Editing 2DStructure/ToroidalCoordinates (section)

==Perform Angular Integration Over Test Mass Distribution==
While, in principle, the pair of nested integrals must be carried out over all of space using the limits, 
<div align="center">
<math>~\xi_1|_\mathrm{min} = 1 \, , ~\xi_1|_\mathrm{max} = \infty \, , ~\theta_\mathrm{min} = - \pi \, ,</math>
&nbsp; &nbsp; and &nbsp; &nbsp; <math>~\theta_\mathrm{max} = + \pi \, ,</math>
</div>
in practice, the limits should be set to reflect the volume boundaries of the mass distribution that is responsible for generating the gravitational potential.  Here we present analytic expressions for these limits in the case of the (pink) [[#Chosen_Test_Mass_Distribution|toroidal test mass distribution specified above]]; details regarding the derivation of these expressions can be found in [[Appendix/Ramblings/ToroidalCoordinates#Yet_Again|our accompanying set of notes]].

===Identifying Limits of Integration===

<table border="1" cellpadding="8" align="right">
<tr><th align="center"><font size="+1">Figure 2</font></th></tr>
<tr>
<td align="center">
[[File:TCoordsE.gif|300px|Diagram of Torus and Toroidal Coordinates]]
</td>
</tr>
</table>
The animation shown here in Figure 2 builds upon the configuration displayed in [[#THH12Figure4|our Figure 1, above]].  It shows a meridional cross-section through the selected (pink) uniform-density, toroidal mass distribution, whose geometric properties are fully determined by specifying values for <math>~\varpi_t</math> and <math>~r_t</math>.  (For the example illustrated in Figure 2, we have specified the same values used by [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Hur&eacute; &amp; Hersant (2012)] to construct their [[#THH12Figure4|Figure 4, as reprinted above]]; namely, <math>~\varpi_t = 3/4</math> and <math>~r_t = 1/4</math>.)  Throughout the Figure 2 animation sequence, these two parameters have been fixed &#8212; thereby fixing the properties of the (pink) torus &#8212; and, in addition, we have fixed the location of the origin of a toroidal coordinate system &#8212; identified by the red-filled circular dot.  We explicitly associate this coordinate-system origin with the (cylindrical) coordinates of the point in space at which we choose to evaluate the gravitational potential, namely, <math>~(R_*, Z_*) = (a, Z_0)</math>. [For illustration purposes, in Figure 2 we have set <math>~(a, Z_0) = (1/3, 3/4)</math>.]

While the values of the four primary model parameters <math>~(a, Z_0,  \varpi_t, r_t)</math> are held fixed, Figure 2 depicts in a quantitatively precise manner how the size of a <math>~\xi_1</math>= constant toroidal surface (the off-center circle traced by the sequence of black dots) varies as the value of the radial coordinate, <math>~\xi_1</math>, is varied.  In each frame of the animation sequence, the value of <math>~\xi_1</math> that was used to define the (black) <math>\xi_1</math>-circle is printed in the lower-right corner of the image; additional quantitative details associated with each animation frame can be obtained from the [[Appendix/Ramblings/ToroidalCoordinates#Example2|table titled, "Example 2" in our accompanying notes]].


As the value of <math>~\xi_1</math> is varied from large values (small black circles) to smaller values (larger black circles), there is a maximum value, <math>~\xi_1|_\mathrm{max}</math>, at which the <math>\xi_1</math>-circle first makes contact with the (pink) equatorial-plane torus, and there is a minimum value, <math>~\xi_1|_\mathrm{min}</math>, at which it makes its final contact.  These are the limiting values of the toroidal radial coordinate to be used in the integration that produces <math>~q_0</math>.  At all values within the parameter range,
<div align="center">
<math>~\xi_1|_\mathrm{max} > \xi_1 > ~\xi_1|_\mathrm{min} \, ,</math>
</div>
the <math>\xi_1</math>-circle intersects the surface of the torus in two locations, defined by two different values of the associated angular coordinate, <math>~\xi_2</math>.  In each frame of the animation, points of intersection are marked with small yellow diamonds; the coordinates of these points of intersection are listed in the table associated with [[Appendix/Ramblings/ToroidalCoordinates#Example2|example 2, in our accompanying notes]].  For each relevant value of <math>~\xi_1</math>, these are the limiting values of the toroidal angular coordinate to be used in the integration that produces <math>~q_0</math>.  It should be realized that, ''at'' the first and final points of contact, the two values of <math>~\xi_2</math> are degenerate.  
<!-- COMMENT OUT
Next we derive the mathematical relations that give the values of <math>~\xi_1|_\mathrm{max}</math> and <math>~\xi_1|_\mathrm{min}</math> for all .

