View source for User:Tohline/Appendix/Mathematics/ToroidalSynopsis01

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=Synopsis of Toroidal Coordinate Approach=


{{LSU_HBook_header}}

==Basics==
Here we attempt to bring together &#8212; in as succinct a manner as possible &#8212; [[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|our approach]] and [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong's (1973)]  approach to determining the gravitational potential of an axisymmetric, uniform-density torus that has a major radius, <math>~R</math>, and a minor, cross-sectional radius, <math>~d</math>.  The relevant toroidal coordinate system is one based on an ''anchor ring'' of major radius, 
<div align="center">
<math>~a^2 \equiv R^2 - d^2 \, .</math>
</div>
If the meridional-plane location of the ''anchor ring'' &#8212; as written in cylindrical coordinates &#8212; is, <math>~(\varpi, z) = (a,Z_0)</math>, then the preferred toroidal-coordinate system has meridional-plane coordinates, <math>~(\eta, \theta)</math>, defined such that,
<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>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <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>
</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>~(\varpi + a)^2 + (z-Z_0)^2 \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~r_2^2 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(\varpi - 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>.  Mapping the other direction, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a \sinh\eta }{(\cosh\eta - \cos\theta)} \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <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>
<span id="DiffVolElement">The three-dimensional differential volume element is,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d^3 r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="center">
<math>\varpi d\varpi ~dz ~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>
</div>

Note that, if <math>~\eta_0</math> identifies the surface of the uniform-density torus, then,
<div align="center">
<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>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\sinh\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{d} \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~\coth\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R}{a} \, ;</math>
  </td>
</tr>
</table>
</div>

and when the integral over the volume element is completed &#8212; that is, over all <math>~\psi</math>, over all <math>~\theta</math>, and over the "radial" interval, <math>~\eta_0 \le \eta \le \infty</math> &#8212; the resulting volume is,
<div align="center">
<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>~\frac{2\pi^2 \cosh\eta_0}{\sinh^3\eta_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi^2 Rd^2 \, .</math>
  </td>
</tr>
</table>
</div>

Also, given that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cosh\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl[ e^\eta + e^{-\eta} \biggr]</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~\sinh\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl[ e^\eta - e^{-\eta} \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\coth\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ e^\eta + e^{-\eta} \biggr]\biggl[ e^\eta - e^{-\eta} \biggr]^{-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{r_1}{r_2} +  \frac{r_2}{r_1} \biggr]\biggl[  \frac{r_1}{r_2} - \frac{r_2}{r_1} \biggr]^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" colspan="1">
<math>~\biggl[ \frac{r_1^2 + r_2^2}{r_1 r_2}  \biggr]\biggl[  \frac{r_1^2 - r_2^2}{r_1 r_2} \biggr]^{-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" colspan="1">
<math>~\biggl[ \frac{r_1^2 + r_2^2}{r_1^2 - r_2^2}  \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" colspan="3">
<math>~
\frac{  \varpi^2 + a^2 + (z - Z_0)^2 }{ 2a\varpi } \, .
</math>
  </td>
</tr>
</table>
</div>

==Arguments of Q and K==

Want to explore argument of <math>~Q_{-1 / 2}(\Chi)</math>, namely,
<div align="center">
<math>
\Chi \equiv \frac{(\varpi^')^2 + \varpi^2 + (z^' - z)^2}{2\varpi^'  \varpi} .
</math>
</div>
Therefore,

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

<tr>
  <td align="right">
<math>~2\varpi \biggl[ \varpi^' \Chi - a\coth\eta\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\varpi^')^2 + \varpi^2 + (z^' - z)^2 - 
[\varpi^2 + a^2 + (z - Z_0)^2 ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\varpi^')^2 - a^2 +  [ (z^')^2 - 2z^' z + z^2]- 
[z^2 - 2zZ_0 + Z_0^2]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\varpi^')^2 - a^2 +  (z^')^2- Z_0^2   +2z(Z_0 - z^' )
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~2a\biggl[ \frac{\sinh\eta }{(\cosh\eta - \cos\theta)}  \biggr]\biggl[ \varpi^' \Chi - a\coth\eta\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\varpi^')^2 - a^2 +  (z^')^2- Z_0^2   +2(Z_0 - z^' )\biggl[ Z_0 + \frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2aC_0
+2a(Z_0 - z^' )\biggl[ \frac{\sin\theta}{(\cosh\eta - \cos\theta)} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \sinh\eta \biggl[ \varpi^' \Chi - a\coth\eta\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
C_0 (\cosh\eta - \cos\theta) +
(Z_0 - z^' ) \sin\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \varpi^' \Chi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\sinh\eta} \biggl[ C_0 (\cosh\eta - \cos\theta) + (Z_0 - z^' ) \sin\theta + a\cosh\eta\biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \Chi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\varpi^' \sinh\eta} \biggl[ (C_0 + a)\cosh\eta + (Z_0 - z^' ) \sin\theta - C_0 \cos\theta \biggr]  
</math>
  </td>
</tr>
</table>
</div>

