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=Using Toroidal Coordinates to Determine the Gravitational Potential=

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

{{LSU_HBook_header}}


Here we build upon our [[User:Tohline/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>.)


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>.

<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>

==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#Torus|accompanying ''Equations'' appendix]].

<table border="0" align="center" cellpadding="10" width="90%">
<tr>
  <td align="left">
Beginning with the identified "Key Equation",
{{ User:Tohline/Math/EQ_Toroidal02 }}

we'll identify  <math>~x</math> with <math>~\cosh\eta</math> &#8212; in which case we have <math>~\lambda = \coth\eta</math> &#8212; and switch the index notations, <math>~n \leftrightarrow m</math>.  This gives,

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

<tr>
  <td align="right">
<math>~Q_{m-1 / 2}^n (\coth\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(-1)^m
\frac{\pi^{3/2}}{\sqrt{2} ~\Gamma(m-n+\frac{1}{2})} (\sinh\eta)^{1 / 2} P_{n-1 / 2}^m(\cosh\eta) \, .
</math>
  </td>
</tr>
</table>

Drawing upon the ''Euler reflection formula for gamma functions'', namely,
{{ User:Tohline/Math/EQ_Gamma01 }}

where it is understood that <math>~m</math> and <math>~n</math> are each either zero or a positive integer, this toroidal-function relation becomes,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~Q_{m-1 / 2}^n (\coth\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(-1)^m~
\frac{\pi^{3/2}}{\sqrt{2} } \biggl[ \frac{ \Gamma(n - m +\frac{1}{2}) }{ \pi (-1)^{m-n} } \biggr] (\sinh\eta)^{1 / 2} P_{n-1 / 2}^m(\cosh\eta) 
</math>
  </td>
  <td align="right" width="15%">
&nbsp;
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(-1)^n \sqrt{ \frac{\pi}{2} } ~\Gamma(n - m + \tfrac{1}{2} )(\sinh\eta)^{1 / 2} P_{n-1 / 2}^m(\cosh\eta) \, .
</math>
  </td>
  <td align="right" width="15%">
<font color="green" size="+3">&#x2460;</font>
  </td>
</tr>
</table>

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

<table border="0" align="center" cellpadding="10" width="90%">
<tr>
  <td align="left">
Again, beginning with the identified "Key Equation",
{{ User:Tohline/Math/EQ_Toroidal02 }}

this time, without switching index notations, we'll identify  <math>~x</math> with <math>~\coth\eta</math> &#8212; in which case we have <math>~\lambda = \cosh\eta</math>.  This gives,

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

<tr>
  <td align="right">
<math>~Q_{n-1 / 2}^m (\cosh\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(-1)^n
\frac{\pi^{3/2}}{\sqrt{2} ~\Gamma(n-m+\frac{1}{2})} \biggl( \frac{1}{\sinh\eta} \biggr)^{1 / 2} P_{m-1 / 2}^n(\coth\eta) \, .
</math>
  </td>
</tr>
</table>

Drawing upon the same ''Euler reflection formula for gamma functions'', as quoted above, this toroidal function relation can be rewritten as,

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

<tr>
  <td align="right">
<math>~Q_{n-1 / 2}^m (\cosh\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(-1)^n
\frac{\pi^{3/2}}{\sqrt{2} } \biggl[ \frac{\Gamma(m-n+\frac{1}{2})}{\pi (-1)^{m-n}} \biggr] \biggl( \frac{1}{\sinh\eta} \biggr)^{1 / 2} P_{m-1 / 2}^n(\coth\eta) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(-1)^{-m}~
\sqrt{\frac{\pi}{2}} \biggl[ \frac{\Gamma(m-n+\frac{1}{2})}{\sqrt{\sinh\eta}} \biggr]  P_{m-1 / 2}^n(\coth\eta) \, .
</math>
  </td>
</tr>
</table>
Finally, calling upon the "Key Equation" relation,

{{ User:Tohline/Math/EQ_Toroidal00 }}

making the index notation substitution, <math>~\nu \rightarrow (m-\tfrac{1}{2})</math>, and associating <math>~z</math> with <math>~ \coth\eta</math> gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P^n_{m-\frac{1}{2}}(\coth\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\Gamma(m+n+\frac{1}{2})}{\Gamma(m-n+\frac{1}{2})} \biggr]P^{-n}_{m-\frac{1}{2}}(\coth\eta)  \, .</math>
  </td>
</tr>
</table>
As a result, we can write,

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

<tr>
  <td align="right">
<math>~Q_{n-1 / 2}^m (\cosh\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(-1)^{-m}~
\sqrt{\frac{\pi}{2}} \biggl[ \frac{\Gamma(m+n+\frac{1}{2})}{\sqrt{\sinh\eta}} \biggr]  P_{m-1 / 2}^{-n}(\coth\eta) 
</math>
  </td>
  <td align="right" width="15%">
&nbsp;
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
P_{m-1 / 2}^{-n}(\coth\eta) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(-1)^m~\sqrt{\frac{2}{\pi}} \biggl[ \frac{\sqrt{\sinh\eta}}{\Gamma(m+n+\frac{1}{2})} \biggr]  
Q_{n-1 / 2}^m (\cosh\eta) \, .
</math>
  </td>
  <td align="right" width="15%">
<font color="green" size="+3">&#x2461;</font>
  </td>
</tr>
</table>
  </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 order <math>~n - \tfrac{1}{2}</math> and degree <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.

===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>, above, 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>.  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,

{{ User:Tohline/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)], 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 as well as 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===

<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>
</table>
where, as above,


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

Employing 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} \int d\eta^' \int d\theta^' \int d\psi^'
\biggl[ \frac{\rho(\eta^',\theta^',\psi^') ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr] 
\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>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], p. 293, Eq. (2.53)
  </td>
</tr>
</table>

=See Also=


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