Editing Appendix/Ramblings/NonlinarOscillation (section)

===Inferred Displacement Function===

<table border="1" cellpadding="8" width="80%" align="center">
<tr><td>
<font color="red"><b>ASIDE:</b></font>&nbsp; The various "shorthand" variables that have been introduced throughout this chapter should be viewed as consistent with one another in the following sense.  We understand that, in each pairing of models, one will be associated with a value of <math>~\tilde\xi > 3</math> and the other will be associated with a value of <math>~\tilde\xi < 3</math>.  The model having the ''smaller'' value of <math>~\tilde\xi</math>, and that has been tagged with the "plus" subscript <math>~(\xi_+)</math>, also corresponds to the model that:
* Has the larger equilibrium radius <math>~(\chi_+)</math> &#8212; that is, <math>~\chi_+/r_\mathrm{crit} > 1</math>;
* Has the larger value of <math>~\tilde{C}</math> &#8212; that is, <math>~\tilde{C} > 4</math>;
* In connection with the ''Delta Profiles'', is tagged with the subscript "2"; hence, <math>~\tilde{C}_2 > 4</math>;
* Has a ''positive'' value of the small parameter, <math>~\epsilon</math> &#8212; that is, <math>~\epsilon_+ > 0</math>, whereas, <math>~\epsilon_- < 0</math>.

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


====Foundation====

From our [[#Run_of_Mass|discussion, below]], for any value of the truncation radius, <math>~\tilde\xi</math>, the fractional mass <math>~(0 \le m_\xi \le 1)</math> that lies interior to <math>~\xi</math> is given by the expression,

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

<tr>
  <td align="right">
<math>~m_\xi \equiv \frac{M(\xi)}{M_\mathrm{tot}}</math>
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~
\biggl(\frac{\xi}{\tilde\xi}\biggr)^3 \biggl(1 + \frac{\xi^2}{3}\biggr)^{-3/2} 
\biggl(1 + \frac{\tilde\xi^2}{3}\biggr)^{3/2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~\biggl(\frac{\tilde{C}}{3}\biggr)^{3 / 2}
\xi^3 \biggl(3 + \xi^2\biggr)^{-3/2} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tilde{C} \equiv \frac{3^2}{\tilde\xi^2}\biggl( 1 + \frac{\tilde\xi^2}{3} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~
{\tilde\xi}^2 = \frac{9}{\tilde{C} - 3} \, .
</math>
  </td>
</tr>

</table>
</div>
And, when normalized to <math>~R_\mathrm{SWS}</math>, the corresponding radius is,

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

<tr>
  <td align="right">
<math>~r_\mathrm{SWS}(\xi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\xi}{\tilde\xi} \biggr) \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\xi}{\tilde\xi} \biggr) \biggl\{ \frac{3\cdot 5}{2^2 \pi} \biggl[ \frac{ \tilde\xi^2/3}{(1+ \tilde\xi^2/3)^{2}} \biggr] \biggr\}^{1/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi\biggl[ \frac{3^2 \cdot 5}{2^2 \pi} \biggr]^{1 / 2} (3+{\tilde\xi}^2)^{-1}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi\biggl[ \frac{3^2 \cdot 5}{2^2 \pi} \biggr]^{1 / 2} \biggl[ \frac{\tilde{C} - 3}{3\tilde{C}} \biggr]  \, .</math>
  </td>
</tr>
</table>
</div>

Now, this works fine in the sense that, for any choice of <math>~\tilde\xi</math>, and therefore <math>~\tilde{C}</math>, this pair of parametric relations can be used to generate a plot of <math>~r_\mathrm{SWS}</math> versus <math>~m_\xi</math> that correctly displays how the mass enclosed within a given radius varies with radial location throughout the spherical configuration.  But, in order to ''compare'' one of these configurations to another, we really need to identify how this function varies across a Lagrangian mass grid that is the same for both configurations.  The easiest way to accomplish this is to derive an expression for <math>~r_\mathrm{SWS}</math> that is directly a function of <math>~m_\xi</math>.  Fortunately, this can be done analytically. First, we invert the mass expression to find,

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

<tr>
  <td align="right">
<math>~m_\xi^{2/3}</math>
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~\biggl(\frac{\tilde{C}}{3}\biggr)
\xi^2 (3 + \xi^2)^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 3 + \xi^2</math>
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~
m_\xi^{-2/3}\biggl(\frac{\tilde{C}}{3}\biggr)\xi^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \xi^2\biggl[ 1 - m_\xi^{-2/3}\biggl(\frac{\tilde{C}}{3}\biggr) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~
-3
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \xi^2</math>
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~
3^2 [ \tilde{C}~m_\xi^{-2/3} -3 ]^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>
<span id="ExactProfile">Inserting this into the radial equation, then, gives,</span>

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

<tr>
  <td align="right">
<math>~r_\mathrm{SWS}(m_\xi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{3^2 \cdot 5}{2^2 \pi} \biggr]^{1 / 2} \biggl[ \frac{\tilde{C} - 3}{\tilde{C}} \biggr] \biggl[ \tilde{C}~m_\xi^{-2/3} -3 \biggr]^{-1 / 2} \, .</math>
  </td>
</tr>
</table>
</div>

====Analytic, Marginally Unstable Eigenfunction====
In terms of <math>~\xi</math>, we know that the [[SSC/Stability/n5PolytropeLAWE#Eureka_Moment|eigenfunction of the marginally unstable model]] &#8212; see also [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|a more general discussion]] &#8212; is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_P = \frac{\delta r}{r_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{\xi^2}{15} \, .</math>
  </td>
</tr>
</table>
</div>
We can now rewrite this eigenfunction in terms of the fractional mass, <math>~m_\xi</math>.  Specifically, given that <math>~\tilde{C} = 4</math> in the marginally unstable configuration, we find that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_P </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{3}{5} \biggl[ 4~m_\xi^{-2/3} -3 \biggr]^{-1} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{5} \biggl[ \frac{10~m_\xi^{-2/3} -9}{4~m_\xi^{-2/3} -3} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{5} \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3}} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

It is important to remember that, although the leading factor of this expression is <math>~\tfrac{2}{5}</math>, in general the overall amplitude of this eigenfunction can be set arbitrarily.  In order to allow for this, we will introduce an overall scaling factor, <math>~A_0</math>, and write,

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

<tr>
  <td align="right">
<math>~x_P </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2A_0}{5} \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3}} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Then our originating expression for <math>~x_P</math> is retrieved by setting <math>~A_0 = 1</math>, in which case the amplitude of the eigenfunction is unity at the center <math>~(m_\xi = 0)</math> and it is <math>~\tfrac{2}{5}</math> at the surface  <math>~(m_\xi = 1)</math>.