For reference purposes, Figure 2 has been displayed here, again, in the lefthand panel of Figure 4; the animation sequence presented in the righthand panel illustrates how the <math>\xi_1</math>-circle (depicted by the locus of small black dots) intersects the surface of the (pink) equatorial-plane torus as the value of <math>~\xi_1</math> is varied over the parameter range,
<div align="center">
<math>~\xi_1|_\mathrm{max} \geq \xi_1 \geq ~\xi_1|_\mathrm{min} \, ,</math>
</div>
for a toroidal coordinate system whose origin (filled, red dot) remains fixed at the (cylindrical) coordinate location, <math>~(\varpi, z) = (a, Z_0) = (\tfrac{1}{3}, \tfrac{3}{4})</math>.  For a toroidal coordinate system with this specified origin and an equatorial-plane torus having <math>~\varpi_t = \tfrac{3}{4}</math> and <math>~r_t = \tfrac{1}{4}</math> &#8212; as recorded in the top row of numbers in the Table, below &#8212; the <math>\xi_1</math>-circle makes ''first contact'' with the torus when <math>~\xi_1 = \xi_1|_\mathrm{max} = 1.1927843</math> and it makes ''final contact'' when <math>~\xi_1 = \xi_1|_\mathrm{min} = 1.0449467</math>.  The animation sequence contains ten unique frames: The value of <math>~\xi_1</math> that is associated with the <math>\xi_1</math>-circle in each case appears near the bottom-right corner of the animation frame.  These parameter values have also been recorded in the first column of ten separate rows in the following table, along with other relevant parameter values.  For example, in each frame of the animation, the points of intersection between the surface of the torus and the <math>\xi_1</math>-circle are identified by filled, green diamonds; the (cylindrical) coordinates associated with these points of intersection, <math>~(\varpi_i, z_i)</math>, are listed in each table row, along with the corresponding value of the toroidal coordinate system's angular, <math>~\xi_2</math> coordinate.

Specific values for these parameters are tabulated in a [[Appendix/Ramblings/ToroidalCoordinates#Example2|Table titled, ''Example 2'']] in our accompanying notes.
END DELETED COMMENT -->

Notice in the animation that, while the origin of the selected toroidal coordinate system (the filled red dot) remains fixed, the ''center'' of the <math>\xi_1</math>-circle does not remain fixed.  In order to highlight this behavior, the location of the center of the <math>\xi_1</math>-circle has been marked by a filled, light-blue square and, in keeping with the [[#THH12Figure4|earlier Figure 1 diagram]], a vertical, light-blue line connects this center to the equatorial plane of the cylindrical coordinate system.

===Associated Analytic Expressions===
We define the following terms that are functions only of the four principal model parameters, <math>~(a, Z_0,  \varpi_t, r_t)</math>, and therefore can be treated as constants while carrying out the pair of nested integrals that determine <math>~q_0</math>:

<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="4">&nbsp;</td>
  <td align="center">Parameters evaluated for Figure 2</td>
</tr>
<tr>
  <td align="right">
<math>~\kappa</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
Z_0^2 + a^2  - (\varpi_t^2 - r_t^2) 
</math>
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>~
\frac{5^2}{2^4\cdot 3^2} \approx 0.17361111
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~C</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~1 + \biggl( \frac{2Z_0}{\kappa}\biggr)^2 ( \varpi_t^2 - r_t^2)
</math>
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>~\frac{17 \cdot 1409}{5^4} \approx 38.3248 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta_\pm</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
- \frac{\kappa}{2} \biggl[ \frac{\varpi_t \mp r_t \sqrt{C}}{(\varpi_t + r_t)(\varpi_t - r_t)} \biggr] 
</math>
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>~
-~\frac{5^2}{2^6\cdot 3}\biggl[ 1\mp \sqrt{ \frac{17\cdot 1409}{3^2\cdot 5^4}} \biggr] 
</math>
  </td>
</tr>
</table>

