Editing Appendix/Mathematics/ToroidalFunctions (section)

==Wong Toroidal Coordinates==

===Wong's Expression for the Potential===
This chapter has been put together in an effort to lay the groundwork for an evaluation of [[Apps/DysonWongTori#Wong_.281973.2C_1974.29|Wong's (1973)]] derived expression for the gravitational potential both inside and outside of a uniform-density, axisymmetric torus.  After multiplying his expression by the negative of {{ Math/C_GravitationalConstant }}, then replacing his total charge, <math>~q</math>, with the total mass, <math>~M</math>, Wong's ''interior'' (i.e., <math>~\eta^' > \eta_0</math>) solution is,

<div align="center" id="Wong1973Interior">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~U(\eta^',\theta^')\biggr|_{\mathrm{for}~\eta^' \ge \eta_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{2^{5 / 2} a^2 G}{3} \biggl[ \frac{1}{2\pi^2 a^2}\biggl(\frac{M}{a}\biggr) \frac{\sinh^3 \eta_0}{\cosh\eta_0} \biggr] \biggl\{ 
- \frac{3\pi^2}{2^{5/ 2}} \biggl[ \frac{\sinh^2\eta^'}{(\cosh \eta^' - \cos \theta^')^2} \biggr]
+~ (\cosh \eta^' - \cos \theta^')^{1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sum\limits_{n=0}^\infty \epsilon_n \cos(n\theta^') 
Q_{n-1 / 2}(\cosh\eta^') B_n(\cosh\eta_0) 
\biggr\} \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.65)
  </td>
</tr>
</table>
</div>
where,

<div align="center" id="Wong1973Interior">
<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+1/2} (\cosh\eta_0)Q^2_{n-1/2} (\cosh\eta_0)
- (n-\tfrac{3}{2})P_{n-1/2} (\cosh\eta_0)Q^2_{n+1/2} (\cosh\eta_0) \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.62)
  </td>
</tr>
</table>
</div>

===Summary of Toroidal Coordinates and Toroidal Functions===

<table border="1" cellpadding="8" width="85%" align="center">
<tr>
  <th align="center" colspan="1">
Summary
  </th>
</tr>
<tr>
  <td align="left">
Suppose you want to evaluate the potential of a uniform-density torus whose major radius is, <math>~R</math>, and minor cross-sectional radius is, <math>~d</math>.  Evaluation of the potential can be relatively easily expressed in terms of a toroidal coordinate system, <math>~(\eta,\theta)</math>, whose "origin" is at a distance, <math>~a</math>, from the symmetry axis, where,

<div align="center">
<math>~a^2 \equiv R^2 - d^2 ~~~\Rightarrow ~~~ \frac{a^2}{d^2} = \frac{R^2}{d^2} - 1\, .</math>
</div>


When expressed in terms of cylindrical coordinates, the meridional-plane location at which the potential is to be evaluated is, <math>~(\varpi, z)</math>, and in toroidal coordinates the location is determined as follows:
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>~\rho_1^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(\varpi - a)^2 + z^2</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\rho_2^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(\varpi + a)^2 + z^2</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ln\biggl( \frac{\rho_2}{\rho_1} \biggr)</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ (\rho_1^2 + \rho_2^2) -4a^2 }{ 2\rho_1 \rho_2 }</math>
  </td>
</tr>
</table>

The ''surface'' of the uniform-density torus is defined by the toroidal "radial" coordinate, <math>~\eta_0</math>, such that,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\cosh\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R}{d} \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\sinh\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[ \cosh^2\eta_0 - 1 ]^{1 / 2} = \frac{a}{d} \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\tanh\eta_0 = \frac{\sinh\eta_0}{\cosh\eta_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{R} \, .</math>
  </td>
</tr>
</table>
</tr>

<tr>
  <td align="left">
The volume of a torus is,
<div align="center">
<math>~V = 2\pi R(\pi d^2) \, .</math>
</div>
When this is rewritten in terms of our toroidal coordinate system, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~V</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi^2 a^3 \biggl( \frac{R}{a} \biggr)  \biggl( \frac{d}{a} \biggr)^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi^2 a^3 \biggl( \frac{\cosh\eta_0}{\sinh^3\eta_0} \biggr)  \, .</math>
  </td>
</tr>
</table>

