Editing 2DStructure/ToroidalGreenFunction (section)

=Using Toroidal Coordinates to Determine the Gravitational Potential=

NOTE: &nbsp; An [[2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|earlier version of this chapter]] has been shifted to our "Ramblings" Appendix.


Here we build upon our [[AxisymmetricConfigurations/PoissonEq#Solving_the_.28Multi-dimensional.29_Poisson_Equation_Numerically|accompanying review]] of the types of numerical techniques that various astrophysics research groups have developed to solve for the Newtonian gravitational potential, <math>~\Phi(\vec{x})</math>, given a specified, three-dimensional mass distribution, <math>~\rho(\vec{x})</math>.  Our focus is on the use of toroidal coordinates to solve the ''integral'' formulation of the Poisson equation, namely,
<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi(\vec{x})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -G \iiint \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
  </td>
</tr>
</table>
For the most part, we will adopt the notation used by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279)]; in an accompanying discussion, we review additional results from this insightful 1973 paper, as well as a paper of his that was published the following year in ''The Astrophysical Journal'', namely, [http://adsabs.harvard.edu/abs/1974ApJ...190..675W Wong (1974)].

In order to accomplish this task, we first present the expressions that define how toroidal coordinates, <math>~(\eta,\theta,\psi)</math>, map to and from Cartesian coordinates <math>~(x, y, z)</math>, and present the toroidal-coordinate expression for the differential volume element, <math>~d^3 x</math>.

==Basic Elements of a Toroidal Coordinate System==

Given the meridional-plane coordinate location of a toroidal-coordinate system's axisymmetric ''anchor ring'', <math>~(\varpi,z) = (a,Z_0)</math>, the relationship between toroidal coordinates <math>~(\eta,\theta,\psi) </math>and Cartesian coordinates <math>~(x, y, z)</math> is,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a \sinh\eta \sin\psi}{(\cosh\eta - \cos\theta)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z - Z_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \, .</math>
  </td>
</tr>
</table>
</div>
This set of coordinate relations appears as equations 2.1 - 2.3 in [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)].  This set of relations may also be found, for example, on p. 1301 within eq. (10.3.75) of [<b>[[User:Tohline/Appendix/References#MF53|<font color="red">MF53</font>]]</b>]; in [http://dlmf.nist.gov/14.19 &sect;14.19 of NIST's ''Digital Library of Mathematical Functions'']; or even within [https://en.wikipedia.org/wiki/Toroidal_coordinates#Definition Wikipedia].  (In most cases the implicit assumption is that <math>~Z_0 = 0</math>.)  It is clear, of course, that the cylindrical radial coordinate is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi = (x^2 + y^2)^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a \sinh\eta}{(\cosh\eta - \cos\theta)} \, .</math>
  </td>
</tr>
</table>


Mapping the other direction [see equations 2.13 - 2.15 of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] ], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\tan\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{y}{x} \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_1^2 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[(x^2 + y^2)^{1 / 2} + a]^2 + (z-Z_0)^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r_2^2 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[(x^2 + y^2)^{1 / 2} - a]^2 + (z-Z_0)^2 \, ,</math>
  </td>
</tr>
</table>
</div>
and <math>~\theta</math> has the same sign as <math>~(z-Z_0)</math>.

<table border="1" cellpadding="10" align="center" width="85%">
<tr>
  <td align="center">
<!-- [[File:Hicks1881TitlePage.png|500px|Title Page of Hicks (1881)]] -->
[https://ui.adsabs.harvard.edu/abs/1881RSPT..172..609H/abstract W. M. Hicks (1881)]<br />
''"On Toroidal Functions"''<br />
Philosophical Transactions of the Royal Society of London, vol. 172, pp. 609-652
</td>
</tr>
<tr><td align="left">
[http://rstl.royalsocietypublishing.org/content/172/609.full.pdf+html W. M. Hicks (1881)] presents one of the first &#8212; if not ''the'' first &#8212; discussions of toroidal coordinates and associated toroidal functions.  Equation (4) on p. 614 of his article provides the following definition of the pair of meridional-plane coordinates <math>~(u,v)</math>, written in terms of the traditional cylindrical-coordinate pair <math>~(\rho,z)</math> and the specified ''anchor ring'' radius, <math>~a</math>:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~u</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2} \ln \biggl[ \frac{z^2 + (\rho+a)^2}{z^2 + (\rho - a)^2} \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~v</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tan^{-1} \biggl[ \frac{2az}{\rho^2 + z^2 - a^2} \biggr] \, .
</math>
  </td>
</tr>
</table>
His equation (5) presents the reverse mapping, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a\sinh u}{\cosh u - \cos v} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a\sin v}{\cosh u - \cos v} \, .
</math>
  </td>
</tr>
</table>

