Editing Appendix/Ramblings/ForCohlHoward (section)

==Evaluation of Index Symbols==
===Three Lowest-Order Expressions===
In our [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Derivation_of_Expressions_for_Ai|accompanying derivation of expressions]] for the three lowest-order index symbols <math>A_i</math>, we have used subscripts <math>(\ell, m, s)</math> instead of <math>(1, 2, 3)</math> in order to identify which associated semi-axis length is (largest, medium-length, smallest).  We have derived the following expressions:

<table border="1" align="center" cellpadding="8"><tr><td align="left">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{A_\ell}{a_\ell a_m a_s}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{a_\ell^3 ~p^2 \sin^3\alpha} 
\biggl[ F(\alpha, p) - E(\alpha, p) \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{A_m}{a_\ell a_m a_s} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{ 2}{a_\ell^3 }
\biggl[ \frac{
E(\alpha, p) 
-~(1-p^2)
F(\alpha, p)
-~(a_s/a_m)p^2\sin\alpha}{p^2 (1-p^2)\sin^3\alpha}
\biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{A_s}{a_\ell a_m a_s}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{ 2}{a_\ell^3 }
\biggl[\frac{
(a_m/a_s) \sin\alpha
- E(\alpha, p)}{ (1-p^2) \sin^3\alpha } 
\biggr] \, .
</math>
  </td>
</tr>

</table>

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

The corresponding expressions that appear in Howard's Mathematica notebook are:

<table border="1" align="center" cellpadding="8"><tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>A_1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2 m \sin^3(\phi) } \biggl[
\mathrm{EllipticF}[\phi, m] - \mathrm{EllipticE}[\phi, m] 
\biggr]
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2 m(1-m) \sin^3(\phi) } \biggl[
\mathrm{EllipticE}[\phi, m] - \cos^2\theta \cdot \mathrm{EllipticF}[\phi, m] - \frac{a_3}{a_2}\cdot\sin^2\theta \sin\phi
\biggr]
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A_3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2 (1-m) \sin^3(\phi) } \biggl[
\frac{a_2}{a_3}\cdot \sin(\phi) - \mathrm{EllipticE}[\phi, m] 
\biggr]
\, .
</math>
  </td>
</tr>
</table>
</td></tr></table>
With a little study it should be clear that our derived expressions for <math>A_i</math> precisely match Howard's Mathematica-notebook expressions when <math>\ell = 1</math>, <math>m = 2</math>, and <math>s = 3</math>, that is, in all cases for which <math>a_1 > a_2 > a_3</math>.  But there will be models to consider (for example, in the uppermost region of the so-called [[ThreeDimensionalConfigurations/Stability/RiemannEllipsoids#Background|"horn-shaped" region for S-Type Riemann Ellipsoids]]) for which <math>a_1 > a_3 > a_2</math>, in which case care must be taken in assigning the proper expressions to <math>A_2</math> and <math>A_3</math>.  Similarly note that most of the Riemann models of [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids|Type I]], II, and III &#8212; see, for example, Figure 16 (p. 161) in Chapter 7 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; have either <math>a_2 > a_1</math> or <math>a_3 > a_1</math>.

===Determination of Higher-Order Expressions===
Howard's Mathematica notebook performs brute-force integrations to evaluate various higher-order index-symbol expressions.  Why doesn't he instead use recurrence relations, which point back to the elliptic-integral-based expressions for <math>A_1, A_2, A_3</math>? Specifically &hellip;

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3">
<font color="#770000">'''Index-Symbol Recurrence Relations'''</font>
  </td>
</tr>
<tr>
  <td align="right">
<math>B_{ijk\ell\ldots}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
A_{jkl\ldots} - a_i^2 A_{ijkl\ldots}
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], §21, p. 54, Eq. (105)
  </td>
</tr>

<tr>
  <td align="right">
<math>a_i^2A_{ikl\ldots} - a_j^2 A_{jkl\ldots}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
+ (a_i^2 - a_j^2) B_{ijk\ell\ldots}
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], §21, p. 54, Eq. (106)
  </td>
</tr>

<tr>
  <td align="right">
<math>A_{ikl\ldots} - A_{jkl\ldots}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- (a_i^2 - a_j^2) A_{ijk\ell\ldots}
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], §21, p. 54, Eq. (107)
  </td>
</tr>
</table>

For example, setting <math>i = 1</math> and <math>j = 2</math> in the third of these expressions gives,

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

<tr>
  <td align="right">
<math>A_{1} - A_{2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- (a_1^2 - a_2^2) A_{12}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ A_{12}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{A_{2} - A_{1}}{(a_1^2 - a_2^2)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ a_1^2 A_{12}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{A_{2} - A_{1}}{(1 - a_2^2/a_1^2)} \, ;
</math>
  </td>
</tr>
</table>
and, from the first of the relations,

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

<tr>
  <td align="right">
<math>B_{12}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
A_{2} - a_1^2 A_{12}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
A_{2} - a_1^2 \biggl[ \frac{A_{2} - A_{1}}{(a_1^2 - a_2^2)} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{1}{(a_1^2 - a_2^2)} \biggl[(a_1^2 - a_2^2)A_2
- a_1^2 ( A_{2} - A_{1} )\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{1}{(a_1^2 - a_2^2)} \biggl[a_1^2 A_1 - a_2^2A_2\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{1}{(1 - a_2^2/a_1^2)} \biggl[A_1 - \frac{a_2^2}{a_1^2} \cdot A_2\biggr]\, .
</math>
  </td>
</tr>
</table>


Also, consider using the set of relations labeled "LEMMA 7" on p. 54 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].

===Example Test Evaluations===

Some of Howard's 20-digit-precision evaluations of various index symbols have been recorded, for comparison with our separate lower-precision evaluations, as follows:
<ul>
  <li>Values of <math>(A_1, A_2, A_3)</math> are recorded for a model with <math>(a_1, a_2, a_3) = (1, 0.9, 0.641)</math> in the [[ThreeDimensionalConfigurations/RiemannStype#TestPart1|table titled, ''TEST (part 1)'', near the top of our chapter on Riemann S-Type ellipsoids]].</li>
  <li>Values of <math>(A_{12}, B_{12})</math> are recorded for a model with <math>(a_1, a_2, a_3) = (1, 0.9, 0.641)</math> in the [[ThreeDimensionalConfigurations/RiemannStype#TestPart2|table titled, ''TEST (part 2)'' in our chapter on Riemann S-Type ellipsoids]].</li>
</ul>