====Delta Profiles====
=====Layout=====
Next, let's define a fractional difference in configuration profiles.  
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{x} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\tfrac{1}{2}\Delta r_\mathrm{SWS}}{\langle r_\mathrm{SWS} \rangle} =
\frac{ \tfrac{1}{2} [r_2(m_\xi) - r_1(m_\xi) ]}{\tfrac{1}{2} [ r_2(m_\xi) + r_1(m_\xi)]}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{\biggl[ \frac{\tilde{C}_2 - 3}{\tilde{C}_2} \biggr] \biggl[ \tilde{C}_2~m_\xi^{-2/3} -3 \biggr]^{-1 / 2}  
- \biggl[ \frac{\tilde{C}_1 - 3}{\tilde{C}_1} \biggr] \biggl[ \tilde{C}_1~m_\xi^{-2/3} -3 \biggr]^{-1 / 2} \biggr\}
\biggl\{\biggl[ \frac{\tilde{C}_2 - 3}{\tilde{C}_2} \biggr] \biggl[ \tilde{C}_2~m_\xi^{-2/3} -3 \biggr]^{-1 / 2}  
+ \biggl[ \frac{\tilde{C}_1 - 3}{\tilde{C}_1} \biggr] \biggl[ \tilde{C}_1~m_\xi^{-2/3} -3 \biggr]^{-1 / 2} \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \tilde{C}_1 ( \tilde{C}_2 - 3 ) \biggl[ \tilde{C}_1~m_\xi^{-2/3} -3 \biggr]^{1 / 2}  
- \tilde{C}_2 ( \tilde{C}_1 - 3 ) \biggl[ \tilde{C}_2~m_\xi^{-2/3} -3 \biggr]^{1 / 2} \biggr\}
\biggl\{ \tilde{C}_1 ( \tilde{C}_2 - 3 ) \biggl[ \tilde{C}_1~m_\xi^{-2/3} -3 \biggr]^{1 / 2}  
+ \tilde{C}_2 ( \tilde{C}_1 - 3 ) \biggl[ \tilde{C}_2~m_\xi^{-2/3} -3 \biggr]^{1 / 2} \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \tilde{C}_1 ( \tilde{C}_2 - 3 ) \biggl[ \tilde{C}_1 -3~m_\xi^{2/3} \biggr]^{1 / 2}  
- \tilde{C}_2 ( \tilde{C}_1 - 3 ) \biggl[ \tilde{C}_2 -3 ~m_\xi^{2/3}\biggr]^{1 / 2} \biggr\}
\biggl\{ \tilde{C}_1 ( \tilde{C}_2 - 3 ) \biggl[ \tilde{C}_1 -3~m_\xi^{2/3} \biggr]^{1 / 2}  
+ \tilde{C}_2 ( \tilde{C}_1 - 3 ) \biggl[ \tilde{C}_2 -3~m_\xi^{2/3} \biggr]^{1 / 2} \biggr\}^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>
Next, let's define,
<div align="center">
<math>~\Delta C_i \equiv {\tilde{C}}_i - 4 </math> &nbsp; &nbsp; &nbsp;<math>~\Rightarrow</math>  &nbsp; &nbsp; &nbsp;<math>~\tilde{C}_i = \Delta C_i + 4 \, ,</math>
</div>
in which case, 

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

<tr>
  <td align="right">
<math>~\mathfrak{x} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ (\Delta C_1 + 4) ( \Delta C_2 + 1 ) \biggl[ \Delta C_1 + 4 -3~m_\xi^{2/3} \biggr]^{1 / 2}  
- ( \Delta C_2 + 4)  ( \Delta C_1 + 1 ) \biggl[ \Delta C_2 + 4 -3 ~m_\xi^{2/3}\biggr]^{1 / 2} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\biggl\{ (\Delta C_1 + 4)  ( \Delta C_2 + 1 ) \biggl[ \Delta C_1 + 4 -3~m_\xi^{2/3} \biggr]^{1 / 2}  
+ ( \Delta C_2 + 4 ) ( \Delta C_1 + 1 ) \biggl[ \Delta C_2 + 4 -3~m_\xi^{2/3} \biggr]^{1 / 2} \biggr\}^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>

Now define,
<div align="center">
<math>~\beta \equiv (4 -3~m_\xi^{2/3})^{-1 / 2} \, ,</math>
</div>
in which case,

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

<tr>
  <td align="right">
<math>~\mathfrak{x} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ (1 + \Delta C_1/4 ) (1 + \Delta C_2 ) (1 + \beta^2 \Delta C_1  )^{1 / 2}  
- (1 + \Delta C_2/4 )  (1+ \Delta C_1 ) ( 1+\beta^2 \Delta C_2 )^{1 / 2} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\biggl\{ (1 + \Delta C_1/4)  (1 + \Delta C_2 ) ( 1 + \beta^2\Delta C_1 )^{1 / 2}  
+ (1 + \Delta C_2/4 ) (1 + \Delta C_1) ( 1 + \beta^2\Delta C_2 )^{1 / 2} \biggr\}^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl[ 1 + \frac{\Delta C_1}{4} + \Delta C_2 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr](1 + \beta^2 \Delta C_1  )^{1 / 2}  
- \biggl[ 1 + \frac{\Delta C_2}{4} + \Delta C_1 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr] ( 1+\beta^2 \Delta C_2 )^{1 / 2} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\biggl\{ \biggl[ 1 + \frac{\Delta C_1}{4} + \Delta C_2 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr] ( 1 + \beta^2\Delta C_1 )^{1 / 2}  
+ \biggl[ 1 + \frac{\Delta C_2}{4} + \Delta C_1 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr] ( 1 + \beta^2\Delta C_2 )^{1 / 2} \biggr\}^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>