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

Then,
<div align="center">
<math>\xi_1|_\mathrm{max} = 
\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2  \biggr]^{-1/2}
\approx 1.1927843</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>\xi_1|_\mathrm{min} = 
\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2  \biggr]^{-1/2}
\approx 1.0449467\, ,</math>
</div>

which implies,
<div align="center">
<math>x_\mathrm{max} = \tanh^{-1}\biggl( \frac{a}{\varpi_t-\beta_+} \biggr) 
\approx 0.61137548 </math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>x_\mathrm{min} = \tanh^{-1}\biggl( \frac{a}{\varpi_t-\beta_-} \biggr) 
\approx 0.29871048 \, .</math>
</div>

----


Also let,
<div align="center">
<math>~\xi_1 \rightarrow \cosh x \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp; in which case,
&nbsp; &nbsp; &nbsp; &nbsp; 
<math>~(\xi_1^2 - 1)^{1/2} \rightarrow \sinh x</math>
&nbsp; &nbsp; and,
&nbsp; &nbsp;  
<math>~(1-\xi_1^{-2})^{-1/2} = \frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} \rightarrow \coth x\, ,</math>
</div>
and define,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr><td align="center">
<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="4">&nbsp;</td><td align="center" colspan="3">Evaluated for</td></tr>
<tr>
  <td align="center" colspan="4">&nbsp;</td>
  <td align="center"><math>~x_\mathrm{max} ~(i.e.~\xi_1|_\mathrm{max})</math><p></p>
----
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="center"><math>~x_\mathrm{min} ~(i.e.~\xi_1|_\mathrm{min})</math><p></p>
----
  </td>
</tr>
<tr>
  <td align="right">
<math>~A(\xi_1)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{Z_0}{a} \biggr)^2 + \biggl[ \coth x - \frac{\varpi_t}{a} \biggr]^2 
</math> 
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="center">
<math>~ \approx 5.23510376 </math> 
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="center">
<math>~ \approx 6.49460545</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~B(\xi_1)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\coth x - \frac{\varpi_t}{a} + \biggl( \frac{2\varpi_t Z_0^2}{a\kappa} \biggr) </math>
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="center">
<math>~\approx 14.1645439</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="center">
<math>~\approx 15.77670608</math>
  </td>
</tr>
</table>

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

<table border="1" cellpadding="8" align="right">
<tr><th align="center"><font size="+1">Figure 3</font></th></tr>
<tr>
<td align="center">
[[File:ConstantXi2.png|300px|Diagram of Torus and xi_2-constant Toroidal Coordinate curve]]
</td>
</tr>
</table>
Then, for each value of the radial coordinate, <math>~\xi_1 = \cosh x</math>, within the range,
<div align="center">
<math>~x_\mathrm{max} \geq x \geq x_\mathrm{min} \, ,</math>
</div>
the limiting values <math>~(\pm)</math> of the angular coordinate are,

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

<tr>
  <td align="right">
<math>~\xi_2\biggr|_\pm</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \cosh x - \sinh x \biggl( \frac{2a^2}{\kappa}\cdot \frac{A}{B} \biggr)
\biggl[1 \pm \sqrt{1-\frac{AC}{B^2}}\biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>

This provides the desired analytic expression for the two limits of integration over the angular coordinate that must be carried out in order to evaluate the key function, <math>~q_0</math>, in our determination of the gravitational potential at the cylindrical-coordinate location, <math>~(R_*, Z_*) = (a, Z_0)</math>.  

