Editing Appendix/Mathematics/ToroidalFunctions (section)

==Second Dabbling==

If <math>~\eta = \ln(r_1/r_2)</math>, as defined by Wong, then we can show that,
<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>~
\frac{ \varpi^2 + a^2 + z^2 }{ 2a\varpi  } \, .
</math>
  </td>
</tr>
</table>
</div>
This expression matches the expression for <math>~\Chi</math>, as defined in the CT99; see, for example, [[2DStructure/ToroidalCoordinates#Statement_of_the_Problem|our accompanying discussion]],
<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{ \varpi^2 + R_*^2 + (Z_* -Z)^2 }{ 2R_*\varpi  } \, .
</math>
  </td>
</tr>
</table>
</div>

It is the context of the CT99 derivation that we state,
<div align="center">
<math>Q_{-1/2}(\Chi)= Q_{-1/2}(\coth\eta) = \mu K(\mu) \, ,</math>
</div>
where the argument of the elliptic integral is related to <math>~\Chi</math> via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{2}{1+\Chi}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{4R_*\varpi}{(R_* + \varpi)^2 + (Z_* - Z)^2} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>


<table border="1" align="center" cellpadding="15" width="85%">
<tr>
  <td align="left">
In a [[Appendix/Mathematics/ToroidalSynopsis01#SummaryTable|separate discussion]], we have derived expressions for the quantities, <math>~\Chi</math> and <math>~\mu^2</math>, in terms of Wong's toroidal coordinates.
  </td>
</tr>
</table>


We note that, while the argument of the CT99 toroidal function is <math>~\coth\eta</math>, the argument of Wong's toroidal function is <math>~\cosh\eta</math>.  It should be useful to keep in mind, therefore, that you can move back and forth between <math>~\coth\eta</math> and <math>~\cosh\eta</math> via the mapping (in either direction), 
<div align="center">
<table border="0" cellpadding="5" align="center">

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

Hence, for example, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cosh^2\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\coth^2\eta \biggl[ \coth^2\eta-1  \biggr]^{-1}</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{\varpi^2 + a^2 + z^2}{\cosh\eta} \biggr]^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4a^2\varpi^2  \biggl\{ \biggl[\frac{ \varpi^2 + a^2 + z^2 }{ 2a\varpi  }  \biggr]^{2}-1  \biggr\}</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 ]^{2} - 4a^2\varpi^2 </math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\cosh\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl\{ \frac{  [ \varpi^2 + a^2 + z^2 ]^2  }{ [ (\varpi + a)^2 + z^2 ] [ (\varpi - a)^2 + z^2 ] } \biggr\}^{1 / 2} \, .</math>
  </td>
</tr>
</table>
</div>

So, according to Wong, for <math>~\eta^' > \eta</math>, the ''axisymmetric'' potential is,

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

<tr>
  <td align="right">
<math>~U(\eta^',\theta^')\biggr|_\mathrm{ axi }</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>

<tr>
  <td align="right">
<math>~U(\eta^',\theta^')\biggr|_\mathrm{ axi }</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  
\int_{-\pi}^{\pi} \biggl[\frac{\cos[n(\theta - \theta^')]}{(\cosh\eta - \cos\theta)^{5 / 2}} \biggr] \sinh\eta ~P_{n-1 / 2}(\cosh\eta) ~Q_{n-1 / 2}(\cosh\eta^') ~d\theta ~d\eta
</math>
  </td>
</tr>
</table>
</div>

Drawing from equations (2.7), (2.17) and (2.18) of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], we see that the volume,</span> <math>~V</math>, of a torus that is bounded by surface <math>~\eta_s</math> is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{V}{a^3} = \frac{1}{a^3} \iiint\limits_{\eta_s}d^3 r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint\limits_{\eta_s} \biggl[ \frac{\sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta~ d\psi~ = \frac{2\pi^2\cosh{\eta_s}}{\sinh^3\eta_s} \, .</math>
  </td>
</tr>
</table>
</div>
This means that, in toroidal coordinates, just the integration over the azimuthal angle,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{a^3} \iiint\limits_{\eta_s}d^3 r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\pi \iint\limits \biggl[ \frac{\sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta  \, .</math>
  </td>
</tr>
</table>
</div>
To be compared with the same expression in cylindrical coordinates,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{a^3} \iiint\limits_{\eta_s}d^3 r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2\pi}{a^3} \iint\limits \varpi ~d\varpi ~dz  \, .</math>
  </td>
</tr>
</table>
</div>
This means that the coordinate mapping is,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

This means that the CT99 axisymmetric potential is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(R_*,Z_*)\biggr|_\mathrm{ axi }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 2G  \iint \frac{Q_{-1/2}(\coth\eta)}{ (R_* \varpi)^{1 / 2}}  \rho(\varpi, Z) ~\varpi~d\varpi dZ</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 2Ga^3  \iint \frac{Q_{-1/2}(\coth\eta)}{ (R_* \varpi)^{1 / 2}}  \biggl[ \frac{\sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] \rho~d\eta~ d\theta</math>
  </td>
</tr>
</table>
</div>
Given that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{ 1 }{ \varpi }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(\cosh\eta - \cos\theta)}{a \sinh\eta} \, ,</math>
  </td>
</tr>
</table>
</div>
we therefore have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(R_*,Z_*)\biggr|_\mathrm{ axi }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 2Ga^2\biggl( \frac{a}{R_*}\biggr)^{1 / 2}  \iint Q_{-1/2}(\coth\eta)  \biggl[ \frac{\sinh\eta}{(\cosh\eta - \cos\theta)^5} \biggr]^{1 / 2} \rho~d\eta~ d\theta</math>
  </td>
</tr>
</table>
</div>