=====Original Manipulation=====
Given that we are only interested in examining configurations very near <math>~m_\mathrm{max}</math> for which, <math>~|\Delta C_i| \ll 1</math>, we draw guidance from the binomial expansion and make the substitution,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(1 + \beta^2 \Delta C_i)^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1 + \frac{\beta^2 \Delta C_i}{2} -  \frac{\beta^4 (\Delta C_i)^2 }{8} + \cdots \biggr]
</math>
  </td>
</tr>
</table>
</div>
This gives,

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

<tr>
  <td align="right">
<math>~\mathfrak{x} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl[ 1 + \frac{\Delta C_1}{4} + \Delta C_2 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr] \biggl[ 1 + \frac{\beta^2 \Delta C_1}{2} -  \frac{\beta^4 (\Delta C_1)^2 }{8} + \cdots \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[ 1 + \frac{\Delta C_2}{4} + \Delta C_1 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr] \biggl[ 1 + \frac{\beta^2 \Delta C_2}{2} -  \frac{\beta^4 (\Delta C_2)^2 }{8} + \cdots \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\biggl\{ \biggl[ 1 + \frac{\Delta C_1}{4} + \Delta C_2 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr] \biggl[ 1 + \frac{\beta^2 \Delta C_1}{2} -  \frac{\beta^4 (\Delta C_1)^2 }{8} + \cdots \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ 1 + \frac{\Delta C_2}{4} + \Delta C_1 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr] \biggl[ 1 + \frac{\beta^2 \Delta C_2}{2} -  \frac{\beta^4 (\Delta C_2)^2 }{8} + \cdots \biggr] \biggr\}^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
\biggl[ 1 + \frac{\Delta C_1}{4} + \Delta C_2 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr]   
+\biggl[ 1 + \frac{\Delta C_1}{4} + \Delta C_2 \biggr] \biggl[ \frac{\beta^2 \Delta C_1}{2} \biggr]  
- \biggl[  \frac{\beta^4 (\Delta C_1)^2 }{8} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[ 1 + \frac{\Delta C_2}{4} + \Delta C_1 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr]  
- \biggl[ 1 + \frac{\Delta C_2}{4} + \Delta C_1 \biggr] \biggl[ \frac{\beta^2 \Delta C_2}{2} \biggr] 
+ \biggl[  \frac{\beta^4 (\Delta C_2)^2 }{8}  \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\biggl\{ 
\biggl[ 1 + \frac{\Delta C_1}{4} + \Delta C_2 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr]   
+\biggl[ 1 + \frac{\Delta C_1}{4} + \Delta C_2 \biggr] \biggl[ \frac{\beta^2 \Delta C_1}{2} \biggr]  
- \biggl[  \frac{\beta^4 (\Delta C_1)^2 }{8} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ 1 + \frac{\Delta C_2}{4} + \Delta C_1 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr]  
+ \biggl[ 1 + \frac{\Delta C_2}{4} + \Delta C_1 \biggr] \biggl[ \frac{\beta^2 \Delta C_2}{2} \biggr] 
- \biggl[  \frac{\beta^4 (\Delta C_2)^2 }{8}  \biggr] 
\biggr\}^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
\biggl[ \frac{\Delta C_1}{4} + \Delta C_2 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr]   
- \biggl[ \frac{\Delta C_2}{4} + \Delta C_1 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\biggl[ 1 + \frac{\Delta C_1}{4} + \Delta C_2 \biggr] \biggl[ \frac{\beta^2 \Delta C_1}{2} \biggr]  
- \biggl[ 1 + \frac{\Delta C_2}{4} + \Delta C_1 \biggr] \biggl[ \frac{\beta^2 \Delta C_2}{2} \biggr] 
+ \biggl[  \frac{\beta^4 (\Delta C_2)^2 }{8}  \biggr] 
- \biggl[  \frac{\beta^4 (\Delta C_1)^2 }{8} \biggr]  
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\biggl\{ 
\biggl[ 2 + \frac{\Delta C_1}{4} + \Delta C_2 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr]   
+ \biggl[ \frac{\Delta C_2}{4} + \Delta C_1 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ 1 + \frac{\Delta C_2}{4} + \Delta C_1 \biggr] \biggl[ \frac{\beta^2 \Delta C_2}{2} \biggr] 
+\biggl[ 1 + \frac{\Delta C_1}{4} + \Delta C_2 \biggr] \biggl[ \frac{\beta^2 \Delta C_1}{2} \biggr]  
- \biggl[  \frac{\beta^4 (\Delta C_1)^2 }{8} \biggr]  
- \biggl[  \frac{\beta^4 (\Delta C_2)^2 }{8}  \biggr] 
\biggr\}^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
\biggl[ \frac{3}{4}- \frac{\beta^2}{2}\biggr] \biggl( \Delta C_2 - \Delta C_1\biggr) + \frac{\beta^4}{8}\biggl[ (\Delta C_2)^2 - (\Delta C_1)^2 \biggr]
+ \frac{\beta^2}{2}\biggl[ \biggl( \frac{\Delta C_1}{4} + \Delta C_2 \biggr)\Delta C_1    
- \biggl(\frac{\Delta C_2}{4} + \Delta C_1 \biggr)\Delta C_2  \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\biggl\{ 
2 +  \biggl(\frac{5}{4} + \frac{\beta^2}{2} \biggr) \biggr( \Delta C_1 + \Delta C_2\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{\Delta C_1 \cdot \Delta C_2}{2}  
+ \biggl[ \frac{\Delta C_2}{4} + \Delta C_1 \biggr] \biggl[ \frac{\beta^2 \Delta C_2}{2} \biggr] 
+\biggl[ \frac{\Delta C_1}{4} + \Delta C_2 \biggr] \biggl[ \frac{\beta^2 \Delta C_1}{2} \biggr]  
- \biggl[  \frac{\beta^4 (\Delta C_1)^2 }{8} \biggr]  
- \biggl[  \frac{\beta^4 (\Delta C_2)^2 }{8}  \biggr] 
\biggr\}^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{16}\biggl\{ 
(6-4\beta^2)  \biggl( \Delta C_2 - \Delta C_1\biggr) + (\beta^4- \beta^2)\biggl[ (\Delta C_2)^2 - (\Delta C_1)^2 \biggr]
\biggr\}\biggl\{ 
1 + \biggl( \frac{5 + 2\beta^2}{8} \biggr) \biggr( \Delta C_1 + \Delta C_2\biggr) 
\biggr\}^{-1} 
</math>
  </td>
</tr>
</table>
</div>