We should note that at both limits &#8212; that is, ''at'' the two values of <math>~x</math> for which the functions <math>~A</math> and <math>~B</math> have been evaluated, immediately above &#8212; the ratio under the square-root,
<div align="center">
<math>~\frac{AC}{B^2} = 1 \, .</math>
</div>
Hence, at both radial-coordinate limits, the two angular-coordinate limits, <math>~\xi_2|_\pm</math>, are degenerate.  Furthermore, we have discovered that the value of this degeneracy angle is exactly the same at both radial-coordinate limits; specifically, we find that, for our chosen test-mass distribution,
<div align="center">
<math>~\xi_2\biggr|_\mathrm{deg} \approx 0.885198 \, .</math>
</div>
This means that exactly the same <math>\xi_2</math>-constant, toroidal-coordinate "line" intersects the surface of the (pink) test-mass torus at both the point of first contact <math>~(\xi_1|_\mathrm{max})</math> and the point of final contact <math>~(\xi_1|_\mathrm{min})</math>.  In order to illustrate this, this particular <math>\xi_2</math>-constant "line" has been traced by a sequence of red dots in Figure 3.  As just described, it passes smoothly through the points on the surface of the (pink) torus where the <math>\xi_1</math>-circles make first (small black-dotted circle) and final (large black-dotted circle) contact.

===Integrate===

<!--  OLD WAY
Because the density is uniform throughout our test model torus, it can be pulled out of both integrals.  In combination with the limits of integration just derived, we can therefore write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~q_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{1/2} a^{5/2}\rho_0
\int\limits_{\tanh^{-1}[a/(\varpi_t-\beta_-)]}^{\tanh^{-1}[a/(\varpi_t-\beta_+)]} \frac{K(\mu) \sinh x ~dx}{( \sinh x+\cosh x )^{1/2}}  
\int\limits_{\sin^{-1}(\xi_2|_-)}^{\sin^{-1}(\xi_2|_+)} 
\biggl[ \frac{d\theta}{(\xi_1 - \sin\theta)^{5/2}} \biggr]  \, .
</math>
  </td>
</tr>
</table>
</div>

Now, using WolframAlpha's online integrator, we find &hellip;
<div align="center">
[[File:TorusIntegration.png|450px|WolframAlpha Integration]]
</div>

Hence, the inner integral over our toroidal system's angular coordinate gives,

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

<tr>
  <td align="right">
<math>~\int\limits_{\sin^{-1}(\xi_2|_-)}^{\sin^{-1}(\xi_2|_+)} 
\biggl[ \frac{d\theta}{(\xi_1 - \sin\theta)^{5/2}} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ \frac{2}{3(\xi_1^2-1)^2 (\xi_1-\sin\theta)^{3/2}}
\biggl[
\cos\theta ( -5\xi_1^2 + 4\xi_1\sin\theta + 1) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (\xi_1+1)(\xi_1-1)^2 \biggl( \frac{\xi_1-\sin\theta}{\xi_1-1} \biggr)^{3/2} F\biggl(\frac{\pi - 2\theta}{4} \biggr| \frac{-2}{\xi_1-1}  \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 4\xi_1(\xi_1-1)(\sin\theta-\xi_1) \biggl( \frac{\xi_1-\sin\theta}{\xi_1-1} \biggr)^{1/2} E\biggl(\frac{\pi - 2\theta}{4} \biggr| \frac{-2}{\xi_1-1}  \biggr)
\biggr] \biggr\}_{\sin^{-1}(\xi_2|_-)}^{\sin^{-1}(\xi_2|_+)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ 
\biggl[ \frac{2[1-(\xi_2|_+)^2] ( -5\xi_1^2 + 4\xi_1\xi_2|_+ + 1)}{3(\xi_1^2-1)^2 (\xi_1-\xi_2|_+)^{3/2}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \tfrac{2}{3} (\xi_1-1)^{-3/2}\biggl[ (\xi_1+1) F\biggl(\frac{\cos^{-1}(\xi_2|_+)}{2} \biggr| \frac{2}{1-\xi_1}  \biggr)
- 4 \xi_1 E\biggl(\frac{\cos^{-1}(\xi_2|_+)}{2}\biggr| \frac{2}{1-\xi_1}  \biggr) \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~- \biggl\{ 
\biggl[ \frac{2[1-(\xi_2|_-)^2] ( -5\xi_1^2 + 4\xi_1\xi_2|_- + 1)}{3(\xi_1^2-1)^2 (\xi_1-\xi_2|_-)^{3/2}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \tfrac{2}{3} (\xi_1-1)^{-3/2}\biggl[ (\xi_1+1) F\biggl(\frac{\cos^{-1}(\xi_2|_-)}{2} \biggr| \frac{2}{1-\xi_1}  \biggr)
- 4 \xi_1 E\biggl(\frac{\cos^{-1}(\xi_2|_-)}{2}\biggr| \frac{2}{1-\xi_1}  \biggr) \biggr] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{2}{3} (\xi_1-1)^{-3/2}\biggl\{ 
\biggl[ \frac{[1-(\xi_2|_+)^2] ( -5\xi_1^2 + 4\xi_1\xi_2|_+ + 1)}{(\xi_1-1)^{1/2} (\xi_1+1)^2 (\xi_1-\xi_2|_+)^{3/2}} \biggr]
- \biggl[ \frac{[1-(\xi_2|_-)^2] ( -5\xi_1^2 + 4\xi_1\xi_2|_- + 1)}{(\xi_1-1)^{1/2} (\xi_1+1)^2(\xi_1-\xi_2|_-)^{3/2}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ (\xi_1+1) F\biggl(\sin^{-1}\biggl[\frac{1-\xi_2|_+}{2} \biggr] \biggr| \frac{2}{1-\xi_1}  \biggr)
- 4 \xi_1 E\biggl(\sin^{-1}\biggl[\frac{1-\xi_2|_+}{2} \biggr]\biggr| \frac{2}{1-\xi_1}  \biggr) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[ (\xi_1+1) F\biggl(\sin^{-1}\biggl[\frac{1-\xi_2|_-}{2} \biggr]\biggr| \frac{2}{1-\xi_1}  \biggr)
- 4 \xi_1 E\biggl(\sin^{-1}\biggl[\frac{1-\xi_2|_-}{2} \biggr]\biggr| \frac{2}{1-\xi_1}  \biggr) \biggr] \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

