}{2}}^2(\cosh{\eta_0}) 
- (n - \tfrac{3}{2}) ~ P_{n - \frac{1}{2}}(\cosh{\eta_0})~Q^2_{n + \frac{1}{2}}(\cosh{\eta_0})
\biggr] - Q_{n - \frac{1}{2}}^2(\cosh\eta) \biggr\} \, .
</math>
  </td>
</tr>
</table>

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


This expression matches the potential for the ''interior region'' that is obtained by combining Eqs. (2.59), (2.60) and (2.62) from {{ Wong73 }}.  Finally, drawing on yet another identified "Key Equation," namely,

{{ Math/EQ_Toroidal08 }}

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 2 July 2018:  Note that this matches Eq. (2.64) in {{ Wong73 }}, except that the numerator on the right-hand side of his equation contains  the hyperbolic sine function raised to the first power whereas in our expression it is squared.  We are confident that it should be squared, in part, because the term ''is'' squared when it reappears in Eq. (2.65) of {{ Wong73 }}.]]and adopting the associations, <math>\mu \rightarrow 2</math>, <math>\nu \rightarrow \theta</math>, and <math>z \rightarrow \cosh\eta</math>, we see that,

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

<tr>
  <td align="right">
<math>
\sum_{n=0}^\infty \epsilon_n  \cos(n\theta) ~Q^2_{n-\frac{1}{2}}(\cosh\eta)
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(-1)^2~\biggl(\frac{\pi}{2} \biggr)^{1 / 2} \Gamma(2 + \tfrac{1}{2})\biggl[ \frac{ \sinh^2\eta }{(\cosh\eta - \cos\theta)^{\frac{5}{2}}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\pi}{2} \biggr)^{1 / 2} \biggl[ \frac{4! \sqrt{\pi}}{ 4^2 \cdot 2! } \biggr] \biggl[ \frac{ \sinh^2\eta }{(\cosh\eta - \cos\theta)^{\frac{5}{2}} } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{1}{2} \biggr)^{1 / 2} \biggl[ \frac{ 3\pi}{ 2^2} \biggr] \biggl[ \frac{ \sinh^2\eta }{(\cosh\eta - \cos\theta)^{\frac{5}{2}} } \biggr] \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Wong73 }}, Eq. (2.64)<br />except, see accompanying (pink, scroll-over balloon) COMMENT
  </td>
</tr>
</table>

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 2 July 2018:  Note that this matches Eq. (2.65) of {{ Wong73 }}, except that in his expression the first term inside the curly braces on the right-hand side contains an extra factor of &pi;.  We are confident that our derived expression is correct because it is consistent with the immediately preceding expression &#8212; labeled Eq. (2.64) in {{ Wong73 }}.]]Hence, the ''interior solution'' can be rewritten as,

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

<tr>
  <td align="right">
<math>\Phi_\mathrm{W}(\eta,\theta)\bigr|_\mathrm{interior}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{ GM}{2\pi a}  \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] \biggl\{ \biggl[ \frac{ \sinh\eta }{(\cosh\eta - \cos\theta) } \biggr]^2
~ -  ~ \frac{2^2\sqrt{2}~ }{3\pi}  (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times 
Q_{n-1 / 2}(\cosh\eta) 
\biggl[
 (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh{\eta_0}) Q_{n - \frac{1}{2}}^2(\cosh{\eta_0}) 
- (n - \tfrac{3}{2}) ~ P_{n - \frac{1}{2}}(\cosh{\eta_0})~Q^2_{n + \frac{1}{2}}(\cosh{\eta_0})
\biggr] \biggr\} \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Wong73 }}, Eq. (2.65)<br />except, see accompanying (pink, scroll-over balloon) COMMENT
  </td>
</tr>
</table>