<span id="First">Again, employing the binomial expansion to approximate the numerator, we have,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{x} </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{16}\biggl\{ 
(6-4\beta^2)  \biggl( \Delta C_2 - \Delta C_1\biggr) + (\beta^4- \beta^2)\biggl[ (\Delta C_2)^2 - (\Delta C_1)^2 \biggr]
\biggr\}\biggl\{ 
1 - \biggl( \frac{5 + 2\beta^2}{8} \biggr) \biggr( \Delta C_1 + \Delta C_2\biggr) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{16}\biggl\{ 
(6-4\beta^2)  \biggl( \Delta C_2 - \Delta C_1\biggr) + (\beta^4- \beta^2)\biggl[ (\Delta C_2)^2 - (\Delta C_1)^2 \biggr]
- (6-4\beta^2)  \biggl[ (\Delta C_2)^2 - (\Delta C_1)^2 \biggr]  
\biggl( \frac{5 + 2\beta^2}{8} \biggr)  
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{16}\biggl\{ 
(6-4\beta^2)  ( \Delta C_2 - \Delta C_1 ) + \frac{1}{8}\biggl[ (\Delta C_2)^2 - (\Delta C_1)^2 \biggr] (8\beta^4- 8\beta^2)
-   \frac{1}{8}\biggl[ (\Delta C_2)^2 - (\Delta C_1)^2 \biggr]  
(30 -8\beta^2 -8\beta^4)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{16}\biggl\{ 
(6-4\beta^2)  ( \Delta C_2 - \Delta C_1 ) + \frac{1}{8}\biggl[ (\Delta C_2)^2 - (\Delta C_1)^2 \biggr] (16\beta^4 - 30 )
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{8}( \Delta C_2 - \Delta C_1 )\biggl[ 
(3-2\beta^2)   + \frac{1}{8}\biggl( \Delta C_2 + \Delta C_1 \biggr) (8\beta^4 - 15 )
\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Compare this result with an earlier derivation that kept only the lowest-order term:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}
\biggl[ (1 + \Delta C_1/4 + \Delta C_2 + \beta^2 \Delta C_1/2  )  
- (1 + \Delta C_2/4 + \Delta C_1 +\beta^2 \Delta C_2/2 ) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}
\biggl[ \Delta C_1/4 + \Delta C_2 + \beta^2 \Delta C_1/2  - \Delta C_2/4 - \Delta C_1 -\beta^2 \Delta C_2/2  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}
\biggl[ \frac{1}{4}(\Delta C_1 - \Delta C_2) - (\Delta C_1 - \Delta C_2) + \frac{\beta^2}{2}( \Delta C_1 - \Delta C_2)  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{8} (\Delta C_1 - \Delta C_2) [2\beta^2 -3] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{8} (\Delta C_2 - \Delta C_1) \biggl[ \frac{10-9m_\xi^{2 / 3}}{4-3m_\xi^{2 / 3}} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

=====New Approach=====
We begin with the derived expression for <math>~\mathfrak{r}</math>, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{x} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \tilde{C}_1 ( \tilde{C}_2 - 3 ) \biggl[ \tilde{C}_1 -3~m_\xi^{2/3} \biggr]^{1 / 2}  
- \tilde{C}_2 ( \tilde{C}_1 - 3 ) \biggl[ \tilde{C}_2 -3 ~m_\xi^{2/3}\biggr]^{1 / 2} \biggr\}
\biggl\{ \tilde{C}_1 ( \tilde{C}_2 - 3 ) \biggl[ \tilde{C}_1 -3~m_\xi^{2/3} \biggr]^{1 / 2}  
+ \tilde{C}_2 ( \tilde{C}_1 - 3 ) \biggl[ \tilde{C}_2 -3~m_\xi^{2/3} \biggr]^{1 / 2} \biggr\}^{-1} \, ,
</math>
  </td>
</tr>
</table>
</div>

and define,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta \equiv \Delta C_2 - \Delta C_1</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\Sigma \equiv \Delta C_2 + \Delta C_1 \, .</math>
  </td>
</tr>
</table>
</div>
Then, on the whiteboard, I have shown that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
4\beta \biggl\{ \tilde{C}_1 ( \tilde{C}_2 - 3 ) \biggl[ \tilde{C}_1 -3~m_\xi^{2/3} \biggr]^{1 / 2}  \biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[16 + 10\Sigma + 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl[1 + \frac{1}{2}(\Sigma -\delta) \beta^2\biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[16 + 10\Sigma + 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{1 
+ \frac{\beta^2}{2^2}(\Sigma -\delta) 
- \frac{\beta^4}{2^5}(\Sigma -\delta)^2 
+ \frac{\beta^6}{2^7}(\Sigma -\delta)^3
+ \cdots 
\biggr\}
</math>
  </td>
</tr>
</table>
</div>
and,

