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=Wong's (1973) Analytic Potential=


{{LSU_HBook_header}}

==Technical Background Review==

Drawing principally from the published works of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279)] and [http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl &amp; J. E. Tohline (1999, ApJ, 527, 86)], in an [[User:Tohline/2DStructure/ToroidalGreenFunction#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|accompanying discussion]] we have detailed how the gravitational potential of any finite mass distribution can be determined using toroidal coordinates, rather than, say, cartesian, spherical or cylindrical coordinates.  Specifically for [[User:Tohline/2DStructure/ToroidalGreenFunction#Axisymmetric_Systems|axisymmetric systems]], the relevant expression derived by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] &#8212; involving a pair of integrals over the meridional-plane coordinates, <math>~(\eta,\theta)</math>, and a summation over all ''polar angle'' modes &#8212; is,

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

<tr>
  <td align="right">
<math>~\Phi(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2Ga^2 (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n 
\int d\eta^' ~\sinh\eta^'~P^0_{n-1 / 2}(\cosh\eta_<) ~Q^0_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl\{ \frac{ ~\cos[n(\theta - \theta^')]}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr\}
\rho(\eta^',\theta^') \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], p. 293, Eq. (2.55)
  </td>
</tr>
</table>
while the expression derived by [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)] &#8212; involving different integrand expressions and no summation &#8212; is,

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

<tr>
  <td align="right">
<math>~\Phi(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 2Ga^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} 
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{- 1 / 2}(\Chi) \rho(\eta^',\theta^') \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], p. 88, Eqs. (31) &amp; (32a)
  </td>
</tr>
</table>
where, <math>~a</math> is the radius of the toroidal-coordinate system's ''anchor'' ring, <math>~P^0_{n-\frac{1}{2}}</math> and <math>~Q^0_{n-\frac{1}{2}}</math> are zero-order half-integer-degree associated Legendre functions of, respectively, the first and second kind, and
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Chi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^'} \, .
</math>
  </td>
</tr>
</table>

In the subsection of our [[User:Tohline/2DStructure/ToroidalGreenFunction#Rearranging_Terms_and_Incorporating_Special-Function_Relations|accompanying discussion]] titled, "Rearranging Terms and Incorporating Special-Function Relations," we have explicitly demonstrated that these two expressions lead to identical evaluations of the gravitational potential.

Here we build upon this technical foundation and detail how [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] was able to complete both integrals to derive an analytic expression for the potential (inside as well as outside) of axisymmetric, uniform-density tori having an arbitrarily specified ratio of the major to minor (cross-sectional) radii, <math>~R/d</math>.  This is an outstanding accomplishment that has received little attention in the astrophysics literature and, therefore, is underappreciated.


==Uniform-Density Torus==

From an [[User:Tohline/Apps/DysonWongTori#Introducing_Toroidal_Coordinates|accompanying review]], we see that for a torus of major radius <math>~R</math> and minor radius <math>~d</math>, the most accommodating (cylindrical-coordinate-based) radial  coordinate location of the toroidal coordinate system's ''anchor ring'' is defined such that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~R^2 - d^2 \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.8)
  </td>
</tr>
</table>
</div>
Then the corresponding (toroidal-coordinate-based) radial coordinate location <math>~\eta_0</math> of the surface of the torus is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cosh^{-1}\biggl(\frac{R}{d}\biggr) \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.9)
  </td>
</tr>
</table>
</div>
Alternatively, given <math>~\eta_0</math> and the value of the parameter <math>~a</math>, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a \coth\eta_0 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~d</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{\sinh\eta_0} \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eqs. (2.10) &amp; (2.11)
  </td>
</tr>
</table>
</div>

If such an axisymmetric torus has a uniform density, <math>~\rho_0</math>, then according to the above-cited Cohl &amp; Tohline (1999) expression, the gravitational potential is,

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

<tr>
  <td align="right">
<math>~\Phi_\mathrm{CT}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 2G\rho_0 a^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} 
\int_{\eta_0}^\infty d\eta^' \int_{-\pi}^\pi d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{- 1 / 2}(\Chi)  \, .
</math>
  </td>
</tr>
</table>

In contrast, according to Wong's (1973) expression, the gravitational potential is,

<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_>)
\int_{-\pi}^\pi \frac{ \cos[n(\theta - \theta^')] d\theta^' }{(\cosh\eta^' - \cos\theta^')^{5/2}} 
\, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], p. 293, Eq. (2.55)
  </td>
</tr>
</table>

===Which Is Easier to Evaluate?===
As has already been alluded to, upon evaluation, we should expect the expressions for <math>~\Phi_\mathrm{CT}</math> and for <math>~\Phi_\mathrm{W}</math> to give identical values for the gravitational potential at all coordinate locations <math>~(\eta,\theta)</math> across the entire meridional plane.  But which of the two expressions is easier to evaluate?

