Editing Appendix/Ramblings/ToHadleyAndImamura (section)

==Radial Eigenfunction==
===Recognition #1===
[[User:Tohline/Appendix/Ramblings/Azimuthal_Distortions#Radial_Eigenfunction|It occurred to me, first]], that the blue curve displayed in the left-hand panel of Figure 1 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,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

Recognizing that the figure depicting 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,
<div align="center">
<math>~f_1(\varpi) = e^{f_\ln(\varpi)} \, .</math>
</div>

===Recognition #2===

Given that, in general, the following mathematical relation holds,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tanh^{-1}x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>
</table>
</div>

<span id="SquareRoot">[[User:Tohline/Appendix/Ramblings/Azimuthal_Distortions#SwitchToLog|we can write]] for the innermost region of the toroidal configuration &#8212; that is, over the lower radial-coordinate range</span> &#8212;
<div align="center">
<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>
</div>
Similarly, we find that, over the upper radial-coordinate range,
<div align="center">
<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>
</div>

===Recognition #3===

[[User:Tohline/Appendix/Ramblings/Azimuthal_Distortions#Playing_with_the_Radial_Eigenfunction|After a bit more experimentation]], we recognized that it is advantageous to replace the square root &#8212; that is, the exponent, &frac12; &#8212; with a variable exponent, <math>~p</math>, that can serve as an adjustable fitting parameter; and, in order to facilitate a degree of radial overlap between the two ATANH functions, we introduced different values of <math>~r_\mathrm{mid}</math> on the left and on the right.  This led to a two-piece radial eigenfunction of the form,

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

<tr>
  <td align="center" bgcolor="blue">&nbsp;</td>
  <td align="right">
<math>~f_\mathrm{blue}(\varpi) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{r_\mathrm{blue} - \varpi}{\varpi - r_-} \biggr)^{p}
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_- < \varpi < r_\mathrm{blue} </math>
&nbsp; &nbsp; &hellip; &nbsp; &nbsp; 
<math>~\biggl[ f_\mathrm{blue}(\varpi) = 0</math> &nbsp;  &nbsp; otherwise<math>~\biggr]</math>,
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center" bgcolor="green">&nbsp;</td>
  <td align="right">
<math>~f_\mathrm{green}(\varpi) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{r_\mathrm{green} - \varpi}{\varpi - r_+} \biggr)^{p}
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_\mathrm{green} < \varpi < r_+ </math>
&nbsp; &nbsp; &hellip; &nbsp; &nbsp; 
<math>~\biggl[ f_\mathrm{green}(\varpi) = 0</math> &nbsp;  &nbsp; otherwise<math>~\biggr]</math>,
  </td>
</tr>
</table>
</div>
where, <math>~r_\mathrm{mid}|_\mathrm{green} \le r_\mathrm{mid}|_\mathrm{blue}</math>.

===Summary===

The expression that we are currently using for the radial eigenfunction is a sum of these two pieces, that is,

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~f_1(\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~f_\mathrm{green}(\varpi) + f_\mathrm{blue}(\varpi) \, .
</math>
  </td>
</tr>
</table>

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

For later use, we define from this function the minimum and maximum values,

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

<tr>
  <td align="right">
<math>~[f_\ln]_\mathrm{min} \equiv \mathrm{min}[\ln(f_1)]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
and
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~[f_\ln]_\mathrm{max} \equiv \mathrm{max}[\ln(f_1)] \, .</math>
  </td>
</tr>
</table>
</div>