where,
<div align="center">
<math>~
C_0 \equiv 
\frac{1}{2a}\biggl[ (\varpi^')^2 - a^2 +  (z^')^2- Z_0^2   +2Z_0 (Z_0 - z^' ) \biggr]
=
\frac{1}{2a}\biggl[ (\varpi^')^2 - a^2 +  (z^')^2 +Z_0^2 - 2Z_0 z^'  \biggr]
=
\frac{1}{2a}\biggl[ (\varpi^')^2 - a^2 + (z^' - Z_0)^2  \biggr] \, .
</math>
</div>

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

<tr>
  <td align="right">
<math>~
( \varpi^')^2 + a^2 + (z^' - Z_0)^2  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" colspan="3">
<math>~
2a\varpi^'~\coth\eta^'
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
( \varpi^')^2 - a^2 + (z^' - Z_0)^2  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" colspan="3">
<math>~
2a\varpi^'~\coth\eta^' - 2a^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
C_0  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" colspan="3">
<math>~
\varpi^'~\coth\eta^' - a
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" colspan="3">
<math>~
\biggl[ \frac{a \sinh\eta^' }{(\cosh\eta^' - \cos\theta^')} \biggr] ~\coth\eta^' - a
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" colspan="3">
<math>~
\biggl[ \frac{a \cosh\eta^' }{(\cosh\eta^' - \cos\theta^')} \biggr] - a \, .
</math>
  </td>
</tr>
</table>
</div>

Hence,

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

<tr>
  <td align="right">
<math>~ \Chi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\cosh\eta}{\varpi^' \sinh\eta}  \biggl[ \varpi^' \coth\eta^' \biggr]  
+
\frac{1}{\sinh\eta}  \biggl[ \frac{(\cosh\eta^' - \cos\theta^')}{a \sinh\eta^' } \biggr]  \biggl[  (Z_0 - z^' ) \sin\theta - C_0 \cos\theta \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\coth\eta \cdot \coth\eta^' 
+
 \biggl[ \frac{(\cosh\eta^' - \cos\theta^')}{a \sinh\eta \cdot \sinh\eta^' } \biggr]  \biggl[  (Z_0 - z^' ) \sin\theta - C_0 \cos\theta \biggr]  
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\coth\eta \cdot \coth\eta^' 
-
 \biggl[ \frac{\sin\theta^' \sin\theta +\cos\theta^'\cos\theta }{ \sinh\eta \cdot \sinh\eta^' } \biggr] 
</math>
  </td>
</tr>

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

[[User:Tohline/2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential|Also]],

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

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

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

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

<span id="SummaryTable">NOTE by Tohline:  On 5 June 2018, I used Excel to test the validity of the toroidal-coordinate-based expressions that have been derived here, and summarized in the following table.</span>

<table border="1" align="center" cellpadding="10">
<tr>
  <th align="center" colspan="3">
Summary Table
  </th>
</tr>
<tr>
  <th align="center">
Quantity
  </th>
  <th align="center">
Raw Expression in Cylindrical Coordinates
  </th>
  <th align="center">
Expression in Terms of Toroidal Coordinates
  </th>
</tr>
<tr>
  <td align="center">
<math>~\Chi</math>
  </td>
  <td align="center">
<math>
 \frac{(\varpi^')^2 + \varpi^2 + (z^' - z)^2}{2\varpi^'  \varpi} 
</math>
  </td>
  <td align="center">
<math>~
\frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } 
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\mu^2 \equiv \frac{2}{\Chi + 1}</math>
  </td>
  <td align="center">
<math>
\frac{4\varpi^' \varpi}{(\varpi^' + \varpi)^2 + (z^' - z)^2} 
</math>
  </td>
  <td align="center">
<math>~
\frac{2 \sinh\eta \cdot \sinh\eta^' }{ \cosh(\eta^' + \eta) - \cos(\theta^' - \theta) } 
</math>
  </td>
</tr>
</table>

==Potential==

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>

</table>
</div>

===Volume Element===

See [[#DiffVolElement|above]].