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

<span id="DiffVolumeElement">&nbsp;</span>
[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 16 August 2017:  In equation (2.17) of his &sect;IIB &#8212; when Wong (1973) introduces the differential volume element &#8212; the variable used to represent the azimuthal coordinate angle switches from &psi; to &#x003A6;.  We will stick with the &psi; notation, here.]]According to p. 1301, eq. (10.3.75) of [<b>[[User:Tohline/Appendix/References#MF53|<font color="red">MF53</font>]]</b>] &#8212; or, for example, as found in [https://en.wikipedia.org/wiki/Toroidal_coordinates#Scale_factors Wikipedia] &#8212; in toroidal coordinates the differential volume element is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d^3x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~h_\eta h_\theta h_\psi d\eta d\theta d\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{a^3 \sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta~ d\psi \, .</math>
  </td>
</tr>
</table>

==Green's Function Expression==

===As presented by Wong (1973)===
Referencing [<b>[[User:Tohline/Appendix/References#MF53|<font color="red">MF53</font>]]</b>], [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] states that, in toroidal coordinates, the Green's function is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{|~\vec{x} - {\vec{x}}^{~'} ~|} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 }
\sum\limits^\infty_{m,n=0} (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \cos[m(\psi - \psi^')]\cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta
\end{cases}\, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], p. 293, Eq. (2.53)<br />
[see also: [http://adsabs.harvard.edu/abs/1997JMP....38.3679B J. W. Bates (1997)], p. 3685, Eq. (31)]<br />
[see also: [http://adsabs.harvard.edu/abs/2000AN....321..363C Cohl, Tohline, Rau, &amp; Srivastava (2000)], &sect;6.2, Eq. (48)]
  </td>
</tr>
</table>
</div>
where, <math>~P^m_{n-1 / 2}, Q^m_{n-1 / 2}</math> are ''Associated Legendre Functions'' of the first and second kind with degree <math>~n - \tfrac{1}{2}</math> and order <math>~m</math> (toroidal harmonics), and <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0 = 1</math> and <math>~\epsilon_m = 2</math> for all <math>~m \ge 1</math>.  This Green's function expression can indeed be found as eq. (10.3.81) on p. 1304 of [<b>[[User:Tohline/Appendix/References#MF53|<font color="red">MF53</font>]]</b>], but it should be noted that the MF53 expression differs from Wong's in two respects (see footnote 2 on p. 370 of [http://adsabs.harvard.edu/abs/2000AN....321..363C Cohl et al. (2000)] for a proposed explanation):  First, the factor, <math>~(-1)^m</math>, appears as <math>~(-i)^m</math> in MF53; and, second, in the term that is composed of a ratio of gamma functions, the denominator appears in MF53 as <math>~\Gamma(n - m + \tfrac{1}{2})</math>, whereas it should be <math>~\Gamma(n + m + \tfrac{1}{2})</math>, as presented here.
<!--
