Editing Appendix/Ramblings/AzimuthalDistortions (section)

===Simpler Connection Between Radial and Phase Eigenfunctions===

When it is used as an argument of ATAN, the function, <math>~D(\varpi)</math>, [[#Constant_Phase_Loci|as defined above]] smoothly steers <math>~\phi_1</math> through a phase shift of approximately <math>~\pi</math> radians principally because the function itself smoothly varies between <math>~+1</math> and <math>~0</math> (then back again).  Let's play with some simpler expressions for this governing function that also vary smoothly between these limits.  

Define,
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  <td align="right">
<math>~\Delta_\mathrm{inner}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; has range of &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~+0 ~~\rightarrow ~~ +1 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Delta_\mathrm{outer}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\varpi - r_+}{r_\mathrm{mid}-r_+} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; has range of &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~+1 ~~\rightarrow ~~ +0 \, .</math>
  </td>
</tr>
</table>
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The argument of ATANH, [[#Radial_Eigenfunction|as presented above]], was obtained from this relatively simple expression via the shift,
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<tr>
  <td align="right">
<math>~1-2\Delta_\mathrm{inner}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; has range of &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~+1 ~~\rightarrow ~~ -1 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~1-2\Delta_\mathrm{outer}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\varpi - r_+}{r_\mathrm{mid}-r_+} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; has range of &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~-1 ~~\rightarrow ~~ +1 \, .</math>
  </td>
</tr>
</table>
</div>

For the argument of <math>~D(\varpi)</math> function, we could therefore use,
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<tr>
  <td align="right">
<math>~1-\Delta_\mathrm{inner}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; has range of &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~+1 ~~\rightarrow ~~ 0 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~1-\Delta_\mathrm{outer}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\varpi - r_+}{r_\mathrm{mid}-r_+} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; has range of &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~0 ~~\rightarrow ~~ +1 \, .</math>
  </td>
</tr>
</table>
</div>