Editing Appendix/Ramblings/ToHadleyAndImamura (section)

====Nice Features====
After summarizing, above, our efforts to develop (by empirical techniques) mathematical expressions that qualitatively match the shape of unstable eigenfunctions in toroidal configurations, we put together a subsection titled, "[[#Additional_Comments|Additional Comments]]," to highlight ways in which the empirical fits were successful and ways in which they fell short of expectations.  Here, following our presentation of Blaes's (1985) analytically derived eigenfunction for slim PP tori, we highlight elements of his (physically justified) eigenfunction that explain the origin of many features that were highlighted, above.

* The "amplitude" plot and a plot of the "constant phase locus" ''should'', indeed, appear to be interdependent because they both depend on the functional forms of <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>; in the end, however, the two plots are formally independent of one another.
* The square of the modulus &#8212; that is, the "amplitude" plot &#8212; should always be the sum of two independent functions, both of which are intrinsically positive.
* If either <math>~\mathcal{A}</math> or <math>~\mathcal{B}</math> is defined by a function that crosses zero (perhaps multiple times) with a naturally continuous derivative, that function can quite naturally give rise to a "log-amplitude" plot that shoots toward minus infinity and exhibits a discontinuous derivative after the function is squared to become a piece of the "modulus" expression.
* It is now easy to understand why the earlier "empirically derived" constant phase loci were phrased in terms of the arctangent function.
* Each "constant phase locus" plot should naturally be composed of two (smoothly joined) pieces:  One defined over the inner portion of the torus &#8212; <math>~\eta</math> goes from zero to 1 while <math>~\theta = 0</math>; and one defined over the outer portion of the torus &#8212; <math>~\eta</math> goes from zero to 1 while <math>~\theta = \pi</math>.
* Because (at least for slim PP-tori) the ratio, <math>~\mathcal{A}/\mathcal{B}</math> contains an overall factor that is an odd power of <math>~\cos\theta</math>, the argument of the arctangent function will automatically flip signs as we move from the "inner" region of the torus to the "outer" region of the torus.
* We now appreciate that a plot of "constant phase locus" is smooth across the mid-point (across the cross-sectional center) of the torus because the definition of <math>~\phi_\mathrm{max}</math> naturally contains a phase shift of <math>~k\theta</math>.