For later reference, note that after drawing from, for example, [https://en.wikipedia.org/wiki/Gamma_function#General Wikipedia's account of the general properties of gamma functions],  the collection of factors immediately inside the double summation may be more explicitly written as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~L_n^m \equiv (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(-1)^m \epsilon_m \epsilon_n \biggl[ \frac{\sqrt{\pi}[2(n-m)]!}{4^{n-m}(n-m)!} \biggr]
\biggl[ \frac{4^{n+m}(n+m)!}{\sqrt{\pi}[2(n+m)]!} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(-1)^m \epsilon_m \epsilon_n \biggl[ \frac{[2(n-m)]!}{(n-m)!} \biggr]
\biggl[ \frac{(n+m)!}{[2(n+m)]!} \biggr]2^{4m}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(-1)^m \epsilon_m \epsilon_n \biggl[ \frac{\cdots [2(n-m)-4]\cdot [2(n-m)-3]\cdot[2(n-m)-2]\cdot[2(n-m)-1]\cdot[2(n-m)] }{\cdots [(n-m)-4]\cdot [(n-m)-3]\cdot [(n-m)-2]\cdot [(n-m)-1]\cdot (n-m) } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\biggl[ \frac{\cdots [(n+m)-4]\cdot [(n+m)-3]\cdot [(n+m)-2]\cdot [(n+m)-1]\cdot (n+m) }{\cdots [2(n+m)-4]\cdot [2(n+m)-3]\cdot [2(n+m)-2]\cdot [2(n+m)-1]\cdot [2(n+m)] } \biggr]2^{4m}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(-1)^m \epsilon_m \epsilon_n \biggl[ \frac{\cdots 2[(n-m)-2]\cdot [2(n-m)-3]\cdot 2[(n-m)-1]\cdot [2(n-m)-1] \cdot 2(n-m) }{\cdots [(n-m)-4]\cdot [(n-m)-3]\cdot [(n-m)-2]\cdot [(n-m)-1]\cdot (n-m) } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\biggl[ \frac{\cdots [(n+m)-4]\cdot [(n+m)-3]\cdot [(n+m)-2]\cdot [(n+m)-1]\cdot (n+m) }{\cdots 2[(n+m)-2]\cdot [2(n+m)-3]\cdot 2[(n+m)-1]\cdot [2(n+m)-1]\cdot 2[(n+m)] } \biggr]2^{4m}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(-1)^m \epsilon_m \epsilon_n \biggl[ \frac{\cdots 2^{n-m}  [2(n-m)-3]\cdot  [2(n-m)-1] }{1} \biggr]
\biggl[ \frac{1}{\cdots 2^{n+m}  [2(n+m)-3]\cdot [2(n+m)-1]] } \biggr]2^{4m}
</math>
  </td>
</tr>
</table>
-->
===As Presented by Hicks (1881)===
Notably, more than 135 years ago, [http://rstl.royalsocietypublishing.org/content/172/609.full.pdf+html W. M. Hicks (1881)] already had constructed a Green's function expression for the reciprocal distance between two points in toroidal coordinates.   The following boxed-in table contains a snapshot image of  equation (35) from Hicks (1881), along with its associated text; notice that he refers to <math>~\phi</math> as the "potential."  Although notations are different &#8212; for example, <math>~C</math> is shorthand for <math>~\cosh u</math> and <math>~c</math> is shorthand for <math>~\cos v</math> &#8212; one can easily see factor-by-factor agreement when compared with the Green's function [[#As_presented_by_Wong_.281973.29|presented by Wong (1973)]]. 
<table border="1" cellpadding="10" align="center">
<tr>
  <td align="center">
<!-- [[File:Hicks1881TitlePage.png|500px|Title Page of Hicks (1881)]] -->
[https://ui.adsabs.harvard.edu/abs/1881RSPT..172..609H/abstract W. M. Hicks (1881)]<br />
''"On Toroidal Functions"''<br />
Philosophical Transactions of the Royal Society of London, vol. 172, pp. 609-652
</td>
</tr>
<tr><td align="center">
[[File:Hicks1881GreenFunction.png|750px|To be inserted:  Eq. (35) from Hicks (1881)]]
</td></tr></table>