  </td>
</tr>

<tr>
  <td align="left">
Expressions for the relevant toroidal functions are as follows:

{{ Math/EQ_PminusHalf01 }}

{{ Math/EQ_PplusHalf01 }}

<table align="center" cellpadding="5" border="1">
<tr>
  <td align="center"><math>~P^0_{-\frac{1}{2}}(z)</math></td>
  <td align="center"><math>~P^0_{+\frac{1}{2}}(z)</math></td>
</tr>
<tr>
  <td align="center">
[[File:P0minusHalf.png|250px|P0minusHalf]]
  </td>
  <td align="center">
[[File:P0plusHalf.png|250px|P0plusHalf]]
  </td>
</tr>
<tr>
  <td align="left" colspan="2">
See [[Appendix/EquationTemplates#Caption|relevant caption]].
  </td>
</tr>
</table>

{{ Math/EQ_QminusHalf01 }}

{{ Math/EQ_QplusHalf01 }}
where, <math>~K</math> and <math>~E</math> are complete elliptic integrals of the first and second kind, respectively.  In equation (23) of [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], this last expression has been written in the more compact form, 

<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>~
\chi \mu K(\mu) - (1+\chi)\mu E(\mu) \, 
</math>
  </td>
</tr>
</table>
where, <math>~\mu \equiv [2/(\chi + 1)]^{1 / 2}</math> and, for example, <math>~\chi = \cosh\eta</math>.  

<table align="center" cellpadding="5" border="1">
<tr>
  <td align="center"><math>~Q^0_{-\frac{1}{2}}(z)</math></td>
  <td align="center"><math>~Q^0_{+\frac{1}{2}}(z)</math></td>
  <td align="center"><math>~Q^0_{+\frac{3}{2}}(z)</math></td>
</tr>
<tr>
  <td align="center">
[[File:Q0minus1Half.png|250px|Q0minus1Half]]
  </td>
  <td align="center">
[[File:Q0plus1Half.png|250px|Q0plus1Half]]
  </td>
  <td align="center">
[[File:Q0plus3Half.png|250px|Q0plus3Half]]
  </td>
</tr>
<tr>
  <td align="left" colspan="3">
See [[Appendix/EquationTemplates#Caption|relevant caption]].
  </td>
</tr>
</table>

We also will employ the so-called ''recurrence relation'',

{{ Math/EQ_Toroidal04 }}

After setting, <math>~\mu = 0</math>, and making the association, <math>~\nu \rightarrow (m - \tfrac{3}{2})</math>, for example, this gives, 

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

<tr>
  <td align="right">
<math>~Q_{m-\frac{1}{2} }(\chi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\biggl[ \frac{m-1}{2m-1}\biggr] \chi Q_{m-\frac{3}{2}}(\chi) - \biggl[ \frac{2m-3}{2m-1}\biggr] Q_{m-\frac{5}{2}}(\chi) \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], &sect;2.2.2, eq. (25)
  </td>
</tr>
</table>
which, for all <math>~m \ge 2</math> provides a means by which an expression for the associated toroidal function, <math>~Q_{m - \frac{1}{2}}</math>, can be generated from the foundation pair of expressions given above for <math>~Q_{- \frac{1}{2}}</math> and <math>~Q_{+ \frac{1}{2}}</math>.
  </td>
</tr>
</table>

===Asymptotic Behavior===

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

<tr>
  <td align="right">
<math>~\lim_{\chi\rightarrow \infty} Q_{m-\frac{1}{2}}(\chi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\Gamma(m+\frac{1}{2}) \sqrt{\pi} }{ \Gamma(m+1) (2\chi)^{m+\frac{1}{2}} }
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\sqrt{\pi} }{ (2\chi)^{m+\frac{1}{2}} } \cdot \biggl[ \Gamma(m+\frac{1}{2}) \biggr] \biggl[ \Gamma(m+1) \biggr]^{-1}
</math>
  </td>
</tr>