====CT====
If the desire is to perform the evaluation numerically, then the <math>~\Phi_\mathrm{CT}</math> expression is almost certainly easier to contend with.  It ''only'' requires a double integration; and as has been detailed in an [[User:Tohline/2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential|accompanying discussion]], the only relatively unfamiliar special function that appears in the integrand, <math>~Q_{-\frac{1}{2}}(\Chi)</math>, can be re-expressed in terms of (the more familiar) complete elliptic integral of the first kind, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q_{-\frac{1}{2}}(\Chi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mu K(\mu) \, ,
</math>
  </td>
</tr>
</table>
where,
<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>~\biggl[ \frac{2}{\Chi + 1} \biggr]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2 \sinh\eta \cdot \sinh\eta^' }{ \cosh(\eta^' + \eta) - \cos(\theta^' - \theta) } \biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>

The task is daunting, however, if the desire is to evaluate the expression for <math>~\Phi_\mathrm{CT}</math> ''analytically.'' To our knowledge, no one has yet been able to start from the expression as presented here for <math>~\Phi_\mathrm{CT}</math> and complete the integral over the angular coordinate, <math>~\theta^'</math>, analytically, let alone analytically evaluate the second, ''radial'' integral.  This is principally because the integrand of the first (angular) integral contains a nontrivial function multiplied by a special function whose argument is, itself, a nontrivial function of both <math>~\theta^'</math> and <math>~\eta^'</math>.

====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>.  [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] 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">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], 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 [[User:Tohline/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">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], 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">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], 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 [[User:Tohline/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 [[User:Tohline/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 [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] Eqs. (2.59), (2.61) and (2.63).

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,

{{ User:Tohline/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 [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] Eqs. (2.59), (2.60) and (2.62).  Finally, drawing on yet another identified "Key Equation," namely,

{{ User:Tohline/Math/EQ_Toroidal08 }}

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 2 July 2018:  Note that this matches [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] Eq. (2.64), 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 Wong's Eq. (2.65).]]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">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], 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 [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] Eq. (2.65), 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 [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)].]]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">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.65)<br />except, see accompanying (pink, scroll-over balloon) COMMENT
  </td>
</tr>
</table>

===Evaluation===

====Composite Gravitational Potential====
In order to illustrate how the potential behaves both inside and outside (in the immediate vicinity) of a uniform-density torus, [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] set <math>~R/d = \cosh\eta_0 = 3</math> and evaluated the pair of boxed-in expressions for <math>~\Phi_\mathrm{W}(\eta,\theta)</math> given above, over most of the cylindrical-coordinate range, <math>~0 \le \varpi/a \le 2.0</math> and <math>~0 \le z/a \le 1.0</math>.  (He certainly truncated the series summation at a finite number of terms, but his article does not state how many terms he included.)  The meridional-plane contour plot that resulted from this evaluation appears as Figure 7 in [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] and has been reprinted here, unaltered, in the top panel of our Figure 1.


<table border="1" align="center" cellpadding="8" width="65%">
<tr><th align="center">Figure 1</th></tr>
<tr><td align="center">
Figure 7 extracted without modification from p. 297 of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)]<p></p>
"''Toroidal and Spherical Bubble Nuclei''"<p></p>
Annals of Physics, vol. 77, pp. 279-353 &copy; Elsevier Science

&nbsp; ----- &nbsp; &nbsp;[[File:Wong1973Fig7.png|500px|Wong's Figure 7]]
</td></tr><tr><td align="center" bgcolor="#D0FFFF">
[[File:Wong111N4.png|380px|Our composite to compare with Wong's Figure 7]]
</td></tr><tr><td align="left">
'''Figure Caption:'''  Contours of the Coulomb (gravitational) potential in the meridional plane for a uniformly charged (uniform-density) torus with an aspect ratio, <math>~R/d = \cosh\eta_0 = 3</math>.  (Top) Figure 7 extracted without modification from [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)].  (Bottom) Contour plot resulting from our numerical evaluation of Wong's analytically specified potential, <math>~\Phi_\mathrm{w}</math>.   