-->

Because the density is uniform throughout our test model torus, it can be pulled out of both integrals.  In combination with the limits of integration just derived, we can therefore write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~q_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-2^{1/2} a^{5/2}\rho_0
\int\limits_{\tanh^{-1}[a/(\varpi_t-\beta_-)]}^{\tanh^{-1}[a/(\varpi_t-\beta_+)]} \frac{K(\mu) \sinh x ~dx}{( \sinh x+\cosh x )^{1/2}}  
\int\limits_{\cos^{-1}(\xi_2|_-)}^{\cos^{-1}(\xi_2|_+)} 
\biggl[ \frac{d\theta}{(\xi_1 - \cos\theta)^{5/2}} \biggr]  \, .
</math>
  </td>
</tr>
</table>
</div>

Now, using WolframAlpha's online integrator, we find &hellip;
<div align="center">
[[File:TorusIntegration2.png|500px|WolframAlpha Integration]]
</div>

Hence, the inner integral over our toroidal system's angular coordinate gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\int\limits_{\cos^{-1}(\xi_2|_-)}^{\cos^{-1}(\xi_2|_+)} 
\biggl[ \frac{d\theta}{(\xi_1 - \cos\theta)^{5/2}} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl\{ \frac{2}{3(\xi_1^2 - 1)^2 (\xi_1 - \cos \theta)^{3/2}} \biggr[
\sin \theta(-5\xi_1^2 + 4\xi_1 \cos \theta + 1) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ (\xi_1+1)(\xi_1-1)^2 \biggl( \frac{\xi_1 - \cos \theta}{\xi_1-1} \biggr)^{3/2}
F\biggl( \frac{\theta}{2} \biggr| \frac{-2}{\xi_1-1}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ 4\xi_1 (\xi_1-1)\biggl( \frac{\xi_1-\cos \theta}{\xi_1-1}\biggr)^{1/2} (\cos \theta - \xi_1) 
E\biggl( \frac{\theta}{2} \biggr| \frac{-2}{\xi_1-1}\biggr) \biggr] \biggr\}_{\cos^{-1}(\xi_2|_-)}^{\cos^{-1}(\xi_2|_+)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{3(\xi_1^2 - 1)^2 } \biggr[ \frac{
\sin \theta(5\xi_1^2 - 4\xi_1 \cos \theta - 1)}{(\xi_1 - \cos \theta)^{3/2}} \biggr]_{\cos^{-1}(\xi_2|_-)}^{\cos^{-1}(\xi_2|_+)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ + \frac{2(\xi_1-1)^{1/2}}{3(\xi_1^2 - 1)^2 } \biggr[ 4\xi_1 E\biggl( \frac{\theta}{2} \biggr| \frac{-2}{\xi_1-1}\biggr) 
- (\xi_1+1) F\biggl( \frac{\theta}{2} \biggr| \frac{-2}{\xi_1-1}\biggr) 
\biggr]_{\cos^{-1}(\xi_2|_-)}^{\cos^{-1}(\xi_2|_+)} 
</math>
  </td>
</tr>
</table>
</div>