===Rearranging Terms and Incorporating Special-Function Relations===

Let's focus on the situation when <math>~\eta^' > \eta</math>, and begin rearranging or substituting terms.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{|~\vec{x} - {\vec{x}}^{~'} ~|} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 }
\sum\limits^\infty_{m=0} (-1)^m \epsilon_m \cos[m(\psi - \psi^')]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sum\limits^\infty_{n=0} \epsilon_n \cos[n(\theta - \theta^')]  ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
~P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^')
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ [ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^')]^{1 /2 } }{\pi a \sqrt{\sinh\eta^'} \sqrt{\sinh\eta} }
\sum\limits^\infty_{m=0} (-1)^m \epsilon_m \cos[m(\psi - \psi^')]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sum\limits^\infty_{n=0} \epsilon_n \cos[n(\theta - \theta^')] \biggl\{ ~
\sqrt{ \frac{\pi}{2} }~\Gamma(n-m+\tfrac{1}{2}) \sqrt{\sinh\eta}~P^m_{n-1 / 2}(\cosh\eta) 
\biggl\}\biggr\{
~\sqrt{ \frac{2}{\pi} }~\frac{\sqrt{\sinh\eta^'}}{\Gamma(n + m + \tfrac{1}{2})} Q^m_{n-1 / 2}(\cosh\eta^')
\biggr\}
</math>
  </td>
</tr>
</table>
</div>

The term contained within the first set of curly braces on the right-hand side of this expression can be replaced by the derived expression labeled <font color="green" size="+1">&#x2460;</font> in the [[#A.1|Appendix, below]], and simultaneously the term contained within the second set of curly braces can be replaced by the derived expression labeled <font color="green" size="+1">&#x2461;</font> in the [[#A.2|same Appendix]].  After making these substitutions, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{|~\vec{x} - {\vec{x}}^{~'} ~|} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ [ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^')]^{1 /2 } }{\pi a \sqrt{\sinh\eta^'} \sqrt{\sinh\eta} }
\sum\limits^\infty_{m=0} (-1)^m \epsilon_m \cos[m(\psi - \psi^')]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sum\limits^\infty_{n=0} \epsilon_n \cos[n(\theta - \theta^')] \biggl\{ ~
(-1)^{-n}Q^n_{m-1 / 2}(\coth\eta) 
\biggl\}\biggr\{
~(-1)^{-m} P^{-n}_{m-1 / 2}(\coth\eta^')
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ [ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^')]^{1 /2 } }{\pi a \sqrt{\sinh\eta^'} \sqrt{\sinh\eta} }
\sum\limits^\infty_{m=0} \epsilon_m \cos[m(\psi - \psi^')]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sum\limits^\infty_{n=0} (-1)^{n} \epsilon_n \cos[n(\theta - \theta^')] 
Q^n_{m-1 / 2}(\coth\eta) P^{-n}_{m-1 / 2}(\coth\eta^') \, ,
</math>
  </td>
</tr>
</table>

where, in writing this last expression we have acknowledged that, since <math>~n</math> is either zero or a positive integer, <math>~(-1)^{-n} = (-1)^n</math>.  Next we draw upon the "Key Equation" relation,

{{ Math/EQ_Toroidal01 }} 

which, after making the substitutions, <math>~\nu \rightarrow (m - \tfrac{1}{2})</math> and <math>~\psi \rightarrow (\theta - \theta^')</math>, and incorporating the Neumann factor, <math>~\epsilon_n</math>, becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
Q_{m - \frac{1}{2} }\ [t t^' - (t^2-1)^{1 / 2} (t^{'2} - 1)^{1 / 2} \cos(\theta- \theta^') ]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sum_{n=0}^\infty (-1)^n \epsilon_n Q^n_{m - \frac{1}{2} }(t) P^{-n}_{m - \frac{1}{2} }(t^') \cos[(n(\theta- \theta^')] \, .
</math>
  </td>
</tr>
</table>