The toroidal coordinate system's ''anchor ring'' is located at position <math>~(\varpi/a, z/a) = (1.0,0.0)</math>, as explicitly identified by the axis label (top panel), and as marked by a small purple circular marker (bottom panel).  The circular cross-section of the torus is identified by the thick-black semicircle (top panel) and by the black circle (bottom panel).   The center of this toroidal cross-section lies just outside the location of the ''anchor ring'' at the radial-coordinate position, <math>~R/a = [1 - (R/d)^{-2}]^{- 1 / 2} = 3/2^{3 / 2} </math>; this position is marked by a small white circular marker in the bottom panel.  Notice that the absolute potential minimum is positioned just ''inside'' the location of the ''anchor ring''.
</td></tr></table>


We have also evaluated [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] derived potential function, <math>~\Phi_\mathrm{W}(\eta,\theta)</math>, both inside and outside (in the immediate vicinity) of a uniform-density torus whose aspect ratio is, <math>~R/d = \cosh\eta_0 = 3</math>.  (As is [[#Contribution_from_Individual_Modes|discussed further, below]], we included only the terms, n = 0, 1, 2, &amp; 3, in the series summation.)  A colored contour plot resulting from our evaluation is displayed in the bottom panel of our Figure 1; the height and width of this rectangular image have been linearly scaled to lengths that allow straightforward &#8212; although regrettably not exact &#8212; comparison with [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] published Figure 7.  The Figure 1 caption highlights some specific points of comparison.  Qualitatively, at least, our plotted contours match quite well the contours presented by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)].

In designing a numerical algorithm to evaluate <math>~\Phi_\mathrm{W}</math>, we first followed [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] lead and rewrote the interior/exterior expressions in a more compact form.  After defining the leading amplitude coefficient, 
<div align="center">
<math>~D_0 \equiv \frac{2^{3/2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] \, ,</math>
</div>
and the pair of parameters,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B_n(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ (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}) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~C_n(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(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_{n - \frac{1}{2}}(\cosh \eta_0)~Q^2_{n + \frac{1}{2}}(\cosh \eta_0) \, ,
</math>
  </td>
</tr>
</table>

each of which is only a function of the axis ratio of the torus (via the parameter <math>~\eta_0</math>) and the ''polar angle'' index, <math>~n</math>, the (dimensionless) expression for the potential becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\sum_{n=0}^{\mathrm{nmax}} \epsilon_n \cos(n\theta) A_n(\cosh\eta) \, ,
</math>
  </td>
</tr>
</table>
<table border="1" cellpadding="10" align="right"><tr><th align="center">Figure 2:<br />3D Animated Depiction of the Warped Surface of<br />Wong's Toroidal Potential</th></tr><tr><td align="center" bgcolor="#D0FFFF">[[File:MovieWongComposite.gif|center|300px|3D Depiction of Wong's Toroidal Potential Well]]</td></tr></table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A_n(\cosh\eta)\biggr|_\mathrm{interior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~B_n(\cosh\eta_0)Q_{n-\frac{1}{2}}(\cosh\eta) - Q^2_{n-\frac{1}{2}}(\cosh\eta)</math>
  </td>
</tr>

<tr><td colspan="3" align="center">and</td></tr>

<tr>
  <td align="right">
<math>~A_n(\cosh\eta)\biggr|_\mathrm{exterior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~C_n(\cosh\eta_0)P_{n-\frac{1}{2}}(\cosh\eta) \, .</math>
  </td>
</tr>
</table>


Next, in FORTRAN, we wrote a group of functions/subroutines that allowed us to evaluate the toroidal functions, <math>~P^0_{n-\frac{1}{2}}</math> and <math>~Q^m_{n-\frac{1}{2}}</math>, individually for m = 0, 1, &amp; 2 and for all desired ''polar angle'' index values; a (double-precision version of a) set of ''Numerical Recipes'' algorithms was used to evaluate complete elliptic integrals of the first and second kind.  An [[User:Tohline/Appendix/Equation_templates#Toroidal_Function_Evaluations|accompanying appendix]] provides details regarding some key tests that we conducted in order to demonstrate the accuracy of these various functions/subroutines.  Then, as has already been stated, the (composite) meridional-plane contour diagram displayed in the bottom panel of Figure 1 was generated by setting <math>~R/d = \cosh\eta_0 = 3</math> and <math>~\mathrm{nmax} = 3</math>.  A three-dimensional ''animated'' depiction of this same potential contour surface is displayed here, on the right.  In an effort to better illustrate key features of the warped surface, different frames of the animation present the same surface as viewed from different lines of sight.