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

<tr>
  <td align="right">
<math>~
4\beta \biggl\{ \tilde{C}_2 ( \tilde{C}_1 - 3 ) \biggl[ \tilde{C}_2 -3 ~m_\xi^{2/3}\biggr]^{1 / 2}\biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[16 + 10\Sigma - 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl[1 + \frac{1}{2}(\Sigma +\delta) \beta^2\biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[16 + 10\Sigma - 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{1 
+ \frac{\beta^2}{2^2}(\Sigma +\delta) 
- \frac{\beta^4}{2^5}(\Sigma +\delta)^2 
+ \frac{\beta^6}{2^7}(\Sigma +\delta)^3
+ \cdots 
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, the numerator and denominator of the expression for <math>~\mathfrak{r}</math> are, respectively,

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

<tr>
  <td align="right">
Numerator
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[16 + 10\Sigma + 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{1 
+ \frac{\beta^2}{2^2}(\Sigma -\delta) 
- \frac{\beta^4}{2^5}(\Sigma -\delta)^2 
+ \frac{\beta^6}{2^7}(\Sigma -\delta)^3
+ \cdots 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-
\biggl[16 + 10\Sigma - 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{1 
+ \frac{\beta^2}{2^2}(\Sigma +\delta) 
- \frac{\beta^4}{2^5}(\Sigma +\delta)^2 
+ \frac{\beta^6}{2^7}(\Sigma +\delta)^3
+ \cdots 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[16 + 10\Sigma + 6\delta + (\Sigma^2 - \delta^2)\biggr]
-
\biggl[16 + 10\Sigma - 6\delta + (\Sigma^2 - \delta^2)\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[16 + 10\Sigma + 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{\frac{\beta^2}{2^2}(\Sigma -\delta) 
\biggr\}
-
\biggl[16 + 10\Sigma - 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{\frac{\beta^2}{2^2}(\Sigma +\delta) 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[16 + 10\Sigma + 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{ 
- \frac{\beta^4}{2^5}(\Sigma -\delta)^2 
\biggr\}
-
\biggl[16 + 10\Sigma - 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{ 
- \frac{\beta^4}{2^5}(\Sigma +\delta)^2 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[16 + 10\Sigma + 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{ 
\frac{\beta^6}{2^7}(\Sigma -\delta)^3
\biggr\}
-
\biggl[16 + 10\Sigma - 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{ 
\frac{\beta^6}{2^7}(\Sigma +\delta)^3
\biggr\} 
+ \cdots
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
12\delta
+ \frac{\beta^2}{2^2} \biggl\{ 
12\delta \Sigma  
- 2\delta \biggl[16 + 10\Sigma + (\Sigma^2 - \delta^2)\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{\beta^4}{2^5} \biggl\{ \biggl[16 + 10\Sigma + (\Sigma^2 - \delta^2)\biggr] 
\biggl[ (\Sigma -\delta)^2 - (\Sigma +\delta)^2 \biggr]
+ 6\delta \biggl[(\Sigma -\delta)^2 
+(\Sigma +\delta)^2 \biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{\beta^6}{2^7} \biggl\{ \biggl[16 + 10\Sigma + (\Sigma^2 - \delta^2)\biggr] 
\biggl[ (\Sigma -\delta)^3 - (\Sigma +\delta)^3 \biggr]
+ 6\delta \biggl[(\Sigma -\delta)^3 
+(\Sigma +\delta)^3 \biggr]
\biggr\} + \cdots
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\delta \biggl\{ 12
- 8\beta^2 \biggl[ 
1 + \frac{1}{4} \Sigma  + \frac{1}{16} \biggl(\Sigma^2 - \delta^2 \biggr)\biggr]
+ \frac{\beta^4 }{2^3} \biggl[  16\Sigma + 7\Sigma^2 -3 \delta^2 + \Sigma(\Sigma^2 - \delta^2) \biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{\beta^6\delta}{2^6} \biggl\{ 
6\Sigma^3 + 18\delta^2\Sigma 
-\biggl[16 + 10\Sigma + \Sigma^2 - \delta^2\biggr] 
\biggl[3\Sigma^2 +\delta^2 \biggr]
\biggr\} + \cdots
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\delta \biggl\{ 12
- 8\beta^2 \biggl[ 
1 + \frac{1}{4} \Sigma  + \frac{1}{16} \biggl(\Sigma^2 - \delta^2 \biggr)\biggr]
+ \frac{\beta^4 }{2^3} \biggl[  16\Sigma + 7\Sigma^2 -3 \delta^2 + \Sigma(\Sigma^2 - \delta^2) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{\beta^6}{2^6} \biggl[ 
-48\Sigma^2 - 16\delta^2  + 8\delta^2\Sigma 
- 24\Sigma^3 - 3\Sigma^4 +2\delta^2\Sigma^2 +\delta^4
\biggr] \biggr\} + \cdots
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\delta \biggl\{ \biggl[12 - 8\beta^2 \biggr] -2 \Sigma\biggl[\beta^2 - \beta^4 \biggr] 
+ \delta^2\biggl[ \frac{\beta^2}{2} - \frac{3\beta^4}{2^3} -\frac{16\beta^6}{2^6}\biggr] 
+ \Sigma^2\biggl[- \frac{\beta^2}{2} + \frac{7\beta^4}{2^3} - \frac{48\beta^6}{2^6}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{\beta^4 }{2^3} \biggl[    \Sigma(\Sigma^2 - \delta^2) \biggr]
+ \frac{\beta^6}{2^6} \biggl[ 
8\delta^2\Sigma 
- 24\Sigma^3 - 3\Sigma^4 +2\delta^2\Sigma^2 +\delta^4
\biggr] \biggr\} + \cdots
</math>
  </td>
</tr>
</table>
</div>
and,