Finally, after making the associations, <math>~t \rightarrow \coth\eta</math> and <math>~t^' \rightarrow \coth\eta^'</math>, this last expression allows us to rewrite Wong's (1973) Green's function in a significantly more compact form, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{|~\vec{x} - {\vec{x}}^{~'} ~|} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ [ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^')]^{1 /2 } }{\pi a \sqrt{\sinh\eta^'} \sqrt{\sinh\eta} }
\sum\limits^\infty_{m=0} \epsilon_m \cos[m(\psi - \psi^')]  Q_{m - \frac{1}{2}}(\Chi) \, ,
</math>
  </td>
</tr>
</table>
where the argument, <math>~\Chi</math>, of the toroidal function, <math>~Q_{m - \frac{1}{2}}</math>, is,

<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>~
t t^' - (t^2-1)^{1 / 2} (t^{'2} - 1)^{1 / 2} \cos(\theta- \theta^') 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\coth\eta \coth\eta^' - (\coth^2\eta-1)^{1 / 2} (\coth^2\eta'- 1)^{1 / 2} \cos(\theta- \theta^') 
</math>
  </td>
</tr>

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

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

===As Presented in Cohl &amp; Tohline (1999)===

This last, compact Green's function expression &#8212; which we have derived, here, from [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] published Green's function by drawing strategically upon a variety of special function relations &#8212; precisely matches the "compact cylindrical Green's function expression" that has been derived independently by [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)] via a less tortuous route, namely,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) 
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="2">&nbsp;</td>
  <td align="left" colspan="1">
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], p. 88, Eq. (17)
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{a\pi} 
\biggl[  \frac{(\cosh\eta^' - \cos\theta^')}{\sinh\eta^' } \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2}
\sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) \, ,
</math>
  </td>
</tr>
</table>
where,
<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{(\varpi^')^2 + \varpi^2 + (z^' - z)^2}{2\varpi^' \varpi}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^'} \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="2">
&nbsp; &nbsp;
  </td>
  <td align="left" colspan="3">
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], p. 88, Eq. (16)
  </td>
</tr>
</table>


'''Note from J. E. Tohline (June, 2018)''':  &nbsp;This is the first time that I have been able to formally demonstrate to myself that these two separately derived Green's function expressions are identical.  See, however, the earlier identification of ''new'' addition theorems in association with equations (49) and (50) of [http://adsabs.harvard.edu/abs/2000AN....321..363C Cohl et al. (2000)].

==Gravitational Potential==

Quite generally, then, the gravitational potential can be obtained at any coordinate location, <math>~(\eta,\theta,\psi)</math> &#8212; both inside and outside of a specified mass distribution &#8212; by carrying out three nested spatial integrals over the product of:&nbsp; <math>~\rho(\vec{x}^{~'})</math>, the [[#DiffVolumeElement|differential volume element]], and the Green's function as specified ''either'' by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] or by [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)].  

In what follows we will make an effort to elucidate the pros and cons of adopting one Green's function expression over the other.  In each case we begin by writing the expression for the potential in such a way that variations in the azimuthal coordinate, <math>~\psi</math>, are described by ''Fourier components,'' <math>~\Phi_m^{(1)}(\eta,\theta)</math> and <math>~\Phi_m^{(2)}(\eta,\theta)</math>, of the potential, such that,
<div align="center">
<math>~\Phi(\vec{x}) = \tfrac{1}{2}\Phi_0^{(1)}(\eta,\theta) + \sum_{m=1}^\infty \cos (m\psi) \Phi_m^{(1)}(\eta,\theta)  + \sum_{m=1}^\infty \sin (m\psi) \Phi_m^{(2)}(\eta,\theta) \, .</math>
</div>

Each Fourier component of the potential depends explicitly on the corresponding ''Fourier component'' of the density distribution, defined such that,  
<div align="center">
<math>~\rho(\vec{x}) = \tfrac{1}{2}\rho_0^{(1)}(\eta,\theta) + \sum_{m=1}^\infty \cos (m\psi) \rho_m^{(1)}(\eta,\theta)  + \sum_{m=1}^\infty \sin (m\psi) \rho_m^{(2)}(\eta,\theta) \, .</math>
</div>