Elements of the accompanying 3D animated depiction:
<ul>
<li>The pair of (cylindrical-coordinate based) meridional-plane axes are (red) <math>~\varpi/R_0</math> and (green) <math>~Z/R_0</math>, where <math>~R_0</math> is the distance in the equatorial plane from the symmetry axis to the center of the circular cross-section of the torus.</li>
<li>The scalar value of the potential is plotted along the third (blue) axis; the surface, as a whole, has been shifted vertically in order to place the potential minimum at zero.</li>
<li>The pink, translucent cylinder identifies the meridional-plane location of the surface of the torus; because the model being illustrated has an axis ratio, <math>~R_0/d = \cosh\eta_0 =3</math>, the radius of the pink cylinder is <math>~d/R_0 = 1/3</math>.  This same cylindrical surface of the torus is identified by the thick black circle in the 2D ''projection'' of the warped contour surface that is displayed in the bottom panel of our Figure 1.  A thin, pink vertical rod identifies the center of the circular cross-section of the (pink) torus; it intersects the red, radial-coordinate axis at the location that is identified in the middle panel of Figure 1 by a small white circular marker.</li>
<li>As in the bottom panel of Figure 1, a small spherical purple marker identifies the radial location of the ''anchor ring'' associated with the adopted toroidal coordinate system.</li>
</ul>

====Contribution from Individual Modes====

It is instructive to examine how large a contribution to the ''composite'' potential, <math>~\Phi_\mathrm{W}</math>, is made by each term in the series summation.  Letting <math>~\Phi_{\mathrm{W}0}</math> represent the dimensionless amplitude of the zeroth-order "mode", we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_{\mathrm{W}0} (\eta,\theta) \equiv \biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)\biggr|_{n=0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~A_0(\cosh\eta) \, .
</math>
  </td>
</tr>
</table>
The 2D contour plot generated by this function is displayed &#8212; along with the label, n = 0 &#8212; in the upper-right corner of the central panel of Figure 3.  For comparison and reference, the ''composite'' 2D contour plot that appears in the bottom panel of Figure 1 has been redisplayed in the upper-left corner of the same Figure 3 panel.  Immediately we appreciate that the zeroth-order contribution to the potential, <math>~\Phi_{\mathrm{W}0}</math>, on its own plays a dominant role in determining the overall structure of the composite potential well.  Notice that this component depends on the ''polar angle'', <math>~\theta</math>, only through the coefficient, <math>~(\cosh\eta - \cos\theta)^{1 / 2}</math>, which it shares in common with all other terms in the series.


<table align="center" border="1" cellpadding="10">
<tr><th align="center" colspan="3">Figure 3: &nbsp; Contribution from Various ''Polar Angle'' "Modes"</th></tr>
<tr>
  <td align="center">n = 1 (magnification 2)</td>
<td align="center" rowspan="2" bgcolor="#D0FFFF">
[[File:Montage04.png|350px|Mode Ensemble]]
</td>
  <td align="center">n = 3 (magnification 100)</td>
</tr>
<tr>
<td align="center" bgcolor="#D0FFFF">
[[File:MovieWongN2.gif|300px|Contribution to potential by mode n = 1 (magnified by 2)]]
</td>
<td align="center" bgcolor="#D0FFFF">
[[File:MovieWongN4b.gif|300px|Contribution to potential by mode n = 3 (magnified by 100)]]
</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) \equiv \biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)\biggr|_{n=1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-2D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\cos(\theta) A_1(\cosh\eta) \, .
</math>
  </td>
</tr>
</table>
The 2D contour plot generated by this function is displayed &#8212; along with the label, n = 1 &#8212; in the left-hand column of the central panel of Figure 3.  By analogy with Figure 2, above, a three-dimensional ''animated'' depiction of this same potential contour surface is displayed in the left-hand panel of Figure 3.  Note, however, that the overall amplitude has been multiplied by a factor of two &#8212; that is, the warped 3D surface has been generated by the function, <math>~2\Phi_{\mathrm{W}1}</math> &#8212; in order to more vividly illustrate the structural modification to the potential that arises when this term is added to the leading, zeroth-order term.