===Green's Function===
Wong (1973) points out 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>.  According to [http://adsabs.harvard.edu/abs/1999ApJ...527...86C CT99], 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=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{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>

Things to note:
<ol>
  <li>The argument of <math>~Q_{m - 1 / 2}</math> in the CT99 expression is very different from the argument of <math>~Q^m_{n - 1 / 2}</math> (or <math>~P^m_{n - 1 / 2}</math>) in Wong's expression.</li>
  <li>In both expressions, <math>~m</math> is the integer multiplying the azimuthal angle, <math>~\psi</math>, but in the CT99 expression this index serves as the ''subscript'' index of the function, <math>~Q</math>, whereas in Wong's expression it serves as the ''superscript'' index of both functions, <math>~Q</math> and <math>~P</math>.  In this context, note that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q^m_{n-\frac{1}{2}}(\cosh\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(-1)^m
\sqrt{\frac{\pi}{2}} ~\Gamma(m-n+\tfrac{1}{2}) \biggl[ \frac{1}{ \sqrt{\sinh\eta}} \biggr] P^{n}_{m - \frac{1}{2}} (\coth\eta) \, .
</math>
  </td>
</tr>
</table>
  </li>
  <li>Wong's expression contains not only a summation over the index, <math>~m</math>, but also an explicit summation over the index, <math>~n</math>, which multiplies the "polar" angle, <math>~\theta</math>; no such additional summation appears in the CT99 expression, indicating that the summation over <math>~n</math> has implicitly already been completed.  In this context, note that the [[User:Tohline/Appendix/Mathematics/ToroidalConfusion#Joel.27s_Additional_Manipulations|summation expression]] gives,

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

<tr>
  <td align="right">
<math>~
Q^{\mu}_{-\frac{1}{2}}\left(\cosh\xi\right)
+
2\sum_{n=1}^{\infty}
Q^{\mu}_{n-\frac{1}{2}}\left(\cosh\xi\right) \cos\left[ n (\theta - \theta^') \right] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ e^{\mu\pi i} \Gamma\left(\mu+ \tfrac{1}{2} \right) \biggl[
\dfrac{\left(\frac{1}{2}\pi\right)^{1/2}\left(\sinh\xi\right)^{\mu
}}{\left\{ \cosh\xi -\cos\left[ n (\theta - \theta^') \right] \right\}^{\mu+(1/2)}}\biggr]
\, ;
</math>
  </td>
</tr>
</table>
or, specifically for the case of <math>~\mu = 0</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\sum_{n=0}^{\infty} \epsilon_n
Q_{n-\frac{1}{2}}\left(\cosh\xi\right) \cos\left[ n(\theta - \theta^') \right] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\dfrac{ \pi/\sqrt{2} }{\left[ \cosh\xi-\cos(\theta - \theta^') \right]^{\frac{1}{2}}  } \, .
</math>
  </td>
</tr>
</table>
  </li>
  <li>Next thought &hellip;</li>
</ol>

==New Insight==

===Identical Green's Function Expressions===

[https://authors.library.caltech.edu/43491/ Caltech's electronic version] of [https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erd&eacute;lyi's (1953)] ''Higher Transcendental Functions''; in particular, &sect;3.11, p. 169 of Volume I gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
Q_\nu[t t^' - (t^2-1)^{1 / 2} (t^{'2} - 1)^{1 / 2} \cos\psi]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
Q_\nu(t) P_\nu(t^') + 2\sum_{n=1}^\infty (-1)^n Q^n_\nu(t) P^{-n}_\nu(t^') \cos(n\psi) \, .
</math>
  </td>
</tr>
</table>
</div>
This is valid for,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~t, t^'</math>&nbsp; real
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~1 < t^' < t</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\nu \ne -1, -2, -3, </math> &hellip;
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\psi</math> &nbsp; real.
  </td>
</tr>
</table>
</div>