This last expression exactly matches the expression found in the integral tables published by [http://www.mathtable.com/gr/ Gradshteyn &amp; Ryzhik (1965)] &#8212; specifically, relation '''2.575'''(5) &#8212; after it is realized that the second argument of both elliptic integral functions, as published in GR65, is the square-root of the parameter, <math>~m = k^2</math>, provided by WolframAlpha.

We now have to deal with the realization that, for our particular problem, <math>~\xi_1 \geq 1</math>, which means that the second argument, <math>~m = 2/(1-\xi_1)</math>, of both elliptical integral functions is negative (if not zero).  Following [http://www.mymathlib.com/functions/elliptic_integrals.html the discussion provided at mymathlib.com], we define,

<div align="center">
<math>~k_1 \equiv \sqrt{\frac{2}{\xi_1-1}} </math>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;
<math>~k_1^' \equiv \sqrt{1-(ik_1)^2} = \sqrt{1 + \frac{2}{\xi_1-1}} = \sqrt{\frac{\xi_1 + 1}{\xi_1 - 1}} \, ,</math>
</div>

in which case,

<div align="center">
<math>~\frac{k_1}{k_1^'} = \sqrt{\frac{2}{\xi_1 + 1}} \, ,</math>
</div>

and we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~F\biggl( \frac{\theta}{2} \biggr| \frac{-2}{\xi_1-1}\biggr) ~~\rightarrow ~~ F\biggl( \frac{\theta}{2} , ik_1\biggr) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{k_1^'} \biggl[K\biggl( \frac{k_1}{k_1^'} \biggr) - F\biggl( \frac{\pi-\theta}{2} \, , \frac{k_1}{k_1^'} \biggr) \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~E\biggl( \frac{\theta}{2} \biggr| \frac{-2}{\xi_1-1}\biggr) ~~\rightarrow ~~ E\biggl( \frac{\theta}{2} , ik_1\biggr) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~k_1^' \biggl[E\biggl( \frac{k_1}{k_1^'} \biggr) - E\biggl( \frac{\pi-\theta}{2} \, , \frac{k_1}{k_1^'} \biggr) \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

<!-- COMMENT MADE ABOUT NEGATIVE PARAMETERs

While I am pleased that I am able to obtain the same result after consulting two different sources (WolframAlpha, and GR65), I am troubled that the parameter found in both elliptic integrals, <math>~m = k^2</math>, is negative.  This leads me to believe that the integration still has not been carried out properly.  Another hint that the obtained result is not physically valid comes from the notes that accompany relation '''2.575'''(5) in GR65.  It is supposed to be valid for the following range of coefficient/parameter values:
<div align="center">
<math>0 \leq \theta \leq \pi</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>\xi_1 > -1 > 0 \, .</math>
</div>
Clearly, this last condition is not met, as negative one is not greater than zero.
-->