In a directly analogous fashion, 2D contour plots showing the behavior of <math>~\Phi_{\mathrm{W}2}</math>, <math>~\Phi_{\mathrm{W}3}</math>, and <math>~\Phi_{\mathrm{W}4}</math>, have been generated and displayed &#8212; along with the corresponding labels, n = 2, 3, &amp; 4 &#8212; in the central panel of Figure 3.  A 3D animated depiction of the warped, n = 3 surface is presented in the right-hand panel of Figure 3; note that, in this instance, the overall amplitude has been multiplied by a factor of one-hundred &#8212; that is, the warped 3D surface has been generated by the function, <math>~100\Phi_{\mathrm{W}3}</math>.

A similarly vivid 3D display of the structure arising from the n = 2 and n = 4 terms in the series would have required, respectively, "magnification" of <math>~\Phi_{\mathrm{W}2}</math> by approximately a factor of fifteen and "magnification" of <math>~\Phi_{\mathrm{W}4}</math> by a factor of five-hundred; successively larger magnifications would be required for terms associated with successively higher ''polar angle'' indices, n.  We conclude, therefore, that a ''composite'' potential containing only the first four terms in the series represents the ''exact'' potential to an accuracy better than one percent.


====Equatorial Plane Behavior====
=====Interior Solution (n = 0)=====

Let's examine in more detail how the potential varies with radial position in the equatorial plane, focusing first on the ''interior'' region &#8212; that is, <math>~\infty \ge \eta \ge \eta_0</math> &#8212; and (cylindrical-coordinate-based) radial locations, <math>~\varpi < a</math>, in which case the corresponding value of the (toroidal-coordinate-based) polar angle is, <math>~\theta = \pi</math>.  In this region, the zeroth-order contribution to the potential is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{a}{GM} \biggr) \Phi_{\mathrm{W}0} (\eta,\theta) \biggr|_\mathrm{Interior} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-D_0
[\cosh\eta - \cos\theta]^{1 / 2} ~\biggl\{
Q_{-\frac{1}{2}}(\cosh\eta) [B_0(\cosh\eta_0)]
- Q^2_{-\frac{1}{2}}(\cosh\eta)
\biggr\} \, ,
</math>
  </td>
</tr>
</table>
where, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} ~P_{+\frac{1}{2}}(\cosh\eta_0) \cdot Q_{- \frac{1}{2}}^2(\cosh\eta_0) 
+ \frac{3}{2} ~ P_{- \frac{1}{2}}(\cosh\eta_0) \cdot Q^2_{+ \frac{1}{2}}(\cosh\eta_0) \, .
</math>
  </td>
</tr>
</table>

Now if <math>~\cosh\eta_0 = 3</math>, as in the above example illustration, then, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sinh\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[3^2-1]^{1 / 2} = 2^{3/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ D_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2^{3/2}}{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] 
= \frac{2^{3/2}}{3\pi^2} \biggl[ \frac{2^{9/2}}{3}\biggr]
=\biggl[\frac{2^3}{3\pi}\biggr]^{2} = 0.720506195 \, ;
</math>
  </td>
</tr>
</table>
and, drawing from the derivations and example double-precision evaluations of selected toroidal functions provided in an [[User:Tohline/Appendix/Equation_templates#Toroidal_Function_Evaluations|accompanying appendix]],

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

<tr>
  <td align="right">
<math>~B_0(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} ~P_{+\frac{1}{2}}(3) \cdot Q_{- \frac{1}{2}}^2(3) 
+ \frac{3}{2} ~ P_{- \frac{1}{2}}(3) \cdot Q^2_{+ \frac{1}{2}}(3) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} \biggl[ 1.597386605 \cdot 1.104816977 \biggr]
+ \frac{3}{2} \biggl\{   \frac{0.8346268417 }{\sqrt{3^2-1}} \biggr[ 
-\frac{3}{2} Q^1_{+\frac{1}{2}}(3) - \frac{3}{2} Q^1_{-\frac{1}{2}}(3)
\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0.882409920
+ \frac{3^2}{2^2} \biggl\{   \frac{0.8346268417 }{\sqrt{3^2-1}} \biggr[ 
0.1718911443 + 0.6753219405
\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0.882409920
+ \frac{3^2}{2^2} \biggl\{   
\frac{1}{2^2}\biggr\} = 1.444909920 \, .
</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} (\eta,\theta) \biggr|_\mathrm{Interior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~0.720506195 
[\cosh\eta - \cos\theta]^{1 / 2} ~\biggl\{ 1.444909920
Q_{-\frac{1}{2}}(\cosh\eta) - Q^2_{-\frac{1}{2}}(\cosh\eta)
\biggr\} \, .
</math>
  </td>
</tr>
</table>