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

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


<table border="1" align="center" cellpadding="8">
<tr><th align="center" colspan="2">
Asymptotic behavior:
</th></tr>
<tr>
<td align="center"><math>~m</math></td>
<td align="center" colspan="1">
<math>~\lim_{\chi\rightarrow \infty} Q_{m-\frac{1}{2}}(\chi)</math>
</td>
</tr>
<tr>
  <td align="center">0</td>
  <td align="center">
<math>~
\frac{\Gamma(\frac{1}{2}) \sqrt{\pi} }{ \Gamma(1) (2\chi)^{\frac{1}{2}} }
= \frac{\pi}{ (2\chi)^{1 / 2} } \, .
</math>
  </td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center">
<math>~
\frac{\pi }{2^{5/2} \chi^{3/2} } 
</math>
  </td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">
<math>~
\frac{\pi }{2^{\frac{9}{2}} \chi^{\frac{5}{2}} } \cdot \biggl[ \frac{(3)!! }{ 2  }\biggr]
=
\frac{360~\pi }{2^{9/2} \chi^{5/2} } 
=
\frac{45~\pi }{2^{3/2} \chi^{5/2} } 
</math>
  </td>
</tr>
</table>

===Dimensionless Potential Expression===

Given that the portion of the leading term in Wong's expression that sits inside the square brackets is equivalent to the density, <math>~\rho_0 = M/V</math>, of the torus material &#8212; that is, given that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{2\pi^2 a^2}\biggl(\frac{M}{a}\biggr) \frac{\sinh^3 \eta_0}{\cosh\eta_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\rho_0 V}{2\pi^2 a^3} \biggl[ \frac{\sinh^3 \eta_0}{\cosh\eta_0} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0\, ,</math>
  </td>
</tr>
</table>
&#8212; a reasonable dimensionless version of Wong's expression could be obtained by dividing through by the quantity, <math>~(G\rho_0 a^2) </math>.  We prefer, instead, to normalize Wong's expression to the quantity, <math>~GM/R</math> , in which case the dimensionless version of the expression becomes,

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

<tr>
  <td align="right">
<math>~\frac{U(\eta^',\theta^')}{GM/R} \biggr|_{\mathrm{for}~\eta^' \ge \eta_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{2^{5 / 2}  }{3} \biggl[ \frac{1}{2\pi^2}\biggl(\frac{R}{a}\biggr) \frac{\sinh^3 \eta_0}{\cosh\eta_0} \biggr] \biggl\{ 
- \frac{3\pi^2}{2^{5/ 2}} \biggl[ \frac{\sinh^2\eta^'}{(\cosh \eta^' - \cos \theta^')^2} \biggr]
+~ (\cosh \eta^' - \cos \theta^')^{1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sum\limits_{n=0}^\infty \epsilon_n \cos(n\theta^') 
Q_{n-1 / 2}(\cosh\eta^') B_n(\cosh\eta_0) 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{2^{3 / 2} \sinh^2\eta_0 }{3\pi^2}\biggl\{ 
- \frac{3\pi^2}{2^{5/ 2}} \biggl[ \frac{\sinh^2\eta^'}{(\cosh \eta^' - \cos \theta^')^2} \biggr]
+~ (\cosh \eta^' - \cos \theta^')^{1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sum\limits_{n=0}^\infty \epsilon_n \cos(n\theta^') 
Q_{n-1 / 2}(\cosh\eta^') B_n(\cosh\eta_0) 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sinh^2\eta_0 
\biggl\{ 
\frac{\sinh^2\eta^'}{2 (\cosh \eta^' - \cos \theta^')^2} 
~- ~\frac{2^{3 / 2} }{3\pi^2} (\cosh \eta^' - \cos \theta^')^{1 / 2} 
\sum\limits_{n=0}^\infty \epsilon_n \cos(n\theta^') 
Q_{n-1 / 2}(\cosh\eta^') B_n(\cosh\eta_0) 
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