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

<tr>
  <td align="right">
Denominator
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[16 + 10\Sigma + 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{1 
+ \frac{\beta^2}{2^2}(\Sigma -\delta) 
- \frac{\beta^4}{2^5}(\Sigma -\delta)^2 
+ \frac{\beta^6}{2^7}(\Sigma -\delta)^3
+ \cdots 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\biggl[16 + 10\Sigma - 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{1 
+ \frac{\beta^2}{2^2}(\Sigma +\delta) 
- \frac{\beta^4}{2^5}(\Sigma +\delta)^2 
+ \frac{\beta^6}{2^7}(\Sigma +\delta)^3
+ \cdots 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[16 + 10\Sigma + 6\delta + (\Sigma^2 - \delta^2)\biggr]
+
\biggl[16 + 10\Sigma - 6\delta + (\Sigma^2 - \delta^2)\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\biggl[16 + 10\Sigma + 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{\frac{\beta^2}{2^2}(\Sigma -\delta) 
\biggr\}
+
\biggl[16 + 10\Sigma - 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{\frac{\beta^2}{2^2}(\Sigma +\delta) 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[16 + 10\Sigma + 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{
- \frac{\beta^4}{2^5}(\Sigma -\delta)^2 
\biggr\}
+
\biggl[16 + 10\Sigma - 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{
- \frac{\beta^4}{2^5}(\Sigma +\delta)^2 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\biggl[16 + 10\Sigma + 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{
\frac{\beta^6}{2^7}(\Sigma -\delta)^3
\biggr\}
+
\biggl[16 + 10\Sigma - 6\delta + (\Sigma^2 - \delta^2)\biggr]
\biggl\{ 
\frac{\beta^6}{2^7}(\Sigma +\delta)^3
\biggr\} 
+ \cdots 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\biggl[16 + 10\Sigma + (\Sigma^2 - \delta^2)\biggr]
+ \frac{\beta^2 \Sigma}{2}\biggl[16 + 10\Sigma + (\Sigma^2 - \delta^2)\biggr] -3 \beta^2 \delta^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{\beta^4}{2^4} \biggl[16 + 10\Sigma + (\Sigma^2 - \delta^2)\biggr] (\Sigma^2 + \delta^2)
+ \frac{3\beta^4 \delta^2 \Sigma}{2^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{\beta^6}{2^7} \biggl[16 + 10\Sigma + (\Sigma^2 - \delta^2)\biggr]\biggl\{ (\Sigma -\delta)^3 +  (\Sigma +\delta)^3 \biggr\}
+ \frac{\beta^6}{2^7} \biggl[6\delta \biggr] \biggl\{ (\Sigma -\delta)^3 -  (\Sigma +\delta)^3\biggr\}
+ \cdots 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
32 + \biggl[20 + 8\beta^2\biggr] \Sigma
+ \biggl[ (2+ 5 \beta^2   - \beta^4 )\Sigma^2 
- ( 2 + 3 \beta^2 + \beta^4 )\delta^2
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{\beta^2 \Sigma}{2}\biggl[(\Sigma^2 - \delta^2)\biggr]  
- \frac{5\beta^4\Sigma}{2^3} (\Sigma^2 + \delta^2)
+ \frac{3\beta^4 \delta^2 \Sigma}{2^2} 
- \frac{\beta^4}{2^4} \biggl[ (\Sigma^2 - \delta^2)\biggr] (\Sigma^2 + \delta^2)
+ \frac{\beta^6 \Sigma}{2^6} \biggl[16 + 10\Sigma + (\Sigma^2 - \delta^2)\biggr]
(\Sigma^2 + 3\delta^2 ) 
- \frac{3\beta^6 \delta^2}{2^5} ( 3\Sigma^2 + \delta^2) 
+ \cdots 
</math>
  </td>
</tr>
</table>
</div>

To lowest order in smallness <math>~(\delta~\mathrm{or}~\Sigma)</math>, then, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{r}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\delta \biggl[ \biggl(12 - 8\beta^2 \biggr) -2 \Sigma\biggl(\beta^2 - \beta^4 \biggr) \biggr]
\biggl[ 32 + \biggl(20 + 8\beta^2\biggr) \Sigma \biggr]^{-1} + \mathcal{O}(\delta^3~\mathrm{or}~\delta\Sigma^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{\delta}{32} \biggl[ \biggl(12 - 8\beta^2 \biggr) -2 \Sigma\biggl(\beta^2 - \beta^4 \biggr) \biggr]
\biggl[ 1 - \biggl(\frac{5}{8} + \frac{1}{4}\beta^2\biggr) \Sigma \biggr] + \mathcal{O}(\delta^3~\mathrm{or}~\delta\Sigma^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{\delta}{32} \biggl[ \biggl(12 - 8\beta^2 \biggr) -2 \Sigma\biggl(\beta^2 - \beta^4 \biggr) 
- \biggl(12 - 8\beta^2 \biggr)\biggl(\frac{5}{8} + \frac{1}{4}\beta^2\biggr) \Sigma \biggr] + \mathcal{O}(\delta^3~\mathrm{or}~\delta\Sigma^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\delta}{32} \biggl\{ 4(3 - 2\beta^2)  +  \Sigma\biggl[ \biggl(2\beta^4 - 2\beta^2 \biggr) 
+ \biggl(8\beta^2 -12\biggr)\biggl(\frac{5}{8} + \frac{1}{4}\beta^2\biggr) \biggr] \biggr\} 
+ \mathcal{O}(\delta^3~\mathrm{or}~\delta\Sigma^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\delta}{32} \biggl\{ 4(3 - 2\beta^2)  +  \Sigma\biggl[ \biggl(2\beta^4 - 2\beta^2 \biggr) + 5\beta^2 + 2\beta^4 - \frac{15}{2} -3\beta^2
\biggr] \biggr\} 
+ \mathcal{O}(\delta^3~\mathrm{or}~\delta\Sigma^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\delta}{32} \biggl[ 4(3 - 2\beta^2)  
-  \frac{\Sigma}{2} \biggl(15- 8\beta^4  \biggr) \biggr] 
+ \mathcal{O}(\delta^3~\mathrm{or}~\delta\Sigma^2) \, .
</math>
  </td>
</tr>
</table>
</div>

This matches the result [[#First|derived earlier]].