=====Exterior Solution (n = 0)=====
In the region ''exterior'' to the torus, the zeroth-order contribution to the potential is,

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

<tr>
  <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>~
-D_0
[\cosh\eta - \cos\theta]^{1 / 2} ~\biggl\{
P_{-\frac{1}{2}}(\cosh\eta) [C_0(\cosh\eta_0)]
\biggr\} \, ,
</math>
  </td>
</tr>
</table>
where, 
<table border="0" cellpadding="5" align="center">

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

Drawing again from the derivations and example double-precision evaluations of selected toroidal functions provided in an [[User:Tohline/Appendix/Equation_templates#Toroidal_Function_Evaluations|accompanying appendix]], this means that, for <math>~\cosh\eta_0 = 3</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~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) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} \biggl[ 0.1128885424 \cdot 1.104816977 \biggr]
+ \frac{3}{2} \biggl\{   \frac{1.311028777 }{\sqrt{3^2-1}} \biggr[ 
-\frac{3}{2} Q^1_{+\frac{1}{2}}(3) - \frac{3}{2} Q^1_{-\frac{1}{2}}(3)
\biggr] \biggr\}
</math>
  </td>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0.0623605885
+ \frac{3^2}{2^2} \biggl\{   \frac{1.311028777 }{\sqrt{3^2-1}} \biggr[ 
0.1718911443 + 0.6753219405
\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0.0623605885
+ \frac{3^2}{2^2} \biggl\{  0.3926990811 \biggr\} = 0.945933521 \, .
</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} (\eta,\theta) \biggr|_\mathrm{Exterior} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~0.720506195~
[\cosh\eta - \cos\theta]^{1 / 2} ~\biggl\{0.945933521~
P_{-\frac{1}{2}}(\cosh\eta)
\biggr\} \, .
</math>
  </td>
</tr>
</table>



<!--
=====Second Attempt=====

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

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM}\biggr)\Phi_\mathrm{W}(\eta,\theta)\bigr|_\mathrm{interior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{ 2\pi} \biggl[ \frac{ \sinh\eta }{(\cosh\eta - \cos\theta) } \biggr]^2
~ - ~D_0  (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) 
~B_n(\cosh\eta_0)~Q_{n-1 / 2}(\cosh\eta)  
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B_n(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ (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}) \, ,
</math>
  </td>
</tr>
</table>

For the model illustrated above, <math>~\cosh\eta_0 = 3</math>, in which case,
<div align="center">
<math>~D_0
= \frac{2^{3/2} }{3\pi^2} \cdot  \frac{(\sinh^2\eta_0)^{3/2}}{\cosh\eta_0} 
= \frac{2^{3/2} }{3\pi^2} \cdot  \frac{(3^2-1)^{3/2}}{3} 
= \biggl(\frac{2^3 }{3\pi} \biggr)^2 = 0.720506194 \, .
</math>
</div>
And, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} ~P_{+\frac{1}{2}}(3) \cdot Q_{- \frac{1}{2}}^2(3) 
+ \frac{3}{2} ~ P_{- \frac{1}{2}}(3) \cdot Q^2_{+ \frac{1}{2}}(3) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} ~[1.597386605] \cdot [1.104816977 ]
+ \frac{3}{2} ~ [0.8346268417] \cdot [Q^2_{+ \frac{1}{2}}(3) ]
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~Q_{n - \frac{1}{2}}^{2}(z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(z^2-1)^{-\frac{1}{2}} \{ (n-\tfrac{3}{2}) z Q^1_{n - \frac{1}{2}}(z) - (n+\tfrac{1}{2})Q^1_{n - \frac{3}{2}}(z)\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl( \frac{3}{2^{5/2}}  \biggr)
\biggl[  Q^1_{+\frac{1}{2}}(3) + Q^1_{- \frac{1}{2}}(3) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl( \frac{3}{2^{5/2}}  \biggr)
\biggl[  -0.1718911443 - 0.6753219405 \biggr] = 0.4493025877 \, .
</math>
  </td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~ \frac{1}{2} ~[1.597386605] \cdot [1.104816977 ]
+ \frac{3}{2} ~ [0.8346268417] \cdot [0.4493025877 ]
=
1.444909919 \, .
</math>
  </td>
</tr>
</table>

