Editing Appendix/Ramblings/AzimuthalDistortions (section)

===Radial Eigenfunction===
It occurred to me, first, that the blue curve displayed in the left-hand panel of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H  HI11]'s figure 6 (our Figure 2) might be reasonably well approximated by piecing together a pair of arc-hyperbolic-tangent (ATANH) functions. In an effort to demonstrate this, I began by specifying a "midway" radial location, <math>~r_- < r_\mathrm{mid} < r_+ \, ,</math> at which the two ATANH functions meet and at which the density fluctuation is smallest.  Then I defined a function of the form,
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<tr>
  <td align="center" bgcolor="blue">&nbsp;</td>
  <td align="right">
<math>~f_\ln(\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tanh^{-1}\biggl[1 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr) \biggr]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_- < \varpi < r_\mathrm{mid} \, ;</math>
  </td>
</tr>
<tr><td colspan="6" align="center">and</td></tr>
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  <td align="center" bgcolor="green">&nbsp;</td>
  <td align="right">
<math>~f_\ln(\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tanh^{-1}\biggl[1 - 2 \biggl( \frac{\varpi - r_+}{r_\mathrm{mid}-r_+} \biggr) \biggr]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_\mathrm{mid} < \varpi < r_+ \, .</math>
  </td>
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This empirically specified, two-piece <math>~f_\ln(\varpi)</math> function has been plotted in the left-hand panel of Figure 3.  (To facilitate quantitative comparison with Figure 2, the function has been normalized as explained in the boxed-in ''PRACTICAL IMPLEMENTATION'' remark that follows.)  Blue dots trace the function's behavior over the lower radial-coordinate range while green dots trace its behavior over the upper radial-coordinate range.  This plot of <math>~f_\ln(\varpi)</math> closely resembles the plot of the eigenfunction, <math>~\delta\rho/\rho (\varpi)</math> (see the left-hand panel of our Figure 2) that developed spontaneously via [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H  HI11]'s linear stability analysis. 


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'''<font color="maroon">PRACTICAL IMPLEMENTATION:</FONT>'''  &nbsp; At the two limits, <math>~\varpi = r_-</math> and <math>~\varpi = r_+</math>, the function, <math>~f_\ln(\varpi) \rightarrow +\infty</math>; while, at the limit, <math>~\varpi = r_\mathrm{mid}</math>, the function, <math>~f_\ln(\varpi) \rightarrow -\infty</math>.  In practice, after dividing the relevant radial extent into 100 zones, we stay ''half of a radial zone'' away from these three limiting radial boundaries, so that the maximum and minimum values of <math>~f_\ln(\varpi)</math> are finite; specifically, in the example plotted here, we have set <math>~[f_\ln]_\mathrm{min} = -2.99448</math> and <math>~[f_\ln]_\mathrm{max} = 2.64665</math>.  Then we strategically employ the finite values of the function at these near-boundary limits to rescale the function such that, in the plot shown here, it lies between -3 (minimum amplitude) and 0 (maximum amplitude).
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<span id="SwitchToLog">Recognizing that the figure depicting</span> the [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H  HI11] eigenfunction is a semi-log plot, it seems clear that the relationship between our constructed function, <math>~f_\ln(\varpi)</math>, and the eigenfunction, <math>~f_1(\varpi)</math>, is,
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<math>~f_1(\varpi) = e^{f_\ln(\varpi)} \, .</math>
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Now, in general, the following mathematical relation holds:
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  <td align="right">
<math>~\tanh^{-1}x</math>
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  <td align="center">
<math>~=</math>
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<math>~\ln\biggl( \frac{1+x}{1-x} \biggr)^{1/2} </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>x^2 < 1 \, .</math>
  </td>
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Hence, for the innermost region of the toroidal configuration &#8212; that is, over the lower radial-coordinate range &#8212; we can set,
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  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr) </math>
  </td>
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  <td align="right">
<math>~\Rightarrow ~~~~ \frac{1+x}{1-x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl[2 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)\biggr] 
\biggl[2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)\biggr]^{-1}
</math>
  </td>
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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<math>
~[(r_\mathrm{mid}-r_-) - ( \varpi - r_-)]  [(\varpi - r_-)]^{-1}
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<math>
~\frac{r_\mathrm{mid} - \varpi}{\varpi - r_-} \, .
</math>
  </td>
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<span id="SquareRoot">Therefore we can write,</span>
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center" bgcolor="blue">&nbsp;</td>
  <td align="right">
<math>~f_1(\varpi) = e^{f_\ln(\varpi)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{r_\mathrm{mid} - \varpi}{\varpi - r_-} \biggr)^{1/2}
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_- < \varpi < r_\mathrm{mid} \, .</math>
  </td>
</tr>
</table>
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Similarly, we find that, over the upper radial-coordinate range,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center" bgcolor="green">&nbsp;</td>
  <td align="right">
<math>~f_1(\varpi) = e^{f_\ln(\varpi)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{r_\mathrm{mid} - \varpi}{\varpi - r_+} \biggr)^{1/2}
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_\mathrm{mid} < \varpi < r_+ \, .</math>
  </td>
</tr>
</table>
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