<table border="1" align="center" cellpadding="8" width="70%">
<tr>
  <th align="center" bgcolor="yellow">
LaTeX mathematical expressions cut-and-pasted directly from
<br />
NIST's ''Digital Library of Mathematical Functions''
  </th>
</tr>
<tr>
  <td align="left">
As an additional primary point of reference, note that according to [https://dlmf.nist.gov/1.8 &sect;1.8(i) of NIST's ''Digital Library of Mathematical Functions''], a Fourier Series is defined as follows:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{1}{2}a_{0}+\sum^{\infty}_{n=1}\biggl[ a_{n}\cos\bigl(nx\bigr)+b_{n}\sin\bigl(nx\bigr) \biggr],</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~a_n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\cos\bigl(nx\bigr)\mathrm{d}x,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\sin\bigl(nx\bigr)\mathrm{d}x.</math>
  </td>
</tr>
</table>

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

Notice, therefore, that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho_m^{(1)}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\pi}\int^{\pi}_{-\pi}\rho(\eta,\theta,\psi)\cos\bigl(m\psi\bigr)\mathrm{d}\psi,</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~\rho_m^{(2)}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\pi}\int^{\pi}_{-\pi}\rho(\eta,\theta,\psi)\sin\bigl(m\psi\bigr)\mathrm{d}\psi \, .</math>
  </td>
</tr>
</table>


===The CT99 Expression for the Potential===
====In Three-Dimensional Generality====
Employing the Green's function expression derived by [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], the gravitational potential for any three-dimensional matter distribution is,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi(\eta,\theta,\psi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -G
\iiint \rho(\eta^',\theta^',\psi^') \biggl\{ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr\}  \biggl[ \frac{a^3 \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] d\eta^'~ d\theta^'~ d\psi^'
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\frac{Ga^2}{\pi} \int d\eta^' \int d\theta^' \int d\psi^'
\iiint \rho(\eta^',\theta^',\psi^')  \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ \times
\biggl[  \frac{(\cosh\eta^' - \cos\theta^')}{\sinh\eta^' } \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2}
\sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\frac{Ga^2}{\pi} \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \sum_{m=0}^{\infty} \epsilon_m
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi)
\int d\psi^' \rho(\eta^',\theta^',\psi^')  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ \times
[\cos(m\psi)\cos(m\psi^') + \sin(m\psi)\sin(m\psi^') ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\frac{Ga^2}{\pi} \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \biggl\{
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{- \frac{1}{2}}(\Chi)
\int_{-\pi}^{\pi} d\psi^' \rho(\eta^',\theta^',\psi^')  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ + \sum_{m=1}^{\infty} 2\cos(m\psi)
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi)
\int_{-\pi}^\pi d\psi^' \rho(\eta^',\theta^',\psi^')  \cos(m\psi^')
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ + \sum_{m=1}^{\infty} 2\sin(m\psi)
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi)
\int_{-\pi}^\pi d\psi^' \rho(\eta^',\theta^',\psi^')  \sin(m\psi^')
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - Ga^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \biggl\{
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{- \frac{1}{2}}(\Chi)  \rho_0^{(1)}(\eta^',\theta^')
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ + \sum_{m=1}^{\infty} 2\cos(m\psi)
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi) \rho_m^{(1)}(\eta^',\theta^')
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ + \sum_{m=1}^{\infty} 2\sin(m\psi)
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi) \rho_m^{(2)}(\eta^',\theta^')
\biggr\} \, .
</math>
  </td>
</tr>
</table>

We conclude, therefore, that each one of the Fourier components of the gravitational potential is given by the expression,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~\Phi_m^{(1),(2)}(\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_{m- 1 / 2}(\Chi) \rho_m^{(1),(2)}(\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, Eq. (20)
  </td>
</tr>
</table>
where, [[#As_Presented_in_Cohl_.26_Tohline_.281999.29|as above]],
<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>