As a result, including only the zeroth-order term in the series summation gives,

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

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM}\biggr)\Phi_\mathrm{W0}(\eta,\theta)\bigr|_\mathrm{interior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{ 2\pi} \biggl[ \frac{ \sinh\eta }{(\cosh\eta - \cos\theta) } \biggr]^2
~ - ~D_0  (\cosh\eta - \cos\theta)^{1 / 2}  
~B_0(\cosh\eta_0)~Q_{-1 / 2}(\cosh\eta)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{ 2\pi} \biggl[ \frac{ \sinh\eta }{(\cosh\eta - \cos\theta) } \biggr]^2
~ - ~1.041066547  (\cosh\eta - \cos\theta)^{1 / 2}  ~Q_{-1 / 2}(\cosh\eta)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{ 2\pi} \biggl[ \frac{ \sinh\eta }{(\cosh\eta - \cos\theta) } \biggr]^2
~ - ~1.041066547  ~\sqrt{\frac{\cosh\eta - \cos\theta}{\cosh\eta +1}} ~K\biggl( \sqrt{\frac{1}{\cosh\eta +1}} \biggr) \, .
</math>
  </td>
</tr>
</table>

-->

=Appendix: Selected Toroidal Function Relationships=

Here, we draw from the set of toroidal function relationships that have been identified as "Key Equations" in our [[User:Tohline/Appendix/Equation_templates#Comparison_with_Table_IX_from_MF53|accompanying ''Equations'' appendix]].  Two related appendices &#8212; numbered '''A.1''' and '''A.2''' &#8212; may be found in an [[User:Tohline/2DStructure/ToroidalGreenFunction#Appendix:_Selected_Toroidal_Function_Relationships|accompanying discussion]].

==A.3==

<table border="0" align="center" cellpadding="10" width="90%">
<tr>
  <td align="left">
Our objective is to evaluate the definite integral,

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

<tr>
  <td align="right">
<math>~\int_{-\pi}^{\pi} \frac{\cos[n(\theta - \theta^')] d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} </math>
  </td>
  <td align="right">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cos(n\theta^') \int_{-\pi}^{\pi} \frac{ \cos(n\theta) ~ d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} 
+ \sin(n\theta^') \cancelto{0}{ \int_{-\pi}^{\pi} \frac{ \sin(n\theta)  d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} }
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="right">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \cos(n\theta^') \int_{0}^{\pi} \frac{ \cos(n\theta)~ d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} \, .
</math>
  </td>
</tr>
</table>
</div>
Notice that, since the limits of the integration are <math>~-\pi</math> to <math>~+\pi</math>: &nbsp; The second integral on the right-hand-side goes to zero because the numerator of its integrand &#8212; ''i.e.,'' <math>~\sin(n\theta)</math> &#8212; is an [https://en.wikipedia.org/wiki/Even_and_odd_functions#Odd_functions odd function]; and, with regard to the first integral on the right-hand-side, the lower integration limit can be set to zero and the result doubled because the numerator of its integrand &#8212; ''i.e.,'' <math>~\cos(n\theta)</math> &#8212; is an [https://en.wikipedia.org/wiki/Even_and_odd_functions#Even_functions even function].

Drawing from Volume I of [https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erd&eacute;lyi's (1953)] ''Higher Transcendental Functions,'' we find the following "Key Equation":

{{ User:Tohline/Math/EQ_Toroidal03 }}

Next we adopt the associations, <math>~z \rightarrow \cosh\eta</math>, <math>~t \rightarrow \theta</math>, <math>~\mu \rightarrow 2</math>, and, <math>~\nu \rightarrow n - \tfrac{1}{2}</math>, where <math>~n</math> is zero or a positive integer.  In this case we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q_{n - \frac{1}{2}}^2 (\cosh\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(2\pi)^{-\frac{1}{2}} (\cosh^2\eta-1) ~\Gamma(\tfrac{5}{2})~\biggl\{
\int_0^\pi (\cosh\eta - \cos \theta)^{-\frac{5}{2}} \cos(n\theta) ~d\theta
- \cancelto{0}{\cos[(n-\tfrac{1}{2})\pi] }~~\int_0^\infty (\cosh\eta + \cosh \theta)^{- \frac{5}{2}} e^{-n\theta} ~d\theta
\biggr\} \, ,
</math>
  </td>
</tr>
</table>
where the prefactor of the second term &#8212; that is, <math>~\cos[(n-\tfrac{1}{2})\pi] </math> &#8212; goes to zero for all allowable values of the integer, <math>~n</math>.  Hence, we conclude that,
<table border="0" cellpadding="5" align="center">

