Editing Appendix/Ramblings/AzimuthalDistortions (section)

====Experimenting====
In Figure 5, <math>~r_\mathrm{blue}</math> and <math>~r_\mathrm{green}</math> are fixed, and <math>~p</math> is varied.

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  <th align="center"><font size="+1">Figure 5:</font>&nbsp; Variable exponent, <math>~p</math></th>
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<math>~r_- = 0.5</math>, <math>~r_+ = 1.5</math>
&nbsp; &nbsp; &nbsp; &nbsp; &hellip; &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~r_\mathrm{blue} = 1.25</math>, <math>~r_\mathrm{green}= 1.1</math>
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[[File:MontageAbrief.png|500px|Playing with radial eigenfunction]]
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In this example, the exponent, <math>~p</math>, is varied over the range, <math>~0.25 \le p \le 1.2</math>, as indicated by the numerical values shown in the upper-lefthand corner of each panel.
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Based on the [[#SquareRoot|above discussion]], I expected that the best match to the eigenfunctions found in [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11] would be <math>~p=0.5</math>, that is, a square-root.  However, as illustrated in Figure 5, this and other fractional exponents less than unity generate noncontinuous derivatives at the overlapping edges of our two-piece function.  Instead, a value of <math>~p = 1.2</math> seems to exhibit a more desired, smooth behavior.


In Figure 6, <math>~r_\mathrm{green}</math> and <math>~p</math> are fixed, and <math>~r_\mathrm{blue}</math> is varied.

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  <th align="center"><font size="+1">Figure 6:</font>&nbsp; Variable <math>~r_\mathrm{blue}</math></th>
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<math>~r_- = 0.5</math>, <math>~r_+ = 1.5</math>
&nbsp; &nbsp; &nbsp; &nbsp; &hellip; &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~p = 1.2</math>, <math>~r_\mathrm{green}= 0.9</math>
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[[File:MontageBbrief.png|500px|Playing with radial eigenfunction]]
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In this example, the "blue" edge is varied over the range, <math>~0.91 \le r_\mathrm{blue} \le 1.25</math>, as indicated by the numerical values shown in the upper-lefthand corner of each panel.
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The frames of Figure 6 illustrate the qualitative behavior we have been seeking.  Setting the exponent, <math>~p</math>, to a value greater than unity then varying one of the edges of the two-part eigenfunction provides a natural variation from "pointed" curves that look like adjoined arc-hyperbolic tangents to others that look more like a parabola.