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

<tr>
  <td align="right">
<math>~\int\limits_{\cos^{-1}(\xi_2|_-)}^{\cos^{-1}(\xi_2|_+)} 
\biggl[ \frac{d\theta}{(\xi_1 - \cos\theta)^{5/2}} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{3(\xi_1^2 - 1)^2 } \biggl\{\biggr[ \frac{
\sin \theta(5\xi_1^2 - 4\xi_1 \cos \theta - 1)}{(\xi_1 - \cos \theta)^{3/2}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ + 4\xi_1(\xi_1-1)^{1/2} \sqrt{\frac{\xi_1 + 1}{\xi_1 - 1}}  \biggl[E\biggl( \sqrt{\frac{2}{\xi_1 + 1}} \biggr) - E\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
- (\xi_1-1)^{1/2} (\xi_1+1) \sqrt{\frac{\xi_1 - 1}{\xi_1 + 1}}  \biggl[K\biggl( \sqrt{\frac{2}{\xi_1 + 1}}  \biggr) - 
F\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) \biggr]\biggr\}_{\cos^{-1}(\xi_2|_-)}^{\cos^{-1}(\xi_2|_+)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2(\xi_1+1)^{1/2} }{3(\xi_1^2 - 1)^2 } \biggr[ \frac{
\sin \theta(5\xi_1^2 - 4\xi_1 \cos \theta - 1)}{(\xi_1+1)^{1/2} (\xi_1 - \cos \theta)^{3/2}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ - 4\xi_1 E\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) 
+ (\xi_1-1) F\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) \biggr]_{\cos^{-1}(\xi_2|_-)}^{\cos^{-1}(\xi_2|_+)} \, .
</math>
  </td>
</tr>
</table>
</div>

<span id="Answer">We have therefore succeeded in deriving an expression for the gravitational potential of a uniform-density torus that requires numerical integration over only one dimension, that is, the "radial" dimension, <math>~\xi_1</math>.</span>  The final, combined expression is,

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Phi(a,Z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2^{5/2} G \rho_0 a^{2}}{3}
\int\limits_{\xi_1|_\mathrm{min}}^{\xi_1|_\mathrm{max}}  \frac{(\xi_1+1)^{1/2}K(\mu) d\xi_1}{(\xi_1^2 - 1)^2 [ (\xi_1^2 - 1)^{1/2}+\xi_1 ]^{1/2} } 
\biggr[ \frac{\sin \theta(5\xi_1^2 - 4\xi_1 \cos \theta - 1)}{(\xi_1+1)^{1/2} (\xi_1 - \cos \theta)^{3/2}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ - 4\xi_1 E\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) 
+ (\xi_1-1) F\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) \biggr]_{\cos^{-1}(\xi_2|_-)}^{\cos^{-1}(\xi_2|_+)} \, .
</math>
  </td>
</tr>
</table>

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


<!-- COMMENT OUT EARLIER EXPRESSION FOR PHI0
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(a, Z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2^{5/2} G \rho_0 a^{2}}{3}
\int\limits_{\tanh^{-1}[a/(\varpi_t-\beta_-)]}^{\tanh^{-1}[a/(\varpi_t-\beta_+)]} \frac{K(\mu) ~dx}{\sinh^3 x( \sinh x+\cosh x )^{1/2}}  \cdot
\biggr[ \frac{\sin \theta(5\xi_1^2 - 4\xi_1 \cos \theta - 1)}{ (\xi_1 - \cos \theta)^{3/2}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ - 4\xi_1 (\xi_1+1)^{1/2} E\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) 
+ (\xi_1-1) (\xi_1+1)^{1/2} F\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) \biggr]_{\cos^{-1}(\xi_2|_-)}^{\cos^{-1}(\xi_2|_+)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2^{5/2} G \rho_0 a^{2}}{3}
\int\limits_{\tanh^{-1}[a/(\varpi_t-\beta_-)]}^{\tanh^{-1}[a/(\varpi_t-\beta_+)]} \frac{K(\mu) ~dx}{\sinh^3 x( \sinh x+\cosh x )^{1/2}}  \cdot
\biggr[ \frac{4 \cosh x \sin \theta }{ (\cosh x - \cos \theta)^{1/2}} + \frac{\sinh^2 x \sin \theta}{ (\cosh x - \cos \theta)^{3/2}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ - 4\cosh x (\cosh x+1)^{1/2} E\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\cosh x + 1}} \biggr) 
+ \sinh x (\cosh x -1)^{1/2} F\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\cosh x + 1}} \biggr) \biggr]_{\cos^{-1}(\xi_2|_-)}^{\cos^{-1}(\xi_2|_+)} \, .
</math>
  </td>
</tr>
</table>
</div>

END DELETED COMMENT -->