<tr>
  <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" colspan="2">
<math>~\frac{ 2(2\pi)^{\frac{1}{2}} Q_{n - \frac{1}{2}}^2 (\cosh\eta) \cos(n\theta^')}{ (\cosh^2\eta-1) ~\Gamma(\tfrac{5}{2})~ } 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{ 2^3 \sqrt{2} }{ 3 } \biggr] \frac{  Q_{n - \frac{1}{2}}^2(\cosh\eta) \cos(n\theta^')}{ \sinh^2\eta  }
\, .
</math>
  </td>
  <td align="right" width="15%">
<font color="green" size="+3">&#x2462;</font>
  </td>
</tr>
</table>
where we have set, 
<div align="center">
<math>~
\Gamma(\tfrac{5}{2}) = \Gamma(\tfrac{1}{2} + 2) = \frac{ \sqrt{\pi} \cdot 4! }{4^2 \cdot 2!}
= \frac{\sqrt{\pi} \cdot 2^3\cdot 3}{  2^5 } = \frac{3 \sqrt{\pi}}{2^2} \, .
</math>
</div>

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

==A.4==


<table border="0" align="center" cellpadding="10" width="90%">
<tr>
  <td align="left">
Beginning with the identified "Key Equation",

{{ User:Tohline/Math/EQ_Toroidal05 }}

we will adopt the associations: &nbsp; <math>~z \rightarrow t</math>, <math>~\mu \rightarrow 2</math>, <math>~\nu \rightarrow (n - \tfrac{1}{2})</math>, <math>~\rho \rightarrow 0</math>, and <math>~\sigma \rightarrow ( n - \tfrac{1}{2})</math>.  As a result, [https://authors.library.caltech.edu/43491/1/Volume%201.pdf Erd&eacute;lyi's (1953)] expression becomes,
<table border="0" cellpadding="5" align="center">

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

Drawing upon the ''recurrence'' "Key Equation,"

{{ User:Tohline/Math/EQ_Toroidal04 }}

which means, after making the associations, <math>~z \rightarrow t</math>, <math>~\mu \rightarrow 0</math> and <math>~\nu \rightarrow (n-\tfrac{1}{2})</math>, that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2n t X_{n-\frac{1}{2}}(t) - (n-\tfrac{1}{2})X_{n - \frac{3}{2}}(t)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~(n-\tfrac{1}{2})X_{n - \frac{3}{2}}(t) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2n t X_{n-\frac{1}{2}}(t) - (n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t)</math>
  </td>
</tr>
</table>

the integral can be rewritten as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\int_a^b\biggl[  \frac{Q_{n - \frac{1}{2}}^2(t) ~X_{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\{
\biggl[ 2n t X_{n-\frac{1}{2}}(t) - (n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t) \biggr]  Q_{n - \frac{1}{2}}^2(t) - (n + \tfrac{3}{2}) Q_{n - \frac{3}{2}}^2(t) ~X_{n - \frac{1}{2}}(t)
\biggr\}_a^b 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{4} \biggl\{
\biggl[ 2n t  Q_{n - \frac{1}{2}}^2(t) 
- (n + \tfrac{3}{2}) Q_{n - \frac{3}{2}}^2(t) \biggr] X_{n - \frac{1}{2}}(t)
-  (n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t) Q_{n - \frac{1}{2}}^2(t) 
\biggr\}_a^b \, .
</math>
  </td>
</tr>
</table>
Returning to the same ''recurrence'' "Key Equation," but this time adopting the associations, <math>~z \rightarrow t</math>, <math>~\mu \rightarrow 2</math> and <math>~\nu \rightarrow (n-\tfrac{1}{2})</math>, we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2n t Q^2_{n - \frac{1}{2}}(t) - (n + \tfrac{3}{2}) Q^2_{n - \frac{3}{2}} (t) \, ,
</math>
  </td>
</tr>
</table>
in which case the integral becomes,

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

<tr>
  <td align="right">
<math>~
\int_a^b\biggl[  \frac{Q_{n - \frac{1}{2}}^2(t) ~X_{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) ~ X_{n - \frac{1}{2}}(t)
-  (n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t) Q_{n - \frac{1}{2}}^2(t) 
\biggr\}_a^b \, .
</math>
  </td>
  <td align="right" width="15%">
<font color="green" size="+3">&#x2463;</font>
  </td>
</tr>
</table>

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

=See Also=


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