If we make the association, <math>~t \leftrightarrow \coth\eta</math>, then we also have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\sinh\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{t^2 - 1} \, ,</math>
  </td>
</tr>
</table>
</div>

in which case,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \Chi </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="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
t t^' - (t^2-1)^{1 / 2}(t^{'2}-1)^{1 / 2}\cos(\theta^' - \theta) \, .
</math>
  </td>
</tr>
</table>
</div>

Put together, then, these expressions mean,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
Q_{m - 1 / 2}(\Chi)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
Q_{m-1 / 2}(\coth\eta) P_{m - 1 / 2}(\coth\eta^') + 2\sum_{n=1}^\infty (-1)^n Q^n_{m - 1 / 2}(\coth\eta) P^{-n}_{m - 1 / 2}(\coth\eta^') \cos[n(\theta^' - \theta)] 
</math>
  </td>
</tr>

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

Also, from [[User:Tohline/Appendix/Mathematics/ToroidalConfusion#QPrelation|our derived <math>~Q-P</math> relation]],
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
P^{-n}_{m - \frac{1}{2}} (\coth\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sqrt{\frac{2}{\pi}} ~\frac{(-1)^m \sqrt{\sinh\eta} }{\Gamma(n+m + \tfrac{1}{2})} ~ Q^m_{n-\frac{1}{2}}(\cosh\eta) 
\, .
</math>
  </td>
</tr>
</table>
</div>
we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
Q_{m - 1 / 2}(\Chi)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sum_{n=0}^\infty \epsilon_n (-1)^n Q^n_{m - 1 / 2}(\coth\eta) 
\biggl\{ \sqrt{\frac{2}{\pi}} ~\frac{(-1)^m \sqrt{\sinh\eta^'} }{\Gamma(n+m + \tfrac{1}{2})} ~ Q^m_{n-\frac{1}{2}}(\cosh\eta^') \biggr\} 
\cos[n(\theta^' - \theta)] 
</math>
  </td>
</tr>
</table>
</div>

Next, we pull from the [[User:Tohline/Appendix/Mathematics/ToroidalConfusion#Gil|accompanying discussion of the Gil et al. (2000) expression]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q_{n-1 / 2}^m (\lambda)</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})} (x^2-1)^{1 / 4} P_{m-1 / 2}^n(x) \, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>~\lambda \equiv x/\sqrt{x^2-1}</math>. Identifying  <math>~x</math> with <math>~\cosh\eta</math>, in which case we have <math>~\lambda = \coth\eta</math>, and, switching index notation, <math>~n \leftrightarrow m</math>, gives,
<div align="center">
<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>

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

where, this last step also incorporates the [[User:Tohline/Appendix/Mathematics/ToroidalConfusion#Proper_Interpretation_of_DLMF_Expression|"Euler reflection formula for gamma functions"]], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

So we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
Q_{m - 1 / 2}(\Chi)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sum_{n=0}^\infty \epsilon_n (-1)^n 
\biggl\{(-1)^n \sqrt{ \frac{\pi}{2} } ~\Gamma(n - m + \tfrac{1}{2} )(\sinh\eta)^{1 / 2} P_{n-1 / 2}^m(\cosh\eta)\biggr\}
\biggl\{ \sqrt{\frac{2}{\pi}} ~\frac{(-1)^m \sqrt{\sinh\eta^'} }{\Gamma(n+m + \tfrac{1}{2})} ~ Q^m_{n-\frac{1}{2}}(\cosh\eta^') \biggr\} 
\cos[n(\theta^' - \theta)] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{\sinh\eta^'} \sqrt{\sinh\eta} 
\sum_{n=0}^\infty \epsilon_n  (-1)^m
\frac{ \Gamma(n - m + \tfrac{1}{2})}{\Gamma(n+m + \tfrac{1}{2})} 
P_{n-1 / 2}^m(\cosh\eta) 
Q^m_{n-\frac{1}{2}}(\cosh\eta^') 
\cos[n(\theta^' - \theta)] \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, the CT99 Green's function may be rewritten as,
<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}{a\pi} 
[ (\cosh\eta^' - \cos\theta^') (\cosh\eta - \cos\theta)]^{1 / 2}
\sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] 
\sum_{n=0}^\infty \epsilon_n  (-1)^m
\frac{ \Gamma(n - m + \tfrac{1}{2})}{\Gamma(n+m + \tfrac{1}{2})} 
P_{n-1 / 2}^m(\cosh\eta) 
Q^m_{n-\frac{1}{2}}(\cosh\eta^') 
\cos[n(\theta^' - \theta)] 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{a\pi} 
[ (\cosh\eta^' - \cos\theta^') (\cosh\eta - \cos\theta)]^{1 / 2}
\sum_{m=0}^{\infty} \sum_{n=0}^\infty \epsilon_m\epsilon_n  (-1)^m   
\frac{ \Gamma(n - m + \tfrac{1}{2})}{\Gamma(n+m + \tfrac{1}{2})} 
\cos[m(\psi - \psi^')] \cos[n(\theta^' - \theta)]
P_{n-1 / 2}^m(\cosh\eta) 
Q^m_{n-\frac{1}{2}}(\cosh\eta^') \, .
</math>
  </td>
</tr>
</table>

