Editing Appendix/Ramblings/ToHadleyAndImamura (section)

===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,
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<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>
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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,
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<math>~f_1(\varpi) = e^{f_\ln(\varpi)} \, .</math>
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