====Near the Maximum Mass====
We seek a power-series expression for,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta C_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(1-\frac{\epsilon_+}{3}\biggr)^{-2} -1 \, ,
</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta C_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(1-\frac{\epsilon_-}{3}\biggr)^{-2} -1 \, .
</math>
  </td>
</tr>
</table>
</div>

Via the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial expansion]], we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl(1-\frac{\epsilon_\pm}{3}\biggr)^{-2} - 1
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2}{3}\biggr)\epsilon_\pm + \biggl( \frac{1}{3} \biggr) \epsilon_\pm^2
+ \biggl(\frac{2^2}{3^3} \biggr) \epsilon_\pm^3 + \biggl( \frac{5}{3^4} \biggr)\epsilon_\pm^4 + \mathcal{O}(\epsilon_\pm^5) \, .
</math>
  </td>
</tr>
</table>
</div>

So, given that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\epsilon_\pm</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
\pm 6^{1 / 2}  \mu
- \biggl(\frac{4}{3}\biggr)\mu^2  
\pm \biggl[ \frac{5\cdot 23}{(2^5 \cdot 3^5)^{1 / 2}} \biggr] \mu^3  
- \biggl(\frac{191}{2\cdot 3^4}\biggr) \mu^4
+ \mathcal{O}(\mu^5) \, ,
</math>
  </td>
</tr>
</table>
</div>
we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta C_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{3} \biggl\{ 
6^{1 / 2}  \mu
- \biggl(\frac{4}{3}\biggr)\mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^5 \cdot 3^5)^{1 / 2}} \biggr] \mu^3  
- \biggl(\frac{191}{2\cdot 3^4}\biggr) \mu^4
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{3} \biggl\{ 
6^{1 / 2}  \mu
- \biggl(\frac{4}{3}\biggr)\mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^5 \cdot 3^5)^{1 / 2}} \biggr] \mu^3  
\biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{2^2}{3^3} \biggl\{ 
6^{1 / 2}  \mu
- \biggl(\frac{4}{3}\biggr)\mu^2  
\biggr\}^3 
+ \frac{5}{3^4} \biggl\{ 
6^{1 / 2}  \mu
\biggr\}^4 
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^2}{3^2} \biggr)^{1 / 2} (2\cdot 3)^{1 / 2}  \mu
- \biggl( \frac{2}{3} \biggr) \biggl(\frac{2^2}{3}\biggr)\mu^2  
+ \biggl( \frac{2}{3} \biggr) \biggl[ \frac{5\cdot 23}{(2^5 \cdot 3^5)^{1 / 2}} \biggr] \mu^3  
- \biggl( \frac{2}{3} \biggr) \biggl(\frac{191}{2\cdot 3^4}\biggr) \mu^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{6}{3} \mu^2 \biggl\{ 1
- (2\cdot 3)^{-1 / 2} \biggl(\frac{2^2}{3}\biggr)\mu
+ (2\cdot 3)^{-1 / 2} \biggl[ \frac{5\cdot 23}{(2^5 \cdot 3^5)^{1 / 2}} \biggr] \mu^2  
\biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{2^2}{3^3} (2\cdot 3)^{3 / 2}  \mu^3 \biggl[ 1 
- (2\cdot 3)^{-1 / 2}\biggl(\frac{2^2}{3}\biggr)\mu  
\biggr]^3 
+ \frac{5}{3^4} \biggl[ 
(2\cdot 3)^{2}  \mu^4
\biggr]
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
- \biggl(\frac{2^3}{3^2}\biggr)\mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3  
- \biggl(\frac{191}{3^5}\biggr) \mu^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 2 \mu^2 \biggl\{ 1 + \biggl[
- \biggl(\frac{2^3}{3^3}\biggr)^{1 / 2}\mu 
+ \biggl( \frac{5\cdot 23}{2^3 \cdot 3^3} \biggr) \mu^2  \biggr]
\biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl( \frac{2^7}{3^3} \biggr)^{1 / 2}  \mu^3 \biggl[ 1 
- \biggl(\frac{2^3}{3^3}\biggr)^{1 / 2} \mu  
\biggr]^3 
+ \biggl( \frac{2^2 \cdot 5}{3^2} \biggr) \mu^4
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+  2\mu^2 
- \biggl(\frac{2^3}{3^2}\biggr)\mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3  
- \biggl(\frac{2^7}{3^3}\biggr)^{1 / 2} \mu^3 
+ \biggl( \frac{2^7}{3^3} \biggr)^{1 / 2}  \mu^3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl( \frac{5\cdot 23}{2 \cdot 3^3} \biggr) \mu^4  
+ \biggl(\frac{2^4}{3^3}\biggr) \mu^4 
- \biggl(\frac{191}{3^5}\biggr) \mu^4
- \biggl( \frac{2^{10}}{3^4} \biggr)^{1 / 2}  \mu^4  
+ \biggl( \frac{2^2 \cdot 5}{3^2} \biggr) \mu^4
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+  \biggl[ 2 
- \biggl(\frac{2^3}{3^2}\biggr) \biggr]\mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}}   
\biggr] \mu^3 
+ \biggl[ \frac{5\cdot 23}{2 \cdot 3^3} 
+ \frac{2^4}{3^3}
- \frac{191}{3^5}
- \frac{2^{5}}{3^2}    
+ \frac{2^2 \cdot 5}{3^2} \biggr] \mu^4
+ \mathcal{O}(\mu^5) \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+ \biggl(\frac{10}{9}\biggr) \mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
+ \biggl( \frac{293}{2 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) \, .
</math>
  </td>
</tr>
</table>
</div>
Similarly, we have deduced that,