===Prior to the Integration===

The Green's function written in toroidal coordinates is,
<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=-\infty}^{\infty} e^{im(\phi - \phi^')}Q_{m- 1 / 2}(\chi) \, ,
</math>
  </td>
</tr>
</table>
where,<br />
<div align="center"><math>~\chi \equiv \frac{\varpi^2 + (\varpi^')^2 + (z - z^')^2}{2\varpi \varpi^'} \, ,</math>
</div>
and, <math>~Q_{m - 1 / 2}</math> is the zero-order, half-(odd)integer degree, Llegendre function of the second kind &#8212; also referred to as a ''toroidal'' function of zeroth order.  Hence, a valid expression for the gravitational potential is,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(\varpi,\phi,z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-G \int \frac{1}{|\vec{x}^{~'} - \vec{x}|} ~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-G \int \biggl\{ 
\frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=-\infty}^{\infty} e^{im(\phi - \phi^')}Q_{m- 1 / 2}(\chi)
\biggr\}~
\rho(\varpi^',\phi^',z^') \varpi^' d\varpi^' d\phi^' dz^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{\pi \sqrt{\varpi}} \int \rho(\varpi^',\phi^',z^') \sqrt{\varpi^'} d\varpi^' d\phi^' dz^'  
\sum_{m=0}^{\infty} \epsilon_m \cos[m(\phi - \phi^')] Q_{m- 1 / 2}(\chi) \, ,
</math>
  </td>
</tr>
</table>
where, <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0=1</math> and <math>~\epsilon_m = 2</math> for <math>~m\ge 1</math>.  

----
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{r} - {\vec{r}}^{~'} ~|} </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_{m,n} (-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)], Eq. (2.53)
  </td>
</tr>
</table>
</div>
where, <math>~P^m_{n-1 / 2}, Q^m_{n-1 / 2}</math> are "<font color="darkgreen">Legendre functions of the first and second kind with order <math>~n - \tfrac{1}{2}</math> and degree <math>~m</math> (toroidal harmonics)</font>," 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>,

<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>
and,
<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^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^2 \, ,</math>
  </td>
</tr>
</table>
</div>
and <math>~\theta</math> has the same sign as <math>~z</math>. 


Hence, the potential, <math>~U({\vec{r}}~')</math>, at a point <math>~{\vec{r}}~'</math> due to an arbitrary mass distribution, <math>~\rho({\vec{r}})</math>, is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~U({\vec{r}}~')</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-G \iiint \frac{\rho(\vec{r}) d^3r}{|~\vec{r} - {\vec{r}}^{~'} ~|} \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-G \rho_0 a^3
\iiint \frac{1}{|~\vec{r} - {\vec{r}}^{~'} ~|} \biggl[ \frac{\Theta(\upsilon) \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>~-G \rho_0 a^3\iiint 
\biggl[ \frac{\Theta(\upsilon) \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">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 }
\sum\limits_{m,n} (-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} \biggr\}
</math>
  </td>
</tr>
</table>
</div>

----
This should be compared with,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(\varpi,\phi,z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-G \iiint \frac{1}{|\vec{x}^{~'} - \vec{x}|} ~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-G \iiint \biggl\{ 
\frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=-\infty}^{\infty} e^{im(\phi - \phi^')}Q_{m- 1 / 2}(\chi)
\biggr\}~
\rho(\varpi^',\phi^',z^') \varpi^' d\varpi^' d\phi^' dz^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{\pi \sqrt{\varpi}} \iiint\rho(\varpi^',\phi^',z^') \sqrt{\varpi^'} d\varpi^' d\phi^' dz^'  
\sum_{m=0}^{\infty} \epsilon_m \cos[m(\phi - \phi^')] Q_{m- 1 / 2}(\chi) \, ,
</math>
  </td>
</tr>
</table>

----

The above expression for the potential of a uniform-density torus has been obtained by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] from the (double) integral expression,

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

<tr>
  <td align="right">
<math>~U(\eta^',\theta^')</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\rho_0 a^2 (\cosh \eta^' - \cos \theta^')^{1 / 2} \sum\limits_n \epsilon_n \int_{\eta_0}^\infty d\eta 
\int_{-\pi}^{\pi} \biggl[\frac{\cos[n(\theta - \theta^')]}{(\cosh\eta - \cos\theta)^{5 / 2}} \biggr]d\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sinh\eta ~\begin{cases}P_{n-1 / 2}(\cosh\eta) ~Q_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P_{n-1 / 2}(\cosh\eta^') ~Q_{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)], Eq. (2.55)
  </td>
</tr>
</table>
</div>
which is valid for any azimuthal angle, <math>~\psi^'</math>.  Notice that the step function, <math>~\Theta(\upsilon)</math>, no longer explicitly appears in this expression for the Coulomb (or gravitational) potential; it has been used to establish the specific limits on the "radial" coordinate integration.  Next, he completes the integration over the angle, <math>~\theta</math>, to obtain,