====For Axisymmetric Systems====
For axisymmetric systems, the density distribution has no dependence on the azimuthal coordinate, <math>~\psi</math>.  Hence, for all <math>~m > 0</math>, the ''Fourier components'' of the density, <math>~\rho_m^{(1),(2)}</math>, are zero.  The only nonzero component is, <math>~\rho_0^{(1)}(\eta,\theta) = 2\rho(\eta,\theta)</math>.  For axisymmetric systems, then, the gravitational potential 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>~\tfrac{1}{2}\Phi_0^{(1)}(\eta,\theta)</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </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>

===Wong's Expression for the Potential===

====Fully Three-Dimensional Case====

Employing [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] Green's function expression, the gravitational potential for any three-dimensional matter distribution is,
<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi(\vec{x})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -G
\iiint \rho(\eta^',\theta^',\psi^') \biggl\{ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr\}  \biggl[ \frac{a^3 \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] d\eta^'~ d\theta^'~ d\psi^'
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{a^2G}{\pi} \int d\eta^' \int d\theta^' \int d\psi^'
\biggl[ \frac{\rho(\eta^',\theta^',\psi^') ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] 
\biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 }
\sum\limits^\infty_{n=0} \sum\limits^\infty_{m=0}(-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \cos[m(\psi - \psi^')]\cos[n(\theta - \theta^')] ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{a^2G}{\pi} (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \sum\limits^\infty_{m=0}(-1)^m \epsilon_m ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
\int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times~
\int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')]
\int d\psi^' \rho(\eta^',\theta^',\psi^') [\cos(m\psi)\cos(m\psi^') + \sin(m\psi) \sin(m\psi^')]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{a^2G}{\pi} (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \biggl\{
\int d\eta^' ~P_{n-1 / 2}(\cosh\eta_<) ~Q_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')]
\int_{-\pi}^\pi d\psi^' \rho(\eta^',\theta^',\psi^') 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~
\sum\limits^\infty_{m=1} 2\cos(m\psi)(-1)^m  ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
\int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')]
\int_{-\pi}^\pi d\psi^' \rho(\eta^',\theta^',\psi^') \cos(m\psi^') 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~
\sum\limits^\infty_{m=1} 2 \sin(m\psi)(-1)^m ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
\int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')]
\int_{-\pi}^\pi  d\psi^' \rho(\eta^',\theta^',\psi^') \sin(m\psi^')
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - a^2G (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \biggl\{
\int d\eta^' ~P_{n-1 / 2}(\cosh\eta_<) ~Q_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')]
\rho_0^{(1)}(\eta^',\theta^') 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~
\sum\limits^\infty_{m=1} 2\cos(m\psi)(-1)^m  ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
\int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')]
\rho_m^{(1)}(\eta^',\theta^')
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~
\sum\limits^\infty_{m=1} 2 \sin(m\psi)(-1)^m ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
\int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')]
\rho_m^{(2)} (\eta^',\theta^')
\biggr\} \, .
</math>
  </td>
</tr>
</table>

We conclude, therefore, that each one of the Fourier components of the gravitational potential is given by the expression,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~\Phi_m^{(1),(2)} (\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 (-1)^m  ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\int d\eta^' ~\sinh\eta^' ~ P^m_{n-1 / 2}(\cosh\eta_<) ~ Q^m_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl\{ \frac{ \cos[n(\theta - \theta^')]}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr\} 
\rho_m^{(1),(2)}(\eta^',\theta^') \, .
</math>
  </td>
</tr>
</table>

====Axisymmetric Systems====
For axisymmetric systems, the density distribution has no dependence on the azimuthal coordinate, <math>~\psi</math>.  Hence, for all <math>~m > 0</math>, the ''Fourier components'' of the density, <math>~\rho_m^{(1),(2)}</math>, are zero.  The only nonzero component is, <math>~\rho_0^{(1)}(\eta,\theta) = 2\rho(\eta,\theta)</math>.  For axisymmetric systems, then, the gravitational potential 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>~\tfrac{1}{2}\Phi_0^{(1)}(\eta,\theta)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </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>

==Uniform-Density Torus==

In an [[Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential|accompanying discussion]], we build upon the above 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, has heretofore been underappreciated.