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

<tr>
  <td align="right">
<math>~\Delta C_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+ \biggl(\frac{10}{9}\biggr) \mu^2  
- \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
+ \biggl( \frac{293}{2 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, 

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

<tr>
  <td align="right">
<math>~\Delta C_2 + \Delta C_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[
\biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+ \biggl(\frac{10}{9}\biggr) \mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
+ \biggl( \frac{293}{2 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[
- \biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+ \biggl(\frac{10}{9}\biggr) \mu^2  
- \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
+ \biggl( \frac{293}{2 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{20}{9}\biggr) \mu^2  
+ \biggl( \frac{293}{3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^6) \, ,
</math>
  </td>
</tr>
</table>
</div>

and, 

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

<tr>
  <td align="right">
<math>~\Delta C_2 - \Delta C_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[
\biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+ \biggl(\frac{10}{9}\biggr) \mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
+ \biggl( \frac{293}{2 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[
- \biggl( \frac{2^3}{3} \biggr)^{1 / 2}\mu
+ \biggl(\frac{10}{9}\biggr) \mu^2  
- \biggl[ \frac{5\cdot 23}{(2^3 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
+ \biggl( \frac{293}{2 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^5}{3} \biggr)^{1 / 2}\mu
+ \biggl[ \frac{5\cdot 23}{(2 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
+ \mathcal{O}(\mu^5) \, ,
</math>
  </td>
</tr>
</table>
</div>

so we have,

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

<tr>
  <td align="right">
<math>~\mathfrak{x}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{8}( \Delta C_2 - \Delta C_1 )\biggl[ 
(3-2\beta^2)   + \frac{1}{8}\biggl( \Delta C_2 + \Delta C_1 \biggr) (8\beta^4 - 15 )
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{8}\biggl\{ \biggl( \frac{2^5}{3} \biggr)^{1 / 2}\mu
+ \biggl[ \frac{5\cdot 23}{(2 \cdot 3^7)^{1 / 2}} \biggr] \mu^3 
 \biggr\}
\biggl\{ 
(3-2\beta^2)   + \frac{1}{8}\biggl[ \biggl(\frac{20}{9}\biggr) \mu^2  
+ \biggl( \frac{293}{3^5} \biggr) \mu^4
\biggr] (8\beta^4 - 15 )
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl( \frac{1}{2 \cdot 3} \biggr)^{1 / 2}\mu
+ \biggl[ \frac{5\cdot 23}{(2 \cdot 3)^{7 / 2}} \biggr] \mu^3 
 \biggr\}
\biggl\{ 
(3-2\beta^2)   + \biggl[ \biggl(\frac{5}{18}\biggr) \mu^2  
+ \biggl( \frac{293}{2^3 \cdot 3^5} \biggr) \mu^4
\biggr] (8\beta^4 - 15 )
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\mu^2}{2 \cdot 3} \biggr)^{1 / 2} (3-2\beta^2) 
\biggl[ 1 + \biggl( \frac{5\cdot 23}{6^3} \biggr) \mu^2 \biggr]
\biggl\{ 
1   + \biggl[ \biggl(\frac{5}{18}\biggr) \mu^2  
+ \biggl( \frac{293}{2^3 \cdot 3^5} \biggr) \mu^4
\biggr] \frac{(8\beta^4 - 15 )}{(3-2\beta^2)}
\biggr\}
</math>
  </td>
</tr>
</table>
</div>

Finally, remembering that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\beta</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(4 -3~m_\xi^{2/3})^{-1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (3-2\beta^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3(4 -3~m_\xi^{2/3}) - 2}{(4 -3~m_\xi^{2/3})} = \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3} } \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>

and defining an overall normalization,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{x}_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{5}{2} \biggr) \mathfrak{x}\biggr|_{\beta = 1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{5}{2}~\biggl( \frac{\mu^2}{2 \cdot 3} \biggr)^{1 / 2}  
\biggl[ 1 + \biggl( \frac{5\cdot 23}{6^3} \biggr) \mu^2 \biggr]
\biggl\{ 
1   -7 \biggl[ \biggl(\frac{5}{18}\biggr) \mu^2  
+ \biggl( \frac{293}{2^3 \cdot 3^5} \biggr) \mu^4
\biggr]\biggr\} 
\, ,
</math>
  </td>
</tr>
</table>
</div>
such that the normalized amplitude is always <math>~\tfrac{2}{5}</math> at the surface  &#8212; that is, at <math>~m_\xi = 1</math> and, hence, at <math>~\beta = 1</math> &#8212; we have,

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

<tr>
  <td align="right">
<math>~x_\mathfrak{x} \equiv \frac{\mathfrak{x}}{\mathfrak{x}_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{2}{5} \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3} } \biggr] 
\biggl\{ 
1   + \biggl[ \biggl(\frac{5}{18}\biggr) \mu^2  
+ \biggl( \frac{293}{2^3 \cdot 3^5} \biggr) \mu^4
\biggr] \frac{(8\beta^4 - 15 )}{(3-2\beta^2)}
\biggr\}\biggl\{ 
1   -7 \biggl[ \biggl(\frac{5}{18}\biggr) \mu^2  
+ \biggl( \frac{293}{2^3 \cdot 3^5} \biggr) \mu^4
\biggr]\biggr\}^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>
To leading order &#8212; in which case, 
<div align="center">
<math>~\mathfrak{x}_\mathrm{norm} = \biggl( \frac{5^2}{2^3\cdot 3} \biggr)^{1 / 2} \mu \, ,</math>
</div>
&#8212; this exactly matches the [[#Analytic.2C_Marginally_Unstable_Eigenfunction|analytically derived eigenfunction]] for the marginally unstable model, namely,

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

<tr>
  <td align="right">
<math>~x_P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{5} \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3} } \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>