Let's compare this with Wong's (1973) Green's function, namely,
<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>
[June 10, 2018]  <font color="red"><b>Amazing! </b></font> The two expressions match precisely!

===Integral Over Polar Angle===

Returning to [https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erd&eacute;lyi's (1953)] ''Higher Transcendental Functions'' &hellip;  
<ul>
  <li>
Equation (5) in &sect;3.7, p. 155 of Volume I gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q_\nu^\mu(z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{i \mu \pi} ~2^{-\nu - 1} \frac{\Gamma(\nu + \mu + 1) }{\Gamma(\nu + 1) } (z^2 - 1)^{-\mu/2} \int_0^\pi (z+\cos t)^{\mu - \nu - 1} (\sin t)^{2\nu + 1} dt \, .
</math>
  </td>
</tr>
</table>

This is valid for,

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

<tr>
  <td align="right">
<math>~\mathrm{Re} ~\nu > -1</math>&nbsp;
  </td>
  <td align="center">
&nbsp; &nbsp; and &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\mathrm{Re} (\nu + \mu + 1) > 0 \, .</math>
  </td>
</tr>
</table>

  </li>
  <li>
Equation (10) in &sect;3.7, p. 156 of Volume I gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q_\nu^\mu(z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{i \mu \pi} ~ (2\pi)^{-\frac{1}{2}} (z^2-1)^{\mu/2} ~\Gamma(\mu + \tfrac{1}{2})~\biggl\{
\int_0^\pi (z - \cos t)^{-\mu - \frac{1}{2}} \cos[(\nu + \tfrac{1}{2})t] ~dt
-\cos(\nu\pi) \int_0^\infty (z + \cosh t)^{-\mu - \frac{1}{2}} e^{-(\nu + \frac{1}{2})t} ~dt
\biggr\}
</math>
  </td>
</tr>
</table>

This is valid for,

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

<tr>
  <td align="right">
<math>~\mathrm{Re} ~\nu > -\tfrac{1}{2}</math>&nbsp;
  </td>
  <td align="center">
&nbsp; &nbsp; and &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\mathrm{Re} (\nu + \mu + 1) > 0 \, .</math>
  </td>
</tr>
</table>

  </li>
</ul>

We will lean on this integral definition of the Legendre function, <math>~Q^\mu_\nu</math>, to evaluate the definite integral in equation (2.56) of Wong (1973), viz.,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\int_{-\pi}^{\pi} \frac{\cos[n(\theta - \theta^')] d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} </math>
  </td>
  <td align="right">
<math>~=</math>
  </td>
  <td align="right">
<math>~\int_{-\pi}^{\pi} \frac{ \{\cos(n\theta)\cos(n\theta^') + \sin(n\theta)\sin(n\theta^') \} d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} \, .</math>
  </td>
</tr>
</table>
</div>
Therefore, we

=See Also=
* [https://authors.library.caltech.edu/43491/1/Volume%201.pdf Arthur Erd&eacute;lyi (1953, New York:  McGraw-Hill)] &#8212; ''Higher Transcendental Functions'', especially Volume I, Chapter 3 (pp. 120 - 174)
* [http://adsabs.harvard.edu/abs/1940QJMat..11..222C T. G. Cowling (1940, Quart. J. Math. Oxford Ser., 11, 222 - 224)] &#8212; ''On Certain Expansions Involving Products of Legendre Functions''


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